Soil Dynamics and Earthquake Engineering V

cover

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title : Soil Dynamics and Earthquake Engineering Vauthor :

publisher : Taylor & Francis Routledgeisbn10 | asin :print isbn13 : 9780203293096

ebook isbn13 : 9780203215944language : English

subject Soil dynamics--Congresses, Earthquake engineering--Congresses, Soils--Mechanics

publication date : 1991lcc : TA711.A1I57 1991eb

ddc : 624.1/5136subject : Soil dynamics--Congresses, Earthquake engineering--

Congresses, Soils--Mechanics

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< previous page page_i next page >Page iSoil Dynamics and Earthquake Engineering V

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< previous page page_ii next page >Page iiFIFTH INTERNATIONAL CONFERENCE ON SOIL DYNAMICS AND EARTHQUAKE ENGINEERING SDEE 91 KARLSRUHE, GERMANY, SEPTEMBER 23–26, 1991LOCAL SCIENTIFIC COMMITTEE, UNIVERSITY OF KARLSRUHEG.Borm J.Brauns J.Eibl K.Fuchs G.Gudehus E.KeintzelO.NatauE.PlateB.PrangeR.SchererP.Vielsack INTERNATIONAL ADVISORY BOARDH.Antes C.A.Brebbia A.S.Cakmak W.D.L.Finn G.Gazetas D.V.Griffiths V.A.Ilyichev K.Ishihara J.M.Roësset F.J.Sánchez-SesmaS.SavidisG.SchmidG.SchneiderG.SchuëllerP.SpanosG.WaasR.V.WhitmanJ.P.WolfR.W.Woods SPONSORING ORGANIZATIONSGerman Science Foundation (DFG)International Society of Soil Mechanics and Foundation Engineering (ISSMFE)International Journal of Soil Dynamics and Earthquake Engineering (JSDEE)German Geophysical Society (DGG)Alfred Wegener Foundation (AWS)German Committee of the International Decade for Natural Disaster Reduction (IDNDR)Swiss Committee on Earthquake Engineering and Structural Dynamics (SGEB/SIA)Acknowledgement is made to H.Takemiya et al. for the use of Figure 5.2 on p. 147, which appears on the front cover of this book.

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< previous page page_iii next page >Page iiiSoil Dynamics and Earthquake Engineering VEdited by: IBF, Institut für Bodenmechanik und Felsmechanik, Universität Karlsruhe, Germany

Computational Mechanics PublicationsSouthampton Boston Co-published with

Elsevier Applied ScienceLondon New York

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< previous page page_iv next page >Page ivIBF Institut für Bodenmechanik und Felsmechanik Universität Karlsruhe W-7500 Karlsruhe 1 GermanyThis edition published in the Taylor & Francis e-Library, 2006. To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.Co-published byComputational Mechanics Publications Ashurst Lodge, Ashurst, Southampton, UKComputational Mechanics Publications Ltd Sole Distributor in the USA and Canada:Computational Mechanics Inc. 25 Bridge Street, Billerica, MA 01821, USAandElsevier Science Publishers Ltd Crown House, Linton Road, Barking, Essex IG11 8JU, UKElsevier’s Sole Distributor in the USA and Canada:Elsevier Science Publishing Company Inc. 655 Avenue of the Americas, New York, NY 10010, USABritish Library Cataloguing-in-Publication DataA Catalogue record for this book is available from the British LibraryISBN 0-203-21594-X Master e-book ISBNISBN 0-203-29309-6 (OEB Format)ISBN 1-85166-699-0 (Print Edition) Elsevier Applied Science, London, New YorkISBN 1-85312-153-3 (Print Edition) Computational Mechanics Publications, SouthamptonISBN 1-56252-081-4 (Print Edition) Computational Mechanics Publications, Boston, USALibrary of Congress Catalog Card Number 91-74078No responsibility is assumed by the Publishers for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein.©Computational Mechanics Publications 1991All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.

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< previous page page_v next page >Page vPREFACEDespite considerable advances having been made in the fields of Soil Dynamics and Earthquake Engineering during the last two decades, earthquakes still continue to cause loss of life and property. In addition, dynamic excitation due to heavy industry, construction machinery, pile driving, high speed traffic, etc. can cause severe damage to existing structures, especially to those of historical importance.The 5th International Conference on Soil Dynamics and Earthquake Engineering (SDEE ’91) was aimed at a better understanding of the dynamic ground-structure-interaction, to exchange experience and knowledge of the participants and to enhance the efforts of geophysics, soil-, rock- and structural dynamics in the mitigation of risks to people and structures in civil and mining engineering. It provided a forum for the presentation and discussion of new ideas and innovative approaches in Soil Dynamics and Earthquake Engineering in theory and practice. The proceedings, in two volumes, contain selected papers from those submitted to SDEE ’91, and are intended to serve as a permanent reference and as a brief survey of the theoretical, experimental and applied methods and their predictive powers, which are available at the present time to deal with dynamic problems in geotechniques.The scope of the conference is reflected by the following topic areas covered in the proceedings: engineering seismology, earthquake hazards, wave propagation, dynamic soil properties, liquefaction, dynamic response of dams and earth structures and of foundations and piles, earthquake engineering of structures, vibrations, impacts and rock dynamics. The conference was further emphasized on contributions to the International Decade for Natural Disaster Reduction (IDNDR). It offered an opportunity for intensive discussions, particularly on the recent advances in European seismic standards which are relevant in view of the continuing integration of the European Community and the progressive opening of the East European countries.The organizers are grateful to the authors for their contributions and for having shared their knowledge and experience. Acknowledgement is also made to the support given by the German Science Foundation (DFG) and the University of Karlsruhe.O.NatauHead of the Institute forSoil Mechanics and Rock MechanicsG.BormCoordinator of SDEE ’91Karlsruhe, July 1991

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< previous page page_vii next page >Page viiCONTENTS

SECTION 1: ENGINEERING SEISMOLOGY, EARTHQUAKE HAZARDS

Urban Earthquake Hazards, Risk and Mitigation M.Erdik

3

Probabilistic Method in Maximum Earthquake Assessment V.Schenk, P.Kottnauer

15

Study of an Assessment for Site Effect of Seismic Strong Motion E.Kuribayashi, T.Jiang, T.Niiro, H.Nagasaka, S.Kuroiwa, S.Nishioka

23

Site-Response at Foster City and San Francisco Airport—Loma Prieta Studies M.Çelebi, A.McGarr

35

SECTION 2: STRONG GROUND MOTIONS Effects of Earthquake Characteristics on Ground Response Spectra A.M.Ansal, A.M.Lav

49

The Artificial Wave in Earthquake Safety Analysis for Nuclear Plant Shield X.Shen, J.Yu

61

Site Dependent Simulations of Earthquake Time Histories O.Henseleit, M.Kostov

73

Spatial Coherency of the Strong Ground Motions on the SMART 1 Seismic Array I.A.Beresnev

99

SECTION 3: WAVE PROPAGATION

Comparison of 2-D and 3-D Models for Analysis of Surface Wave Tests J.M.Roësset, D.-W.Chang, K.H.Stokoe, II

111

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< previous page page_viii next page >Page viii Inversion of Rayleigh Wave Dispersion Curve for SASW Test N.Gucunski, R.D.Woods

127

Transient Response of Certain Topographical Sites for SH-Wave Incidence H.Takemiya, C.Y.Wang, A.Fujiwara

139

Surface Wave Propagation in Stiff Top Layer Half-Space W.Haupt

151

Wave Transmission at a Multimedia Interface R.S.Steedman, S.P.G.Madabhushi

163

SECTION 4: DYNAMIC SOIL PROPERTIES

In-Situ Dynamic Property Evaluation of Gravelly Soil T.Kokusho, Y.Tanaka, Y.Yoshida

177

Characterization of Material Damping of Soils Using Resonant Column and Torsional Shear Tests D.-S.Kim, K.H.Stokoe, II, J.M.Roësset

189

Effect of Triaxial Stresses on Shear Wave Propagation H.-C.Fei, F.E.Richart, Jr.

201

Stiffness Degradation of Weathered Marl in Cyclic Undrained Loading J.A.Little, N.Hataf

215

Measurements of Material Anisotropy by Ultrasonic Technique S.V.Jagannath, C.S.Desai, T.Kundu

223

Elastic Attenuation in Non-Homogeneous Porous Materials B.Gurevich, S.Lopatnikov

235

SECTION 5: LIQUEFACTION

Liquefaction of Gravelly Soil at Pence Ranch During the 1983 Borah Peak, Idaho Earthquake R.D.Andrus, K.H.Stokoe, II, J.M.Roësset

251

Validation of Procedures for Analysis of Liquefaction of Sandy Soil Deposits J.H.Prevost, C.M.Keane, N.Ohbo, K.Hayashi

263

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< previous page page_ix next page >Page ix Liquefaction of Sands Under Undrained and Non-Undrained Conditions J.Chu

277

The Characteristics of Liquefaction of Silt Soil H.-C.Fei

293

Evaluation of Liquefaction Susceptibility A.M.Ansal

303

Post Initial Liquefaction Behaviour of Soils K.Talaganov

313

Liquefaction Associated with Manjil Earthquake of June 20 1990, Iran S.M.Haeri

325

Countermeasures Against the Permanent Ground Displacement due to Liquefaction S.Yasuda, H.Nagase, H.Kiku, Y.Uchida

341

Soil-Pile Interaction in Liquefied Sand Layer K.Kobayashi, S.Nakamura, K.Sato, N.Yoshida, S.Yao

351

SECTION 6: DYNAMIC RESPONSE OF DAMS AND EARTH STRUCTURES

Dynamic Behavior of Embankment on Locally Compacted Sand Deposits S.Yanagihara, M.Takeuchi, K.Ishihara

365

Three-Dimensional Finite Element Analyses of the Natural Frequencies of Non-Homogeneous Earth Dams P.K.Woodward, D.V.Griffiths

377

Lumped-Parameter Model of Semi-Infinite Uniform Fluid Channel for Time-Domain Analysis of Dam-Reservoir Interaction J.P.Wolf, A.Paronesso

389

Earthquake Resistant Design of Earth Walls—A Probabilistic Approach D.Genske, H.Klapperich, T.Adachi, M.Sugito

403

Passive Earth Pressure Coefficients in Seismic Areas by the Limit Analysis Method A.H.Soubra, R.Kastner

415

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SECTION 7: SOIL-STRUCTURE-INTERACTION, FOUNDATIONS, PILES

Dynamic Stiffness of Unbounded Soil by Finite-Element Multi-Cell Cloning J.P.Wolf, C.Song

429

Application of the Hybrid Frequency-Time-Domain Procedure to the Soil-Structure Interaction Analysis of a Shear Building with Multiple Nonlinearities G.R.Darbre

441

Dynamic Soil-Structure-Interaction of Nonlinear Shells of Revolution in the Time Domain W.Wunderlich, B.Schäpertöns, H.Springer, C.Temme

455

Dynamic Soil-Structure Interaction of Rigid and Flexible Foundations L.Auersch

467

Experimentally Determined Impedance Functions of Surface Foundations B.Verbi•, S.Meler

479

Stiffness and Damping of Closely Spaced Pile Groups B.Boroomand, A.M.Kaynia

491

Chaotic Motions in Pile-Driving M.Storz

503

SECTION 8: EARTHQUAKE ENGINEERING OF STRUCTURES

Seismic Damage Assessment for Reinforced Concrete Structures A.S.Cakmak, S.Rodriguez-Gomez, E.DiPasquale

515

Reduction of Linear Elastic Response Spectra due to Elastoplastic Behaviour of Systems S.E.Ruiz, O.Díaz

545

Problems in the Determination of Input Data for the Seismic Design of Structures in Regions of Low Seismicity J.Eibl, E.Keintzel

555

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< previous page page_xi next page >Page xi Determination of the Behaviour Factors for Brick Masonry Panels Subjected to Earthquake Actions V.Vratsanou

565

Statistical Study of Nonlinear Response Spectra for Aseismic Design of Structures E.Miranda

577

Shear Transfer and Friction across Cracks in Concrete under Monotonic and Alternate Loads C.Karakoç

589

Tests on Upgrading Dynamic Properties of Existing Damaged Structures for a Better Seismic Performance O.Yuzugullu

599

Helical Springs in Base Isolation Systems G.K.Hueffmann

613

Damage Reduction with Controlled Seismic Pounding S.Govil, A.Singhal

627

On-Line Hydraulic Servodrives to Protect Serviceability of Antiseismic Structures—Pre-Design Criteria A.Carotti

639

SECTION 9: VIBRATIONS

Shielding of Structures from Soil Vibrations G.Schmid, N.Chouw, R.Le

651

Vertical Vibration of a Rigid Plate on a Continuously Nonhomogeneous Soil S.Savidis, C.Vrettos, B.Faust

663

The Influence of Thickness Variation of Subway Walls on the Vibration Emission Generated by Subway Traffic R.Thiede, H.G.Natke

673

Vibration Isolation by an Array of Piles B.Boroomand, A.M.Kaynia

683

Numerical Modelling of Stability Cases for Caisson-Type Breakwaters without Through-flow E.Stein, M.Lengnick

693

SECTION 10: ROCK DYNAMICS

Explosion Effects in Jointed Rocks—New Insights F.E.Heuzé, T.R.Butkovich, O.R.Walton, D.M.Maddix

707

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< previous page page_xii next page >Page xii Numerical Analysis and Measurements of the Seismic Response of Galleries H.-J.Alheid, K.-G.Hinzen

719

Dynamic Solution of Poroelastic Column and Borehole Problems of Soil and Rock Mechanics D.E.Beskos, I.Vgenopoulou

731

Fundamentals of a Practical Classification of Mining Induced Seismicity (Rock Bursts) P.Knoll

743

Authors’ Index 757

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< previous page page_1 next page >Page 1SECTION 1: ENGINEERING SEISMOLOGY, EARTHQUAKE HAZARDS

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< previous page page_3 next page >Page 3Urban Earthquake Hazards, Risk and MitigationM.ErdikBo•aziçi University, Kandilli Observatory and Earthquake Research Institute 81220, Çengelköy, Istanbul, TurkeyABSTRACTThe hazard assessment, microzonation, vulnerability, risk and mitigation issues involved with urban centers prone to earthquake disasters are covered. The mitigation efforts should concentrate in the preparedness phase. The microzonation maps and the land use requlations are important long-term tools in the mitigation of earthquake risk. The weak areas in the urban infrastructure and the critical structures may need to be retrofitted. The treatment is supported with case studies from .INTRODUCTIONWith the recent 19.9.1985 Mexico (M8.1), 7.12.1988 Armenia (M7.0), 17.10.1989 Loma Prieta (M7.1), 21.6.1990 Iran (M7.7) and 16.7.1990 Philippines (M7.8) earthquakes the earthquake hazards and the attendant risk in urban areas gained focused attention. Urbanization in earthquake prone countries create an associated increase in the earthquake vulnerabilies and the risk.Assessment of the earthquake hazard is one of the preliminary steps towards the mitigation of the risk. In urban centers the earthquake hazard is usually quantified and portrayed in terms of microzonation maps. The microzonation maps and the land use requlations are important long term tools in the mitigation of earthquake risk. The vulnerabilities and the damage statistics of lives, structures, systems and the socio-economic structure are the main factors influencing the earthquake risk in the urban areas. The mitigation efforts should concentrate in the preparedness phase with emphasis on awareness building and training. The weak areas in the urban infrastructure needs to be retrofitted. The earthquake performance of the critical structures and systems may also need strengthening. This report will review these critical issues with case studies from and suggest solutions.



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< previous page page_4 next page >Page 4ASSESSMENT OF THE EARTHQUAKE HAZARD IN URBAN AREASA rational earthquake hazard assessment methodology should provide for the uncertainties associated with the input parameters and be based on appropriate stochastic models. The purpose of such probabilistic earthquake hazard analysis is to provide a basis for decision making about the design basis ground motions applicable in a metropolis.Probabilistic Assessment of Ground MotionThe earthquake hazard is usually depicted as annual probabilities of exceedance for given ground motion (or intensity) levels. The probabilistic earthquake hazard assessment, in a rigorous way, were probably first initiated by Cornell [9]. Although the basic elements of his methodology remained the same, in the recent decade several researchers have tried to improve by addressing to the issues associated with high uncertainties. The development of criteria for the interpretation of alternative source zones and seismicities, and expert systems have been the focus of these efforts.

Earthquake Hazard in The probabilistic hazard assessment methodology that will be employed for will be an updated version of the one incorporated in [13]. It involves: Acquisition of geotectonic and seismologic data and seismic source modeling; Construction of recurrence relationships; Development of intensity based local attenuation relationships; and Use of a proper stochastic model for recurrence forecasting. For the computational part a computer program entitled SEISRISK III [6] will be utilized. Figure 1 shows the active tectonic elements to be considered for the earthquake hazard assessment for . The epicentral distribution of earthquakes are indicated in Figure 2. For the attenuation of intensities the relationships of Erdik et al. [13] will be used. For the attenuation of peak ground acceleration (PGA) the relationship developed by Campbell [7] are found to be appropriate on the basis of comparisons with Turkish data [13]. Figures 3 and 4 provide the variation of MSK Intensity and the PGA (at competent rock outcrops) for the northern and southern

for different return periods.Modification of the Ground Motion by Site Conditions:Due to urbanization the reclaimed land near the coast has been spread rapidly. The earthquake response of the reclaimed land and the soft alluvium can be much more amplified than that of the consolidated deposits, as observed in 1985 Mexico (great damage in the lake bed region) and 1989 Loma Prieta (collapse of the I–880 Cypress viaduct at the north end founded on bay mud) earthquakes. Several researchers have shown that for layers of given thickness, the relative shaking response will be greatest where the surface layers have the lowest impedance values and where the impedance contrast between the surface layer and the underlying one is the greatest. Joyner and Fumal [18] have incorporated the local shear wave velocity of near surface geologic material in the assessment of site effects for the attenuation of peak ground acceleration (PGA) and velocity (PGV). Their results indicate that, contrary to PGV, the site dependence of the PGA is not statistically significant. However this finding is not shared by others. For example, Fukushima et al. [17] have computed, on the basis of a worldwide strong



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< previous page page_5 next page >Page 5motion data set, the residuals between the observed and the predicted PGA’s and the mean values for each ground classification (rock, hard-, medium—and soft-soil ground). The observed PGA’s are about 40% lower at the rock site and about 40% higher at the soft soil site than those predicted, but the differences are insignificant for other soil sites.Roger et al. [23] using a collection of nuclear test explosion recordings, have correlated the gelogic structure with the mean spectral amplification of ground motion in the Los Angeles region. In the 0.3–2Hz frequency range the mean ground response on crystalline rock has the following values relative to:Age of surficial material: 3.2 for Holocene/Pleistocene, 1.7 for Pliocene/Miocene and 1.4 for Mesozoic;Quaternary thickness: 1.6 for Om, 2.3 for <75m and 3.6 for >75m;Depth to basement rock: 1.1 for Okm, 2.7 for <4km and 3.8 for >4km.As evidenced by in several earthquakes, the immediate vicinity of lateral discontinuities and contact zones between highly contrasting formations are also usually the zones of amplification. Amplification due to topography has been identified in theoretical as well as empirical studies. Celebi [8] has observed the topographical amplification phenomena in 1985 Central Chile and in 1983 Coalinga earthquakes. The top of isolated hills, elongated crests, edges of plateaus and cliffs are usually zones of amplification due to diffraction and focusing.Earthquake Induced Soil Failures and Terrain Movements:The most important earthquake induced soil failures are liquefaction, loss of strength and densification. Liquefaction involves [10]: Flow Failures (massive displacement of completely liquefied soil), Lateral Spreads (lateral displacement of surficial soil layers over a liquefied layer), Slumps (in steep banks underlain by liquefied sediments), Loss of Shear Strength (tipping or bearing failure of above ground structures, buoyant rise of underground structures). Liquefaction tends to begin at an intensity threshold of about MMI V–VI. Soil liquefaction occurred and caused much damage in 1989 Loma Prieta earthquake (Marina district in San Francisco) and in 1990 Philippines earthquake. Techniques to evaluate the liquefaction potential are well established and generally involve the preparation of two types of maps: one showing the liquefaction susceptibility and the other expressing the opportunity for critical levels of shaking. These two maps are merged to depict the liquefaction potential [26]. Tokida [25] lists the following criteria for liquefaction susceptibility (a conglomerate of Japanese criteria for bridges, water supply, sewage and buildings): (1) Saturated alluvial sandy layers within 20m from ground surface, (2) Ground water level within 10m from ground surface, (3) D50 values between 0.02 and 2mm in grain size acccumulation curve, and (4) Standard penetration test blow count N≤30.Earthquake induced terrain movements include landslides, rockfalls, and subsidence. Materials that are susceptible to earthquake induced landslides are: weakly cemented, weathered or fractured rocks; Loose unsaturated sands; saturated sand and gravel with layers sensitive clay. Experience show that most earthquake induced landslides involve mate-



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< previous page page_6 next page >Page 6rials that have not previously failed, and existing landslides are seldomly reactivated [10]. The probability that a landslide will occur on a particular slope during a particular earthquake is a function of both the pre-earthquake stability of the slope and the severity of the earthquake ground motion. According to Newmark [21] one of the measures of slope stability under seismic shaking is the acceleration required to initiate an irreversible displacement of the soil mass. Strength loss in sensitive clay in strong earthquakes may involve failures similar to liquefaction and specifically can initiate large landslides as is the case in 1964 Alaska earthquake. Urban area landslides (rock falls, soil slides, lateral spreads and slumps) can cause massive property damage. Transportations are blocked and lifelines are damaged disrupting the community services. Prudent siting, involving adequate setbacks from steep slopes, flattening cut slopes and avoidance of instability areas can mitigate the hazard. For massive landslide problems the risk can be accepted with appropriate emergency response preparedness.Tectonic Ruptures (Surface Fault Ruptures):The information about the movements and the surface expressions of possible active faults should be included in the microzonation maps to avoid (or to accomodate) their effects on structures and systems. It is generally regarded to be appropriate to consider faults that show evidence of Quaternary motion as active with a possibility of rupture.MICROZONATION MAPSSeismic microzonation maps can be defined as maps providing estimates of parameters needed for the siting and the earthquake resistant design of civil engineering structures and systems. The necessary information that should be conveyed by an earthquake hazard based microzonation map are: (1) Modification of the strong ground motion by site conditions; (2) Earthquake induced soil failures and terrain movement; and (3) Tectonic surface ruptures.Site Specific Ground MotionFor the incorporation of the “Site-Specific” ground motion in the microzonation maps there exists several analytical, empirical and experimental approaches. Analytical procedures range from simple one-dimensional calculations to three-dimensional, linear/non-linear, time/frequency domain and finite difference/element computations. Microzonation maps have been prepared using one-dimensional non-linear analytical procedures. These maps yield the input-motion-amplitude-dependent predominant periods and the peak surface accelerations [24].From analyses of microtremor records obtained at over 1000 locations in a wide variety of soil conditions, Kanai et al. [19] discovered that the time and frequency domain wave shapes of microtremors are distinctly different in different soil conditions and proposed two methods for the purpose of microzoning. One method attempts to delineate the four soil zones on the basis of the largest period and the mean period of the microtremors. The other one does the same using the largest amplitude and the predominant period of the microtremor measurements. The critics of this method claim that the microtremors originate at shallower



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< previous page page_7 next page >Page 7depths, may be nonstationary over different periods of the day and can provide information only on the low-strain behavior of the medium.Medvedev [20] attempted to relate the increments of seismic intensity to seismic site rigidity (product of the longitudinal seismic wave velocity by the density of the geologic layer) and to the elevation of the water table. The intensity increments for the basic categories of the ground with respect to bedrock (granite) is found to vary from 1 (firm ground) to 3 (loose fills). An additional intensity increment of 1 unit are considered for cases where the water table lies directly below the structural foundation level. The microzonation of Bucharest, Romania, which is based on the Medvedev’s method, had the opportunity of being tested by the 1977 (M 7.2) Romanian earthquake [4]. Erdik et al. [15] have compared the microzonation map of Bucharest with the intensity observations of this earthquake and found a negative correlation between the predicted and the observed intensities. Similar to the Mevedev’s method, in western USA tables providing changes of intensity has been used for microzonation purposes [16]. For California the relative intensity values for different ground characteristics vary from −3.0 for Granite and methamorphic rocks to 0.5 for saturated Quaternary sedimentary deposits.

Microzonation for The preliminary microzonation for the Central part of . the area within the ancient city walls, is based on the morphology, geology, the distribution of artificial fills and other geophysical and geotechnical data [15]. Figure 7 illustrates the four identified earthquake hazard zones. The stable rock zone (MSKI VIII) defines some part of the Carboniferious rock (where the artificial cover is little or none) and the late Miocene Mactra Limestone. The semi stable zone (MSKI VIII–IX) represents mostly late Miocene sand and gravel, and clay and marl. In this zone ground shaking hazard is somewhat increased and slopes are prone to land sliding. The zone encompassing the thick artificial cover (MSK IX) will be subjected to increase in the ground shaking. The zone of mud and fill (MSKI IX–X) delineates potential of ground failures such as liquefactions, fissuring and slumping. In Figure 7 the locations of potential earthquake induced landslides are also illustrated. It should be noted that, in the stable rock zone there might be local problems due to fracture planes versus slopes (e.g. fault and joints) and ground shaking may increase depending on the thickness of the weathering zone. A preliminary damage zone map for the 1509 and 1894 earthquakes is presented in Figure 8. The 1509 earthquake [3] was one of the largest in istanbul, killing about 5000 and injuring 10.000. Every single house had some degree of damage, in some places the ground opened up, sand ejected and a 6m high sea wave occurred. In the 1894 earthquake [22] (M=6.7, I=VIII) most damage took place on the Fatih-Beyazit ridge and slumping observed in Eminönü. The Grand Bazaar had heavy damage due to its “loose fill” type ground condition. Both earthquakes were in the Marmara Sea and related to the movement of the North Anatolian Fault.



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< previous page page_8 next page >Page 8EARTHQUAKE VULNERNERABILITY AND RISKIn technical terms earthquake risk is the probability of expected earthquake losses (such as: lives, injuries, physical damage and socio-economic). Indirect damage due to the disruption of industry, commerce and services should also be considered as losses. Seismic risk analysis attempts to calculate the probability of adverse socio-economic effect of an earthquake or series of earthquakes in a given urban center. A probabilistic seismic risk analysis takes into account the uncertainties inherent in the earth sciences and the engineering information. The process of rapid urbanization, the attendant socio-economic development, large scale constructions and the provision of infra-structural services will expose larger populations and valuable elements to earthquake hazards and risks. In many developing countries with increasing populations and inadequate housing the increase in the number of such buildings will bound to aggravate the earthquake related casualties and economic losses over the coming years. Several developing countries spend about 2% of their gross national product for post-earthquake reconstruction [14] and, in some instances, the losses caused by earthquake disasters have completely cancelled out any growth in the GNP [11].Earthquake vulnerability is defined as the degree of loss to a given element(s) at risk resulting from the occurrence of an earthquake. Vulnerabi1ity assessments are usually based on past earthquake damages (observed), on laboratory testing and, to a lesser degree, on analytical investigations (predicted). In addition to these physical vulnerability, the social vulnerability of urban population needs to be assessed for a comprehensive earthquake risk assessment. The past earthquake disaster experience indicate that single parent families, women, handicapped, children and the elderly constitute the most vulnerable social groups.Vulnerability of BuildingsIn urban areas the vulnerability assessments for engineered structures (e.g. residential, governmental and commercial buildings, bridges, dams, port and harbor structures, lifelines and utilities) and non-engineered structures (mainly squatter settlements) needs to be differentiated. The vulnerability of the engineered structures depends on the siting, design and construction essentials and defies generalizations. Generalizations can be made on the earthquake vulnerability of different building classes of non-engineered construction. Several researchers have provided deterministic vulnerability functions for different structures [e.g. 1, 12]. Worldwide data [1] indicates that the average damage ratios (i.e., cost of repair divided by the cost of rebuilding) unreinforced adobe and brick masonry structures are at least 4 to 5 times more vulnerable to receive damage than properly designed reinforced concrete and steel structures under the same earthquake exposure.Vulnerability of Turkish BuildingsIn this century only a limited number earthquakes in Tükiye have effected urban areas. The following vulnerability functions for three different classes of urban buildings have been obtained on the basis of damage and casualty data obtained from these events [5]. For low rise (2–5 story) reinforced concrete buil-dings, the percentage of the building stock to experience damage in an MSKI VIII are: 28±12% (no damage), 32±6% (slight damage), 26±9% (medium damage), 10±5% (heavy damage), 3±3%(collapse). The death rate is 0.45 persons per building. For low-rise unreinforced masonry buildings, the percentages are 36% (medium damage), 37% (heavy damage) and 27% (collapse). The death rate being 0.77 persons per building.



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< previous page page_9 next page >Page 9Vulnerability of Other StructuresMany other engineered urban structures, infra-structures, life-lines and services are vulnerable to the effects of earthquakes. Landslides, rock falls and fault ruptures can block highways and railways or damage pipelines. Strong shaking can cause transmis-sion lines and bridges to fail. Liquefaction can cause failure of port and harbor structures. The earthquake vulnerabilities of these structures and systems are not generally known in explicit formats. In any case these vulnerabilities are highly case- and site- specific and defy the general use. However the following observations acquired from past urban earthquakes can be used as a guide to assess their earthquake performance [10].BridgesSlope instability, liquefaction. settlement can move the abutments of bridges and tilt the piers causing extensive damages to bridges. Liquefaction phenomena can start at intensities as low as MMI V–VI. The bridge girders can fall off of their supports. Seismic restrainers that tie the adjacent simple spans prevent the fall off of the girders. The continuous span bridges should be tied together at the expansion joints.Building FoundationsOutside the zones of faulting, liquefaction and ground failure the foundation failure due to strong shaking is very rare, with the exception of friction piles set in soft clays. In 1985 Mexico earthquake the penetration and/or poll out of such piles caused the tilting of the pile cap and, consequently, the superstructure.Retaining StructuresThe increased lateral soil and water pressures, loss of bearing strength and liquefaction have seen to cause damage to the retaining structures.TunnelsOutside the zones of faulting and landslide, the tunnels generally perform well during the earthquakes. Damage to cut-and-cover type tunnels has been caused mainly by increased lateral pressure from the backfill.Water Supply and SewageGreatest damage to pipeline occurs in zones of faulting, liquefaction and landslide. Ductile pipes and flexible connections have the best earthquake performance.Electrical Transmission and Distribution SystemsHigh voltage porcelain insulators, bushings and supports are most vulnerable to earthquakes. Damage generally occurs in improperly anchored electrical equipment.TelecommunicationsThe anchorage of switching and battery racks against lateral displacement and toppling is the essential measure to avoid earthquake damage.



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< previous page page_10 next page >Page 10Gas and Liquid Fuel LifelinesThese lifelines are vulnerable to large differential ground movements. Quality of weld is the important factor for the earthquake performance of steel pipes. Ruptured gas lines lead to leaks and fire hazard.Ports and harborsLarge scale liquefaction and sliding of the soil (or between the blocks) can lead to damage in port and harbor structures.Building ContentsModern buildings can suffer major functional and economic loss by damage to the equipment and furniture it houses even though the structure experiences little damage. Especially in research laboratories, hospitals and offices the non-anchored equipment are highly vulnerable to earthquake damage. The same also applies to the exhibited pieces in museums and in art galleries.MITIGATION OF URBAN EARTHQUAKE RISKEarthquake risk in urban areas can be reduced by either not building or moving away from the hazardous areas, which in either case is unrealistic. What remains is then the reduction of vulnerability, in terms of casualties, material losses and socio-economic losses. In urban areas, the process of anticipating and planning for damage that a major earthquake would eventually create is termed as “earthquake disaster management”. It is an unbroken chain of concerted actions involving: disaster, response, relief, rehabilitation, reconstruction, risk reduction, mitigation, preparedness and (if possible) warning. The earthquake disaster preparedness and the mitigation constitute the two of the important activities of the earthquake disaster management. For any earthquake disaster management program, the public awareness building, information dissemination and the training of personnel constitutes the fundamental ingredient of success [14].Major losses of life in the past earthquakes in urban areas have occurred due to the collapse of buildings with insufficient earthquake resistance or with inappropriate siting considerations. The facilities provided by the metropolitan governments that are essential for the operation of the socio-economic system (sanitary services, utilities and, health services etc.) should be designed with lowest vulnerabi1ity levels. For example: the collapse of the central fire station in 1972 Managua, Nicaragua earthquake endangered the fire fighting; and the much needed ambulances were damaged under the collapsed canopy in Olive View Hospital in 1971 San Fernando, California earthquake. Transportation facilities (highways, railroads, port and harbors, airports and bridges etc.) are vital for rescue and recovery efforts. Redundancy in transportation networks is essential for rapid restoration. Metropolitan governments should also be responsible and take necessary measures to protect the cultural heritage through the maintenance and retrofitting of monuments and museums. Most of these issues can be addressed through proper planning, microzonation and appropriate construction technologiesThe pre-earthquake restrengthening and retrofitting of critical urban infrastructure, facilities and hazardous buildings is an important physical measure for earthquake risk mitigation. In this respect, the


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< previous page page_11 next page >Page 11action taken by the city of Los Angeles should serve as a model. Noting that “the pre-1934 unreinforced masonry buildings represent the greatest single threat to the life and limb in Los Angeles in the event of a major earthquake”, in 1981, the Los Angeles City Council passed an ordinance (No:154, 807) requiring the strengthening or removal of pre1934 buildings that have bearing walls of unreinforced masonry.For mitigation of urban earthquake disasters the necessary plans, programs and the activities can be listed as follows under the pre-, co- and post-earthquake phases:Preparedness (Pre-earthquake) Planning and ActivitiesInstallation of earthquake data acquisition and monitoring stations and services; Assessment of earthquake hazard (seismo-tectonic mapping); Development of earthquake resistant design codes and construction standards; Pre-disaster planning and management activities and techniques; Disaster awareness, public information, education and training; Development of methods for retrofitting hazardous buildings and facilities; Development of appropriate techniques for repair and strengthening of non-engineered low-strength constructions; Creation and strengthening of programs and organizations for the prevention of earthquake disasters; Hazardous material management; Legislative and regulatory measures; Response readiness; Logistical support; Resource management and stockpiling; Mobile command and communication operations.Emergency (Mid-disaster) Planning and ActivitiesEmergency rescue, evacuation, transportation and communication; Damage assessment, demarcation and condemnation of dangerous buildings and zones; Debris removal; Recovery and disposal of dead bodies; Emergency provision of health care, shelter. food and utilities; Human response and information management; Law enforcement; Quick assessment socioeconomic losses; Planning and coordination of disaster assistance.Post Earthquake Planning and ActivitiesDetailed surveys regarding repair, restoration and condemnation; Assessment of socioeconomic conditions, resources and needs; Measures and policies for relief, resettlement and rehabilitation; Re-establishment of government services; Institutional framework, implementing agencies; Hazard abatement; Disaster accounting; Planning and coordination of rehabilitation and reconstruction assistance; Siting of new settlements and communities; Retrofit of design codes and construction standards; Training and education programs; Reconstruction.REFERENCES1. Akkas, N. and M.Erdik (1984), Considerations on Assessment of Earthquake Resistance of Existing Buildings. Int.Jour. for Hous.Sci., v.8.2. Algermissen, S.T., K.V.Stinbrugge, and H.J.Lagorio (1978),“Estimation of Earthquake Losses to Buildings”, USGS, Open File Report No: 78–441.3. Ambraseys, N.N. and Finkel, C.F. (1990), The Marmara Sea earthquake of 1509. Terra Motae, 167–174.4. Balan, S., V.Cristescu and I.Cornea (1982), Curemurul de pamint din Romania de la 4 martie 1977, Editura Academiiei, Bucharest.



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< previous page page_12 next page >Page 125. Bayülke, N. (1982), Building Types in Bolu Turkey and Their Predicted Earthquake Damages, in Sismic Risk Assessment and Development of Model Code for Seismic Design, UNOP/UNESCO Project RER/79/014, Sofia.6. Bender,B. and D.M.Perkins (1987), SEISRISK III: A Computer Program for Seismic Hazard Estimation, U.S.G.S., Bulletin 1772, 48p.7. Campbell, K.W.(1981). Near-source Attenuation of the Peak Horizontal Acceleration, Bull. Seism. Soc. Am., 68, 828-843.8. Celebi, M. (1988), Topographical Amplification - A Reality?, Proc. 9WCEE, Tokyo-Kyoto, Japan, pp 11-459..9. Cornell, C.A. (1971), Engineering Seismic Risk Analysis, Bull. Seism. Soc. Am., v.58, p.1583.10. EERI (1986), Reducing Earthquake Hazards: Lessons Learned From Earthquakes, EERI Publ. No: 86-02, San Francisco, California11. Einhaus, H. (1988), Emergency Planning and Management for Disaster Mitigation, Regional Development Dialogue, v.9, No.l, UNCRD, Japan.12. Erdik, M. (1987), Training and Education for Disaster Preparedness, Regional Development Dialogue, v.9, No.l, UNCRD, Japan.13. Erdik, M., V.Doyuran, N.Akkas, P.Gülkan (1985), A Probabilistic Assessment of the Seismic Hazard in Turkey, Tectonophysics, 117.14. Erdik, M.(1990), Disaster Management Education on Earthquakes, Proc., IDNDR International Conference 1990, pp.383-388, Yokohama, Japan.15. Erdik, M., A.Barka and B.Ücer(1991), Seismic Zonation Studies in Türkiye: an Overview, Submitted to 4th Conf. Seism. Zonation, San Francisco.16. Evernden, J.F. and J.M.Thomson(1985), Predicting Seismic Intensities, in Evaluating Earthquake Hazards in the Los Angeles Region, USGS Professional Paper No:1360, US Government Printing Office, Washington.17. Fukushima, Y.,T.Tanaka, and S.Kataoka(1988), A New Attenuation Relationship for Peak Ground Acceleration Derived from Strong Motion Accelerograms., Proc. 9th World Conf. on Earthq. Eng., Tokyo, Japan18. Joyner,W.B. and T.E.Fumal (1985), Predictive Mapping of Earthquake Ground Motion, in Evaluating Earthquake Hazards in the Los Angeles Region, USGS Prof. Paper No:1360, US Gov. Printing Office, Washington.19. Kanai.K., T.Tanaka, K.Osada and T.Suzuki (1966), On Microtremors-X, Bull. Earthq. Res. Inst., v.44, Tokyo, Japan.20. Medvedev, S.S. (1965), Engineering Seismology, Translated from Russian, Israel Program for Scientific Translations, Jerusalem, 1965.21. Newmark, N.M. (1965), Effects of Earthquakes on Dams and Embankments, Geotechnique, v.15, pp.139,160.

22. and N.Bayülke (1990), Historical Earthquakes of , Kayseri and Elazig, General Directorate of Disaster Affairs, Ministry of Public Works and Settlement, Ankara, Turkey. 22pp.23. Roger, A.M., J.C.Tinsley and R.D.Borcherdt (1985), Predicting Relative Ground Response, in Evaluating Earthquake Hazards in the Los Angeles Region, USGS Prof. Paper No:1360, US Gov. Printing Office, Washington.24. Sugimura,Y. and I.Ohkawa, (1984), Seismic Microzonation of Tokyo Area, Proc. 8th.WCEE, v.2, pp.721-728, San Francisco, California.25. Tokida, K. (1990), Earthquake Disaster and Approach to Damage Reduction, Proc. ESCAP/UNDRO Regional Symposium on the International Decade for Natural Disaster Reduction, Bangkok.26. Youd, L.T., J.C.Tinsley, D.M.Perkins, E.J.King and R.F.Preston (1979), Liquefaction Potential Map of San Fernando, California, in Progress on Seismic Zonation in the San Francisco Bay Region, USGS Circ. No.807



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< previous page page_13 next page >Page 13

Figure 1. Active Fault Segments in the Marmara Region South of (After Barka and Kadinsky-Cade, 1988)

Figure 2. Epicentral Distribution of Damaging Earthquakes in the Marmara Region, South of .

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Figure 3. probabilistic MSK intensity for North and South

Figure 4 probabilistic PGA(Rock Outcrop) for North and South



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Figure 5. Preliminary Microzoning Map of (after [15]).

Figure 6. Approximate damage zonation map for the 1509 and 1894 earthquakes in (after [15]). (Dashed contours is for 1984, solid contour is for 1509 earthquake)


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< previous page page_15 next page >Page 15Probabilistic Method in Maximum Earthquake Assessment*V.Schenk, P.KottnauerDepartment of Seismology, Geoph. Inst., Czechosl Acad. Sci., CS-141 31 Praha 4—Spo•ilov, CzechoslovakiaINTRODUCTIONThe “maximum possible (or expected) earthquake” is one of the most important input parameters in seismic hazard calculation techniques. Among the different probabilistic approaches usually used in the determination of the “maximum possible earthquake” is the method of extreme values, frequently called the Gumbel method. The application of the third Gumbel distribution and a method of determining the accuracy of the maximum earthquake estimate are presented and discussed on a few examples.APPLICATION OF THE THIRD GUMBEL DISTRIBUTIONAn application of the method of extreme values [1] requires a relatively long sequence of observations of “extreme values”, in our case, e.g. a sequence of “maximum observed earthquakes”. The time series of observations of T years has to be divided into intervals of a certain duration (one year, 5 years, 10 years, 3 months, etc.). From each interval a single extreme value is taken into the calculation: that of the maximum earthquake having occurred in the interval. If there is a sufficient amount of intervals for which the maximum earthquake is known, then the obtained estimate of the maximum earthquake (the asymptotic value of the 3rd Gumbel distribution) is assumed to be a representative value. * Contribution of the Geoph.Inst.,Czechosl.Acad.Sci., No. 21/91.



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< previous page page_16 next page >Page 16We do not doubt that the value is representative indeed, but in some cases it has to be proved and the accuracy of the maximum earthquake determination assessed. As mentioned above, for an application of the Gumbel statistics an interval of a certain duration (years, months, weeks, etc.) has to be defined. Each interval can be further subdivided into shorter and shorter time intervals. It is understandable that the subdivision cannot be applied without a restriction. It is obvious that the number of observed earthquakes and their frequency of occurrence in a given area act controversially.As an example let us take the time series of earthquake observations for the period of 1700–1985, i.e. 286 years, and apply a 10-year interval. It means that for the first interval of 1700–1709 we obtain one value of the maximum earthquake, for the second interval of 1710–1719 the second value of the maximum earthquake, and similarly the other values of maximum earthquakes up to the last one the for interval of 1970–1979. For this case we shall have a sequence of 28 values of maximum earthquakes.However, the original time series of earthquake observations can be divided into other nine possible combinations of 10-year intervals: we can start with the interval of 1701–1710 or with the interval of 1702–1711 and sequentially we can reach the last possible interval 1709–1718 (Fig.1). In this way we obtain ten different more or less similar sets of the maximum earthquake values: seven sets of 28 values of maximum earthquakes and three sets of 27 values of maximum earthquakes.These combinations were created under the shifting interval of one year. Of course if the shifting interval is half a year, then we will obtain fourteen sets with 28 values of maximum earthquakes and six sets with 27 values of maximum earthquakes. It is evident that for one time series of observations there could be a great number of combinations of applied intervals, for which the maximum earthquake has to be found, their overlapping being given by the shifting interval. Such a combinatory approach extends the standard way of the application of the Gumbel statistics, especially its 3rd Gumbel distribution, to another dimension: for an assessment of the approximation of this distribution.



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Figure 1.ACCURACY OF THE MAXIMUM EARTHQUAKE DETERMINATIONFor the whole time series of observations let us introduce for example a shifting interval S equal to one year and divide the time series into one-year intervals D in which the values of maximum earthquakes are determined. We obtain only one Gumbel approximation. If the same (one-year) shifting interval S is applied to the time series which is divided into two-year intervals D, in which the values of maximum earthquakes are determined, then we obtain two Gumbel approximations. Likewise, for the one-year shifting interval S applied to the time series which is divided into ten-year intervals D gives us ten different Gumbel approximations.For a better explanation of this approach the following example is demonstrated. Using the catalogue of



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< previous page page_18 next page >Page 18Italian earthquakes [2], we selected the subcatalogue for the Friuli region (northern Italy). In this catalogue the earthquake size is given in epicentral intensity of MCS. The Friuli region under study was delineated by geographic coordinates from 45°50’N to 46°36’N and from 12°50’E to 13°50’E . The subcatalog of observed earthquakes contains 1764 events from the period of 1700–1980 with the maximum observed earthquake of 9.5°MCS. In our calculations we applied only one combination of shifting interval S equal to 1 year and interval D equal to 30 years. Figure 2 shows the distribution of all approximations (23 cases) for which the 3rd Gumbel extreme value distribution has a convergent character. The fact that for 7 cases the statistical process had not a convergent solution was rather surprising and has to be explained in the near

Figure 2.future. This finding is very important from the point of view of practical applications. It gives evidence that our idea concerning the “representativeness” of the 3rd Gumbel approximation need not be necessarily always valid.The values of the maximum possible earthquakes (MPE) of all 23 convergent Gumbel approximations were analyzed for three different return periods of 120, 1290 and 15500 years and then they were compared with their asymptotes. Figure 3 shows the changes of the MPE values in dependence on the thirty different positions of the beginnings of intervals D with respect to their first positions. We can see that for return period of 120 years minimum values of MPE were obtained for cases in which the beginning of interval



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< previous page page_19 next page >Page 19D is situated approximately in the centre of the original interval D, i.e. for the total shift equal to 10≈18 interval S. This conclusion is not very important because the original beginning of the earthquake time series could be shifted and then we can obtain quite easily an opposite result. But what is extremely important is the fact that for the same cases we obtain quite opposite results for higher return periods, and consequently, for the asymptotes too, e.g. for these cases the maximum values of the MPE are determined. An explanation of this feature does not seem to be very simple and probably special tests have to be made in order to clarify it.

Figure 3.For each set of the MPE values obtained for selected return periods the “mean value of the maximum possible earthquake MMPE” and its “standard deviation MSD” were determined. These quantities obtained for the Friuli seismogenic region by the application of the 3rd Gumbel extreme value distribution are drawn in Figure 4. Such a chart informs us immediately about the representativeness of the 3rd Gumbel approximation for the given subcatalogue of earthquakes. We can see that the best approximations and thus the MMPE value with the highest degree of a representativeness, because of the lowest values of the MSD, belongs to the return periods which are close to the middle of the observation period; in our case about 350 years. For higher return periods the standard deviations increase and the degree of the representativeness of the resulting MMPE values becomes slightly lower. Numerically, the MSD for the return period of 350 years is around 0.5% of the MMPE, but for the return period of 15500 years it is as much as 6.5%, at-



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Figure 4.taining 9% for the asymptote of the MMPE value. We assume that the obtained accuracy of the MMPE can be accepted as a general degree of representativeness for the maximum possible earthquakes determined by the 3rd Gumbel approximations.CONCLUSIONThe described statistical approach allows us to estimate the accuracy of the approximation obtained by the 3rd Gumbel distribution in a prediction of the maximum earthquake for given return periods of earthquake occurrences. These estimates do not only define the resulting predicted values but also give their possible variance. Such characteristics are quite important from the economic point of view, because for example, in tasks of the earthquake hazard assessments the determination of the maximum possible earthquake directly affects the total cost of seismic resistant structures. Such predicted values of maximum earthquakes also help in calculations of the seismic risk and can make a contribution in some questions of earthquake mitigation.



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< previous page page_21 next page >Page 21REFERENCES1. Gumbel E.J., 1954: Statistical Theory of Extreme Values and Some Practical Applications. Nat.Bureau of Standards, Appl.Math.Series 33, U.S.Govt.Print. Office.2. Postpischl D., Ed., 1985: Catalogo dei terremoti italiani dal’anno 1000 al 1980. CNR, PFG, Quad. Ric. Scie. 114–2B, Graficoop, Bologna.



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< previous page page_23 next page >Page 23Study of an Assessment for Site Effect of Seismic Strong MotionE.Kuribayashi (*), T.Jiang (**), T.Niiro (*), H.Nagasaka (***), S.Kuroiwa (****), S.Nishioka (*)(*) Dept. of Civil Engineering/Regional Planning, Toyohashi University of Technology, Toyohashi, Japan(**) Dept. of Geotechnical Engineering, Tonji University, Shanghai, The Peoples Republic of China(***) Kumagai-gumi Co., Ltd., Toyokawa, Japan(****) Nagano Prefectural Government, JapanABSTRACTEffects of sediment-filled valley on seismic ground motions have become of major interest in earthquake engineering throughout this decade. This paper presents interesting phenomena of both analytical and experimental approaches.INTRODOCTIONDisasters caused by earthquakes are generally complicated. The earthquake damage strongly depends on the subsoil conditions and topography from the past experience of severe earthquake damage.In order to prove the effects of topographical and geological conditions in behavior of ground motions, a strong motion observation network so called TASSEM, Toyohashi University of Technology Array System for Strong Earthquake Motions, has been developed since 1989. They are located around Toyohashi city, east part of Aichi prefecture, that is regarded as one of the most vulnerable areas to destructive earthquakes designated by many seismologists.Several records of the strong motion observation have brought a reasonable results among analytical results using one and two dimensional analyses and consequences in microtremor and strong motion observations. From analytical results, amplification would not be influenced very much by the direction and angle of incident wave, but by the topographical conditions.



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< previous page page_24 next page >Page 24In addition, it is clear that Boundary Element Method is an effective tool to estimate the behavior of responses in symmetric valleys subjected to incidental waves.PAST EARTHQUAKES AND THEIR DAMAGE IN AICHI PREFECTURE [1] [2] [3] [4]There are no large-scale mountains in Aichi prefecture. The crustal movement in the area is very complicated and active in Honsyu (the mainland of Japan). In this area, Median Tectonic Line that divide the south-west part of Japan into two areas called Inner Zone and Outer Zone, is lying from direction of N.E. to S.W. Many active faults exist in inland and off or along the ocean coast. This area has suffered great disasters many times caused by great earthquakes and is one of the most vulnerable regions to destructive earthquakes. Fig. 1 shows distribution of the epicenters of past major earthquakes.The earthquakes brought destructive damage to this area in historical time are classified into two types. One is the earthquakes off or along the Pacific coast. Another is the inland earthquakes. In recent decades, typical earthquakes which caused major damage are, the earthquake off or along the Pacific coast, 1944 Tonankai earthquake with magnitude of 8.0 in Richter scale, the inland earthquake, 1945 Mikawa earthquake with magnitude of 7.1 in Richter scale. In Tonankai earthquake, the damage caused by liquefaction was concentrated in alluvial plain. In Mikawa earthquake, the damage was concentrated the south side of Fukouzu fault which continues for 28km from seabed of off Gamagouri in Mikawa Bay to Yahazu River as shown in Fig. 2. In these two earthquakes, severe damage was observed in the 5 or 6th degree on Japan Meteorological Agency scale in Aichi.

Fig. 1 Distribution of the of Past Major Earthquakes


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Fig. 2 Epicenters of Tonankai and Mikawa Earthquake



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< previous page page_25 next page >Page 25In the area around the western half of Suruga Bay, or the Tokai area, there have been no earthquakes since the one which occurred in 1854. This is the only place along the Pacific coast where no large scale earthquake has occurred in the past 100 years. The Japanese Government has prescribed Large-scale Earthquake Countermeasures, and has designated the Tokai region as one of the Areas Under Intensified Measures Against Earthquake Disaster.ARRAY SEISMIC MOTION OBSERVATION SYSTEMIn order to prove the effects of topographical and geological conditions in behavior of ground motions, a strong motion observation network so called TASSEM has been developed around Toyohashi city, Toyohashi University of Technology as a center station, that is regarded as marginal area near one of the most vulnerable areas to destructive earthquakes. Geological and Topographical Aspects [5] [6]The object area of seismic observation is Toyohashi city located east part of Aichi. Topographical aspects of Toyohashi is generally classified into three areas; (1) the hilly land and the terrace area, (2) the alluvial plain and (3) high lands. The geological feature is made up of the Paleozoic, the Quaternary and the alluvium. The sedimentary layers are consisted of marines without any igneous and metamorphic rocks.PaleozoicThis area is composed of the Paleozoic Chichibu zone. It is the base of the diluvial formation widely distributed most part of Toyohashi and reveal at the highland of east of Toyohashi. Paleozoic is composed of chart, mudstone, sandstone, etc. and runs generally in direction from east-north-east to west-south-west and has a tendency to incline towards the north or south vertically. Around the object area, it exists about 200m under sea level.DiluviumDiluvial formations mainly consist of gravel, sand and silt and form the hilly land and terrace distributed most part of Toyohashi. These are almost horizontally laid on. Caused by the crustal movement called “Atsumi upheaval movement” by Kuroda [7], they incline slightly from south to north.AlluviumAlluvium mainly consist of gravel, sand and silt, they have not harden enough yet, and is widely distributed the basin of river.Location of the Observation SystemThe arrangement of the observation points is shown in Fig. 3 and 4. Fig. 5 shows the distribution of standard penetration values, N, at each observation point. Table 1 shows site information.



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< previous page page_26 next page >Page 26Accelerometers were installed in December, 1989 ground surface, actually one meter below the surface, of three sites which geological and topographical features are different respectively. POINT 1 is at Hongo Junior High School located on the center of the valley, Umeda River runs east-west and the ground surface is covered with the soft alluvial deposit. Two points locate in the left side of the river, terrace area composed from diluvial layers and called Tempaku-hara Terrace, Tempaku Elementary School as POINT 2, Toyohashi University of Technology as POINT 3. Moreover, as a POINT 4, at Toyohashi Fire Department located in the right side of the river, terrace area composed from diluvial layers and called Takashi-hara Terrace, supplemental observation is being done. From Fig. 5, it is clear that the thickness of soft layer.

Fig. 3 Arrangement of the Observation Points

Fig. 4 Cross Section of the Observation Site

Fig. 5 Distribution of N values at Each Observation PointTable 1 Site Information of the Observation Points


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Location Specific Height Surface Layer Depth of Accelerometer

POINT 1 N34º42.7′ E137°24.0′ 6.2m Silt, Gravel GL-1m

POINT 2 N34°42.5′ E137º24.9′ 21.0m Clayey Sand, Gravel GL-lm

POINT 3, 3B N34°41.9′ E137°24.7′ 39.7m Sand GL-1m, −60m

POINT 4 N34°43.4′ E137°24.3′ 25.0m Clayey Sand, Gravel GL



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< previous page page_27 next page >Page 27Observation of base rock motion has been carried out at the layer composed of gravely sand under ground -60m at T.U.T. as POINT 3B since January, 1991 in present system.Specifications of the Observation SystemBy employing an advanced electronic technology, seismological recording techniques have made remarkable progress. TASSEM has been ordinarily composed to obtain the fairly distinctive data and easy data management. Table 2 shows the specifications of the system. As a practicable means, the observation center at T.U.T. controls the branch observation points by using of telephone line. Fig. 6 shows outline of TASSEM. TASSEM is provided with the following remarkable functions.1) By an advanced technology, wide dynamic range and frequency range can be acquired in the system.2) By watching the state of system operation constantly, the center can get the certainty and reliability of total system operation.3) Recorded data can be sent to main computer at TUT through the telephone line directly and made visible easily.4) All seismic observation parameters such as trigger level, record length, sampling frequency, delay time, and correction of time can be easily controlled from the center.5) As a counterplan against the power failure, the observation can be continued for three hours or more by use of back-up battery. Table 2 Specifications of the SystemAccelerometer Triaxial Force Balance Servo Type

Frequency Range 0.02–30 Hz overall

Measurement Range ±1000 gal

Dynamic Range 84 dB overall

Low Pass Filter 30 Hz, −18 dB/oct.

A-D Converter 14bits, Sampling Rate 100 Hz

Internal Memory IC memory: 1.25 Mbyte, Froppy Disk: 1.25 Mbyte

Telemetering Public Telephone Line, Data Transfer Rate 4800bps

Fig. 6 Outline of the system



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< previous page page_28 next page >Page 28Observation ResultsAt present time, several ground motion records have been obtained. Max. acceleration of ground motions observed by TASSEM is shown in Table 3. As one of the largest records, 06:13:07 Sep. 24, 1990, Fig. 7 shows Fourier spectra calculated and smoothed by using Hamming type window. From the Fig. 7, peculiar peaks are shown in each observation point. The Fourier spectra in POINT 1 and 2 which thickness of soft layer is similar have a flat and wide peak in a short period range. In POINT 3 which thickness of soft layer is comparatively thick, the spectra has the peculiar peak around 3Hz. Up to now, strong motion data have never been obtain at POINT 3B (GL-60m at TUT).Spectral ratio between the surface and the base, POINT 3B, for microtremor data observed by TASSEM in each observation point is shown in Fig. 8. In case of POINT 3/POINT 3B, peaks are shown in range from 2 to 3Hz and in case of POINT 1/POINT 3B and 2/3B, large amplification is shown in high frequency range more than 5Hz. Table 3 Max.Acceleration Observed by TASSEM (gal)

Date Feb. 20, 1990 Apr. 13, 1990 May 17, 1990 Sep. 24, 1990 Sep. 24, 1990

Epicenter N34º46' N35º9' N34°45' N33º6' N33º8'

E139º14' E136º31' E137°37' E138º38' E138º36'

Depth 6km 40km 33km 60km 42km

Magnitude 6.5 4.4 3.4 6.6 6.0

Direction EW NS UD EW NS UD EW NS UD EW NS UD EW NS UD

POINT 1 – – – 4.1 4.2 1.7 4.3 10.5 2.6 11.2 15.1 3.9 2.3 3.2 1.1

POINT 2 5.5 5.2 2.5 3.7 3.7 2.0 7.1 6.5 4.4 14.9 14.7 6.7 4.1 3.7 1.5

POINT 3 – – – – – – 4.0 2.2 1.7 11.1 15.9 5.4 3.3 4.0 1.3

Fig. 7 Fourier Spectra of the Observation Points


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Fig. 8 Spectral Ratio



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< previous page page_29 next page >Page 29ANALYSES OF LOCAL TOPOGRAPHY AND GEOGRAPHY [8]In order to confirm the effects of topographical and geological conditions in amplification characteristics of ground motions, linear response analyses are carried out by one-dimensional Multiple Reflection Method and two-dimensional Finite Element Method.Soil properties needed in one and two dimensional analyses are obtained from field tests, soil types and N values, of each observation point and shown in Table 4. Damping ratio 5% is used in analyses. S-wave velocity is induced by following equation.

[9]Two-dimensional model is shown in Fig. 9. The model is formed of 7000 nodal points. Viscous boundary as its side boundary and rigid base as its base are regarded to the boundary conditions of the model. Calculated transfer functions in each point of observation are shown in Fig. 10. All results between one and two dimension show difference obviously on account of influence of the surface configration. Peaks of natural frequency of two-dimensional analyses tend to be lower than the one-dimensional’s.As compared with the spectral ratio of observed data, value of magnification is differ in absolute ordinate, similar tendency is however shown in frequency domain.Besides, two-dimensional analyses are carried out with the different incident angles, θ=0°, ±5°, ±10°, ±15°, in order to prove the effects of the direction and angle of incident wave in amplification characteristics of ground motion. The sign+stands for the incident direction from left side of model. Fig. 11 shows the transfer function with different incident angles. From the Fig. 11, the amplification was not influenced very much by the direction and angle of incident wave in any of point within the incident angle ± 10°, but by the topographical conditions.

Fig. 9 Two-dimensional Analytical Model



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< previous page page_30 next page >Page 30Table 4 Soil Properties Used in Analyses(a)POINT 1

Layer (m) Thickness Soil Type υ γ (t/m3) Vs (m/s) G (t/m2)

No.1 1.6 Silt 0.49 1.60 145 3450

2 2.2 0.49 1.65 130 2850

3 2.2 Gravel 0.46 2.10 300 19300

4 2.0 Sand 0.47 2.00 315 20250

5 2.3 0.47 2.00 315 20250

6 1.7 Gravel 0.46 2.10 350 26250

7 2.7 0.44 2.10 450 43400

8 0.8 Sand 0.48 1.85 350 23125

9 1.8 Gravel 0.44 2.10 450 43400(b)POINT 2

Layer (m) Thickness Soil Type υ γ (t/m3) Vs (m/s) G (t/m2)

No.1 2.5 Sand 0.49 1.60 170 4700

2 0.6 0.49 1.95 220 9650

3 0.6 0.49 1.80 205 7700

4 1.3 Silt 0.49 1.75 220 8650

5 7.0 Gravel 0.45 2.10 400 34300

6 7.0 0.45 2.10 400 34300

7 7.1 0.45 2.10 400 34300

8 5.5 0.44 2.10 450 43400(c)POINT 3Layer (m) Thickness Soil Type υ (t/m3) Vs (m/s) G (t/m2)

No.1 2.5 Sand 0.48 1.75 180 5800

2 2.5 0.48 1.75 190 6450

3 1.9 0.49 1.60 145 3450

4 1.9 0.49 1.60 155 3900

5 1.9 0.49 1.70 180 5600

6 2.0 0.48 2.00 255 13250

7 1.9 0.48 2.00 210 9000

8 2.0 0.48 2.00 235 11250

9 0.7 Clay 0.48 1.80 200 7350

10 1.5 Gravel 0.46 2.10 350 26250

11 1.8 Sand 0.48 1.95 280 15600

12 1.1 Gravel 0.44 2.10 450 43400

13 1.3 Sand 0.49 1.80 200 7350

14 0.6 Gravel 0.46 2.10 270 15600

15 0.7 Clay 0.49 1.75 200 7150


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16 7.1 Gravel 0.45 2.10 400 34300

17 7.1 0.45 2.10 400 34300

18 7.1 0.45 2.10 400 34300

19 5.5 0.44 2.10 450 43400

Fig. 10 Result of Response AnalysesFig.11 Results of Response Analyses with Different Incident Angles



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< previous page page_31 next page >Page 31ANALYSES OF SIMPLE MODEL OF MEXICO VALLEY BY BOUNDARY ELEMENT METHODBased on Ref. [10] (Kuribayashi et al.), the boundary element method using the half space fundamental solution is used for the response analyses of symmetric valleys subjected to incident SH waves and the vibration amplification characteristics. Analytical simple model took the case of Mexico Valley and adopted parameter are shown in Fig. 12 and Table 5. Response analyses are carried out with the different incident angles, 0°, 30°, 60°, in both cases that the soft layer exists or not. Fig. 13 shows analysed amplitude ratio between surface and base.As results; (1) because of being with the soft layer, amplitude ratio is distinctly larger than the case without the soft layer, (2) in both sides of valley, amplitude ratio is larger than the other parts of valley, (3) effects of different incidental angle are only a little. These results are equivalent to actual disaster in Mexico Valley in 1985.

Fig. 12 Analytical ModelSL: Soft Layer ML: Middle Layer BL: Bed LayerTable 5 Soil ParametersLayer γ (t/m3) Vs (m/s) h (%)

SL 1.40 70 5

ML 1.80 500 0

BL 2.16 1250 0

Fig. 13 Amplitude Ratio



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< previous page page_32 next page >Page 32CONCLUSIONSAichi prefecture is regarded as one of the most vulnerable area to destructive earthquakes. TASSEM developed in Toyohashi, east part of Aichi, is an earthquake observation network and has the sufficient specifications as the high-fidelity observation system. [11]The amplification characteristics in frequency domain around TASSEM are proved from analytical results and consequences in microtremor and strong motion observations. [11]From analytical results, amplification would not be influenced very much by the direction and angle of incident wave, but by the topographical conditions.In addition, it is clear that Boundary Element Method is an effective tool to estimate the behavior of responses in symmetric valleys subjected to incidental waves.ACKNOWLEDGEMENTSThis research was supported by a grant from subsidy of Science Research Fund, Ministry of Education, Science and Culture through 1989 to 1990 fiscal year: this support is gratefully acknowledged.REFERENCES1. Conference on Prevention of disasters in Aichi Prefecture, “Anti-disaster Plan in Aichi Prefecture”, 19902. Asada, T., 1988, “Earthquake Prediction Study in Japan”, Proc. 9th WCEE, Aug. 2–9, 1988 Tokyo-Kyoto, Japan, Vol 2, pp13–193. Usami, T., “Damage Caused by Major Earthquakes in Japan—New Edition”, University of Tokyo Press, 19874. Ishibashi, K., “Specification of Soon-to-occur Seismic Faulting in the Tokai District, Central Japan, Based upon Seismotectonics”, Earthquake Prediction, An International Preview, 4, 527–532, Amer. Geophys. Union, 1981, Washington D.C.5. Yamashita, N., et al., “Tyubu Region II”, Geology of Japan 5, Kyoritsu Press Co. Ltd., 19886. Secretariat of Conference for Preservation of Subterranean Water in Toyohashi, “Subterranean Water of Toyohashi”, 19867. Kuroda, K., “Diluvium and Tectonics of Atsumi Pen.”, Earth Science Shizuhata, 16, pp38–45, 19588. Lysmer, J., et al., 1975, “FLUSH—A Computer Program for Approximate 3-D Analysis of Soil-Structure Interaction Problems”, EERC, Univ.of California, Barkley, Dec., 1972



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< previous page page_33 next page >Page 339. The Japan Road Association, “Standard Specifications for Highway Bridges Part 5, 198010. Jiang, T., and Kuribayashi, E., “The Three-Dimensional Resonance of Axisymmetric Sediment Filled Valleys”, Soil and Foundations, Vol. 28, No. 4, pp130–146, Dec.198811. Kuribayashi, E. et al., “Engineering Tactics on Lifelines Safety Against Earthquakes”, Proc. 3rd Conference on Lifeline Earthquake Engineering, ASCE, 1991*A11 references are written in English except 1, 3, 5, 6, 7, 9



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< previous page page_35 next page >Page 35Site-Response at Foster City and San Francisco Airport—Loma Prieta StudiesM.Çelebi, A.McGarrU.S. Geological Survey, MS/977, Menlo Park, CA 94025, U.S.A.ABSTRACTStrong motions recorded during the Loma Prieta earthquake and aftershocks recorded thereafter are used to quantify the amplification at several locations within a 20-km strip of the bay side of mid-peninsula, in San Mateo County, south of San Francisco, California. The area, extending from San Francisco International Airport southward to Foster City and Redwood Shores, presents a unique opportunity to quantify the amplification of motions at soft soil sites as compared to hard rock sites and compare them with the recent code site factors and maps based on these factors. The amplifications are calculated for frequency ranges of engineering interest and are shown to exceed 2 (the maximum site factor in the code) in most locations.INTRODUCTIONThe peninsula south of San Francisco experienced significant variation of ground motions during the Loma Prieta earthquake of October 17, 1989 [Ms=7.1]. In this paper, we present evidence of such variation within a 20-km strip of the bay side of the mid-peninsula covering the San Francisco airport (SFO) and extending south to Foster City and Redwood Shores (Figure 1) within the boundaries of San Mateo County (at epicentral distance of 75km or more). The area includes an important lifeline such as SFO, the commercial strip south of SFO, and the urban areas of Foster City, Redwood Shores and vicinity.The damage sustained in the study area during this earthquake, by most



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< previous page page_36 next page >Page 36standards, was moderate to minimal. A two-story structure within the airport grounds built according to pre-1960 codes was extensively damaged and later razed. Two hotels in Burlingame were damaged extensively. Structural damage occurred in the Fluor Building at Redwood Shores. A 22-story steel structure in Foster City was reported to have received minor structural damage. In the majority of cases, residential homes within the study zone were not damaged.Available strong-motion records from the main shock and records of two aftershocks are used to document and discuss the features of the varying ground motion within the described mid-peninsula strip. There are several other aftershock records from these stations; however, due to space limitations, only two are presented and utilized herein. It should be recognized that, as in this case, unless special dense arrays are deployed prior to an earthquake, strong-motion stations in general are sparsely deployed. Therefore, we have relied on temporarily deployed dense arrays to record aftershocks. Figure 1 shows the locations of temporary aftershock stations as well as the strong-motion stations discussed throughout this paper. The strong-motion stations SF1 and AP7 (Figure 1) are maintained by the California Division of Mines and Geology (Shakal and others, [4]) and the MAL (Foster City), AP2 and AP9 stations are maintained by the United States Geological Survey (Maley and others, [3]).Epicentral distances, latitudes, longitudes, elevations, depths to bedrock and relevant descriptions of all stations (and for only the strong-motion stations, the recorded peak accelerations) are provided in Table 1. In addition, specific site factors (S1–S4) are assigned to each station using a recent zoning map by Hensolt and Brabb [2]. The factors S1 through S4 assigned in their map are adopted from Section 2312 (Table No. 23-J : “Site Coefficients” of the Uniform Building Code (UBC) [5]. The site factors are 1.0 for S1, 1.2 for S2, 1.5 for S3 to 2.0 for S4. These are intended to correspond to amplification factors and are to be compared with the spectral ratios found in this work, Ramplification(ω)=A2j (ω)/A1j (ω) where Aij(ω) is the jth component Fourier amplitude spectrum at recording station i. This relationship is valid assuming the differences in distances can be neglected. The amplitude spectra are calculated using 25 seconds of each record (originally 200 samples per second, decimated to 50sps). The spectra and ratios are smoothed with a hanning window of 10 points.The scope of this paper is limited to assessing the engineering implications of the variation of the recorded strong-motion main shock and its aftershocks. Particular attention is given to impact on structural shaking and therefore zonation as a result of the variation of motions within distances



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< previous page page_37 next page >Page 37that are close to one another. Only the horizontal motions are quantified in this work. Table 1. Recording site information. (D epicentral distance)Station D (km) Lat. Deg. N Long. Deg. W Elev. (m) Site Factor Depth to Bedrock

(m)Comments (* Co-sited

Stations)

SF1* 79 37.62 122.39 2 S2 92 Engineering Building, San Francisco Airport Co-sited with CSMIP #58223 strong-motion recorder. [0.33, 0.05, 0.24].**

SF2 79 37.63 122.39 2 S1 0 On pavement 25 m east of northeast corner of United Overhaul Shop. Situated close to bedrock outcrop, free field.

SF3 79 37.62 122.39 2 S3 37 Inside Butler Aviation Building on concrete floor of hangar.

SF4 78 37.61 122.35 1 S4 107 Near east end of Runway 28R on concrete slab, free field.

SBM 80 37.69 122.45 100 S1 0 Near San Bruno Mountain

AMF 72 37.39 121.95 5 S3/S4 107 1492 Old Bayshore Highway, Burlingame. Across from Hyatt (Burlingame).[0.20, 0.12, 0.12].

MAL* 66 37.55 122.24 1 S4 107 Co-sited with USGS #1515 strong-motion recorder. 335 Menhaden Ct., Foster City. [0.12, 0.09, 0.11].

FOX 64 37.51 122.25 1 S2 50 Fox & Carskadon Office, 75 Shoreway Rd., San Carlos.

AP2* 63 37.52 122.25 1 S3 117 Co-sited with USGS #1002 strong-motion recorder. Portside Park, Redwood Shores [0.23, 0.08, 0.28].

AP7* 63 37.48 122.31 108 S1 0 Co-sited with CSMIP #58378 strong-motion recorder. Canada Rd., rural San Mateo County.[0.09, 0.06, 0.16].

AP9 62 37.47 122.32 106 S1 0 USGS strong-motion station (#1161). No aftershocks recorded. Crystal Springs Reservoir.[0.11, 0.06, 0.12].

CRA 66 37.55 122.24 1 S4 107 Same conditions as MAL.

*Note 1: Numbers in brackets are peak accelerations (horizontal (NS), vertical, horizontal (EW)—in g’s) recorded during the Loma Prieta earthquake.**Note 2: Another strong motion station (AP1) in Redwood Shores recorded peak accelerations of [0.29, 0.11, 0.26],



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< previous page page_38 next page >Page 38GEOLOGY AND BACKGROUNDSFO is situated on low-velocity alluvium, including man-made fill, of varying thickness. An important objective of this work is to relate variations in ground motion from site to site to corresponding differences in the alluvial column above bedrock. The site, SF2 (Figure 1), chosen as one reference site, is next to the northern boundary of the airport, and is nominally on an outcrop of bedrock, a Jurassic or Cretaceous sandstone and shale formation (Bonilla, [1]). SF3 was sited as close as possible to the severely-damaged building at SFO (Figure 1) to find out whether this location might have experienced greatly enhanced levels of ground motion. SF4 was sited at the end of runway 28R, partly because of the wooden trestle damage but also owing to the substantial thickness of man-made fill beneath this portion of SFO. Other rock sites, SBM and AP7, are also used as reference sites to calculate the spectral ratios that follow.Stations AMF, AP2, CRA and MAL are on varying thicknesses of bay mud (soft clay), covered with compacted engineered surficial fills (sand mixed with sea-shells). Stations AP7, AP9 are rock stations and are located at or in the vicinity of Cystal Springs Reservoir. No aftershocks were recorded at CRA.RECORDED MOTIONSStrong-Motions, Amplitude Spectra and RatiosIn Figure 2, uncorrected horizontal acceleration time-histories recorded at the strong-motion network stations of both CDMG and USGS that are in the immediate area of concern within the mid-peninsula are shown.Figure 3 shows both the Fourier amplitude spectra and spectral ratios for the N-S and E-W components of the strong motions shown in Figure 2. The spectral ratios are calculated with respect to station AP7. The AP7 spectrum is superimposed in each spectral plot for easy comparison.The spectra from the strong motions indicates clearly that the energy of the motions practically dies out after approximately 4Hz and, in the case of AP2, after 2Hz. The ratios reach as high as 7.9 for AP2, 2–3 for MAL and 3–5 for SF1. The higher ratios are for frequencies that are less than 4Hz.Aftershocks, Amplitude Spectra and RatiosWe make use of two aftershock records (events 3091337 [Ms=3.8] and 3112342 [Ms=4.0]). The equiscaled velocity seismograms of these two events are provided in Figure 4. Only the velocity seismograms will be used



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< previous page page_39 next page >Page 39throughout this work as they give the best signal-to-noise ratio. These two events provide at least one seismogram for each one of the temporary deployments described previously (except for CRA).Figure 5 shows the spectral ratios calculated from Fourier amplitude spectra of the horizontal components for the 7 stations that recorded event 3091337 and normalized to SF2 for the SFO stations and AP7 for all stations. We note first that for those stations for which there are ratios with respect to SF2 and AP7, the ratios with respect to SF2 are larger for frequencies less than 4 because of the low energy at SF2 for the frequencies that are less than 5–6Hz. The amplitude spectra for SF2 (not provided here) also exhibits significantly higher energy at higher frequencies (4–6Hz). Therefore, we conclude that with respect to AP7, the ratios reach as high as 6 (for SF3) and all have prominent peaks at frequencies less than 2Hz.Figure 6 shows the spectral ratios calculated from Fourier amplitude spectra of the horizontal components for the 6 stations that recorded event 3112342 and normalized to SBM for AMF and SF4 and AP7 for all. In this case, we note that the ratios with respect to AP7 are smaller than those with respect to SBM. The Fourier amplitude spectra of both AP7 and SBM (not shown here) are similar in frequencies but an order higher for AP7, therefore, the higher ratios with respect to SBM. This is not apparent from the amplitude of the seismograms of AP7 and SBM which, for larger distances to SBM, give comparable seismogram amplitudes. The ratios are significant in all cases for frequencies less than 4Hz.DISCUSSION OF RESULTSThe peak accelerations of the main shock compiled in Table 1 are recorded at epicentral distances between 62km (AP2, AP7 and AP9) and 79km (SF1). Conservatively ignoring the attenuation due to distance, and the dependence of the motions on frequency, the ratio of peak accelerations at SFl to that at AP7 (0.33/0.09) is 3.67 in the east-west direction and is the maximum ratio for any component of strong-motion within the study area. This is surprising as the Hensolt and Brabb map [2] classifies SF1 as being a site which best fits the description of site type S2 which corresponds to a site factor 1.2. This maximum is followed by that for AP2 to AP7 and is approximately 3.0. The minimum ratio corresponds to MAL versus AP7 and is actually 0.67 for the N-S direction and 1.33 for the E-W direction. The Hensolt and Brabb map classifies MAL as having a site factor S4 which corresponds to 2.



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< previous page page_40 next page >Page 40The frequency-dependent spectral ratios from strong motions show that the AP2/AP7 is as high as 7.9. It is noted herein that the AP2 site is at a green area (public park) within 10 meters of the Belmont Slough in Redwood Shores; therefore, the consolidation of the site may not be as good as that of MAL in Foster City. The ratios from the aftershock motions are not as high as those from the strong-motion.CONCLUSIONSThe spectral ratios calculated from strong motions or aftershocks are comparable, particularly within the frequency ranges of engineering interest. Furthermore, specifically the ratios clearly indicate significant amplification of motions at those sites where the recent zonation maps provide site factors between 1 and 2. In this study, we quantified amplification in various soft soil sites of the peninsula strip that were larger than the 2, the largest site factor in the current codes. This then raises two questions: (1) what are realistic factors with which the motions should be amplified at these sites that will make them both economically feasible and at the same time provide the necessary strength and stiffness to the structures, and (2) will these ratios change for future large earthquakes with epicentral distances that are closer than was the case in the Loma Prieta earthquake.The results presented herein would have been better characterized with borehole logs and pertinent geotechnical data including shear wave velocity values. We are reminded once again that such documentation of strong-motion station sites is needed and essential for better evaluation of the data sets recorded during earthquakes.The quantified values of amplification at different sites of the mid-peninsula tell us that for the soft sites of the region, zonation maps such as those of Hensolt and Brabb [2] are very useful. However we need to add that those amplifications occur particularly at frequencies less than 10Hz and in most cases between 1.0–4.0Hz which correspond to the frequencies of 2–10-story buildings. Such buildings are in the region and may be vulnerable due to the amplification of the motions at these sites. Therefore, the code provisions with site factors are more than justified and may not be conservative enough.



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< previous page page_41 next page >Page 41ACKNOWLEDGEMENTSUse of the computer program MATLAB by MathWorks Inc. is acknowledged. M.G.Bonilla and E.Brabb provided the requisite geological information and the maps. Howard Bundock (USGS) helped with the maps and Carol Sullivan with typing. J.Andrews, M.Bonilla and E.Brabb provided critical comments.REFERENCES1. Bonilla, M.G., Preliminary Geologic Map of the San Francisco South Quadrangle and Part of the Hunters Point Quadrangle, California, Miscellaneous Field Studies Map MF-311, U.S. Geological Survey, 1971.2. Hensolt, W.H. and Brabb, E.E., Maps Showing Elevation of Bedrock and Implications for Design of Engineered Structures to Withstand Earthquake Shaking in San Mateo County, California, U.S. Geol. Surv. Open-File Rep. 90–496, 1990.3. Maley, R. et al., U.S. Geological Survey Strong-Motion Records from the Northern California (Loma Prieta) Earthquake of October 17, 1989, U.S. Geol. Surv. Open-File Rep. 89–568, 1989.4. Shakal, A.F. et al., CSMIP Strong-Motion Records from the Santa Cruz Mountains (Loma Prieta), California Earthquake of October 17. 1989, Report OSM 89–03, 1989.5. Uniform Building Code (UBC), The International Conference of Building Officials, Whittier, CA, 1985, 1988.



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Figure 1. General map of the San Francisco peninsula with insets showing details of SFO and Foster City-Redwood Shores stations.

Figure 2. Unprocessed equiscaled acceleration time-histories for the main shock recorded by the strong-motion stations


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that are used in this study (SF1 and AP7 are CDMG stations and AP2, MAL AP9 are USGS stations).



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Figure 3. Fourier amplitude spectra and spectral ratios calculated from strong-motion records.



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Figure 4. Equiscaled velocity seismograms from two aftershocks of the Loma Prieta earthquake. At least one event triggered the temporary stations deployed (except for CRA).



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Figure 5. Spectral ratios calculated from velocity (north-south and east-west) seismograms of (Ms=3.8) event 3091337 (Julian 309, November 5, 1989, at 13:37 GMT).



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Figure 6. Spectral ratios calculated from velocity (north-south and east-west) seismograms of (Ms=4.0) event 3112342 (Julian 311, November 7, 1989, at 23:42 GMT).



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< previous page page_47 next page >Page 47SECTION 2: STRONG GROUND MOTIONS



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< previous page page_49 next page >Page 49Effects of Earthquake Characteristics on Ground Response SpectraA.M.Ansal, A.M.LavIstanbul Technical University, Civil Engineering Faculty, Department of Geotechnical Engineering, Ayazaga, Istanbul, TurkeyABSTRACTA parametric study was carried out to estimate the effects of statistical variation in the characteristics of an input earthquake. Strong motion records from California and Turkiye were scaled to different peak acceleration levels and a number of soil profiles, within the old sector of Istanbul, were selected to study the influence of input earthquake characteristics on the response of different types of soil profiles. The statistical distribution of the calculated peak ground accelerations, predominant soil periods, and response spectra were evaluated leading to a probability analysis that can be utilized in the selection of a design earthquake for site response analyses and microzonation studies.INTRODUCTIONThe factors controlling structural response during earthquakes may be considered in three groups as; earthquake source characteristics, local soil conditions, and structural features. The earthquake source characteristics represents the effects of geology and tectonic formations of the region. However, even in the case of a widespread city like Istanbul, the influence of these factors are on more macro level and would not be sufficient to explain the variations in structural damage that may be observed within relatively short distances. On the contrary, the local soil conditions which can be very different due to changes in the thickness and properties of soil layers, depth of bedrock and water table would have a more dominant impact on damage distribution. Soil layers as well as modifying properties of earthquake excitations, would also be affected by them and may cause important instabilities as in the case of liquefaction and slope failures. Therefore it is necessary to study and evaluate response of soil layers during earthquakes.



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< previous page page_50 next page >Page 50From an engineering perspective it appears possible to investigate the properties of local soil layers to implement necessary preventive measures and to design structures minimizing the vulnerability. However, at the present age, neither epicenter location, magnitude, and time of an earthquake can be controlled nor can be predicted. This aspect of the problem introduces an uncertainty into the engineering design. One logical way to approach this natural randomness is to adopt a statistical analysis in estimating the probabilities and for selecting risk levels with respect to the importance of the structures and the corresponding financial investment.An ideal way to perform such a statistical evaluation of earthquake variability is to utilize strong motion records taken at same locations during different earthquakes [3]. However, this type of data is very scarce if not completely unavailable. In studying this phenomena to estimate the effects and behavior of soil layers during a possible earthquake, one alternative is to use numerical models developed for site response analysis [6]. In this situation the results obtained will be directly dependent on the characteristics of the input earthquake motion. Therefore one of the important stages in site response studies is the selection of an appropriate and realistic design earthquake. Experience and observations during the past earthquakes have shown that due to regional differences each earthquake would normally possess unique properties representing the local tectonic formations and earthquake source mechanisms. It was observed in some cases that even earthquakes occurring in the same fault zone with epicenters close to each other could have important differences. Therefore a statistical evaluation of this aspect of the problem could introduce a probabilistic interpretation enabling the design engineers to base their decision on better defined risk levels.In some cities like Istanbul which has lived through many strong earthquakes and was demolished severely many times in its history, there may be no representative strong motion record since no major event has taken place during the last century. In these cases one the problems is the selection of a realistic strong motion record that would not yield overconservative and uneconomical results. In some countries due to the limited financial sources and due to the difficulties in applying more sophisticated technologies, there is a very unstable balance between the required safety and financial capabilities. In cases where the outcome of an analysis would necessitate large spending people are inclined to ignore the situation completely. This type of attitude would be much more detrimental in comparison to even limited implementation of preventive measures. Therefore the use of statistical and probabilistic analysis would also allow to establish a relationship between risk levels and corresponding financial investment to



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< previous page page_51 next page >Page 51decide on the level of allowable risk depending on the availability of the sources and the importance of the structures that are being analyzed.The purpose of this study is to evaluate the effects of randomness in strong motion records on site response analysis for different soil profiles. In this way it would be possible to understand the range of the influence of earthquake source characteristics. And depending upon the statistical distribution of these factors a better defined criteria can be adopted for selecting the design earthquake for soil amplification and microzonation investigations.SELECTED STRONG MOTION RECORDSA total of 25 strong motion records (8 from Turkiye with their two components, 1 from Yugoslavia, 7 from California, and 1 synthetic) were used to determine the effects of different earthquake characteristics. All of these strong motion records were scaled to peak acceleration levels of 0.15, 0.30, 0,45g to study the influence of the earthquake magnitude. The predominant periods for all of the selected strong motion records and the acceleration response spectra for the Turkish earthquakes are shown on Figure 1 and Figure 2. The differences are partly due to the differences in the source mechanisms and partly due to the site conditions where these records were obtained. Such broad range of variation in the characteristics of the selected earthquake records is believed to cover all the possibilities concerning a probable earthquake that may affect the city of Istanbul.

Figure 1. The predominant periods for the selected strong motion records



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Figure 2. The acceleration response spectra for the Turkish strong motion records scaled to peak acceleration of 0.45g


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< previous page page_53 next page >Page 53The site response analysis adopted in this study is based on the procedure developed by Schnabel, et al. [5]. A parametric study is conducted using 25 strong motion records scaled to 3 different peak acceleration levels (0.15, 0.30, 0.45g) and different soil profiles representing the soil conditions at various parts of the old sector of Istanbul. An effort is made to choose soil profiles with different characteristics.EFFECTS OF EARTHQUAKE CHARACTERISTICSThe results of the site response analyses conducted indicate the importance of the earthquake characteristics on the response of soil layers as shown on Figure 3, for one of the selected soil profiles. In this case all of the Turkish strong motion records were scaled to have peak acceleration of 0.15g. One alternative under this condition is to draw an outer envelope as a possible design spectrum such that sufficient safety can be achieved in the design and construction of structures against all probable variations in earthquake characteristics. However, as demonstrated on Figure 4 which shows the acceleration response spectra for the same soil profile and for the same strong motion records scaled to the peak acceleration of 0.45g, the properties of such a design spectrum is also very dependent on the magnitude of the earthquake or more specifically on the level of peak acceleration. Since the earthquake induced stresses and resulting strains are larger in the second case, the response of the soil layers given in terms of absolute acceleration response spectra show much larger variation in terms of spectral amplification and predominant soil periods.One other way of demonstrating the effects of earthquake characteristics is to consider the variation of the calculated peak accelerations on the ground surface. As shown on Figures 5 and 6, for soil profile S1 and S4 respectively, the normalized peak acceleration values for different earthquakes and for the selected base rock peak acceleration levels have a significant scatter. On the other hand the degree of amplification in terms of peak accelerations is very dependent on the level of the base rock peak acceleration which in a way represents the magnitude of the earthquake input. As observed by various researchers [2, 4], with the increase in the earthquake magnitude the calculated soil amplifications decrease as shown on Figures 5 and 6.The influence of the characteristics of input earthquake and its magnitude is also reflected in the calculated soil predominant periods as shown on Figures 7 and 8 for the same soil profiles. In the case of stronger earthquake input the predominant soil periods increasès. However, again this phenomena is very dependent on the properties of the soil layers.



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Figure 3. Response spectra for soil profile S1 for Turkish strong motion records scaled to peak acceleration of 0.15g



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Figure 4. Response spectra for soil profile S1 for Turkish strong motion records scaled to peak acceleration of 0.45g



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< previous page page_56 next page >Page 56The thickness of soil deposit in the selected soil profiles are approximately same but the geotechnical properties of the soil layers encountered at both locations are different. As a result the effects of earthquake input are significantly different. It is clearly evident that structures located on these layers would experience different earthquake forces. In order to make a more realistic evaluation of earthquake induced forces on structures, it is very essential to take into account the properties of local soil conditions.

Figure 5. Calculated normalized peak accelerations at ground surface for the soil profile S1 for all of the selected strong motion records

Figure 6. Calculated normalized peak accelerations at ground surface for the soil profile S4 for all of the selected strong motion records



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Figure 7. Calculated predominant soil periods for the soil profile S1 for all of the selected strong motion records

Figure 8. Calculated predominant soil periods for the soil profile S4 for all of the selected strong motion recordsPROBABILITY ANALYSISAn attempt is made to conduct a preliminary statistical analysis based on the calculated variations in peak accelerations at ground surface and predominant soil periods. In this way it is believed that the effect of the



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< previous page page_58 next page >Page 58differences in the input earthquake characteristics can be taken into account with respect to the safety level required for structures located on these layers. It appears realistic to assume that the selected strong motion records represent the range of possible earthquakes that may take place in the near vicinity of Istanbul and the variation of peak ground acceleration and predominant periods can be modelled statistically by a normal distribution. Under these circumstances, it is relatively simple to calculate the probability of exceedence in terms of peak ground accelerations and predominant soil periods as shown on Figure 9. At this stage it is justifiable to consider the variations in the probability separately with respect to the selected base rock peak acceleration levels since they represent approximately the seismicity of the region. The magnitude of a possible earthquake in a region should be estimated based on available seismological data with relation to adopted risk levels or return periods. After these analyses have been conducted then the above mentioned probabilities concerning the characteristics of possible earthquakes can be taken into account. However, in order to be consistent in the final evaluation of the earthquake characteristics at the investigated site, the probability level selected for the peak ground acceleration and predominant soil period should match the risk level adopted in the seismicity study.

Figure 9. Probability of exceedence for the calculated predominant soil periods and peak ground accelerations for the soil profile S4



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< previous page page_59 next page >Page 59CONCLUSIONSAn attempt is made by conducting an analytical study to evaluate the effects of input earthquake characteristics on response of soil layers. Although the dominant factor controlling the response of structures located on the ground surface is local soil conditions, it is evident from this study that earthquake characteristics also may play an important role. From an engineering perspective, since it appears rather difficult, if not impossible, to make a deterministic evaluation concerning the earthquake characteristics, a statistical approach can be utilized to estimate the range of input earthquake effects and to calculate the corresponding probabilities.REFERENCES1. Faccioli, E., Battistella, C., Alemani, P., Lo Presti, D. and Tibaldi, A. Seismic Microzoning and Soil Dynamics Studies in San Salvador, Infuence of Local Soil Conditions on Seismic Response, 12th ISMFE Conf., Rio de Jenerio, Brazil, pp. 21–36, 1984.2. Idriss, I.M. and Seed, H.B. An Analysis of Ground Motion During the 1957 San Francisco Earthquake, Bull.Seismological Soc. America, Vol. 58(6), pp. 2013–2032, 1969.3. Katayama, I., Iwasaki, T., and Seaiki, M. Statistical analysis of earthquake acceleration response spectra, Trans. Japanese Soc.Civ.Eng., Vol. 10, pp. 311–313, 1978.4. Kiremidjian, A. and Shah, H.C. Probability Site-Dependent Response Spectra, ASCE, J. Struc.Div., Vol. 106(ST1), pp. 69–86, 1980.5. Schnabel, P.B., Lysmer, J., and Seed, H.B.Shake—A Computer Program for Earthquake Analysis of Horizontally Layered Sites, EERC Report No. 72–12, Uni.of California, Berkeley, 1972.6. Valera, J.E. and Donovan, N.C. Incorporation of uncertainties in seismic response of soils, Proc.5th World Conf. Earthquake Engng., Rome, Vol. 1, pp370–379, 1973.7. Vinale, F. Microzonazione Sismica di Un’Area Campione di Napoli, Rivista Italiani di Geotechnica, Organo della Associazione Geotecnica Italiana, Vol. XXII (3), pp.141–162, 1989.



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< previous page page_61 next page >Page 61The Artificial Wave in Earthquake Safety Analysis for Nuclear Plant ShieldX.Shen, J.YuInstitute of Structural Theory, Tongji University, Siping Road 1239, Shanghai, 200092 ChinaABSTRACTUsually, the artificial earthquake waves used for the earthquake safety analysis of Nuclear Plant Shield are determined by fit technique based upon the frequency spectrum of NRC RG—1.60 design criteria of USA . In this paper the authors’ basic idea about the design of the artificial earthquake wave is introduced. Finally, 10 groups of time history curves of the artificial earthquake wave are proposed and used in practice. Comparing the 10 groups of curves with the 3 curves from USA, the results are satisfactory for the earthquake reliability analysis of the Nuclear Plant Shield.INTRODUCTIONThe nuclear plant shield is an important protective structure which protects the environment from leakage of nuclear materials in case of an accident. In the earthquake resistance design for the nuclear plant shield, it is necessary to calculate the dynamic response of the structure with acceleration time history inputted in its base. An earthquake safety analysis is also needed for the structure. Affected by various factors, such as source mechanism, wave propagation and local site effect, the earthquake ground motion may involve some uncertainties. For that reason, it is appropriate to treat earthquake signals as a random process. With such an assumption , a certain amount of ground motion waves is needed. The best input signal to the structure is the acceleration history measured at the site where the structure is situated. A simple way to obtain these input signals is to find a real acceleration history whose parameters are similar to that of the real



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< previous page page_62 next page >Page 62site, and make some modifications of the magnitude, time duration and frequency components according to the local conditions. Unfortunately, it is very difficult to have these real acceleration histories. In 1960s, the artificial earthquake wave method was first adopted [1], [2], [3]. Recently, the artificial wave method has developed rapidly due to the development of computer technology and the advantages of this artificial method, which can easily meet the requirements of the target spectrum in magnitude, time duration and frequency characteristic [4], [5], [6], [7].Artificial earthquake waves produced by using the computer are a number of acceleration time histories , which will satisfy some requirements (such as ground peak acceleration, frequency characteristic and duration) and in the meantime, other condition are not controlled. These uncontrolled factors are just the right random characteristics of the earthquake motion. In the artificial earthquake wave method the earthquake is usually regarded as the product of a certain time function and a stable Gauss process. In this way, the non-stable characteristics of earthquake are reflected, and the theory for stable process can be used.ARTIFICIAL EARTHQUAKE WAVES SATISFYING THE DESIGN RESPONSE SPECTRUMUp to now, there have been various ways in the artificial wave method. From the engineering point of view, a practical acceleration history is modeled by superposing some cosine functions with an evenly distributed random phases. The magnitude of every cosine function is determined in such a way that the response spectrum of artificial waves should meet the requirement of the target response spectrum. Certain low frequency noises are introduced in the mathematical model which makes the displacement of ground motion unreasonably large at the end of the earthquake. To avoid such a problem, the baseline correction method is followed to remove the linear drift of the baseline, [8]. In doing so, it will cause a serious side effect that the response spectrum of final artificial wave no longer meets the target spectrum since the linear function may contain various frequency components. The calculation shows the maximum displacement is still large because the final displacement is forced to be zero at the last stage. Analysis indicates that the large displacement of the artificial wave is caused by a small amount of low frequency components in accele-



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< previous page page_63 next page >Page 63ration. These low frequency components in acceleration are greatly amplified in integration. To solve this problem better, we use the digital filter to filter out these low frequency noises during every iteration. The baseline correction technique is also used before filtering . The criterion of the digital filter design is to rule out the frequency components outside of the target spectrum introduced during the calculation. Omsby filter is used, whose band width and cut off frequency can easily be changed to meet the requirement of the target spectrum. The first step of the filtering process is the low—pass filtering. The upper limit frequency of the target spectrum is used as the high cutoff frequency of the filter with the input acceleration a1(t) and the output acceleration a2(t). The second step is called the high-pass filtering which is used to filter out low frequency noises. In fact, this is also done by a low-pass filter to pick out the low frequency noise a3(t). Then we remove a3(t) from a2(t) to obtain the desired acceleration a(t).The desired acceleration function is modeled as follows,

(1)where a(t) is the desired artificial earthquake acceleration. T is the time duration of the acceleration. N is the sampling points of the acceleration.

Fig. 1 The common shape of ψ(t).ψ(t) is the envery function a(t). It is also called shape function. No matter how to choose An and Fn, it enables a(t) to have the same shape . The common shape is shown in Fig. 1. It is expressed as

(2)The computing flow chart is as follows (see Fig. 2).THE ARTIFICIAL WAVE IN THE EARTHQUAKE SAFETY ANALYSIS FOR NUCLEAR PLANT SHIELD



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Fig. 2 flow chart for artificial earthquake waveThe shield of a nuclear plant is a cylindrical structure with a dome. It is made of pre-stressed rein forced concrete as shown in Fig. 3. Three modified EI Centro earthquake accelerations are provided by the design institute for one direction as well as three direction input. Since the earthquake safety analysis needs more accelerations to input, a certain amount of artificial waves


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Fig. 3 The sectional sketch map of shield of a nuclear plant.



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< previous page page_65 next page >Page 65is produced. The requirements for the artificial waves are as follows, 1. Must meet US NRC RG-1.60 design spectrum (Fig. 4 , 5).

Fig. 4 The horizontal design reponse spectra of US NRC RG-1.60

Fig. 5 The vertical design response spectra of US NRC RG-1.602. The bottom plate of the structure is built on tuff.3. The earthquake magnitude of the location is degrees of earthquake intensity. The corresponding horizontal ground peak acceleration is 150gal, the vertical ground peak acceleration is 100gal according to SSE criteria.To meet the above requirement, 10 groups of artificial free ground motion are calculated. The sample interval is chosen to be 0.01 second. Total numbers of sampling points is 2048. The parameters T1, T2 , T3, α1, α2 in shape function ψ(t), are 2.0, 12.0, 20.0, 2.0, 0.8 respectively. One of the results is shown in Fig. 6. The corresponding response spectrum and the target spectrum are shown in Fig. 7. Fig. 8 gives the comparison between the average response spectrum of 10 horizontal and vertical artificial accelerations and the target spectrum. From the results shown in the above figures we can conclude1.The results are satisfactory since we control the iteration numbers according to the criterion that the difference between calculated response spectrum and the target spectrum should be smaller than a given value.2.Their displacements are reasonably small due to the use of baseline correction and filtering technique. A typical displacement curve is shown in Fig. 9.



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Fig. 6 A typical free ground artificial earthquake wave , three directions. unit: cm/s/s

Fig. 7 The corresponding response spectra and target spectra , corresponding to Fig. 6.


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Fig. 8 The comparison between the average response spectrum of 10 horizontal and vertical artificial acceleration and the target spectrum.



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Fig. 9 A typical displacement curve of artificial earthquake wave.3. Most of the control points of the artificial wave response spectrum, except a few, are in good approximation to the target spectrum. It has been found that these points in different groups are not falling into the same frequency range. It is possible for us to avoid these points fall into the interested frequency range by changing the random numbers in iteration.To include the existence of rock foundation and the local topography, the interaction between the structure and rock foundation is also considered, [9]. The artificial free ground acceleration is used as the input of earthquake excitation. Boundary Element Method is adopted to calculate the response of the bottom plate of nuclear plant shield. The responses are in three directions with six freedoms (Tab. 1). A typical acceleration response of the bottom plate is shown in Fig. 10. Next, we use these accelerations as the input of the shield structure to calculate the dynamic responses of the structure with Finite Element Method. The total number of structure inputs is fourteen, whose peak values are shown in Tab. 2. The analysis shows that the stress-concentrating areas are at the joints of the bottom plate and the cylinder and at the boundary of equipment hole. Tab. 3 and 4 show the maximum main stresses of these area. From the table we can see that the stress responses excited by modified EI Centro earthquake waves and by artificial waves are close to each other. The maximum main pull stress is 16.87 kg/cm which is smaller than the standard pull-resistance stress 25.5kg/cm of the concrete. The maximum main compress stress is 101.42kg/cm which is smaller than the standard compress-resistance stress 230 kg/cm of the concrete, this means that the safety of plant shield under SSE excitation is guaranteed.CONCLUSIONFrom the above, it is appropriate and feasible to use artificial earthquake wave by linear superposition of a certain amount of cosine functions with evenly distributed random phases. The response spec-



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UNIT: TRANSLATION cm/s/sTURN 1/100 rad/s/s

FREE GROUND THE BOTTOM OF THE SHIELD STRUCTURE

X Y Z L* X Y Z XX YY ZZ

143.9 150.0 94.5 0 146 .9 146. 7 107. 4 0 .0330 0. 0220 0.0047

30 147 .2 168. 9 125. 6 0 .0345 0. 0274 0.0121

143.9 150.0 97.4 0 150 .4 147. 3 99. 4 0 .0338 0. 0220 0.0045

30 145 .6 167. 3 103. 1 0 .0393 0. 0253 0.0119

148.2 150.0 103.7 0 140 .9 144. 3 109. 9 0 .0230 0. 0250 0.0055

30 159 .7 145. 1 131. 3 0 .0289 0. 0304 0.0144

148.2 150.0 104.5 0 150 .3 145. 3 93. 0 0 .0250 0. 0259 0.0053

30 149 .7 146. 0 112. 3 0 .0306 0. .0293 0.0146

150.0 139.8 95.7 0 131 .1 153. 4 108. 3 0 .0312 0. 0264 0.0043

30 157 .8 183. 3 116. 0 0 .0355 0. 0255 0.0116

150.0 139.8 98.8 0 138 .7 154. 1 99. 2 0 .0287 0. 0273 0.0050

30 156 .7 186. 5 85. 6 0 .0382 0. 0238 0.0113

150.0 149.1 98.4 0 139 .8 147. 2 107. 5 0 .0221 0. 0232 0.0052

30 166 .2 147. 4 125. 1 0 .0281 0. 0229 0.0111

150.0 149.1 106.7 0 144 .5 142. 4 126. 0 0 .0189 0. 0223 0.0054

30 158 .2 146. 4 121. 0 0 .0300 0. 0221 0.0122

150.0 143.8 108.7 0 130 .3 141. 4 110. 3 0 .0367 0. 0197 0.0053

30 189 .3 128. 9 102. 1 0 .0456 0.0221 0.0101

150.0 143.8 94 .5 0 128 .7 138. 8 97. 5 0 .0331 0. 0186 0.0059

30 197 .6 129. 1 98. 9 0 .0424 0. 0234 0.0085

(MEAN VALUE)

148.4 146.5 100.3 0 140.4 146.1 105.9 0.0286 0.0232 0.0052

30 162.7 154.9 112.1 0.0353 0.0252 0.0110

( MEAN SQUALE DEVIATION )

2.37 4.08 4.93 0 7.85 4.62 0.98 .00565 .00273 .00040

30 16.54 19.49 13.49 .00570 .00280 .00171

* L ---- incident angle, unit: degree .Tab. 1 The peak values of the artificial free ground accelerations and these responses in three direction with six freedoms.

UNIT : TRANSLATION cm/s/sTURN rad/s/s

INCIDENT ANGLE 30 degrees

No. X Y Z θx θy θz

1 150.0

2 150. 0

3 100. 00


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4 150 .0 150. 0 100. 00

5 147 .2 168. 9 125. 80 0.0345 0. 0274 0 .0121

6 145 .6 167. 2 103. 20 0.0393 0. 0253 0 .0120

7 158 .7 145. 0 131. 30 0.0289 0. 0304 0 .0144

8 149 .7 146. 0 106. 40 0.0306 0. 0293 0 .0146

9 157 .7 183. 4 116. 00 0.0355 0. 0255 0 .0118

10 156 .6 186. 7 85. 63 0.0382 0. 0238 0 .0113

11 166 .2 147. 2 125. 10 0.0281 0.0229 0 .0111

12 158 .2 146. 2 121. 10 0.0300 0. 0221 0 .0122

13 189 .3 128.9 102. 10 0.0456 0. 0221 0 .0101

14 197.6 129. 0 99. 06 0.0424 0. 0234 0 .0085Tab. 2 14 peak values of acceleration to input to the shield structure .



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Fig. 10 A typical acceleration response of the bottom plant (three directions with six freedoms), corresponding to Fig. 6.



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< previous page page_70 next page >Page 70 UNIT: kg/cm/cm

No.

1 max −36.71 0.86 2.04 0.98 −36.82 −86.90

rain −48.53 −83.75 0.71 −47.34 −84.94 −1.15

2 max −37.22 −2.68 3.16 −2.40 −37.50 −84.81

min −48.39 −83.29 −0.26 −48.57 −83.11 0.43

3 max −43.37 −46.49 −0.02 −43.37 −46.50 0.37

min −43.41 −46.74 0.02 −43.51 −46.64 −0.34

4 max −34.75 15.66 −6.05 16.37 −35.46 83.25

min −50 .84–101.07 −3.63 −50 .49–101.42 4.11

5 max −37.13 −1.93 0.15 −1.93 −37.13 −89.76

min −48.88 −85.68 2.68 −49.48 −85.08 −4.14

6 max −37.01 −1.13 0.92 −1.10 −37.03 −88.53

min −48.82 −85.39 1.96 −49.06 −85.15 −3.06

7 max −39.14 −16.19 −1.00 −16.15 −39.18 87.51

min −47.13 −73.79 −0.91 −48.95 −71.96 1.95

8 max −38.48 −11.61 −1.65 −11.51 −38.58 86.50

min −47.10 −73.86 −0.34 −47.05 −73.92 0.73

9 max −37.71 −5.36 −0.29 −5.36 −37.71 89.49

min 48.11 −80.89 33 −48.28 −80.73 2.32

10 max −37.69 −5.32 0.75 −5.30 −37.70 −88.67

min −47.88 −79.25 −2.47 −47.19 −79.94 4.47

11 max −37.37 −3.93 −0.62 −3.91 −37.38 88.94

min −48.48 −83.14 −1.47 −49.02 −82.59 2.42

12 max −37.65 −5.77 −0.54 −5.76 −37.66 89.03

min −48.59 −83.96 −1.54 −50.26 −82.29 2.49

13 max −40.02 −22.97 1.13 −22.89 −40.09 −86.22

min −45.81 −63.90 −2.21 −46.04 −63.66 6.87

14 max −39.12 −16.50 −1.30 −16.43 −39.19 86.72

min −46.69 −70.46 −0.36 −47.26 −69.89 0.87Tab. 3The stresses concentrated area at the joints of the bottom plate and cylinder. UNIT: kg/cm/cm

No.

1 49.69 3.41 5.41 3.96 −50.24 −84.24

2 −51.78 −4.20 −18.60 2.21 −58.19 70.99

3 −42.11 −0.40 −0.36 −0.39 −42.11 89.50


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4 −58.53 −0.86 −13.25 2.04 −61.43 77.66

5 −45.91 −6.10 −18.33 1.05 −53.06 68.68

6 −46.55 −6.07 −18.59 1.17 −53.79 68.71

7 −39.22 −3.03 −6.24 −1.99 −40.27 00.48

8 −39.16 −2.97 −5.92 −2.02 −40.11 80.94

9 −41.11 5.58 15.75 10.40 −45.93 −73.00

10 −40.55 5.54 16.04 10.57 −45.59 −72.58

11 −39.06 −4.54 −8.47 −2.58 −41.02 76.92

12 −38.72 −4.49 −8.68 −2.41 −40.80 76.55

13 −42.84 −3.09 −7.42 −1.76 −44.17 79.76

14 −38.20 −2.04 −2.45 −1.88 −38.37 86.14Tab. 4 The stresses at the boundary of equipment hole.trum of artificial waves should be in good approximation to the target spectrum. To avoid the interference of low frequency noises which cause unreasonably large ground displacements, it is necessary to use baseline correction and digital filtering technique. This method can easily meet the design criteria which require artificial waves to have a certain spectrum, time duration and acceleration peak value. The method is acceptable to engineers who are familiar with spectra method. The use of the method is simple and one needs only to change one parameter-phase to have a variety of artificial waves having the same characteristics. This example of application shows that the method is practical and satisfactory.



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< previous page page_71 next page >Page 71REFERENCE[1] G.W.Housner and P.C.Jennings: Generation of Artificial Earthquakes. J. Engng Mech. Div., ASCE, Vol. 90, 113–150, 1964.[2] P.C.Jennings, G.W.Housner and C.Tsai: Simulated Earthquake Motions for Deign Purposes. Proc. 4th Wld Conf. Earthquake Engng, Vol. 1, A-1, 145–160, Chilean Association on Seismology and Earthquake.[3] R.N.Iyengar and K.T.Iyengar: A Nonstationary Random Process Model for Earthquake Acceleration. Bull. Seis. Soc. Am. Vol. 59, 1163–1188, 1969.[4] G.R.Saragoni and G.C.Hart: Simulation of Artificial Earthquake. Engineering and Structural Dynamics, Vol. 2, No. 3, Jan–Mar. 1974[5] R.H.Scanlan and K.Sachs: Earthquake Time Histories and Response Spectra . J. Engng Mech. Div., ASCE, Vol. 100, No. EM 4, Aug., 1974.[6] Chang Chen: Some Considerations in the Aseismic Analysis of Nuclear Power Plant. Proc. of Symposium on Structural Design of Nuclear Power Plant Facilities, J.Abrams and J.Stevenson, eds, Department of Civil Engineering, University of Pittsburgh, Pa, Dec., 1972.[7] Nien-Chien Tsai: Spectrum-compatible Motions for Design Purpose. J. Engng Mech. Div., ASCE, Vol. 98, No. EM 2, April 1972.

[8]

[9]



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< previous page page_73 next page >Page 73Site Dependent Simulations of Earthquake Time HistoriesO.Henseleit, M.KostovUniversity of Karlsruhe, Germany, resp. Bulgarian Academy of Sciences, Sofia, BulgariaABSTRACTIn cooperation with the Geophysical Institute of the University of Stuttgart [1] a method was developed to simulate accelerograms approximating German earthquake conditions. The transient part of the time function is modelled by a time dependent intensity function. Haskell matrices were used for wave propagation from the source to the building foundation, assuming a horizontally layered elastic medium with vertically propagating SH-waves.INTRODUCTIONDestructive earthquakes have very long return periods. Therefore our codes allow the elastic limit to be exceeded in design. This requires an analysis by nonlinear methods taking plasticity into account. The enormous progress in numerical mathematics in the last two decades has resulted in methods like “finite elements”, “finite differences”, “boundary elements” etc. However, it is also necessary to predict correctly, time histories of destructive earthquakes—complete accelerograms for example—adapted to site conditions. Hence, all the seismological data describing the source process, the geophysical data of the layers crossed by the propagating waves and the subsoil conditions at the site must be considered.Destructive quakes in various regions have been registered during the past twenty years and collected in data bases. All these accelerograms contain, of course, site dependent data. Therefore they cannot be used without problems for other sites. The main issue of our research was to provide a method for the generation of artificial time histories which correspond to the measured ones containing all significant site dependent properties.



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< previous page page_74 next page >Page 74METHODWe used a method published by Boore [2]. He proposed to generate an acceleration spectrum by the following relation

A(ω)=CMoS(ω,ωc)p(ω,ωm)G(ω) (1)where C is a constant.

(2)G(ω) transfer function of the geology, assuming a horizontally stratified viscous-elastic medium and inclined

propagating SH-waves

radiation pattern

PR reduction factor that accounts for the partitioning of energy into two horizontal components

ρ mass density

β shear wave velocity

Qβ attenuation factor for shear waves

R site distance

S(ω,ωc) source spectrum

(3)ωc corner frequency

p(ω, ωm) high cut filterp(ω, ωm)=[1+(ω/ωm)2s]−1/2 (4)

ωm 15 Hz as proposed by Boore



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< previous page page_75 next page >Page 75where s controls the decay rate at high frequencies. Boore suggested a value of 4. According to our experience, a weaker decay rate is more appropriate. Therefore we also used a value of 2 for s. ωm is the frequency at which the amplitude decay starts. For ωm, we used frequencies significantly higher than the eigenfrequencies of the structures. That leads to a model controlled by two parameters: the seismic moment Mo, and the corner frequency ωc. These two parameters can be related by

fc=4.9 106β(∆σ/Mo)1/3 (5)fc corner frequency [Hz]

β shear wave velocity [km/sec]

∆σ effective stress drop [bar]

Mo seismic moment [dyn·cm]ωc=fc2π (6)

If the effective stress drop ∆σ is known, the model is controlled only by the seismic moment Mo. It can be determined with the following relation.

M=2/3 log Mo−10.7 (7)M moment magnitudeA transient accelerogram is generated by filtering a gaussian white noise process. The stationary process has a spectrum given by the filter function (1). The time dependent window- or intensity- function represents an envelope of the accelerogram, such that

a(t)=W(t)·x(t) (8)where a(t) is the accelerogram and x(t) is a realisation of a shot noise process.The window proposed by Sargoni et al [7] was used.Focal depth is considered in the analysis. Starting from a point source at depth h, the angle of incidence required for the seismic wave to strike a site at distance R on the surface is determined iteratively using Snell’s law.Regarding M, Ms, ML…, we have used only M—a moment magnitude. Because of the relations between different magnitude scales, other magnitude types could be used also.



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< previous page page_76 next page >Page 76All geological and source data have been considered as independent model parameters. A Monte Carlo simulation has been applied to all data which are assumed to be equally distributed. The ranges of these distributions, mean values and variance, are given in the tables.In the final version, the computer code “Simul” provides the possibility to adjust the generated simulations by means of the best approximation to the local seismological and geological conditions.VERIFICATIONData BaseFor the verification we used three different sets of data containing measured time histories of three different earthquakes• Friuli Italy 1976 (Station Forgaria Cornino, Station San Rocco)• Coalinga California (USA) aftershock 1983• The earthquakes used by Keintzel [3] and Hoeflich [4]These earthquakes are well documented. All relevant seismological, geophysical and geological data are well known. With respect to magnitude and epicentral distance they are comparable to German conditions. The focal parameters and the geological data of the layers crossed by the seismic waves were supplied by the Geophysical Institute of the University of Stuttgart.The first data set was chosen from the numerous registered time series of the Friuli Italy 1976 earthquake. Only stations with short epicentral distances, where the direct S-waves govern the character of the ground motions, were selected. Furthermore, earthquakes with magnitudes of M>5.0 were preferred. The stations Forgaria Cornino and San Rocco were chosen, because the subsoil of Forgaria Cornino consists of layers of soft soil, whereas San Rocco is built on rock. The geological data can be found in [5]. The parameters are presented in tables 3 and 4. The complete set of input data used for the simulation of the artificial earthquakes is listed in tab. 1 to 5.The second data set contains selected accelerograms of the Coalinga earthquake (Jul. 22. 1983 2:39 UTC Tab. 6). The generalised geological section and the focal parameter can be seen in tables 6 and 7. Because of the significance, sites with the short epicentral distances of 5, 10 and 17km were chosen.The geophysical parameters of the earthquakes used by Keintzel and Hoeflich are listed in [12]. These accelerograms of older American earthquakes were modified in the frequency content and scaled in the Fourier spectrum



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< previous page page_77 next page >Page 77to correspond to German conditions. The detailed procedure is described in [3]. For the simulations a geological section and focal parameters typical for a site in the Swabian Jura with an epicentral distance of 7, 5km were used. The simulations made with this third data set are not used for verification. They were only used to check the computer code in comparison to the results by Keintzel and Hoeflich.Linear elastic response spectraThe main purpose of the research project was to examine statistically the parameters of the measured and simulated earthquake time histories. The simpliest way is to use linear elastic response spectra. We started with the maxima of the displacements, velocities and accelerations. However, it is well known that the maxima of the response do not always describe the seismic load significantly. It is also well known that the duration of the strong motion phase is very important. Therefore also energetic expressions were analysed. Starting with the differential equation

(9)whereω eigenfrequency

D damping

acceleration of the soil

it is possible to obtain an equation of equilibrium of energy by multiplying Eq.(9) with the velocity and integrating

(10)or

Ekin+Evis+Eel=Einp (11)where

(12)

(13)



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(14)

(15)Time series of these integrals can be seen in the addendum A 2 of [12] for different eigenvalues of the single degree of freedom oscillator. Maxima shown in a diagram are similar to the well known response spectra. For the verification of the simulation procedure we usedmax a maximum acceleration

max ν maximum velocity

max d maximum displacement

max Einp maximum input energyAn example can be seen in Fig. 6.Elastic multi degree of freedom systemsLinear response spectra show the reaction of a single degree of freedom system due to base acceleration. For multi degree of freedom systems the mode superposition method can be used. Hence, all the results of the previous chapter are also valid for multi degree of freedom systems. Therefore it is not necessary to treat this problem separately.Nonlinear response spectraFor the nonlinear system of Fig. 2 the same analysis is made as for linear elastic response spectra. The results are similar spectra but they reflect the characteristics of the nonlinear oscillator. In the differential equation (9) the term with the spring constant is now replaced by the elastic plastic resistance of the system.

(16)where

with



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Rel=max R=xelk=const. The yield load was defined by 80% of the ultimate load and the critical value xel by the elastic displacement at that load. Possible strain hardening was neglected as a horizontal yield plateau was assumed. It was appropriate for our analysis to divide the potential energy in two parts as shown by the following equations

(17)

(18)xpl(t)=x(t)−xel (19)

The elastic energy Eel is restored to the system and is available for future oscillations. The hysteretic energy Ehyst is dissipated and therefore not available for future oscillations. The response is reduced.The following values are compared for verification.max a maximum acceleration

max ν maximum velocity

max d maximum displacement

max dpl maximum plastic displacement

max Einp maximum input energy

max duc maximum ductility factor

max Ehys maximum hysteretic energy

max Evis maximum dissipated energyThe detailed results are given in tab. 13 and 14 of [12].VERIFICATION (BUILDINGS)An engineer does not ask for simulated earthquake time histories fitting given spectra. He is interested in other questions, e. g.: Is the nonlinear performance of a building with respect to its plastic deformations in critical regions comparable for real (measured) and artificial (simulated) earthquakes? To obtain relevant results we have chosen two types of typical structures consisting of moment resisting frames or of shear walls with 5, 10 or 20 stories. We used only computer codes with reliable and experimentally verified nonlinear models, as for example, the Takeda model.



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< previous page page_80 next page >Page 80Moment resisting framesRegular and symmetric frame type structures with two bays and five or ten storeys (Fig. 3) were analysed. These frames had already been designed by Hoeflich [4] for a distributed load of 10 kN/m2. The earthquake loads corresponded to the earthquake zone 4 of the German code DIN 4149 [8].The German earthquake standard DIN 4149 sets two limits for the axial load in columns

(n≤0.23 or n≤0.5) (20)

(21)wheren nondimensional axial load

N axial load in the column

Ac concrete cross section

Rc nominal concrete strengthn≤0.23 can be used without restrictions while n≤0.5 is only allowed in connection with special requirements for a higher ductility. 4 types of frames were obtained. The sections are given by tables 15 and 16 of [12]. In tables 17 and 18 of [12] the three lowest eigenvalues are presented. The frames were loaded with the base accelerations of the measured and simulated earthquake time histories.For all relevant sections, the moment curvature relation for the different reinforcement was calculated with the computer code ZAEH 1 [3]. The constitutive laws for concrete and reinforcing steel were assumed according to German standards. Strain hardening was not taken into account.The nonlinear analysis was made with the computer code SAKE [9] especially developed for moment resisting plane frames with degrading stiffness according to Takeda [10] as shown in Fig. 5. Damping is considered as



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< previous page page_81 next page >Page 81Rayleigh-damping so that the damping of the two lowest modes is D=0.05. P-∆ -effect, i.e. the increase of moments due to large deflections is taken into account.For the verification, we compared the following values:• maximum storey acceleration• maximum storey deflection • maximum bending moment at the base of columns• maximum base shear• maximum ductility factors required for girders, inner and outer columns.The results can be seen in tables 19 to 22 of [12] and for one example in Fig. 7. Generally, the structural response of real and simulated earthquakes corresponds very well to each other.Shear wall structuresOnly two-dimensional systems with uncoupled shear walls and 5, 10 and 20 stories were considered (Fig. 4). Due to the slenderness of the walls the predominant part of the elastic and plastic deflections are caused by bending deformations. However, shear deformation was included in the analysis. It was assumed that the shear reinforcement remains in the elastic range so that only linear elastic shear deformations need to be considered. This assumption can always be realized in design by using adequate shear reinforcement.Maximum stresses and strains of slender shear walls under horizontal load always occur near the base. Therefore only the overturning moment with earthquake loads according to the German standard DIN 4149 was calculated. The systems shown in Fig. 4 with constant section and mass distribution were assumed. The distributed load of 10kN/m2 (including dead and life load) for one storey yields to a storey mass of 60kN s2/m. The stiffness of the wall was determined so that periods of 0.4s, 0.8s, 1.6s and 2.4s were obtained.



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< previous page page_82 next page >Page 82For the verification, the following values were considered:• required ductility factor at the base• accumulated plastic hinge rotation at the base.The analysis was made with the nonlinear finite element program DRAIN-2D [11] with the degrading stiffness as in the Takeda model (Fig. 5). Damping was assumed as Rayleigh damping corresponding to a damping ratio D=0.05 in the first two modes.The results can be seen in tables 23 to 25 of [12] or in Fig. 8. It is obvious that the stresses and strains in the structure of most of the real earthquakes agree well with the simulated ones. But it can also be seen that sometimes the results are over- or underestimated. This tendency for poorer results is not surprising. Contrary to frame type structures, the complete plastic deformation is concentrated in one single hinge at the base of the shear wall. In this hinge all inaccuracies resulting from the basic assumptions of the earthquake simulation up to the approximations in the structure, are expressed in an offset from the true value. These results show very clearly the equalizing and compensating effect of statically indeterminate structures where the plastic deformations are distributed over many plastic zones in the structure.CONCLUSIONNowadays a great number of measured strong motion data of earthquakes exist. They are collected in databases. However, these databases must contain gaps. With our site dependent simulation model it is possible to close these gaps in two different ways. First, all geometrical parameters concerning properties of soil or rock layers, epicentral distance, focal depth can be chosen arbitrarily. Second, it is possible to interpolate over all magnitudes up to about Ms=6.5. Further on, it is surely possible to extrapolate up to Ms=7.0. So it is possible to simulate the earthquakes with the highest damage potential. However, the model does not allow to simulate earthquakes with• surface faulting because of the importance of the neglected surface waves• magnitudes higher than Ms=6.5–7.0 because of the enormous dimensions of the focus, which cannot be assumed as a point source.Earthquakes of this type are not important for German conditions. Therefore it can be safely assumed that the proposed simulation model covers nearly all cases occurring in Germany.



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< previous page page_83 next page >Page 83In detail, we found the following degree of agreement between real and simulated earthquakes: linear response spectra: good agreement nonlinear response spectra: relatively good agreement frame type buildings: good agreement shear wall buildings– mean value: agreement– maxima: poorer agreementIf the plastic hinges are distributed more or less uniformly over the structural system, the response of real buildings can be well predicted.The advantages of this method can be summarized as follows: Generally, good results can be expected Input data is relatively simple Individual site data such as geological layers, subsoil properties, focal depth and epicentral distance can be prescribed arbitrarily The simulation covers earthquake magnitudes from microseisms up to M≥6.5 and includes the possibility to extrapolate up to All current data normally derived from earthquake measurements, such as accelerations, velocities, displacements, energies, different types of magnitudes, duration of the strong motion phase and all the related statistical data can be calculated.The disadvantages of the method in the present version are the following: The influence of surface waves is not taken into account. The method is not appropriate for earthquakes with surface faulting or for great epicentral distances. In principle, however, it is also possible to include surface waves. The model is completely linear. Nonlinear effects are not regarded but are approximately considered by equivalent linearisation.



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< previous page page_84 next page >Page 84It should be mentioned that the proposed simulation model is not a phenomenological but a physical one. This means that the focal process is physically modelled in a statistical sense. Also, the energy transmission by waves through the geological layers up to the surface is physically modelled with all reflections and refractions at the layer interfaces. The seismic moment is used for scaling on a geophysical basis.ACKNOWLEDGEMENTThe scientific support of Dr. F.Scherbaum and Prof. G.Schneider, University of Stuttgart, in the computer program arrangement and in selecting data for model verification is highly appreciated. This work was sponsored in parts by DFG (Deutsche Forschungsgemeinschaft) and DAAD (Deutscher Akademischer Austauschdienst).REFERENCES1. Schneider, G., Scherbaum, F.: DFG-Forschungsvorhaben “Erdbebengrundlagen”, Nr. 354/23, Bericht zum 1. Nov. 19822. Boore, D.: Stochastic Simulation of High-Frequency Ground Motions Based on Seismological Models of the Radiated Spectra, Bull. Seism. Soc. Am.. 73/61, pp. 1865–1894, December 19833. Keintzel, E.: Zähigkeitskriterien für Stahlbetonhochbauten in deutschen Erdbebengebieten, Dissertation, Universität Karlsruhe (TH), 19814. Hoeflich, S.G.: Nichtlineares Verhalten von Stahlbetonbauten unter Erdbebenbelastung, Dissertation, Universität Karlsruhe (TH), 19835. Scherer, R.J., Schueller, G.I.: Friuli Earthquake Sequence of 1976. Records and Power Specta of Corrected and Integrated Strong Motion Earthquake Data, Insbruck-Munich, May 19856. Housner, G.W.: Intensity of Ground Shaking Near the Causative Fault, Proceed, III WCEE, pp. 81–94, 19657. Sargoni, G.R., Hart, G.C.: Simulation of Artificial Earthquakes, Earthquake Eng. and Structural Dynamics, Vol. 2, 1974 pp. 249–2678. DIN 4149 Bauten in deutschen Erdbebengebieten Lastannahmen, Bemessung und Ausführung üblicher Hochbauten (German earthquake standard).



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< previous page page_85 next page >Page 859. Otani, S.: SAKE, a Computer Program for Inelastic Response of R/C Frames to Earthquakes, University of Illinois, Urbana, 197410. Takeda, T., Sozen, M.A., Nielsen, N.S.: Reinforced Concrete Response to Simulated Earthquakes, ASCE, Journal of the Structural Division, Vol. 96, 197011. Kanaan, A.E., Powell, G.H.: DRAIN-2D, A General Purpose Computer Programm for Dynamic Analysis of Inelastic Plane Structures. EERC, Rep. 73–6 and 73–22, University of California. Berkeley, California, 197312. Eibl, J., Henseleit, 0. and Kostow, M.: DFG-Forschungsvorhaben “Erdbebengrundlagen” Nr. Mu 354/23 Teil 2. Bericht zum 31. Dezember 1986



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< previous page page_86 next page >Page 86Tab. 1. Friuli Earthquake 1976, Station Forgaria Cornino

N time ML R[km] h[km] comp. max. a [g/10]

1 76.09.11 5.3 20.6 6 NS 1.0000

16.31.10 WE 1.1429

2 76.09.11 5.6 19.4 6 NS 1.3131

16.35.01 WE 2.3382

3 76.09.15 5.9 15.9 5 NS 2.6367

03.15.19 WE 2.1934

4 76.09.15 6.0 15.8 7 NS 3.5404

09.21.18 WE 3.3505

5 77.09.16 5.3 6.1 8 NS 2.4557

23.48.07 WE 2.0139

6 76.05.11 5.0 7.6 6 NS 1.9192

22.43.60 WE 3.1185



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< previous page page_87 next page >Page 87Tab. 2. Friuli Earthquake 1976, Station San Rocco

N time ML R[km] h[km] comp. max. a [g/10]

1 76.09.11 5.3 20.6 6 NS 0.3403

16.31.10 WE 1.7151

2 76.09.11 5.6 19.4 6 NS 0.8024

16.35.01 WE 0.9555

3 76.09.15 5.9 15.9 5 NS 0.6045

03.15.19 WE 1.3556

4 76.09.15 6.0 15.8 7 NS 1.3092

09.21.18 WE 2.5106Tab. 3. Geological Cross Section, Forgaria-Cornino

layer no. layer thickness S-wave velocity density quality-factor

[m] [m/s] [kg/m3]

σ σ σ σ1 5 1 200 50 1800 100 20 5

2 21 2 600 100 2100 100 50 5

3 500 50 900 100 2100 100 100 10

4* 9000 — 2500 — 2500 — 200 —

*half space



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< previous page page_88 next page >Page 88Tab. 4. Geological Cross Section, San Rocco


[m] [m/s] [kg/m3]

1 30 5 600 100 2000 200 50 5

2 2000 200 2500 200 2200 100 100 10

3* 9000 — 3000 — 2400 — 200 —

* half spaceTab. 5. Focal Parameter, Friuli EarthquakeN earthquake date/time focal depth radiation pattern stress drop moment magnitude

[km] [MPa]

1 76.09.1116.31.10

6 0.7 0.1 1.5 0.5 5.2 0.25

2 76.09.1116.35.01

6 0.7 0.1 1.7 0.5 5.5 0.25

3 76.09.1503.15.19

5 0.7 0.1 1.5 0.5 5.85 0.25

4 76.09.1509.21.18

7 0.7 0.1 1.5 0.5 6.0 0.25

5 77.09.1623.48.07

8 0.7 0.1 1.5 0.5 5.2 0.25

6 76.05.1122.43.60

6 0.7 0.1 1.8 0.5 5.0 0.15



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< previous page page_89 next page >Page 89Tab. 6. Dataset Coalinga. Characteristics of the Accelerogramsno. station distance

[km]comp. max. acceler.

[cm/s2]max. velocity

[cm/s]max. displacement

[cm]

1 Coalinga, Burnett Construction 11.3 360 330.48 17.54 1.35

270 −251.58 15.70 2.39

2 Coalinga, Oil city 5.0 360 −454.72 34.13 −9.45

270 −920.76 38.20 −6.22

3 Coalinga, Oil Fields, Fire Station, Freef. 9.4 360 189.45 −15.88 −3.50

270 −212.05 −16.78 3.39

4 Coalinga, Oil Fields, Fire Station, Pad 9.4 360 215.64 −16.76 −3.71

270 −209.95 −16.91 3.52

5 Coalinga Palmer Avenue 10.0 360 312.52 −21.32 −3.42

270 283.94 −13.20 2.22

6 Pleasant Valley, Pump Plant, Basement 17.40 135 −141.38 5.61 −1.63

095 430.65 −25.93 4.59

7 Pleasant Valley, Pump Plant, Basement 17.40 360 419.99 −21.84 −2.99

270 −230.55 19.44 −4.12

8 Coalinga, Skunke Hollow 11.10 360 229.84 14.86 −3.65

270 365.33 16.23 −3.27

9 Coalinga, Transmitter Hill 6.90 360 1145.73 −46.82 −4.40

270 −834.62 46.58 6.15



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< previous page page_90 next page >Page 90Tab. 7. Geological Cross Section, Coalinga, Californien


[m] [m/s] [kg/m3]

1 500 100 600 100 1800 100 50 5

2 500 100 850 200 1800 100 50 5

3 1150 150 1750 200 2100 100 150 50

4 850 150 2050 200 2200 100 150 50

5 1000 100 2300 200 2300 100 200 50

6 900 100 2650 200 2300 100 200 50

7 2400 — 2800 — 2600 — 200 —



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< previous page page_91 next page >Page 91Fig. 1. Flow chart of computer code “SIMUL”



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Fig. 2. Plane frames for numerical comparisons with computer code SAKE

Fig. 3. Numerical model for SAKE



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Fig. 4. Deformation and numerical model of a horizontally loaded shear wall

Fig. 5. Determination of accumulated plastic deformation



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< previous page page_94 next page >Page 94Fig. 6. FRIULI EARTHQUAKE, FORGARIA CORNIND RESPONSE SPECTRA, DAMPING 0. AND 0.10———REAL DATA- - - - - SIMULATED DATA


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< previous page page_95 next page >Page 95Fig. 7. FRAME A, 5 STOREY, 2 BAY, N.GT.-0.23 BASE MOTION: FRIULI EARTHQUAKE, FORGARIA CORNINO SOLID LINE: REAL DATA STRUCTURAL RESPONSE ENVELOPES DASHED LINE: SIMUL DATA



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< previous page page_96 next page >Page 96Fig. 8.BASE MOTION: FRIULI EARTHQUAKE. FORGARIA CORNINDSINGLE SHEAR WALL 20 STOREYMEAN, MEAN+1.SIG AND PEAK-HOLD ACCUMULATED PLASTIC ROTATIONSOLID LINE: REAL DATADASHED LINE: SIMULATED DATA



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< previous page page_97 next page >Page 97Fig. 9.BASE MOTION: FRIULI EARTHQUAKE, SAN ROCCOSINGLE SHEAR WALL 5 STOREYMEAN, MEAN+1.SIG AND PEAK-HOLD REQUIRED DUCTILITYSOLID LINE: REAL DATADASHED LINE: SIMULATED DATA



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< previous page page_99 next page >Page 99Spatial Coherency of the Strong Ground Motions on the SMART 1 Seismic ArrayI.A.BeresnevInstitute of Physics of the Earth, USSR Academy of Sciences, Bolshaya Gruzinskaya 10, Moscow 123810, USSRABSTRACTThe SMART 1 Seismic Array data are used to study characteristics of spatial coherency of strong ground motions such as. correlation coefficients and differences between traces. It is shown that these characteristics are clearly dependent on earthquake magnitude. To explain this fact a dependence of the predominant period of seismograms on magnitude and hypocentral distance is analyzed. We show that the period is increasing as magnitude increases, and there is no such evidence of its dependence on nypocentral distance. Thus, the explanation of the strong motions spatial coherency increase with magnitude lies in the fact that the wave field period is also increasing with magni tude.INTRODUCTIONThe SMART 1 Seismic Array is located in the north-east corner of Taiwan. It is specially designed for a three-component recording of the strong local seismic motions. The array consists of 39 accelerometers configured in three concentric circles of radii 200m, 1000m and 2000m. There are twelve equally spaced stations on each ring and a central station. The detailed information about the array design, the geology of the site and characteristics of most significant recordings can be found in [1].We used in our analysis a collection of the first 24 events recorded by the SMART 1 Array. The local magnitudes in this data set are ranging from 3.4 to 7.2, and hypocentral distances are -from



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< previous page page_100 next page >Page 1001.8 to 120.5km. The peak recorded acceleration is about 250Gals.The study of the spatial coherency of strong ground motions and the factors governing it was in the spotlight of this work. Previous investigators have noted the dependence of the spatial coherency on the local magnitude of earthquake. Abrahamson et al. [1], Abrahamson [2] Studied the standard deviation of the natural logarithm of peak ground accelerations as a function of magnitude. They observed that the standard deviation decreases as magnitude increases. This indicated the increase of the spatial coherency of strong motions with magnitude.In this investigation we analyze the other characteristics of spatial coherency such as the correlation coefficients and the relative differences between records of pairs of stations in the array, as a function of local magnitude and hypocentral distance. The frequencies of the Fourier acceleration spectra maximum values are also analyzed in order to account for the spatial coherency dependence on magnitude. We give new evidence of the coherency increase with magnitude and associate this fact with a frequency content of the wave field radiated by the earthquakes with different magnitudes.ANALYSIS OF DATAWe calculated the correlation coefficients between records of central station COO of the array and the stations in the inner and middle rings. The procedure was the following. Let us consider two records of the same event recorded by two different stations. Because of the triggering mode in which all accelerometers operate the time origins of these records are generally different. That is why we used the origin of the time series equal to the time of the latest triggering among two stations. Moreover, because of the delay in the same wave arrivals to the neighboring stations due to their finite propagation velocity we successively shifted two series one with respect to another to better fit the waveforms. The shift was within the limits of 0.5s for “central station—inner ring” pairs and 1.0s for “central station—middle ring” ones. The correlation coefficients were calculated for each individual shift, and the maximum value was taken as a result.


file:///F|/LIbros%20de%20ingenieria_21/LIbros%20de%20ing...ynamics_and_Earthquake_Engineering_V/files/page_100.html12/05/2010 06:30:47 a.m.

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< previous page page_101 next page >Page 101Note that in all following calculations the full records were used: the separate analysis of individual wavetrains in corresponding time windows has not been performed.For a single event we obtain at most 12 values of the correlation coefficients on each ring, this number depending on the total number of stations triggered. We averaged all these values thus obtaining the only one coefficient for an individual event, separately for inner and middle rings.The Fig. 1 shows the average correlation coefficient rav as a function of the local magnitude ML for an EW horizontal component of accelerations (all data shown below are calculated for this component:) . we see that almost linear dependence exists. The correlation coefficients increase rapidly as magnitude increases, reaching the values of 0.8 for magnitudes about 7.The linear regression is also shown in Fig.1 satisfying the equation

rav =−0.205+0. 124 ML, the standard deviation of the coefficient at ML being σcoeff=0.0167.We remind that the distance between stations for which the Fig. 1 was calculated is 200m. The Fig. 2 shows the correlation coefficients vs magnitude for the “COO—middle ring” pairs with a distance between stations equal to 1000m. It can be seen that in this case there is no systematic distribution of points with respect to magnitude, and the absolute values of the coefficients rav are approximately half as large as in the previous case. The conclusion is that neither high coherency nor its dependence on magnitude are observed as distance between stations increases up to 1km.We mentioned before that a number of values of correlation coefficients have been calculated for a single event on each ring, depending on the number of triggered stations. The Figs. 1 and 2 show the behavior of their average values. For each event we also calculated the ratio rmax/rmin of maximum and minimum values in



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Fig. 1. The dependence of the average correlation coefficient on the local magnitude for “central station—inner ring” pairs. The straight line is a linear regression.

Fig. 2. The dependence of the average correlation coefficient on the local magnitude for “central station—middle ring” pairs.

Fig.3. The dependence of the rmax/rmin ratio on magnitude, where rmax and rmin are the maximum and the minimum values, respectively, of the correlation coefficients calculated for all stations on the ring triggered by a single event. The “central station—inner ring” pairs have been used.



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< previous page page_103 next page >Page 103this range and looked how it depends on magnitude. This ratio describes in some way the dispersion of the correlation coefficient values for a single event caused by different local effects. The data are shown in Fig. 3. At low magnitudes (up to 5) the scattering of data is rather high, the ratios rmax/rmin varying from 1.5 to 4, approximately. But at larger magnitudes (above 6) there is far less scattering, scattering, and the ratios rmax/rmin are in the range of 1.1–1.6.We also investagated the absolute differences between accelerations in the “central station—inner ring” pairs. The average difference can be calculated as follows. At each moment the difference between the corresponding samples of the central station record and the station in the ring is taken, the calculation being continued up to physical end of the shortest trace. Then the maximum difference is selected, divided by the maximum acceleration at the central station for normal ization. The last step is averaging all obtained single values of MD/COO (maximum difference/maximum value at the central station) ratios over the whole ensemble of stations triggered in the ring, giving an AMD/COO (average max difference/COO) .The data obtained are shown in Fig. 4. The pattern similar to the previous one can be observed: the dispersion of the data points tends to decrease with increasing magnitude. At the same time the tendency is evident to decrease of the normalized AMD.The previously given results clearly prove that the spatial coherency of the ground motions strongly depends on earthquake magnitude, so that it increases as magnitude increases.We tried to arrange data versus hypocentral distance ∆H of the events in order to look for correlation. The example is given in Fig. 5 where rav as in Fig. 1 are plotted vs ∆H. A clear dependence like in Fig. 1 is not observed here because of the greater scattering of data points.To account for the observed statistical dependence of the spatial coherency of strong ground motions on magnitude let us consider the relation



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Fig. 4. The normalized maximum differences between records of central station and inner ring, averaged over all triggered pairs, as a function of magnitude.

Fig. 5. The dependence of the average correlation coefficient on the hypocentral distance for the “central station—inner ring” pairs



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< previous page page_105 next page >Page 105between the predominant period Tmax of the seismograms and magnitude. The predominant periods were calculated from the Fourier acceleration spectra at every single station. In Fig. 6 the Tmax’s are plotted vs ML for the central station. The data distribution shows that the systematic relationship exists between these quantities consisting in the obvious increase of Tmax with magnitude. Note that it is typical with strong motion data sets that the correlation exists between magnitude and distance because the large magnitude events trigger the stations at larger distances than smaller events [2]. It can be naturally supposed therefore that the result obtained in Fig. 6 is simply explained by this fact: larger magnitude events are more distant and have therefore the longer periods because of the trivial attenuation of short periods. To avoid such misinterpretation we plotted Tmax together with hypocentral distance ∆H. The result is given in Fig.7. The correlation is much less clear than in Fig.6.DISCUSSIONIt is evident that the mechanism of the strong motions coherency dependence on magnitude lies in the fact that the increase of magnitude is accompanied by the substantial increase of the predominant period (and wavelength) of seismic field. Hence, the influence of the local inhomogeneities ties of the medium within an array becomes weaker, which results in the increasing coherency.Two factors determine the predominant period as Figs. 6 and 7 show: the local magnitude and the hypocentral distance. The effect of the first one is connected with the focal mechanism of earthquake and is much more significant (Fig. 6). There is also some weaker influence of the second factor acting through the attenuation of high frequencies with distance. Thus, the spatial coherency is more clearly related with magnitude than with hypocentral distance, as we see reflected in Figs. 1 and 5.ACKNOWLEDGMENTSThe SMART 1 data are made available by the Seismographic Station of the University of California



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Fig. 6. The predominant period of the seismograms at central station as a function of magnitude.

Fig. 7. The predominant period of the seismograms at central station as a function of hypocentral distance.at Berkeley and the Institute of Earth Sciences of the Academia Sinica in Taipei. I would like to thank Professor B.A.Bolt for delivering data and the benevolent correspondence, and Professor A.V.Nikolaev for his numerous discussions and support of this work.



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< previous page page_107 next page >Page 107REFERENCES1. Abrahamson, N.A., Bolt, B.A., Darragh, R.B., Penzien, J. and Tsai, Y.B. The SMART 1 accelerograph array (1980–1987): a review, Earthquake Spectra, Vol. 3, pp. 263–287, 1987.2. Abrahamson, N.A. Statistical properties of peak ground accelerations recorded by the SMART 1 Array, Bull. Seism. Soc. Am., Vol. 78, pp. 26–41, 1988.



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< previous page page_109 next page >Page 109SECTION 3: WAVE PROPAGATION



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< previous page page_111 next page >Page 111Comparison of 2-D and 3-D Models for Analysis of Surface Wave TestsJ.M.Roësset, D.-W.Chang, K.H.Stokoe, IIDepartment of Civil Engineering, The University of Texas at Austin, TX 78712, U.S.A.ABSTRACTIn situ measurement of wave propagation velocities can be used, directly or indirectly, to evaluate the material properties at a site. Traditional seismic methods such as the crosshole or downhole methods are expensive and often difficult to use because of the need to drill boreholes. The Spectral Analysis of Surface Waves (SASW) technique, on the other hand, operates entirely from the surface. The inversion process needed to estimate shear wave velocities and their variation with depth from the measured data is normally based on a two dimensional solution in which only plane Rayleigh waves are considered. This solution is sufficient when dealing with soil profiles where properties vary smoothly with depth. A more accurate three dimensional solution must be used, however, when there are sharp charges in the properties of the layers which give rise to wave reflections or refractions. In this paper the results of both types of solutions are presented and discussed for a number of cases.GENERAL BACKGROUNDEvolution of the Surface Wave TestThe existence of surface waves was reported by Rayleigh in 1887. It was not however until 1938 that his theoretical and experimental work found an application in the studies of foundation vibrations conducted by the German Society of Soil Mechanics (DEGEBO). Applications to pavements were performed next by Bergerstorm and Linderholm (1946), Van der Poel (1951) and Nijboer and Van der Poel (1953). Further studies of the dispersion curves for soil and pavement profiles were conducted by Jones (1958, 1962), Heukelom and Foster (1960), Ballard (1964) and Fry (1965). In nearly all of these studies the excitation consisted of a steady state harmonic vertical force and the method became known as the steady state Rayleigh wave technique. In this early work the equipment was bulky and the interpretation of the data was based on relatively simple empirical rules which could result in erroneous results for complicated but realistic material profiles. As a consequence, the method failed to gain widespread acceptance. With the development of portable and sophisticated electronic equipment capable of performing accurate high frequency data acquisition and complex mathematical manipulations rapidly in the field and with the establishment of a theoretically sound basis for data analysis the surface wave test has been improved and simplified as the Spectral Analysis of Surface Waves (SASW) method (Heisey, et. al. 1982, Stokoe and Nazarian 1983). In recent years a considerable amount of theoretical and experimental research work has been conducted at the University of Texas at Austin in order to understand better and improve the applicability of the method (Shao 1985, Sanchez Salinero 1987, Sheu 1987, Rix 1988, Roesset et. al. 1990, Kang 1990).



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< previous page page_112 next page >Page 112Equipment and Field TestingThe general arrangement of the source, receivers (accelerometers), and recording equipment in a SASW test is shown schematically in Fig. 1. No boreholes are required because both the source and receivers are placed on the surface. A piezoelectric shaker can be used effectively as a source to generate surface waves over frequencies ranging from about 1kHz to 50kHz. The high frequencies are necessary to sample the surface layer of stiff pavements. A digital waveform analyzer coupled with a microcomputer is used to capture and process the output from the receivers.The vertical accelerometers and source are arranged in a linear array. The distance, D, between receivers (see Fig. 1) may be varied by the operator to optimize the test results for a particular site. The distance between the source and the first receiver, d1, is usually kept equal to D but may also be increased by the operator to minimize destructive interference from body wave reflections. However, d1/D=1.0 is normally a good arrangement, as shown in a number of analytical studies.Surface Wave DispersionIn the original technique, a steady-state vibrator acting vertically on the surface of the soil produced a harmonic excitation at a known frequency. A vertically oriented sensor was moved away from the source until the recorded motion was in phase with the excitation. The distance between any two of these successive positions was assumed to correspond to one wavelength L of a Rayleigh wave propagating along the surface. So, for a frequency ω (in radians/sec), or f (in Hz), the phase velocity of the surface wave would be

ν=Lf=Lω/2π (1)Repeating this process for different excitation frequencies f a plot of velocity versus frequency (or wavelength) was obtained. Such a plot is known as a dispersion curve. In the SASW method, instead of using a steady state vibrator at a fixed frequency, an impulsive or random-noise load is applied at the surface of the soil deposit. A variety of sources can be used to generate the impact, from hand held hammers of different sizes (small hammers are sufficient for high frequency excitation), to drop weights (heavier weights for low frequency excitation). The passage of the wave train generated by the impact is monitored by two vertical receivers, located also on the surface, as shown in Figure 1. The electrical signals recorded by the receivers are digitized and transformed to the frequency domain, using a Fast Fourier Transform algorithm, by a dynamic spectral analyzer, which provides also automatically the cross spectrum and the coherence function of the two records. The phase difference between the signals is obtained directly from the cross

spectrum as a function of frequency. If is the phase difference in radians at a frequency ω, and D is the distance between the two receivers, the travel time is



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(2)and the velocity of propagation is

(3)The corresponding wavelength is then

(4)The shape of the dispersion curve depends on the variation of soil properties with depth. For a given frequency and wavelength the particle motion is restricted to a soil depth of the same order of the wavelength and the velocity of propagation of the surface waves depends almost exclusively on the soil properties over that depth. For high frequencies, and short wavelengths, the phase velocity reflects thus the properties of the soil near the surface and as the frequency decreases the properties of deeper and deeper layers get into play (Fig. 2).To estimate the soil properties from the experimental dispersion curve the original steady state Rayleigh wave method assumed that the measured propagation velocity was equal to the shear wave velocity of the soil deposit at a depth of one or half a wavelength. In the past decade this procedure has been modified to account for the relationship between the shear wave velocity and the Rayleigh wave velocity for a half space. The Rayleigh wave velocity Vr varies from 0.874 to 0.955 Vs depending on Poisson’s ratio. For values of Poisson’s ratio ν larger than 0.1 we can write approximately

Vs=C Vr (5)with

C=1.135–0.182v (6)then

(7)where G=shear modulus, E=Young’s modulus, γ=total unit weight, and g =acceleration of gravity.With the development of the SASW method, an additional modification was made to this procedure by Heisey et. al. (1982) who considered that the propagation velocity was the Rayleigh wave velocity of the material at a depth of 1/3 of the wavelength. This approach has been used recently by Vrettos and Prange (1990) in the study of dispersion curves at various sites where the material stiffness increased gradually with depth with excellent results.A more accurate and more sophisticated procedure to backcalculate the material properties from the experimental dispersion curves is to assume a given soil profile, to conduct an analytical study to obtain the dispersion curve corresponding to that profile, to compare this theoretical curve with the experimental one, to introduce appropriate modifications to the profile and to repeat the process until a satisfactory agreement is reached. The main objective of this paper is to discuss two alternative ways in which the analytical determination of the dispersion curve can be performed.



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< previous page page_114 next page >Page 114ANALYTICAL MODELINGModeling of the SASW test to obtain theoretical dispersion curves for a given soil profile can be accomplished with two approaches: considering only plane generalized Rayleigh waves (a two dimensional solution) or accounting for three dimensional wave propagation effects and attempting to model exactly the actual experimental set up. The first method provides a simple and expedient basis to understand the results of the test, while the second simulates more realistically the physical process including possible reflections and/or for refractions when there are abrupt changes in properties. Two-Dimensional AnalysesThe objective of these analyses is to obtain the theoretical dispersion curve for plane Rayleigh waves propagating in a horizontally layered deposit with known properties. The mathematical model consists thus of a horizontally layered half space with homogeneous properties within each layer. The solution of the differential equations of motion for each layer allows the stresses and displacements at the top of the layer to be related to those at the bottom as a function of the frequency of vibration and the wave number (or wavelength) through a “transfer” matrix (Thomson 1950, Haskell 1953). Imposing compatibility of displacements and equilibrium at the interfaces between layers and multiplying the transfer matrices of the different layers a relationship is obtained between the stresses and displacements at the free surface and those at any depth, or the amplitudes of the waves travelling up and down in an underlying half space. With no excitation (zero stresses) at the top and no waves propagating upward from the bottom the amplitude of the waves travelling down is given by a system of homogeneous equations

TA=0 (8)To have any waves and therefore nontrivial motions the determinant of the 2×2 matrix T must be equal to 0. The terms of this matrix are function of the frequency ω and the wave number k. For a fixed ω the values of k that make the determinant zero (eigenvalues) are the wave numbers of the generalized Rayleigh waves. For each value of k one can then find a phase velocity ν=ω/k and a wavelength λ=2π/k.Alternatively the tractions at the top and bottom of each layer can be expressed in terms of the displacements at the same levels through a dynamic stiffness matrix (Kausel and Roesset 1981). The stiffness matrices of the individual layers (as well as that of a half space) can be assembled following the normal rules of matrix structural analysis leading to a system of equations of the form

(9)

where K is the dynamic stiffness matrix of the complete soil deposit, are the amplitudes of the displacements at the various layer interfaces (starting at the free surface) and P is the vector of applied loads at these interfaces. The terms of K are again function of ω and k (frequency and wave number). To have nontrivial displacements without any external loads the determinant of K must vanish. It



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< previous page page_115 next page >Page 115should be noticed that K is a tridiagonal matrix in terms of 2×2 submatrices and the determinant can be easily evaluated for any value of k and a fixed ωWhen dealing with a soil profile with properties increasing with depth there will always be at least one real eigenvalue (wave number) k. In most cases in fact there will be many real eigenvalues. A question may then arise as to which one corresponds to the wave propagation velocity that would be measured in the field. When the soil properties increase smoothly with depth the first eigenvalue (smallest value of k) is the one of interest. When soil properties vary in a more complex way, however, this may not be always the case and one may find that the measured propagation velocities are in better agreement with the phase velocities of the second, third or fourth eigenvalue. When the modulus of the underlying half space is smaller than those of the upper layers (typical situtation for a pavement profile) there will be a maximum frequency above which there are no real wave numbers. A proper solution in this case would require the determination of the complex eigenvalues, leading also to complex phase velocities. A simpler alternative often used is to assume that the half space is made of air (plate theory) which will yield again real roots (Jones 1962, Nazarian 1984). Another alternative is to select as an approximation real values of k that make the real part of the determinant of the matrix K equal to 0 (Shao, 1985). In any case, whichever alternative is used, the question of whether the smallest eigenvalue or a higher one is the appropriate one remains open and can only be answered by modelling the actural three dimensional problem.Three Dimensional AnalysesTo simulate the dynamic response of a soil profile to a vertical disk load applied on the surface the solution can be expressed in cylindrical coordinates. Displacements and stresses (or tractions) on a horizontal surface can be expanded in Fourier series in the circumferentical direction and in terms of cylindrical functions (Bessel, Neuman or Hankel functions) in the radial direction as shown by Kausel and Roesset, 1981. For application to cases with an axisymmetric loading, only one term of the Fourier series is needed (the 0 term), and the radial and vertical displacements U, W can be expressed as

(10)where J0 and J1 are the zero and first order Bessel functions; k is the wave number; r is the radial distance from the source; R is the radius of the disk and q is the magnitude of the uniformly distributed load. u and w are functions of k and can be obtained finding the solution to Eq. 9 for a harmonic load at the surface with wavelength 2π/k. Two different approaches can be followed: Continuous Formulation The solution of the problem requires assembling the dynamic stiffness matrix of the soil profile, solving it for various values of k (wave number) and evaluating the integrals of Eq. 10. The computation is relatively time consuming when there is a large number of layers and the integration requires special precautions if there is no damping in the soil. Typically, a small amount of material damping is always assumed for computational purposes. For a simple layered system with few layers (n≤3) or a half space, this type of solution can be very efficient.



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< previous page page_116 next page >Page 116Discrete Formulation An alternative to the continuous model can be obtained by expanding the terms of the dynamic stiffness matrix of a layer in terms of k and keeping only up to second degree terms. This is equivalent to the assumption that the displacements have a linear variation with depth over each layer. The stiffness matrices of each layer, and of the total profile can then be expressed in the form

K=Ak2+Bk+C−Ω2M (11)where the expressions for the matrices A, B, C and M can be founded in Waas 1972, Kausel 1974. The wave numbers and in-plane modes of propagation are now the solution of a quadratic eigenvalue problem. In this case one will obtain both real and complex wave numbers and the analysis could be stopped at this point with the determination of the 2-D dispersion curve. The question still remains, however, as to which eigenvalue is the most appropriate one. Kausel has shown that the displacements , in Equation 10 can be expressed as

(12)for a system of n layers over a halfspace, where ui1 and wi1 denote the horizontal and vertical displacements at the surface in the ith mode and can be found from the corresponding mode shape. Substituting Eq. 12 in Eq. 10, the integral can be evaluated analytically in closed form (Kausel, 1981). This solution is particularly convenient when dealing with a large number of layers as is the case when it is desired to obtain a detailed variation of soil properties with depth. However, owing to the assumption of linear variation of the displacements within a layer, a large number of sublayers must be used to obtain satisfactory results. Shao (1985) suggested both static and dynamic rules to divide the physical layers into finer sublayers to provide appropriate thicknesses. Determination of Dispersion Curve Once the complex response at the surface of the soil deposit has been obtained, there are two ways to compute the corresponding dispersion curve:1) Phase Spectrum in Spatial DomainFor a given excitation frequency, the phase of the response can be plotted versus distance to the source leading to what may be referred to as a spatial domain phase spectrum. The apparent velocity of propagation of the waves between any two point is

(13)

where v is the wave velocity, f is the excitation frequency, D is the distance between the two receivers, and is the difference of the phases (in radians/sec) at the two receivers (values of the spatial phase spectrum at the two points).



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< previous page page_117 next page >Page 117This procedure allows to observe the variation in the wave propagation velocity with distance. On the other hand, since the solution is based on the spatial information, a varying number of receivers must be simulated within an appropriate distance from the source and the analytical setup does not truly reflect the in-situ arrangement of the SASW test.2) Phase Spectrum in Frequency DomainInstead of computing the phases at various distances from the source for each frequency, the phase information can be obtained for a fixed distance between receivers as a function of frequency. In a typical SASW setup, the phase spectrum between two receivers at a given distance from the source is used to calculate the propagating velocities over a certain frequency range. Several receiver spacings are used to cover the complete frequency range of interest yielding a composite dispersion curve as illustrated in Fig. 3.A comparison of the results obtained using the 2-D dispersion curves of generalized Rayleigh waves and the 3-D solution was initially performed by Shao in 1985. Further parametric studies were conducted by Sanchez-Salinero, 1987 and Roesset, et. al, 1990, investigating the application of the SASW method to pavements. These studies indicated that the best agreement between the two dispersion curves is obtained when the distance between receivers is equal to the distance from the source to the first receiver and when this distance, which is the main controlling parameter, is of the order of two wavelengths.ExamplesTo show the dispersion results obtained from the 2-D and 3-D analyses, two artificial profiles were studied. Profile No.1 has a shear wave velocity increasing gradually with depth up to 80m as shown in Fig. 4a. The corresponding 2-D and 3-D dispersion curves are shown in Fig. 4b. Excellent agreement between the two solutions is obtained taking the 1st mode of the 2-D solution. Fig. 4c shows the variation of properties with depth that would result from the use of the simplified inversion procedure with 1/3 of the wavelength and the actual properties. It should be noticed that the predictions are very good in this case particularly up to a depth of 40 m. Even beyond this point the error is small. Artificial profile No. 2 has a shear wave velocity decreasing with depth over the top 40 m as shown in Fig. 5a and its dispersion curves obtained from 2-D and 3-D analyses are plotted in Fig. 5b. The difference between the two solutions is more apparent in this case particularly for wavelengths between 15 and 65m. Fig. 5c shows the predicted properties from the simplified inversion. The predictions are still very good up to a depth of 20m, but the errors are important between 20 and 50m.CASE STUDIESTo illustrate further the difference between the 2-D and the 3-D solutions and to verify the applicability of the simplified inversion procedure, three actual sites A (soil deposit), B and C (pavements) were studied. For each site, the experimental dispersion curve was obtained using the SASW method. Approximate soil properties were computed using the simplified procedure and iterative analyses assuming a profile, computing the theoretical dispersion curves, comparing them to the experimental data, and modifying the assumed properties to improve the fit.



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< previous page page_118 next page >Page 118For a soil deposit with stiffness gradually increasing with depth like site A, a set of properties which gave an excellent agreement with the experimental dispersion curve could be easily obtained with very few iterations. Fig. 6a shows the soil properties obtained by the simplified and the analytical procedures. The agreement is very good up to 10 or 12m but deteriorates for larger depths. Fig. 6b shows the theoretical dispersion curves corresponding to the soil profile obtained with the simplified inversion procedure. The agreement with the experimental data is very good up to a wavelength of approximately 30 m, but the theoretical results are noticeably smaller than the experimental ones beyond that point. Fig. 6c compares the experimental dispersion curves to the 2-D and 3-D theoretical curves corresponding to the fitted profile. The 2-D solution is a smooth curve which matches very well the experimental data except for the range of wavelengths between 35 and 50m. In this range, the experimental data show a sharp change in slope instead of the smooth behavior of the 2-D solution. The 3-D solution, on the other hand, reproduces very well this abrupt change in slope.Site B represents an asphalt concrete pavement. Fig. 7a shows the material properties obtained with the simplified and the iterative procedure. The two profiles have now some marked differences. Fig. 7b shows the theoretical dispersion curves corresponding to the simplified profile. It should be noticed that the results of the 2-D analysis with the first mode appear to be in better agreement with the experimental data for wavelengths up to 3 m, although they are in fact less accurate. The phase velocities obtained with the 3-D solution are larger than the experimental data. Fig. 7c shows the corresponding results for the fitted profile. Even in this case for the number of iteration performed, the agreement between the theoretical dispersion curves and the experimental data is not perfect particularly for wavelengths between 1.5 and 3m. It is however much better than that of Fig. 7b.Similar results for the material profile at site C are shown in Fig. 8a. It is apparent that the approximate profile obtained from the simplified inversion is significantly different from the fitted one for depths below 0.5m. Differences between the dispersion curves of the approximate profile and the experimental data as shown in Fig. 8b are viewed at wavelengths from .5m to 1.5m. Dispersion results for the profile with analytical fitting are shown in Fig. 8c. The 3-D solution reproduces very well the experimental data. The use of the first mode in the 2-D solution is again inappropriate for the this site.CONCLUSIONSThe results of the cases shown here and a large number of other studies conducted in recent years indicate that the simplified inversion procedure based on assigning a Rayleigh wave velocity at each depth equal to the phase velocity for a wavelength of 3 times that depth will produce very good results when dealing with a soil profile where the properties increase smoothly with depth. When there are layers of soil with very abrupt and marked changes in properties or when the properties decrease with depth, as in the case of pavements, the procedure can be used to obtain a preliminary estimate but must be refined through a series of analyses.



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< previous page page_119 next page >Page 119The use of the 2-D solution with the dispersion curve corresponding to the first mode of propagation (smallest eigenvalue) is reasonable for soil deposits with gradual variation of properties but cannot reflect sudden jumps and discontinuities in the slope of the curve caused by wave reflections or refractions. For more complicated soil profiles and most pavement systems, the use of the more accurate three dimensional solution is recommended since this approach can reproduce the true wave propagation phenomena involved in the test. Even so some smoothing of the experimental data, which is not without error or noise, and even of the theoretical curves, may be necessary to get good fits with a reasonable amount of work. This is due to the fact that the results of the 3-D analyses depend on the actual position of the points at which the phases are computed. Varying these points will introduce modifications in the dispersion curves. To obtain a reasonable agreement, the position of the receivers should thus be similar in the field and in the analyses. REFERENCES1. DEGEBO,“Deutsche Gesellschaft fur Bodenmechanik,” Vol. 4, Springer, Berlin, 1938.2. Bergstorm, S.G. and Linderholm, S.,“Dynamic Method att Utrona Ultiga Marklagers Genomsnittliga Elasticitetsegens Kaper,” Handlinger No. 7, Svenska Forsknings-Institutet for Cement och Betong Vid. Kungl. Tekniska, Hogskolan, 1946.3. Van der Poel, C., “Dynamic Testing of Road Construction,” Journal of Applied Chemistry, Vol. 1, 1951, pp. 281–290.4. Nijboer, L.W. and Van der Poel, C.,“A Study of Vibration Phenomena in Asphaltic Road Construction,” Proceedings, Association of Asphaltic Pavement Technology, 1953, pp. 197–231.5. Jones, R., “In-Situ Measurement of the Dynamic Properties of Soil by Vibration Methods,” Geotechnique, Vol. 8, No. 1, March, 1958, pp. 1–21.6. Jones, R.,“Surface Wave Technique for Measuring the Elastic Properties and Thickness of Roads: Theoretical Developement,” British Journal of Applied Physics, Vol. 13, 1962, pp. 21–29.7. Heukelom, W. and Foster, C.R., “Dynamic Testing of Pavements,” Journal of Soil Mech. and Found. Div., Proc. ASCE, Vol. 86, No. SM1, Part 1, February, 1960.8. Ballard, R.F., “Determination of Soil Shear Moduli al Depth by In-Situ Vibratory Techniques,” Miscellaneous Paper No. 4–691, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, 1964.9. Fry, Z.B.,“Dynamic Soil Investigations Project Buggy, Buckboard Mesa Neveda Test Site, Mercury, Neveda,” Miscellaneous Paper No. 4–666, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, 1965.10. Heisey, J.S., Stokoc, K.H., II, and Meyer, A.H., “Moduli of pavement Systems from Spectral Analysis of Surface Waves,” Transportation Research Record, No. 852, Washington, D.C., January, 1982.11. Slokoe, K.H., II, and Nazarian, S., “Effectiveness of Ground Improvement from Spectral Analysis of Surface Waves,” Proceedings, 8th European Conference on Soil Mechanics and Foundation Engineerin, Helsinki, Finland, May 1983.12. Shao, K.-Y., “Dynamic Interpretation of Dynaflect, Falling Weight Deflectometer and Spectral Analysis of Surface Waves Tests on Pavement System”, Ph.D. Dissertation, The University of Texas al Austin, December 1985.13. Sanchez-Salinero, I.,“Analytical Investigation of Seismic Methods Used for Engineering Applications”, Ph.D. Dissertation, The University of Tcxas at Austin, May 1987.14. Sheu, J.C., “Applications and Limitations of the Spectral-Analysis-of-Surface-Waves Method”, Ph.D. Dissertation, The University of Texas at Austin, August 198715. Rix, G.J., “Experimental Study of Factors Affecting the Spectral-Analysis-of-Surface-Waves Methods, Ph.D. Dissertation, The University of Texas at Austin, August 1988.



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< previous page page_120 next page >Page 12016. Roesset, J.M., Chang, D.-W., Stokoe, K.H., II and Aouad, M., “Modulus and Thickness of the Pavement Surface Layer from SASW Tests,” Transportation Research Record, Vol. 1260., January 1990, pp. 53–63.17. Kang, Y.V., “The Effect of Finite Width on Dynamic Deflections of Pavements”, Ph.D. Dissertation, The University of Texas at Austin, May 1990.18. Vrettos, C. and Prange, B.,“Evaluation of In Situ Effective Shear Modulus from Dispersion Measurements,” Journal of Geotechnical Engineering, Vol. 116, No. 10, October, 1990, pp. 1581–1585.19. Thomson, W.T., “Transmission of Elastic Waves Through a Stratified Soil Medium,” Journal of Applied Physics, Vol. 21, February 1950., pp. 89–93.20. Haskell, N.A.,“The Dispersion of Surface Waves on Multilayer Media,” Bulletin of the Seismological Society of America, Vol. 43, 1953, pp. 17–34.21. Kausel, E. and Roesset, J.M., “Stiffness Matrices for Layered Soils,” Bulletin of the Seismological Society of America, Vol. 71, 1981, pp. 1743–1761.22. Nazarian, S., “In Situ Determination of Elastic Moduli of Soil Deposits and Pavement Systems by Spectral-Analysis-of-Surface-Waves Method”, Ph.D. Dissertation, The University of Texas at Austin, August 1984.23. Waas, G., “Linear Two-Dimensional Analysis of Soil Dynamics Problems in Semi-Infinite Layered Media,” Ph.D. Dissertation, The University of California at Berkeley, 1972.24. Kausel, E., “Forced Vibration of Circular Foundations on Layered Media”, Research Report R74–11, Department of Civil Engineering, M.I.T., 1974.25. Kausel, E., “An Explicit Solution for the Green Functions for Dynamic Loads in Layered Media”, Research Report S81–13, Department of Civil Engineering, M.I.T., 1981.



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Figure 1 General configuration of equipment used in SASW test

Figure 2 Approximate distribution of vertical particle motion with depth subjected to various excitation frequencies (or wavelengths)

Figure 3 Composite dispersion curve from data of individual receiver spacing



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Figure 4 Artificial profile No.1, a) material property, b) 2-D and 3-D dispersion curves, c) comparison of profile and solution from simplified inversion



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Figure 5 Artificial profile No.2, a) material property, b) 2-D and 3-D dispersion curves, c) comparison of profile and solution from simplified inversion



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Figure 6 Site A, a) approximate profile and computed Profile, b) dispersion curves of approximate profile, c) dispersion curves of computed profile



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Figure 7 Site B, a) approximate profile and computed Profile, b) dispersion curves of approximate profile, c) dispersion curves of computed profile



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Figure 8 Site C, a) approximate profile and computed profile, b) dispersion curves of approximate profile, c) dispersion curves of computed profile



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< previous page page_127 next page >Page 127Inversion of Rayleigh Wave Dispersion Curve for SASW TestN.Gucunski (*), R.D.Woods (**)(*) Department of Civil & Environmental Engineering, Rutgers University, P.O. Box 909, Piscataway, NJ 08855–0909, U.S.A.(**) Department of Civil Engineering, The University of Michigan, Ann Arbor, MI 48109–2125, U.S.A.INTRODUCTIONThe Spectral-Analysis-of-Surface-Waves method is a seismic technique for measuring in situ elastic moduli and thicknesses of layered systems, like soils and pavements. The advantages of the method are: it is performed from the surface and therefore does not require boreholes, it is nondestructive, it is in most cases highly accurate, and it utilizies a simple procedure and test setup with the prospect of being fully automized. The major deficiency of the method at this moment is the process of inversion of the dispersion curve.The inversion process is a simple task and provides reliable results in cases of regular soil profiles where the shear wave velocity increases with depth. Experience which the authors have from SASW measurements in which lower velocity layers were found below higher velocity layers by a complementary crosshole test, is that the inversion process can be an ambiguous process.Results of the study of Rayleigh wave dispersion in soil profiles where a softer layer is trapped between a harder surface layer and a harder half-space have indicated (Gucunski and Woods, 1991) that the uniqueness of a derived soil profile is attributed to a strong influence of higher Rayleigh modes on the overall wave propagation pattern. This paper presents results on Rayleigh wave dispersion for several cases of soil layering which can be characterized as irregular patterns of soil stratification. The cases include: • A softer layer trapped between a harder surface layer and a half-space,• A hard surface layer, and• A harder layer trapped between a softer surface layer and a half-space.The goal of the study is to explore alternative ways for the improvement of the inversion process and to provide guidelines for identification and interpretation of results in the field.



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< previous page page_128 next page >Page 128THE SPECTRAL-ANALYSIS-OF-SURFACE-WAVES METHODThe SASW method is based on the dispersive character of Rayleigh waves in layered media. The method can be viewed as an extension of a steady-state technique which has principally solved problems of efficient simultaneous generation and detection of a broad spectrum of surface waves, and based on the collected data the determination of a shear wave velocity profile.The SASW testing procedure can be divided into three phases: 1) collection of data in the field, 2) evaluation of the Rayleigh wave dispersion curve, and 3) inversion of the dispersion curve to obtain the shear wave velocity profile. Figure 1 shows a schematic of the current testing configuration. Surface waves are generated by an impact source, detected by a pair of receivers, and recorded on an appropriate recording device. A very convenient device for the spectral analysis is a Wave Form Analyzer which can perform operations either in the time or the frequency domain. The test is repeated for several receiver spacings to cover a desired range of wavelengths, and in two directions to cover effects of dipping layers and any internal phase shift due to receivers and instrumentation, as described by Nazarian and Stokoe (1983). Figure 2 presents the commonly used Common Receivers Midpoint Geometry.

Figure 1. Schematic of experimental arrangement for SASW test.Among several spectral functions the phase of the cross power spectrum and the coherence are of the greatest importance. The phase of the cross-power spectrum, shown in figure 3a, contains information on the frequency-phase velocity relations, while the coherence, shown in figure 3b, contains information on quality of the recorded signal. Based on the relationships presented in figure 4, where β represents the phase in degrees and f frequency in Hz, the dispersion curve for a single receiver spacing can be constructed. The dispersion curve needs to be filtered. For the source to near reciver spacing, S, equal to the receiver spacing, X (see Fig. 1), Heisey et al. (1982), based on an experimental study, suggested the following filter criteria

λph/3<X<2λph (1)where λph is the phase wavelength. Sheu et al. (1986) combined experimental results with results of a theoretical study (Sanchez-Salinero et al, 1987) and defined that the usable



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Figure 2. Common Receivers Midpoint Geometry.wavelength should be less than 1.5 receiver spacing. Results for several receiver spacings and both directions are finally statistically combined to derive an average dispersion curve, as shown in figure 5a.Inversion of the Rayleigh wave dispersion curve is a process of determination of the shear wave velocity profile from the wavelength-phase velocity relationship. A currently used inversion process is based on the comparison of theoretical dispersion curves for an assumed profile with the experimental dispersion curve, as described by Nazarian, 1984. Once a good match between those is achieved the assumed shear wave profile is accepted as a solution. Figure 5b respresents an experimental dispersion curve marked with asterisks and theoretical dispersion curves for the first two Rayleigh modes marked by crosses, A perfect match between the experimental and the theoretical first Rayleigh mode is marked by zeros. The question that was raised at the beginning of this study was whether the inversion process should be guided solely by the comparison of the experimental dispersion curve with the theoretical first Rayleigh mode or should higher Rayleigh modes be considered. This problem was identified previously by Nazarian (1984), Nazarian and Stokoe (1986) and Sanchez-Salinero et al. (1987).NUMERICAL SIMULATION OF THE SASW TESTThe SASW test can be described as an axisymmetric problem in which the impact source is represented by a circular loading at the center of the system. As described in Gucunski (1991) and Gucunski and Woods (1991), the problem can be, by utilizing Fourier and Hankel’s transforms, presented in the frequency-wave number domain in the form

(2)where represents the system stiffness matrix, the vector of layer interface displacements, and the vector of external layer interface loadings. Because the Rayleigh wave represents a natural mode of wave propagation, it can be found from the solution of an eigenvalue problem


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(3)in which the eigenvalues represent phase velocities of the Rayleigh wave while the vector u represents its shape as a function of depth.

Figure 3. (a) The phase of the cross power spectrum. (b) The coherence.An extensive numerical study of the influence of soil stratification on Rayleigh wave dispersion was conducted which included evaluation of: • Wave propagation field due to three-dimensional simulation of the SASW test,• Dispersion curves of several lowest modes for plane Rayleigh waves according to equation (3),• Rate of energy transmission in the horizontal direction,• Modal displacements in the frequency-wave number and the spatial domains, and• Mode shapes.The results presented later include only some of the above elements.CASE 1: A SOFTER LAYER TRAPPED BETWEEN A HARDER LAYER AND A HALF-SPACERayleigh wave dispersion in a system where a softer layer is trapped between a harder surface layer and a half-space will be illustrated by displacements in the spatial domain, modal displacements and the comparison of the “simulated” dispersion curve and the curves for several modes of plane Rayleigh waves.Figure 6 represents displacements in the frequency-wave number domain for frequencies 20 to 50Hz. R0 represents the radius of the loading plate, d1 and d2, and Vs1 and Vs2 represent the thicknesses and the shear wave velocities of the first and the second layer,



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Figure 4. Evaluation of the phase velocity of a R-wave.

Figure 5. (a) A typical set of dispersion curves for a soil site and an average dispersion curve. (b) Inversion of the experimental dispersion curve.respectively, ν and ξ are the Poisson’s and damping ratios of the soil, respectively, ρp is the mass density of the plate, and p the intensity of loading. The sharp peaks represent Rayleigh waves. The figure clearly indicates the dominant influence of the first mode at 20Hz, the second mode at 30 and 40Hz, and the third mode at 50Hz.Displacements for the first three modes in the spatial domain, calculated by applying the inverse Hankel’s transform with the interval of integration over the vicinity of each of the modes, are shown in figure 7. The figure cofirms the transition of dominant influence from the



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Figure 6. Vertical surface displacements in the frequency-wave number domain.first to the second and the third mode, between 20 and 30, and between 40 and 50 Hz, respectively.Figure 8 represents the comparison between the dispersion curves of the first three modes of plane Rayleigh waves and the “simulated” dispersion curve. The dispersion curves for plane Rayleigh waves are evaluated according to equation (3). The “simulated” curve represents the total solution for all the waves, and therefore is calculated by applying the inverse Hankel’s transform over the entire frequency-wave number domain. The “simulated” curve follows well the first Rayleigh mode from 0 to about 20Hz, and afterwards moves towards the second and the third mode curves. Zones of transition of the “simulated” curve from one two another mode are characterized by localized approaches of dispersion curves for plane Rayleigh waves.CASE 2: A HARD SURFACE LAYERA surface layer stiffer than soil below can exist in situations where soil improvement measures were applied like: grouting, compaction or freezing, or where soil is naturally frozen or dessicated. The results presented include modal displacements, the rate of energy transmission in the horizontal direction, and the comparison of the “simulated” dispersion curve and the curves for several modes of plane Rayleigh waves.Figure 9 represents modal displacements for the first five modes. A clear transition of dominant influence from the first towards higher modes as frequency increases can be observed. Peaks of the second and higher modes match well corresponding natural frequencies of vertical oscillations of the surface layer, slightly modified by the presence of the underlying softer layer. This was confirmed by the results on the rate of energy transmission in the horizontal direction. The rate represents the amount of energy propagating through a layer and averaged over a period 2π/ω. Figure 10 presents normalized rates, rates divided by the rate for the entire



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Figure 7. Vertical surface displacements in the spatial domain at the distance r=2Vs1/f from the source.

Figure 8. Theoretical curves of the first three modes for plane Rayleigh waves and the “simulated” dispersion curve.system, for the first four Rayleigh modes. Frequency at which peaks of the rate for the surface layer occur correspond well to the energy is transmitted completely by the underlying softer layer. The comparison of the “simulated” dispersion curve and curves for the plane Rayleigh waves is shown in figure 11. As a comparison to the case presented in figure 8, points of transition are not characterized as strongly by the approaches of dispersion curves, as by the change in curvature. The “simulated” curve follows very well the “valleys” created at dispersion curves. The other important fact is that at high frequencies the dispersion curves approach the shear wave velocity of the underlying layer. This contradicts the known fact that the phase



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Figure 9. Surface vertical displacements in the spatial domain at the distance r=2Vs1/f from the source.velocity of the first mode should approach the Rayleigh wave velocity of the surface layer, while the phase velocity of higher modes should approach the shear wave velocity of the surface layer. These results are in an excellent agreement with results on the rate of energy transmission presented in figure 10, which indicate propagation within the second layer only.CASE 3: A STIFF LAYER BETWEEN SOFTER LAYERSA stiffer layer trapped between softer layers can be a result of natural soil deposition, or can be artiffically created in cases where a system with a stiff surface layer was covered by a fill. Results from several tests have indicated that only in cases where a stiff layer is underlain by a much thicker softer layer higher Rayleigh modes can have a significant role. Therefore the results presented include only a case in which a stiff layer is underlain by a four times thicker soft layer and a half-space.Figure 12 shows a comparison of the “simulated” dispersion curve and dispersion curves for the plane Rayleigh waves. A localized transition of the “simulated” curve from the first to the second mode in the range 8 to 16Hz is characterized by the approach of two curves. It can be observed also from modal displacements presented in figure 13, where the second mode has a dominant amplitude in the same frequency range. Results from all the cases studied indicate that at higher frequencies transmission of energy occurs completely either through the surface softer layer or the layer below the stiffer layer. In the case of a thick soft layer energy of the first mode is transmitted through the surface layer, while energy of all the other modes it is transmitted through the underlying layer. This is illustrated in figure 14 for the first four modes. Dispersion curves in figure 12 indicate that the phase velocity of the first mode approaches the Rayleigh wave velocity of the surface layer as frequency increases. This is in good agreement with the rate of energy transmission for the first mode. Dispersion curves for all higher modes approach the shear wave velocity of the third layer. This is again in good agreement with the rate of energy transmission and indicates behavior similar to the one in the case of a stiff surface layer.



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Figure 10. Normalized rate of energy transmission for the first four Rayleigh modes.

Figure 11. Theoretical dispersion curves for the first five modes for the plane Rayleigh waves and the “simulated” dispersion curve.Both the modal displacements in figure 13 and the rate of energy transmission for the second mode in figure 14 indicate that the stronger participation of the second mode was initiated by the fundamental frequencies of vertical oscillations of the third layer around 7.5Hz and of the surface layer around 15Hz. It is obvious from the first mode plot that below the fundamental frequency of the third layer, the stiff layer does not allow significant transmission of energy to the system below. Slightly pronounced displacements of the fourth


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Figure 12. Theoretical dispersion curves for the first five modes for the plane Rayleigh waves and the “simulated” dispersion curve.

Figure 13. Surface vertical displacements in the spatial domain at the distance r=2Vs1/f from the source.and the fifth mode (not shown), can be attributed to the excitation of the first mode of vertical oscillations of the surface layer.ALTERNATIVE INVERSION PROCESSESFrom all the above results it is obvious that the inversion process must not be guided solely by the comparison of the experimental dispersion curve and the dispersion curve for the theoretical first Rayleigh mode. Influence of higher Rayleigh modes should be considered through evaluation of the modal displacements, the rate of energy transmission or mode shapes.



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Figure 14. Normalized rate of energy transmission for the first four Rayleigh modes.Two procedures at this moment seem to represent the simpliest approaches. The first one is based on the direct comparison of the experimental and the “simulated” dispersion curves. This approach provides a comparison of the experimental dispersion curve, in which Rayleigh waves are contaminated by body waves, with an equivalent numerically simulated one. The second approach is based on the comparison of the experimental dispersion curve and the “average phase velocity” dispersion curve. The “average phase velocity” curve represents a weighted average of the phase velocity of several modes, in which the weighting factors are represented by modal displacements. The “simulated” and the “average phase velocity” dispersion curves are compared to the theoretical dispersion curves for the plane Rayleigh waves in figure 15.CONCLUSIONSConclusions from the study on Rayleigh wave dispersion in soil profiles where the shear wave velocity does not generally increase with depth are:1) Higher Rayleigh modes provide a significant, and in many cases, a dominant influence on the overall wave propagation pattern along the surface of a system. Therefore the inversion of the experimental dispersion curve should not be guided solely by the theoretical first Rayleigh mode.2) Transition of influence from one Rayleigh mode to another is characterized either by localized approaches or by significant changes in curvature of the corresponding dispersion curves.3) The presented results suggest that the inversion process should be based on the direct comparison of the experimental either with the “simulated” or with the “average phase velocity” dipersion curve.



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Figure 15. Theoretical dispersion curves for the first four modes for plane Rayleigh waves, the “simulated” and the “average phase velocity” curve.REFERENCESGucunski, N. (1991), Generation of Low Frequency Rayleigh Waves for the Spectral-Analysis-of-Surface-Waves Method, Ph.D. Dissertation, Department of Civil Engineering, The University of Michigan, Ann Arbor.Gucunski, N. and Woods, R.D. (1991), “Use of Rayleigh Modes in Interpretation of SASW Test,” Proceedings of the Second International Conference on Recent Advances in Geotechnical Earthquake Engineering in Soil Dynamics, Vol. II, St. Louis, Missouri, March 11–15, pp. 1399–1408.Heisey, J.S., Stokoe, K.H.II, Hudson, W.R., and Meyer, A.H. (1982), Determination of In Situ Shear Wave Velocities from Spectral Analysis of Surface Waves, Research Report No. 256–2, Center for Transportation Research, The University of Texas at Austin, December, 277 pp.Nazarian, S. and Stokoe, K.H.II (1983), Evaluation of Moduli and Thicknesses of Pavement Systems by Spectral-Analysis-of-Surface-Waves Method, Research Report No. 256–4, Center for Transportation Research, The University of Texas at Austin, December, 123 pp.Nazarian, S. (1984), In Situ Determination of Elastic Moduli of Soil Deposits and Pavement Systems by Spectral-Analysis-of-Surface-Waves Method, Ph.D. Dissertation, Civil Engineering Department, The University of Texas at Austin.Nazarian, S. and Stokoe, K.H.II (1986), “Use of Surface Waves in Pavement Evaluation,” Transportation Research Record, No. 1070, pp. 132–144.Sanchez-Salinero, I., Roesset, J.M., Shao, K.-Y., Stokoe, K.H.II and Rix, G.J. (1987), “Analytical Evaluation of Variables Affecting Surface Wave Testing of Pavements,” Transportation Research Record, No. 1136, pp. 86–95.



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< previous page page_139 next page >Page 139Transient Response of Certain Topographical Sites for SH-Wave IncidenceH.Takemiya, C.Y.Wang, A.FujiwaraDepartment of Civil Engineering, Okayama University, Okayama, JapanABSTRACTIn view of the past seismic damages the soil amplification with modified predominant period is pointed out as a crucial factor. This paper is concerned with the analysis of the wave propagation and scattering through a certainly topographical-shaped alluvium.The time domain boundary element method is applied with use of elementwise analytical double integrals over space and time domains. For the SH wave incidence, the time history response of canyon/ alluvium surface are computed and interpreted from the engineering viewpoint.INTRODUCTIONObservations of past earthquake damages are very indicative of the seismic wave amplification by alluvium from the rock-like base level. The vertical shear wave propagation (1-D theory), has been dominantly employed for evaluating such surface soil amplification. In view of the surface/subsurface irregularities like a canyon or an alluvial valley, the 2-D modeling should be made at least to interpret the scattered wave field.The steady state harmonic analyses for certain topographical site condition have extensively been conducted. The amplification/reduction due to such topographies in comparison with the far field without it has been investigated with respect to type of the incident wave, angle of the incidence, and nondimensional frequency such defined as the ratio between the width of the site and the incident wave length. This response characteristic gives a meaningful interpretation of the topographical effect on seismic waves. Frequency domain analysis was first started on the antiplane motion for its simplicity. The investigation on the in-plane motion was then followed by many researchers. The methodologies for these studies are a series of expansion by wave functions, the finite element method (FEM), the so-called Aki-Larner method which assumes the discrete wave number expansion, the boundary element method (BEM), and the hybrid method of the FEM and the BEM, depending the complexities of the topography (e.g., Takemiya [1]). Extensive reviews of the topic are seen in the works by Sanchez-Sesma, et al. [2], Mossessian and Dravinski[3].



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< previous page page_140 next page >Page 140Another important aspect is the response time history or the transient response, which gives the phase characteristic for wave propagation and scattering due to the presence of the surface/subsurface soil irregularities. Reviewing past works concerned, the finite difference technique was taken by Ohtsuki [4]; the discrete wave number boundary element method was developed by Kawase [5]. The Fourier synthesis method was the straightforward approach to get it from the above steady state solution through the Fast Fourier Transform (FFT) algorithm as Mossessian and Dravinski [6] showed. However, care should be taken for the phase variation since the FFT presumes a certain periodic duration. The direct time domain BEM analysis is a promising approach for transient response problems. Discretization both in space and time for the boundary integral equation needs the elementwise double singular integrals over these domains. The author succeeded in getting such an explicit solution (Wang and Takemiya [7]). The time stepping algorithm facilitates the transient response computation for a prescribed incident wave.Herein, the out-of-plane motion of a canyon/alluvium on a uniform elastic halfplane base is analyzed for the SH wave incidence. The point of interest is placed on the wave scattering to be characterized by the phase by taking the Ricker wavelet as an incident wave function. Also, interested is the soil amplification for sinusoidal incident wave function to compute until the would-be steady state response is attained.FORMULATIONSuperposition of wave fieldsThe soil domain which includes a certain topographical alluvium is substructured into the surface soil deposits and the surrounding far field. The input seismic motion is prescribed as an incident wave to be defined at the far field (denoted by superscript F). The presence of the subsurface of soil deposits, reflecting the incident waves at the interface with the far field, generates scattered waves (denoted by the superscript S) in the far field. The near field is composed of the transmitted waves and the reflected waves from the free surface. See Fig. 1.

Fig. 1 Wave field for an alluvium desposit



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< previous page page_141 next page >Page 141Free field motionThe SH wave incidence is considered. The displacement is prescribed by

uI(x,z,t)=a f(αI) H(αI) (1)in which f(αI) denotes a certain function to describe the wave form. Herein, the Ricker wavelet of representative wave length λc=VsT in which Vs denotes the shear velocity and T is the representative period is used. Then,

uI(αI)=[2(πλcαI)2−1] exp−(πλcαI)2 for Ricker wavelet (2.1)uI(αI)=1−cos(2πηαI) for sinusoidal wave (2.2)

with dimensionless frequency η=2a/λ (2a=width of the soil deposit, λ=wave length). H() stands for the Heaviside step function and a is a constant. The argument αI indicates the phase to be specified at location (x0, z0) at time to as

αI=Vs(t−t0)−sinθ(x−x0)+cosθ(z−z0) (3)The reflected wave at the free surface keeps the wave form but changes the phase

αR=Vs(t−t0)−sinθ(x−x0)−cosθ(z+z0) (4)The total free field response is then

UF=uI+UR (5)Boundary Integral Equation for Wave ScatteringThe scattered wave propagation of the far field is computed by the BEM. The boundary integral equation representation in time domain for elastic waves scattering is formulated from the reciprocal theorem.

(6)in which u and t are the displacement and traction of the concerned body; u* and t* are the Green functions (or the fundamental solutions) for displacement and traction at field location y at time t due to a unit impulse force at location x at initial time equal to zero; the symbol (*) denotes the convolution integral operator with respect to time. For a 2-dimensional full space of an isotopic, homogeneous, elastic solid, the fundamental solution, when the out-of-plane motion is concerned, is given by

(7), (8)in which τ=Vst, r=|x-y|, n(y) is the outward normal of the boundary S(y) for the concerned domain. The integral p.v.∫( ) is interpreted by the Cauchy’s principal value and c(x)u(x) is the so-called free term.Spatial and Temporal DiscretizationThe boundary is discretized into the E linear segments within each the (M+1)



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< previous page page_142 next page >Page 142nodes exist. The displacement and traction in the e-elment can be expressed approximately in terms of the nodal values

with the aid of the assumed interpolation function, .

(9)in which ξ is the local coordinate within the element. Substituting this into Eq.(6) yields

(10)or in a matrix form

CU(t)=G(t)*T(t)−H(t)*U(t) (10)′The time axis is divided into a sequence of equal increment ∆t so that the time tk= k∆t (k=1,.., K; K refers the current time). The time variation of response is also approximated by use of the interpolation function φNu for displacement and φNt for traction as

(11)Substitution of Eq.(11) into Eq.(10) results in

(12)Time Stepping algorithmUnder the assumption of an identical interpolation function for every step, which spans ∆t only, Eq.(12) is rewritten as

(13)in which

(14), (15)with δk=1 for k=0 and δk=0 otherwise. Eq.(13) is solved stepwise for the unknown quantities based on the known quantities at previous times.In the above formulation the crucial part lies in the execution of the elementwise double integral operation as defined by

(16)which is singular about t* when xl is at the e-element. The analytical solution has



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< previous page page_143 next page >Page 143been obtained for a combination of representative space and time interpolation functions by use of the Cagniard-de Hoop method (Wang and Takemiya, 1991). The simplest case is for the 0-th order element or the constant element whichleads

(17)For other elements the reader refer to the above publication.Substructure formulationFor the wave field analysis for an alluvium on a uniform halfspace, the substructure procedure is effectively used. Referring to the illustration in Fig. 2, the separated alluvium deposits are characterized by the discretized form of the boundary equations.

(18)The exterior halfspace should be treated only for the scattering wave, not including waves as the free field, so that it is governed by

(19)Condensing out other variables than those related to their interface, the governing equations for the respective domain are expressed with the interface variables as unknown quantities.

(20), (21)The continuity condition is claimed to make an original total couped domains such that

(22), (23)

Fig. 2 Substructure formulation



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< previous page page_144 next page >Page 144VALIDATION FOR TWO-LAYERED HALFSPACEA two-layered halfspace is treated analytically for the purpose to give a validation to the present BEM numerical solution. The closed form solution can be obtained by considering the multiple wave reflections both at the free surface and the bottom of the top layer together with the transmission across it upward and downward. See Fig. 3. The

displacement in the top layer (denoted by 1)/ halfspace base (denoted by 2) is expressed as the sum of up-going wave

and down-going wave . Thus,

Fig. 3 Wave field in two-layered halfspace

(24)in which

(25), (26)

(27), (28)with the Tij giving the transmission coefficient from the j-th layer into the i-th layer and Rij the reflection coefficients into the i-th layer due to the presence of the j-th layer, which are defined respectively as

(29), (30)The phases indicated by the superfix U for the up-going and by D for the down-going waves in the top layer are specified by

(31), (32)and those for waves in the base halfspace by

(33), (34), (35)



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Fig. 4 Transient response for two-layered halfspace, SH Ricker wavelet incidence, λc=1



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< previous page page_146 next page >Page 146Fig. 4 shows the comparison of the above solution for the Ricker wavelet incidence with the BEM numerical one for a flat rectangular soil deposit configuration. The size of elements is determined by analyzing first a uniform halfplane. We note that an excellent agreement was attained for the surface location far from the boundary while some difference is observed near the boundary due to the wave diffraction.NUMERICAL COMPUTATION AND DISCUSSIONNumerical computation was conducted on different site topographies; canyons and alluvium deposits. The cross sections considered are a triangle, a half-circule and a trapezoid. As an input wave to the far field, a Ricker wavelet is used to investigate the wave propagation at transient state and a sinusoidal one is to know the amplification at steady state. The angle of incidence is varied for both cases.Transient responses:In Figs. 5. are shown the scattering waves due to the wave reflection and refraction at the half-circle canyon surface. The models for the numerical computations are illustrated in the same figures. We note that the first wave arrival time is determined by the distance from the incident wave front to the free surface. The late arrivals appear, like originating from the edges of the canyon, and propagate outward and inward the canyon. The outside-going waves lasts for longer distance while the inside-going waves diminish and vanish after they meet from both sides.The difference by the canyon configuration is appreciable. The triangle section (not shown here) gives rise to the most remarkable such late arrival waves. The trapezoidal section tends to confine such a wave diffraction at the nearby of the edges so that the late arrivals diminish at the middle part as the canyon width becomes wide. The biggest peak appears at the canyon edge, followed by the nearby surface outside the canyon, and the smallest peaks inside of it. The effect of angle of incidence results in amplified peaks of late arrival waves at the front surface and the reduced peaks at the rear surface of the canyon.Figs. 6 give the scattering wave field for alluviums for the numerical models attached. We observe that the response features are quite different from those for canyons. Due to the successive wave reflections at both the free surface and the bottom of the alluvium layer, the more complex waves are generated within it, and the longer wave duration results. Significantly amplified peak responses are attained at the alluvium center. The outside going waves are no more significant as for the canyon.The difference of alluvium configuration leads to significantly different response features. The trapezoidal section yield a very remarkable response, which is characterized by the up-going and down-going waves assorted by the surface traveling waves, resulting in a comparable peak response after a certain time elapse from the first Ricker wave arrival. Almost no outside-going waves directly from the edge of the alluvium exist. However, the inside-going waves turn out to be the outgoing waves after they meet. The triangle and the half-circular sections, on the other hand, do no show such sharp overlapping of waves, so that relatively smaller peaks appear at a certain time after the first Ricker wavelet arrival.The outside-going waves are observed appreciably.



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Fig. 5.1 Transient response for a half-circle canyon, Vertical incidence of SH Ricker wavelet λc=1

Fig. 5.2 Transient response for a trapezoidal canyon, Vertical incidence of SH Ricker wavelet, λc=1



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Fig. 6.1 Transient response for a half-circle alluvium, Vertical incidence of SH Ricker wavelet, λc=1

Fig. 6.2 Transient response for a trapezoidal alluvium Vertical incidence of SH Ricker wavelet, λc=1


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Fig. 7.1 Maximaum response amplitude of a half-circle canyon, after several cycle for a sinusoidal SH wave incidence,

Fig. 7.2 Maximaum response amplitude of a half-circle alluvium, after several cycle for a sinusoidal SH wave incidence,



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< previous page page_150 next page >Page 150The biggest amplification of the first Ricker wavelet arrival is noted for the triangle section and followed by the half-circular section. That of the trapezoidal is modest. For the third case, the comparison of the results with those for the horizontally layered situation makes clear the zone where the diffraction waves exist from the alluvium edge.Steady state responses:The steady-state harmonic response is of interest to understand the amplification of seismic waves by the site topography. The 2-dimesional frequency domain studies have extensively been made [1], from which we note the significance of the site topography on the seismic amplification with respect to the incident wave type, angle of incidence and the dimensionless frequency 11=2a//λ. In Figs. 7, the would-be steady state responses after several cycles are compared with the previous results [8, 9] to check the transient period and the associated accuracy. The comparison between these results indicates an excellent agreement for deep angle of incidence while some small discrepance appears at the rear free surface for shallow angle of incidence. However, this is imporved by taking a longer distance for free field nodes.CONCLUSIONThe transient time domain analysis in constrast to the frequency domain analysis made clear the response features of the topographical site condition like canyon or alluvium on a halfspace with respect to the phase for wave propagation besides the amplification.Through the numerical results with use of a Ricker wavelet for representative sectional configuration, the following findings are pointed out. For canyons, the scattering waves going outside are seen from the edge of it, most remarkably for triangle section, and modestly for the trapezoidal section. For alluviums, the longer duration of waves is observed for trapezoidal section than others and comparable peak value is noted for the late arrival waves. The amplification at the steady state response gives a good agreement with the available frequency domain solution.REFERENCES1) Takemiya, H., Ono, M. and Suda, K., BEM-FEM hybrid analysis for topographical site response characteristics, Proc. 2nd Int. Conf. on Recent Advances in Geot. Earthq. Eng. and Soil Dyn., St. Louis., USA, 1991.2) Mossessian, T.K. and Dravinski, M., Application of a hybrid method for scattering of P, SV and Rayleigh waves by near-surface irregularities, Bull. Seis. Soc. Am., 77, 1784–1803, 1987.3) Sanchez-Sesma, F.J., et. al., Surface motion of topographical irregularities for incident P, SV, and Rayleigh waves, Bull. Seis. Soc. Am., 75, 263–269, 1985.4) Ohtsuki, A. and Harumi, K., Effect of topography and subsurface inhomogeneities on seismic SV waves, Earthq. Eng. & Struc. Dyn. Vol.11, 441–462, 1983.5) Kawase, H., Time domain response of a semi-circular canyon for incident SV,P, and Rayleigh waves calculated by discrete wavenumber boundary element method, Bull. Seis. Soc. Am., 78.4, 1415–1432, 1988.6) Mossessian, T.K. and Dravinski, M., Amplification of elastic waves by a three dimensional valley. Part 2: Transient response, Earthq. Eng. & Struc. Dyn., 19, 681–691, 1990.7) Wang, C.Y. and Takemiya, H., Analytical elements of time domain BEM for Two-dimensional scalar wave problems, Int. J. Numr. Methd. Eng. (to appear)8) Trafunac, M.D., Scattering of plane SH waves by a semi-cylindrical canyon, Earthq. Eng. Struc. Dyn., Vol. 1, 267–281, 1973.9) Trafunac, M.D., Surface Motion of a semi-cylindrical alluvial valley for incident plane SH waves, Bull. Seis. Soc. Am., 61.6, 1755–1770, 1971,



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< previous page page_151 next page >Page 151Surface Wave Propagation in Stiff Top Layer Half-SpaceW.HauptGrundbauinstitut, Landesgewerbeanstalt Bayern, 8500 Nürnberg, GermanyABSTRACTThe propagation of surface waves in a half-space with a stiff top layer is investigated by FE-analyses as a plane, steady-state problem. The results show, that the top layers are causing only a relatively small increase of the wave velocity. The attenuation of the vibration amplitudes with distance at the surface is much greater than it can be due to material damping only. The reason for this behaviour might be radiation of energy from the top layer into the half-space.INTRODUCTIONThe propagation of vibrations in the ground generated by a dynamically loaded foundation, for instance a machine foundation, becomes increasingly an environmental problem. Also, such vibrations can impair the proper function of sensitive installations in the neighbourhood. The magnitude of the vibrations does not only depend on the dynamic forces and on the kind of excitation, but the dynamic properties of the subsoil do also play an essential role.Very often, the soil is more or less stratified, and usually the wave velocity increases with depth. However, there are cases where a stiff layer is situated at the surface. This condition is typically encountered in frozen soil over long periods of the year near the arctic regions of the earth. The investigations reported in the present paper, are dealing with such conditions, analyzing the propagation of vibrations in a half-space with a stiff top layer. Different situations are consid-



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< previous page page_152 next page >Page 152ered, and the homogeneous half-space is included in the study for comparison.Analytical and numerical solutions for the propagation of waves at the surface of a layered half-space can be found in various publications, for example [1–4]. However, these investigations are treating systems where a soft layer, e.g. a layer with smaller wave-velocity is located above a stiffer half-space. This is a common situation. Nevertheless, from all these publications it can be concluded, that the stratification of the subsurface soil is of extreme importance to the wave propagation.Among the experimental investigations, the one reported by Rao [5] is of special interest. Rao measured the wave propagation characteristic of an air-field pavement, a system with a stiff top layer above a relatively soft subsoil. The results indicate a distinct increase of the wave velocity with increasing ratio of d/λR (d=thickness of layer, λR =R-wave length). However, at Rao’s measurements the values of this ratio were not in the range that is analysed in the present study.The dynamic properties of frozen soil depend to a great extent on the soil type and on the degree of saturation. Investigations were performed by Vinson [6] and others [7, 8]. In general, it can be concluded that the wave velocity increases with an increasing degree of freezing and with decreasing temperature. The dynamic stiffness of frozen soil is also influenced by the frequency of the cyclic loading, this observation indicates the influence of the viscous behaviour of ice.PERFORMANCE OF INVESTIGATIONSDescription of calculationsThe investigations were performed by means of FE-analyses for plane strain conditions. The soil was assumed to be a linearly visco-elastic material, e.g. a Voigt-Kelvin-body. The considered cross section is a rectangular vertical plane, the upper boundary of which represents the free surface of the half-space. The system containing 5680 triangular elements is shown in fig. 1. The dimensions are 48 m ×6 m which corresponds to about 8λr×2λr, λr being the Rayleigh-wave length

of the half-space material. The element density decreases from at the surface to near to the lower boundary (fig. 1).



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Figure 1. FE-mesh with boundary conditions and levels.At the left upper corner of the FE-mesh a concrete strip foundation of width/depth=1.8m/0.6m with the center at x=1.5m acts as the wave source. It is excited to steady-state harmonical vibrations in vertical direction by a force

P=Po eiωt (1)with the amplitude po=333 kN/m2. Since the excitation is steady-state, the calculation is performed for quasi-static conditions with complex unknowns.To simulate the infinite half-space at depth and at the sides of the considered FE-mesh, appropriate boundary conditions have to be applied. At the lower horizontal boundary this is best accomplished by the well-known “dash-pot” boundary condition by Lysmer/Waas [9].For the vertical boundaries at the sides however, this condition can not be applied, because the longitudinal and the transverse displacements due to Rayleigh-waves are coupled. On the other hand, the R-wave boundary condition will initiate reflections if this boundary is not located far enough away from the wave source, that the theoretical far-field Rayleigh-wave can be developed. For the calculations presented here, the problem was solved by the so-called Influence-Matrix boundary condition (IM-BC) which consists essentially of a virtual lateral extension of the FE-field by about 6 Rayleigh-wave lengths. At the outer vertical boundary of this virtual area, the analytically calculated Rayleighwave boundary condition can now be used without the initiation of any reflections. The properties of the virtual area with respect to the propagation of waves are represented by the influence-matrix, which



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< previous page page_154 next page >Page 154can easily be calculated by standard FE-procedure. If the IM-BC now is applied to the lateral boundaries of the original FE-grid it perfectly simulates the surrounding half-space. The good results, easy handling and considerable advantages with respect to calculation performance of this boundary condition are described in detail by Haupt [10, 11].Material propertiesThree different types of soil are considered. The type A material represents the unfrozen soil and is used as reference material. For the frozen soil types B and C two different shear moduli were selected, to take into account variations in the degree of freezing and temperature. The material properties and the resulting dynamic quantities are listed in table 1.Table 1: Material propertiesquality dimension soil type concrete

Aunfrozen

Bfrozen

Cfrozen

Gdyn MN/m2 132 3300 825 9500

Poisson ratio – 0.25 0.25 0.25 0.33

density ρ t/m3 1 .937 1.937 1.937 2.548

R-wave velocity CR m/s 240 1200 600 1800

R-wave length λR m 6.0 30.0 15.0 45.0

Damping D % 2.5 2.5 2.5 1 .0

αs 1/m 0.0241 0.0048 0.0096 0.0013The coefficient of attenuation for the shear-wave αs, which is given in the table 1, is calculated from:

(2)with f=frequency and cs=shear-wave velocity.The frequency of the exciting force being f= 40Hz yields a Rayleigh-wave length of λr=240/40= 6, 0m in a homogeneous half-space of soil type A.



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< previous page page_155 next page >Page 155Although damping is presumably much smaller in a frozen soil than in a non-frozen soil, in this study the degree of damping D was assumed to be the same for all soil types. This assumption allows to observe the influence of the stiff top layer on the wave propagation without any additional effects of other parameters. For the same reason the density was taken the same in the frozen and unfrozen soil respectively.Four different depth profiles of the wave velocity are considered. The profil 0 represents the homogeneous half-space of soil type A and is used as reference system. The profiles with a stiff top layer number I to III are shown in fig. 2.

Figure 2. Velocity profiles in terms of R-wave velocity.RESULTSPresentation of the resultsThe function of an elastic wave propagating in x-direction in a soil body is generally given by the expression:

(3)with ω=2πf, t=time and x=horizontal coordinate. The last term represents a harmonic vibration, whereas the first two terms are functions of the distance. In the case of a wave generated by a wave source at the surface, the wave number k is not a constant but it depends on x: the phase function



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< previous page page_156 next page >Page 156k(x). In the following presentation and discussion of the results the considerations are focussed on the amplitude function vo(x) and on the phase function k(x) at the surface of the half-space and at horizontal levels at different depths below the surface. These levels are (fig. 1)No. 1 2 3 4

depth below the surface 0.0m 0.6m 1.2m 3.0mHence, in profile II the level 2 is situated at the interface between the top layer and the half-space below, in the profiles I and III this applies for level 3.

Figure 3. Amplitude and phase functions at profile III and amplitude at reference profile 0.Wave lengthA typical result of the calculations is shown in fig. 3. The amplitude function vo(x) at levels 1 and 2 for profile III and the corresponding curve at level 1 (surface) for the reference case (homogeneous half-space, material A) are plotted. Furthermore, in this diagram the curves k(x) for profile III are presented. Since in reality these two lines almost coincide, they are plotted separately for clearness. The slope of the phase function is a scale for the wave length: the steeper the slope, the smaller the wave length, e.g. the wave velocity.The curves k(x) at the levels 1 (surface) and 4 (3.0m depth) respectively for the four considered profiles are together presented in fig. 4, again



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< previous page page_157 next page >Page 157they are plotted separately. It may be seen that the slopes of the curves for each case, e.g. the wave lengths, are the same at the surface and within the half-space. It can be shown that this is true also for greater depth, at least at some distance from the wave source [12].The following wave lengths and wave velocities of the surface-wave ar resulting from the average slopes of the curves in fig. 4:prof ile 0 I II III

λ [m] 6.08 9.23 7.94 7.86

CR [m/s] 243.2 369.2 317.6 314.4The slight increase of the Rayleigh-wave velocity for the reference case as compared to the value in table 1 is due to the stiffening of the system by reducing the infinite number of degrees of freedom to a finite one in the FE-calculation.If one takes into consideration that in the homogeneous half-space about 36 % of the energy of the Rayleigh-wave is transmitted within a surface layer of thickness of 20 % of the Rayleigh-wave length—which in this case corresponds to 1.2m the increase in surface-wave length at profile I by a factor of 9.23/6.08=1.52 is surprisingly small. At the two other profiles, the increase is even smaller. The effect of the reduced wave velocity in the top layer at profile III equals about the smaller thickness of this layer at profile II.

Figure 4. Phase functions k(x) at all profiles, levels 1 and 4.



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Figure 5. Normalized amplitudes at profiles I to III, levels 1 and 2.Amplitude functionsIn fig. 3 it can be observed that the amplitude at the surface of the homogeneous half-space (profile 0) does not decrease monotonously with distance but that it shows some variation due to the interference of the Rayleigh-wave with body-waves. This phenomenon has been dealt with in detail in [10]. The amplitude functions in the cases of a half-space with stiff top layer do not show these interference pattern or at least they do to a much smaller extent.In a plane strain system the attenuation of the amplitude of the Rayleigh-wave or surface-wave with distance is not caused by the propagation of the wave away from the source, if far enough from the wave source, but it is due to the damping only. Hence, if a reference point xo is defined, the amplitude at some point x is

(4)



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< previous page page_159 next page >Page 159If at profile 0 an averaging exponential function is drawn through the amplitude curve at the surface (fig. 3) from the values of this function at xo=12m and x=48m the coefficient of attenuation of the Rayleigh-wave is found to be a=0.0139. This value is considerably smaller than that given in table 1 for the shear-wave. This can be explained by two reasons:– It has been assumed that damping takes place only at shear deformations, which means that at pure compression no dissipation of energy occurs.– The stiffening of the system due to the reduction of the degrees of freedom implies also a reduction of the energy dissipation.Since all results of this study were obtained under the same conditions concerning damping, they can be compared with each other.Normalized amplitudesIf the curves vo(x) in the cases of profiles I to III are normalized on the corresponding amplitude functions of the reference case 0, the curves in fig. 5 can be plotted. First of all it is to state that the vibration amplitudes of the foundations at the stiff top layer profiles are considerably reduces as compared to the reference case. The ratios vf/vr are 0.45 (profil I), 0.68 (profile II) and 0.59 (profile III). The index f refers to the profiles I to III, the index r to the reference profile 0. This reduction is easy to understand by keeping in mind the stiffening of the foundation-subsoil system due to the top layers.Furthermore, attention is drawn to the attenuation of the normalized surface-wave amplitudes with distance. If again an averaging exponential function is plotted through the normalized amplitude curves and the values of this functions at xo=12m and x =48m are considered, a reduction is found by a factor, which is given in line 1 of table 2. However, at equal values of D, the coefficient of attenuation a decreases with increasing wave length. The values of a, calculated by equation (2) and using surface-wave lengths found at the velocity profiles I to III are presented in line 3 of the table. It follows from this, that the normalized amplitudes should increase between the two points by the factor given in line 4 of table 2. Hence, the attenuation of the amplitudes found from the FE-calculation is about 2.6–3.2 times greater than the one corresponding to the above assessment.



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< previous page page_160 next page >Page 160Table 2: Attenuation of surface-wave amplitudes with distance

Profile I Profile II Profile III

1 0.39 0.35 0.44

2 λf/λr 1.52 1.31 1 .29

3 αf 0.0092 0.0106 0.0108

4 1.19 1.13 1.12

5 Zeile 4/ Zeile 1 3.1 3.2 2.6This unexpected result may possibly be explained by the following process: Part of the energy transferred by the top layer is continuously emitted downwards into the half-space. By this the amplitude at the surface is more attenuated than it should be expected from pure material damping effects. On the other hand, the amplitudes in the subsoil then should be increased as compared to theory. This, in fact, is the case as may be seen from fig. 6. It shows the amplitude of the vertical vibration component depending on depth at x=42m and x=48m respectively for profile I.For comparison the amplitude in the reference case (profile 0, material A) is presented as it is obtained from the FE-calculation. It coincides very closely with the analytically calculated function. However, this amplitude is much greater, by a factor of about 10. The different scales of the plots should be noticed.The discussed result is not necessarily in contradiction to the experience that vibrations are transmitted better and to a greater distance in a frozen soil than in an unfrozen soil. In reality, the damping in the frozen soil will be much smaller than assumed in this study. There are also indications that in the stiff top layer the horizontal vibration component is transferred as a body-wave rather than as a surface-wave [12].



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Figure 6. Amplitude of the vertical component depending on depth, profiles 0 and I.CONCLUSIONThe investigations described in this report are dealing with:– the influence of a stiff top layer on the propagation velocity of surface-waves;– the attenuation of the wave amplitudes with distance at the surface.The findings will help for a better understanding of wave propagation processes in the soil. However, the detected phenomena still demand for more detailed explanations and further investigations will be necessary.



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< previous page page_162 next page >Page 162REFERENCES1. Ewing, W.M., Jardetzky, W.S., Press, F. Elastic Waves in Layered Media, Intern. Series on Earth Science, McGraw-Hill, New York, Toronto, London, 1957.2. Rücker, W. Schwingungsausbreitung im Untergrund, Bautechnik 66, 1989.3. Chow, N., Le, R., Schmied, G. Ausbreitung von Erschütterungen im inhomogenen Boden, Bauingenieur 65, 1990.4. Haupt, W. Ausbreitung von Wellen im Boden. Chapter 3, Bodendynamik, Grundlagen und Anwendung, Vieweg-Verlag, Braunschweig, Wiesbaden, 1986.5. Rao, H.A.B. Nondestructive Evaluation of Airfield Pavements (Phase I), Techn. Rep. No. AFWLTR-71–75, Air Force Weapons Lab., Albuquerque, N.M., 1971.6. Vinson, T.S. Parameter Effects on Dynamic Properties of Frozen Soil, Proc. ASCE, No. GT 10, Oct. 1978.7. Finn, W.D.L., Yong, R.N. Seismic Response of Frozen Ground, Proc. ASCE, No. GT 10, Oct. 1978.8. Czajkowski, R.L., Vinson, T.S. Dynamic Properties of Frozen Silt under Cyclic Loading, Proc. ASCE, No. GT 9, Sept. 1980.9. Lysmer, J., Kuhlemeyer, R.L., Finite Dynamic Model for Infinite Media, Proc. ASCE, No. EM 4, Aug. 1969.10. Haupt, W. Verhalten von Oberflächenwellen im inhomogenen Halbraum mit besonderer Berücksich-tigung der Wellenabschirmung, Veröff. des Inst. f. Bodenmech. und Felsmech., Universität Karls-ruhe, Heft 74, 1978.11. Haupt, W. Numerical method for the computation of steady-state harmonic wave fields. Proc. Dyn. Meth. in Soil and Rock Mech. (DMSR 77), Vol. I, A.A.Balkema, Rotterdam, 1978.12. Haupt, W. Erschütterungsabschirmung in gefrorenem Boden, Veröffentlichung des Grundbauinstituts der LGA, Heft 43, 1985.



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< previous page page_163 next page >Page 163Wave Transmission at a Multimedia InterfaceR.S.Steedman (*), S.P.G.Madabhushi (**)(*) Geotechnics and Special Projects Division, BEQE, Science Park, Cambridge, CB4 4 WE, U.K.(**) Cambridge University Engineering Department, Trumpington Street, Cambridge, CB2 1PZ, U.K.ABSTRACTIn the physical or numerical modelling of a dynamic soil structure interaction problem the semi-infinite extent of the soil medium must be simulated. In the physical modelling of a dynamic problem, for example using a geotechnical centrifuge, this can be achieved using an energy absorbing boundary made from a clay-like material. In the numerical modelling using finite element techniques the free field condition of the soil deposit can be simulated by superposition of two solutions carried out in a narrow boundary region with ‘Neumann’ and ‘Dirichlet’ boundary conditions. This scheme, often termed as the Smith-Cundall boundary, involves an interface between three media. Stress waves impinging on such an interface are partially reflected back into the main mesh and are partially transmitted into the connecting boundary regions. In this paper a relation for achieving complete transmission of the incident stress waves into any number of media connected by the interface is derived. The validity of this relation is demonstrated for interfaces joining three and five elastic media.INTRODUCTIONThe tectonic plate movements during an earthquake induce stress waves in the overlying soil layers. These stress waves, usually in the form of compression waves (P waves) and shear waves (S waves) are propagated through the soil medium in all directions. In analysing soil-structure systems subjected to earthquake vibrations it is important to simulate the semi-infinite extent of the soil medium. In dynamic centrifuge tests the technique of using an energy absorbing boundary made from a clay-like



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< previous page page_164 next page >Page 164commercial sealant (Duxseal) to simulate the free field conditions has been developed at the Princeton University (Coe et al, 1983) and has now been widely adopted. The performance of duxseal as an absorbing boundary was evaluated recently by Steedman and Madabhushi (1991).In numerical analyses using the finite element method non reflecting boundaries need to be used to simulate the semi-infinite extent of the soil. Smith (1973) proposed the superposition of two solutions for the same boundary value problem with ‘Neumann’ and ‘Dirichlet’ boundary conditions. Cundall et al (1981) have suggested that multiple analyses and superposition can be carried out in two narrow boundary regions attached to a main mesh. This technique, often termed as the Smith-Cundall boundary involves an interface between three media as shown in Figure 1.

Figure 1 Two independent overlapping boundary zones connected to main mesh (after Wolf, 1988)Stress waves impinging on such an interface will be partially reflected back into the main mesh and are partially transmitted into the boundary regions. However, the reflected portion may interfere with the oncoming stress waves in the main mesh resulting in erroneous solutions. This can be avoided if the incident waves are completely transmitted into the two boundary regions. In this paper a relation between the properties of the main medium and the boundary media is derived for complete transmission of stress waves to occur. Continuity of stress conditions and displacement conditions along the interface must be assumed.FORMULATIONThe mathematical formulation is first established for an interface between two elastic media and then it is extended for an interface connecting more than two media.



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< previous page page_165 next page >Page 165Incident Sh waveConsider an incident Sh wave impinging on an interface between two media as shown in Fig. 2. Let the velocity of this wave be VSa and VSb in the two media. Let the displacement amplitude of the incident wave be unity and those of the reflected and the transmitted waves be ‘A’ and ‘B’ respectively. Let the incident wave be a harmonic which can be represented as

u(t)=exp (−i ω t) (1)where u(t) is the displacement at any time t and ω is real.The displacement amplitude of the reflected wave in the region y<0 is given as

(2)and the displacement amplitude of refracted wave in the region y>0 is given by

Figure 2 Reflection and Refraction of Sh wave incident on an interface

(3)where



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mx=sin(θs) (4)

(5)

(6)

(7)Continuity of displacement and stress at the interface leads to the following equations for A and B

1+A=B (8)

(9)where µa and µb are the elastic moduli of rigidity of the two media. Following Hudson (1980), the solution of these two equations gives

(10)

(11)where

(12)and ρa and ρb are the densities of the two media.If however, more than two media are connected by the interface then equation 12 for ψ has to be modified. For an interface connecting medium a with media l, 2,…, n the equation for ψ can be written as

(13)



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< previous page page_167 next page >Page 167The amplitudes of the reflected and the refracted waves can be computed using equations 10 and 11. For a normal incidence of the SH wave Eq. 13 reduces to

(14)In order to achieve the complete transmisson of a wave at the interface, the value of ψ should be unity. This will reduce the reflected wave amplitude to zero. Thus, a condition for the complete wave transmisson can be written as

ρaVs=ρ1VS1+ρ2VS2+…ρnVsn (15)This implies that the impedence of medium a must be equal to the sum of the impedences of the media connected by the interface for complete transmisson of the wave to occur.For any general angle of incidence, an equation similar to Eq. 15 can be obtained by cross multiplying the terms in Eq. 13.Incident Sv waveIf the incident wave is an Sv wave then the reflected wave and refracted wave will have P wave components in addition to the Sv wave components as shown in Fig. 3. Equation 14 in this case can be modified to take into account the additional P-wave components.

Figure 3 Reflection and Refraction of Sv wave incident on an interface



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(16)where

(17)

(18)However, for a normally incident Sν wave, Eq. 16 reduces to

(19)where VPa is P-wave velocity in medium a and VP1, VP2,….,Vpn are the P wave velocities in media 1,2,..,n respectively.The condition for the complete wave transmission of a wave into the media connected by the interface can be written in this case as

ρa(VPa+VSa)=ρ1(VP1+VS1)+ρ2(VP2+VS2)+….ρn(VPn+VSn) (20)The validity of Eq. 15 is here checked initially for the problem of an interface connecting three elastic media and then for a interface connecting five media.INTERFACE CONNECTING THREE ELASTIC MEDIAThe transmission of a shear wave at an interface connecting a main mesh to two or more boundary zones has been studied. The analyses were carried out using a finite element program called ‘SWANDYNE’, Chan (1988), which is a general purpose program for static and dynamic problems in Geomechanics.The finite element discretisation of the case of three media connected at an interface (as in the Smith-Cundall boundary) is shown in Fig. 4.



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Figure 4 Schematic representation of the finite element discretisationTable 1 Material properties of zones A, B and C

CASE Zone A Zone B Zone C

Density Modulus Density Modulus Density Modulus

1 1.0 1.0E8 0.5 0.5E8 0.5 0.5E8 6593.8 6593.8

2 1.0 1.0E8 0.75 0.5E8 0.301 0.5E8 6593.8 6593.8

3 1.0 1.0E8 0.25 0.5E8 0.835 0.5E8 6593.8 6593.8

4 1.0 1.0E8 0.5 0.7E8 0.5 0.334E8 6593.8 6593.8

5 0.5 2.27E8 0.4 0.709E8 0.4 0.709E8 7024.8 7022.9

6+ 1.0 1.0E8 2.0 0.5E8 2.0 0.5E8 6593.8 13187.6

+I1 is not equal to I2



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< previous page page_170 next page >Page 170Standard four noded quadrilateral elements were used. The end conditions of the boundary zones B and C are maintained as ‘fixed’ and ‘free’ respectively. A single pulse S wave excitation is given at node 8 and its propagation is monitored at various nodes in the main mesh as well as the boundary zones as shown in Fig. 4. In the analyses shown here a general isotropic elastic constitutive law was used.

Figure 5 S wave propagation in the horizontal sand layer


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< previous page page_171 next page >Page 171Material propertiesEquation 15 was then investigated by modifying the impedences of the three zones shown in Fig. 4. In this paper six different analyses were carried out varying material properties as shown in Table 1. For cases 1 to 5 the impedence of the main mesh is equal to the sum of the impedences of the boundary zones. Finally, in case 6, a mismatch of the impedence of the main mesh and the boundary zones was considered.Results of the analysesThe propagation of the Sh wave through the horizontal sand bed (Zone A) is monitored by observing the displacement-time histories at the nodes shown in Fig. 4. The length of the sand bed from the point of excitation to the interface was 75.0 metres. Figure 5 presents the results of the analysis using the first set of material properties (case 1). The shear wave velocity

Figure 6 Displacement time history at node 92 (case 1 to case 5 parameters)



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< previous page page_172 next page >Page 172is computed as 6593.8m/s for the three zones from the material properties. The arrival time of the Sh wave at nodes 8, 62, 92 and 152 (each seperated by 15.0m) may be estimated as 2.3, 4.5, 6.8 and 9.1 milliseconds. From Fig. 5 it can be seen that the arrival time at these nodes is indeed close to these values. However, any reflection from the interface should show up on the time history as a second pulse as the reflected wave arrived back at the node. For example, this should have occured at node 8 around 20 ms. From Fig. 5 it can be seen that no portion of the incident wave was reflected back from the interface.In Fig. 6 the displacement time histories at node 8 for each of cases 2 to 5 in Table 1 are presented. In all the cases there is an impedence match between the main mesh and the boundary zones and as a result complete stress wave transmission at the interface was achieved. In Fig. 7

Figure 7 Reflection from the interfaceresults of the analysis without an impedence match (case 6) are show. Clearly, a portion of the incident wave is reflected back from the interface (for example, arriving at node 8 after 20.0ms).Further more, using equations 10, 11 and 14 it is possible to predict the ratio of the amplitude of the reflected Sh wave from the interface to that of the incident wave and to compare it with the observations of the



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< previous page page_173 next page >Page 173numerical model. For the parameters of case 6 this ratio was computed as 33.3 %. From Fig.6 it is possible to estimate

the same ratio at any node as which is close to the computed value.INTERFACE CONNECTING FIVE ELASTIC MEDIAAn interface connecting four boundary zones to the main mesh is considered next. The finite element discretisation of such a configuration is shown in Fig. 8. The density and modulus of rigidity of the main mesh

Figure 8 Schematic diagram of an interface joining five media

Figure 9 S wave transmission at an interface joining five media



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< previous page page_174 next page >Page 174are chosen as 1.0 and 1.0E8. For each of the boundary zones these values were 0.25 and 0.25E8 respectively so that the impedence of the main mesh is equal to the sum of the impedences of the four boundary zones.The displacement-time history at node 92, 413 and 584 are shown in Fig. 9. The S wave was completely transmitted into the four boundary zones and no reflection from the interface was observed at node 92.CONCLUSIONSA relation for complete stress wave transmission at an interface joining one medium to any number of boundary zones has been derived for any combination of P and S waves at any incident angle. In the specific case of normal incidence this result can be interpreted as requiring an impedence match between the main mesh and the sum of impedences of the boundary zones. The validity of this relation was demonstrated using a general finite element code for Sh wave transmission at an interface joining three elastic media as in the case of the Smith-Cundall boundary, and for the case of an interface connecting five elastic media.REFERENCES1. Chan A.H.C., User Manual for DIANA SWANDYNE-II, Institute for Numerical Methods in Engineering, University College of Swansea, also, Department of Civil Engineering, University of Glasgow, 1989.2. Coe, C.J., Prevost, J.H. and Scanlan, R.H., Dynamic stress waves/attenuation: Earthquake simulation in Centrifuge soil models, Earthquake Eng. and Structural Dynamics, Vol. 13, 1985.3. Cundall, P.A., Kunar, R.R., Carpenter, P.C., Marti, J., Solution of infinite dynamic problems by finite modelling in the time domain, 2nd Int. Conf. on Applied Numerical Modelling, Madrid, 1978.4. Hudson, J.A., The excitation and propagation of elastic waves, Cambridge University Press, 19805. Kuhlemeyer, R.L. and Lysmer, J., Finite dynamic model for infinite media, ASCE Jnl. of Engg. Mech. Div., Vol. EM4., 1969.6. Steedman, R.S. and Madabhushi, S.P.G., Wave propagation in sand medium, Proc. Int. Conf. on Seismic Zonation, Stanford University, Standford, 1991.7. Smith, W.D., A non reflecting plane boundary for wave propagation problems, Jnl. of Computational Physics, Vol. 15.,1973.8. Wolf. J.P., Soil-Structure interaction analysis in time domain, Prentice Hall Inc., 1988.



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< previous page page_175 next page >Page 175SECTION 4: DYNAMIC SOIL PROPERTIES



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< previous page page_177 next page >Page 177In-Situ Dynamic Property Evaluation of Gravelly SoilT.Kokusho, Y.Tanaka, Y.YoshidaCentral Research Institute of Electric Power Industry, JapanABSTRACTA new evaluation method; in-situ freezing sampling and laboratory testing, has been developed in order to evaluate reliable in-situ dynamic properties of gravelly soil which are of considerable importance to the seismic safety of a nuclear power plant to be located on Pleistocene gravelly soil. The undrained cyclic strength based on this method has given much higher value than that based on conventional sampling or reconsituted samples, thus demonstrating the importance of recovering a sample as intact as possible to adequately evaluate the in-situ strength. These reliable strength data have led to a correlation allowing one to estimate the in-situ undrained cyclic strength from the result of dynamic penetration tests without resorting to the costly sampling method.INTRODUCTIONNuclear power plants which are currently on rock base are expected to situate themselves also on Pleistocene soils in the near future in Japan. In this case, the seismic stability of the foundation ground is of utmost importance despite its relatively high stiffness because very large seismic motions are employed for the design compared to normal civil engineering structures.The Pleistocene layers which are widespread as subsurface layers in plain areas normally consist of stiff gravelly soils. The dynamic properties; the dynamic shear modulus, the hysteretic damping and the undrained cyclic strength, which determine the seismic stability of the soils under a given seismic motion have



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< previous page page_178 next page >Page 178scarcely drawn engineers’ attention.In the past decade, lots of research efforts have been made to evaluate in-situ dynamic properties for various kinds of soils, demonstrating the significant effect of sample disturbances introduced during sampling, transporting and laboratory handling procedures on the properties measured in laboratory tests. Kokusho1 pointed out that this effect is more pronounced for stiffer soils like dense sands and may sometimes reduce the undrained cyclic strength evaluated in the lavoratory to one half or one third.In the light of these previous studies, it is essential to develop special procedures for sampling and laboratory testing in order to evaluate as reliably as possible the dynamic properties of the stiff Pleistocene gravelly soils. A new technology; in -situ freezing sampling and laboratory testing, has therefore been developed in this research. This has been applied to several gravelly soils in the field to measure reliable dynamic properties to be used for seismic stability evaluations of foundation soils under strong design earthquake motions.IN-SITU FREEZING SAMPLING TECHNIQUE FOR GRAVELSGravelly soils often pose a great problem for evaluating reliable soil properties because of the difficulty of obtaining their intact samples. The only probable way for taking an intact sample may be the in-situ freezing technique which has been successfully employed for dense sands.In this technique freezing pipes are first installed into gravelly soil as illustrated in Fig. 1 with as small disturbance effect on the soil as possible. Then, the liquid nitrogen or other coolants is circulated through the double-walled pipes, eventually making saturated ambient soils into a frozen soil column of approximately 180cm in diameter. If the soil is above the ground water table, special steps are needed to saturate the soil for successful freezing. The temperature in the frozen soil to be sampled is kept below about −10ºC at the highest for a successful sampling.A triple tube sampler with an inner tube diameter of either 10cm or 30cm specially developed for gravelly soils is used to core out the frozen soil as shown in Fig 2. The outer tube tip is equipped with a diamond bit and lubricated with cooled mud flow of −10°C in temperature. With the larger sampler, samples of



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Fig. 1 Arrangement of freezing pipes and casing pipes for coring

Fig. 2 A core obtained by In-situ freezing sampling30cm in diameter and of 150cm in maximum length are cored, which are stored in ice boxes and transported for the laboratory test.



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< previous page page_180 next page >Page 180SITE INVESTIGATIONS FOR GRAVELLY SOILSIn order to establish a geotechnical data base for the gravelly soils, site investigations have been conducted at four different sites of Pleistocene grounds in Japan. The soil profiles of these sites are available in Fig. 3 along with the data of dynamic penetration tests and the depths where soil samples were recovered by the in-situ freezing technique.

Fig. 3 Soil profiles at four sites



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< previous page page_181 next page >Page 181Two kinds of the penetration test were conducted in this investigation; the Standard Penetration Test (SPT) and the Large Penetration Test (LPT), the specifications of which are shown in Fig. 4. The latter developed by Kaito2 employs a larger probe diameter and a larger hammer, providing larger driving energy per unit area to overcome the short-comings of the SPT in dense gravelly soils. The number of blow counts N and Nd for the SPT and the LPT respectively are roughly correlated to each other as N=2Nd for gravels and N=1.5Nd for sands (Yoshida et al.)3. The N and Nd values in Fig. 3 obviously indicate that the gravelly layers at the four sites are very stiff and the LPT seems to reflect the soil density more reliably than the SPT which may fluctuate due to the strong locality of gravelly soils.

Fig. 4 Concept of dynamic penetration testFig. 5 summarizes the typical grain size curves obtained from the cored samples. Since the maximum grain size reaches 50 to 100mm and the fines content finer than 0.074mm in particle size is less than 10% and nearly zero in most soils, it is concluded that, the unfavorable effect due to freezing may be insignificant.



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06:31:37 a.m.file:///F|/LIbros%20de%20ingenieria_21/LIbros%20de%20...mics_and_Earthquake_Engineering_V/files/page_182.html (1 of 2)12/05/2010

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Fig. 5 Typical grain size distributions of gravelly soils sampled by in-situ freeze sampling at four sites



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< previous page page_183 next page >Page 183THE CYCLIC TRIAXIAL TESTThe frozen column-shape samples of either 10cm(for Site A only) or 30cm in diameter were cut with a special saw to make the length -to-diameter ratio two or four-thirds(Tanaka et al.)4. Fine sand particles were first attached to the specimen side and then trimmed to leave a thin sand layer at the surface in order to reduce the membrane penetration effect in the undrained cyclic strength (Tanaka et al.)5. The frozen specimen was gradually thawed under a prescribed confining pressure by circulating the pressure-chamber water. Two kinds of cyclic triaxial tests were carried out; firstly, a small to medium strain test to obtain the shear modulus and the material damping and a lso their strain -dependent variations and secondly, a large strain test to obtain the undrained cyclic strength. The former test method initially proposed by Kokusho et al.6 has been extensively used in Japan for soil properties measurements for the strain level as small as 10’5 to substitute the resonant column test especially for large size specimens.SMALL TO MEDIUM STRAIN PROPERTIES OF PLEISTOCENE GLAVELSIn Fig. 6 representative results of the shear modulus and the damping ratio for Pleistocene gravels calculated from the hysteresis curves measured in the cyclic triaxial test are shown for the cyclic strain amplitude of the order of 10−6 to 10−3. Evidently seen is a continuous variation of the properties along the strain amplitude for both small and medium strain levels, indicating this triaxial test can fully substitute the resonant column test which may be difficult to perform for such large specimens.The small strain modulus measured in this test is compared in Fig. 7 with in-situ modulus calculated from the in-situ shear wave velocity test. The laboratory modulus undershoots the in-situ modulus by 50%, despite that all the data are based on the intact specimens obtained by the in-situ freezing sampling. However, it is also noted that the modulus is measured farther smaller for stiff sands and clays based on similar cyclic triaxial tests and resonant column tests for conventionally sampled specimens. Obviously the gap between the field and the laboratory still exists even with the help of the new sampling technique, implying other causes may also be responsible for the gap.



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Fig. 6 G/Go~γ, h~γ relationships of gravelly soils at two sites


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Fig. 7 Comparison of small-strain shear moduli obtained from in-situ shear wave velocity tests and laboratory tests



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< previous page page_185 next page >Page 185UNDRAINED CYCLIC STRENGTHTypical results on the undrained cyclic strength measured for gravels sampled from one of the sites are shown in Fig. 8 against the number of loading cycles. The undrained cyclic strength defined here means the stress ratio τ d/σc’, corresponding to the double amplitude strain of 2%, where τd is the cyclic shear stress and the σc’ is the effective confining stress. It is noted that the strengths based on the freezing sampling are much higher than those of soil specimens taken from the same layer with the conventional non-freezing sampling. Fig. 8 also indicates that the strengths of the intact specimens are greater than those of specimens remoulded in the laboratory having the same density as the frozen sample. Thus the importance of the seismic stability evaluation based on the in-situ freezing sampling technique has been fully demonstrated to make a rational seismic design.

Fig. 8 Undrained cyclic strength of gravelly soils at A-site sampled by different methods



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< previous page page_186 next page >Page 186However, a high cost associated with the in-situ freezing sampling may allow its application only to representative points, leaving the rest to a more simplified evaluation method of the in-situ strength. In order to meet this purpose, an ernpirical correlation between the undrained cyclic strength (in terms of the stress ratio, τd/σc’, corresponding to the double amplitude strain of 2% or 2.5% in 20 cycles of loading) and the LPT blow counts, Nd,, modified for the confining stress σc’=98kPa has been developed as shown in Fig. 9 based on this research results as well as others including stiff sands and gravels. The LPT blow counts were derived from the SPT blow counts whenever necessary based on the previous research (Yoshida et al.)3. Obviously a unique curve can be drawn as shown in the graph to estimate the undrained cyclic strength from the LPT blow counts, although it may be effective only for the effective confining stress, σc’, not far from 98kPa, for which the data collected here are justified.

Fig. 9 Empirical correlation between undrained cyclic strength and LPT blow counts



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< previous page page_187 next page >Page 187To be noted here is that the stress ratio for the Pleistocene stiff soil is not almost constant like loose sands but quite variable depending on the applied confining stress. Fig.10 indicates the stress ratios for the two kinds of tests with two different confining stresses for the same intact sample. It is easy to understand the importance of taking this effect into consideration to evaluate the dynamic soil stability whenever cutting or banking is involved in the design. Therefore, a similar correlation between the LPT blow counts and the undrained cyclic strength which can take account this effect has also been proposed, which is available in another 1iterature (Tanaka et al.)7.

Fig. 10 Effects of confining pressure on undrained cyclic strength of gravelly soilsSUMMARY AND CONCLUSIONSIn this research newly developed in-situ freezing sampling and laboratory testing methods have been applied to Pleistocene stiff gravelly soils to be used for their seismic stability evaluations Dynamic properties; the small to medium strain modulus and the damping as well as the undrained cyclic strength have been measured by means of the cyclic triaxial test, demonstrating an effectiveness of the new sampling method especially for evaluating the undrained cyclic strength.These reliable strength data using the sophisticated sampling along with the dynamic penetration tests in the same soil have led to a simple correlation between the two variables. This correlation may allow one to estimate the in-situ undrained cyclic strength from the penetration resistance without. resorting to the costly sampling method.



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< previous page page_188 next page >Page 188REFERENCES1. Kokusho,T., In-situ dynamic soil properties and their evaluations, Proc. 8th Asian Regional conf. on SMFE, Vol. 2, pp. 215–240, 1987.2. Kaito,T., Large penetration test, Tsuchi-to-Kiso, Vol. 19, No.7, pp. 15–21, 1971, (in Japanese).3. Yoshida,Y., Ikemi, M. and Kokusho,T., Empirical formulas of SPT blow-counts for gravelly soils, Proc.ISOPT., Vol.1, pp. 381–387, 1988.4. Tanaka,Y., Kudo, K., Yoshida,Y., Kataoka,T. and Kokusho,T., A study on the mechanical properties of sandy gravel,—Mechanical properties of undisturbed sample and its simplified evaluation, Report. No.U88021, CRIEPI, 1988, (in Japanese).5. Tanaka,Y., Kokusho,T., Yoshida,Y., and Kudo, K., A method to evaluate system compliance in dynamic strength tests of gravelly soils, Report No.U89040, CRIEPI, 1989, (in Japanese).6. Kokusho, T., Cyclic triaxial test of dynamic soil properties for wide strain range, Soils and Foundations, Vol. 20, No.2, pp. 45–60, 1980.7. Tanaka,Y., Kudo.K., Kokusho,T., Yoshida,Y., Dynamic strength of gravelly soils and i ts relation to the penetration resistance, Proc. 2nd Int. Conf. on Recent Advances in Geotechnical Earthquake Engineering and Soi1 Dynamics, St. Louis Vol.1, pp. 399–406, 1991.8. Yasuda, S., Yamaguchi, I., Dynamic shear modulus obtained by laboratory and in-situ tests, Proc. of Sym. on Evaluation of Deformation and Stability of Sand and Sandy Ground, pp. 115−118, 1984, (in Japanese).9. Yokota, I., Imano, M., Comparison of soil properties obtained by laboratory and in-situ tests, Proc. of Sym. on Evaluation of Deformation and Stability of Sand and Sandy Ground, pp. 111–114, 1984, (in Japanese).



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< previous page page_189 next page >Page 189Characterization of Material Damping of Soils Using Resonant Column and Torsional Shear TestsD.-S.Kim, K.H.Stokoe, II, J.M.RoëssetDepartment of Civil Engineering, The University of Texas at Austin, TX 78712, U.S.A.ABSTRACTA common assumption is that material damping in shear of soils measured by hysteresis loops is zero at small strains, strains less than about 0.001%. To investigate this assumption, a torsional shear/resonant column device was modified so that damping measurements could be made over strains ranging from about 10−4 to 10−1%. A metal specimen was used to calibrate the device and showed that system compliance contributed to material damping measurements above about 1Hz. Damping measurements made on a dry sand and a compacted clay showed nonzero values at small strains in both the torsional shear and resonant column modes (after accounting for system compliance). The effects of frequency, strain amplitude and number of loading cycles on material damping of these soils were also investigated.INTRODUCTIONMaterial damping can be evaluated in the laboratory using torsional resonant column and torsional shear tests. A number of studies have been conducted in the past to compare the results of the two methods for different strain amplitudes (4, 5, 7, 8) and to study the effect of number of loading cycles (5, 7, 8). The accuracy of the damping measurements with the torsional shear test was relatively poor for shearing strains below 0.01% because of difficulties in accurately measuring the hysteresis loops (3).In this study, a single piece of equipment is used for both the resonant column (RC) and torsional shear (TS) tests (Figure 1). The motion monitoring system in the TS test has been improved so that both tests can provide accurate results for the range of strains from 10−4 to 10−1% (2). A metal specimen was first tested to calibrate the equipment over a wide range of frequencies. Material damping in shear for a dry sand and a compacted clay were then obtained by varying the strain level, frequency of the excitation and number of loading cycles in order to investigate the importance of these effects.



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Fig. 1. Configuration of Simplified Resonant Column and Torsional Shear (RCTS) Test Equipment (without Outer Confinement Chamber) (After Ref. 1)

Fig. 2 Determination of Material Damping Ratio from the Hysteresis Loop Measured in the Torsional Shear Test


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< previous page page_191 next page >Page 191MEASUREMENT TECHNIQUESIn the torsional shear test, material damping is determined from the hysteresis loop (Figure 2) as the ratio of the energy dissipated in one cycle of loading (AL) to the peak strain energy stored during the cycle (At) times a factor of 4π:

D=A/(4πAt) (1)In the resonant column test, on the other hand, material damping is evaluated from the dynamic response using either the free-vibration decay curve or the half-power bandwidth. The free-vibration decay curve is recorded by hutting off the driving force after the specimen has been vibrating for a large number of cycles in steady-state motion at the resonant frequency (Figure 3a). The logarithmic decrement (δ) is then defined as:

δ=Ln (z1/z2)=2πD/(1−D2)1/2 (2)where z1 and z2 are the amplitudes of two successive cycles and D is material damping. Material damping can then be expressed as:

D=[δ2/(4π2+δ2)]1/2 (3)The half-power bandwidth method is based on measurement of the width of the frequency response (amplification) curve near resonance (Figure 3b). For small values of damping, one can approximate damping as:

(4)

where f1 and f2 are the two frequencies at which the amplitude is times the amplitude at the resonant frequency (fr).For very small strains (less than 10−3%), background noise can be a problem with the free-vibration decay method. However, background noise has less effect on the frequency response curve. On the other hand, at large strains the assumptions implied in the derivation of Eq. 4 are no longer valid, and serious errors can be induced in the half-power bandwidth method. In this study both methods were used at small-strain levels, but only the free-vibration decay method was applied for large strains.EVALUATION OF RCTS EQUIPMENT WITH A METAL SPECIMENTo evaluate the RCTS equipment, a metal specimen made of a brass tube (2.5-cm outside diameter and 0.81-mm wall thickness) was used. The tube was welded to the top cap and bottom plate of the equipment. It was assumed that the tube should have zero damping over the complete range of frequencies used in these tests (from about 0.05 Hz to 100Hz). Hz).Hysteresis loops from the TS test for a frequency of 0.5Hz are shown in Figure 4a; the stress-strain curve is linear resulting in no damping as expected. On the other hand, Figure 3 shows the results with the RC test that predict a damping of 0.4% from both measurement methods. The results for various



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Fig. 3. Determination of Material Damping Ratio in the Resonant Column Test using Metal Calibration Specimen



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< previous page page_193 next page >Page 193frequencies are plotted in Figure 4b. It can be seen that, for frequencies less than or equal to 1Hz, the damping predicted by the TS test is zero as expected. For larger frequencies, however, nonzero damping values are obtained. These values are considered to be due to a compliance problem with the equipment and are, therefore, subtracted from the damping measurements at the same frequencies with soil specimens.EXPERIMENTAL PROCEDURE USED TO TEST SOILSBoth RC and TS tests were performed on the same soil specimen. To begin testing, ten cycles of torsional shear were applied at a very small strain amplitude and a frequency of 0.5Hz. The RC test at the same strain level followed the TS test. Both tests were then performed in sequence for increasing strain amplitudes. At each strain amplitude, a range of excitation frequencies (typically 0.05Hz to 10Hz) was used in the TS test to investigate frequency effects. When the strain amplitude in the TS test had exceeded the level at which the stiffness started to decrease, another ten cycles of torsional shear were applied after the RC test to check how the deformational characteristics had changed after the numerous cycles (about 1000) applied during RC testing at each strain level.MATERIAL DAMPING OF DRY SANDTypical results for dry sand are shown in Figure 5a. At strains below 0.002%, the damping values are nearly identical for both tests and independent of the number of loading cycles. At higher strains, however, the values of damping decrease with increasing number of cycles. The results from the RC test, with at least 1000 cycles of steady-state response before the free vibration and from the TS test performed afterwards, are nevertheless very similar. It should be noticed that nonzero values of damping are obtained from both tests for very small strains (smaller than 10−3%) even for a frequency of 0.5Hz for which the metal specimen yielded zero damping.The variation of the damping ratio with number of loading cycles is further illustrated in Figure 5b where the solid symbols at 103 cycles are the results from the RC tests and the other points correspond to TS tests. For a strain level of 10−3%, the damping ratio is basically independent of the number of cycles as stated earlier. However, the effect of cycling becomes more and more apparent as the strain level increases. Figure 6 shows the hysteresis loops for a strain level of 0.035% obtained from the first ten cycles of torsional shear test (Figure 6a) and from the ten cycles applied after the RC tests (Figure 6b). The shift in the hysteresis loop is accompanied by a decrease in its area, but the loop eventually stabilizes after a large number of cycles.The effect of frequency on damping is shown in Figure 7. Figure 7a shows the actual measured values. Figure 7b incorporates the corrections at each frequency by subtracting the damping obtained for the metal specimen. It is interesting to notice that once this correction is applied, material damping is essentially independent of frequency as one would expect for dry sand.



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Fig. 4. Material Damping Ratio of Metal Specimen Determined by Both Torsional Shear and Resonant Column Tests


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Fig. 5. Damping Characteristics of Dry Sand Determined by Torsional Shear and Resonant Column Tests



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Fig. 6 Comparison of Hysteresis Loops of Dry Sand from the Virgin Loading Curve with Hysteresis Loops after RC Testing at a Strain Amplitude of 0.035%



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Fig. 7 Variations in Uncorrected and Corrected Damping Ratios of Dry Sand with Loading Frequency


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< previous page page_198 next page >Page 198MATERIAL DAMPING OF COMPACTED CLAYFigure 8a shows the values of material damping for a compacted clay obtained with the RC and TS tests. It can be seen that in this case the results are independent of the number of loading cycles but are quite different for each test, with the measured values of material damping being much larger in the RC test. Values of damping are again nonzero for very small levels of strain. In fact, the small-strain values are substantially larger than those measured for the sand.Figure 8b shows the variation of damping with frequency after applying the correction for system compliance. Contrary to what was observed for the sand specimen, the values of damping increase markedly with frequency, particularly for frequencies higher than 5Hz. This frequency effect explains the large differences between the two tests. The TS tests reported in Figures 8a were conducted at 0.5Hz while the RC tests were conducted at frequencies above 40Hz. CONCLUSIONS1. Material damping measurements using RCTS equipment were evaluated using a metal specimen. Below a loading frequency of 1Hz, damping ratios of the metal specimen were zero as expected. Above 1Hz, damping ratios greater than zero were measured because of system compliance. Damping ratios of soils measured at frequencies above 1Hz were corrected taking this compliance into consideration.2. For a dry sand, damping ratios obtained from resonant column and torsional shear tests match well at shearing strains below about 0.002%. At higher strains, damping ratios are sensitive to number of loading cycles. Damping values obtained from the first TS cycle are much larger than those computed from later cycles or from the RC test. However, damping ratios from TS and RC tests are essentially equal at the same strain level and the same number of cycles. The effect of frequency on material damping is negligible for this dry sand.3. For compacted clay, damping ratios obtained from the TS and RC tests are different over the complete strain range (from 0.0004 to 0.05%). This difference results from the difference in frequencies used in the two methods of testing. However, the effect of frequency in the torsional shear test does not begin to increase material damping until the frequency exceeds about 5Hz. The effect of number of loading cycles on material damping is negligible in this compacted clay.4. Material damping ratios at small strains (less than 0.001%) and low frequencies (less than 0.1Hz) were measured in the torsional shear test for both soils. In these strain and frequency ranges, damping is independent of both strain and frequency. Values of material damping of about 0.9% and 2.2% were measured for the dry sand and compacted clay, respectively.



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Fig. 8 Damping Characteristics of Compacted Clay Determined by Resonant Column and Torsional Shear Tests



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< previous page page_200 next page >Page 200ACKNOWLEDGEMENTSThis work was partially funded by the Texas State Department of Highways and Public Transportation, Project 2/3/10-8-88/0-1177. This support is gratefully acknowledged.REFERENCES1. Isenhower, W.M., Stokoe, K.H., II, and Allen, J.C., “Instrumentation for Torsional Shear/Resonant Column Measurements Under Anisotropic Stresses,” Geotechnical Testing Journal, GTJODF, Vol. 10, No. 4, December, pp. 183–191, 1987.2. Kim, D.-S., “Deformational Characteristics of Subgrades at Small to Intermediate Strains from Cyclic Tests,” Ph.D. Dissertation, The University of Texas at Austin (in progress), 1991.3. Lin, M.L., Ni, S.H., Wight, S.G. and Stokoe, K.H., II, “Characterization of Material Damping in Soil,” Proceedings of the Ninth World Conference on Earthquake Engineering, Tokyo, Japan, 1989.4. Ni, S.-H., “Dynamic Properties of Sand Under True Triaxial Stress States from Resonant Column/torsional Shear Tests,” Ph.D. Dissertation, The University of Texas at Austin, 421 pp, 1987.5. Ray, R.P. and Woods, R.D., “Modulus and Damping Due to Uniform and Variable Cyclic Loading,” Journal of Geotechnical Engineering Division, ASCE, Vol. 114, No. 8, August, pp. 861–876, 1988.6. Richart, J.E., Jr., Hall, J.R., Jr., and Woods, R.O., “Vibration of Soils and Foundations,” Prentice-Hall Inc., Englewood Cliffs, New Jersey, 414 pp, 1970.7. Stokoe, K.H., II, Isenhower, W.M., and Hsu, J.R., “Dynamic Properties of Offshore Silty Samples,” Proceedings 1980 Offshore Technology Conference, OTC 3771, Houston, Texas, pp. 289–302, 1980.8. Tatsuoka, F., Iwasaki, T. and Takagi, Y., “Hysteretic Damping of Sands Under Cyclic Loading and Its Relation to Shear Modulus,” Soil and Foundations, Vol. 18, No. 2, June, pp. 25–40, 1978.



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< previous page page_201 next page >Page 201Effect of Triaxial Stresses on Shear Wave PropagationH.-C. Fei (*), F.E.Richart, Jr. (**)(*) Department of Geotechnical Engineering, Tongji University, Shanghai, People’s Republic of China(**) Department of Civil Engineering, University of Michigan, Ann Arbor, Michigan, U.S.A.ABSTRACTCross-Hole Tests were run in a “Quicksand” tank in which the principal stresses could be controlled separately. For shear wave propagations in one of the principal stress directions, the stresses in the direction of wave propagation, and in the plane of particle motion governed the wave velocity. The third principal stress had no influence, as had been found previously. For wave propagations at an angle to one or more principal stress, all three principal stresses contribute.INTRODUCTIONThe low-amplitude shear modulus, Go, is a basic parameter for establishing the dynamic response of foundations. It is usually evaluated in the laboratory or in the field by measuring the velocity of propagation of the shear wave, designated by Vs, which is related to the shear modulus by

Go=ρ vs2 (1)In Eq. (1), ρ is the mass density of the soil.



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< previous page page_202 next page >Page 202In the early 1960’s [1, 2] it was established that the low-amplitude shear wave velocity for clean, dry sands was primarily a function of the void ratio, e, and the effective confining pressure, σo’. This discussion will treat only unsaturated sands for which σo’=σo, therefore the prime markings will be dropped. The resonant column tests used to establish empirical relationships between these parameters utilized only hydrostatic confining pressures. Subsequent interpretations of these results included the mean effective principal stress, σo, for situations in which the field conditions were not hydrostatic. The empirical equation thus became:

vs(ft/sec)=(170–78.2 e) (σo)m (2)in which m=0.25,

σo(lb/ft2)=( σ1+σ2+σ3)/3 (3)andIn 1979, Roesler [3] found that only TWO of the effective principal stresses were important: those IN THE DIRECTION OF WAVE PROPAGATION, σa, and IN THE DIRECTION OF PARTICLE MOTION, σp. The third principal stress, σs, did not influence the shear wave propagation velocity. Roesler’s “cross-correlation pulse tests” were carried out in a cubic box (30cm.on a side) filled with dry sand and subjected to pressures ranging from 4N/cm2 (5.8 psi.) to 16N/cm2 (23.2 psi). Roesler presented a modification of Eq.(2) in the form

vs ~ (σa0.149)(σp0.107)(σs0) (4)In 1985, Stokoe, et al.[4] evaluated shear wave propagations in a 7 ft. (2.1m) cubical triaxial steel chamber under stress conditions from 10 to 40 psi (6.9 to 27.6N/cm2). They found results similar to those of Roesler. Yu and Richart [5] reported on resonant column tests using dry sands subjected to anisotropic static stress conditions, and noted that σs did not influence the results. They concluded that it is reasonable to modify the Hardin Equation (Eq.3) by using σo2=( σa+σp)/2 . Furthermore, when the the stress ratio (σa/σp) was greater than about 2, the value of the shear wave velocity (or shear modulus) was reduced.



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< previous page page_203 next page >Page 203It was the purpose of this investigation to study the effects of the three principal stresses on shear wave propagation, by use of the cross-hole test (see refs. 6 and 7) including controlled stress conditions.TESTING EQUIPMENT AND PROCEDURESAll tests were run in a 7.5ft. (2.3m) dia. ×10 ft. (3m) high quicksand tank which contained Muskegon Dune Sand (rounded quartz sand, D10=0.18mm, Cu=1.44, e=0.71). Rubber “air bags” were used to load the top surface of the sand (see ref 7), and two pairs of air bags were inserted vertically into the sand (while it was in the liquefied condition) to provide controllable horizontal pressures, as shown in Fig.1. Each vertical air bag was 3ft.(0.9m) high and 5ft.(1.52 m) long. Also shown in Fig.1 are the locations of the two pickup transducers, and the location of the pulse source.

FIG. 1. TEST SET-UP IN QUICKSAND TANK



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< previous page page_204 next page >Page 204A vertical impulse (to generate vertically polarized shear waves), or a torsional impulse (to generate horizontally polarized shear waves) was applied to the sand bed, and the measured time for the pulse to travel the known distance between the pickups permitted evaluation of the shear wave velocity. In each test two of the principal stresses were held constant, and the third principal stress was varied for each cross-hole test, to determine the influence of this third stress on Vs.

FIG. 2. ARRANGEMENTS OF TESTS IN QUICKSAND TANK



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< previous page page_205 next page >Page 205Before each test sequence, the sand bed was reconstituted by making it “quick” by upward water flow, which also permitted installation of the pickups and the impulse exciter. Then, after downward flow had returned the bed to a uniform state, and the pressure system on the surface was reinstalled, a new test series could begin. Resonant column tests have determined that no significant capillary effects exist in this sand at the drained water content.

FIG. 3. CASE SV I



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< previous page page_206 next page >Page 206WAVE PROPAGATION ALONG PRINCIPAL STRESS DIRECTIONSFigure 2 (a), CASE SV 1, shows the test set-up used for the vertically polarized wave propagating along the σ2 direction. Figure 3 shows the effects on Vs caused by variations of one principal stress while holding the other two principal stresses constant. In Fig.3(a), variations of σ2(=σa, the stress in the direction of wave propagation) cause changes in Vs according to σa0.12, while Fig.3 (b) shows that changes in σ1(=σp, the stress in the plane of particle motion) cause Vs to vary according to σp0.16. Fig. 3 (c) shows essentially no changes in Vs as σ3 (=σ3, the stress on the plane perpendicular to the plane containing the directions of particle motion and wave propagation) is changed. Note that Vs no longer follows the straight line relationship when σmax/σmin becomes greater than about 2.The test arrangement for the tests, Case SH 1, is shown in Fig. 2 (b). Here the horizontally polarized shear wave is in the σ2-σ3 plane. Figure 4 (a) shows that Vs changes according to σ2(=σa) to the 0.10 power. Figure 4 (b) shows no changes in Vs with variations in σ1(=σs), while Fig.4(c) shows that Vs changes according to σ3(=σp) to the 0.13 power.The results shown in Figs. 3 and 4 are consistent with those obtained by Roesler and by Stokoe, et al., for wave propagations along principal stress directions. Table 1 contains a brief summary of these test results.lt is evident that the influences of σa and σp are essentially equal, and that σs has no effect. Undoubtedly, the method of placing the sand, the type of sand, the levels of confining pressures developed, and other testing parameters contribute to the differences in absolute values.The test data illustrated in Fig.3 is shown numerically in Table 2. The first four columns include the basic data from the tests in which the pressures were measured, and the shear wave velocities were calculated from the measured distances and time differences



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< previous page page_207 next page >Page 207between signal arrivals at the first and second pickups. Consequently, the values of shear wave velocities as shown to three significant figures are calculations, and we usually anticipate about 5% variations—even in repeated tests. Columns (5) and (6) of Table 2 are values of σo and σo2 which have been calculated from the measured pressures, and which have been used with Eq.(2) to calculate shear wave velocities shown in Columns (7) and (8).These values are quite similar, except for the third group of data based on variations

FIG. 4. CASE SH 1



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< previous page page_208 next page >Page 208TABLE 1. EFFECTS OF PRINCIPAL STRESS CHANGES ON SHEAR WAVE VELOCITYRoesler Stokoe et al Fei and Richart

σa0.149(Vert.) σa0.10-0.11 σ10.16 (Vert.) σ10

σp0.107(Horiz.) σb0.09-0.10 σ20.12 (Horiz.) σ20.10

σs0(Horiz.) σc0 σ30(Horiz.) σ30.13

SH Wave SH & SV Waves SV Wave SH Wave

σ1=σp σ1=σs

σ2=σa σ2=σa

σ3=σs σ3=σpof σ3, while holding σ1 and σ2 constant. In these tests, the variations in σ3 have no influence on the measured values, but they do affect the calculated values of Vs Finally, Columns (9) and (10) show values of the exponent, m (which is 0.25 in Eq.(2)) as calculated using Vs (meas.) and σo or σo2. Note that when the calculated value of Vs is greater than Vs (meas.), the calculated value of m is less than 0.25, and vice versa.WAVE PROPAGATION UNDER ISOTROPIC STRESS CONDITIONSFrom the five test conditions shown in Fig. 2, there were 16 tests in which the three principal stresses were all very close to 1.7 psi.—which produced isotropic stress conditions. From these tests, the average measured shear wave velocity was 459ft/sec., and the average m was 0.252, which compare quite well with the values of 454ft/sec.and m=0.25 which come from Eq.(2). Therefore we may conclude that the cross-hole tests in the quicksand tank are in good agreement with the test results obtained from resonant column tests. From these 16 tests, the standard deviation was S=14.3ft/sec., or +/−2S=28.6ft/sec. which corresponds to +/−6.2 % of the average shear wave velocity. This is close to the +/−5 % that we usually anticipate from repeated tests.



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< previous page page_209 next page >Page 209TABLE 2. TEST RESULTS FOR CASE SV 1 (σ—psi; vs—ft/sec.)(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

σ1 σ2 σ3 vs σo σo2 vs(σo) vs(σo2) m m

(=σp) (=σa) (=σs) (meas.) (calc.) (calc.) (σo) (σo2)

1.69 0.64 1.69 424 1.34 1.17 427 413 0.248 0.255

1.69 1.00 1.69 436 1.46 1.35 437 428 0.250 0.253

1.69 1.18 1.69 444 1.53 1.44 442 435 0.251 .254

1.69 1.62 1.69 467 1.67 1.66 451 451 0.256 0.256

1.69 1.92 1.69 476 1.77 1.81 458 461 0.257 0.256

1.69 2.35 1.69 481 1.91 2.02 467 474 0.255 0.253

1.69 3.21 1.69 501 2.20 2.45 484 497 0.256 0.251

1.69 3.67 1.69 506 2.35 2.68 492 509 0.255 0.249

1.69 4.14 1.69 467 2.51 2.92 500 519 0.238 0.232

1.01 1.69 1.69 427 1.46 1.35 437 428 0.246 0.249

1.30 1.69 1.69 441 1.56 1.49 444 439 0.249 0.251

1.48 1.69 1.69 444 1.62 1.58 448 446 0.248 0.249

1.69 1.69 1.69 449 1.69 1.69 453 453 0.248 0.248

1.71 1.69 1.69 463 1.70 1.70 453 453 0.254 0.254

2.07 1.69 1.69 470 1.81 1.88 461 465 0.253 0.252

2.42 1.69 1.69 486 1.93 2.05 468 476 0.256 0.254

2.97 1.69 1.69 501 2.12 2.33 479 491 0.258 0.254

3.40 1.69 1.69 515 2.26 2.54 487 502 0.260 0.254

1.69 1.69 0.94 458 1.44 1.69 435 453 0.260 0.252

1.69 1.69 1.69 469 1.69 1.69 453 453 0.256 0.256

1.69 1.69 1.69 448 1.69 1.69 453 453 0.247 0.248

1.69 1.69 2.07 465 1.82 1.69 461 453 0.251 0.255

1.69 1.69 2.31 465 1.90 1.69 466 453 0.249 0.255

1.69 1.69 2.56 452 1.98 1.69 471 453 0.243 0.250

1.69 1.69 2.93 452 2.10 1.69 479 453 0.240 0.250

1.69 1.69 3.19 448 2.19 1.69 483 453 0.237 0.248

1.69 1.69 3.27 441 2.22 1.69 485 453 0.233 0.245

1.69 1.69 3.78 420 2.39 1.69 494 453 0.222 0.236

1.69 1.69 4.62 390 2.67 1.69 508 453 0.206 0.222



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FIG. 5. CASE SV 2WAVE PROPAGATION AT AN ANGLE TO THE PRINCIPAL STRESSESFigures 5–7 illustrate the effects on the shear wave velocity caused by varying one principal stress while keeping the other two principal stresses constant, when the wave is propagating at an angle to one or more principal stress. The relationships are summarized in Table 3. Note that the sum of the exponents would be high, if the results had been expressed as shown by Eq.(4).Another way of interpreting the data is to calculate σo and σo2 and evaluate the “m—values”, using the measured values of the shear wave velocities in Eq.(2) The results of these calculations, for which σa and σp were obtained by Mohr circle relationships, are shown in Table 4. For both sets of calculations, the calculated exponent, m, was very close to 0.25.



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FIG. 6. CASE SH 2TABLE 3. INFLUENCE OF VARYING ONE PRINCIPAL STRESSTest Series Vertical Horizontal Horizontal

SV—2 σ10.16 σ20.12 σ30.05

SV—3 σ10.14 σ20.09 σ30.08

SH—2 σ10.085 σ20.10 σ30.13TABLE 4. SUMMARY OF AVERAGE EXPONENT “m” VALUESTest Series No. Tests m(σo) m(σo2)

SV—1 29 0.247* 0.252

SV—2 22 0.254 0.253

SV—3 11 0.243 0.244

SH—1 27 0.245 0.246

SH—2 10 0.259 0.261

* For variations in σ3, m(σo)= 0.240



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FIG. 7. CASE SV 3CONCLUSIONSThe tests in the quicksand tank using isotropic stress conditions gave results comparable to those obtained before in resonant column tests. For conditions in which the shear wave propagated along a principal stress direction, and had particle motions



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< previous page page_213 next page >Page 213in a second principal stress direction, only these two stresses influenced the value of propagation velocity. The third principal stress had no influence, as had been pointed out previously. It should be noted that the pressure levels were considerably smaller for these tests in the quicksand tank, because of the pressure capacity of the air bags involved. It had been noticed previously [2] that the “m”-values for (Eq.(2)) tended to be larger at low pressures than at higher pressures, which might explain some of the differences between these test results and those noted in refs. [3] and [4].From 16 tests under isotropic stress conditions, the average shear wave velocity was 459 ft/sec, and the average “m”-value was 0.252, which compared well with the values calculated from Eq.(2) of 454ft/sec. and 0.25. Thus, the results from cross-hole tests in the quicksand tank were comparable to those obtained by resonant column tests in the laboratory.For the three series of tests in which the wave propa-gated at an angle to one or more principal stress, the exponent, “m”, from Eq.(2) was reasonably close to 0.25 when calculated using either σo or σo2. However, when the results were expressed in the form indicated by Eq. (4), the exponents indicated very high values. For example, under isotropic stress conditions, the sum of the exponents shown in Table 3 would amount to: 0.33 for SV-2, 0.31 for SV-3, and 0.315 for SH-2. For this kind of wave propagation, it appears satisfactory to calculate shear wave velocities from Eq. (2), using σo and m=0.25.ACKNOWLEDGEMENTS.Thanks are extended to Prof. R.D.Woods for his help and advice in preparing the test program, and to Mr.Kevin Hoppe and Mr.Kevin Scmidt who assisted in the tests. The Wang Educational Foundation has provided funds for Prof. Fei to attend the SDEE’91, and their support is appreciated



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< previous page page_214 next page >Page 214REFERENCES1. Hardin, B.O., (1961), “Study of Elastic Wave Propagation and Damping in Granular Materials,” dissertation presented to the University of Florida, in partial fulfilment of the requirements for the PhD.2. Hardin, B.O., and Richart, F.E., Jr., (1963), “Elastic Wave Velocities in Granular Soils,” J. Soil Mechanics and Foundations Div., Proc.ASCE, v. 89, No.SM1, Feb., pp.33–653. Roesler, S.K.,(1979), “Anisotropic Shear Modulus due to Stress Anisotropy,” J. Geotechnical Eng. Div., Proc. ASCE, v. 105, No. GT7, July, pp.871–880.4. Stokoe K.H., II, Lee, S.H.H., and Knox, D.P., (1985), “Shear Moduli Measurements under True Triaxial Stresses,” Advances in the Art of Testing Soils Under Cyclic Conditions, (Ed.by V.Khosla), Proc. of a session sponsored by the Geotech. Eng.Div., ASCE Convention in Detroit, Mich., Oct.24, pp. 166–185.5. Yu, Peiji, and Richart, F.E., Jr. (1984), “Stress Ratio Effects on Shear Modulus of Dry Sands,” J. Geotech Div., Proc. ASCE, v.110, No.3, Mar., pp. 331–345.6. Stokoe, K.H., II, and Woods, R.D., (1972), “In Situ Shear Wave Velocity by Cross-Hole Method,” J. Soil Mechanics & Foundations Div., Proc. ASCE, v.98, No. SM5, May, pp. 443–460.7. Woods, R.D., and Henke, R.,(1981), “Seismic Techniques in the Laboratory,” J. Geotechnical Div., Proc.ASCE, v.107, No. GT10, Oct. pp. 1309–1325–



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< previous page page_215 next page >Page 215Stiffness Degradation of Weathered Marl in Cyclic Undrained LoadingJ.A.Little, N.HatafDepartment of Civil Engineering, Heriot- Watt University, Riccarton, Edinburgh, U.K.ABSTRACTThe results of a study on the effect of remoulding, stress-history, cyclic strain amplitude and number of cycles on the cyclic secant modulus degradation of weathered Keuper Marl, a calcareous mudstone formed mainly from clay minerals, are presented. The research is based on a series of monotonic and low frequency undrained strain-controlled cyclic triaxial tests on undisturbed and reconstituted specimens with overconsolidation ratios 1 to 30. An automated triaxial system especially developed for this research has been used to conduct the tests. It is shown that the rate of stiffness degradation is a function of number of cycles and cyclic strain amplitude. The difference between undisturbed specimens’ behaviour with the reconstituted samples’ characteristics have been used to investigate the effect of remoulding on the cyclic properties of the marl.INTRODUCTIONThe imposed stresses on elements of soils due to earthquake and wave loadings are cyclic in nature. Knowledge of soil behaviour underlying structures, built-in seismically active areas, and offshore structures subjected to cyclic shear loading is therefore significant for engineering purposes.Previous investigations on the effects of cyclic loading on both granular and cohesive soils under simulated earthquake and wave loading have demonstrated reductions in both soil stiffness and strength, (e.g. Seed and Lee [1], Seed and Chan [2], Idriss et al [3], Vucetic and Dobry [4]).Differences in dynamic characteristics of reconstituted and intact spec-



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< previous page page_216 next page >Page 216imens of soils subjected to cyclic loading have also been reported by a number of researchers, (e.g. Vrymoed et al [5]). There are very few published findings describing the effects of cyclic loading on the stress-strain behaviour of weathered rocks.This paper therefore describes the results of a series of undrained cyclic strain-controlled triaxial tests conducted on reconstituted and undisturbed specimens of a weathered marl.TEST EQUIPMENTAn automated triaxial system, consisting of an IBM microcomputer and modified conventional strain-controlled laboratory equipment, in conjunction with in-house software, were used to perform strain-controlled monotonic and cyclic loading tests. The cyclic strain-time pattern was of a triangular form.Pore pressures were measured at the mid-plane using miniature transducers implanted in the soil; strain measurements were made using high resolution, submersible, displacement transducers. MATERIAL TESTEDThe word marl is used to describe calcareous mudstones, which are clayey rocks, formed from clay minerals, quartz dust, fine dolomite crystals and gypsum aggregates. Keuper Marl, a Triassic sediment, consists usually of red-brown mudstone with subordinate siltstone and sandstone.Table 1: Basic engineering properties of reconstituted and undisturbed Keuper MarlProperty reconstituted undisturbed

Bulk density (Mg/m3) 1.96 2.14

Specific gravity 2.72

Liquid Limit (%) 45

Plastic limit (%) 30 The conditions of deposition of Keuper Marl were complex. A depositional mode for Keuper Marl has been described by Chandler and Davis [6]. Keuper Marl outcrops throughout central and south western England and it has formed the foundation for many important civil structures in that area.Since its deposition, Keuper Marl has been subjected to different degrees of weathering at different depths. The difference in weathering conditions has resulted in different mechanical behaviour of the material. Ke-



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< previous page page_217 next page >Page 217uper Marl has been divided into different zones relating to the degree of weathering in different ways, Chandler and Davis [6]; Birch [7].The specimens of Keuper Marl used in this study were considered as partially weathered, Chandler and Davis [6]. Table 1 shows some physical properties of the Keuper Marl examined.TESTING PROGRAMMEThe testing programme was designed to examine the behaviour of the weathered marl in its undisturbed state and in its reconstituted remoulded state under monotonic and cyclic, strain-controlled, undrained loading conditions. The programme of work consisted of: (i) preliminary tests, (ii) monotonic tests, (iii) cyclic loading tests and (iv) post-cyclic monotonic loading tests.For cyclic testing the specimens were initially isotopically consolidated and overconsolidated, having previously undergone incremental back saturation. Samples were divided into six types depending on stress-history, with overconsolidation ratio (R0) ranging from 1 to 30.The number of cycles applied to the test sample varied according to the cyclic strain amplitude selected to prevent sample failure during cyclic straining. The tests were carried out undrained and strain-controlled. The maximum strain amplitude used was 3.0 %. Each test took, on average, six days to complete. To allow pore water pressure equalization within the specimen, during cyclic loading, a frequency of 0.01Hz was used.SAMPLE PREPARATIONUndisturbed specimens were recovered from 100mm tubed samples and trimmed carefully to the prespecified diameter and height.Reconstituted specimens were obtained by crumbling tubed samples into small lumps. These were then air dried. The dried soil was then ground by pestle and passed through a 0.425mm sieve. The powder was then mixed with distilled water to produce a workable paste with a water content close to the natural water content. The paste was then wrapped in ‘Cling film’ and sealed in a plastic bag and kept in a humid box for at least 24 hrs to allow equilibration. All specimens underwent saturation before triaxial testing and were consolidated and overconsolidated against a back pressure, at least equal to 200kPa. The length to diameter ratio for specimens was two, except those with lubricated ends, for which a ratio of one was used (Rowe and Barden [8]).STIFFNESS DEGRADATION DURING CYCLIC LOADINGThe behaviour of normally consolidated reconstituted and intact speci-



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< previous page page_218 next page >Page 218mens under strain-controlled cyclic loading is illustrated in Figs. 1 and 2. Secant modulus (Es), defined as the ratio of the maximum cyclic devia-

Figure 1: Variation of secant modulus with number of cycles for normally consolidated reconstituted specimenstoric stress over the corresponding cyclic shear strain, εc, normalised with respect to the initial consolidation pressure

is plotted as a function of number of cycles (N) in these diagrams.

It can be observed that there is a similar trend of reducing with increasing N for both types of specimens. When normalised with respect

Figure 2: Variation of secant modulus with number of cycles for normally consolidated undisturbed specimens

to the undisturbed specimens usually showed higher cyclic strength at a given cyclic strain amplitude and number of loading cycle han the reconstituted specimens. Vrymoed et al [5] reported that the lower cyclic strength of reconstituted specimens can usually be attributed to the absence of any bonding at the particle contacts due to cementation. For Keper Marl the higher cyclic strength of the undisturbed specimens might also be related to inter-particle cementing or aggregating. It has been



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< previous page page_219 next page >Page 219shown (Sherwood [9], Davis [10], Lees [11]), that the undisturbed Keuper Marl has larger grain size due to aggregation of the clay particle into larger silt-size units by some cementing or aggregating agent. Such bonding would be broken down in the process of reconstitution.For the undisturbed specimens the data show linear but different relationships corresponding to the different cyclic shear strains employed (Fig. 2). This would indicate that steady state values had not been reached under the applied number of cycles. This is in agreement with published test results on undisturbed normally consolidated clays reported by others (e.g. Idriss et al [3], Vucetic and Dobry [4]).For reconstituted specimens, however, the data indicate that the soil reached constant modulus values under high cyclic shear strains and “softening” was continuing more slowly with increasing number of cycles for the lower cyclic strain applied. Taylor and Hughes [12] observed the same phenomenon and indicated a lower limit for soil strength (and therefore modulus) below which the strength of the soil at a given moisture content cannot fall. Procter and Khaffaf [13] also observed a decreasing rate of softening with increasing number of cycles toward a limit, termed the fully-weakened state.It would appear that the undisturbed Keuper Marl has to experience more strain cycles to reach an equilibrium state than does the reconstituted Keuper Marl. This may be due to the local remoulding which the undisturbed material undergoes during load reversals, a process which the reconstituted soil has already been subjected to during its preparation.

Figure 3: Cyclic stress-strain behaviour of overconsolidated reconstituted specimensThe results of the cyclic loading on overconsolidated specimens are



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< previous page page_220 next page >Page 220illustrated in Fig. 3. In this figure the normalised stress-strain curves obtained from cyclic loading tests on reconstituted normally consolidated and overconsolidated specimens are presented. The normalised deviatoric stresses related to cyclic loading tests are those corresponding to the first cycle, N=1. The pattern of stress-strain behaviour for cyclic loading is similar to that observed during monotonic loading. The higher was the overconsolidation ratio the larger was the stiffness. Similar behaviour was observed for intact specimens.CONCLUSIONSUndrained strain-controlled cyclic loading tests have been carried out on reconstituted and undisturbed specimens of weathered Keuper Marl. The test results have shown differences in cyclic behaviour of reconstituted and intact specimens which are attributed to inter-particle bonding.A reduction in stiffness due to cyclic straining is shown, for both intact and reconstituted specimens. This degradation of stiffness was found to be strongly dependent, on cyclic strain amplitude for both types of soils. The peak strength loss due to cyclic loading has been found significant even for the limited number of cycles applied.Higher stiffness at a given cyclic strain amplitude and number of loading cycles was observed for more overconsolidated specimens indicating the overconsolidation effect on soil behaviour.ACKNOWLEDGEMENTSThe authors would like to thank Wykeham Farrance Engineering Ltd., Slough, Bucks, for provision of laboratory equipment used in this study.REFERENCES1. Seed, H.B. and Lee, K.L., Liquefaction of saturated sands during cyclic loading conditions. J. Soil Mech. and Found. Div., ASCE, Vol. 92, No. SM6, pp. 105–34, 1966.2. Seed, H.B. and Chan, C.K., Clay strength under earthquake loading conditions. J. Soil Mech. and Found. Div., Proc. ASCE, Vol. 92, No. SM2, pp. 53–78, 1966.3. Idriss, I.M., Dobry, R. and Singh, R.D., Nonlinear behaviour of soft clays during cyclic loading. J. Geotechnical Eng. Div., ASCE, Vol. 104, No. GT12, pp. 1427–47, 1978.4. Vucetic, M. and Dobry, R., Degradation of marine clays under cyclic loading. J. Geotechnical Eng. Div., ASCE, Vol. 114, No. 2, pp. 133–49, 1988.



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< previous page page_221 next page >Page 2215. Vrymoed, J., Bennett, W., Jafroudi, S. and Shen, C.K., Cyclic strength and shear modulus as a function of time. Int. Symp. Soils under Cyclic and Transient Loading, Swansea, pp. 135–142, 1980.6. Chandler, R.J. and Davis, A.G., Further work on the engineering properties of Keuper Marl. CIRIA Report No. 47, 1973.7. Birch, N., Keuper Marl. Proc. Midland Soil Mechanics and Foundation Engineering Society, Vol. 6, No.31, pp. 41–84, 1966.8. Rowe, P.W. and Barden, L., Importance of free ends in triaxial testing. J. Soil Mech. and Found. Div., Proc. ASCE, Vol. 90, No. SM1, pp. 1–27, 1964.9. Sherwood, P.T., Classification tests on African red clays and Keuper Marl. Q.J. Engng Geol. Vol. 1, pp. 47–55, 1967.10. Davis, A.G., The mineralogy and phase equilibrium of Keuper Marl. Q. J. Engng Geol. Vol. 1, pp. 25–38, 1967.11. Lees, G., Geology of the Keuper Marl. Proc. Geol. Soc. London., No.1621, 46, 1965.12. Taylor, P.W. and Hughes, J.M.O., Dynamic properties of foundation subsoils as determined from laboratory tests. Proc. 3rd World Conf. Earthquake Eng., Auckland and Wellington Vol.1 pp. 196–211, 1965.13. Procter, D.C. and Khaffaf, J.H., Cyclic triaxial tests on remoulded clays. J. Geotechnical Eng., Vol. 110, No. 10, pp. 1431–1445, 1984.



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< previous page page_223 next page >Page 223Measurements of Material Anisotropy by Ultrasonic TechniqueS.V.Jagannath, C.S.Desai, T.KunduDepartment of Civil Engineering and Engineering Mechanics, The University of Arizona, Tucson, Arizona 85721, U.S.A.ABSTRACTMany geologic materials when subjected to sequences of loading, unloading and re-loading, experiences induced anisotropy, due to factors such as microstructural changes and damage. Herein, an experimental procedure has been developed, which enables multi-axial mechanical (destructive) and ultrasonic (non-destructive) testing of geologic materials. The paper describes the experimental scheme, data acquisition system and test results on a cemented sand under hydrostatic stress path of loading, unloading and re-loading. Examination of the stress-strain, stress-velocity and stress-attenuation data provides insight into the material anisotropy, its quantification, and correlation between mechanically defined anisotropy and non-destructively measured anisotropy.INTRODUCTIONFor realistic modelling of the response of geologic materials under cyclic (dynamic) loading, various constitutive models have been proposed (Desai and Siriwardane [1]). In the past, emphasis has been given to the development and verification of such models by performing laboratory (destructive) tests. An important area which requires further study is the development of Non Destructive Testing (NDT) procedures and toward correlation of mechanical and non-destructive behavior.It is observed that some geologic materials when subjected to sequences of loading, unloading and reverse loading, exhibit anisotropic (stress-strain) response. In the literature, it is often called stress induced anisotropy and is attributed to reorientation of particles (e.g., Casagrande and Carillo [2], Arthur et al. [3]) and to the formation of micro-cracks



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< previous page page_224 next page >Page 224and damage (e.g., Desai et al. [4]). Some of the constitutive models such as Hierarchical Single Surface (HISS) models based on the phenomenological approach ([4], [5]), can account for these effects. However, for better understanding and to quantify induced anisotropy, alternative testing procedures are desirable. Ultrasonic technique is one such NDT technique to measure induced anisotropy (Varadarajan and Desai [6]). Very few studies have been performed for the characterization of highly attenuating (geologic) material, using the ultrasonic technique (Krautkramer and Krautkramer [7]).The material presented herein is a part of ongoing research on correlation between mechanical and ultrasonic responses for anisotropic behavior of geologic materials. In this paper, detailed description of the experimental scheme and testing procedure is given. Also, preliminary results pertaining to anisotropy from cyclic hydrostatic loading tests are presented. Test results corresponding to shear stress paths of loading, unloading and re-loading are given elsewhere (Jagannath et al. [8], [9]).The proposed mechanical and ultrasonic measurements, as described subsequently, are performed while test samples are subjected to loading under various stress paths in the modified truly triaxial device. Ultrasonic P-waves pass through a sample of a geologic material in the X, Y and Z directions in the through-transmission mode, and the received signals are recorded. The stress-strain, stress-velocity and stress-attenuation data are examined to define stress-induced anisotropy.ULTRASONIC AND MECHANICAL TESTINGFigure 1 shows the setup for ultrasonic and mechanical testings. It consists of the truly triaxial device, which needs to be modified, an ultrasonic unit and a data acquisition system. In the following sections, brief descriptions of the instrumentation and the procedure for the sample preparation are given:1. Ultrasonic Device:This device is a broadband ultrasonic pulser/receiver unit supplied by the Panametrics Inc.. This unit has two-pole, three-position manual coaxial switch and ports for transducer connections. The unit provides a digital output of the transit time (in µsec) of P-waves through the sample.HP 54501A Oscilloscope: The received pulse from the pulser/receiver unit can be viewed on an oscilloscope. The oscilloscope is portable and programmable. The oscilloscope is connected to the central data acquisition and control unit. The output signature is digitized and stored in the HP 300 computer. Thus, it enables on-line monitoring of the entire experiment.



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Figure 1 Setup for Ultrasonic and Mechanical TestingVT—101 Transducers: In the present study, transducers of 1.25 in. (31.25mm.) diameter and 0.63in. (15.75mm.) thick, with a frequency of 0.5MHz are used. This transducer is found to satisfy the requirements, such as, near-field effects, material grain size effects etc. ([8], [9]).2. Modification of the Multi-Axial Device:Existing truly triaxial device has a cavity of 4×4×4 in. (100×100× 100 mm), in which the cubical specimen is placed. The sample is stressed in three principal directions using air pressure, contained by rubber membranes. The deformations of the sample are measured using Linear Variable Differential Transformers (LVDTs) in three principal directions. This device needs to be modified for performing the ultrasonic testing in combination with the mechanical testing.Figure 2 shows the schematic for installing the transducers in the cubical device. The transducers are pressed against the face of the specimen on all the six faces, and the contact is ensured by applying copious supply of vacuum grease (Dow Corning grease), and through connection to a pre-compressed spring. In order to make the ultrasonic measurements, during the deformed state, the transducers are housed in a special device.



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Figure 2 Exploded View of the Cubical Device and Details of MountingThis device consists of a core and a sleeve made up of a smooth material. The core houses the transducer and moves inside a frictionless sleeve during the specimen deformation. The base of the core is supported by a pre-compressed spring of very low stiffness, so as to ensure proper contact between the sample and the transducer. The outer sleeve is connected to the base plate of the cubical device by screws and is placed concentric with the other three (existing) LVDTs on each face. A special connector is used to carry the co-axial cable of the transducer through the end cap of the LVDT protection cylinder.



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< previous page page_227 next page >Page 2273. Data Acquisition System:The data acquisition system consists of a HP 9000 model 300 series computer, a HP 3852A mainframe controller and a power supply for 18 modified Schaevitz GCA-121–250 LVDTs. During a test, the controller takes the voltage readings from the power supply, digitized waveform from the oscilloscope and transfers the data on to the computer. All the instruments are communicated through the computer, by running the data acquisition program (written in BASIC).

Figure 3 Schematic of Ultrasonic TestingFigure 3 shows a schematic of the proposed ultrasonic testing. A cubical sample of the material of dimensions Lx×Ly×Lz is enclosed within the modified truly triaxial device. Tx,Ty,Tz are the transmitting transducers which send in a negative spike pulse of known amplitude, through the sample in the X, Y, and Z directions, respectively. After travelling through the sample, the pulse is received by receiving transducers Rx, Ry, Rz, respectively. Figure 3 also shows the schematic output wave signatures in the three directions as recorded by the oscilloscope. In this investigation, we use two parameters to study anisotropy. First is the velocity (Vi m/sec), which is related to relative orientation of the



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< previous page page_228 next page >Page 228particles, and the second is the attenuation (Ai=20 log (Output voltage /Input voltage)), related to the energy of the P-wave. Here, the subscript i corresponds to X,Y and Z directions, respectively.4. Material and Sample Preperation:The geologic material used in this investigation consists of a cemented sand. The cemented sand has been selected for two reasons: (a) a purely granular material enclosed in a rubber membrane causes scattering of ultrasonic waves, and (b) a cemented sand imparts cohesion which keeps the sample stable and intact and permit passage of the waves with very little scattering. The sample is (4×4×4in.) (100×100×100mm) size and consists of Leighton-Buzzard sand, 5% by weight of Burke stone (quick setting cement) and 14% by weight of water. Dry sand, cement and water are mixed thoroughly in a tray and the mixture is placed in a cubical mold. The inner surface of the mold is greased before adding the mixture. The mixture is then compacted in four equal layers using a compacting rod. The mold is removed after 24 hours and a thin coat of quick cement paste is applied on all faces, to insure a smooth surface for ultrasonic testing.TESTING PROCEDURECalibration of the Ultrasonic DeviceA steel test piece (4 in. (100mm) long and 1.5 in. (37.5mm) dia.), for which the exact transit time of the P-wave is known, is selected. The test piece is placed between a pair of transducers and the test is performed in the through-transmission mode. The resulting transit time, as displayed by the pulser/receiver unit is matched with the exact transit time, by adjusting the zero adjustment knob.Specimen Installation and TestingThe prepared sample is installed in the cubical cavity of the reaction frame with flexible membranes on each side. The sample is transferred to the apparatus in such a manner that the vertical direction during sample preperation coincides with the vertical Z-direction of the apparatus. Further, the principal directions of loading and deformations are defined to coincide with each other. A uniform (normal) stress on each face of the soil sample is applied by pneumatically pressurizing the membranes. The applied pressure in each of the three directions is controlled independently. The air pressure system consists of an air compressor, pressure regulators and Bourdon tube gages. The loading is quasi-static. Deformations of the sample are measured by means of LVDTs. An unique feature of the data acquisition unit is that at the end of the experiment, both mechanical (stress and deformation) and ultrasonic (wave signatures and velocities) data are available in respective files for further analysis.



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< previous page page_229 next page >Page 229TEST RESULTSIn this section, typical test results on cemented sand under hydrostatic loading are presented. In this test, the confining pressure is applied in small increments up to a maximum of 120 psi (827kPa). During the test, the sample is subjected to two cycles of loading, unloading and re-loading. At each stress level, the strains in the material, and the velocity and the attenuation of the P-wave through the sample are recorded. Also, wave signatures at these points are recorded.Figures 4, 5 and 6 show the stress-strain, stress-velocity and stressattenuation data, respectively. It is observed that the strains, the velocities and the attenuations change with the applied confining stress. Further, at any given stress level, each of these value is different in the X, Y and the Z directions. This is attributed to the material anisotropy. However, at very large confining stress, the incremental change in the values of strain, velocity and attenuation in each direction are almost same, indicating that the material approaches the isotropic state. It is evident that the sample is initially anisotropic (due to sample preperation procedure) and the magnitude of anisotropy decreases with the applied hydrostatic stress. Any definition leading to the quantification of anisotropy should consider this effect.Figures 7 and 8 show the wave signatures at stress levels of 0, 30, 60 and 90 psi. (0, 207, 414, and 621kPa) along the virgin loading curve. It is observed that the received wave signatures at lower values of confining stress, are different (for the same input wave) in three principal directions. However, they tend to become similar in three directions at high confining stress, indicating convergence toward the isotropic state.QUANTIFICATION OF ANISOTROPYIn order to quantify the material anisotropy, we identify two forms of anisotropy, namely, mechanical and ultrasonic anisotropy. The mechanical anisotropy is a function of (plastic) strains developed within the material due to the application of stress. While, the ultrasonic anisotropy is a physical measure, related to the variations in the values of attenuation and velocity due to the applied stress.At any given stress state, let , Vi and Ai be the strains, the velocities and the attenuations in the three directions,

respectively. Let , and be the average values of the strains, the velocities and attenuations, at that stress state, respectively.



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Figure 4 Stress—Strain Plot

Figure 5 Stress—Velocity Plot

Figure 6 Stress—Attenuation Plot



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Figure 7 Wave Signatures along X, Y and Z directions at Confining Pressures of (a) 0 psi and (b) 30 psi (207Kpa)



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Figure 8 Wave Signatures along X, Y and Z directions at Confining Pressures of (a) 60 psi (414 Kpa) and (b) 90 psi (621 Kpa)



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< previous page page_233 next page >Page 233We define the mechanical anisotropy (Manis), the velocity anisotropy (Vanis) and the attenuation anisotropy (Aanis) as follows:

(1)

(2)

(3)Figure 9 shows the variations of Manis, Vanis and Aanis with the applied confining stress. It is seen that all the three measures of anisotropy show similar trends and provides a guideline for the development of suitable correlation functions. The magnitudes of these measures are larger at low value of confining stress and they decrease and attain constant values at large values of confining stress.

Figure 9 Variation of Mechanical and Ultrasonic Anisotropies with Confinig PrssureCONCLUSIONSA combination of mechanical (destructive) tests and ultrasonic (NDT) provides information about enhanced understanding of the material anisotropy. The quantification of material anisotropy suggests that the mechanical anisotropy (due to strains) compares very well with the ultrasonic anisotropy. Development of correlation functions between the two is therefore feasible and a subject of continuing investigation.



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< previous page page_234 next page >Page 234ACKNOWLEDGEMENTSThe research results presented herein were supported through Grant No. CES 8711764 from the National Science Foundation, Washington, D.C.REFERENCES1. Desai, C.S. and Siriwardane, H.J. Constitutive Laws for Engineering Materials, Prentice-Hall, Inc., New Jersey, 1984.2. Casagrande, A. and Carillo, N. Shear Failure of Anisotropic Material, Proceedings of the Boston Society of Civil Engineers, Vol. 31, pp. 74–87, 1944.3. Arthur, J.R.F., Chua, K.S. and Dunstan, T. Induced Anisotropy in A Sand, Geotechnique, Vol. 29, No. 1, pp. 13–30, 1977.4. Desai, C.S., Somasundaram, S and Frantziskonis, G.A Hierarchical Approach for Constitutive Modelling of Geologic Materials, Int. JI. Numer. Anal. Methods Geotech., Vol. 10, pp. 225–257, 1986.5. Desai, C.S. A General Basis for Yield, Failure and Potential Functions in Plasticity, Int. JI. Numer. Anal. Methods Geotech., Vol. 4, pp. 361–375, 1980.6. Varadarajan, A. and Desai, C.S. Multi-axial Testing and Constitutive Modelling of a Rock Salt, Constitutive Laws for Engineering Materials; Theory and Applications., Vol. I, Desai C.S. et al. (Edt.), Elsevier Science Publishing Co., Inc., pp. 465–473, 1987.7. Krautkramer, J. and Krautkramer, H. Ultrasonic Testing Materials, Springer-Verlag, 2nd ed., New York., 1977.8. Jagannath, S.V., Desai, C.S. and Kundu, T. Correlation Between Mechanical and Ultrasonic Responses For Anisotropic Behavior of Soils Under Cyclic Loading, Report to National Science Foundation, Washington, D.C., Dec. 1990.9. Jagannath, S.V. Correlation Between Mechanical and Ultrasonic Responses For Anisotropic Behavior of Soils Under Cyclic Loading, Ph. D thesis submitted to the University of Arizona, Tucson, Arizona, (Under Preperation), 1991.



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< previous page page_235 next page >Page 235Elastic Attenuation in Non-Homogeneous Porous MaterialsB.Gurevich, S.LopatnikovLaboratory of Mathematical Modeling, VNIIGeoinformsystem, Varshavskoe shosse, 8, Moscow, 113105, USSRABSTRACTElastic wave attenuation due to the Biot slow wave generation on inhomogeneities in saturated porous media is studied both theoretically and numerically. Theoretical results for one-dimensional case show essential dependence of the attenuation on the autocorrelation function of the inhomogeneities. Concrete calculations are made for two kinds of inhomogeneities: random (with exponential autocorrelation function) and periodic. For the numerical calculations the standard matrix technique is extended to the waves in poro-elastic media. The numerical results show good agreement with the theoretical predictions. Comparison with experimental data for frequency dependence of attenuation is also presented.INTRODUCTIONThe most often used theory of elastic waves in fluid-saturated porous media was proposed by Frenkel [1] and Biot [2, 3] forty years ago. However, the majority of applications appears to be concerned with ultrasonics, while in low-frequency dynamics it is used rather rarely, mainly because of its wrong predictions



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< previous page page_236 next page >Page 236for the compressional and shear waves attenuation. For low frequencies Biot theory predicts the attenuation coefficients a of the fast compressional wave to be proportional to f2, while laboratory measurements commonly show much larger values of α and less steep frequency dependence (α∝fn, 1<n<1.6).Our approach to the problem is based on the idea (first proposed by White et al [7]) that the discrepancy between measured and theoretically predicted attenuation might be related to the energy transfer from the ordinary elastic wave to the Biot slow wave on the inhomogeneities. The simplest and most important kind of non-homogeneity is stratification. Several attempts have been made to study the effect of stratification of porous media on attenuation for certain particular cases [8–11]. Here we propose a systematic approach to this problem based on the low-frequency version of the Biot model [2] and the statistical wave theory. The effect of stratification is studied both theoretically and numerically. For the theoretical calculations the Biot theory is applied to non-homogeneous (random and periodic) porous media leading to the Biot equations with variable coefficients. These equations are analyzed by means of the statistical perturbation technique widely used for the waves in random media (see e.g. Karal, Keller [12]).The numerical calculations were carried out to verify analytical results. Thomson-Haskell matrix technique for elastic layered medium was extended to the waves in saturated porous materials. The extended matrix procedure takes into account both fast and Biot slow compressional waves leading in 1-d case to (4×4)- matrices instead of (2×2) for classical elastic situation. Unlike the former approaches to the problem (Barzam, [13]; Allard et al, [14]) our algorithm is applicable



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< previous page page_237 next page >Page 237not only to comparably high frequencies (in terms of Biot [2, 3]), but also to the lower frequencies. In this case the Biot slow wave is non-propagating and appears only within a thin boundary layer near the interface, that is why the formerly proposed solutions become unstable. To overcome these difficulties the new approximate interface conditions for the fast wave have been derived resulting in new efficient (2×2) matrix procedure. This procedure is similar to the classical elastic one, but also takes into account the dissipation effect associated with the energy transfer into the Biot slow wave.By means of these algorithms effective attenuation coefficients for both random and periodic sequences of porous layers are computed. The results of these computations are presented together with the theoretical predictions. We also compare our theoretical results for the frequency dependence of attenuation with the corresponding laboratory data.THEORYBasic EquationsWe start with the low-frequency Biot equations [2, 15] for a saturated porous medium. The parameters of the medium are assumed to depend on x coordinate only. For compressional waves propagating along x axis these equations can be written in the form [16]:

(1)



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< previous page page_238 next page >Page 238Here the components uf=m(uf—us) and us of the vector-function u are defined by the displacements us and uf of the fluid and solid phase respectively, m—porosity, k—permeability, η—fluid viscosity, ρ1, β1, i=s, f are the densities and compressibilities of each phase, indicated by the corresponding subscript, θ, βo and µo denote tortuosity, compressibility and shear modulus of the dry (unsaturated) solid matrix respectively.Here and below we assume for simplicity that ρs, ρf, m, k and η are constants, while elastic properties of both phases

and dry skeleton may depend on x. Furthermore, we introduce the displacement potentials φs and φf: ,

. Considering monochromatic situation (φs=vs exp(−iωt), φf= vf exp(−iωt)) for vector-function v=vf, vs we can rewrite the equation (1) in the form:

B1(x)vxx+B2v=0, (2)where the elements of the matrix B2 depend on the constant medium parameters and frequency ω. Let’s express B1(x) as a sum of the constant and fluctuating terms:

B1(x)=Bo+ε(x)B′. (3)

Premultiplying the equation (2) by and introducing the notations , , we rewrite (2) in the form:

vxx+Cv+ε(x)Bvxx=0. (4)The characteristic equation of the matrix C is a familiar dispersion equation for the compressional waves in a saturated porous medium [1, 2, 15]. The eigenvalues of C are squared wave numbers k1 and k2 of the fast and slow Biot compressional waves, and the eigenvectors e1 and e2 define the “polarization” of these waves, i.e. the relationship

between solid and fluid velocities in each wave respectively. The components of the eigenvectors (j—denotes the wave type and i indicates



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< previous page page_239 next page >Page 239its component) form a matrix R, which enables to transform the first two terms of the equation (4) to the diagonal form

wxx+Kw+ε(x)Swxx=0, (5)

while v=w1e1+w2e2 =Rw. For a homogeneous medium (ε(x)≡0) the equations (5) describes the independent propagation of the fast and slow compressional waves

(6)

where in the low-frequency limit k1 and k2 may be expressed in the form: ,

For the general situation the third term in the left-hand side of the equation (5) describes the effects associated with the inhomogeneity of the medium and is responsible for the energy exchange between the fast and slow waves.Statistical approachFor the study of the effect of the medium inhomogeneity on the attenuation we assume that inhomogeneities are distributed uniformly along x axis. In other words, we consider ε(x) to be a stationary random function with the autocorrelation function ψ(x)=<ε(x′+x)ε(x′)>Then the equation (5) may be interpreted as a stochastic equation

L0w+L1w=0 (7)Here L0=<L> is the operator of the “homogeneous” system (6) and L1 is a random operator with a zero average (L1 is of the same order as ε(x). Then in Bourret approximation [12] an equation for the mean field <w> can be written in the form:



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[L0—<L1L0L1 >] <w>=0, (8)where angle brackets denote averaging over the set of realizations. In particular, an equation for the normal (fast) compressional wave is

(9)

where s2=s12s21; G1 and G2 are Green’s functions for the Helmholtz equation with the coefficients

and respectively

(10)Seeking the solution of (9) in the form

<w1>∝exp(ikx), (11)we obtain the following dispersion relation for the fast wave

(12)where σ2=ψ(0) is the dimensionless dispersion of the fluctuations. In the case of the sufficiently small fluctuations

for the attenuation coefficient α=Im k and inverse quality factor

(13)(c=ω/Re k1) we finally obtain

(14)1/Q=soσ2g(ω) (15)



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(16)Therefore we see that the frequency dependence of Q−1 is defined by the form of the autocorrelation function . In the next section we consider two particular correlation functions, that seem to be of natural interest.Results for random and periodic fluctuationsThe first case to be considered is concerned with the exponential correlation function

(17)Such correlation properties are common for the Markov random functions that are considered to represent a satisfactory description of the real rocks stratification [17] from theoretical point of view as well as from empirical data. Substituting (17) into (16) we get

(18)

where δ=ω/ω0 is dimensionless frequency, while is a frequency which makes wavelength of the Biot

slow wave equal to the mean inhomogeneity size (1/k2(ω0)=b/2). From (18) for the limits of the thin

and thick layers respectively we deduce:

(19)

(20)Another typical situation corresponds to the periodic layering of the medium. This situation leads to the “serrate” correlation function

(21)



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Fig. 1. Normalized transformational attenuation as a function of dimensionless frequency solid line—random layering (eq. (18) dashed line—periodic layering (eq. (22)

Fig. 2. Comparison of two theoretical models of attenuation in porous media with laboratory data (o—Tittmann et al [5]) solid line—present work (eq. 18); dashed line—O’Connel [19]



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< previous page page_243 next page >Page 243For such fluctuations the equation (16) gives:

(22)Asymptotic relations for the low and high frequencies similarly to (19) and (20) are

(23)

(24)By comparing these results with the relations (19) and (20) (fig. 1) one can see that the structure of the porous medium inhomogeneity (random or periodic) plays an important role at the low frequencies and doesn’t influence the attenuation Q−1 at the higher frequencies. Therefore, in order to calculate precise acoustic properties of the real rocks it is necessary to study common laws of the stratification.NUMERICAL EXPERIMENTSAlgorithmTo verify the theoretical results derived we’ve performed a number of numerical experiments. Their goal was to calculate numerically the apparent attenuation as a function of frequency for the stratified porous medium by means of the computer program designed for synthesis of compressional wave seismograms in a given sequence of homogeneous porous layers. For the calculations in such problems matrix methods [13, 14, 18] are widely used. However, at high frequencies, when the Biot slow wave attenuation length l2=1/Im k2 is less than layer thicknesses, the algorithms available appear to be unstable. Such difficulties can be overcome when all the layers are thick compared to l2. In this case the Biot slow wave exists only near each interface, that is why it doesn’t influence the processes at any other interface. This effect is used in the numerical procedure, which deals with the normal P-waves with the help of interface conditions modified with the view of the Biot slow wave generation at the interface. This procedure will be refereed to as a reduced algorithm.



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< previous page page_244 next page >Page 244Results of computationsIn fig. 1 we show the individual and average results of the calculations for both random and periodic sequences of 100 layers with a total thickness of 10m and the following parameters: m=0.3, K=10−12 m2, ρs=3.103 kg/m3, ρf=1.103 kg/

m3, pa, µs=2.7·1010 Pa, βf=4.8.10−10 Pa, ηf=1.10−3 Pa.s, , Here we’ve also plotted the curves corresponding to our theoretical results (15), (18) and (22). One can see sufficiently good agreement between the theory and the numerical experiments.DISCUSSION AND COMPARISON WITH EXPERIMENTAL DATAWe are now ready to discuss certain questions concerning applicability of the theoretical results obtained. First, we want to point out that from the two types of the stratification considered (random and periodic) only the first one is valid for natural rocks. In particular, we must emphasize the importance of the asymptotic result (19), which gives quite a unique frequency dependence of attenuation in the low-frequency limit(α ∝ ω3/2) This result as well as more general relation (18) seems to be in conformity with common knowledge concerning frequency dependence of attenuation at frequencies 10–1000Hz.To make more definite conclusions we need to compare our theoretical results with the measured data for the rocks with the given statistical parameters. If we assume a rock to fit



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< previous page page_245 next page >Page 245the correlation function (18), we must know at least two parameters—σ2=ψ (0) and the correlation radius 2b. No data is now supplied with such information. However, we tried to make at least qualitative comparison by seeking the best agreement of our result (18) with the laboratory data of Tittmann et al [5] (fig. 2). The agreement obtained seems to be quite good. Here we also show the theoretical result of O’Connel [19] obtained on the basis of a particular version of the general theory of a local fluid flow [20]. As was reported [19], this result also fits the data, but unlike our result it seems not to follow the main tendency of the data in the low-frequency limit. In any case, fig. 2 demonstrates that our theoretical predictions on the frequency dependence of attenuation are quite reasonable.CONCLUSIONSThe analytical formulae and numerical results obtained above show frequency dependence of the normal (fast) compressional wave attenuation due to its energy transfer into the strongly dissipative Biot slow wave at each interface inside a porous medium (transformational attenuation mechanism). The dynamics of the porous medium was considered under the conditions of the low-frequency limit of the Frenkel-Biot theory [1, 2]. At higher frequencies it is necessary to

take into account the Biot correction function [3] and higher terms in the expansion of and in powers of ω. In

general case total attenuation is the sum of the usual Biot attenuation and the transformational component Q−1 caused by medium inhomogeneity.Finally, we want to refer to Yegorov [16], who gave proof of our results by using a rigorous mathematical asymptotic procedure known as homogenization (e.g. [21]).



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< previous page page_246 next page >Page 246REFERENCES1. Frenkel, Ya.I. On the theory of seismic and seismoelectric phenomena in moist soil, Izvestiia Academy of Sciences, USSR, Geographical and geophysical series, Vol. 8, pp. 133–150, 1944 (in Russian).2. Biot, M.A. Propagation of elastic waves in a fluid-saturated porous solid. I. Low frequency range, J. Acoust. Soc. Am., 28, pp. 168–178, 1956.3. Biot, M.A. Propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range, J. Acoust. Soc. Am., 28, pp. 179–191, 1956.4. Jones, T.D. Pore Fluids and Seismic Attenuation in Rocks, Geophysics, 51, pp. 1939–1953, 1986.5. Tittmann, B.R., Bulau, J.R. and Abdel-Gawad, M. Dissipation of Elastic Waves in Fluid Saturated Rocks, in Physics and Chemistry of Porous Media (Ed. Johnson, D.L. and Sen, P.N.), pp. 131–143, Proceedings of AIP Conf. on Physics and Chemistry of Porous Media. American Institute of Physics, New York, 1984.6. Töksöz, M.N., Johnston, D.H. and Timur, A. Attenuation of Seismic Waves in Dry and Saturated Rocks. I. Laboratory Measurements, Geophysics, 44, pp. 681–690, 1979.7. White, J.E., Mikhaylova, N.G. and Lyakhovitsky, F.M. Low-frequency Seismic Waves in Fluid Saturated Layered Rocks, Izvestiia Academy of sciences USSR, Physics of the Solid Earth, pp. 654–659, 1975.8. Yumatov, A.Ju. and Markov, M.G. Elastic Waves in Periodically Stratified Saturated Porous Medium, Soviet Geology and Geophysics, No 3, pp. 93–104, 1987.9. Markov, M.G. and Yumatov, A.Ju. Acoustic Properties of a Layered Porous Medium, J. of Appl. Mech. and Tech. Phys., No 1, pp. 115–119, 1988.10. Gurevich, B. and Lopatnikov, S.L. Attenuation of Longitudinal Waves in Saturated Porous Medium with Random Inhomogeneities, Doklady Earth Science Sections, 281, No. 2, pp. 47–50, 1985.11. Lopatnikov, S.L. and Gurevich, B. Transformational Mechanism of Elastic Waves Attenuation in Saturated Porous Media, Izvestiia Academy of Sciences USSR, Physics of the solid Earth, 24, pp. 151–154, 1988.



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< previous page page_247 next page >Page 24712. Karal, F.C. and Keller, J.B. Elastic, Electromagnetic and Other Waves in Random Media, J. Math. Phys., 5, pp. 537–547, 1964.13. Barzam, V.A. Calculation of the Dynamic Characteristics of Waves in Finely Stratified Porous media. Izvestiia Academy of Sciences USSR, Physics of the Solid Earth, 15, No 12, 1979.14. Allard, J.-F., Bourdier, R. and Depollier, C. Biot Waves in Layered Media, J. Appl. Phys., 60, pp. 1926–1929, 1986.15. Bourbie, T., Coussy, O and Zinszner, B. Acoustics of Porous Media, Gulf Publications, Houston, 1987.16. Yegorov, A.G. Attenuation of Elastic Waves in Finely Laminated Saturated Porous Media, Applied Mathematics and Mechanics, 53, pp. 911–918, 1989.17. Waters, K. Reflection Seismology, Wiley & Sons, New York, 1981.18. Kosachevsky, L.Ya. On Reflection of Sound Waves from Layered Two-Component Media, Applied Mathematics and Mechanics, 25, pp. 1076–1082, 1961.19. O’Connel, R.J. A Viscoelastic Model of Anelasticity of Fluid Saturated Rocks, in Physics and Chemistry of Porous Media (Ed. Johnson, D.L. and Sen, P.N.), pp. 166–175, Proceedings of AIP Conf. on Physics and Chemistry of Porous Media. American Institute of Physics, New York, 1984.20. O’Connel, R.J. and Budiansky, B. Viscoelastic Properties of Fluid Saturated Cracked Solid, J. Geophys. Res., 82, pp. 5719–5736, 1977.21. Sanches-Palencia, E. Non-homogeneous Media and Vibration Theory, Springer-Verlag, New York, 1980.



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< previous page page_249 next page >Page 249SECTION 5: LIQUEFACTION



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< previous page page_251 next page >Page 251Liquefaction of Gravelly Soil at Pence Ranch During the 1983 Borah Peak, Idaho EarthquakeR.D.Andrus, K.H. Stokoe, II, J.M. RoëssetDepartment of Civil Engineering, The University of Texas at Austin, U.S.A.ABSTRACTEfforts in evaluating the liquefaction susceptibility of soils have been directed towards sands. Little information has been gathered on the characteristics and field performance of gravelly soils. As a result, generally accepted guidelines have not been developed for assessing the liquefaction susceptibility of gravelly soils. Investigations were conducted at Pence Ranch where liquefaction had occurred during the 1983 Borah Peak, Idaho earthquake. Sediments in the loosest layer range from clean gravelly sand to sandy gravel. A low permeablity cap appears to have aided in the development of high pore water pressures. Liquefaction assessment methods for sands directly applied to SPT, BPT and Vs measurements in these gravelly soils agree well with observed field behavior. By extending the relationship between qc/N60 and D50 into the gravel range, a susceptibility boundary based on the CPT was constructed for gravelly soils. Liquefaction susceptibility was then correctly predicted with this new boundary.INTRODUCTIONA magnitude 7.3 (Ms) earthquake occurred in the Borah Peak area of central Idaho on October 28, 1983. The earthquake created a 37-km-long surface rupture along the Lost River Fault and caused many of the saturated granular sediments in the lower areas of the Big Lost River and Thousand Springs Valleys to liquefy. Reported liquefaction effects included numerous sand boils and lateral spreading failures (Youd et al. [19]). In their reconnaissance report, Youd et al. described liquefaction-induced lateral spreading of a terrace at the Pence Ranch, located approximately 8km southeast of the southern terminus of the surface rupture. This terrace lies about 2m above and 300m away from the present-day channel of the Big Lost River. As illustrated in the map shown in Fig. 1, liquefaction beneath the terrace generated a zone of lateral spreading over 240m long and 30m wide. Large curved fissures and gravelly sand boils marked the head of the lateral spread. The fissures were as wide as 0.3m, with scarps as high as 0.3m. Horizontal movement was northward into the marsh. Damage included the distortion of the house and steel-frame barn shown in Fig. 1, and disruption of farm roadways. The wire fence enclosing the hay yard was pulled apart about 75cm by the lateral movement.



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Fig. 1—Map of the Pence Ranch site showing liquefaction effects.

Fig. 2—Location of testing and sampling near the hay yard shown in Fig. 1



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< previous page page_253 next page >Page 253In 1984, 1985 and 1990, field investigations were conducted across the the lateral spread at Pence Ranch (Stokoe et al. [14]; Harder [6]). The field work consisted of trenching, drilling, sampling and seismic testing. The investigations conducted by the authors in 1985 have been briefly summarized in earlier publications (Stokoe et al. [14 and 16]; Andrus and Youd [2]). A more complete discussion of the investigation performed in the vicinity of the hay yard is present in this paper, including the recent studies conducted in 1990.FIELD INVESTIGATIONSSampling and testing locations near the hay yard are shown in Fig. 2. Drilling included standard penetration tests (SPT), cone penetration soundings (CPT) and Becker penetration soundings (BPT) at three principal test areas. SPT tests were conducted using a 5-cm (2-in.) outside diameter splitspoon tube following the procedures outlined in ASTM D-1586–67. SPT boreholes were advanced with hollow-stem augers. CPT soundings were made with 15-cm2 and 10-cm2 electric cone penetrometers following procedures outlined by ASTM D-3441–79. Soundings made with the 15-cm2 cone were performed by the U.S. Bureau of Reclamation (USBR) in 1985. Soundings made with the smaller cone were pushed behind the drill rig employed in 1990. Although different SPT and CPT equipment was employed in 1985 and 1990, the penetration data agrees well. BPT soundings were made with the Becker drill AP-1000 following procedures outlined by Harder [6]. Sampling was attempted in the boreholes using a 3.5-cm inside diameter splitspoon sampler and 12.7-cm inside diameter, continuous hollow-stem augers. Finally, a trench was excavated across the major fissure. The trench was used to log the sediment profile, to describe fissures, and to perform in-situ densities. In-situ density tests in the gravelly sediment were conducted using a 1.2-m diameter ring following procedures outlined by the U.S. Bureau of Reclamation [18]. Bulk samples were collected in test pits just below the trench with the aid of a backhoe.In situ seismic testing was performed by the Spectral-Analysis-of-Surface-Waves (SASW) method. The SASW method requires no boreholes and thus is well-suited for undisturbed testing of gravelly soils. Two principals of the SASW method are that the velocity of propagation of a surface wave depends on the frequency (or wavelength) of the wave and that waves of different frequency sample different parts of the the layered medium. Procedures described by Stokoe et al. [15] were followed. Hammers, dropped weights and a bulldozer were used as sources. For each receiver spacing, the time delay between receivers was calculated as a function of frequency from the phase of the cross-power spectrum between the two receivers. The surface wave phase velocity was determined by dividing the distance between receivers by the time delay for various frequencies. The results from all receiver spacings along a test line were assembled together into a dispersion curve. Shear wave velocity, Vs, profiles were then obtained through an iterative process of matching the assembled dispersion curve to theoretical dispersion curves using a two-dimensional computer model discussed by Roesset et al. [10] to this conference. FIELD RESULTSThe Pence Ranch is covered by a thin mantle of loess consisting of fine silty sand. Beneath this mantle lies a complex sequence of braided, fluvial sediment.



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< previous page page_254 next page >Page 254A simplified profile of sediments exposed in the trench near the hay yard is shown in Fig. 3. Trench sediments can be divided into silty sand facies (units A and A1) and clean sandy gravel facies (units B and C). Descriptions of each unit are summarized in Table I. Units A and A1 form a cap of relatively low permeability and of varying thickness over the site. At the north and south ends of the trench, the cap is thin and is underlain by clean, loose to dense sandy gravel. In the central portion of the trench, the cap is as much as 1.4m thick. During the 1983 earthquake, saturated sediments just below the trench developed very high pore water pressures and liquefied. Loss of shear strength within the liquefying material caused part of the ground to move northward, opening the major fissure shown in Fig. 3. Water carrying sand and gravel was ejected up through the major fissure onto the ground surface. As pore water pressures dissipated, the major fissure filled with some of the ejecting sand and gravel. Minor amounts of water and sand also flowed up to the ground surface through much smaller cracks, also shown in Fig. 3.Many of the SPT, CPT and SASW profiles along the test alignment are included in the cross section in Fig. 4. Grain-size, penetration, and velocity data for the loosest layer, unit C, are tabulated in Table II. SPT N-values within unit C range from 6 to 16. CPT tip resistance is lowest just below the water table, and varies from 0.4 to 17.5 MPa. Uncorrected Becker blowcounts within unit C range from 7 to 13 blows per 0.3m. Similar Becker blowcounts were recorded by Harder [6] within the enclosed hay yard. Shear wave velocities range from 90 to 158m/sec. Sediments within unit C range from gravelly sand (SP-GP) to sandy gravel (GP) containing less than a few percent fines. The occasional high fiction ratios in the cone soundings suggest thin lenses of silty material within unit C. Grain-size distribution curves of bulk samples collected in unit C are shown in Fig. 5. The amount of sand contained in these samples suggests the gravel to be predominantly clast-supported (stone-on-stone). Sediments beneath unit C (units D and E) have much higher penetration and velocity properties.A qualitative estimate of in situ density has been suggested by Seed et al. [12] using the following relationship:

Gmax=1000 K2(σ′m)0.5 (1)where Gmax is the low-amplitude shear modulus, K2 is an empirical constant which reflects the density of the material, and σ′m is the mean effective stress. Since Gmax is directly related to shear wave velocity, shear wave velocity was used to estimate the qualitatively in situ density. Values of K2, determined for unit C range from 15 to 38. These very low values of K2 indicate that unit C is composed of very loose material.Based on the low penetration resistances and low shear wave velocities, it is proposed that liquefaction occurred within unit C. Unit A1, located so close to the water table and lying above very loose gravelly material, appears to have made test area 2 the most vulnerable to liquefaction and provides an explanation for the location of the major fissure. Although the sediment of unit C is much coarser than the gravelly sand boil deposits in the hay yard area (as shown in Fig. 5), the coarser particles segregated out during upward transport. Further evidence to support this segregation was observed in the trench.



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Fig. 3—Sediment profile exposed in trench at Pence Ranch.Table I: Description of Trench Sediments at Pence RanchUnit A. This unit is a dry to slightly damp siltya sand with gravel dispersed throughout (SM)b. The finer fraction is very dark grayish brown (10 YR 3/2)c, non-plastic, with strong to moderate reaction with HCl. Contact with lower unit A1 is gradational; Contact with lower units B and C is sharp.

Unit A1. Deposit grades upward from a sand with trace of silt (SW) to a silty sand (SM) with occasional charcoal fragments. The finer fraction is dark grayish brown (10 YR 4/2), non-plastic, and does not react with HCl. In-situ density is about 1.65 g/cm3 and moisture is 11 to 16%. Contact with upper unit B is sharp; contact with lower unit C is sharp to gradational.

Unit B. This unit is a sandy gravel (GP) consisting of about 56% fine to coarse, hard, subrounded gravel with low sphericity; 43% fine to coarse, hard, subangular sand; less than 1% dark grayish brown (10 YR 4/2) fincs; less than 1% subrounded cobbles, maximum dimensions 100mm. The finer faction in the upper few feet reacts weakly with HCl. In-situ density is about 2.19g/cm3 and moisture is 3%. Gravels are clast-supported (stone-on-stone) with a filled framework of sand to sand matrix-supported (gravel floating in sand), and appear to have more than one mode of imbrication. Internal stratification is crude, characterized by beds, less than about 0.3m thick, of more densely packed gravel and by thin (2cm), low angle (0 to 14 degrees), planer forsets having a higher sand content.

Unit C. Deposit grades from a gravelly sand (SP) to sandy gravel (GW-GP). Gravels are hard, subrounded with low sphericity; sand is hard and subangular; less than 1% dark grayish brown (10 YR 4/2) fincs, which react weakly to HCl. In-situ density of the gravelly sand is about 1.99g/cm3 and moisture is 5 to 14%. Gravels are sand matrix-supported to clast-supported; clongated axis of several gravel particles is oriented in an east-west direction. Internal stratification is crude, defined by very low angle (about 2 degrees) planar beds with few planar crossbeds.

a Particle size defined according to ASTM D2487–83; cobbles are 75 to 300mm, gravel is from 4.75 to 75mm, sand is from 0.75 to 4.75mm, and silt and clay (fines) are <0.75mm (<200 mesh).b Unified Soil Classification System, ASTM D2487–83.c Color based on wet specimen and Munsell color chart; (huc value/chroma). Color of dry specimen is generally two value units higher.



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Fig. 4—Cross section of the lateral spread near the hay yard. Cone penetration sounding CPA, surface wave test SA A, and Becker penetration logs have been omitted for clarity.



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< previous page page_257 next page >Page 257Table II: Summary of Grain-Size, Penetration and Velocity Data for Unit CTestArea

MeanaGrain-Size

(mm)

PENETRATION DATAb VELOCITYDATA

ConeHole

Averageqc

(MPa)

DrillHole

AverageN60

(blows/ft)

TestArray

AverageVs

(m/sec)

SPT Auger Bulk

1 4.4 5.4 11.3 CP1 5.6 SP1 8 SA1 103

CPA 6.2 SPA 7

BP1 5

2 6.0 9 15.5 CP2 6.4 SP2 6 SA2 92

CPB 5.4 SPB 7 SAA 132

SPD 9

BP2 5

3 4.8 7.1 CP3 6.6 SPC 9 SA3 97

CPC 6.6 BP3 5

aSPT—5-cm (2-in.) I.D. splitspoon, Auger—127-mm I.D. auger, Bulk—test pit sampleb Tests performed in 1985: CP1, CP2, CP3 with 15-cm2 cone and SP1, SP2 with safety hammer; tests performed in 1990: CPA, CPB, CPC with 10-cm2cone; SPA, SPB, SPC, SPD with “pin hammer”; average based on between water table and unit D. CPT tip resistance, qc; 1 ton/ft2=96 kPa. Corrected N-value based on procedures of Seed et al. [10]; energy ratio approximately 60%, based on SPT load cell measurements made in 1990, x 1.0; splitspoon barrel without liner, x 1.0 (loose sand) and x 1.15 (medium dense sand); short rods, x 0.75 (testing depths < 3m). BP N-values are equivalent SPT N60-values based on BPT and procedures by Harder [6]ANALYSISAt present, there are no generally accepted procedures for evaluating the liquefaction susceptibility of soils containing gravel. Therefore, simplified procedures developed for clean sands were initially used and evaluated.Stress-Based ApproachThe most widely used approach for assessing liquefaction susceptibility of sands is the simplified procedure developed by Seed and his colleagues based upon SPT and CPT test results (Seed et al. [11]; Seed and de Alba [13]). To use this method, the in situ cyclic stress ratio and the modified penetration resistance are calculated. The cyclic stress ratio is calculated using the following expression:

τav/σ′o=0.65 (amax/g) (σo/σ′o)rd (2)in which amax=maximum ground acceleration, σo=total overburden pressure,

, and rd=a stress reduction coefficient. Based on several predictive approaches for stiff soil sites, amax was estimated to be 0.35g. Analytical studies suggest (Bierschwale and Stokoe [3]) that amax on top of sites which liquefy is less than on top of stiff sites, therefore, 0.3g was used in the analysis. Overburden pressures were estimated from the densities given in Table I. Stress reduction factors, rd, estimated at the center of unit C range from 0.99 and 0.97.



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< previous page page_258 next page >Page 258The SPT has not been recommended for liquefaction assessment in gravelly soils (National Research Council, [7]; Finn, [5]). According to the National Research Council (p. 104):“It is not possible to evaluate the liquefaction susceptibility of [soils containing gravel] using the SPT; the presence of small quantity of gravel can increase greatly the penetration resistance without having much influence upon the susceptibility to liquefaction.”Based on SPT and BPT data at both sand and gravel sites, Harder [6] suggested that gravel will increase the SPT N-value by 1.6 times for loose to medium dense gravelly soils. A similar difference exists between the SPT N-values and equivalent Becker N-values presented in Table II. However, Finn [5] has pointed out that the correlation between SPT and BPT may be dependent on grain-size, just as the correlation between SPT and CPT is dependent on mean grain-size for sand (Seed and de Alba [13] and Robertson et al. [8]) and for gravelly materials as shown in Fig. 6. The low blowcounts in unit C suggest a lack of significant influence of gravel particles and could not have been increased much due to gravel content. In addition, no abrupt irregularities occur in plots of penetration verses number of blows, as shown in Fig. 7. Based on these findings and without established guidelines to correct for gravel, the influence of gravel on the SPT was ignored. The simplified procedure of Seed et al. [11] for sands was used directly to modify the SPT N-value (see footnote of Table II). Any effect of gravel on the normalizing overburden coefficient, Cn, and the cyclic stress ratio was also ignored. Cyclic stress ratios are plotted versus the average modified SPT and BPT penetration resistance in Fig. 8 for each test area. Also shown is the liquefaction potential curve for sands containing less than 5 percent fines. By applying the criteria of Seed and his colleagues directly, unit C is predicted to liquefy and has significant shear deformation potential.Liquefaction assessment charts based on modified cone tip resistance, qc1, can be generated from the assessment chart based on (N1)60 (Fig. 8) and the relationship between qc/N60 and mean grain-size (Fig. 6). Seed and de Alba [13] proposed the susceptibility boundary shown in Fig. 9 for sands with a mean grain-diameter of 0.8mm. Also plotted in Fig. 9 are cyclic stress ratios versus the average normalized tip resistance, qc1, for unit C. By applying the criteria of Seed and de Alba directly and using the boundary for mean grain-size of 0.8mm, unit C is predicted to have a marginal liquefaction potential. However, a possible susceptibility boundariy could be constructed for gravelly soil by extending the relationship between qc/N60 and mean grain-size into the gravel range. Such an extension was proposed by Andrus and Youd [1] and later modified by Stokoe et al. [14] using 15-cm2 cone data and borehole samples. Subsequent testing with a 10-cm2 cone has shown little variance in tip resistance with the 15-cm2 cone soundings (see Table II). Further sampling in test pits has shown that the sediment beneath Pence Ranch is somewhat coarser than was previously assumed based on borehole samples. The relationship of Seed and de Alba has been extended into the gravel range through these coarser samples, as shown in Fig. 6. With this extended relationship and neglecting any effect of gravel on the assessment chart in Fig. 8, the susceptibility boundary for a mean grain-size of 12mm has been added to the assessment chart of Fig. 9. Based on this boundary for mean grain-size of 12mm, an assessment similar to the assessment based on N-value can be made, where unit C is predicted to



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Fig. 5—Grain-size distribution curves of test-pit samples taken from unit C and a gravelly sand boil sample collected near the hay yard fence.

Fig. 6—Relationship between qc/N60 and mean grain size with data from Pence Ranch and Whiskey Springs (modified from Stokoe et al.[13]; after Andrus and Youd [1]; and Seed and de Alba [13]).

Fig. 7—Plots of penetration per blow for a few SPT tests conducted in unit C.


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Fig. 8—Liquefaction assessment chart based on modified N-value (Seed et al. [11]) with SPT results from Pence Ranch.

Fig. 9—Liquefaction assessment chart based on Normalized CPT (Seed and de Alba [13]) with results from Pence Ranch.


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Fig. 10—Proposed Liquefaction assessment chart based on normalized shear wave velocity (Finn [5]) with results from Pence Ranch.

Fig. 11—Liquefaction assessment chart based on shear wave velocity (Stokoe et al. [17]) with results from Pence Ranch.



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< previous page page_261 next page >Page 261liquefy. Additional data from other gravelly soils which have and have not liquefied are still needed to verify this procedure.A third stress-based approach for liquefaction assessment has been proposed based on normalized shear wave velocity (Robertson [9]; Finn [5]). The normalized shear wave velocity, Vs1, is calculated as follows:

Vs1=Vs(Pa/σ′o)0.25 (3)were Pa=reference stress, typically 100 kPa and σ′o=effective overburden stress in same units as Pa. Normalized shear wave velocity data from unit C are plotted on the proposed assessment chart in Fig. 10. The plotted velocity data from unit C lie within the liquefiable region, and a high liquefaction susceptibility is correctly predicted.Strain-Based ApproachAnother method of evaluating the liquefaction potential of sands is based upon analytical studies by Bierschwale and Stokoe [3] and Stokoe et al. [17] using measured shear wave velocity and maximum ground acceleration. This method has evolved from the strain approach proposed by Dobry and his colleagues [4]. The liquefaction potential is estimated using Fig. 11 and by plotting the shear wave velocity versus the maximum ground acceleration estimated for a stiff site at the candidate-site location. The lowest value of shear wave velocity from within unit C are shown in Fig. 11. Unit C lies within the zone where liquefaction is predicted to occur which agrees with the field performance.CONCLUSIONSLiquefaction at Pence Ranch occurred in loose gravelly soils containing only a few percent fines. The degree of pore water pressure generation appears to have been controlled by a thick soil cap of low permeability that lies just above the loose gravel. Four simplified liquefaction assessment methods developed for sands based on SPT, CPT, and Vs were applied directly to the field data. Assessment methods based on SPT and Vs measurements correctly predicted observed field behavior. On the other hand, the method based on the CPT predicted marginal liquefaction susceptibility. By using SPT, CPT and test pit samples, the relationship between the ratio of qc1 to N60 and mean grain-size was extended. This extended relationship was then used to generated a possible susceptibility boundary for gravelly soil. Based on this new boundary, liquefaction behavior was also correctly predicted using the CPT method.ACKNOWLEDGEMENTSThe authors are grateful to the U.S. Geological Survey for supporting this work. Professor T. Leslie Youd of Brigham Young University graciously provided information about the performance of Pence Ranch during the 1983 earthquake. Glenn J. Rix, Ignacio Sanchez-Salinero, Jiun-Chyuan Sheu, Young-Jin Mok, James A. Bay, Dong-Soo Kim, and Byungsik Lee assisted with field testing.



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< previous page page_262 next page >Page 262REFERENCES1. Andrus, R.D. and Youd, T.L. Subsurface Investigation of a Liquefaction-Induce Lateral Spread, Thousand Springs Valley, Idaho. U.S. Army Corps of Engineers, Geotech. Laboratory Misc. Paper GL-87–8, 1987.2. Andrus, R.D. and Youd, T.L. Penetration Tests in Liquefiable Gravels, in Proceedings of the 12th Int. Conference on Soil Mechanics and Foundation Eng., Rio de Janeiro, Brazil, pp. 679–682, 19893. Bierschwale, J.G., and Stokoe, K.H., II. Analytical Evaluation of Liquefaction Potential of Sands Subjected to the 1981 Westmorland Earthquake. Geotech. Eng. Rpt. GR84–15, Univ. of TX at Austin, 1984.4. Dobry, R., Ladd, R.S., Yokel, F.Y., Chung, R.M., and Powell, D. Prediction of Pore Water Pressure Buildup and Liquefaction of Sands During Earthquakes by the Cyclic Strain Method, N.B.S. Bldg. Science Series 138, U.S. Dept. of Commerce.5. Finn, W.D.L. Assessment of Liquefaction Potential and Post-Liquefaction Behavior of Earth Structures: Developments 1981–1991, in Proceedings of the 2nd Int. Conference on Recent Advances in Geotech. Earthquake Eng. and Soil Dyn. (Ed. Prakash, S.), St. Louis, MI, Vol II, pp. 1833–1850, 1991.6. Harder, L.F. Use of Penetration Tests to Determine the Liquefaction Potential of Soils During Earthquake Shaking. Ph.D. Dissertation, Univ. of CA, Berkeley, 1988.7. National Research Council. Liquefaction of Soils During Earthquakes, National Academy Press, Washington, D.C., 1985.8. Robertson, P.K., Campanella, R.G., and Wightman, A. SPT-CPT Correlations, ASCE J. Geotech. Div., Vol. 109, No. 3, pp. 1449–1459, 1985.9. Robertson, P.K. Seismic Cone Penetration Testing for Evaluating Liquefaction Potential. Proceedings, Symposium on Recent Advances in Earthquake Design Using Laboratory and In Situ Tests, ConeTec Investigations Ltd., Burnaby, B.C., 1990.10. Roesset, J.M., Chang, D.W., and Stokoe, K.H., II. Comparison of 2-D and 3-D Models of Analysis of Surface Wave Tests, Proceedings of Fifth Int. Conference on Soil Dyn. and Earthquake Eng., Karlsruhe, Germany, 1991 (in these proceedings).11. Seed, H.B., Tokimatsu, K., Harder, L.F., and Chung, R.M. Influencc of SPT Procedures in Soil Liquefaction Resistance Evaluations. ASCE J.Geotcch.Div., Vol. 111, No. 12, pp. 1425–1445, 1985.12. Seed, H.B., Wong, R.T., Idriss, I.M., and Tokimatsu, K. Moduli and Damping Fators for Dynamic Analyses of Cohesionless Soils, J. Geotech. Eng. Div., ASCE, Vol. 112, No. 11, pp. 1016–1032, 1985.13. Seed, H.B., and de Alba, P. Use of SPT and CPT Tests for Evaluating the Liquefaction Resistance of Sands. Use of In Situ Tests in Geotechnical Engineering, ASCE Geotech. Special Pub. No. 6, pp. 1249–1273, 1986.14. Stokoe, K.H., II, Andrus, R.D., Rix, G.J., Sanchez-Salinero, I., Sheu, J.C., and Mok, Y.J. Field Investigation of Gravelly Soils Which Did and Did Not Liquefy During the 1983 Borah Peak, ID, Earthquake. Geotech. Eng. Rpt. GR87–1, Univ. of TX, 1988.15. Stokoe, K.H., II, Nazarian, S., Rix, G.J., Sanchez-Salinero, I., Sheu, J.C., and Mok, Y.J. In Situ Testing of Hard-To-Sample Soils by Surface Wave Method, in Earthquake Eng. and Soil Dyn. II, ASCE Geotech. Special Publication. No. 20, pp. 264–278, 1988.16. Stokoe, K.H., II, Rix, G.J., Sanchez-Salinero, I., Andrus, R.D. and Mok, Y.J. Liquefaction of Gravelly Soils During the 1983 Borah Peak, Idaho Earthquake, Proceedings of Ninth World Conference on Earthquake Eng., Tokyo, Japan, III, pp. 183–188, 1989.17. Stokoe, K.H., II, Roesset, J.M., Bierschwale, J.G., and Aouad, M. Liquefaction Potential of Sands from Shear Wave Velocity, Proceedings of Ninth World Conference on Earthquake Eng., Tokyo, Japan, Vol. III, pp. 213–218, 1989.18. U.S. Bureau of Reclamation. Procedure for Determining Unit Weight of Soils In-Place by the Water Replacement Method in a Test Pit. Laboratory and Field Procedures for Soils Engineering, Geotech. Branch, USBR 7221–86, pp. 153–169, 1988.19. Youd, T.L., Harp, E.L., Keefer, D.K., and Wilson, R.C. The Borah Peak, Idaho Earthquake of October 28, 1983- Liquefaction, Earthquake Spectra, 2, No.4, 71–89, 1985.



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< previous page page_263 next page >Page 263Validation of Procedures for Analysis of Liquefaction of Sandy Soil DepositsJ.H.Prevost (*), C.M.Keane (*), N.Ohbo (**), K.Hayashi (**)(*) Department of Civil Engineering and Operations Research, Princeton University, Princeton, New Jersey 08540, U.S.A.(**) Civil Engineering Department, Kajima Institute of Construction Technology, 19–1 Tobitakyu 2-Chome, Chofu-Shi, Tokyo 182, JapanABSTRACTNumerous constitutive laws have been proposed by researchers for expressing the behavior of sand as an elastoplastic material in analyses of liquefaction which occur in sandy soil deposits during earthquakes. Sophisticated engineering judgment is often necessary in defining the required material constitutive parameters for such analyses. The one-dimensional liquefaction analysis program DYNA1D is based on the three-dimensional multi-surface plasticity theory and required material parameters can be obtained from standard soil tests.In this paper, the basic components of the analysis procedure are presented, together with its validations. Firstly, the results of liquefaction experiments using a shaking table and computed results obtained in the simulation of the liquefaction tests by DYNA1D are presented and discussed. Then, DYNA1D is used to simulate the Superstition Hills (California) earthquake of November 24, 1987. Computed and recorded excess pore-water pressure histories are compared and discussed in detail.INTRODUCTIONDYNA1D (Prevost [8]) is a finite element computer program for nonlinear seismic site response analysis. Dry and saturated, cohesive and cohesionless soil deposits can be analyzed. DYNA1D has been developed to allow site response analyses to be performed taking into account: (1) the nonlinear, anisotropic, and hysteretic stress strain behavior of the soil materials, and (2) the effects of the transient flow of the pore-water through the soil strata. The procedures used (field (Biot [3]) and constitutive equations (Prevost [7, 9, 10]) are general and applicable to multi-dimensional situations.



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< previous page page_264 next page >Page 264PLASTICITY MODEL FOR FRICTIONAL SOILSThe yield function is selected of the following form:

(1)where

(2a)s=σ−pδ deviatoric stress tensor (2b)

(2c)

(2d)

with c=cohesion; α=kinematic deviatoric tensor defining the coordinates of the yield surface center in deviatoric stress subspace; M=material parameter. The yield function plots as a conical yield surface in stress space with its apex located along the hydrostatic axis at the attraction. For cohesionless soils, a=0 and the apex of the cone is at the origin. Unless α=0, the axis of the cone does not coincide with the space

diagonal. The cross section of the yield surface by any deviatoric plane is circular with radius

. Its center does not generally coincide with the origin but is shifted by the amount . The is illustrated by Figure 1 in principal stress space. A non-associative flow rule is for the dilatational component, and a purely deviatoric kinematic hardening rule is adopted. In order to allow for the adjustment of the plastic hardening rule to any kind of experimental data, a collection of nested yield surface (Mroz [6]) is used. To avoid overlappings of the surfaces (which would lead to a non-unique definition of the constitutive theory), the direction of translation µ is selected such that:

(3)where M′ and α′ are the plastic parameters associated with the next outer surface (M‘>M). This is illustrated in Figure 2. Further details of the plasticity model can be found in Prevost [7, 9, 10].The constitutive parameters required to characterize the behavior of any given soil are determined by fitting the model to available experimental soil test data. The (hypo-)elastic shear G and bulk B moduli (low strain moduli) are best determined through seismic (wave velocity)-type measurements. Their dependence on the mean stress is empirical in nature, and is suggested by Richard et al. [12]. Correlation formula (relating moduli to initial void ratio, confining stress, overconsolidation ratio, etc.) based on the results of resonant column tests are also available (see, e.g., Hardin and Drnevich [4]). Typically, B=G[2(1+ν)/ 3(1–2ν)], where ν=Poisson’s ratio.All required plastic model parameters can be derived entirely from the results of conventional soil tests (e.g., “triaxial” or simple shear soil tests). In the absence of detailed laboratory soil test data, the plastic parameters may be generated by the procedure reported in Prevost and Keane [11].



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< previous page page_265 next page >Page 265SIMULATION OF LIQUEFACTION TESTS BY DYNA1DLiquefaction tests performed using a shaking table (base dimensions of 2.5m× 1.0m and height of 1.0m) were utilized to investigate the validity of DYNA1D in solving liquefaction problems observed in such tests. A sinusoidal wave (4 Hz, 200gal) input, in which the excess pore-water pressure ratio reached 1.0, was selected as the test case for the purpose of simulating the liquefaction phenomena by use of the one-dimensionsal liquefaction analysis program DYNA1D (Prevost [8]).The one-dimensional finite element mesh consists of 21 elements and 22 nodes. The mesh is fixed at the base, free at the top, and semi-infinite at the sides. Input to the model consists of the acceleration response wave observed at point A1. The required material constitutive parameters were determined from the methods listed in Table 1. The total number of yield surfaces was set equal to 20.Profiles of the excess pore-water pressure ratios shown at one second intervals for a particular cross-section are presented in Figure 3. The computed vertical distributions of the excess pore-water pressure ratios at 1 second intervals are presented in Figure 4. The excess pore-water pressure ratios are negative for the first 4 seconds of shaking, with the maximum negative value occurring at about 20cm in depth. Liquefaction has been nearly reached after 10 seconds of excitation from 40 to 60cm depth. The experimental and numerical liquefaction analysis procedures agree in the occurrence of the negative excess pore-water pressure, the rise of the excess pore-water pressure, and the vertical distribution of the maximum excess pore-water pressure.Figure 5 is a comparison of the excess pore-water pressure time histories at observation points W2 and W3 obtained in the experiment and in the analysis. The computed results compare quite well with the experimental results.ANALYSIS OF EARTHQUAKE DATA USING DYNA1DA special instrumentation array, hereafter referred to as the Wildlife Liquefaction Array, was installed in 1982 in Imperial County, California to monitor earthquake-induced pore-water pressures in a cohesionless soil deposit (Youd and Wieczorek [13]). The array consists of surface and downhole (depth=7.5m) accelerometers and six pore-water pressure transducers. The earthquake in Imperial County, CA, of November 24, 1987 included an event (MS=6.6) for which pore-water pressure increases accompanied by reduction of the vertical effective stress were recorded for this event. Pore-water pressure increased for 97 sec after recordings began, although most increases occurred during the first 40sec of the event.A geotechnical investigation of the site is contained in Bennett et al., [1], and a cross-section of the array site is shown in Figure 6 which includes shear-wave velocity values (Bierschwale [2]). Table 2 contains material parameters for each layer. The finite element mesh used to represent the semi-infinite soil deposit consists of 29 one-dimensional elements (30 nodes). Nodal spacing varies for each geologic layer with the largest spacing of 0.3m used for layer 5 (the bottom-most layer) and the smallest spacing of 0.175m used for layer 1 (the top-most



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< previous page page_266 next page >Page 266layer). Layers below the water table are assumed fully saturated, whereas layers above the water table are assumed dry.Two approaches are used to study the response of the soil column to base excitation. In the first analysis, 3 dimensional kinematics are used, and six degrees of freedom are assigned to each node at or below the water table; three for the solid phase and three for the fluid phase (for the N-S motion, vertical motion, and E-W motion, respectively). Nodes above the water table are assigned three degrees of freedom for the solid phase only. Using this approach, all downhole acceleration components are input at the same time and the response obtained in one set of computations.In the second analysis, 2 dimensional kinematics are used, and each node at or below the water table is assigned four degrees of freedom; two for the solid phase and two for the fluid phase (for horizontal motion and vertical motion). Nodes above the water table are assigned two degrees of freedom for the solid phase only. Downhole accelerations in each horizontal direction, N-S and E-W, are input individually along with the vertical component of downhole acceleration, and two separate computations are performed, referred to as 2D Analysis A and 2D Analysis B discussed hereafter.Input motions for the 3D kinematical analysis consist of the downhole acceleration components in the N-S, vertical, and E-W directions. Computed and recorded pore-water pressure time histories at the locations of the three porewater pressure transducers (3m, 5m, and 6.6m) are shown in Figures 7a, 7b, and 7c, respectively. Good agreement is found between computed and recorded pore-water pressure time histories. For example, all plots display the same starting time of pore-water pressure increase, similar rate of pore-water pressure increase once it has begun, similar times at which the rates of pore-water pressure increase begin to slow down, and good agreement in the final value of excess pore-water pressures.Input motions for the 2D kinematical analysis consist of the downhole acceleration components in the: (a) N-S and vertical directions for 2D Analysis A, and (b) E-W and vertical directions for 2D Analysis B. Computed and recorded plots of the pore-water pressure time histories at depths of 3m, 5m, and 6.6m as a result 2D Analysis A and 2D Analysis B are shown in Figures 8a, 8b, and 8c, respectively. It seems that the pore-water pressure increases are largely the result of the N-S acceleration component, which is not unexpected since it is a stronger signal than the E-W motion. The recorded and computed pore-water pressure time histories have the same similarities as in the 3D Analysis.It is clear in this case that the 3D Analysis provides marginally better improvements than either 2D Analysis. This point is further emphasized by the vertical effective stress variations with depth as a result of the 3D Analysis, 2D Analysis A, and 2D Analysis B (Figures 9a, 9b, and 9c, respectively). The stress variation with depth is displayed every 2 sec; the solid line on each plot represents the initial stress condition. Soil strata and location of recording devices are also shown. Note that only three pore-water pressure transducers are located within the soil deposit of interest. It is of interest to note the strong likeness between the results of the 3D Analysis (Figure 9a) and 2D Analysis A (Figure 9b). Clearly, the N-S component dominates the response.



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< previous page page_267 next page >Page 267CONCLUSIONSThe validity of the liquefaction analysis program DYNA1D has been studied for two types of data: (1) data obtained from shaking table tests, and (2) data recorded during the Superstition Hills earthquake of November 24, 1987.As a result, it was confirmed that DYNA1D is capable of closely simulating the details of an experimental shaking table liquefaction test, such as: the early occurrence of negative excess pore-water pressure, the vertical distribution of maximum excess pore-water pressures, the trends in vertical distribution of the maximum shear strains, and the time history response of excess pore-water pressures.The records for the MS=6.6 event of the Superstition Hills earthquake of Nov. 1987 have been used to compute the response of a soil deposit as observed in-situ using a nonlinear one-dimensional analysis (i.e., DYNA1D) of the semi-infinite soil deposit. Comparisons of pore-water pressure variations have been discussed in detail and found to be in good agreement. Also, the results of the 3-dimensional kinematical analysis and the 2-dimensional kinematical analysis are compared and discussed.ACKNOWLEDGEMENTSThis research was supported in part by the National Science Foundation under Grant ECE 85–12311 via sub-contracts under the auspices of the National Center for Earthquake Engineering Research, and by Kajima Corporation (Japan) as part of a collaborative research program in Earthquake Engineering with Princeton University. These supports are gratefully acknowledged.REFERENCES1. BENNETT, M.J., MCLAUGHLIN, P.V., SARMIENTO, J.S., AND YOUD, T.L., “Geotechnical investigation of liquefaction sites, Imperial Valley, California,” U.S. Geological Survey Open-File Report 84–252, 1984, 103 p.2. BIERSCHWALE, J.G.,, “Analytical evaluation of liquefaction potential of sands subjected to the 1981 Westmorland earthquake,” University of Texas Geotechnical Engineering Report GR-84–15, 1984,231 p.3. BIOT, M.A., “Mechanics of Deformation and Acoustic Propagation in Porous Media,” J. App. Phys., Vol. 33, No. 4, 1962, pp. 1482–1498.4. HARDIN, B.O. AND DRNEVICH, V.P., “Shear Modulus and Damping in Soils: Design Equations and Curves,” J. Soil Mech. Found, Div., ASCE, Vol. 98, No. SM7, 1972, pp. 667–692.5. HOLZER, T.L., YOUD, T.L., AND BENNETT, M.J., “In situ mea-surement of pore pressure build-up during liquefaction,” Presented at 20th Joint Meeting of United States-Japan Panel on Wind and Seismic Effects, Gaithersburg, MD, 1988.6. MROZ, Z., “On the Description of Anisotropic Work-Hardening,”. Mech. Phys. Solids, Vol. 15, 1967, pp. 163–175.



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< previous page page_268 next page >Page 2687. PREVOST, J.H., “A Simple Plasticity Theory for Frictional Cohesionless Soils,” Soil Dynamics and Earthquake Engineering, Vol. 4, No. 1, 1985, pp. 9–17.8. PREVOST, J.H., “DYNA1D: A Computer Program for NonlinearSeismic Site Response Analysis,” Report No. NCEER-89–0025, Dept. of Civil Eng. and Operations Research, Princeton University, 1988.9. PREVOST, J.H., “Mathematical Modeling of Monotonic and Cyclic Undrained Clay Behavior,” Int. J. Num. Meth. Geom., Vol. 1, No. 2, 1977, pp. 195–216.10. PREVOST, J.H., “Two-Surface vs. Multi-Surface Plasticity Theories,” Int. J. Num. Meth. Geom., Vol. 6, 1982, pp. 323–338.11. PREVOST, J.H. AND KEANE, C.M., “Shear Stress-Strain Curve Generation from Simple Material Parameters,” J. Geotech. Engrg. Div., ASCE, Vol. 116, No. 8, 1990, 1255–1263.12. RICHARD, R.E., WOODS, R.D. AND HALL, J.R., Vibrations of Soils and Foundations, Prentice-hall, N.J., 1970.13. YOUD, T.L., AND WIECZOREK, G.F., “Liquefaction during the 1981 and previous earthquakes near Westmorland, California,” U.S. Geological Survey Open-File Report 84–680, 1984, 36 p.



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< previous page page_269 next page >Page 269Table 1: Input dataEquations in DYNA1D Input Data Value Test Method

2-PhaseEffectiveStressAnalysis

•Specific Gravity of Soil ρs •Specific Gravity of Water pw •Bulk Modulus of Water Bw• Porosi ty n w •Coefficient of Permeability k

2.71.02.0×109N/m238 %0.01cm/sec

Specific Gravity testKnown QuantityKnown QuantityMoisture Content TestPermiablity Test

Multi-Surface Theory •Shear Modulus of Soil Gs•Bulk Modulus of Soil Bs•Initial Effective Stress pθ•Number of Yield Surface•Cohesion C•Friction Angle ø•Max. Com. Shear Strain •Max. Ext. Shear Strain •Dilatancy Parameter χp

1.5×107 N/m23.3×107 N/m21.0×104 N/m2200 N/m238°%0.02

PS-LoggingTriaxial Test (Drained)Known QuantityAssumptionTriaxial Test (Drained)″″″″



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Figure 1: Yield surface in principal stress space

Figure 2: Yield surface translation by the stress point in deviatoric stress space



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Figure 3: Recorded excess pore-water pressure ratio distribution

Figure 4: Computed excess pore-water pressure ratio distribution


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Figure 5: Comparison between the recorded and computed results



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Figure 6: Profile view of layered soil deposit at Wildlife site



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< previous page page_273 next page >Page 273Table 2: Material parameters for layered soil deposit at Wildlife site

Layer 1 2 3 4 5

Depth (m) 0.0 to 1.2 1.2 to 2.5 2.5 to 3.5 3.5 to 6.8 6.8 to 7.5

Shear WaveVetocity(1)

(m/s)

99.0 99.0 116.0 116.0 130.0

TotalDensity(2)

(kg/m3)

1600.0 1940.0 1970.0 1970.0 2000.0

Shear Modulus(solid)(N/m2)

1.57×107 1.47×107 2.08×107 2.08×107 2.70×107

Poisson’s Ratio 0.25 0.25 0.30 0.30 0.30

BuIk Modulus(solid)(N/m2)

2.61×107 2.44×107 4.50×107 4.50×107 5.83×107

Void Ratio(2) 0.6799 0.7955 0.7400 0.7400 0.6878

Porosity 0.4047 0.4431 0.4253 0.4253 0.4075

Reference MeanNormal Stress

(N/m2)

1.15×104 2.95×104 4.10×104 6.10×104 8.00×104

Friction Angle(3) 21.3º 20.0° 22.0º 22.0° 35.0°

Dilation Angle 21.3º 20.0º 19.0º 18.0º 5.0°

Coefficient ofPermeability

(m/sec)

— 1.0×10−5 1.0×10−5 1.0×10−4 1.0×10−6

xp, Dilatancy parameter 0.0833 0.0833 0.0833 0.0833 0.0833

(1) From Bierschwale (1984).(2) From Holzer, et al.. (1988).(3) From Bennett, et al., (1984).Other parameter values assumed or computed for



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Figure 7: 3D Analysis—Pore-water pressure time histories (a) Depth=3m (b) Depth=5m (c) Depth=6.6m



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Figure 8: 2D Analysis—Pore-water pressure time histories (a) Depth=3m (b) Depth=5m (c)Depth=6.6m



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Figure 9: Vertical effective stress vs. depth (a) 3D Analysis (b) 2D Analysis A (c) 2D Analysis B



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< previous page page_277 next page >Page 277Liquefaction of Sands Under Undrained and Non-Undrained ConditionsJ.ChuDepartment of Civil and Maritime Engineering, University College, University of New South Wales, Canberra AustraliaABSTRACTIn the first part of the paper, liquefaction of granular soils under the undrained condition is studied. It is emphasized that the flow strength of liquefied soil may be influenced by the initial effective confining stress. Hence, the steady state line may not be unique for some soils, but depending on both the void ratio and the initial effective stress. To consider such influences, a new procedure for the interpretation of undrained test data for soil liquefaction is proposed. In the second part, a kind of liquefaction which occurs under non-undrained conditions is investigated. The experimental results show that under non-undrained conditions, liquefaction is also possible for dense granular soil. The implication of this kind of liquefaction is discussed.INTRODUCTIONLaboratory studies on liquefaction of soils are almost exclusively based on undrained test results conducted under either monotonic or cyclic loading conditions. A key concept in the interpretation of undrained test data for liquefaction is the so called steady flow state [3, 4], Based on this concept, procedures for evaluation of liquefaction of soil have been proposed [15].Despite of intensive studies over the last 20 years, a thorough understanding on the steady state behaviour of granular soil is still yet to be achieved. It has been pointed out by Seed [16] that the available experience seems to indicate that in many cases the existing procedure for the evaluation of liquefaction may lead to significantly higher values of



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< previous page page_278 next page >Page 278steady-state than those observed on actual liquefied sands in the field. Considering the above fact, it may be necessary to re-examine the basic assumptions of the steady-state method.Two major assumptions have been made in the steady state methods:1). a unique steady state line exists in the void ratio and effective stress space and this line is a function of the initial void ratio of the soil only [3];2). the undrained test condition inherently implies that the void ratio of a sand deposit, after it liquefies, is the same as that of soil before it liquefied.The first assumption was made by Castro [3] based on his experimental studies. However, Based on his undrained test results, Konrad [10, 11] recently pointed out that the steady state strength of a soil is influenced not only by the void ratio of the soil but also by the initial effective confining stress. Konrad’s finding is further supported by Chu [6] and Chu et al [9]. Considering the influence of the initial effective confining stress, the existing methods may no longer be applicable for interpreting the undrained test data for liquefaction of soils [9]. The adequacy of the second assumption has also been questioned. Seed [16] has indicated that “it is possible that there is a redistribution of water content in sand samples in the laboratory and in sand layers in the field.” Seed’s argument is supported by the results of shaking table test [16]. Consequently, it requires that liquefaction of soil needs also to be studied under the non-undrained, that is, the other than undrained, conditions.The objectives of this paper are to address the above issues. In the first part of the paper, a method which can consider the effective stress dependent behaviour of steady state is proposed to evaluate the liquefaction potential and the steady state strength. In the second part, a kind of liquefaction occurs under non-undrained conditions is studied. The discussion is confined to static liquefaction only. However, the approach presented in this paper can be extended to situations where liquefaction occurs under cyclic or dynamic loading conditions.LIQUEFACTION UNDER UNDRAINED CONDITIONStress dependency of steady stateWhen a soil sample liquefies in an undrained test, the volume of the sample, the shear stress, and the pore water pressure become constant while the axial strain keeps on increase as in a flow structure. Such a phenomenon has been referred to as steady state [3], Another flow phe-



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< previous page page_279 next page >Page 279nomenon in an undrained test is the so called limited flow state, in which the steady flow is only developed over a limited range of strain. Kramer and Seed [12] suggested that because large deformation may still induce potential damage to engineering structures, limited flow over a significant strain range should also be considered in liquefaction study.

A steady state or a limited flow state can be plotted on an plane as a point (e is the void ratio and

is the first invariant of effective stress). Such points can, in turn, form a steady state line [3]. The data published in the past appears to indicate that the steady state line for both steady flow and limited flow is unique and is entirely independent of the initial effective confining stress. However, there are also data published recently which clearly show that the initial effective confining stress does affect the steady flow behaviour. Such evidences have been given by Alarcon and Leonards [1] and Konrad [10]. In fact, even Castro’s own data also indicate that the initial effective confining stress affects the steady flow behaviour. Fig. 1 presents the results of two isotropic consolidated undrained tests for a loose sand with the same void ratio but different initial effective confining stresses conducted by Castro [3], It clearly shows that the two steady state points were not identical. The difference between the two points are so large that it cannot be explained as testing error only. The initial effective stress affects the flow strength of soil even within the same kind of flow behaviour. More evidences were given by Chu [6] and Chu et al [9]. These studies have indicated that for some soils the steady state line may not be unique.

Figure 1. Influence of Initial Stress on Flow Strength (Data from [3])



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< previous page page_280 next page >Page 280It has been reported [18] that for some granular soils, the critical state depends on the initial effective stress, especially at a low stress range. A number of studies have shown that the steady state line can be regarded as identical to the critical state line for granular soils [17]. Thus, the dependence of the flow strength on the initial effective stress may be related to the dependence of the critical state on the initial effective stress [9].InterpretationMost of the existing methods for evaluation of liquefaction potential are based on the assumption that the steady state line is a function of the void ratio only. In considering the scatter of some data used for obtaining a unique steady line, Konrad [10] argued that the steady state line may not be unique, but should be bounded by an upper bound and a lower bound. These two bounds have been denoted as the UF and LF lines in Konrad’s papers [10, 11]. However, the determination of the UF and LF lines is quite arbitrary and the lines so determined are lack of physical meanings, as discussed by Chu [5, 6]. Therefore, the real difference between Konrad’s and the previous methods is whether a unique

band or a unique line exists in the plane. The influence of the initial effective stress still cannot be considered properly [6].To overcome the shortcomings in the existing methods, a new procedure is proposed for interpreting the undrained test data for the study of static liquefaction of granular soils.The assumption that the steady state line is unique has offered a great convenience in studying the liquefaction behaviour of soils. Without this assumption, the analysis would inevitably become much more complicated. The procedure suggested in the following is intended to be applied for general situation.In undrained tests, the liquefied state is referred to as a steady flow state. A non-liquefied state may be referred to as non-flow state. A limit flow state may be regarded as a transition from the non-flow state to the flow state. The occurrence of flow behaviour under the undrained condition for a given soil depends on the void ratio e and the initial effective

confining stress . A relationship between the occurrence of flow and the initial conditions of tests (e and ) can be established experimentally. Given the initial conditions of a test, the flow behaviour of the soil under the undrained condition can be predicted by the established relationship. In the following, conditions for the occurrence of steady flow or limited flow behaviour and its strength characteristics will be discussed separately.



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Figure 2. Flow States versus Initial ConditionsLiquefaction potentialThe influence of void ratio on the occurrence of flow can be stated as that the looser the sample, the higher the tendency to flow [10] [12]. Supposedly four undrained tests for samples having four specially chosen void ratios are tested under

the same initial effective confining stress Schematically, the stress-strain curves of the four tests can be presented in Fig. 2(a) in which curves 1 and 2 represent non-flow and steady flow states. Correspondingly, their initial conditions can be shown in Fig. 2(b) as point 1 and point 2. For the other two curves, curve l represents the state where the limited flow just occurs and curve u the state where the steady flow just occurs. The initial conditions for curves l and u can also be shown in Fig. 2(b) as point l and point u. The region bounded by the two points represents transition states between the steady flow and non-flow states. The points l and u also define the two boundaries of the transition zone. They are called lower point and upper point respectively in the following discussion.



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Figure 3. Initial Conditions for Liquefaction

Figure 4. Critical Flow State and Steady Flow Lines


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Since the initial effective confining stress affects the flow behaviour of soil, corresponding to different there can be different u and l points. Schematically they are shown in Fig. 3 as U1 and L1, U2 and L2,…, Un and Ln. By joining all

these points, a upper curve and lower curve can be obtained. These form a band on plane, which can be called I-band. This band represents the transition state in which limited flow occurs. The band divides the plane into two zones, the non-flow zone and the flow zone as shown in Fig. 3. For an undrained test, if the initial condition falls in the flow zone, liquefaction will be manifested in an undrained test. Similarly, if the initial condition is in the non-flow zone, liquefaction will not occur.It is noted that the determination of the I-band does not require the data regarding the flow strength and whether the steady state can be uniquely determined or not. It is entirely dependent on the initial test conditions. Thus, it is a general and objective approach for the evaluation of the liquefaction potential.Critical flow statesTests under the initial conditions given by the upper points and lower points of the I-band correspondingly lead to critical steady states and critical limited states. It is the critical values which may have more engineering interest. These states

can also be plotted on the plane as points Su and Sl, as shown in Fig. 4. Su and Sl represent the critical steady flow state and the critical limited flow state respectively. All these points also form a band, called T-band. The T-band describes the strength characteristics during the transition from the non-flow state to the steady flow state.Flow strengthAs the steady state strength is different from the limited flow state strength for the same initial effective confining stress, the steady state strength and the limited flow strength need to be considered separately. Corresponding to one initial effective confining stress, samples liquefied at different void ratios will result in different steady state points and so will limited flow state points, as shown in Fig. 4. These points form a pair of steady state line and limited flow state line. They can be specially denoted as SS line and LS line. It needs to be noted that the SS line and the LS line are different from the UF line and the LF line defined by Konrad [10], although they are quite similar. The SS line and LS line defined here have distinct physical meanings. However, the UF line and LF line are determined arbitrarily and do not have physical meanings [8, 9]. Since the steady state and the limited flow state are influenced by the initial effective stress state, the SS line and LS line



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< previous page page_284 next page >Page 284so defined are not unique, but depends on the initial effective confining stress. Strictly speaking, undrained tests conducted under different initial effective confining stress will have different pairs of SS line and LS line, as schematically illustrated by Fig. 4. As the bounding points of F-band, points of Su and Sl form the starting points of SS line and LS line. These SS and LS lines represent the strength characteristics of soil in the steady flow and limited flow state under a certain initial effective confining stress. Different SS lines and LS lines reflect the influence of the initial effective confining stress on the steady flow and limited flow behaviour.The above procedure for determining the steady flow or limited flow state lines, although general and feasible, is too complicated to be used for a normal liquefaction analysis. It has been discussed in [9] that the influence of the initial effective confining stress is only considerable when the stress level is low. Furthermore, by examining the published data it is noted that in some cases the data for steady states generated from different initial effective confining stresses can still be fitted into one line within certain accuracy. Therefore, provided the steady state and the limited flow state are modelled separated, SS line or LS line can be approximately modelled as a unique line for some soils.

Figure 5. Interpretation of Undrained Tests for Dune Sand [10]It needs to be pointed out that the SS line and the LS line defined above may not be parallel to each other. However, there is no reason why they should be parallel to each other. Comparison of the proposed



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< previous page page_285 next page >Page 285procedure with that suggested by Been and Jefferies [2] and Konrad [10] is made in [9].ExampleAs an example to illustrate the application of the above proposed method, the test results on Dune sand [10] are interpreted in Fig. 5. By taking points L14 and L18 and L8 and L13 as two pairs of upper and lower points, the I-band and the T-band can be determined. Using the other data points, the SS and LS lines corresponding to

can also be determined in Fig. 5. This diagram provides a systematic presentation for both the prediction of the liquefaction potential by the initial conditions of an undrained test and strength characteristics at the steady or limited flow state.LIQUEFACTION UNDER NON-UNDRAINED CONDITIONThe need for studying liquefaction of soil under non-undrained conditions has been discussed in Section 1. A non-

undrained condition can be modelled as with representing the undrained condition.

Thus, the response of soil under the non-undrained conditions can be studied by controlling the ratio of . When

the deformation of the sample is in compression and water flows out of the sample. On the other hand,

when the sample will dilate and water will flow into the sample. To control a non-undrained condition, strain path testing technique developed by Chu and Lo [7] was adopted to restraint the deformation of the sample, if any,

along a prescribed constant path.Testing methodsSydney sand, a uniformly graded quartz sand with the average grain diameter of 0.3mm, was used for the study. The dry sand was re-constituted into saturated samples of 100-mm-diameter by 100-mm-height.The test arrangement is schematically shown in Fig. 6. An internal load cell was used for measuring the vertical force. Two submergible LVDTs which were mounted directly over the top platen were used for measuring the axial deformation. Free-ends with enlarged platens were used to effectively reduce the platen restraint. In a strain path test, both bedding and membrane penetration errors were present. Thus, the ‘liquid rubber technique’, as detailed by Lo et al [13], was adopted in all the tests to reduce these errors to an insignificant magnitudes. The elimination of the above errors is essential for a precise control of strain path.



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Figure 6. Test ArrangementIn a load controlled loading mode, the vertical force was control by a digital pressure/volume controller (DPVC) via a hydraulic bellofram actuator. Strain path control was achieved by using a second DPVC to control the volume change of the sample.The conducted test involves maintaining the vertical load to be constant and control the drainage condition. They were achieved by a micro computer controlled data-logger system. The axial load was maintained constant by continuously scanning the load cell. If there was any discrepancy, computer would instruct the DPVC #1 which connected to the actuator to regulate the pressure. The drainage condition was controlled by activating strain path control using DPVC #2. The technique of strain path testing has been detailed in Chu and Lo [7].During a strain path control, the volumetric change of the sample was controlled, thus a change in pore water pressure was resulted, which was registered by the DPVC #2. The cell pressure was maintained constant during a test. However, the change in pore water pressure led to a change in effective confining stress.



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The testing procedures are as follows. An isotropic consolidated sample was first sheared along a path to the required stress level (which was smaller than the failure stress level). The external loading (including both axial force and cell pressure) were then maintained constant and the volume change of the sample was restrained at a certain strain increment ratio in accordance with the increment in the axial deformation. Whether liquefaction would occur or not was observed in this stage.Testing resultsA group of tests were conducted. However, it is impossible to have a detailed discussion here due to the space limitation. Interested readers can refer to [8]. For illustrative purpose, only the tests results of tests #01 and #02 are presented in this paper. Tests #01 and #02 were two dense samples with void ratios of 0.598 and 0.60 respectively. The effective stress

paths followed by the two tests are presented in Fig. 7(a). In both tests, the samples were sheared along a path commencing from an isotropic effective stress state of 150kPa. In each test, when the prescribed stress level was reached, computer control was activated to maintain the external load at a constant level and to switch the drained

condition to strain path control defined by . The stress ratios at which strain path control took over were 3.5 and 3.0 for test #01 and #02 respectively. For both tests, a run-away increase in pore water pressure was manifested, thus leading to effective stress paths plummeting towards the failure surface. Both samples ‘crashed’ in a matter of seconds. The pore water pressure generation curve of test #01 is given in Fig. 7(b).Discussion of test resultsThe observed instability has the following two characteristics:1). it losses a large percentage of its shear resistance;2). it flows in a manner resembling a liquid. Evidently, it is a kind of liquefaction according to the definition of Sladen et al [17].Although both of this kind of liquefaction and the liquefaction studied in undrained tests exhibit a flow state, the failure behaviours of the two kinds of liquefaction are quite different. The liquefied soil in an ordinary undrained test shows a steady state. However, the sample liquefied in the above discussed test crashed suddenly with the pore water pressure increased to the cell pressure level and a steady state was not followed. Consequently, the existing methods developed for evaluation of the liquefied behaviour of soil, such as the steady state method [15] or the collapse surface approach [17] does not apply to this new kind of liquefaction.



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Figure 7(a). Effective Stress Paths for Tests #01&#02

Figure 7(b). Pore Water Pressure Generation in Test #01



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< previous page page_289 next page >Page 289There are a number of features in the liquefaction observed in the above tests. First, it occurred at a stress state well below failure. Second, the liquefaction occurred under a non-undrained condition. Lastly, the liquefaction occurred for dense, thus, dilative soil. The factors influencing this kind of liquefaction and the conditions for the initiation of liquefaction are discussed in detail in [8].One of the most important implications which can be drawn from the above experimental study is that liquefaction can also occur for dense (that is dilative) soil. It indicates that the volumetric change pattern can be a very important factor affecting the occurrence of liquefaction. It has been generally believed that “only soils that tend to decrease in volume during shear, i.e. contractive soils can suffer the necessary loss of shear resistance to result liquefaction” [15]. However, it should be borne in mind that the above knowledge is based on the studies conducted under undrained condition. The discovery that liquefaction can occur for dense sand has put new challenges to our existing understanding of liquefaction.SUMMARYIn this paper, the main assumptions inherent in the steady state method is examined. The influence of the initial effective confining stress on the steady flow behaviour of granular soils is discussed. The study shows that the steady state line for a soil at a given void ratio is not unique but depends on the initial effective confining stress.The existing methods for evaluation of the liquefaction potential, however, are not applicable to the above general situation because most of them rely on the assumption that there exists a unique steady state line.A new procedure for interpreting the undrained test data for static liquefaction is proposed. This new method can be used to identify the initial test conditions under which static liquefaction may potentially occur and to evaluate the strength at the steady state or limited flow state.The need for studying the liquefaction of soil under non-undrained conditions is addressed. A kind of liquefaction occurred under the volumetric change controlled condition is manifested experimentally. This kind of liquefaction can occur at a stress level well below failure and can even occur for dense sands. It indicates that the volumetric change condition can be an important factor influencing the liquefaction of soils.



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< previous page page_290 next page >Page 290REFERENCES1. Alarcon, A. and Leonards, G.A. Discussion on Liquefaction Evaluation Procedure, J. Geot. Eng., ASCE, Vol. 115, pp. 232–236, 1988.2. Been, K. and Jefferies, M.G. A State Parameter for Sands, Geotechnique, Vol. 35, pp. 99–112, 1985.3. Castro, G. Liquefaction of Sands, Soil Mechanics Series No. 81, Harvard Univ., Cambridge, Mass, 1969.4. Castro, G. and Poulos, S.J. Factors Affecting Liquefaction and Cyclic Mobility, J. Geot. Eng., ASCE, Vol. 103, pp. 501–505, 1977.5. Chu, J. Discussion on Minimum Undrained Strength of Two Sands, submitted to J. Geot. Eng., ASCE, 1990.6. Chu, J. Discussion on Minimum Undrained Strength versus Steady-State Strength of Sands, submitted to J. Geot. Eng., ASCE, 1990.7. Chu, J. and Lo, S-C.R. On the Implementation of Strain Path Testing, Proc. 10th European Conf. Soil Mech. Found. Eng., Italy, May, 1991.8. Chu, J., Lee, I.K, and Lo, S-C.R. A New Kind of Liquefaction Occurred for Dilative Granular Soils, Research Report, Dept. Civil and Maritime Eng., Univ. College, Univ. New South Wales, Australia, 1991.9. Chu, J., Lee, I.K., and Lo, S-C.R. Interpretation of Undrained Test Data for Liquefaction of Sands, Research Report, Dept. Civil and Maritime Eng., Univ. College, Univ. New South Wales, Australia, 1991.10. Konrad, J.-M. Minimum Undrained Strength of Two Sands, J. Geot. Eng., ASCE, Vol. 116, pp. 932–947, 1990.11. Konrad, J.-M. Minimum Undrained Strength versus Steady State Strength of Sands, J. Geot. Eng., ASCE, Vol. 116, pp. 948–963, 1990.12. Kramer, S.L. and Seed, H.B. Initiation of Soil Liquefaction under



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< previous page page_291 next page >Page 291 Static Loading Conditions, J. Geot. Eng., ASCE, Vol. 114, pp. 412–430, 1988.13. Lo, S-C.R., Chu, J., and Lee, I.K.A Technique for Reducing Membrane Penetration and Bedding Errors, Geot. Testing J., ASTM, Vol. 12, pp. 311–316, 1989.14. Poulos, S.J. The Steady State of Deformation, J. Geot. Eng., ASCE, Vol. 107, pp. 553–562, 1981.15. Poulos, S.J., Castro, G., and France, J. Liquefaction Evaluation Procedure, J. Geot. Eng., ASCE, Vol. 111, pp. 772–792, 1985.16. Seed, H.B. Design Problems in Soil Liquefaction, J. Geot. Eng., ASCE, Vol. 113, pp. 827–845, 1987.17. Sladen, J.A., D’Hollander, R.D., and Krahn, J. The Liquefaction of Sands, A Collapse Approach, Can. Geotech. J., Vol. 22, pp. 564–578, 1985. 18. Wu, W. Discussion on The Behaviour of Very Loose Sand in the Triaxial Compression Test, Can. Geotech. J., Vol. 27, 159–162, 1990.



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< previous page page_293 next page >Page 293The Characteristics of Liquefaction of Silt SoilH.-C.FeiDepartment of Geotechnical Engineering, Tongji University, Siping Road. 1239, Shanghai 200092, ChinaABSTRACTSilt soil is a kind of special soil in China. On July 28 1976, the city of Tianjin was damaged due to the liquefaction of silt soil caused by the Tangshan earthquake whose magnitude was 7.8. This paper presents the effect of fine clay particles on liquefaction potential, the residual strength of silt soil under liquefaction condition and the empirical relationship to predict the liquefaction potential of silt soil using the CPT test.INTRODUCTIONIn China, silt soil is defined as a soil whose fine particle (d<0.005mm) content is from 3 % to 15 % and whose plastic index Ip is less than or equal to 10.It has been demonstrated by earthquakes in China, in the recent years, that not only does the liquefaction of fine sands develop easily, but also some types of silt will liquefy under earthquake loads, e.g. the Tangshan earthquake in 1976 caused the liquefaction of silt soil in large areas in Tianjin city. The seismic intensity was at Tianjin. Based upon the results of the field investigation in seismic areas and the laboratory test, it can be said that the liquefaction potential of silt soil is strongly influenced by the fine particle contents, besides the geological and geographical conditions as well as the ground water table. Fig. 1 shows the statistic relationship between the fine particle contents Pc and liquefied events of silt soil in areas of different seismic intensities in Tangshan Earthquake.



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Fig. 1 Relationship Relationship between Fines Content and Seismic Intensity in Tangshan EarthquakeThis empirical relationship has been adopted by the new Chinese Building Aseismic Design Code published in 1989, see Table 1.Table 1. Boundary value of Pc versus I

Seismic Intensity Fine Particle Content

(I) (Pc %)

7° 10

8° 13

9° 16

10° 19 It can be seen clearly from Fig. 1 and Table 1 that the liquefaction resistance of silt soil increases with an increase in the fine particle content.SAMPLE MATERIAL AND EQUIPMENTSilt soil has a wide difference in grain size distribution. All silt samples taken from nature in Shanghai or remolded for this research project were limited in the distribution of grain sizes. The general distribution of grain size of silt is listed in Table 2.A dynamic triaxial device was employed to determine the liguefaction potential of silt soil. Samples were 8cm in height and 3.91cm in diameter.



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< previous page page_295 next page >Page 295Table 2

Grain size (mm) 0.05−0.005 <0.005

content (%) >65 3–5A new device—superimposed ring dynamic/static shear device (Fig. 2), manufactured by the Tongji University Factory was employed to measure the residual strength of silt soil under liquefaction condition. A soil sample 6cm in diameter and 2cm in thickness was tested.

Fig. 2 Sketch of the Dynamic—Static Shearing Device.THE EFFECT OF FINE PARTICLE CONTENTS ON THE DEVELOPMENT OF PORE PRESSURE IN SILT SOIL

The test results presenting the, relationship between the pore pressure ratio and cyclic number ratio are shown in Fig. 3, 4, 5, and 6, for 4 different samples of silt soil containing diffeient quantities of the fine particle Pc (Pc= 2.8, 7, 12, 15). It is obvious that the curve family in each figure can be divided into two groups. The slope of the first group of curves is steeper than that of the second group. The first group is obtained under high dynamic loads and a few cyclic

numbers, to reach . The curves in the second group have a different shape and the slope is more gentle in the middle section of the curve. They are obtained under a small dynamic load applied on the sample and with large cyclic load numbers. The shape of the curve in the second group is close to that of the empirical equation of the pore

pressure ratio, arcsin



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Fig. 3 The Curve of Versus N/NL




for sands suggested by Prof. B.H.Seed. However, the curves in the second group are insignificant for earthquake engineering applications, because the dynamic load applied on the sample is too small and the cyclic number too large (N=60–1422, cyclic load period being 1sec.). Therefore the first group of the curves is of interest . After a statistic analysis, the relationship between the pore pressure and the cyclic number for 4 different samples of silt is shown in Fig. 7.


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Fig. 7 The Curve of Versus N/NL for Pc=2.8, 7, 12, 15It can be seen from the figure that the higher the content of fine particle of silt , the greater is the cyclic number for the

pore pressure ratio . This means that it is very difficult for the pore pressure to build up in soil

with a, high content of fine particle and to reach under the same cyclic numbers see Fig. 8. In Fig. 7, the curves for the samples with the fine particle Pc=7 and Pc=12 almost meet. The



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< previous page page_297 next page >Page 297empirical relationship between the pore pressure ratio and the cyclic number ratio of these curves can be written as follows

(1)where NL—cyclic number when silt liquefiesTHE RESISTANCE OF LIQUEFACTION OF SILT SOIL

Usually, the pore pressure ratio is used as the criterion to evaluate the initial liquefaction of sands in the laboratory test. It can be used as a base for the evaluation of initial liquefaction of silts. Fig. 9 shows the relationship between the fine particle content Pc

Fig. 8 The Curve of the Maximum Versus Pc

Fig. 9 The Curve of Versus Pc For Nature & Remodelled Soil.

and the stress ratio when and N=20. The dashed line represents the test results of remodeled samples of silt and the solid line is for natural soil samples. For the remodelled sample the stress ratio decreases linearly with the increase of the fine particle content but the test results of natural samples tend to re-rise when Pc is larger than 10 %.

This shows that on the one hand the effect of structure strength on is not obvious when Pc <10 %, but on the other hand the resistance of liquefaction of silt increases very much when Pc>10 %. The question is whether the resistance of liquefaction goes down or not when the stress ratio of-silt decreases within the range



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< previous page page_298 next page >Page 298of 3 %—10 % of Pc. In order to clarify the fact, it is necessary to determine the residual strength of silt under the

condition .THE RESIDUAL STRENGTH OF SOIL UNDER THE LIQUEFACTION CONDITIONWhen the pore pressure u in silt samples set in the superimposed ring shear device reaches under the dynamic load, static shear will be applied to the samples under undrained condition. Some results are shown in Fig. 10 a and b. With the increase of the static shear strain, the pore pressure in the sample decreases and the shear stress of the sample increases steadily. This means that the shear dilation phenomenon occurs in

Fig. 10 Relationship between pore pressure or Shearing Stress and Displacementthe sample. When the pore pressure stops decreasing, the shear stress reaches its maximum value, and the soil sample is in a plastic state The maximum shear stress can be taken as the ultimate residual strength τm. It can be observed that the ultimate residual strength is controlled by pore pressure, and that the variation of pore pressure is influenced by Pc. If Pc is over 14 %, only a slight shear dilatancy occurs. It is very difficult for pore pressure to dissipate. Fig. 11 shows the


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relationship between the shearing stress and the ratio of pore pressure to overburden



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pressure . Three groups of data show a linear relation with the same slope. To avoid the

Fig. 11 Relationship between Shearing Stress and Pore Pressureeffect of the loss of pore pressure by shear dilatancy on residual strength, additional water head through a tube connected to the ring box is applied to the sample. The test result is shown in Fig. 12. Three curves meet together and form a line. This can be expressed by the following equation.

τ=τo+A(Uo−U) (2)

Fig. 12 Relationship between Shearing Stress and Ratio of Pore Pressure to Overburden Pressure



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< previous page page_300 next page >Page 300where:τ – the residual strength at a given u.

τo – initial residual strength at .

u – pore pressure in soil.

u – critical pore pressure, equal to .

A – slope of the curve.Eq. 2 provides a possibility to predict the residual strength under any pore pressure, if τo and A can be obtained in advance.The relationship between A and Pc is shown in Fig. 13, and can be expressed as

A=1.31 exp(1.61/Pc) (3)(Pc=3 %–15 %)

Fig. 13 Relationship between A and Fines ComentA plot of o versus Pc is given in Fig. 14. The empirical equation is as follows:

τo=22.4 exp(−4.42/Pc) (4)(Pc=3 %–14 %)

THE PREDICTION OF THE LIQUEFACTION POTENTIAL OF SILTS USING CPTBecause the saturated silt soil is one kind of soft soil, it is very difficult to determine the potential of liquefaction with S. P. T, brow count N63.5 (the footnote 63.5 is the hammer weight 63.5 kg of SPT). The higher the fine particle content , the smaller will be the SPT brow count N.As we know, the existence of the fine particle in



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Fig. 14 Relationship between Shearing Stress and Fines Contentsilts will increase the resistance of liquefaction, But, the small brow count of SPT will give a liquefaction conclusion. Therefore it is very convenient to determine the potential of liquefaction of silts by using CPT. The advantage is obvious: 1) The records of CPT can be used to classify the soil easily.2) The mechanical properties of the soil can be represented clearly with the cone resistance of CPT. The problem is, how to built a standard for the estimation of the liquefaction of silts with CPT. The simple practical solution is to make a empirical equation based on the statistic relationship between gc and N63.5. The empirical relation between gc and N63.5 is diffeient in diffeient areas. For Shanghai city, it can be given as

(5)Based on eq. 5, eq. 6 is employed to predicate the potential of liquefaction of silt in Shanghai,

gc′= gc [1+0.125(ds-3)–0.05(dw-2)−0.1(Pc-3)]

(6)

Where: gc′ the critical resistance of CPT to divide the liquefaction and unliquefaction in silts.



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< previous page page_302 next page >Page 302c – the fundamental resistance of CPT see Table 3.

ds – the depth of silt deposit.

dw – the ground water table.Table 3. The fundamental value of CPT and SPTSeismic Intensity 7° 8° 9°

SPT blow count (blow) 6 10 16

CPT resistance c (bar) 46 76 121CONCLUSIONThe fine particles in silt is an important factor influencing the potential of liquefaction of silt. Dynamic stress ratio for

always has a minimum value around Pc=10 %, and the residual strength of silts will increase with the increasing of Pc. The regularity of the variation of the residual strength with Pc is identical to that of the field investigation of the liquefaction events. It demonstrates that the resistance of liquefaction of silt will increase with the increasing of Pc.ACKNOWLEDGEThe authors wish to express their appreciation to the Wang Education Foundation for its support to enable Prof. Fei to participate in the SDEE’91 in Germany.REFERENCES1. Zhao-jie Shi (1982): The Characteristics of Liquefaction of Silt and the Prediction in the Field. Hydraulic Geology and Engineering Geology. No. 32. Zhao-jie Shi (1984): The Prediction of LIquefaction Potential of Saturated Silt Foundation. Earthquake Engineering and Engineering Vibration, Vol. 4. No. 3.3. Chinese Building Aseismic Design Code (1989).



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< previous page page_303 next page >Page 303Evaluation of Liquefaction SusceptibilityA.M.AnsalIstanbul Technical University, Civil Engineering Faculty, Department of Geotechnical Engineering, Ayazaga, Istanbul, TurkeyABSTRACTLiquefaction susceptibility of natural soil deposits composed of silty sands and sandy silts encountered in an area with high seismic activity have been evaluated. A parametric study was carried out based on semi-empirical procedures developed in terms of SPT blow counts and grain size distributions and cyclic simple shear tests were performed on undisturbed samples.INTRODUCTIONAn investigation was conducted to determine effects of local soil conditions and liquefaction potential for a factory site located in Western Anatolia. The seismicity of the region with respect to historical and instrumental records was evaluated adopting a conventional probability analysis. However, due to intrinsic differences between two types of data sets, an averaging procedure was implemented and variation of return periods as well as the probability of exceedence were calculated in terms of earthquake magnitudes.Taking into consideration geological, tectonic and seismological aspects of the region, and selecting a probable epicenter for a strong earthquake, the peak base rock acceleration at the site for different earthquake magnitudes were calculated with respect to return periods as given in Table 1.Table 1. Expected Earthquake Magnitude and Peak AccelerationReturn Period (years) 100 200 500 1000

Magnitude (M) 6.4 6.9 7.6 8.1

Peak Horz. Acc. (g) 0.12 0.17 0.26 0.35Due to regional similarities, the N-S component of 1971 Montenegro (Yugoslavia) earthquake recorded at Ulcinj, Albatros was selected as a



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< previous page page_304 next page >Page 304possible design earthquake and used in site response analyses to study the behavior and the effects of local soil layers under earthquake excitations.The liquefaction susceptibility of shallow sandy soil layers encountered along the soil profile were evaluated by carrying out a parametric study using seven different semi-empirical procedures developed based on SPT blow counts and grain size distributions. In addition cyclic simple shear tests were performed on undisturbed soil samples obtained by special sampler. Safety factors for liquefaction were determined with respect to earthquake induced average shear stresses along soil profiles calculated from site response analyses.SITE RESPONSE ANALYSISThe procedure developed by Schnabel, Seed and Lysmer [7] was used to estimate earthquake characteristics on ground surface for free field conditions, to evaluate effects of local soil stratification and to calculate variation of peak horizontal acceleration as well as maximum shear stress with depth for the selected boring profiles. Adopting the conservative estimate of return period of 1000 years, the selected earthquake time history record was scaled such that the peak horizontal acceleration is ap=0.35g. Site response analyses using this scaled record as the input motion on the base rock were carried out.Dynamic properties of soil layers in boring locations were estimated based on field SPT blow counts and classification tests carried out in the Soil Mechanics Laboratory of Istanbul Technical University. The shear wave velocity profiles for each boring location were determined utilizing the relationship proposed by Ohta and Goto [6]. The SPT blow counts were corrected according to the energy efficiency ratio generally valid for the testing systems and techniques used in the region, based on the suggestions of Skempton[9].The results of site response analyses carried out for six locations which were selected to represent the encountered variations in the soil profile, are shown on Figure 1 in terms of acceleration response spectra. As expected the calculated soil amplification and predominant soil periods are very dependent on the thickness and properties of soil layers. However, from an engineering perspective, it appears realistic to consider an outer envelope of the calculated acceleration spectra as the design earthquake spectrum for the site.The variation of maximum horizontal acceleration with depth were also determined and depending on the depth of bedrock and properties of soil layers a significant deamplification was observed at some locations. No amplification of peak acceleration was obtained in any of the response analyses performed for the investigated site.



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< previous page page_305 next page >Page 305This aspect of the problem appears to agree with the results reported in literature concerning deamplification of earthquake waves as they pass through soil layers. According to Kiremidjian and Shah [5] and Idriss and Seed [2] deamplification may become as high as 50%, similar to the analytical results obtained in the present study. This type of response is partly due to increased strain dependent damping and partly due to viscoplastic nature of soil behavior at large strain levels that may develop during strong earthquakes. Therefore it was considered appropriate to use a reduced peak acceleration value at the ground surface in the empirical methods to evaluate liquefaction potential.

Figure 1. Acceleration response spectra for representative soil profilesPARAMETRIC STUDYThe liquefaction susceptibility of sandy silt and silty sand layers encountered at the investigated site are evaluated based on methods proposed in the literature by:Method 1—Seed, Tokimatsu, Harder, and Chung [8]Method 2—Taiping, Chenchun, Lunian, and Hoishan [10]Method 3—Iwasaki, Tatsuoka, Tokida, and Yasuda [4]Method 4—Ishihara and Perlea [3]Method 5—Yokota [11]Method 6—Yuqing, Fang, Quingyu, and Guoxin [12]Method 7—Atkinson, Liam Finn, and Charlwood [1]



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< previous page page_306 next page >Page 306The safety factors against liquefaction susceptibility were evaluated using these 7 semi-empirical procedures. In the case of Method-3, three alternative formulations and in the case of Method-6, two alternative procedures were carried out. As a result, liquefaction susceptibility is evaluated based on 10 different semi-empirical methods for the total of 244 locations where sandy layers were encountered in boring profiles.One purpose of this study is to obtain a comprehensive picture about the liquefaction susceptibility of sandy soil layers, while the other purpose to conduct such a parametric study concerning 10 different methods proposed by different researchers, is to demonstrate the subjective nature of the liquefaction evaluation procedures. One of the major reasons for discrepancies among different methods is due to variations in the SPT testing procedures or more precisely due to deviations in the impact energy between the different SPT testing systems and techniques since the above cited semi-empirical liquefaction evaluation methods were developed in different countries where SPT testing procedures may differ significantly. In addition dissimilarities in data bases (site conditions and soil types) used to develop the empirical correlations play an important role in the divergence of the results obtained.Variation of calculated safety factors utilizing two of the above mentioned procedures for all boring profiles are given on Figure 2. The large scatter observed in the safety factors within each method as well as differences between two methods are clearly visible. Even though the presence of safety factors smaller than one indicate a liquefaction susceptibility, it is apparent that it may not be very dominant and wide spread.In order to give a more comprehensive picture, considering 244 locations, the number of locations where safety factor is larger than 1.0 with its percentage in respect to total number of locations and the average safety factors are given on Table 2 for all 10 semi-empirical procedures separately and for all of them together.As can be observed from this table, even the lowest average safety factor (0.936) does not indicate a high liquefaction susceptibility. Six procedures out of ten gave average safety factors larger than 1.0. It appears realistic at this stage to evaluate the results obtained from all methods together as given in the last row of Table 2. The average safety factor calculated using all the results from all methods is larger than 1.2 and the locations with safety factors larger than 1.0 is 57.7% with respect to the total number.



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Figure 2. Variation of safety factors based on (a) Method 1 (b) Method 3 and 3a


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< previous page page_308 next page >Page 308Table 2. Summary of the Results of Semi-Empirical MethodsMethod Number Number of Locations with Safety Factor>1 Average Safety Factor

Method-1 89 (36.5%) 0.971

Method-2 94 (38.5%) 1.062

Method-3a 242 (99.2%) 1.310

Method-3b 162 (66.4%) 1.146

Method-3c 97 (39.8%) 0.966

Method-4 205 (84.0%) 1.285

Method-5 64 (26.2%) 0.936

Method-6a 200 (82.0%) 2.294

Method-6b 160 (65.6%) 1.340

Method-7 95 (38.9%) 0.956

All Methods 1408 (57.7%) 1.227From an engineering point, this general statistical evaluation of all methods separately and together indicates relatively limited liquefaction susceptibility at the site during a strong earthquake. In addition presence of gravel, silt and clay particles and pockets in sandy soil layers as observed in samples obtained from the site will be another important factor decreasing the influence of liquefaction.LABORATORY CYCLIC SIMPLE SHEAR TESTSA more accurate and realistic approach to determine the liquefaction potential of saturated sand deposits is to conduct a set of cyclic tests, preferably, on undisturbed samples. For this purpose, sets of consolidated undrained, cyclic simple shear tests were performed on undisturbed soil samples obtained from the site. The samples were taken by a special sampler and were frozen before they were transported to the laboratory. The frozen samples were trimmed in the laboratory and placed in the simple shear test cell and allowed to melt under a relatively low confining pressure. After sufficient time, back pressure was applied to the samples to assure saturation and confining pressure was increased incrementally to a predetermined level which is slightly higher then the calculated in-situ effective stresses.The cyclic simple shear tests were carried out on samples with different grain size distributions and different percentage of fines. The grain size characteristics of sandy soil layers encountered at the site vary within a large range. At some locations the fines content may decrease below 5% where the sand samples can be classified as SW or SP and at some locations fines content may increase as high as 50% where the sand samples can be classified as SC or SM. An effort is made to conduct sets of tests on samples with



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< previous page page_309 next page >Page 309different fines content in order to have a general picture to evaluate the liquefaction susceptibility of the sandy soil layers encountered at the site.Initial liquefaction was observed in 4 sets of tests conducted on samples which had fines content varying between 11% to 29%. The results of these sets of tests are utilized to determine the liquefaction strength of the sandy soil samples as shown on Figure 3. As expected the increase in the fines content increased the liquefaction resistance. In 2 sets of tests pore pressure accumulation were limited and no liquefaction was observed even after large number of cycles or at large shear strain amplitudes. The main reason for this type of response is the presence of high percentage of fines in these samples. Taking into consideration the effects of testing technique and sample disturbance and in order to be on the safe side, it appears justifiable to use the lower bound curve given on Figure 3 in the calculation of the safety factors for liquefaction

Figure 3. Cyclic simple shear results for initial liquefactionLIQUEFACTION SUSCEPTIBILITYAt the north part of the site, the base rock is relatively shallow overlain by medium dense silty gravelly sand layers of approximate thickness of 10 to 12m. The SPT blow counts varies between 3 and 12 along the depth of the sand layers. The grain size analysis performed on the samples from the upper part of these sand layers indicate that the fines content are around 10% and the gravel content is around 20% while for the samples obtained from the lower part the fines content is around 15% and gravel content is around 10 %. The safety factors calculated from the empirical procedures are shown only with the symbols while the safety factors calculated based on simple shear tests and site response analysis are shown with a continuous line on Figure 4. As can be observed safety factors based on laboratory tests are larger than 1 for the whole depth of the sand layers.



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Figure 4. Safety factors based on all methods for the soil profile A1Around the south part of the site, the depth of the base rock increases from approximately 16m to 30m. The sand layers encountered in the soil profile are medium dense and contains high percentage of fines and some gravel at various depths. The SPT blow counts were between 3 and 14 along the top 15m. As can be observed from Figure 5, the safety factors based on laboratory tests are larger than 1 for the whole depth of the sand layers. However, the value of the calculated safety factor is around 1.05 indicating a medium liquefaction susceptibility between the depths of 6 to 8m.

Figure 5. Safety factors based on all methods for the soil profile A2



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< previous page page_311 next page >Page 311Similar results were observed for other boring locations and according to these analyses summarized above based on both semi-empirical procedures and laboratory test results, the liquefaction susceptibility of the sand layers at investigated site was considered to be marginal. Even though the laboratory determined strength values were reduced significantly (in some cases reduction is in the order of 100%), the calculated safety factors for all boring locations were all larger than 1.0 indicating a low or no liquefaction potential.CONCLUSIONSA detailed investigation was conducted to evaluate the seismicity of the region and to determine the effects of local soil conditions on the earthquake characteristics at the ground surface as well as the liquefaction potential for the site based on parametric and experimental studies. The liquefaction potential of the shallow sandy soil layers were studied in detail utilizing 7 (with their alternatives 10) different semi-empirical methods and based on laboratory cyclic simple shear tests conducted on undisturbed samples. The results obtained from semi-empirical procedures show a large scatter, however, the overall evaluation of these findings indicate only marginal liquefaction susceptibility. The more sophisticated evaluation based on cyclic simple shear tests and site response analysis supports this conclusion that the effect of liquefaction at the site would be negligible. This is mostly due to the relatively large percentage of fines and gravel present in these layers. In addition the presence of gravel pockets will lead to a faster dissipation of pore pressure preventing liquefaction. Considering the types of structures to be build at the site, it was decided that it would be adequate to use pile foundations, which was also necessary due to conventional bearing capacity and settlement problems, for major part of the factory to achieve the sufficient safety against marginal liquefaction susceptibility that may exist at some locations.REFERENCES1. Atkinson, G.M., Finn, L.W.D. and Charlwood, R.G. Simple Computation of Liquefaction Probability for Seismic Hazard Application, Earthquake Spectra, Vol. 1(1) pp.107–123,1984.2. Idriss, I.M. and Seed, H.B. An Analysis of Ground Motion During the 1957 San Francisco Earthquake, Bull. Seismological Soc. America, Vol. 58(6), pp. 2013–2032, 1969.3. Ishihara, K. and Perlea, V. Liquefaction-Associated Ground Damage During the Vrancea Earthquake of March 4, 1977, Soils and Foundations, Vol. 24(1), pp.99–112, 1984.



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< previous page page_312 next page >Page 3124. Iwasaki, T., Tatsuoka, F., Tokida, K., and Yasuda, S.A Practical Method for Assessing Soil Liquefaction Potential Based on Case Studies at Various Sites in Japan, pp. 885–896, Proc. 2nd Int. Con. on Microzonation for Safer Construction-Research and Application, San Francisco, 1978.5. Kiremidjian, A. and Shah, H.C. Probability Site-Dependent Response Spectra, ASCE, J. Struc.Div., Vol. 106(ST1), pp. 69–86, 1980.6. Ohta, Y. and Goto, N. Estimation of S-Wave Velocities in Terms of Characteristic Indices of Soil, Butsuri-Tanko, Vol. 29, No.4, pp. 34–41, 1976.7. Schnabel, P.B., Lysmer, J.,and Seed, H.B. Shake—A Computer Program for Earthquake Analysis of Horizontally Layered Sites, EERC Report No. 72–12, Un i.of California, Berkeley, 1972.8. Seed, H.B., Tokimatsu, K., Harder, L.F., and Chung, R.M. Influence of SPT Procedures in Soil Liquefaction Resistance Evaluations, ASCE, J Geotech. Engng. Div., Vol. 111 (GT12) pp. 1425–1445, 1985.9. Skempton, A.W. Standard penetration test procedure and the effects in sands of overburden pressure, relative density, particle size, ageing and overconsolidation, Geotechnique, Vol. 36, No.3, pp. 425–447, 1986.10. Taiping, Q., Chenchun,W., Lunian,W., and Hoishan, L. Liquefaction Risk Evaluation During Earthquakes, Vol. 1, pp. 445–454, Proc. Int.Con. on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, St. Louis, 1984.11. Yokota, K. Evaluation of Liquefaction Strength of Sandy Soils, Vol. 3, pp. 121–124, Proc. 7th WCEE, Istanbul, 1980.12. Yuqinq, W., Fang, L.,Quingyu, H., and Guoxin, L. Formulae for Predicting Liquefaction Potential of Clayey Silt as Derived from Statistical Method, Vol. 1, pp. 227–234, Proc. 7th WCEE, Istanbul, 1980.



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< previous page page_313 next page >Page 313Post Initial Liquefaction Behaviour of SoilsK.TalaganovInstitute of Earthquake Engineering and Engineering Seismology, University “Cyril and Methodius”, 91000 Skopje, YugoslaviaABSTRACTThe problem of liquefaction of cohesionless soils has been so far, generally investigated with the stress approach. Laboratory tests, which form the basis for this approach, enable investigation only until the occurrence of initial liquefaction. After that, the investigation is limited. However, the problem can be further investigated by the strain approach. The laboratory tests with the strain approach enable investigations in the post initial liquefaction conditions. Here, results of the dynamic laboratory tests of sands with application of cyclic shear strain during the whole process of liquefaction, including both pre initial and after initial phase, are presented. Based on the laboratory results, the constitutive laws for the pore water pressure buildup have been defined as well as the decrease of shear stress and transformation of non-linear stress-strain relationships. It has been concluded that the post initial liquefaction behaviour of soils is of extreme importance in the investigation of the liquefaction problem.INTRODUCTIONFor the assessment of the liquefaction potential of cohesionless water-saturated soils under the effect of strong earthquakes, methodologies have been developped, which are based on in-situ investigations of sites where liquefaction has occurred, as well as on laboratory tests and analytical studies. In these methodologies, the dynamic soil strength and

the dynamic excitation are usially expressed in terms of shear stress τ, or stress ratio (where is the initial stress) and the number of cycles N, producing initial liquefaction. The increase of pore pressure is also a function of Rτ and N. The details of the methodologies can be found in Seed [1], Ishihara [2] and [3] , and others.



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< previous page page_314 next page >Page 314To a relatively limited number are carried out investigations and methods are developed, in which the dynamic soil strength and the dynamic excitation are expressed in terms of shear strain γ. This approach is defined as shear strain approach compared to the previous defined as shear stress approach. The details can be found in Dobry [4], [3] and Talaganov [5].The methods for the analysis of the liquefaction potential are based, to a great extent, on the laboratory tests which enable investigation of the phenomenon in detail, taking into consideration more influential factors.MAIN CHARACTERISTICS OF THE LABORATORY TESTSThe laboratory tests for the definition of the conditions for occurrence of liquefaction in the soil samples are performed mainly by inducing cyclic excitation. The soil samples are water saturated and the water drainage is prevented. Cyclic Stress TestsIt is a general practice to use cyclic shear stress with amplitude τ for the cyclic excitation in the laboratory tests. As a

result of this, the pore pressure U increases and, parallelly, the effective initial stresses in the samples decrease. A characteristic of the phenomenon of the pore pressure increase is that it is cyclic as well with a frequency twice bigger than the excitation one. The pore pressure amplitudes are significantly increased with the increase of the relative densitty Dr of the tested samples.This kind of tests primarily enable definition of the conditions of the initial liquefaction occurrence through the relationships between τ and the number of cycles N1. The initial liquefaction is defined as the first occurrence of U with value equal to the initial effective stress. It causes first occurrence of , where is effective stress. Besides that it is also defined the number of cycles N, which causes a certain level of shear strain, for example 5%, 10% etc’.However, the difference between N1 and Nγ is very small. It is because of the application of cyclic stress with constant amplitudes during the whole test. The meaning of such an excitation for the behaviour of the material is different for the beginning and for the end of the test. At the beginning, the material normally bears the excitation. However, towards the end, close to the initial liquefaction occurrence, due to the increase of U and the parallel reduction of the strength of the material is significantly reduced. Under such conditions τ shows a very high excitation, close to the strength of the material. Due to this, very often, the total amount of the amplitude τ as an excitation practically cannot be applied. This causes abrupt occurrence of shear strain increase, which cannot be under a control. Thus, parallely or immediately after the occurrence of initial liquefaction a destruction of the sample takes place and the experiment has to be stopped.



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< previous page page_315 next page >Page 315Cyclic Strain TestsIn the laboratory tests it is rearly applied excitation with constant amplitudes of shear strain γ. Main characteristics of this kind of tests is the water saturated soil samples to be exposed to excitations with constant strain amplitudes γ. In this case, as well as a result of the dynamic excitation, the pore pressure U increases and is cyclic as well, with frequency mainly double than the excitation frequency. Parallel with the increase of U a reduction of takes place, which is accompanied with the phenomenon of softening of the material. Thus, the shear strain excitation with constant amplitudes γ causes a different level of shear stress in the material during the experiment. At the beginning, the shear stresses have the highest level, while during the experiment they decrease with the softening of the material. At the moment of occurrence of the initial liquefaction which in this case, as well, is defined as the first occurrence of

, i.e. , the equivalent τ is decreased and is proportional to the soil characteristics. Due to this, in the phase around the initial liquefaction, the application of excitation does not cause destruction of the soil so that the experiment may continue. Such a behaviour of the soil enables testing even under post initial liquefaction conditions, so that the investigation of the phenomenon can be fully completed.PERFORMED LABORATORY TESTS BY CYCLIC STRAINTestsAt the Dynamic soil testing laboratory of the Institute of Earthquake Engineering and Engineering Seismology in Skopje, Yugoslavia, laboratory tests have been carried out on soil samples taken from sites, Bijela and Baosic, where soil liquefaction occurred during the 1979 Montenegro earthquake. The grain size distribution curves of these sands are shown in Fig. 1. Reconstituted sand samples with relative densities Dr, including loose and dense sands, have been tested. Direct cyclic simple shear laboratory equipment (Dames&Moore type) was used, applying a specific procedure for dry samples and constant volume. The sand samples were excited by a series of cyclic strains, with an amplitude of γ , in accordance with the scheme shown in Fig.2. During the test performance, in addition to the strains, the decrease in the initial vertical stress as well as the shear stress τ were recorded. The decreases was taken to be equal to the

pore pressure increase in water saturated samples. The values of the applied initial were 100, 200 and 300 kN/M2. The τ—γ relationship and their transformation with the number of cycles N were also recorded.Increase in Pore Water PressireAs a result of the applied excitation with cyclic shear strains γ,



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Fig.1. Grain-size distribution curves for the tested sandsFig. 2. Stress-stram state of sand sampleappears a decrease of , i.e. a build-up of U. Typical result of it is shown in Fig. 3, where the time histories of γ and U are presented. It is characteristic that U has a cyclic shape and starts to increase immediately with the application of γ. The peak value

of U from different cycles gradually reaches the value of and then remains constant. The amplitudes of U

are the largest in the part upto occurrence of . Then, the amplitudes start to decrease with a tendency to disappear completely when the pore pressure U from a cyclic one transforms into a -constant phenomenon with a value

In Fig. 3 is also shown the time history of τ. It can be seen that the amplitudes of γ decrease permanently from the initial maximal value of τo. At the first appearance of the amplitudes, although reduced, have a certain finite nonzero level. Then they continue to drop with a tendency towards a complete reduction.Transformation of τ—γ RelationshipsThe decrease of τ can be presented clearly through the τ—γ relationships, which were permanently recorded during the tests. A typical result is shown in Fig. 4 which results from the same test shown in Fig. 3. It can be seen that the τ-γ relationships are in permanent transformation. In the starting cycles they have characteristics of softening curves, while in the firther cycles they are characterized by hardening curves. However, regardless of that, the extreme values of τ permanently



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Fig. 3. Typical results of laboratory testsdecrease. If the regular definition for a secant modulus Gs is taken to be a mean modulus which corresponds to the straight line (linear τ—γ relationship) passing through the extreme points of the nonlinear relationships τ—γ, it becomes easily evident that the moduli Gs permanently decrease. The secants through the extreme points, from cycle to cycle, “rotate” towards the horizontal γ—axis with a tendency to get a horizontal position. Then the area surrounded by τ—γ decreases from cycle to cycle and has a tendency of a complete reduction. The condition to which the τ—γ relationship transformations aim is a straight horizontal line, corresponding to a total reduction of the shear strength of the soil with development of a shear strain practically without resistance of the material. It is important to state that the tendency towards such transformation began from the very beginning of the excitation, only that it does not get practical

realisation with the first appearance of but in the phase after that moment.LiquefactionBased on the previously mentioned results from the tests several basic definitions for the soil behaviour can be applied.In the range upto the occurrence of the initial liquefaction, the state is preinitial liquefaction. In this phase, the peak

values of U from separate cycles gradually approach . The stresses τ permanently decrease, but their extreme values are still with finite nonzero values. The relationship τ—γ perma-



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Fig. 4. Typical τ-γ relationships from the performed tests

Fig. 5. Liquefaction definitions nently transform when both Gs and damping decrease but still keep finite nonzero values.

The condition of the initial liquefaction is defined at the moment of the first occurrence of . This happens in the cycle N1. However, this condition appears only as one moment during the whole process.The state after the occurrence of the initial liquefaction is postinitial liquefaction. In this phase, the peak values of U from separate cycles are permanently at the level of , while the minimal Umin is still increasing. The stresses τ continue to decrease with their extreme values tending to zero. The τ—γ relationships continue to transform tending towards a straight horizontal line.

A state of total liquefaction is defined when U occurres as noncyclic with constant value of , and when τ, Gs and the damping get zero values. During this state the τ—γ relationship appears as horizontal line. This happens in the cycle N11. Between N11 and N1 the difference can be a multiple depending on the relative density of the material Dr. Idealized state, very close to total liquefaction, is obtained during a number of experiments. These definitions are schematically presented in Fig.5.ANALYSIS OF THE TEST RESULTSStatistical processing. of a large number of test results was performed in order to define the basic analytical relationships for the main liquefaction parameters. As a first step, relationships between the strain level and the number of cycles producing the first occurrence of peak pore water pressure equal to the initial effective stresses were established. Then the relationship of peak pore water pressure increasing as a function of strain level, as well as the relationship of the decrease of initial shear stress as a result of the pore water pressure increase have been defined. The obtained relationships have been normalized by means of a number of cycles necessary for initial liquefaction occurrence, separately for each strain level. In this way general functions have been obtained.



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< previous page page_319 next page >Page 319Initial LiquefactionThe number of cycles N1 producing initial liquefaction for the investigated levels of shear strain amplitudes in the range

from 0,2% to 2% are shown in Fig. 6. In it are jointly shown the results for different Dr and . From the analysis of

the results it can be stated that the effects of both Dr and are not specifically emphasized. Instead of having a

tendency of separating the different relationships between γ and N1 related to the separate Dr and , there is a

tendency of forming a single relationship range between γ and N1 for all Dr and . Conclusions like these have also been drawn after the rare investigations where excitation with cyclic shear strain is applied as in [3], [4] .Having in mind the self-grouping of the results in a single range, a regression analysis is preformed to define the mean γ-N1 relationship. The following analytical expression, which is thought to make good consideration of the results, is applied:

Y=AeBlnX (1)where:

Y=γ in % X=N1

A, B=parameters The best solution has been obtained for the following values of A and B:

A=1, 9838 B =−0, 4126 (2)The relationship (1), for the values A and B according to (2) are also -shown in Fig. 6.

Fig.6. γ-N for initial liquefaction

Fig.7 Normalized U-N for initial liquefaction



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< previous page page_320 next page >Page 320Peak Water Pressur Increase

The increase of the peak pore water pressure U upto the level is the function of γ and N. To the different

values of U correspond equivalent relationships of the increase of U, so that series of relationships between and γ can be obtained. However, by analysis of that relationship series it has been stated that in case N is normalized by dividing it with N1 in the series of harmonized relationships appears a tendency to form a single relationships range, independent from γ . The results harmonized in this way are shown in Fig. 7.Having in mind the above, a regression analysis with the normalized results was performed, with the purpose to define a

mean relationship between and N/N1. For that purpose, the following analytical expression, which, as it has been stated, considers the results quite well, was applied:

Y=X(A+BX) / (C+DX) (3)

where: X=N/N1

A, B, C, D=parameters The best solution has been obtained for the following parameter values:

A=1,1218 B=0,1143 C=0,1400 D=1,2809 (4)The relationship (3) for the values A, B, C, and D of (4) is also shown in Fig. 7.

In the range after N1 the peak pore water pressure retains the value , so that for that range the equation (3) transforms imto:

(5)As it has been presented with the results in the previous chapter, the minimal values of the pore pressure also increase from cyrcle to cyrcle. However, the Umin in the monent N1 do not reach the value of . In the range after N1, Umin

continues to increase tending to reach the value in a certain cyrcle N11. The analysis of the results shows that, however, different from the increase of U, the increase of Umin is a function of Dr. In the range after the initial liquefaction, the increase of Umin and the decrease of τ and Gs are related and express the solid behaviour. For this purpose, the soil behaviour in the further procedure will be analysed through the decrease of τ. Shear Stress DecreaseIn all phases of the liquefaction process the shear stress decreases tending to total reduction in the moment N11. By



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< previous page page_321 next page >Page 321analysis of the results, series of relationships between τ and N was obtained. These relationships are functions of γ and Dr. The initial amplitudes of the shear stress, as the maximal ones, are defined with τo, while the amplitudes of the different cycles are defined with τ. The further analysis showed that if a normalization of τ is applied in a way that it will be divided by τo, and if N is normalized with N1, the relationships tend to form single ranges of relationships, which become independent from. One such normalized series of results is presented in Fig. 8. They show that in the N1 moment τ is not totally reduced and that it continues to decrease. In Fig. 8 is also shown the mean analytical relationship between τ/τo and N/N1, which was obtained by applying of the expression:

Y=1–X(A+BX)/(C+DX) (6)where: Y=τ/τoX=N/N1A, B, C, D=parametersThe best solution was obtained for the following values of the parameters:

A=1, 1209 B=0, 1137 C=0, 1384 D=1, 2753 (7)From the results shown in Fig. 8 it can be seen that the decrease of τ in the post initial liquefaction phase tends towards total reduction. This happens in the moment N11, which corresponds to a beginning of a complete liquefaction. The comparison of N1 with N11 shows that in this particular example N11=2 N1, i.e. that for occurrence of a complete liquefaction is needed a double number of cycles of excitation with γ, compare to the cycles which cause initial liquefaction.

Fig. 8. Typical τ/τo—N/N1 relationship



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< previous page page_322 next page >Page 322Based on the regression analyses of a great number of experimental results of samples with different Dr series of normalized function between τ/τo and N /N1 have been obtained. In Fig. 9 are shown the functions between τ/τo and N/N1 for characteristic Dr. If we analyse these functions it can be seen that N11 is a function of Dr and that it increases with the increasing of Dr. Compared to N1 the N11 is several times bigger. Due to all that in case of dense sands, for example, the post initial liquefaction phase is longer than the initial liquefaction phase.

Fig. 9If we consider the approach of N11 towards N1 by a decrease of Dr, it can be assumed that the very loose sands will tend towards leveling between N11 and N1, i.e. will tend towards occurrence of a complete liquefaction together with or immediately after the occurrence of the initial liquefaction. Due to the difficulties in building-up of extremely loose sands, during the laboratory tests the above was not experimentally proved.Opposite to that, having in mind the drifting apart of N11 and N1 by the increase of Dr, it can be assumed that the very dense sands will tend towards very high values of N11, which, practically, is impossible to reach. This means that the liquefaction ends with the cyclic mobility of the soil.MODELS FOR CONSTITUTIVE LAWSThe analytical expressions, defined as described in the previous capters, for the basic liquefaction parameters can be taken as basis for defining of the constitutive laws models for the stress-strain soil behaviour, which can be applied in all phases of the liquefaction process. In the further text is given the definition of one such model.As basis for modelling of the relationship τ—γ is used the Martin-Davidenko’s model [7] , which is given with:

(8)



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< previous page page_323 next page >Page 323where: the first equation is for loadingthe second equation is for unloading

Go=maximum shear modulus τ1, γ1, τ2−γ2=coordinates of the extreme points

H (γ)=characteristic function expressed by:

(9)where: γo, A, B=are parametersThe parameters γo, A, B are defined from the experimental results. Information regarding their values can be found in the research work of some authors, eg. [7], [8]For series of relationship τ—γ , which, from a semy-cycle n (n=N/2) change to a semi-cycle n+1 due to the increase of U, can be applied the relationships (8) where Go, H(γ), τ1, τ2, γ1 and γ2 appear as variables dependednt from the semi-cycle n. According to this concept the equation (8) can be expressed with:

For defining of the variable G in the equation (10) can be applied the relations between τ /τo and N/N1 from Fig. 9, defined with the equation (6), in case the following approximation is accepted:

τ/τo=G/Go (11)The presented model has been practically applied for analysis of the dynamic response of soils with occurred liquefaction during the earthquake in Montenegro, Yugoslavia in 1979. It was proved that it gives satisfactory results [6].CONCLUSIONSThe liquefaction process in. water-saturated sands under the effect of rather extensive dynamic excitations is developped in characteristic phases, which can be summarized as: pre-initial liquefaction, initial liquefaction, post-initial liquefaction and complete liquefaction. The post-initial liquefaction phase was not investigated in details by laboratory tests. However, that phase is of a great importance for a complete explanation of the liquefaction prenomenon. Thus, it is necessary to include in the laboratory tests this last phase as well, in order to complete the image of the liquefaction process. The laboratory tests with cyclic shear stran enable complete testing of this phase.



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< previous page page_324 next page >Page 324REFERENCES1. Seed, H.B. Soil Liquefaction and Cyclic Mobility Evaluation for Level Ground During Earthquakes. Journal of the Geotechnical Engineering Division, ASCE, Vol. 105, No. GT2 19792. Ishihara, K. Stability of Natural Deposits During Earthquakes. Proceedings of the Seventh International Conference on Soil Mechanics and Foundation Engineering, A.A. Balkema Publishers, Roterdam, 19853. Committee on Earthquake Engineering, Commission on Engineering and Technical Systems, National Research Council. Liquefaction of Soils During Earthquakes. National Academy Press, Washington D.C., 19854. Dobry, R., Ladd, R.S., Yokel, F.Y. and Chung, R.M. Prediction of Pore Water Pressure Build-up and Liquefaction of Sands During Earthquakes by the Cyclic Strain Method. NBS Building Science Series 138, Washington, 19825. Talaganov, K. Determination of Liquefaction Potential of Level Site by Cyclic Strain. Institute of earthquake Engineering and Engineering Seismology, Report IZIIS 86–129, Skopje, Yugoslavia. 19866. Talaganov, K. Geotechnical Aspects of Montenegro 1979 Earthquake. Earthquake Geotechnical Engineering, The Japonese Society of Coil Mechanics and Foundation Engineering, Universal Academy Press, Inc. Tokyo, 19897. Martin, P.P. and Seed, H.B. MASH a Computer Program for Non-Linear Analysis of Vertically Propagating Shear Waves in Horizontally Layered Deposits. EERC, Report No.UCB/EERC—78/23, Berkeley, 19788. Talaganov, K., Zafirova, I. and •ubrinovski, M., Non-Linear Soil Dynamic Models Based on Performed Laboratory Tests. European Conference on Structural Dynamics, Eurodyn ’90, Bochum Germany, June 5–7, 1990.



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< previous page page_325 next page >Page 325Liquefaction Associated with Manjil Earthquake of June 20 1990, IranS.M.HaeriCivil Engineering Department, K.N.TOOSSI University, Tehran, IranABSTRACTManjil earthquake of June 20 1990 caused extensive damage and loss of life throughout the relatively populated epicentral region mainly in the towns of Manjil and Rudbar and their suburbs. Considerable additional damage occurred further north and west in Gilan and Zanjan provinces; especially in Rasht and Astaneh. Liquef action occurred mostly in Astaneh and Rudbaneh some 75km northeast of epicenter. Soil liquefaction caused extensive damages to buildings, farms and lifelines in a vast area.A general view on the earthquake characteristics and the geotechnical aspects of this earthquake is reviewed. The extent of the liquefaction in addition to the general subsurface soil condition in affected area, as much as available, is described. Results of a preliminary subsurface soil investigation performed after the earthquake in Astaneh are presented and discussed. The reason why liquefaction did occur only in two sections of the city of Astaneh is clarified.INTRODUCTIONManjil earthquake of June 20 1990 caused extensive life lost and damage to buildings and substructures in a relatively populated region of provinces of Gilan and Zanjan. Geographic location of the region including the epicenter is shown in Fig. 1. Due to the extent of damage occuring in this area a number of researchers from all over the world visited the affected area. In this respect a geotechnical team leading by the author visited the site a few days after the earthquake to investigate the geotechnical aspects of this earthquake.



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< previous page page_326 next page >Page 326Geotechnical aspects of this strong earthquake is a complete set of all possible events associated with any strong ground shaking, i.e., liquefaction, landslide, rockfall, local site effects and soil amplification and foundation problems. Location of any of these aspects occurred during the Manjil earthquake is given in Fig. 2. A comprehensive report in this respect is given elsewhere [2].One of the most important geotechnical earthquake engineering considerations of this earthquake was liquefaction of level ground. Liquefaction of level ground has shown to be responsible for many damages incurred to structures and lifelines during moderate to strong earthquakes. Site and laboratory studies on the behavior of loose sands under dynamic loading in recent decades resulted in various theoretical and experimental expressions and design charts to evaluate the potentiality of the liquefaction [eg. 3, 5, 6, 7].To evaluate the reasons for earthquake induced liquefaction in Gilan and especially in Astaneh a careful surface study of the site was performed to map the zone of liquefaction. The program of a comprehensive study of this liquefaction was planned in two phases. The preliminary part includes subsurface investigation and in situ testing in Astaneh. This part of study has been completed and the results are presented in this paper. The continuation of the study involves both in situ and laboratory dynamic testing in all liquefaction affected area.MANJIL EARTHQUAKE OF JUNE 20 1990Just 30 minutes after midnight of June 21 1990, local time (21 hour June 20 GMT), a disastrous and destructive earthquake occurred in the provinces of Gilan and Zanjan, IRAN (Fig. 1). The magnitudes of Mb=7.3 and Ms=7.6 and the maximum intensity of X in modified Mercali scale are reported. The epicenter is reported to be about 200km northwest of Tehran, between Manjil and Rudbar and with the focal depth of about 10km. The approximate intensity distribution given by Building and Housing Research Center is shown in Fig. 3, [4] The rupture zone defined by site investigation is reported to be about 100km long with a major trend of northwest-southeast direction; the same as that of the major faults in the region. The main aftershocks defined the same trend as mentioned above for the rupture zone.



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Fig. 1 Geographical location of epicenter


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Fig. 2 Places of Geotechnical considerations with respect to Manjil earthquake of June 20 1990



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< previous page page_328 next page >Page 328Earthquake induced damage exrtended to many cities and villages in the provinces of Gilan an Zanjan covering and area of more than 10000km. The damage occurred in the city of Rasht some 60km north of epicenter is mainly due to the soil amplification resulting in destruction and sever damage to buildings taller than four stories. Liquefaction was solely responsible for the damage occurred in the town of Astaneh and nearby villages some 75km northeast of epicenter. Earthquake induced landslides ruined villages, farms and roads. Sefidrud Dam located at the vicinity of epicenter experienced only minor damages.A number of accelerographs installed in the region recorded the main shock. Unfortunately seismograph installed at Sefidrud Dam was out of order at the time of the ground shaking. The nearest instrument to the epicenter in the region is an accelerograph installed on a rock site at Abbar some 40km west of Manjil and 10km far from the extent of the major fault (Fig. 2). The maximum acceleration recorded by this accelerograph is 0.65g in horizontal direction and 0.23g in vertical direction [4]. The maximum horizontal acceleration recorded at Lahijan some 75 km far from epicenter, and some 10km southwest of Astaneh, was 0.17g. The latter accelerograph installed on alluvial plain.LIQUEFACTION INDUCED GROUND DAMAGELiquefaction caused extensive damage in a vast area of fluvial deposit of Sefidrud and i ts tributaries. The distance between observed liquefaction in this region and epicenter is between 50km and 90km (Fig. 2). The most extensive damage occurred in Astaneh and Rudbaneh some 75km northeast of epicenter. Study of the geological map of the region (Fig. 4) indicates that the liquefaction occurred mostly in levee deposits of the present and abandoned channels of Sefidrud river. The Sefidrud channel has changed its coarse of movement several times due to the fall of water level in Caspian sea [1]. The latest course of Sefidrud bed before its present channel are Heshmatrud and Aliakbari rivers passing through Astaneh and Rudbaneh and meandering and flowing towards the Caspian sea. As shown in Fig. 4, levee deposit shown by dark color covers a wide area and earthquake induced liquefaction occurred mostly in such a soil formation containing loose sand and silt. The water table generally is high being at about 1 to 2 meter below ground level at the time of earthquake. Most of affected lands are rice farms and



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Fig. 3 Approximate Intensity distribution of June 20 1990 earthquake of Manjil

Fig. 4 Pleistocene and Recent deposits of the Sefidrud delta and adjoining parts of the Caspian plain



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< previous page page_330 next page >Page 330therefore it was not an easy task to distinguish between the liquefied and not liquefied sites in such a wide area. The places with the signs of liquefaction are within the zone shown in Fig. 5. This does not mean that any point in this zone is a liquefied site. However, the observed places with clear signs of liquefaction are shown by dark circles in this figure.Sand liquefaction caused extensive damage such as foundation bearing failure, total and differential settlement, destruction of houses, disposition of irrigation canal linings, damage to pipes and buried utilities, damage to pavements and roads, opening and cracking of the ground surface, uprooting of large trees, sand boils in rice farms, and boiling and jetting of sand and water from ground and water wells resulting in sand fill in wells. Damage induced by liquefaction was mainly concentrated in Astaneh, Rudbaneh, Pahmedan, and Naserkiadeh to Rudposht along the abandoned Sefidrud channel (Heshmatrud and Aliakbari rivers in Fig. 5). The depth of level ground liquefaction in this area seams to be shallow. This conclusion is based on surface evidence and testimony of the local inhabitants about the method of their private water well excavation, the material they have been involved at the time of excavation, the normal depth of water table and the depth of water table just before earthquake. The wells are mainly for domestic water consumption and are dug by hand to a depth of at the most two meters below water table and lined by concrete rings. The water table is normally at about 2 meters below ground surface and the soil in places of liquefaction consists of 1 to 2 meters silty clay and clayey silt overlaying loose sand. In the liquefied zone most of these wells filled with sand which was boiled out during the earthquake. Shallow subsurface soil condition could be seen f’rom excavations being underway for pipeline repair.Within the city of Astaneh, liquefaction caused extensive damages to buildings in two particular parts of the city as shown in Fig. 6. Within these two particular sections the houses are collapsed, torn apart and experienced differential settlement. Sand boiled from room floors, water wells and everywhere that could penetrate to release the earthquake induced pore water pressure. Two private houses separated by a short wall is shown in Fig. 7a which are torn apart and the wal 1 tilted due to the heave of the ground. The heave as shown in Fig. 7a indicates that the ground shaking has resulted in loosening and



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Fig. 5 Zone of Liquefaction in Gilan

Fig. 6 Zone of Liquefaction in Astaneh


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< previous page page_332 next page >Page 332densifying different parts of the subsurface soil. Thus any other earthquake of similar intensity can repeatedly cause liquefaction. This sort of unconformity can be seen in many places in these two sections of the city. The fact that in other parts of the city no such a damage could be seen reveals that liquefaction of level ground is the sole responsible for incurred damage in the city of Astaneh.There has been no accelerograph installed in the city of Astaneh. However, an accelerograph installed in the city of Lahijan some 10km southeast of Astaneh recorded a maximum horizontal acceleration of 0.17g as mentioned before. The instrument installed at Lahijan is on an almost similar soil condition as that of Astaneh and thus one can assume that the maximum ground acceleration in Astaneh is of the same order as that recorded in Lahijan. To have a better understanding of liquefaction mechanism in Astaneh, a preliminary subsurface soil investigation was performed in this city. Results of this investigation are described in the next section.Damage incurred in Rudbaneh was almost of the same intensity as that of Astaneh. The main road passing through Rudbaneh is mainly parallel to a river located at the north part of the village. The soil condition in Rudbaneh is also similar to the general soil condition of places with the signs of liquefaction in this area; i.e. a top layer of maximum 2m of clayey silt overlaying loose sand. Liquefaction of loose sand caused the hardpan to break and a sort of flow slide occurred towards the river. As a result very long open cracks, mostly filled by sand, appeared at the ground surface especially along the road shoulders. The cracks along the road shoulder and uprooting of large trees are shown in Fig. 7b. Destruction of houses and damage to pavements and roads are other visible kinds of damages induced by liquefaction in Rudbaneh (Fig. 7c).A section of Pahmedan was also damaged by earthquake induced liquefaction. Other part did not experience damage. Shallow soil condition in these two sections are quite different. Loose sand underlies a top cohesive soil of about 2m in liquefied part, and the soil in other part consists of 1 to 2m sand overlaying a thick deposit of clay. The water table in the liquefied part has been about 2.5m below ground surface and the depth of hand dug wells is about 4m. There are a few bore holes excavated within few kilometers from Pahmedan for irrigation canal design. Location of bore holes are given in Fig. 5.



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Fig. 7 Liquefaction induced damages: (a) Damage to houses in Astaneh, (b) Open cracks along the road and uprooting of trees near Rudbaneh (c) Damage to pavements and houses in Rudbaneh (d) Damage to irrigation canals near Astaneh



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< previous page page_334 next page >Page 334The log of one of these holes containing SPT information is shown in Fig. 8 (bore hole no. 17). Apparently at that location liquefaction di d not occur. On the other hand, soil mechanics report associated with these bore hole logs indicates that the drilling is performed without using bentonite or casing. In this respect the operator could not drill the hole no. 18 (Fig. 5) at the right place due to the tendency of the sand to fill the drill hole. Thus the original place of boring was changed to a point about 200m off road to be able to drill the hole in a cohesive soil. The heterogeneity of the subsurface soil condition in the fluvial deposit of abandoned Sefidrud channel can be seen from this soil mechanics report. Presence of loose saturated sandy silt and silty sand with N values of less than 5 is indicative of possible mobilization of liquefaction due to a moderate or strong ground shaking. A number of subsurface investigations had been performed by different organizations for various projects in this region. The location of some of bore holes are shown in Fig. 5. None of the bore holes are located in Astaneh and few of them contain geotechnical parameters.Liquefaction induced damages in other places within the zone shown in Fig. 5. are more or less of the same intensity. The damage incurred to the unlined as well as lined irrigation canals was very heavy. An example of damage to lined irrigation canals near Astaneh is shown in Fig. 7d. Buried utilities such as culverts and pipelines experienced damage as well. An example is the water pipeline transmitting water from Astaneh to Kiashahr which damaged along 4km of its length. STUDY OF LIQUEFACTION IN ASTANEHEvidences from the damage incurred to the ground induced by liquefaction in Astaneh indicates that the depth of liquefaction is shallow. The liquefaction induced damage is concentrated in two sections of the city as shown in Fig. 6. No subsurface investigation had been performed in the city and no engineering subsurface soil information was available. To understand the liquefaction mechanism and answer questions concerning the liquefaction process in this city, a preliminary subsurface investigation performed composed of nine bore holes drilled, three in northeast and four in southwest zones of liquefaction and two in others part of the city (Fig. 6). Although the soil condition changed during and after the earthquake, the heave and settlement of the ground surface in different locations tell us



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Fig. 8 A bore hole log showing the soil condition in Raiat Mahaleh near Pahmedan

Fig. 9 Soil profile in a place located within the zone of liquefaction in Astaneh (BH3)



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< previous page page_336 next page >Page 336that both loosening and densif ication of the soil have occurred as a result of liquefaction. Thus, the subsurface investigation could clarify questions as to the liquefaction in Astaneh.The cyclic stress ratio developed in the field due to earthquake shaking can be computed from an equation of the form as below given by Seed and Idriss [5]:

(1)in which amax=maximum acceleration at the ground surface; overburden pressure on sand layer under consideration; effective overburden pressure on sand layer under consideration; and rd=stress reduction factor varying from 1 at the ground surface to about 0.9 at a depth of about 10m. The stress ratio determined by Eq. 1, then compared with the stress ratio required to cause liquefaction of the soil determined by implementing charts given by Seed et al (1983).Results of site and laboratory tests on soils taken from bore holes no. 3 (BH3) located in one of the zones of liquefaction and no. 4 (BH4), located in the part of the city without indication of liquefaction are shown in Figs. 9 & 10 respectively (see Fig. 6 for the location of these bore holes). SPT was performed carefully at every meter of the bore holes total length implementing exactly the same procedure as that recommended by Seed et al [7]. The hammer used was U.S. type safety hammer. The information given in Figs. 9 & 10 are plotted in charts given by Seed et al [6]. Fig 11 includes the results of this analyses. The circles plotted in Fig. 11 indicate the stress ratio caused by an earthquake with a maximum acceleration of 0.17g in a soil condition as that shown in Fig. 9. Each circle is representative of stress ratio of the soil at every meter of the bore hole depth. The comparison of the stress ratio caused by such an earthquake with the stress ratio required to cause liquefaction by an earthquake of Magnitude of 7.5 indicates that under the condition as that of Manjil earthquake of June 20 1990, liquefaction occurs almost along the total depth of such a soil profile as shown in Fig. 9. This means that under such a circumstances as that mentioned above the thickness of liquefied layer is about 8m. The considerable upward thrust caused by pore water pressure generalized by ground shaking could break through the thin cohesive upper part of the soil and liquefaction



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Fig. 10 Soil profile in a place located outside the zone of liquefaction in Astaneh (BH4)

Fig. 11 Results of liquefaction analyses for soils at BH3 and BH4


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< previous page page_338 next page >Page 338became visible.The same procedure described for BH3 accomplished for BH4 located in the section of the city with no sign of liquefaction. As seen from Fig. 10 at least three meters of cohesive material is present at the upper part of the soil, at this location, underlaying a compacted granular material of pavement. Results of the analyses for sandy parts of this cross section are plotted in Fig. 11, shown by triangles. Liquefaction can occur in a thin layer of loose sand and silt at a depth of 5m below ground surface and in layer of a medium sand at a depth of about 8m. These two narrow and relatively deep strata did not have enough thrust to break through the dense sand present at 6 to 7m deep and the thick cohesive soil present near the ground surface.CONCLUSIONSCharacteristics and geotechnical considerations of Manjil earthquake of June 20 1990 briefly reviewed. Earthquake induced liquefaction occurred in province of Gilan especially in Astaneh and Rudbaneh described. Geological and ground damage maps indicate that most of the earthquake induced liquefaction occurred in heterogeneous fluvial deposit of Sefidrud and its tributaries, mainly in levee deposits of Sefidrud abandoned channels containing loose sand and silty sand with high water table.Liquefaction occurred only in two sections of Astaneh. To understand the mechanism of liquefaction in this city a preliminary subsurface soil investigation performed. This investigation reveals that the presence of a relatively thick loose sand near ground surface and high water table was responsible for observed earthquake induced liquefaction. Places with no signs of liquefaction at the ground surface, contain mostly a thick layer of cohesive soil near the ground surface underlaid mostly by a thin layer of sand which might be prone to liquefaction.ACKNOWLEDGEMENTThe financial support granted by Housing Foundation and International Institute of Earthquake Engineering and Seismology of Islamic Republic of Iran is gratefully acknowledged.



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< previous page page_339 next page >Page 339REFERENCES1. Annells, R.N., Arthrton, R.S., Bazley, R.A. and Davies, R.G. Explanatory of the Qazvin and Rasht Quadrangles Map. Geological Survey of Iran, Tehran, 1975.2. Haeri, S.M. Geotechnical Aspects of Manjil Earthquake of June 20 1990. Report of International Institute of Earthquake Engineering and Seismology of Iran, 1991.3. Ishihara, K. Simple Method of Analysis for Liquefaction of Sand Deposits During Earthquakes. Soils and Foundations, Vol. 17, pp 1–17, 1977.4. Moinfar, A.A., and Naderzadeh, A. An immediate and preliminary report on the Manjil, Iran earthquake of June 20 1990. Building & Housing Research Center, Tehran, 1990.5. Seed, H.B. and Idriss, I.M. Simplified procedure for evaluating soil liquefaction potential. Journal of SMFE Division, ASCE, Vol. 97, No. SM9, 1971.6. Seed, H.B., Idriss, I.M., and Arango, I. Evaluation on of liquefaction potential using field performance data. Journal of Geot. Eng. Division, ASCE, Vol. 109, No. 3, pp 458–482, 1983.7. Seed, H.B., Tokimatsu, K., Harder , L.F. and Chung, R.M. Influence of SPT procedures in soil liquefaction Resistance Evaluation. Journal of Geotechnical Engineering Division, ASCE, Vol. 111, No.12, pp 1425–1445, 1985.



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< previous page page_341 next page >Page 341Countermeasures Against the Permanent Ground Displacement due to LiquefactionS.Yasuda, H.Nagase, H.Kiku, Y.UchidaDepartment of Civil Engineering, Kyushu Institute of Technology, Tobata, Kitakyushu, 804, JapanABSTRACTAppropriate countermeasures against the permanent ground displacement due to soil liquefaction were studied based on shaking table tests and analyses. Four types of countermeasures were tested in the shaking table tests: (1) sand compaction, (2) steel pile, (3) compaction of the ground with a band, and (4) continuous underground concrete or steel wall. In all tests, displacement of some area close to the countermeasure on the upstream side was decreased due to the countermeasure. The continuous wall method was the most effective among the four methods.INTRODUCTIONRecently, permanent ground displacements caused by the 1964 Niigata Earthquake and the 1983 Nihonkai-chubu Earthquake were measured by pre- and post-earthquake aerial surveys (Hamada et al. 1,2), and clarified that extremely large ground displacements, up to several meters, occurred in the ground liquefied during the two earthquakes though the ground surface was almost flat. The authors conducted shaking table tests, vane tests and cyclic shear tests to study the mechanism of the permanent ground displacement and to ascertain the rate of decrease of the shear modulus and the shear strength (Yasuda et al. 3). Based on these tests, a simplified procedure for forecasting permanent ground displacement was proposed.In the next step, appropriate countermeasures against the permanent ground displacement were studied based on shaking table tests and analyses. Countermeasures by strengthening the ground with sand piles, steel piles, densification at a narrow band or continuous walls were studied. The effectiveness and the limitation were clarified. Moreover, effectiveness of the countermeasure in the full scale ground was studied by some analyses.



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< previous page page_342 next page >Page 342In this paper, results of shaking table tests to know the mechanism, and proposed simplified procedure for the analysis are shown briefly. Then, shaking table tests and analyses for countermeasures are shown in detail.SHAKING TABLE TESTS TO KNOW THE MECHANISM OF PERMANENT GROUND DISPLACEMENTShaking table tests were conducted on 24 soil models shown in Fig. 1 to study the mechanism of permanent ground displacement due to soil liquefaction (Yasuda et al.3). The shaking table used was 1m in length and 1m in width in plane. The soil container, used for series A, B, C, D, E and F, was 80cm in length, 65cm in depth and 60cm in width. For series G, H and I, a slightly bigger container, of 100cm in length, 65cm in depth and 60cm in width, was used.A soil model consisted of two sand layers: an upper layer of loose sand which would liquefy during shaking, and a lower layer of dense sand which would not liquefy during shaking. The same kind of sand was used for both sand layers; however, the method of compaction used for each layer was different. First, the lower sand layer was compacted by shaking at 300 gals of acceleration for two minutes. Then, the upper sand layer, which will be called the “liquefied layer” hereafter, was passed through a sieve in air or in water. Three types of sand were used for the tests: (1) very clean, (2) sand with fines and (3) Toyoura sand. Grain size distribution curves of the sands are shown in Fig. 2. The fine contents of the three sands were 0 %, 6 % and 0 %, respectively. Models tested were classified into nine series, as shown in Fig. 1. Several slopes of the surface or the bottom surface of the liquefied layer and three densities were adopted for each series.Shaking motion was applied in one direction parallel to the

Fig. 1 Model Types



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Fig. 2 Grain Size Distribution Curveshorizontal axis in Fig. 1 except series G, at a frequency of 3 Hz and with appropriate accelerations to induce liquefaction after several cycles of shaking. The shaking was finished 10 seconds after liquefaction. Displacement at several points in the liquefied layer was measured by two methods: (1) deformations of nine to sixteen noodles, placed vertically in the soil at the front of the model, just behind the glass, were measured by photo at intervals of 2.5 seconds after liquefaction, and (2) displacement of 18 to 30 pins on the ground surface was measured by a scale after stopping the shaking.In this paper, only some of the main results are shown, because detailed test results have presented already by the authors (Yasuda et al.3).(1) According to observation of the noodles, displacement developed gradually after the occurrence of liquefaction. The displacement increased almost linearly in a vertical direction from zero at the bottom surface of the liquefied layer to a maximum value at the ground surface. This means that displacement did not occur at the boundary between the liquefied layer and the non-liquefied layer, but occurred with a constant shear strain in the liquefied layer because of a fall of shear strength and shear modulus due to liquefaction.(2) In series A, displacements were not so large. However, it seems that displacement was induced toward the direction of slope of the bottom surface of the liquefied layer.(3) In series G, displacement also developed gradually after the occurrence of liquefaction, almost as it did in series B. In series H, displacement occurred not toward the slope of the bottom surface of the liquefied layer, but toward the slope of the filled layers.(4) The displacement of loose sand was greater than that of medium dense sand, and the displacement of sand with fines was less than that of clean sand.CYCLIC TORSIONAL SHEAR TESTSCyclic torsional shear tests were conducted to clarify the rate of decrease of the shear modulus due to liquefaction (Yasuda et al.3) The sand used for these tests was also Toyoura Sand. After saturated specimens were consolidated a

cyclic shear stress of 0.1Hz was applied until the excess pore pressure ratio, reached a prescribed amount. Then, static shear stress was applied under a constant strain rate of 0.1 % per minute. The prescribed amount of the excess pore pressure



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Fig. 3 Comparison between Calculated Displacements and Measured Displacements at Niigata Railway Station

Fig. 4 Three Categories of to Countermeasure against Permanent Ground Displacement

ratio, , was changed for each specimen from 0 to 1.0. A specimen of is a fully liquefied specimen.The rate of reduction of shear modulus, G0, 1/G0, i decreases rapidly with an increase in excess pore pressure

ratio, , and reaches a very small value, almost 0.001, if a specimen is fully liquefied.A SIMPLIFIED PROCEDURE FOR THE ANALYSIS OF THE PERMANENT GROUND DISPLACEMENTBased on the test results mentioned above and case studies conducted by Hamada et al. [1], the authors proposed a simplified procedure for the analysis of permanent ground displacement (Yasuda, et al. 3). In this procedure, the authors assumed that permanent ground displacement would occur in liquefied softened ground due to shear stress present before


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liquefaction. The finite element method was applied twice as follows: (1) In the first stage, the distribution of stresses in the ground is calculated by the finite element method using the elastic modulus before the earthquake. In the calculation, model layers must be made in several steps, because the soil layers in natural ground have filled gradually.(2) Then, holding the stress constant, the finite element method is conducted again using the decreased modulus due liquefaction by the earthquake.(3) The difference in deformation measured by the two



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< previous page page_345 next page >Page 345analyses is supposed to equal the permanent ground displacement.To confirm the accuracy of this procedure, several analyses of the soil models used for the shaking table tests were conducted. The analyzed results coincide fairly well with the test results.In the next step, permanent ground displacements at the site of Showa Bridge and around Niigata Railway Station in Niigata City during the 1964 Niigata Earthquake were analyzed. These results were compared with the results measured by Hamada et al. [1]. In these analyses, elastic modulus, E, was estimated from SPT N-value by using the formula, E=28N. Poisson’s ratios during filling and during an earthquake were assumed as 0.35 and 0.499, respectively, as in the analyses of soil models used for shaking table tests. Three rates of elastic modulus decrease were assumed: 1/500, 1/1000 and 1/2000. Fig. 3 compares the results of analysis of ground around Niigata Railway Station with the displacements measured in that area. In Fig. 3(a), analyzed displacements and measured displacements have the same tendency: displacement increases with the thickness of the liquefied layer. By comparing the amount of deformation analyzed with that measured, it can be said that the analysis assuming an elastic modulus of decrease rate of almost 1/1000 is appropriate. This decreasing ratio coincides well with the result of the cyclic torslonal shear test.KIND OF COUNTERMEASURES AGAINST PERMANENT GROUND DISPLACEMENTIt is not clear what kind of countermeasures are effective against permanent ground displacement due to liquefaction, because no countermeasures have been applied. However, based on the tests, analyses and case studies, the following three categories of countermeasures, as shown in Fig. 4, seem to be effective: (1) improving the ground in all area by densification to prevent liquefaction, (2) strengthening structures to prevent damage, and (3) strengthening the ground with walls or steel piles, sand piles, densification at narrow bands, to prevent large ground displacement if liquefaction occurs. Ground densification in all area is generally considered uneconomical, because it must be applied to a wide area. Different methods must be used to strengthen different structures making this approach somewhat impractical. Therefore, strengthening the ground by walls or steel piles, sand piles, densification through a narrow band was studied by shaking table tests and analyses.SHAKING TABLE TESTS ON COUNTERMEASURES AGAINST PERMANENT GROUND DISPLACEMENTShaking table tests to ascertain effective countermeasures against permanent ground displacement due to liquefaction were



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< previous page page_346 next page >Page 346carried out by using the same soil container for series G, H and I as described before. Sand used was Toyoura Sand and the relative density of the loose layer, which is the liquefied layer, was arranged as 30 %. Four types of countermeasures were applied to the model ground, as shown in Fig. 5: (1) sand compaction, (2) steel piles, (3) compaction of the ground with a band, and (4) continuous underground concrete or steel wall. The following models were used for the four types of countermeasures in considering scale effects:(1) In the sand compaction method, aluminum piles of 2cm in outer diameter were stood in the dense layer, which is the not-liquefied layer, with a depth of 5cm. Then the loose layer, which is the liquefied layer, was filled with the pipes erect. After filling the loose layer, the pipes were pulled out and some Toyoura Sand was poured into the holes. The sand in the holes was compacted by pushing a rod to a relative density of almost 90 % to 100 %. Tests were conducted under three conditions. The number of the compacted sand piles and rate of replacement in each case is shown in Table 1(a).(2) In the steel pile method, vinyl chloride piles of 1.8cm in outer diameter and 2.5mm in thickness were used. The method of erecting the piles and of filling the loose layer were also the same. Young’s modulus of the piles was 32,000kgf/cm2. In this method, piles were stood in one row or in two rows with triangle alignment. Test conditions are shown in Table 1(b).(3) Instead of vinyl chloride pipes, two sheets of walls made of aluminum, with a thickness of 2mm, were used in the ground compaction with a band method. The depth of installation of the walls, method of filling loose layer, and method of compacting the sand in the trench after pulling out the walls were the same as in the sand compaction method. Four thicknesses of the compaction band were tested, as shown in Table 1(c).(4) In the continuous underground concrete or steel wall method, an acryloyl wall of 2mm or 3mm in thickness was used, as shown in Table 1(d). The wall was stood on the bottom surface of the soil container. Eight pieces of strain gauges were pasted on the wal1 to measure the bending strain of the wall due to earth pressure. Installation of the wall and method of filling loose layer were the same as in the steel pile method.In all tests, the thickness of the loose layer was 20cm, and slopes of the ground surface and bottom surface of the loose layer were 3 %. Models were shaken in the perpendicular direction to the horizontal axis, the same as series G in Fig. 1, according to a 3Hz sine wave up to 10 seconds after the occurrence of liquefaction.Fig. 6 shows the measured displacements on the ground surface after stopping the shaking. Without countermeasures, displacements of 2 to 2.5cm occurred on the ground surface, with the maximum value at the center. In contrast, displacements with countermeasures decreased to 2cm to 2mm,



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Fig. 5 Models of Countermeasures in Shaking Table TestsTable 1 Test Conditions of Countermeasures (a) Sand Compaction

Case No. Number of piles Rate of replacement As (%)

S-1 6 3.1

S-2 8 5.6

S-3 10 8.7

(b) Steel pile

Case No. Number of piles Pitch of piles(cm) Number of rows

P-1 10 6 1

P-2 12 5 1

P-3 15 4 1

P-4 20 3 1

PT-1 15 7.5 2

PT-2 20 5.8 2

(c) Compaction of the ground with a band

Case No. Thickness of the compacted band

W-1 0.5


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W-2 1.0

w-3 1.5

W-4 2.0

(d) Continuous underground concrete or steel wa

Case No. Thickness of the wall (cm)

A-1 0.2

A-2 0.3



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Fig. 6 Measured Displacement in Shaking Table Testswith the minimum value on the upper side, on the left side in the figure of the countermeasures. Fig. 7 compares the displacements with rate of replacement, pitch of the steel piles, thickness of the compacted band, and thickness of the wall, in four types of countermeasures. It can be seen that displacements changed with these factors. In the steel pile method, an alignment with two rows was more effective than an alignment with one row if the numbers of piles were the same. In case of the continuous underground wall method, the distribution of earth pressure acting on the wall was estimated as shown in Fig. 8 based on the measured strain and Young’s modulus of the wall. The distribution curve was almost triangular. Displacements with an underground wall were the smallest among the four types of countermeasures, as shown in Fig. 7. In the sand compaction method or steel pile method, some soil-flow through the piles was induced. And, in the ground compaction with a band, some bending of the compacted band occurred due to inadequate stiffness of the compacted band. Therefore, it can be said that the continuous wall method is the most effective among the four methods. However, stress induced in the wall must be evaluated during the design of the wall.



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Fig. 7 Effect of Countermeasures in Shaking Table Tests


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Fig. 8 Distribution of Earth PressureANALYSES FOR THE EFFECTIVENESS OF COUNTERMEASURES IN THE GROUNDTo know the effectiveness of the countermeasures, mentioned above, in the ground, several analyses were performed, assuming different countermeasure parameters, on a ground model of 100m in length, with a liquefied layer 10m in thickness and a 3 % slope of the ground surface. Among the four types of countermeasures by the continuous underground concrete or steel wall and the compaction of the ground with a band were selected for the analyses. The SPT-N values of liquefied layer and the non-liquefied layer were assumed as 3 and 30, respectively. The rate of decrease of the elastic modulus due to liquefaction was supposed as 1/1000.Five of the results of analysis are shown in Fig. 9. Analysis showed that the amount of ground displacement was



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Fig. 9 Analyzed Deformation of Model Grounds with Countermeasuredecreased by installing continuous wall, or by compacting the ground. Moreover, the effectiveness of the countermeasures decreases if the compacted zone does not reach the bottom of the liquefied layer, or if the continuous wal1 is weak.CONCLUSIONSPermanent ground displacement due to soil liquefaction brings severe damages to many structures. To study the effectiveness of countermeasures by strengthening the ground, shaking table tests and analyses were conducted. Four type of strengthening method were selected: (1) sand compaction, (2) steel pile, (3) compaction of the ground with a band, and (4) continuous underground concrete or steel wall. In all shaking table tests, the amount of the ground displacement of some area close to the countermeasures was decreased. The most effective method was the continuous wall method. Analyses on a ground model of 100m in length also showed that the amount of the displacement was decreased by installing the continuous wall.REFERENCES1. Hamada, M., Yasuda, S., Isoyama, R. and Emoto, K. “Study on Liquefaction Induced Permanent Ground Displacements,” Association for the Development of Earthquake Prediction, 1986.2. Hamada, M., Yasuda, S. and Isoyama, R. “Liquefaction-induced Permanent Ground Displacement During Earthquakes,” Proc. of the Pacific Conf. on Earthquake Eng. Vol., pp. 37–47, 1987.3. Yasuda, S., Nagase, H. and Kiku, H. “Shaking Table Tests on Permanent Ground Displacement Due to Liquefaction,” Proc. of the 2nd Int. Conf. on Recent Advances in Geotechnical Earthquake Eng. and Soil Dynamics, Vol. 1, pp. 245–252, 1991.



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< previous page page_351 next page >Page 351Soil-Pile Interaction in Liquefied Sand LayerK.Kobayashi (*), S.Nakamura (*), K.Sato (*), N.Yoshida (*), S.Yao (**)(*) Engineering Research Center, Sato Kogyo., Co., Ltd., Kanagawa, Japan(**) Faculty of Engineering, Kansai University, Osaka, JapanABSTRACTBehavior of a pile foundation in a liquefied sand layer during an earthquake is investigated to make rational aseismic design method of pile-structure system. Shaking table tests by using large scale shear bin (4m in length, 2m in width and 2m in height) were conducted. A 2-story structure model supported by four piles was set on the saturated sand layer in large scale shear bin.From the obtained dynamic restoring force characteristics at the pile top, it is recognized that the secant modulus and equivalent damping ratio depend on the excess porewater pressure in the sand layer.INTRODUCTIONMuch research has been performed related to the liquefaction after the 1964 Niigata earthquake at which severe damage to the structures such as subsidence and differential settlement firstly occurred due to soil liquefaction. These researches [1, 2, 3]was to recognize the mechanism of the liquefaction and to know the behavior of ground during earthquake.Lately, many buildings in Japan were constructed or are going to be constructed with pile foundation at the liquefiable soft ground. In the design of the piles in the liquefiable



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< previous page page_352 next page >Page 352ground, a method based on the Japan Highway Bridge Code is frequently used in the earthquake-resistant design of a pile-foundation system. In this method, a pile-foundation system is modeled to a beam on an elastic foundation, in which the coefficient of horizontal subgrade reaction is estimated considering the generation of the excess porewater pressure. On the other hand, the effect of the liquefaction or porewater pressure generation is not usually considered in the design of the superstructure. When the ground liquefy or porewater pressure generates, the behavior of piles will be affected, hence the behavior of the superstructure will also be changed. Therefore it is important to know the soil-pile interaction behavior in the design of the superstructure.However analytical or experimental studies were hardly carried out to obtain soil-pile dynamic behavior in liquefied sand layer, hence there remains some uncertainty on modeling the soil-pile interaction in liquefied sand layer.This paper deals with the dynamic behavior of a soil-pile-superstructure system in the liquefied sand layer. Shaking table tests by using large scale shear bins were carried out to obtain it. Dynamic restoring force characteristics at the pile top are investigated in relation with the excess porewater pressure of the saturated sand layer.TEST APPARATUS AND TEST PLANFigure 1 shows longitudinal cross-section of the large scale shear bin (4m in length, 2m in width and 2m in height) put on the shaking table. It is composed of 25-frame steps (80mm in height), and designed so as to move without friction in the horizontal direction.After setting a pile-superstructure model at the center of the shear bin, a saturated sand layer is made by water-pluviation method by means of a power-bucket. A sand layer is composed of siliceous sand No. 6, whose material properties are shown in Table 1. Relative densitiy of the sand layer, Dr, is calculated based on the depth of sand layer, the weight of deposited sand and the water content. It varies between 45 and 90%. The unit weight of saturated sand varies between 1.88 and 1.98gf/cm3.



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< previous page page_353 next page >Page 353S-wave velocities in the sand layer, Vs, are calculated by an elastic wave exploration test. They are between 50 and 90 m/sec.A 2-story structure model supported by four piles is used in the test, which is a scaled model of a middle size R/C building commonly constructed in Japan considering a similarity rule[4]. The piles are aluminium square pipes whose dimensions of the pile is shown in Table 2. The weight of the 2-story structure model is 320 kgf. The following measuring instruments were set: 6 servo accelerometers on each floor of the structure, 2 servo accelerometers and 2 inductance type displacement gauges on the ground, 8 inductance type displacement gages on the side of the shear bin, 5 piezometers. Locations of these instruments are also shown in Fig. 1. In addition, a total of 18 strain gauges are put on a pile at the depths of 10, 30, 50, 70, 90, 110, 130, 150 and 170cm.Sinusoidal vibrations are applied in the horizontal direction. The acceleration of the vibrations is increased gradually keeping the frequency constant, which is called sweep up method. Several series of tests were carried out in which the frequency of the external load is different to each other. The same sand deposit is used 3 times in average. It is changed from 1 to 7Hz. Among these tests, only 4 series of the tests is introduced in this paper.TEST RESULTExcess porewater pressure-time histories are shown in Fig. 2. The degrees of excess porewater pressure generation in the tests of case 1 and 2 are different from those of case 3 and case 4. Excess porewater pressure reach the initial effective overburden pressure, which is in definition of liquefaction level, at the same time in the former cases, but in the latter cases excess porewater pressure at upper layer liquified earlier.Figure 3 shows horizontal displacement-time histories of displacement gauges (D1, D6 and D8) In the test of case 3. The displacement have both residual and cyclic components, which is known because displacements shift to one side as the acceleration increases. The same tendencies are observed in the other cases. The instantaneous displacement distributions in the



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< previous page page_354 next page >Page 354soil layer are shown in Fig. 4. In general, the soil moves in one direction at the beginning of loading, but the direction at the upper layer becomes opposite to one at the lower layer, which may be the effect of porewater pressure generation.Figure 5 shows the pile bending moment-time histories in the test of case 3. The deformation of the pile is similar to that of the soil shown in Fig. 3. The same tendencies are observed in the tests of other cases. The instantaneous bending moment distribution along the pile axis in case 4 is shown in Fig. 6. The bending moments at the depths of 30 and 60cm are larger than those at other depths at 9.18 seconds, when porewater pressure generation is small. However the bending moments at the depth of 70cm becomes larger after liquefaction.DISCUSSIONTo obtain the dynamic restoring force characteristics at the pile top, the inertia force at the pile top, P, is computed as the sum of the inertia force at two stories, the product of the mass and measured acceleration, and displacement at the pile top, δ, is computed by substracting the displacement at the bottom of shear bin from the displacement at the first floor. Figure 7 shows the outline of alternations of the P versus δ relationships in the test of case 3 according to the generation of excess porewater pressure.It is clearly observed in Fig. 7 that stiffness becomes low and the hysteresis damping becomes large as excess porewater pressure generates. For understanding these tendency quantitatively, Figure 8 shows the relationship between the secant modulus and the excess porewater pressure ratio and the relationship between the equivalent damping ratio and the excess porewater pressure ratio. Here K denotes the ratio between the secant modulus K and the initial modulus K0, and

h denotes equivalent damping ratio, denotes the ratio of excess porewater pressure to the initial effective overburden pressure. It is noted that the maximum porewater pressure among those at 5 depths are employed in the horizontal axis in Fig. 8. The secant modulus ratio becomes smaller than 0.3, as the excess porewater pressure ratio exceeds 0.8 at which soils are just to liquefy. The equivalent damping ratio is smaller than 0.3 when the excess



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< previous page page_355 next page >Page 355porewater pressure ratio is smaller than 0.8. The secant modulus versus excess porewater pressure ratio relationship as well as the damping versus excess porewater pressure ratio relationship in each case is a little different to each other.The differences between the change of secant modulus and damping ratio against excess porewater pressure ratio seem to depend on the behavior of the soil before the liquefaction.CONCLUDING REMARKSThe behavior of a soil-pile-superstructure system is investigated from the shaking table tests using a large scale shear bin. From the obtained dynamic restoring force characteristics at the pile top, the secant modulus decreases and the damping ratio increases as the excess porewater pressure ratio increases. They are strongly related to the excess porewater pressure generation. In other words, the dynamic behavior of soil before liquefaction at every depth takes a significant effect on the restoring force characteristics of the pile top during the liquefaction.ACKNOWLEGEMENTSThe test facilities is owned and operated by Takechi Engineering Co., Ltd., Osaka, Japan. The test was supported by H.Matsuo ( former graduate student of Kansai University), M.Abe, T.Otawara, T.Tsujikawa, S.Yamamoto (former student of Kansai University). Their support and cooperation are gratefully acknowledged.REFERENCES1. Lee, K.L. and Seed, H.B. Cyclic Stress Conditions Causing Liquefaction of Sand, Journal of Soil Mechanics and Foundation Engineering Division, ASCE, Vol. 92, No. SM1, pp. 47–70, 1967.2. Seed, H.B. and Idriss, I.M. Simplified Procedure for Evaluating Soil Liquefaction Potential, Journal of Soil Mechanics and Foundation Engineering Division, ASCE, Vol.97, No. SM9, pp. 1249–1273, 1971.3. Finn, W.D.L., Lee, K.W. and Martin, G.R. An Effective Stress



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< previous page page_356 next page >Page 356 Model for Liquefaction, Journal of Geotechnical Division, ASCE, Vol. 103, No. GT6, pp. 517–533, 1977.4. Yao, S. Pile Foundation Behaviors in Liquefied Sandy Layer of Large Scale Model (Part 1 Model Rules), 23rd Japan Natinal Conference on Soil Mechanics and Foundation Engineering, Vol. 1, pp. 863–864, 1988. (in Japanese)Table 1 Constants of soil propertiesSoil Type D50 (mm) D10 (mm) UC GS emax emin

Siliceous Sand (No.6) 0.254 0.16 1.81 2.66 1.046 0.654Table 2 Dimensions of piles

Length 175.0 cm

Cross-section 2.5×5.0 cm

Young’s modulus 0.725×106(kgf/cm2)

Cross-section area 2.84 cm2

2nd moment of inertia 2.96 cmTable 3 Test planNo. Frequency (Hz) Dr (%) Vs (cm/sec)

Case 1 2 74 66

Case 2 3 83 69

Case 3 5 76 78

Case 4 7 45 70



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Figure 1 Longitudinal cross-section of shear bin

Figure 2 Time histories of excess porewater pressures during loading


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Figure 3 Time histories of cyclic load, horizontal displacements during loading (Case 3)


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Figure 4 Vertical distribution of horizontal displacements of shear bin



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Figure 5 Time histories of cyclic load, pile bending moments during loading (Case 3)

Figure 6 Vertical distribution of pile moments (Case 4)



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Figure 7 Dynamic restoring force characteristics at the pile top (Case 3)



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Figure 8(a) The relationship between the secant modulus and excess porewater pressure ratio

Figure 8(b) The relationship between the equivalent damping ratio and excess porewater pressure ratio



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< previous page page_363 next page >Page 363SECTION 6: DYNAMIC RESPONSE OF DAMS AND EARTH STRUCTURES



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< previous page page_365 next page >Page 365Dynamic Behavior of Embankment on Locally Compacted Sand DepositsS.Yanagihara, M.Takeuchi, K.IshiharaEngineering and Development, Okumura Corporation, Tokyo, JapanABSTRACTThree types of model tests using large-scale shaking tables were conducted to observe dynamic behavior of embankment(dike) on locally compacted sand deposits. The model tests have clearly demonstrated that a flow type of deformation could develop and has progressed through a long distance away from the dike. Furthermore, provision of a compacted zone beneath the dike was shown to be an effective countermeasure to prevent the progression of the flow slide in front of the dike.INTRODUCTIONThere are several reported evidences of lateral flow having taken place during recent major earthquakes such as 1948 Fukui earthquake in Japan by Hamada et al.[1], the 1923 Kanto earthquake in Japan by Wakamatsu et al.[2], and the 1971 San Fernando earthquake by O'Rourke et al.[3] . As a result of the above case studies, it is known that the amount of permanent deformations in the liquefied sand deposits are generally on the order of several meters and they can develop over a wide area extending several tens of meters from the place where the lateral flow is initiated.In this paper three types of shaking table tests were introduced. The first in the series was a test in which the model dike was placed on a uniform deposit of loose sand. The second and the third tests were identical to the first one except the provision of a compacted zone beneath or in front of



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Figure 1: Layouts of the sand deposits for shaking tablesthe dike. The purpose of this paper is to observe the occurrence of liquefaction and consequent flow-type deformation, if it occurs, in loose deposits of sand. Furthermore, the effectiveness of a preventive countermeasures against the flow slide was examined.PROCEDURE OF TESTSThree types of shaking table tests were conducted in the laboratory of Okumura Co. in Japan. A large box 4.5m long, 0.9m high and 2m wide with transparent lucite-made side walls was placed on a shaking table having a maximum dynamic load capacity of 60 ton with 1g acceleration. At one end of the bin, a rigid steel plate was bolt—fixed to provide an inclined boundary for minimizing the adverse effects of end constraint which may inhibit the smooth continuous deformation of sand deposits through the length of the box. The test arrangements are schematically described in Fig.1. The



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< previous page page_367 next page >Page 367first test in the series was a test in which the model dike was placed on a uniform deposit of loose sand with a relative density of 40% (Fig. 1(a)). The second test was identical to the first one except for the provision of a compacted zone beneath the slope of the dike. The dense sand zone 75cm wide compacted to a relative density of 80% was shown in Fig. 1(b). As the third in the series, a test was conducted in which a compacted zone was provided in the deposit in front of the dike toe as shown in Fig. 1(c). Other conditions were identical to those in the previous two tests except for this shift in the location of the compacted zone.Exact locations of pickups for piezometers are shown in Fig. 2. Sand from Sengenyama in Chiba was used to provide loose sediments of sand. The mean diameter of the sand is 0.30mm. To construct a model dike on top of the sand deposit, a clayey silt material was prepared by blending cement, kaolinite and feleit with 1:2:1.5 proportion in weight. Owing to heavy-weight of the feleit material, the unit weight of the dike was as much as 1.88 ton/m3.The test bin was pooled with water and then the dry sand was rained through a narrow slit of a sliding hopper atop the box as shown in Fig. 3. By this procedure, a uniform sand deposit was obtained with a relative density of 40%. In some tests where it was necessary to provide a compacted zone, two steel plates were first placed vertically across the test bin to provide an enclosure and the sand was poured from the hopper. Every time the sand was piled up about 10cm, the loose deposit enclosed within these twc steel plates was compacted with a rammer and then the plates were pulled up by about 10cm. By repeating this procedure successively, a compacted zone with a relative density of 80% was provided in the middle portion of the test bin.To facilitate visual observation of deformation progress in the sand deposit, a lattice of white-colored sand was provided on the face of the transparent front wall. At each stage of the sand placement as above, sets of piezometers and accelerometers were installed at pre-determined locations within the fill. After completing the sand filling, a 30cm high model dike with a frontal slope 1:2.5 was placed at one end of the deposit. The water table was maintained at the level of the ground surface. The displacement gauges were then placed on the surface along the center line of the model



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Figure 2: Locations of pickups


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Figure 3: Sliding hopper



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Figure 4: Monitored pore water pressures (case 1)dike and the deposit.In all the tests, the test bin was shaken by a sinusoidal motion with a frequency of 2Hz having an acceleration amplitude of 200gal. The shaking was continued as long as necessary until the model deposit and dike deformed significantly.RESULTS OF TESTSIn the first test, time histories of pore water pressures monitored at key points along the mid-depth of the model fill are presented in Fig. 4. It may be observed that the pore water pressures at P57 and P63 distant from the dike indicated a sharp rise to a level of initial overburden pressure within the first one and half cycles, whereas there was some delay in the pore pressure build up at P45 and P51 located in the vicinity of the dike. The pore water pressures as above indicates the fact that the liquefaction developed first in the portion of the sand deposit free from any constraint of the dike and then progressed to the deposit near the dike where the resistance to liquefaction



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Figure 5: Monitored pore water pressures (case 2)


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Figure 6: Monitored pore water pressures (case 3)



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< previous page page_371 next page >Page 371was stronger owing to sustained shear stress produced by the surcharge of the dike.In the second test, the pore water pressures monitored at four locations along the mid-depth of the deposit are displayed in Fig. 5, where it may be seen that the build-up of pore pressures was significantly small and slow at points P45 and P48 within the compacted zone, in contrast to the immediate build-up at point P57 in the uncompacted loose part far from the dike.In the third test, the pore pressures monitored during the shaking at four representative spots are displayed in Fig. 6. It may be observed that the piezometer at P45 in the loose zone below the dike indicated a sharp rise in the early part of the shaking. Similar rapid build-up of pore pressures was also recorded in the loose zone far away from the dike. In contrast to this, the piezometer within the compacted zone indicated native pore pressures, because of the dilatant tendency of the dense sand by the shear stress generated by the weight of the dike. The distributions of final pore water pressure ratio are indicated in Fig. 7.The progression of lateral deformation at main 6 points in the sand fill during shaking are indicated in Fig. 8. In the first test as indicated in Fig. 8(a), it may be observed that the lateral deformations increased constantly until 10 seconds from the beginning of shaking, and after ten seconds the deformation progressions were ceased. This tendency was similar at all 6 points. In the second and the third tests, the deformation in the sand fill were on the order of 0 to 2cm. From this result, it may be mentioned that the compacted zone prevents the progression of the lateral deformations due to the lateral flow of the liquefied sand.The pattern of deformation obtained from the visual observation of the latticed white-colored sand in the first test is demonstrated in Fig. 9(a) for the final stage where the dike completely slumped accompanied with large lateral deformation. While the dike and its toe settled, the free field portio away from the dike experienced a heave on the order of 20 to 50mm. It is to be noticed here that the lateral deformation and heave has progressed through a long distance away from the toe of the dike. This observation is indicative of the fact that the influence of a driving force caused by the presence of an object such as dikes and buildings on the liquefied level



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Figure 7: Distribution of pore water pressure ratioground could spread farther out and create a large- scale lateral flow of the liquefied sand.The pattern of the deformation in the second test are demonstrated in Fig. 9(b). With the crest settlement as small as 22mm and without any spread of lateral deformation toward the loose zone in front of the dike toe, it may be mentioned that the compacted zone, being properly positioned, acted most effectively toward reducing the distress caused by the excessive lateral deformation of the loose sand deposit which would otherwise had occurred as evidenced by the first test in the series. The loose zone outside the compacted zone appears to have behaved on its own characteristics without being influenced by the dike.The pattern of deformation in the third test are demonstrated in Fig. 9(c). Settlements of the dike crest was on the order of 80 to 125mm which is a fairly large amount as compared to 22mm observed in the second test. In contrast to this, the settlements away from the dike were much smaller. And it may be seen that, while the compacted part did not move appreciably, the dike settled significantly. As shown in Fig. 10, severe cracking near the toe of the dike slope was seen developing at the end of the



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Figure 8: Lateral displacements of sand deposits



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Figure 9: Patterns of permanent deformation at the end of the test

Figure 10: Deformation of embankment



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< previous page page_375 next page >Page 375shaking where the liquefied sand and water oozed or spurted out of the loose zone beneath the dike. It appears that the pore pressures developed under the dike pushed up the portion of small surcharge and broke out the toe expelling the liquefied sand. The amount of the dike settlement was equal to the volume of water and sand which had been expelled out of the underlying liquefied zone.The observation as above thus appears to indicate that, if the surcharge is not thick enough, there remains a high possibility for the underlying liquefied sand coming out on the surface, thereby causing destruction due to intorelable settlements of the dike. It is obvious that the presence of the compacted zone had acted favorably as a stopper to prevent a complete slumping of the dike due to the lateral flow of the liquefied sand. In this sense, some beneficial effects could be expected from the installation of a compacted zone, but at a sacrifice of an adverse consequence as described above. Thus it may be mentioned that the installation of a compacted zone would be completely effective only when it is executed in front of a liquefiable zone which is covered by a sufficiently thick surcharge. Installing a compaction zone just beneath the dike such as that executed in the second test satisfies this requirement and hence successfully achieved an intended purpose in minimizing the crest settlement.RESIDUAL STRENGTHIn the first test, by assuming an approximate tri-linear sliding surface as shown in Fig. 11, a stability analysis by Janbu method was made to back-calculate the residual strength of the liquefied sand which appears to have been mobilized along the horizontal portion of the sliding plane in the middle. In this analysis, the static driving force was estimated by consider-

Figure 11: Slip surface for calculating residual strength



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< previous page page_376 next page >Page 376ing the completely slumped configuration of the model dike. The residual strength thus calculated was 0.0038 ton/m2. As compared with the other case studies by Ishihara et al. [4], [5], and Seed [6], the above value is smaller. However, this value is enough to induce the flow failure of any liquefied sand.CONCLUSIONThe results obtained from the tests are as follows. (1) In the case without compacted zone, the dike completely slumped accompanied with large lateral deformation in sand deposits, which has progressed through a long distance away from the toe of the dike. (2) The provision of a compacted zone beneath the dike was shown to be an effective countermeasures to prevent a complete slumping of the dike due to the lateral flow of the liquefied sand. (3) If the surcharge on the compacted zone is not thick enough, there remains a high possibility of boiling the underlying liquefied sand, thereby causing destruction due to the lateral flow of the liquefied sand.REFERENCES1. Hamada, M., Yasuda, S., Isoyama, R. and Emoto, K. (1986), “Observation of Permanent Ground Displacements Induced by Soil Liquefaction,” Proc. Japan Society of Civil Engineers, No.336, III-6, pp. 211–220.2. Wakamatsu, K., Hamada, M., Yasuda, S. and Morimoto, I. (1989) “Liquefaction Induced Ground Displacement during the 1923 Kanto Earthquake”, Proc. 2nd U.S.-Japan Workshop on Liquefaction, Large Ground Deformation, and Their Effects on Lifeline Facilities, Niagara Falls.3. O’Rourke, T.D., Roth, B.L. and Hamada, M. (1989), “A Case Study of Large Ground Deformation during the 1971 San Fernando Earthquake,” Proc. 2nd U.S.-Japan Workshop on Liquefaction Large Ground Deformation, and Their Effects on Lifeline Facilities, Niagara Falls.4. Ishihara, K., Yasuda, S., Yosida, Y., (1990) “Liquefaction—Induced Flow Failure of Embankments and Residual Strength of Silty Sand,”, Soils and Foundations, Vol. 30, No. 3, pp. 69–80.5. Ishihara, K., Okusa, S., Oyagi, N., and Ischuk, A. (1991), “Liquefaction -Induced Flow Slide in the Collapsible Loess Deposit in Soviet Tajik”, Soils and Foundations, Vol. 30, No.4, pp. 73–89.6. Seed.,H.B. (1987), “Design Problem in Soil Liquefaction,” ASCE, Vol.113, No.8, pp. 827–845



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< previous page page_377 next page >Page 377Three-Dimensional Finite Element Analyses of the Natural Frequencies of Non-Homogeneous Earth DamsP.K.Woodward, D.V.GriffithsDepartment of Engineering, University of Manchester, Manchester, M13 9PL, U.K.ABSTRACTFinite element analyses have been performed in order to assess the effects of three-dimensionality on the natural frequencies of earth dams. Parametric studies are performed in which the shape of the valley floor is varied systematically. Results are compared with both 2-d finite element analyses and values measured at the site of an actual earth dam.INTRODUCTIONRecently there has been a renewed interest in understanding the dynamic behaviour of earth dams. The assumption of plane-strain conditions is only valid when considering dams of infinite length. However, many dams situated in seismic regions, are built in narrow valleys where the rigid valley sides create a stiffening effect. This stiffening effect increases the earth dams natural frequencies. The effect of three-dimensionality was first investigated by Hatanka [6] and Ambraseys [1], through the shear beam, or shear-wedge concept. This concept was further developed by Dakoulas and Gazetas [2]. By considering valleys of different shapes Gazetas [3] reported the effects on the natural frequencies of homogeneous earth dams due a change in the valley geometry. Comparisons between two-dimensional and three-dimensional analyses of earth dams has also been performed by Mejia and Seed [7] for the Oroville Dam.The purpose of this paper is to quantify and assess further the stiffening effect of narrow valleys on the natural frequencies of non-homogeneous trapezoidal earth dams, and to clearly show the three dimensional mode shapes in high quality output. The analyses have also given the opportunity to test a quite simple 3-D mesh generator for dams in arbitrarily shaped canyons.



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< previous page page_378 next page >Page 378In order to assess further the three-dimensional effect of the valley geometry finite element analyses have been performed in which the valley is systematically varied in the following ways;1. by varying the valley slope angle2. by varying the valley floor widthParametric studies have also been performed to assess the effect of varying the dam height. The notation used for the geometry of the dam is shown in Figure 1. The results are compared to two-dimensional and three-dimensional analyses, in which the three-dimensional mesh is made to simulate plane-strain conditions. The frequencies are also compared to values obtained at the site of the Long Valley Dam in the Mammoth Lake area of California (Griffiths and Prevost [4]).NUMERICAL ANALYSISFour noded quadrilateral elements were used in the two-dimensional analysis, and eight noded brick elements were used in the three-dimensional analyses.Both types of analyses incorporated selective reduced integration in the formation of the stiffness matrices. A lumped mass approximation was implemented. The QR method was used to calculate eigenvalues and the eigenvectors were found by inverse iteration.The soil properties used are taken from the Long Valley Dam Analysis, (Griffiths and Prevost [4]).Three-dimensional mesh generatorThe three-dimensional mesh was generated by first specifying a parent section (Figure 2), similar to the two-dimensional mesh. The parent section is divided into groups, each group being assigned its own material properties. It is worth noting that each element could be assigned its own material properties if required. The parent section is extrapolated horizontally into three dimensions by specifying the coordinates of a series of sections. Each of these sections has a minimum vertical height corresponding to the rising valley floor at that point. Thus the dam is generated by a series of brick elements which have their nodes at adjacent sections as shown in Figures 4, 5 and 6 (Griffiths and Woodward [5]). Although this method is very flexible, ie any shape of valley floor/side can be considered, caution must be exercised in specifying the section heights, in order to generate the required mesh characteristics. Although this paper does not consider asymmetrical dams, Figure 7 shows how a typical asymmetric mesh could be generated, to reflect non-symmetrical valleys.



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< previous page page_379 next page >Page 379TWO DIMENSIONAL AND THREE DIMENSIONAL PLANE STRAIN ANALYSESThe mesh used in the two-dimensional finite element analyses is the same as that used by Griffiths and Prevost [4] and shown in Figure 3. The mesh consisted of 215 nodes, 178 elements and 352 degrees of freedom. To reflect the spatial variation in stiffness, nine material properties were used in the analysis as listed in Table 1.Group E kPa ν1: drained 1.6E5 0.3

2: drained 2.1E5 0.3

3: undrained 4.0E5 0.45






9: elastic 4.9E6 0.3Table 1: Material Properties for Two-Dimensional AnalysesThe results of the eigenvalue analysis are presented in Table 3. Figures 8 and 9 show the first two mode shapes. The three-dimensional plane strain analysis gave the fundamental frequency within 2% of the two-dimensional analysis.THREE DIMENSIONAL ANALYSESThe meshes used for the three-dimensional analysis varied between 9 and 51 sections. This gave a variation in degrees of freedom between 315 and 4413. The meshes used only five material properties (see Table 2). Typical meshes are shown in Figure 4 and 6. Group EkPa ν1: drained 1.9E5 0.3




5: elastic 4.9E6 0.3Table 2: Material Properties for Three-Dimensional Analyses



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< previous page page_380 next page >Page 380Variation of the valley slope angleIn order to assess more closely the effect of the non-dimensional parameter L/H, commonly used when considering three-dimensional effects on earth dams, finite elements were used to analyse the effect of changing the valley slope angle (ie L/H ratio) while keeping the valley floor width and height constant. Figure 10 shows the results of the eigenvalue analyses. Although there are a few discrepencies, the results are generally in acceptable agreement with those presented by Mejia and Seed [7] and Gazetas [3].In this work the valley geometry is also defined by a second nondimensional parameter W/H. A value of W/H<1.5 would represent a very narrow valley floor. From Figure 10 it is evident that the parameter W/H is an important factor when considering the effects of a narrowing valley geometry.Variation of the valley floor widthThe most interesting case is when the width of the valley is systematically varied while the valley slope and height is kept constant. Figure 11 shows the results of the eigenvalue analyses by the changing parameter L/H. Again these results clearly show three-dimensional effect due to the valley geometry. Figure 12 shows the effect on the fundamental period due to the variation of the parameter W/H. The graphs converge at a W/H ratio of approximately 3.75. This suggests that the length of the crest (ie the L/H ratio) is unimportant (for this particular value of H) on the fundamental frequency after this value of W/H is exceeded. To demonstrate this further, a moderate valley slope, (L/H ratio of 5.2) was chosen. This ratio was kept constant while the valley floor width was systematically varied. Again it was found that the graphs converged at a W/H ratio of approximately 3.75.Variation of the dam heightTo investigate further the W/H ratio at convergence the valley height was varied. When the valley height is decreased by 50%, the W/H ratio at convergence increases by approximately 50% . Conversely when the height is increased by 50% the W/H ratio decreases by approximately 50%. This indicates that the W/H ratio at convergence is inversely proportional to the height of the dam. A graph can thus be drawn (Figure 13) which shows the W/H ratio at convergence for a given height of the valley.A typical narrow valley (L/H=3.63,W/H=1.53) was chosen to demonstrate the mode shapes. The undeformed mesh is shown in Figure 4 and 5. The first mode shape (Figure 14) is clearly an up/downstream motion, similar to the first two-dimensional mode shape, however modes 2 and 3 (Figure’s 15, 16) do not coincide with those obtained from the 2-D solution.The 3-D plots incorporate both perspective and depth viewing.



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< previous page page_381 next page >Page 381COMPARISONS TO THE LONG VALLEY ANALYSISThe results from the Long Valley analysis are shown in Table 3. These results are given by Griffiths and Prevost [4]. The three- dimensional frequency is obtained from the ‘Variation of the valley floor width’ analyses, for a width of 80m. The measured frequencies are obtained from a spectral analysis. The computed and measured frequencies are in acceptable agreement. Griffiths and Prevost [4] Present Work

Mode Number Spectral Analysis: Hz 2-D Analysis Hz 3-D Analysis Hz 2-D Analysis Hz 3-D Analysis Hz

1 1.85 1.76 1.95 1.79 2.14

2 2.15 2.58 2.20 2.63 2.63

3 2.45 3.00 2.25 3.06 2.67Table 3: Comparison of Results to the Long Valley DamCOMPARISON BETWEEN HOMOGENEOUS AND NON-HOMOGENEOUS EARTH DAMSOne of the objects for this paper is to study the effects of dam non-homogeneity. However it was felt necessary to perform a simple test in order to classify any difference between homogeneous and non-homogeneous dams. By considering a typical narrow valley (L/H=3.63, W/H=1.56) both homogeneous and non-homogeneous finite element analyses were performed. It was found that the ratios of the time periods between the three-dimensional results and the two-dimensional plane-strain results for the homogeneous and non-homogeneous dams were not consistent. The difference being of the order of 9%. It was also found that reducing the height of the valley by 50% had little effect on these ratios. In order to classify further the effects of non-homogeneity the same analyses were performed but with the rate of stiffness increase with depth doubled. It was found that the difference between the homogeneous and non-homogeneous ratios was now of the order of 15%.This shows that as the rate of increase in stiffness in the dam becomes greater, the effects of three-dimensionality on the fundamental frequency reduce. Table 4 shows the results of these analyses.



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< previous page page_382 next page >Page 382T/T∞ (HOM) T/T∞ (NON-HOM) T/T∞ (NON-HOM) [INCREASED STIFFNESS]

0.76 0.83 0.89Table 4: Comparisons of T/T∞ for homogeneous and non-homogeneous earth damsCONCLUSIONThe results presented in this paper are intended to build upon the work already performed in this area of research. The three-dimensional effect of the narrowing valley geometry on the natural frequencies for trapezoidal dams has been extensively examined. It has been shown that when considering non-homogeneous trapezoidal earth dams the width to height ratio is very important when assessing the natural frequencies and hence the seismic response of the earth dam. It has also been shown that as the degree of non-homogeneity increases the three-dimensional effect of the valley geometry, on the natural frequencies decreases. The computed frequencies for the Long Valley Dam compare very favourably with those obtained by Griffiths and Prevost [4].



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Figure 1. Geometry notation

Figure 2. Three-dimensional Parent Section

Figure 3. Two-dimensional mesh

Figure 4. Typical three-dimensional narrow valley dam (aerial view)



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Figure 5. Typical three-dimensional narrow valley dam (underside)

Figure 6. Typical three-dimensional wide valley dam (aerial view)

Figure 7. Typical non-symmetrical dam (aerial view)



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Figure 8. First two-dimensional mode

Figure 9. Second two-dimensional mode

Figure 10. Eigenvalue results for the variation of the valley slope angle

Figure 11. Eigenvalue results for the variation of the valley door width



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Figure 12. Eigenvalue results for the variation of the valley floor width showing the effect of the parameter W/H


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Figure 13. Graph showing the variation of W/H at convergence for various values of dam height



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Figure 14. First three-dimensional mode (up/down-stream motion)

Figure 15. Second three-dimensional mode (up/down motion of the crest)

Figure 16. Third three-dimensional mode (S-shape along the crest)



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< previous page page_388 next page >Page 388REFERENCES1. Ambraseys, N.N. The seismic stability of earth dams, Proc. 2nd World Conf. on Earthquake Engineering, Tokyo, III, 1960, 1345–1363.2. Dakoulas, P. and Gazetas, G. Seismic lateral vibration of embankment dams in semi-cylindrical valleys, Earthquake Engineering and Soil Dynamics, 1986, No. 14, 19–40.3. Gazetas, G. Seismic response of earth dams: some recent developments, Soil Dynamics and Earthquake Engineering, Vol. 6, No. 1, Jan 1987.4. Griffiths, D.V. and Prevost, J.H. Two and three dimensional finite element analyses of the Long Valley Dam, Geotechnique, 38, No. 3, 367–3885. Griffiths, D.V. and Woodward, P.K. Mesh generation for three-dimensional earth dams, Manc. Univ. Internal Report, 1991.6. Hatanaka, M. Fundemental considerations on the earthquake resistant properties of the earth dam, Bull No. 11, Disaster Prevention Research Inst, Kyoto Univ, 1955.7. Mejia, L.H., and Seed, H.B., Comparison of 2-D and 3-D dynamic analyses of earth dams, J. Geotechnical Engineering, ASCE 1983, 109, GT11, 1383–1398.



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< previous page page_389 next page >Page 389Lumped-Parameter Model of Semi-Infinite Uniform Fluid Channel for Time-Domain Analysis of Dam-Reservoir InteractionJ.P.Wolf, A.ParonessoInstitute of Hydraulics and Energy, Department of Civil Engineering, Swiss Federal Institute of Technology, CH-1015 Lausanne, SwitzerlandABSTRACTThe dynamic stiffness of a semi-infinite uniform fluid channel can be represented for each mode by a lumped-parameter model with a small number of additional internal degrees of freedom. The modal dynamic stiffness is approximated by a ratio of two polynomials which then leads to the frequency-independent coefficients of the springs, dashpots and masses of the model. Only elementary mathematics such as curve-fitting and partial fraction expansion are used. Assembling the corresponding property matrices with those of the irregular region of the reservoir allows the hydrodynamic forces acting on the dam to be calculated directly in the time domain. Nonlinear response of the dam for seismic excitation can thus be calculated straightforwardly.INTRODUCTIONTo determine the hydrodynamic forces acting on the upstream face of a dam during an earthquake, the reservoir’s region of irregular geometry adjacent to the dam (part A—B—C—D—A in Fig. 1) is modelled with an assemblage of finite elements connected to a semi-infinite uniform channel along the upstream direction. This representation of the reservoir is applied in the seismic analysis of two-dimensional gravity dams [1] and three-dimensional arch dams [2] and the idealization based on a semi-infinite channel is used



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Fig. 1 Dam-Reservoir System.also to calculate three-dimensional short-length gravity dams [3].Widely used computer programs for the seismic analysis of dams such as EADFS [1, 2] and EACD-3D [4] are also based on this concept. Alternatively, the total reservoir could be modelled based on the boundary-element method. Better results are achieved with this procedure, however, when a fictitious boundary C—D is introduced, i.e. when the channel of constant cross section is modelled separately using analytical solutions. This is demonstrated in Ref. [5] and confirmed in Ref. [6]. Thus, dividing the reservoir into an irregular region and a semi-infinite uniform channel for modelling purposes is appropriate.To model the channel analytical solutions at least for the variation in the upstream direction are used. They have to satisfy the radiation condition at infinity which is much easier to formulate in the frequency domain than in the time domain. This leads to the absurd situation that because of the semi-infinite fluid channel the total dynamic analysis of the dam-reservoir interaction is performed in the frequency domain. A time-domain analysis is far superior, as this is the natural approach for a dynamic calculation and with which the structural analyst is familiar. In addition, a nonlinear analysis in the time domain with nonlinearities arising for instance in joints in a concrete dam or from the constitutive law in embankment dams is feasible in a straightforward way. Such a time-domain procedure to model the semi-infinite fluid channel is discussed in this paper.The dynamic-stiffness matrix of the semi-infinite fluid channel relating the pressures to their normal derivatives on the interface with the irregular region (C—D in Fig. 1) is frequency dependent, as the dispersion relationships for the various modes appear. This dynamic-stiffness coefficient of the input-output description in the frequency domain for each mode can be approximated as a ratio of two polynomials in frequency, for which a lumped-parameter model consisting of springs, dashpots and



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< previous page page_391 next page >Page 391masses with frequency-independent coefficients can be constructed. The corresponding structural property matrices can then be used directly in a time-domain analysis. This procedure, which can be regarded as a so-called realization in system theory, is developed for a semi-infinite soil in Refs. [7] and [8]. Actually, the results of the out-of-plane motion of the layer built in at its base can be used directly for the analysis of the infinite fluid domain of constant depth. In particular, the procedure based on modal coordinates described in the last paragraph on page 31 of Ref. [7] is applied in this paper. A few results of the lumped-parameter models to represent the fluid are published in the appendix of Ref. [9]. A mathematically very demanding realization is sketched in Ref. [10].The dynamic-stiffness coefficient in the frequency domain for each mode of the uniform semi-infinite fluid channel is the same as that for the semi-infinite bar on elastic foundation. For the latter many other realizations allowing a dynamic analysis directly in the time domain to be performed are described in Refs. [11] and [12] .They are also summarized in Sections 6.8 and 6.9 of Ref. [13]. Although lacking the appeal of physical insight of the lumped-parameter model, the recursive evaluation of the convolution integrals at discrete time stations used in the computational algorithm is a more direct method.SEMI-INFINITE FLUID DOMAIN OF CONSTANT DEPTHThe concept of determining the lumped-parameter model for each modal response can be explained using the two-dimensional semi-infinite fluid channel of constant depth d shown in Fig. 1.Dynamic-stiffness matrixThe following boundary value problem is defined in the frequency domain. The amplitude of the hydrodynamic pressure p (x,z,ω) is governed by

(1)with c denoting the wave velocity. The boundary conditions for 0<x<∞ are equal to

z=0: p(x,ω)=0 (2a)z=d: p(x,ω), y=0 (2b)

and the solution is subjected to the radiation condition. Using the technique of separation of variables [14, 1] leads to



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(3)

where sin λjz are the normalized eigenvectors with amplitudes qj(ω) corresponding to the eigenvalues λj

(4a)and only the wave number kj depends on the frequency

(4b)After determining the modal amplitudes qj from p(z,ω) at x=0 the following distributed “force-displacement” relationship can be formulated at x=0

(5)Expressing p(z,ω) as a function of the nodal values P(ω) based on the shape function [N(z)]

p(z, ω)=[N(z)] P(ω) (6)and applying the principle of virtual work leads to the concentrated “force-displacement” relationship

(7)with the dimensionless frequency a0=ωd/cThe dynamic-stiffness matrix [S(a0)] contains the coefficients on the right-hand side of equation (7). For each mode j only the term under the square root dkj depends on a0. The expression starting with the second integral represents qj. The modal input-output description equals

qj(a0),x=Sj(a0)qj(a0) (8)with the modal dynamic-stiffness coefficient

(9)The same formulation also applies to the semi-infinite uniform channel in a three-dimensional problem as arising with arch dams [2]. In this case, the eigenvalues λj and the corresponding eigenvectors are determined from an eigenvalue problem involving a finite-element discretization of the cross-section of the channel. The modal equations (8) and (9) are not affected, and thus the modal lumped-parameter model discussed below can still be used.



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< previous page page_393 next page >Page 393Lumped-parameter modelA lumped-parameter model is determined for each mode, representing a realization of the corresponding dynamic-stiffness coefficient Sj (a0). The method is described in detail in Ref. [7]. The curve fitting and the calculation of the

coefficients of the springs, dashpots and masses have to be performed only once for the coefficient with

as the dynamic-stiffnesses for all modes Sj, equation (9), can be determined by scaling.To be able to understand the examples presented further on, it is necessary to define some notation. To achieve this, a summary of the procedure is given. At first, the so-called singular part of the dynamic-stiffness cocoefficient

which is equal to its value at is calculated. This determines the singular term of the lumped-parameter model, consisting of a dashpot (Fig 2a). The singular part is then subtracted from the total coefficient, resulting in the regular part. The latter is then

Fig. 2 Discrete Models Serving as Building Blocks for the Lumped-Parameter Model.



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< previous page page_394 next page >Page 394approximated using a curve-fitting technique as a ratio of two polynomials in ia0. The degrees of the polynomials in the numerator and in the denominator are equal to M—1 and M, respectively. Performing a partial-fraction expansion, the regular part can be written as a sum of first-order and second-order terms, for which the discrete models shown in Fig. 2b and 2c follow without introducing any additional approximation. K is the static value, the κ′s, γ′s and µ are dimensionless coefficients specified in Ref. [7]. In the actual calculation, the frequency-independent stiffness, damping and mass matrices corresponding to the lumped-parameter model are used, which are determined by assembling the corresponding discrete models shown in Fig. 2 in parallel.Direct modelling in physical coordinatesTo illustrate the construction of the lumped-parameter model, the “force-displacement” relationship specified in equation (7) is at first modelled directly (without explicit transformation to modal coordinates). One degree of freedom with a shape function of the pressure varying linearly over the depth is selected. Substituting N(z)=z/d in equation (7) leads to the stiffness coefficient

(10)which is non-dimensionalized as

S(a0)=K[k(a0)+ia0c(a0)] (11)with the static-stiffness coefficient

(12)The spring and damping coefficients k(a0) and c(a0) are plotted as a solid line in Fig. 3. c(a0) vanishes up to the cutoff frequency=π/2. The curve-fitting is performed for 0<a0<3.5. For M=10, i.e. 5 building blocks shown in Fig. 2c arranged in parallel, excellent agreement is achieved (Fig. 3b). Also for M=3, i.e. the one building block of fig. 2b in parallel with the one in Fig. 2c, an acceptable accuracy results (Fig. 3a), while the results for M=2, i.e. the building block of Fig. 2c,

should not be used. The non-dimensional time step equals 0.03.At first the harmonic response of the lumped-parameter model is calculated for M=3. Applying a harmonic load P,x with a0=1 and=2.5 to



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Fig. 3 Dynamic-Stiffness Coefficient in Frequency Domain for Linearly Varying Pressure.

Fig. 4 Harmonic Response; (a) a0=1; (b) a0=2.5

the model at rest leads to the resultant pressure P as a function of plotted as solid lines in Fig. 4a and 4b. The scale is selected in such a way that the exact harmonic solution leads to an amplitude equal to 1. The applied load is shown as a dotted line. After the initial phase influenced by the initial



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Fig. 5 Dynamic-Stiffness Coefficient in Time Domain.

Fig. 6 Dynamic-Flexibility Coefficient in Time Domainconditions the agreement is good. In particular for a0=1, the pressure is in phase with the load, which indicates that no radiation damping occurs for this frequency below the cutoff frequency while for a0=2.5 the phase angle is close to 90°.

Then the dynamic-stiffness coefficient in the time-domain is addressed, i.e. the load as a function of time which is generated by a unit-impulse resultant pressure P acting at time zero. This (non-zero) prescribed pressure is applied over the first four time steps. The exact solution of P,x for the rest of the time can be calculated analogously as in [13, p.

340]. This (so-called regular part) of the dynamic-stiffness coefficient equals

(13)

where J1 is the Bessel function of the first kind of order one. The value is plotted in Fig. 5. The results of the lumped-parameter model with M=10 agree very well.

Finally, the dynamic-flexibility coefficient in the time domain is addressed, i.e. the resultant pressure caused by a unit-impulse load P,x applied at time zero. For the lumped-parameter model the load is applied over one time step. An exact solution does not exist. If the first term only is considered in equation (10), the dynamic-flexibility coefficient

equals

(14)where J0 is the Bessel function of the first kind of order zero. The value



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is plotted in Fig. 6 together with the results of the lumped-parameter model for M=10 and M=3.Modelling in modal coordinatesTurning to the construction of the lumped-parameter model for each mode, the normalized dynamic-stiffness coefficient

is processed (Fig. 7). The curve-fitting is performed for 0<a0<3.5 M=3 (i.e. the building block of Fig. 2b in parallel to that of Fig. 2c) and M=6 (i.e. 3 building blocks of Fig. 2c in parallel) are selected.

Fig. 7 Dynamic-Stiffness Coefficient in Frequency Domain for Normalized Mode.Four modes each with its lumped-parameter model are chosen, whose coefficients are calculated by scaling those corresponding to the normalized dynamic stiffness.To study the lumped-parameter models for the modes, equations (5) and (8) are used. For a prescribed p (z,t),x at x=0, the contribution for mode j qj (t),x is calculated as

(15)which represents the load acting on the lumped-parameter model of mode j. After determining qj (t), p (z,t) follows analogous to equation (3) as

(16)A unit impulse load p(z),x constant over the height is applied at x=0 at time t=0. This load acts over one time step

. The resultant



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Fig. 8 Dynamic-Flexibility Coefficient in Time Domain at for Lumped—Parameter Model.

Fig. 9 Pressure Distribution at .

pressure, which is equal to the dynamic flexibility is calculated taking the first 4 modes with M=6. It is plotted in non-dimensional form in Fig. 8. The exact solution with infinitely many modes equals [9]

(17)

Excellent agreement results. At the resultant hydrodynamic pressure exhibits a negative minimum. The corresponding non-dimensional pressure distribution over the height using the lumped-parameter model is compared to the analytical solution in Fig. 9.

(18)SEMI-INFINITE FLUID DOMAIN OF IRREGULAR GEOMETRYThe reservoir’s region of irregular geometry is modelled with finite elements [1, 2]. The corresponding “static-stiffness” and “mass” matrices relating the pressures at the nodes to the integrated normal derivatives of the pressures are straightforwardly established. Assembling these matrices with the realization of the dynamic-stiffness matrix of the semi-infinite region (equation (7)) leads to the discretized system of motion of the total reservoir in the time domain. An explicit algorithm with lumped masses based on the Newmark scheme is applied.To evaluate the performance of the coupled system, the semi-infinite


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Fig. 10 Dynamic-Flexibility Coefficient in Time Domain for Coupled System.fluid domain of constant depth is again addressed. The first part adjacent to the dam is modelled with square finite elements with linear shape functions over a length of twice the depth. Over the depth 12 finite elements are selected. The second part up to infinity is modelled using the lumpedparameter model for each mode. 4 modes and M=6 are chosen in the realization of equation (7). As a stringent test, a unit impulse (applied over one ∆t) of p,x (corresponding to a

horizontal acceleration of the dam) constant over the depth is applied. The explicit algorithm with and the parameter γ=0.65 is processed. The resultant pressure as a function of time is plotted together with the exact solution in Fig. 10. This figure should be compared to Fig. 8, where the same result calculated without a finiteelement region is shown. A very satisfactory behaviour is observed.Finally, the semi-infinite irregular reservoir with an inclined bottom as shown in Fig. 11 is examined. The irregular region is discretized with square finite elements; the semi-infinite regular part is again represented with 4 modes each with its lumped-parameter model with M=6.

Fig. 11 Semi-Infinite Irregular Reservoir with Inclined Bottom.



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Fig. 12 Hydrodynamic Resultant Pressure for Impulse-Acceleration Loading.A unit impulse acceleration of the rigid dam (introduced as a rectangle over 0.00173 s) is applied. ∆t=0.0009 s and γ=0.5 are selected. The resultant pressure acting on the dam R(t), divided by the static water pressure Rst, is plotted as a function of time in Fig. 12. An acceptable agreement is obtained with a time-domain boundary element solution [6], whereby only 4 boundary elements are used over the depth d.CONCLUSIONSTo calculate the hydrodynamic forces of the reservoir directly in the time-domain, each mode of the semi-infinite uniform fluid region is represented by a lumped-parameter model with frequency-independent coefficients of the springs, dashpots and masses and with only a few additional internal degrees of freedom. The only approximation introduced at the very beginning of the procedure consists of replacing the modal dynamic-stiffness coefficient by a ratio of two polynomials in frequency. This is performed by a simple curve-fitting scheme based on the least-squares′ method, resulting in a linear system of equations. No unfamiliar discrete time manipulations such as the z—transformation are necessary. By selecting different degrees of the polynomial, a whole family of lumped-parameter models with easily verifiable accuracy is constructed. The stability is guaranteed and the corresponding property matrices are automatically symmetrical.REFERENCES[1] Hall, J.F. and Chopra, A.K. Two-Dimensional Dynamic Analysis of Concrete Gravity and Embankment Dams Including Hydrodynamic Effects. Earthquake Engineering and Structural Dynamics, 10, 305–332 (1982).



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< previous page page_401 next page >Page 401[2] Hall, J.F. and Chopra, A.K. Dynamic Analysis of Arch Dams Including Hydrodynamic Effects. Journal of Engineering Mechanics, 109, 149–167 (1983).[3] Rashed, A.A. and Iwan, W.D. Dynamic Analysis of Short-Length Gravity Dams. Journal of Engineering Mechanics, 111, 1067–1083 (1985).[4] Fok, K.L., Hall J.F. and Chopra, A.K. EACD-3D, a Computer Program for Three-Dimensional Earthquake Analysis of Concrete Dams. Earthquake Engineering Research Center, Report No UCB/EERC—86/09, University of California, Berkeley, 1986.[5] Lin, P.L.-F. and Cheng, A.H.-D. Boundary Solutions for Fluid-Structure Interaction, Journal of Hydraulic Engineering, 110, 51–64 (1984).[6] Wepf, D.H., Wolf, J.P. and Bachmann, H. Hydrodynamic-Stiffness Matrix Based on Boundary Elements for Time-Domain Dam-Reservoir-Soil Analysis. Earthquake Engineering and Structural Dynamics, 16, 417–432 (1988).[7] Wolf, J.P. Consistent Lumped-Parameter Models for Unbounded Soil: Physical Representation. Earthquake Engineering and Structural Dynamics, 20, 11–32 (1991).[8] Wolf, J.P. Consistent Lumped-Parameter Models for Unbounded Soil: Frequency-Independent Stiffness, Damping and Mass Matrices. Earthquake Engineering and Structural Dynamics, 20, 33–41 (1991).[9] Wolf, J.P. and Paronesso, A. Lumped-Parameter Model for Foundation on Layer in Proceedings Second International Conference on Recent Advance in Geotechnical Earthquake Engineering and Soil Dynamics (Ed. Prakash), St. Louis, Missouri, 1, 895–905, 1991.[10] Weber, B. Fluid-Structure Interaction for Arch Dams in Structural Dynamics (Ed. Krätzig et al), 2, 851–858, Proceedings of European Conference on Structural Dynamics Eurodyn’90, Bochum, Germany, June 1990, A.A.Balkema, Rotterdam 1991.[11] Wolf, J.P. and Motosaka, M. Recursive Evaluation of Interaction Forces of Unbounded Soil in the Time Domain. Earthquake Engineering and Structural Dynamics, 18, 345–363 (1989).[12] Wolf, J.P. and Motosaka, M. Recursive Evaluation of Interaction Forces of Unbounded Soil in the Time Domain from Dynamic-Stiffness Coefficients in the Frequency Domain. Earthquake Engineering and Structural Dynamics, 18, 365–376 (1989).[13] Wolf, J.P. Soil-Structure Interaction Analysis in Time Domain. Prentice-Hall. Englewood Cliffs, N.J., 1988.[14] Chopra, A.K. Hydrodynamic Pressures on Dams During Earthquakes. Journal of Engineering Mechanics, 93, 205–223 (1967).



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< previous page page_403 next page >Page 403Earthquake Resistant Design of Earth Walls—A Probabilistic ApproachD.Genske (*), H.Klapperich (*), T.Adachi (**), M.Sugito (**)(*) DMT Deutsche Montan Technologie, Institut für Wasser und Bodenschutz-Baugrundinstitut, 4630 Bochum, Germany(**) School of Civil Engineering, Kyoto University, Kyoto 606, JapanABSTRACTThis paper applies a probabilistic safety analysis to a special problem in soil mechanics, a seismic loaded reinforced earth retaining structure. The failure model considered is a special internal one with the failure plain cutting through the geotextile layers. Parameter studies indicate that for the failure model considered the toe of the reinforced wall is its most sensitive part. Based on the probabilistic safety concept a reliability analysis is carried out and design factors are derived, which permit the assessment of the safety of the structure in an easy and convenient way.INTRODUCTIONThe design of reinforced earth retaining walls [1] depends strongly on the safety factors used for the stability analysis. The question, how large the safety factors should be and whether a global safety factor or partial safety factors should be applied, has not thoroughly been answered yet. This subject becomes even more complicated when dynamic forces such as earthquake loadings are included in the analysis. A problem of this kind cannot be solved by empirical evaluation, since we are talking about a fairly new development in geotechnical engineering. In fact, both the mechanical behavior of the geotextile and the failure mechanism are still subjected to research. Thus, the best way to derive safety factors for this type of geotechnical structure is a probabilistic approach.MODELSAn earth retaining structure reinforced by geotextiles can fail due to a variety of failure modes. We have to distinguish between the external failure modes—toppling/sliding/external circular failure/bearing capacity failure and internal failure modes—breaking of the geotextile/partial pull out of geotextile/global failure of the structure with a slip surface cutting through the geotextiles.Reinforced earth retaining structures usually also experience considerable settlement. Therefore, special attention should be paid to the settlement potential.



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< previous page page_404 next page >Page 404But not only the vertical settlement appears to be important. In accordance with BROMS [2] the potential lateral displacement also has to be considered as a relevant design criterion.PERFORMANCE FUNCTIONSThe external failure modes refer to conventional mechanisms, which have been investigated already. The internal failure modes, on the other hand, still require further investigation. Recent studies by GUDEHUS and SCHWING [3] help to understand this particular type of failure mechanism. Based on laboratory tests a 2-block failure mechanism was identified (fig. 1) and the following performance function was derived:

(1)with

where φ=angle of internal friction , φg=angle of friction between fabric and soil , γ=unit weight of soil [kN/m3],

b=1/h [-], νg=critical sliding angle n=number of geotextile layers above base layer [-], he=h (1−b tan νg) [m].For this performance function it was assumed that:–There is no vertical loading of the wall other than caused by gravity of the soil.– The wall is completely drained. – The earth pressure behind the reinforced wall is calculated in accordance with COULOMBs earth pressure theory.– The slip surfaces of the two blocks meet where the reinforcement ends.– The variation of the slip surface of the block 1 yields the minimum of the performance function.As indicated in the above mentioned article [3], this 2-block failure mechanism yields the most realistic results.If a reinforced earth retaining structure is subjected to earthquake loading, the two blocks of the global internal failure model will be exposed to an earthquake acceleration. Horizontal earthquake accelerations directed towards the open side of the retaining wall will destabilize the system considerably. The horizontal accelerations causes horizontal forces as depicted in figure 2. They are a function of the mass of each block and the horizontal acceleration due to the earthquake.



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Fig. 1 The 2-block failure mechanism—statical loading conditions.


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Fig. 2 The 2-block failure mechanism—earthquake loading conditions.



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< previous page page_406 next page >Page 406In order to derive a performance function for the system, the following assumptions were made: – No additional vertical loading of the wall itself and the area behind the wall is considered.– Only horizontal earthquake accelerations were applied.– No liquefaction effect or related phenomena where taken into account.– The geotextiles in the lower 10 % of the wall do not contribute to the overall stability of the wall [3].–Due to the settlement of the wall, a pre-existing sliding surface for block 2 is assumed. Its inclination was calculated from COULOMBs earth pressure theory.Concerning the last assumption it will be shown, however, by the end of this article that the safety factors derived can also be applied to the case of an earthquake induced sliding plane as described by venerable MONONOBE-OKABEs formula [4]. As commonly known the equation for earth pressure developed in Japan in the 1920’s has not worked badly for basic design. However, the generally used seismic coefficients are only a fraction of the peak acceleration of the design earthquake—which means some yielding of the wall should be expected [14, 15]. Extensive research has shown that the actual dynamic response of retaining walls is much more complex than outlined in MONONOBE-OKABEs equation which foremost is developed for gravity retaining walls. This holds especially for new types of earth retainment as described in this paper. As a first attempt the outlined failure mechanism for reinforced walls based on sliding is considered.Based on the above assumptions, which represent a simplified statical approach, the following performance function was derived (ADACHI & GENSKE [5]):

(2)with, besides the parameters mentioned already in equation 1:

ã=a/g=horizontal earthquake acceleration over g [-]

A detailed analysis of this performance function is given by GENSKE, ADACHI & SUGITO [6]. Fig. 3 illustrates how the stability of the structure decreases, if an earthquake occurs and with it a horizontal acceleration. This decrease of stability is especially drastic in the vicinity of the toe of the wall.



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Fig. 3 The effect of an increase of the normalized earthquake acceleration a/g on the performance function (φ=φg=40°, ∆h/h=0.05, b=0.5).RELIABILITY ANALYSISAmong the possible procedures to derive the failure probability and appropriate safety factors, the Invariant Second Moment Approach by HASOFER & LIND [7] has proven to be very effective and easy to understand. A good explanation of this method is given in ANG & TANG [8].In the paper of GUDEHUS & SCHWING [3] mentioned above, this method was applied in order to calculate partial safety factors. They were derived for the internal friction and the friction angle between the fabric and the soil, since these two parameters show a considerable dispersion. The friction coefficient T between the fabric and the soil is interpreted as GAUSS-normal distributed with a coefficient of variation (COV) ranging between 0.10≤COVT≤0.15 (T=tan φg). The internal friction angle is considered to be lognormally distributed. It’s COV may range, in accordance with GUDEHUS & SCHWING [3] and GENSKE & WALZ [9] between 0.10≤COVφ≤0.15. For simplicity, the probability density function is considered to be bound at zero to the left hand side.The stochastic model for the seismic load is much more complicated. In accordance with research carried out in Japan (e.g. SUGITO [10], SUGITO & KAMEDA [11]) a stochastic model for the earthquake induced ground acceleration which, as will be shown later, controls the earthquake loading can be derived as follows: Based on an eight hundred year earthquake record



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< previous page page_408 next page >Page 408(USAMI [12]) the annual probability of occurrence can be deduced for every region in Japan. Figure 4 gives an example for the annual probability of occurrence U(M, ∆) for the Tokyo region. U(M, ∆) is a function of the magnitude M and the distance ∆ of the epicenter to the region considered. These characteristic local earthquake probabilities allow the derivation of earthquake hazard curves.

Fig. 4 Annual probability of occurrence U(M, ∆) as a function of the magnitude M and the distance from the epicenter ∆ for Tokyo.With the attenuation equation for the horizontal peak acceleration [10]

(3)

, M=Magnitude on the RICHTER-Scale [-], ∆=distance of location considered’to epicenter [km], ∆o=∆o(M)=1.06·100.242M−30; (M≥6.0) [km]the annual probability of exceedance for a given peak acceleration a can be calculated:

(4)

For the peak acceleration in a lognormal distribution is assumed. Plotting P(a) over the peak acceleration gives the hazard curves for



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< previous page page_409 next page >Page 409Tokyo, Osaka and Kyoto (figure 5). By means of these hazard curves, peak accelerations for given time intervals, also called return periods, can be estimated. For design purpose, the life time of the structure will specify the return period. For Tokyo, Osaka and Kyoto figure 5 yields the following normalized peak accelerations ã=a/g (a=peak acceleration, g=gravitational acceleration):Annual Exceedance Probability [-] Osaka

ã=Kyotoã=

Tokyoã=

1/10 0.07 0.07 0.20

1/50 0.18 0.23 0.43

1/100 0.24 0.35 0.53

Fig 5. Earthquake hazard curves for Tokyo, Osaka and Kyoto.It should be mentioned that in this approach only the peak acceleration was considered. For further studies on this topic it is recommended to also include the effect of the duration of the ground motion and the spectral intensity. Since hazard curve which include these effects can also be derived their consideration should be possible.In order to ensure the safety of a structure, appropriate safety factors are needed. On the basis of the probabilistic safety concept partial safety factors can



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< previous page page_410 next page >Page 410be derived. These partial safety factors are applied directly to the parameter which have been recognized as statistically dispersive ones.As to the peak acceleration, which is used to calculate the horizontal earthquake loads, it depends on the annual probability of exceedance and thus on the return period. It is therefore used to specify the live time of the structure. In this paper the earthquake statistics only assists in deriving the design earthquake loads. Since these design loads already include a statistical model, they are interpreted as deterministic (i.e. not dispersive) in the subsequent reliability analysis. Although this procedure includes some approximations it has the advantage of being easy to apply and flexible as to the local characteristics of the earthquake record.Thus, for the derivation of design factors only the internal friction angle and the friction coefficient are considered to be dispersive. Probabilistic design factors ensure a certain probability of failure not to be exceeded. In accordance with the First Order Second Moment Approach [7] the design factor for the GAUSS-normal distributed friction coefficient T and the lognormal distributed internal fiiction angle φ is determined by

(5a)and

(5b)withγT,γφ=design factors for T and φ [-]µT=mean value of T [-]

T*=design point of T=µT—αT β σT [-]

β=safety index [-], pf=Φ(−β)=probability of failure [-], Φ()=CumulativeGAUSS-Normal Probability Function [-]σT=standard deviation of T [-]

with

and



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These equations show that in order to calculate the partial safety factors the values of a have to be determined, which are functions of the partial derivatives of the performance function to the dispersive parameters. The results can only be found by means of an iterative procedure. From the definition of the α-values it follows that

(6)This means, that with increasing a the relative effect of the dispersive parameter associated with this a on the failure probability will also increase, and so will the partial safety factor. In figure 6 the α2-values are plotted over the normalized height of the reinforced wall. The failure probability in this study was fixed to 1/1000, which refers to a safety index of β=3.1.

Fig. 6 α2-values as a function of the normalized height of the wall h′/h. Constant failure probability maintained at 1/1000. µφ=40°, µT=0.839 = tan (40°), b=0.5, ∆h/h=0.05.In the lowest part of the wall where the geotextiles do not contribute to the safety of the 2-block system the a for the internal friction angle is 1.0 since φ is the only dispersive parameter contributing to the safety of the structure.



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< previous page page_412 next page >Page 412Consequently, the partial safety factor must have a constant value. As soon as the geotextile starts supporting the system an a for the friction coefficient T generates. The increase of αT will be especially large, if the COV of the friction coefficient T is also large relative to the COV of φ. The region where both dispersive parameters have a considerable effect on the safety of the wall is rather limited, however. The αT will with increasing heights h′/h soon dominate the failure probability, even if its COV is at the lowest possible limit If figure 6 is now compared with figure 3 we have to realize that the region of uncertain α′s approximately refers to the most sensitive part of the wall where the deterministic safety already has a minimum. Therefore, no attempt has been made to monitor the behavior of the α′s with further parameter studies. It is rather suggested to approximate both α′s with their possible upper value, which is 1.0. With this assumption we achieve to be on the safe side as to this critical part of the wall.Utilizing the relationships pointed out in equation 5 and 6, the partial safety factors necessary to ensure a certain probability of failure (expressed by β) can be calculated [6]. Figure 7 gives the partial safety factors as a function of the safety index β With increasing β which refers to a decrease in the tolerable failure probability, the safety factors increase as well. The upper and lower margins of the COV of both parameters are given. For a large COV the required safety factor must be large, too. Also indicated are the recommended failure probabilities for geotechnical structures [13]. If the failure of the wall has only a minor economical impact the lower boundary is appropriate, whereas if losses of live have to be expected in case of a failure the upper limit is recommended.

Fig. 7 Conservative design factors for φ and T with increasing failure probability.



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< previous page page_413 next page >Page 413In figure 8 the critical geotextile length 1 (normalized with the wall heights h) in the toe zone is plotted over the earthquake return period. Whereas for Osaka and Kyoto long return periods still yield feasible reinforcement length, a geotextile structure in Tokyo allows, from the practical point of view, only rather short return periods.

Fig. 8 The critical geometry b=1/h as a function of the annual exceedance probability. µφ=40°, µT=0.839=tan (40°), COVφ=0.075, COVT= 0.125, γφ=1.2, γT=1.7, ∆h/h=0.05.SUMMARYIn order to establish design criteria for an internal 2-block failure mechanism of reinforced earth retaining walls a reliability analysis was carried out and design factors were derived. Parameter studies indicated that the internal friction angle and the friction coefficient between the geotextile and the soil are appropriate for the application of design factors. These design factors are a function of the probability of failure considered to be tolerable. For the earthquake induced horizontal forces the derivation of stochastic design loads was treated separately, since the earthquake loading is in contrast to all other parameters a time depended parameter. The seismic design loads depend strongly on the regional earthquake record and the return period presumed for the design of the earth wall.Further research should be directed towards the reliability analysis for other failure modes of the reinforced wall, the consideration of a lateral vertical loading of the upper part of the wall and a more detailed analysis of the dispersive character of the local peak acceleration and its influence on the reliability of the structure. Model tests using shaking tables and centrifuges will be helpful to study this complex interaction problem. Basic experimental findings in terms of failure modes in relation to seismic input (harmonic, stochastic) will lead to further modifications of the theoretical model and finally to satisfactory design.



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< previous page page_414 next page >Page 414ACKNOWLEDGMENTWe are very grateful for the financial support which was granted to us from the Japanese Society for the Promotion of Science (JSPS), Tokyo. REFERENCES1. VIDAL, H. La Terre Armee, Annales de l’Institut Technique de Batiment et des Travaux Publics, Nos. 223–229,, 888–939, Paris, 1966.2. BROMS, B.B. Fabric Reinforced Retaining Walls, Proceedings of the International Geotechnical Symposium on Theory and Practice of Earth Reinforcement, Fukuoka, Japan, 1988.3. GUDEHUS, G. & E.SCHWING Standsicherheit kunstoffbewehrter Erdbauwerke an Geländesprüngen, Baugrundtagung, 129–147, Nürnberg, 1986.4. JSCE (Japan Society of Civil Engineers), Earthquake Resistant Design for Civil Engineering Structures in Japan, Tokyo, 1984.5. ADACHI, T & D.D.GENSKE Earthquake Loaded Reinforced Earth Retaining Structures, JSPS Research Report (unpublished), Kyoto, 1990.6. GENSKE, D., ADACHI, T. & M.SUGITO Reliability Analysis of Reinforced Retaining Structures Subjected to Earthquake Loading, Soils and Foundations (in press).7. HASOFER, A.M. & N.C. LIND Exact and Invariant Second Moment Code Format., J. Engineering Mechanics, Vol 100, No EM1, 1974.8. ANG, A.H.S. & W.H.TANG Probability Concepts in Engineering Planing and Design, vol. 2, Wiley & Sons, New York, 1984.9. GENSKE, D.D. & B. WALZ Anwendung der probabilistischen Sicherheitstheorie auf Grundbruchberechnungen, Geotechnik 10, vol 2, 53–66, 1987.10. SUGITO, M. Earthquake Motion Prediction, Microzonation and Buried Pipe Response for Urban Seismic Damage Assessment.—Thesis, School of Civil Engineering, Kyoto University, Japan, 1986.11. SUGITO, M. & H.KAMEDA Nonlinear Soil Amplification Model with Verification by Vertical Strong Motion Array Records, 4th National Conference on Earthquake Engineering, Palm Springs, USA, 1990.12. USAMI, T. Descriptive Catalog of Disaster Earthquakes in Japan (in Japanese), University of Tokyo Press, Tokyo, Japan, 1975.13. MEYERHOF, G.G. Limit State Design in Geotechnical Engineering, Structural Safety, 1, 67–71, 1982.14. WHITMAN, R.V. & H.KLAPPERICH Model Tests for Earthquake Simulation in Geotechnical Problems, 3rd International Conf. on Soil Dynamics and Earthquake Engineering, Princeton, 1987.15. WHITMAN, R.V. Seismic Design of Gravity Retaining Walls, Earthquake Resistant Construction and Design (ERCAD), Berlin 1989, Savidis (ed.), Balkema 1990,,



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< previous page page_415 next page >Page 415Passive Earth Pressure Coefficients in Seismic Areas by the Limit Analysis MethodA.H.Soubra (*), R.Kastner (**)(*) Ecole Nationale Supérieure des Arts et Industries de Strasbourg, 24, Bd de la Victoire, 67084 Strasbourg Cédex, France(**) Institut National des Sciences Appliquées de Lyon, 20, Av. Albert. Einstein, 69621 Villeurbanne Cédex, FranceABSTRACTThe upper-bound method in limit analysis is applied to the log-spiral rotational mechanism for calculating the passive earth presure coefficients in seismic areas. Numerical results are discussed and compared with other authors’ results.INTRODUCTIONEarthquake can endanger the stability of a soil-wall system by either increasing (or reducing) the active (passive) earth pressures acting on the wall. Thus, the dimensioning of deep sheet piling stuctures in seismic areas requires the determination of active and passive earth pressures acting on these structures taking into account the earthquake forces. So, a rational analysis of these pressures is of great interest in geotechnical engineering.Traditionally, the determination of active earth pressures acting on a retaining wall and taking into account the earthquake forces, is made using the classical method introduced by Mononobe-Okabe [4]. In his method, this author used an extension of the Coulomb’s sliding wedge theory [3] in which earthquake effects are taken into account by the addition of horizontal and vertical inertia terms.In this paper, we present a more rational and simple method which makes it possible to calculate the earth pressure taking into consideration the earthquake forces. The approach presented is a rigorous one in regard to the limit



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< previous page page_416 next page >Page 416equilibrium method since it makes no assumptions concerning the shape of the slip surface and the normal stress distribution along this surface.It was shown by Soubra [15], that the variational limit equilibrium method is equivalent to the upper-bound method in limit analysis for a rotational mechanism. Hence, the solution obtained is an upper bound one for a rigid perfectly plastic material obeying Hill’s maximal work principle.VARIATIONAL LIMIT EQUILIBRIUM METHODThe classical method introduced by Mononobe-Okabe [4] is a limit equilibrium method giving unsafe solutions since it is based on Coulomb’s approach [3] which highly overestimates the passive earth pressure coefficients: This fact is due to the a priori hypothesis concerning the shape of the slip surface. In this paper, we look for the shape of the mechanism giving the minimum value of the passive earth force Pp and for which the three limiting equilibrium equations are satisfied. This problem is formalized mathematically using a variational approach.Mathematical formulation of the problemIt is well known that a rigorous limit equilibrium method is one for which the following conditions are satisfied: a. The shape of the slip surface y (x) and the normal stress distribution σ (x) will give the minimum value of the passive earth force Pp.b. The three equations of the static equilibrium are satisfied for the soil mass ABC (fig. 1).

Figure 1. Slip surface and normal stress distribution for passive earth pressure analysis.

Figure 2. Free body diagram.Notice that a mass is in a state of limit equilibrium when the Mohr-Coulomb criterion is satisfied along the slip surface AB. Writing the three equations of the



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< previous page page_417 next page >Page 417static equilibrium for the soil mass (fig. 2), and combining these equations with the Mohr-Coulomb criterion; one obtains the three limiting equilibrium equations as follows

(1a)

(1b)

(1c)where all parameters of these equations are defined in figure (2). From these equations, it is easy to see that the passive earth force Pp is a functional of two functions y (x) and σ (x). Thus, the rigourous passive limit equilibrium problem is a variational one of the isoperimetric type as follows

subject to

where F(x, y, y’, σ) is simply obtained through one of the equations (1). Gi(x, y, y’, σ) and ai can be obtained from the two remaining equilibrium equations. It was shown (Petrov [8]) that the solution of such a problem is obtained using the Euler equations as follows

(2a)

(2b)Where H is an intermediate functional given by

H=F+λi·Gi (i=i, 2) Notice that H can be written as



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H=σ.f(x, y, y′)+g(x, y, y′) (3)Hence, equation (2a) is equivalent to: f(x, y, y′)=0. Solving this equation, one obtains the equation of the slip surface

which is a log-spiral in the case of a constant . Replacing this equation into equation (3), one can see that H becomes independent of the normal stress distribution. This result is a direct consequence of the shape of the slip surface. It was shown (Soubra [15]) that any equation of the normal stress distribution having at least two degrees of freedom will satisfy the three equations of static equilibrium and that, only the equation of moments around the centre of the log-spiral is sufficient to calculate the passive earth force Pp. It is easy to see that the moment equation of the rotational log-spiral mechanism around the centre is identical to the work equation for the same mechanism in the upper-bound method in limit analysis. Thus, solving the passive earth pressure problem by the upper-bound method in limit analysis for a rotational mechanism will give a rigorous solution in regard to the limit equilibrium method. This method is detailed in the following section.UPPER-BOUND METHODThe equation of the rotational log-spiral mechanism (fig. 3) is given as

(4)

Figure 3. Log-spiral mechanism for passive earth pressure analysis.



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< previous page page_419 next page >Page 419For a rigid body rotation, this mechanism is kinematically admissible since the velocity V along the plastically deformed

surface AB (fig. 3) makes an angle with the transition layer according to the normality condition for an associated flow rule material.According to the upper-bound theorem in limit analysis, for a kinematically admissible mechanism, the rate of external work exceeds the internal rate of dissipation of energy along the plastically deformed region. Thus, equating the external rate of work of all external forces to the internal rate of dissipation of energy gives an upper-bound of the exact solution for an associated flow rule material.Rate of external workThe external forces acting on the soil mass are shown on the free body diagram shown in figure (3). These forces consist of: a. The weight of the soil mass between the log-spiral surface and the ground surface.b. The passive earth force which is inclined at δ to the normal of the sheet piling wall.c. The force K.W which simulates the inertial force due to the earthquake effect.Notice that the seismic vector K has two components: The horizontal seismic coefficient Kh whose value is dominating and the vertical seismic coefficient Kv which is often disregarded. The currently used values of Kh ly between 0.05 and 0.15 in the United States and between 0.15 and 0.25 in Japan. Notice that the choice of the seismic coefficient is completely empirical (Seed [11, 12, 13]). Seed [13] showed that a value of Kh which lies between 0.1 and 0.17 describes very well the failure of the Sheffield dam in California: This dam was subjected to a maximal base acceleration of 0.15g. This author has also shown that for higher accelerations (0.4–0.5g) which describe the Californian earthquakes, a minimum value of 0.3 is necessary for the horizontal seismic coefficient.When studying the stability of slopes, Taniguchi and Sasaki [16] have analysed the failure which occured for a slope subjected to the Naganokon Seibu earthquake in 1984 in Japan. These authors have shown that the seismic coefficient can be described by either the following formulas

Finally, it is interesting to notice that the real value of the seismic coefficient requires the analysis of actual failure cases.



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< previous page page_420 next page >Page 420The weight of the soil mass ABC is given as

(5)where f is the penetration depth and y represents the equation of the slip surface in the coordinate system (ox, oy). Based on equation (4), one can easily show that

y=r.sinθ-r0.sinθ0

Replacing these equations into equation (5), it can be shown that

W=γ.r02.f1(θ0, θ1) where f1(θ0, θ1) is given elsewhere (Soubra [15]). Having established the weight of the soil mass, one can calculate the rate of external work done by the weight of the soil mass as the product of the weight by the vertical component of the velocity of the soil mass. The vertical component of the velocity is given as

where represents the distance between the y axis and the line of action of the weight force, and Ω being the angular

velocity of the soil mass. is simply calculated as follows

where f2(θ0, θ1) is given elsewhere (Soubra [15]).The rate of external work done by the passive earth force is given as

The rate of external work done by the horizontal inertial force Kh.W is the product of this force by the horizontal velocity of the soil mass ABC as follows



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where and f3 (θ0, θ1) is given by Soubra [15]Rate of internal dissipationThe internal dissipation of energy along the log-spiral surface is simply calculated by first calculating the differential

energy dissipation along AB which is the product of the surface element by the cohesion c by the tangential

velocity and then by integrating over the surface AB as follows

Replacing V by Ω.r and integrating, one obtains

D=c.r02.Ω.f4 (θ0,θ1) where f4 (θ0, θ1) is also given by Soubra [15]. For a cohesionless soil, this dissipation is vanishing.Work equationEquating the total external work done by the weight, the inertial force and the passive force Pp, to the internal rate of dissipation of energy, one gets

(6)Notice here that the passive earth force Pp is assumed to act at the bottom third of the penetration depth. This hypothesis depends on problem kinematics and it will be discussed later. Due to this hypothesis, one can write Pp=Kp.γ.f2/2. The most critical Kp-value can be obtained by minimizing with respect to θ0 and θ1 angles shown in figure (3). The θ0 and θ1 at which the Kp-value is minimum determine the most critical sliding surface. A FORTRAN computer program for assessing seismic passive earth pressures has been developed with equation (6) as a basis.NUMERICAL RESULTSEffect of the point of action of the passive earth force on the passive earth pressure coefficientsIn fact, the point of action of the passive earth force depends greatly on problem kinematics. This point was the subject of great controversy in literature. Prakash



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< previous page page_422 next page >Page 422and Basavanna [9] showed that the point of action of the active earth force lies between 0.4f and 0.5f when the seismic coefficient varies between 0.1 and 0.3. Wood [17] was based on the elastic soil hypothesis and suggested a force acting at the middle of wall height. Aubry and Chouvet [1] made a finite element analysis and suggested a point of action lying slightly higher than the bottom third of the wall height. The present analysis have shown that the passive earth pressure coefficient is increased when the passive force goes up. Thus, a conservative approach concerning the Kp-value is to adopt the bottom third distance.Seismic effect on the passive earth pressure coefficientsIt is known that earthquakes have the unfavorable effect of increasing active and decreasing passive lateral earth pressures. An earthquake can also reduce the shearing resistance of a soil. The reduction in the shearing resistance of a soil during an earthquake is only effective when the magnitude of the earthquake exceeds a certain limit and the ground conditions are favorable for such a reduction. The evaluation of such a reduction requires considerable knowledge in earthquake engineering and soil dynamics. Research conducted by Okamoto [7] indicated that when the average ground acceleration is larger than 0.3g, there is a considerable reduction in strength for most soils. However, he claimed that in many cases, the ground acceleration is less than 0.3g and the mechanical properties of most soils do not change significantly in these cases. In this paper, the shear strength of the soil is assumed to remain unaffected as the result of the seismic loading.To investigate how the passive earth pressures are affected, numerical results based on the above mentioned upper-bound method in limit analysis for a rotational mechanism are presented in dimensionless form (figure 4). As mentioned previously, the present limit analysis solutions are valid when there is no reduction in soil strength due to an earthquake.

Due to figure (4), it is easy to see that for ; ; the reduction in the passive earth pressure coefficient is about 16.5% when the horizontal seismic coefficient increases from zero to 0.3. Thus, the calculation of the coefficients of passive earth pressure taking into account the earthquake forces is of great interest in areas of high earthquake risks.Comparison with authors’ resultsThe best upper-bound solution in limit analysis is given by Chang and Chen[2] for the translational log-sandwich mechanism. His results have shown that the Mononobe-Okabe approach seriously overestimates the Kp-value. This is especially the case when the wall is rough.



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< previous page page_423 next page >Page 423The results obtained by the present upper-bound method in limit analysis for a rotational log-spiral mechanism are compared with the above mentioned upper-bound solutions (figure 4). It is interesting to remember here that the log-sandwich translational mechanism is the best mechanism available in literature since it gives the smallest upper-bound solution. Our approach gives better solutions than the Chang and Chen log-sandwich ones since our passive earth pressure coefficients are smaller than those of Chang and Chen [2] for δ>0. However, when δ=0; we obtain a planar surface and our passive earth pressure coefficients are the same as those of Chang and Chen since both the log-spiral and the log-sandwich mechanisms degenerate to a planar surface when δ=0.

Figure 4. Some Kp-value by the present analysis and the Chang-Chen’s one.

For δ>0 ; the passive earth pressure coefficient as calculated by the present approach is 3.7% smaller than the Chang and Chen’s one when Kh=0.This difference decreases with the increasing of the Kh-value. This difference is about 0.9% when Kh=0.3.

In order to braket the collapse load, our solution is compared with the lower-bound solution for available in literature.This comparison



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< previous page page_424 next page >Page 424shows that our upper-bound solution (Kp=9.81) is 2.8% greater than the Lysmer [5] lower bound solution (Kp=9.54) which indicates that the upper-bound solution in limit analysis for a rotational log-spiral mechanism is very close to the exact solution for an associated flow rule material.Seismic effect on the critical slip surfaceThe seismic acceleration generated by earthquakes not only imposes extra loading to a soil mass but also shifts the sliding surface to less favorable positions. Consequently, in addition to the change in the passive earth pressures, the most critical sliding surface is also altered. The numerical results given by the Fortran computer program have shown that the slip surface approaches a planar surface due to the increase in the Kh-value in the case of a rough wall (δ>0). Whereas, in the case of a smooth wall (δ=0); when the Kh-value is equal to zero, the slip surface is planar making an

angle equal to with the horizontal direction: This is in accordance with the Rankine solution. For higher values of Kh, the slip surface remains planar, but it is inclined at smaller angles than the Kh=0 case. Figure (5) shows some typical changes in the critical sliding surface as the result of an earthquake.

Figure 5. Effect of seismic forces on failure mechanism.



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< previous page page_425 next page >Page 425Finally, it is interesting to notice that the critical sliding surface becomes more extended when earthquakes occur. This conforms with the experimental results of Murphy [6]. The change in the critical sliding surface as the result of earthquake has also been noted by Sabzevari and Ghahramani [14].CONCLUSIONThe upper-bound technique of limit analysis for a rotational log-spiral mechanism is used for determining the seismic passive earth pressure coefficients in a quasi-static manner. The approach presented is interesting since the passive earth pressure coefficients so obtained are smaller than the ones given by the best upper-bound solution available in literature concerning the translational log-sandwich mechanism (Chang and Chen [2]) and the difference with the lower-bound

solution (available only when Kh=0) is less than 3% in the case.REFERENCES1. Aubry, D. and Chouvet D. Calcul sismique des murs de soutènement, Génie Parasismique, chap. VIII.3, presses de l’E.N.P.C., Paris.2. Chang, M.F. and Chen, W.F. Lateral Earth Pressures on Rigid Retaining Walls subjected to Earthquake Forces, Solid Mechanics Archives, Vol. 7, Martinus Nijhoff Publishers, The Hague, The Netherlands, pp. 315–362, 1982.3. Coulomb, C.A. Essais sur une application des règles de maximis et minimis à quelques problèmes de statique relatifs a l’architecture, Mém. Acad. R. Pres. Sav. Etr., Vol. 7, pp. 343–382, 1776.4. Mononobe, N. and Matsuo, H. On the Determination of Earth Pressures during Earthquakes, Proc. World Engineering Conference, Vol. 9, 176p., 1929.5. Lysmer, J. Limit Analysis of Plane Problems in Soil Mechanics, J. Soil Mechanics Foundation DIV., ASCE, Proc. Pap. N° 7416, Vol. 96, N° SM4, pp. 1311–1334, 1970.6. Murphy, V.A. The Effect of Ground Characteristics on the Aseismic Design of Structures, Proc. 2nd World Conf. on Earthquake Engineering, Tokyo, pp. 231–247, 1960.7. Okamoto, S. Bearing Capacity of Sandy Soil and Lateral Earth Pressure during Earthquakes, Proc. 1st World Conference on Earthquake Engineering, California, pp. 27–1 to 27–26, 1956.8. Petrov, I. Variational Methods in Optimum Control Theory, Academic Press, New-York and London, 216 p., 1968.



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< previous page page_426 next page >Page 4269. Prakash, S. and Basavanna, B.M. Earth Pressure Distribution behind Retaining walls during Earthquake, Proc. 4th World Conference on Earthquake Engineering, Chile, pp. 133–148, 1969.10. Prakash, S. Analysis of Rigid Retaining Walls During Earthquakes, International Conference on Recent Advances in Geotechnical Earthquake Engineetring and Soil Dynamics, Rolla, Missouri, pp. 993–1020, 1981.11. Seed, H.B. A Method for Earthquake Resistant Design of Earth Dams, Journal of the Soil Mechanics and Foundation Division, ASCE, Vol. XCII, N° SM1, 1966.12. Seed, H.B. and Martin G.R. The seismic coefficient in Earth Dam Design, Journal of the Soil Mechanics and Foundation Division, ASCE, Vol. XCII, N° SM3, 1966.13. Seed, H.B., Lee, K.L. and Idriss, I.M. Analysis of the Sheffield Dam Failure, Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. XCV, N° SM6, 1969.14. Sabzevari, A. and Ghahramani, A. Dynamic Passive Earth Pressure Problem, J. Geotechnical Eng. Div., ASCE, Vol. 100 (GT1), pp. 15–30, 1974.15. Soubra, A.H. Application de la Méthode Variationnelle au Problème de Détermination des Pressions Passives des Terres. Influence des Forces d’Ecoulement, Thesis, INSA Lyon, 200 p., 1989.16. Taniguchi, E. and Sasaki, Y. Back Analysis of a Landslide due to the Naganoken Seibu Earthquake of September 14, 11th ICSMFE, Speciality Session on Seismic Stability of Slopes, San Francisco, 1985.17. Wood, J.H. Earthquake-Induced Soil Pressures on Structures, Report N° EERL-73–05, Earthquake Engineering Research Laboratory, California Institute of Technology, Pasadena.



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< previous page page_427 next page >Page 427SECTION 7: SOIL-STRUCTURE-INTERACTION, FOUNDATIONS, PILES



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< previous page page_429 next page >Page 429Dynamic Stiffness of Unbounded Soil by Finite-Element Multi-Cell CloningJ.P.Wolf, C.SongInstitute of Hydraulics and Energy, Department ofCivil Engineering, Swiss Federal Institute of Technology, 1015 Lausanne, SwitzerlandABSTRACTThe ingenious concept of the standard cloning algorithm to calculate the dynamic stiffness is examined in depth using the scalar cases of a spherical cavity embedded in a full space and a two-dimensional wedge. It is shown that this one-cell cloning works only for special cases. The concept can, however, be expanded to multi-cell cloning, leading to a practical approach at least in the scalar case.INTRODUCTIONIn the cloning concept the essential notion of infinity is captured by stating that adding a finite part to an infinite quantity does not change its value. The fundamental idea of cloning is illustrated in Fig. 1 for the semi-infinite soil taking the embedment into account. Adding the bounded cell of finite elements to the semi-infinite domain with the characteristic length re results in a similar semi-infinite domain with length ri. The concept can be applied to their dynamic-stiffness matrices in the frequency domain by assembling the known dynamic-stiffness matrix of the cell and the unknown matrix of the unbounded soil referenced by the length re, which results in the unknown dynamic-stiffness matrix of the unbounded soil with length ri. As a relationship for the dynamic-stiffness matrices referenced by different lengths exists, the cloning algorithm leads to an expression for the dynamic-stiffness matrix of the unbounded soil as a function of that of the cell. Potentially, this method is a stand-alone finite-element formulation competent to capture the radiation condition at infinity without using analytical solutions.In the standard cloning algorithm pioneered by Dasgupta [1] over ten



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Figure 1: Fundamental concept of cloning algorithmyears ago, it is assumed that an average value of the characteristic lengths of the inner and outer boundaries ri, and re of the cell can be used in defining the dimensionless frequency of which the dynamic stiffness of the unbounded soil is a function. Or in other words the dynamic stiffnesses of the unbounded soil referred to the outer and inner boundaries are assumed to be equal. This is not consistent with the derivation of the dynamic stiffness of the cell. With the exception of cases where the dimensionless frequencies are the same at the inner and outer boundaries (such as for a soil layer or in the static case) incorrect results are obtained outside the high-frequency range. In particular, an artificial cutoff frequency exists below which no radiation of waves takes place. This is demonstrated in Wolf and Weber [2]. In the same reference, the procedure has been extended to take into account the variation of the dimensionless frequency from the inner to the outer boundary of the cell. This generalized cloning method results in ordinary nonlinear first-order differential equations for the dynamic stiffness with the dimensionless frequency as the independent variable. The system can be solved numerically (e.g. by Euler’s method) provided a starting value is known. The static stiffness cannot be used for this propose. Taking the limit of zero width of the cell results in differential equations which lead to the exact solution of the dynamic stiffness.A new procedure called multi-cell cloning is introduced in this paper. For n cells, the basic cloning equation can be formulated n times. An additional equation is introduced stating that the n+1 dynamic stiffnesses referred to all boundaries form a n−1 degree polynomial of the dimensionless frequency. The standard cloning of Dasgupta [1] corresponds to one-cell cloning.For the sake of illustration, all the different cloning algorithms are applied to the analysis of the dynamic stiffness of the spherical cavity in a full space with symmetric waves. In this scalar case for which analytical expres-sions can be formulated, valuable physical insight is gained. In addition, the dynamic stiffness with one coefficient of the out-of-plane motion of a wedge of a two-dimensional problem is addressed as an example.



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< previous page page_431 next page >Page 431ANALYTICAL SOLUTION AND BASIC CLONING EQUATIONA spherical cavity of radius a with uniform pressure of amplitude p embedded in a full space with shear modulus G, Poisson’s ratio ν and mass density ρ which leads to symmetric P-waves is used as the benchmark problem (Fig. 2). The dynamic-stiffness coefficient in the frequency domain S∞

Figure 2: Spherical cavity with uniform pressure (section)

Figure 3: Interior and exterior infinite domains and cell of spherical cavityis defined as

(1)with u0 denoting the amplitude of the radial displacement of the cavity’s wall. The governing scalar differential equation of motion equals

(2)where r is the radial coordinate and with the dimensionless frequency a0= ωa/cp (cp=dilatational-wave velocity). Considering only the outwardly propagating wave, S∞ is formulated as (see Wolf [3] pp. 16–20 for a detailed derivation)

(3)with the dimensionless dynamic-stiffness coefficient X

(4)



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< previous page page_432 next page >Page 432X is normalized with the static value

X(a0)=Xst[k(a0)+ia0c(a0)] (5)where k(ao) and c(a0) are the spring and damping coefficients. They are shown in Fig. 4 for ν=1/3, which is used throughout this paper.The concept of cloning is applied to the one-dimensional case of the spherical cavity (Fig. 3). The force-displacement relationship of the cell located between the interior and exterior boundaries is written as

(6)where [S] denotes the dynamic-stiffness matrix of the cell. The corresponding equations for the interior and exterior infinite domains are formulated as

(7)

(8)Eliminating Pi and Pe from equations (6), (7) and (8) results in

(9)It is convenient to introduce nondimensional stiffness coefficients as follows

(10)

(11)

(12)This transformation substituted in equation (9) leads to the basic cloning equation relating the stiffness coefficients of the infinite domains referenced to the interior and exterior boundaries to those of the cell

Xi=Dii−Die(Xe+Dee)−1Dei (13)where for a cell with a finite element based on a linear shape function in the radial direction

(14)



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< previous page page_433 next page >Page 433γ is the aspect ratio re/ri and a0i=ωri/cp. For a parabolic shape function, [D] is calculated using numerical quadrature. The degree of freedom corresponding to the middle node is condensed out. For accurate results, the aspect ratio γ should not exceed 1.25 and 1.6 for the linear and the parabolic finite elements, respectively. In addition, the cell width cannot be larger than a fraction of the wave length (e.g. 0.1 and 0.2 for linear and parabolic elements).Equation (13) can be used to calculate Xi (corresponding to ri) for a known Xe (corresponding to re). Applying this procedure repeatedly, whereby a0i in equation (14) changes, the variation of X for r less than the starting value can be determined. This actually corresponds to dynamic condensation. Alternatively, equation (13) can be solved for Xe as a function of Xi, which allows the variation of X for r larger than the starting value to be calculated. This procedure is called substructure deletion. Figure 4

Figure 4: Direct application of basic cloning equation

Figure 5:1-cell cloning (γ=1)shows the variation of the non-dimensional dynamic-stiffness coefficient X decomposed as specified in equation (5) starting with X=−0.5+ia00.997 at a0=20 for dynamic condensation, and starting with X=0.9904+ia00.0105 at a0=0.103 for substructure deletion. γ=1.03. The two curves coincide.ONE-CELL CLONINGAs explained in Dasgupta [1], Xi and Xe are assumed to be equal

Xi=Xe=X (15)Substituting in equation (13) leads to the quadratic equation in X

X2+(Dee−Dii)X−DiiDee+DieDei=0 (16)



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< previous page page_434 next page >Page 434Of the two roots the one corresponding to outwardly propagating waves exhibits a positive imaginary part (damping coefficient) and when the imaginary part vanishes, the real part has to be positive.In the limit γ→1 equation (16) is transformed to

(17)with the solution

(18)When compared to the analytical solution in equation (4), it can be seen that the static value and the limit for a0→∞ calculated from cloning are exact. X from equation (18) is plotted in Fig. 5. c vanishes up to the artificial cutoff frequency=1.5. Large discrepancies arise. The results for γ ≠1 exhibit the same tendency (not shown).The assumption of one-cell cloning expressed in equation (15) implies that a0 is independent of r. It is worthwhile to determine the corresponding physical system. Postulating the mass density to vary as

(19)leads to a linear variation of cp with r and thus to a constant a0. G and ν remain constant. The differential equation of motion of this system equals

(20)which leads to the dimensionless dynamic-stiffness coefficient specified in equation (18). This means that the one-cell cloning algorithm is actually solving another physical system with the mass density varying as specified in equation (19).GENERALIZED CLONINGTo distinguish between the X at the exterior and interior boundaries, Xe is expanded at Xi into a Taylor series (Wolf and Weber [2]) as

(21)Substituting into equation (13) leads to the generalized cloning equation (with X=Xi and a0=a0i)

(22)



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< previous page page_435 next page >Page 435which is a nonlinear ordinary first-order differential equation for X (a0). This differential equation can be solved numerically for increasing and decreasing a0, e.g., using the Euler scheme, provided a starting value is known. The behaviour is the same as that described using substructure deletion and dynamic condensation (Fig. 4).Valuable insight can be gained after taking the limit γ→1. Equation (22) is transformed to

(23)Its solution equals

(24)Equation (24) can be derived starting from equation (2) by including both the outwardly and inwardly propagating waves and by using equation (1). The integration constant of the first-order differential equation equals c2/c1 for c1≠0 and c1/c2 for c2≠0. Actually, c2=0 and c1=0 correspond to outwardly propagating and inwardly propagating waves, respectively.The limit of X for a0→0 is investigated. Equation (24) results in

(25)

(26)Equation (25) does not correspond to the static value Xst. As all other c1, c2 lead to the static value (equation(26)), Xst cannot be used as the starting value to solve the generalized cloning differential equation.In contrast, the generalized cloning equation can be solved starting from a0=∞. Transforming both the independent and dependent variables

(27)

(28)equation (23) is written as

(29)



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< previous page page_436 next page >Page 436where

(30)

(31)

(32)For a0→∞ and thus b0→1 and for dY/db0=o((1−b0)−2), Y=±i results. For outwardly propagating waves, the plus sign applies. For the limit a0→∞ the (transformed) generalized cloning differential equation contains the starting value. An adaptive Euler scheme starting from b0=1 and Y=i; is used to calculate the dynamic-stiffness coefficient down to the static value (Fig. 6). Excellent agreement results using, however, a very large number of steps.

Figure 6: Generalized cloning (γ=1)

Figure 7: Multi-cell cloningMULTI-CELL CLONINGThe basic cloning equation can be formulated repeatedly leading to multi-cell cloning.For n cells, with the same aspect ratio γ and each with its own exterior and interior boundaries, n+1 boundaries (each with its own dynamic-stiffness coefficient) arise (Fig. 7). Formulating the basic cloning equation (13) for cell j with the interior boundary j and the exterior boundary j+1 (j=1,…, n) leads to n equations in n+1 unknowns Xj

(33)To determine the n+1 equations needed to supplement equation (33), it is assumed that the Xj (j=1,…, n+1) form a polynomial of order n−1


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< previous page page_437 next page >Page 437(and not of order n) in the coordinate r. This results in

(34)where

lj(rj)=(rj−r1)…(rj−rj−1)(rj−rj+1)…(rj−rn+1) (35)The system of nonlinear equations is solved (for each α0) by the Newton-Raphson method. The starting value for this iterative algorithm is selected for each j as

Xj=Xst+iα0 (36)Multi-cell cloning can be used to calculate the dynamic-stiffness coefficient for any specific α0. Parabolic elements with an aspect ratio γ less than 1.6 and satisfying the fraction-of-the-wave-length criterion are used for the 2- and 3-cell cloning (Figs. 8 and 9). For two-cell cloning, a small

Figure 8: Two-cell cloning

Figure 9: Three-cell cloningcutoff frequency is still observed. Compared to the 1-cell cloning (Fig.5) leading to useless results in the intermediate frequency range, the results are highly accurate, making the multi-cell cloning procedure an attractive tool to calculate the dynamic-stiffness coefficient.For the limit γ→1, n-cell cloning will result in a polynomial for X of n+1 degree. For instance, for the case of 2-cell cloning the following cubic equation is obtained.


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(37)



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< previous page page_438 next page >Page 438When this equation has one real and one pair of complex conjugate roots, the complex root with the positive imaginary part is selected. The real root is always negative. When this equation has three real roots, the largest one is selected.Addressing the limit a0→∞ in equation (37), the high-frequency limit of X equals (1−3ν)/(1−ν)+ia0. This value is one-order more accurate than that of one-cell cloning in the higher-frequency range (compare k(ao) in Figs. 8 and 5).Solving equation (37), which is the same as applying the 2-cell cloning for γ→1, leads to inferior results (Fig. 10). In particular, the fictitious cutoff frequency is increased. This phenomenon can be explained qualitatively as follows. For non-vanishing c1 and c2 the solution for X specified in equation (24) oscillates with a0. For c1=0 and c2≠0 the solution is smooth, as it is for c2=0 and c1≠0. Multi-cell cloning sets the n-th derivative equal to zero. When the width of the cells is large (γ>1), the oscillating solutions are suppressed to a larger extent than when the width is small (γ→1). Selecting a larger γ is thus more effective in favouring the smooth solutions, resulting in higher accuracy.

Figure 10: Two-cell cloning (γ=1)

Figure 11: Semi-infinite wedge with prescribed linear displacementOUT-OF-PLANE MOTION OF SEMI-INFINITE WEDGEThe out-of-plane motion (SH-waves only) of a wedge (shear modulus G, mass density ρ with an opening angle of α=π/6) with a free and a fixed boundary extending to infinity is examined as a two-dimensional problem (Fig. 11). The dynamic-stiffness coefficient corresponding to an out-of-plane motion ν(θ) prescribed as a linear function in the circumferential direction



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< previous page page_439 next page >Page 439on the arc is addressed.Using the technique of separation of variables, the corresponding dynamic-stiffness coefficient S∞ in the frequency domain (i.e. the force amplitude at point O corresponding to a unit displacement amplitude) equals

(38)where

(39)

and with is the second-kind Hankel function of order λj. For the presentation of the results S∞ is formulated analogous to equation (5). The static value equals 0.5427G. The spring coefficient k(a0) and damping coefficient c(a0) are plotted in Fig. 12.

Figure 12: Dynamic-stiffness coefficient of wedgeFor all cloning computations one or more cells composed of one finite element with the shape of an annular wedge and with a bi-linear shape function for the displacement are used. The aspect ratio γ is equal to 1.1 if not forced to be smaller to satisfy the fraction-of-the-wave-length criterion. Again, 1-cell cloning leads to a significant fictitious cut-off frequency and useless results in the important intermediate-frequency range. 2- and 3-cell cloning perform well for all frequencies.CONCLUDING REMARKS1. The standard cloning algorithm (1-cell cloning) actually determines the dynamic stiffness of a different physical system for which the density decreases proportionally to the square of the radial coordinate.



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< previous page page_440 next page >Page 440 2. Multi-cell cloning (2- or 3-cells) as applied to the scalar case (associated with even a two-dimensional homogeneous problem) leads to a highly accurate dynamic stiffness for any specific frequency.3. When the geometry of the boundary of the semi-infinite domain permits the transformation to independent scalar equations (as e.g. for a circle by using a Fourier series in the circumferential direction), each one can be solved using cloning.4. The real challenge consists of determining the dynamic-stiffness matrix for an arbitrary geometry of the boundary using multi-cell cloning. In particular, the criterion for selecting the roots of the nonlinear equations must be established.ACKNOWLEDGMENTThe authors wish to express their gratitude to Professor J.Descloux and Mr. H.Débonnaire for in-depth discussions.REFERENCE[1] Dasgupta, G. A Finite Element Formulation for Unbounded Homogeneous Continua. Journal of Applied Mechanics. Vol. 49, pp. 136–140 March 1982.[2] Wolf, J.R and Weber, B. On Calculating the Dynamic-Stiffness Matrix of the Unbounded Soil by Cloning, (Ed. Dungar, R. et al.) pp. 486–494, International Symposium on Numerical Models in Geomechanics, Zurich, Switzerland, 1982. A.A.Balkema, Rotterdam.[3] Wolf, J.P. Soil-Structure-Interaction Analysis in Time Domain, Prentice-Hall, Englewood Cliffs, N.J., 1988.



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< previous page page_441 next page >Page 441Application of the Hybrid Frequency-Time-Domain Procedure to the Soil-Structure Interaction Analysis of a Shear Building with Multiple NonlinearitiesG.R.DarbreSwiss Division of Safety of Dams, P.O.Box 2743, CH-3001 Bern, SwitzerlandABSTRACTThe hybrid frequency-time-domain procedure is applied to the nonlinear seismic analysis of a 6-story shear building interacting with the supporting soil. Both the frequency dependence of the foundation stiffness coefficients and the nonlinear hysteretic characteristics of the individual stories are retained in the analysis. The reliability of the analysis results is confirmed by way of comparison with the results of a time-stepping algorithm for the specialized case of constant soil-stiffness coefficients.The influence of the frequency dependence of the foundation stiffness coefficients on the seismic response is less important in the nonlinear case than in the linear case. For all practical purposes, it may be disregarded in the nonlinear case.1. INTRODUCTIONThe hybrid frequency-time-domain (hftd) procedure is an attractive computational tool aimed at performing the dynamic analysis of nonlinear systems with frequency-dependent characteristics. Such systems include nonlinear structures which interact with infinite or semi-infinite media (nonlinear soil-structure and fluid-structure interacting systems) and systems which have substructures whose internal degrees of freedom are dynamically condensed out of the equation of motion.The hftd procedure consists of replacing the nonlinear system being analyzed by a pseudo-linear system obtained by transferring the nonlinear component of the internal forces to the right-hand side of the equation of motion. This latter force component depends on the—unknown—motion and is obtained iteratively. The resulting left-hand side of the equation of motion



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< previous page page_442 next page >Page 442is processed in the frequency domain and frequency-dependent characteristics are accounted for directly (see Refs. 1 to 3 for details).The feasibility of such an approach has been demonstrated analytically and by the numerical analysis of an uplifting rigid block interacting with the supporting soil in Ref. 1. More recently, the hftd procedure has been applied to the seismic analysis of a reactor building on a sliding-type isolation (Ref. 2). In this study, the influence of the frequency dependence of the foundation stiffness coefficients on the nonlinear seismic response of the base-isolated reactor building has been evaluated. This influence was found to be negligible for all practical purposes.The latter analysis of a system of practical importance is a significant contribution to the field of nonlinear soil-structure interaction analysis as no approximations, being the use of constant foundation stiffness coefficients anchored at a specific frequency or of linearized isolation characteristics, are introduced. Rather, both the frequency dependence of the foundation stiffness coefficients and the nonlinear characteristics of the isolation are retained in the analysis.The applications of Refs. 1 & 2 are limited to the seismic analysis of soil-structure interacting systems with a single nonlinearity. The present study is aimed at extending the area of application of the hftd procedure so as to encompass dynamic soil-structure interacting systems with multiple nonlinearities. Although theoretically feasible (Ref. 1), such applications have not been performed yet.The 6-story shear building with hysteretic elasto-plastic story characteristics used in this study is introduced in Section 2. The seismic response of the building interacting with the supporting soil is presented in Section 3. Five stories simultaneously perform in the nonlinear range during the seismic excitation. The occurrence of these multiple nonlinearities does not prevent the hftd procedure of performing in a satisfactory manner. Special issues are addressed in Section 4. They include an evaluation of the accuracy of the analysis results by way of comparison with the results obtained by a time-stepping algorithm, a discussion on what effects changing the length of the time segments of integration and reducing the Nyquist frequency have on the response, and a brief discussion on the spectral radius to which the criterion of stability of the hftd procedure is related (Ref. 1).2. SYSTEM INVESTIGATED AND FUNDAMENTAL RELATIONS2.1 System investigatedThe system investigated consists in the 6-story shear building interacting with the supporting soil shown in Fig. 1.



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< previous page page_443 next page >Page 443Shear building The building has a story height of 3m, a floor mass of 150*103kg and a floor mass moment of inertia of 2.7*106 kg*m2. The force-displacement relation of the individual stories is elasto plastic, with a linear stiffness coefficient of 300*106 N/m and a yielding force varying linearly over the height of the building from 1.5*106 N in the first story to 0.5*106 N in the sixth story. The case of a constant yielding force of 1.5*106 N in all stories is also investigated. The small-amplitude energy dissipation within the building is stiffness proportional with 2% critical damping at 1.7Hz (lowest fixed-base natural frequency, corresponds to a period of 0.6 seconds).

Fig. 1—Shear Building investigatedSoil The soil consists in a layer of 7 meter depth resting on a semi-infinite halfspace. The properties of the layer are: shear modulus of 20* 106 N/m2, mass density of 2*103kg/m3 (Cs=100 m/sec), Poisson’s ratio of 1/3 and hysteretic damping ratio of 5%. The properties of the halfspace differ only in the value of the shear modulus equal to 80*106 N/m2 (Cs=200m/sec).The dynamic-stiffness matrix [Sgbb(ω)]=[Kgbb(ω)]+iω[Cgbbω)] of the massless circular rigid foundation in welded contact with the layer is frequency dependent. The elements of [Kgbb(ω)] are shown in Fig. 2a. They are normalized with respect to the corresponding static values of Khh=889*106 N/m, Krr=33*109 Nm and Khr=Krh=236*106 N (the coupling coefficient is divided by 5 for better representation). The elements of [Cgbb(ω)], shown in Fig. 2b, are normalized through multiplication by Cs/a (shear wave velocity



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< previous page page_444 next page >Page 444of layer divided by foundation radius of 7m) and division by the static value of the corresponding coefficient.The lowest natural frequency of the flexible-base building is 1.4Hz (small amplitude, based on static foundation spring coefficients).Excitation The seismic excitation is given by a horizontal free-field acceleration with a peak value of 25% g. The time history is identical to that used in Refs. 4 and 2 in which its trace and response spectrum are presented.

Fig. 2—Dynamic stiffness coefficients of massless circular rigid foundation a) spring coefficients b) damping coefficients2.2 Fundamental relationsThe nonlinear seismic analysis of the shear building interacting with the supporting soil is performed with the hybrid frequency-time-domain procedure. The equation of motion in the frequency domain is (in total displacements)

(1)The elements of the dynamic-stiffness matrices identified by the superscript s refer to the pseudo-linear structure (shear building with linear stories) and those identified by the superscript g refer to the massless foundation in welded contact with the soil. The subscript s refers to the degrees of freedom associated with floors 1 to 6 and the subscript b refers to those associated with the foundation floor 0. ugb(ω) identifies the input motion. Q(ω) is the Fourier transform of the vector of correcting forces Q(t)=F(t)Pseudolinear-F(t))linear which represents the difference between the internal forces associated with the pseudo-linear system and the internal forces associated with the actual nonlinear system. Denoting by Si



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the internal shear force in story i, the correction force in the horizontal direction at floor j is

(2a)The correction force Q(t)r0 in the rocking direction is

(2b)The internal story forces S are obtained as described in Ref. 4.2.3 Implementation of hybrid frequency-time-domain procedureThe hftd procedure is implemented based on the flow chart of Ref. 2 (because all degrees of freedom can be affected by the occurrence of nonlinearities, no dynamic condensation is performed here). The implementation of the procedure is straightforward and the analysis is performed without major difficulties. It must be emphasized that the frequency dependence of both the real and imaginary parts of the foundation stiffness coefficients is duly accounted for in the analysis, although the problem is nonlinear. This application demonstrates how efficiently the hftd procedure can be used in the dynamic analysis of systems with multiple nonlinearities which have frequency-dependent characteristics. This is a significant result as, to the author’s knowledge, no other similar study has been successfully completed previously with such efficiency using the hftd procedure or any other method. A major exception consists in the study of Ref. 5 in which the nonlinear seismic analysis of a one-story building interacting with the supporting halfspace is performed in the time domain using approximate analytical impulse response functions for the contribution of the halfspace to the equation of motion, Ref. 6.3. SEISMIC RESPONSEThe peak values of floor acceleration ü and of interstory deformation δ occurring during the seismic excitation of 25 seconds duration are presented in Tables 1a and 1b, respectively. The values obtained by disregarding the frequency dependence of the foundation stiffness coefficients are also indicated. They are obtained by introducing the zero-frequency foundation spring coefficients (static coefficients) and the infinite-frequency foundation damping coefficients in the analysis. The peak values obtained for the linear building (infinite yielding forces) are included for comparison purposes as well.



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< previous page page_446 next page >Page 446Table 1a—Peak total accelerations [m/sec2]

system frequency dependence ü0 ü1 ü2 ü3 ü4 ü5 ü6

nonlinear, yes 2.55 2.37 2.57 2.15 2.55 2.58 3.30 0.55

varying no 2.46 2.45 2.65 2.22 2.51 2.68 3.32 0.46

nonlinear, yes 2.56 2.70 2.73 2.72 2.98 3.54 4.56 0.49

constant no 2.43 3.07 3.15 2.72 3.00 3.63 4.85 0.44

linear yes 2.56 2.37 3.62 4.13 3.82 4.35 4.88 0.44

no 2.39 2.62 3.95 4.46 4.57 5.04 5.60 0.39Table 1b—Peak interstory deformations [cm]system frequency dependence δ1 δ2 δ3 δ4 δ5 δ6

nonlinear, yes 0 .88 1 .45 1 .24 1.01 0.46 0.16

varying no 1.12 1.86 1 .23 1 .07 0.44 0.16

nonlinear, yes 1 .28 0 .87 0 .62 0 .50 0.40 0.23

constant no 2 .05 0 .92 0 .75 0 .52 0.42 0.24

linear yes 0 .98 0 .91 0 .79 0 .64 0.45 0.24

no 1 .18 1 .08 0 .93 0 .74 0.52 0.28The level of nonlinearities occuring during the excitation is appreciated from Fig. 3a in which the total number of stories performing in the plastic range is indicated as a function of time for the case of the varying yielding forces. Up to 5 stories simultaneously perform in the plastic range. The same information is presented in Fig. 3b for the case of the constant yielding forces.

Fig. 3—Number of stories performing in plastic range a) varying yielding forces b) constant yielding forces



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< previous page page_447 next page >Page 447Up to 4 stories simultaneously perform in the plastic range in this latter case. The repeated incursions in the plastic range result in a decrease in the acceleration response of the building. This decrease is the largest for the case of the varying yielding forces.The influence of the frequency dependence of the foundation stiffness coefficients on the acceleration response is the largest for the linear building and the smallest for the nonlinear building with varying yielding forces. This comes from the fact that the increase in structural flexibility and structural damping associated with the nonlinear hysteretic action of the individual stories reduces the relative overall importance of the soil flexibility and of the soil energy dissipation. Consequently, the influence on the response of the variation of the foundation dynamic-stiffness coefficients with frequency is also reduced.The deformation response is more sensitive to the soil modeling than the acceleration response. The ductility requirement of the individual stories is thus affected by the frequency dependence of the foundation dynamic-stiffness coefficients in all cases. This influence must however be critically assessed as the calculated deformation values of bilinear systems are sensitive to the length of the time step used in the time integration, Ref. 7 (a time increment of 0.01 second is used here in the fast Fourier transform). Part of the differences observed between the deformation response calculated by duly considering the frequency dependence of the foundation dynamic-stiffness coefficients and that calculated under the approximation of constant foundation spring and damping coefficients is believed to be due to the time discretization (see also Section 4.1).From a practical point a view the differences in response observed in the nonlinear cases hardly justify using the much more elaborate analysis which accounts for the frequency dependence of the foundation dynamic-stiffness coefficients and for the structural nonlinearities rather than performing a simpler time-domain nonlinear analysis using constant foundation spring and damping coefficients. A similar conclusion was reached in the case of the reactor building on sliding-type isolation, Ref. 2. (The study of Ref. 5 led to the conclusion that consideration of the interaction effect is not as important in the design of yielding structures as in the design of elastic structures).4. SPECIAL STUDIES4.1 Comparison with time-stepping algorithmThe nonlinear equation of motion can be solved directly in the time domain when the foundation dynamic-stiffness coefficients are approximated by the constant zero-frequency spring coefficients and the infinite-frequency damping coefficients. The associated maximum response values obtained by an



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< previous page page_448 next page >Page 448explicit integration are compared in Table 2 with the values obtained by the hftd procedure for the same foundation stiffness coefficients.Table 2a—Peak total accelerations [m/sec2]

system scheme ü0 ü1 ü2 ü3 ü4 ü5 ü6

nonlinear, hftd 2.46 2.45 2.65 2.22 2.51 2.68 3.32 0.46

varying expl. 2.43 2.45 2.65 2.23 2.50 2.70 3.33 0.47

nonlinear, hftd 2.43 3.07 3.15 2.72 3.00 3.63 4.85 0.44

constant expl. 2.39 3.15 3.10 2.71 2.99 3.67 4.78 0.44

linear hftd 2.39 2.62 3.95 4.46 4.57 5.04 5.60 0.39

expl. 2.35 2.61 3.96 4.45 4.58 5.04 5.59 0.39Table 2b—Peak interstory deformations [cm]system scheme δ1 δ2 δ3 δ4 δ5 δ6

nonlinear, hftd 1.12 1.86 1.23 1.07 0.44 0.16

varying explicit 0.99 1.93 1.35 1.08 0.43 0.17

nonlinear, hftd 2.05 0.92 0.75 0.52 0.42 0.24

constant explicit 1.51 0.96 0.75 0.52 0.42 0.24

linear hftd 1.18 1.08 0.93 0.74 0.52 0.28

explicit 1.18 1.08 0.93 0.74 0.52 0.28Excellent agreement in maximum acceleration values is found considering that two different methods of time integration are used (hftd procedure and explicit scheme) each having a different length of time step (0.01 second for the hftd procedure and 0.0025 second for the explicit scheme). The discrepancies which occur in the peak deformation values are due partly to the bilinear nature of the nonlinearities. The determination of yielding inception and of yielding conclusion occurs at time values which are multiples of the length of the time step of integration. No account is made of the fact that yielding inception or yielding conclusion can occur at intermediate time values. A loss of accuracy which differs for each procedure of time integration ensues. This is illustrated by comparing the number of stories performing in the plastic range (Fig. 4) and the deformation time history of the first story (Fig. 5) as obtained by applying the hftd procedure and the explicit time-integration scheme for the case of constant yielding forces. The differences in interstory deformation initiate at time values of 7.1, 10.1, 13.7, 14.4 and 23.1 seconds. At these particular time values, the hftd procedure and the explicit scheme predict incursions in the plastic range which are one time step apart.



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Fig. 4—Number of stories performing in plastic range a) obtained by hftd procedure b) obtained by explicit integration

Fig. 5—Deformation of first story a) obtained by hftd procedure b) obtained by explicit integrationIn view of these remarks the reliability of the results obtained by application of the hftd procedure is confirmed by the comparison.4.2 Length of time segmentsOne deficiency of the hftd procedure is that no rules exist at present to assist the analyst in selecting the length of the time segments which must be introduced in the analysis, Refs. 1 & 2. In order to gain some insight into this question, the analysis has been repeated for various lengths of time segments for the case of the varying yielding forces. The associated number of iterations is shown in Table 3a and the maximum response values in Tables 3b and 3c.



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< previous page page_450 next page >Page 450Table 3a—Total number of iterations

number of segments time steps per segment total number of time steps total number of iterations

45 or less 56 or more about 2′500 diverge

50 50 2′500 502

60 42 2′520 816

75 34 2′550 646

100 25 2′500 923Table 3b—Peak total accelerations [m/sec2]

number of segments ü0 ü1 ü2 ü3 ü4 ü5 ü6

50 2.55 2.37 2.57 2.15 2.55 2.58 3.30 0.55

60 2.54 2.37 2.57 2.14 2.60 2.64 3.34 0.53

75 2.54 2.37 2.57 2.15 2.56 2.58 3.29 0.55

100 2.54 2.37 2.57 2.14 2.57 2.71 3.42 0.50Table 3c—Peak interstory deformations [cm]

number of segments δ1 δ2 δ3 δ4 δ5 δ6

50 0.88 1.45 1.24 1.01 0.46 0.16

60 1.04 1.86 1.73 0.87 0.41 0.17

75 0.83 1.52 1.21 1.03 0.44 0.16

100 1.05 2.14 2.19 1.26 0.44 0.17The pattern observed here is slightly different from the one observed in the studies of Refs. 1 and 2 where it was recognized that selecting too many time segments wastes computer time, that selecting too few time segments leads to a divergent solution, and that the total number of iterations remains essentially constant over a rather wide range of selection of lengths of time segments. In this present application, selecting too few time segments still leads to a divergent solution. However, the total number of iterations is more largely affected by the number of time segments introduced in the analysis; introducing 60 time segments surprisingly requires more iterations than introducing 50 or 75 time segments. Also, a new effect is identified for the cases in which a large number of iterations are required (100 and 60 time segments). Aliasing effects apparently become more important thus affecting the quality of the deformation response (see also Section 4.1). While this might not be of great concern in this application as the acceleration response is still fairly accurate, this points out a possible source of inaccuracies in future applications of the hftd procedure to systems whose analysis requires



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< previous page page_451 next page >Page 451a very large number of iterations. Overcoming this unwelcome phenomenon could be attempted by way of recovering the appropriate initial conditions by adding a free-vibration component of response after each iteration (see e.g. Ref. 8). For the time being, it is recommended that an analysis involving multiple nonlinearities be repeated with various integration parameters and that the accuracy of the calculated response be verified by comparing the results of the various analyses.Unless indicated otherwise, the results presented in this study have been obtained using 50 segments of 50 time steps of length ∆t=0.01 second and Tfft=4’096*∆t. Decaying loading functions of 1.5 seconds duration have been appended at the end of the segments (Ref. 2).4.3 Spectral radius and Nyquist frequencyThe variation of the spectral radius with real frequencies is shown in Fig. 6a for the most severe nonlinear case which can possibly occur in the building investigated, namely that of all stories performing simultaneously in the plastic range. It is shown in Fig. 6b for the most severe case which does occur in this study, namely that of the five lowest stories performing simultaneously in the plastic range while the sixth story remains elastic. The spectral peaks occurring near the structural natural frequencies of the linear building are clearly recognized in Fig. 6. (The spectral radius of the undamped soil-structure system is infinite at the natural frequencies as the dynamic stiffness matrix of the pseudo-linear system is zero at these frequencies).

Fig. 6—Variation of spectral radius with real frequenciesa) 6 stories performing simultaneously in plastic rangeb) 5 stories performing simultaneously in plastic rangeWhile the criterion of stability for the hftd procedure depends on the value of the spectral radius evaluated at the frequency value of -iΩ (minus unit imaginary number times Nyquist frequency) rather than Ω Ref. 1, it is still interesting to note that the spectral radius at the Nyquist frequency of 50 Hz is well below the critical value of unity as it equals 0.08 (5 stories simultaneously in the plastic range). In the present case, the spectral radius at iΩ cannot be evaluated readily because the soil stiffness coefficients are given



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< previous page page_452 next page >Page 452as functions of real frequencies rather than as functions of imaginary frequencies.The analysis for the varying yielding forces has been repeated using a Nyquist frequency of 25Hz (∆t=0.02 second) with the objective of assessing to which extent the calculation is affected by such a change as the spectral radius at 25Hz is 0.41. The corresponding results are compared in Table 4 with the results obtained for Ω=50Hz (∆t=0.01 second). They are essentially identical, considering that a time step of 0.02 second is somewhat too long to follow adequately the hysteretic paths of the various stories. 510 iterations are required to process the 50 segments of 25 time steps of 0.02 second duration (versus 502 for the 50 segments of 50 time steps of 0.01 second duration).Table 4a—Peak total accelerations [m/sec2]

Nyquist frequency ü0 ü1 ü2 ü3 ü4 ü5 ü6

25 [Hz] 2.55 2.37 2.57 2.15 2.55 2.58 3.30 0.55

50 [Hz] 2.79 2.38 2.61 2.23 2.49 2.57 3.23 0.56Table 4b—Peak interstory deformations [cm]Nyquist frequency δ1 δ2 δ3 δ4 δ5 δ6

25 [Hz] 0.88 1.45 1.24 1.01 0.46 0.16

50 [Hz] 0.94 1.46 1.00 1.07 0.48 0.165. CONCLUSIONSThis application of the hybrid frequency-time-domain procedure to the soil-structure interaction analysis of a building with multiple nonlinearities demonstrates the effectiveness of the hftd procedure as an analytical/numerical tool aimed at performing the dynamic analysis of nonlinear systems with frequency-dependent characteristics. The analysis of this system of practical importance is a significant contribution to the field of nonlinear soil-structure interaction as no approximations, being the use of constant foundation stiffness coefficients anchored at a specific frequency or of linearized story characteristics, are introduced. Rather, both the frequency dependence of the foundation stiffness coefficients and the nonlinear hysteretic characteristics of the individual stories are retained in the analysis.



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< previous page page_453 next page >Page 453The implementation of the procedure and the ensuing calculation do not present any difficulties. The calculated response is reliable, as demonstrated by way of comparison with the results obtained by a time-stepping algorithm for the specialized case of constant soil-stiffness coefficients. Slightly deficient results in deformation response occur for those integration parameters which lead to a very large number of iterations. This is believed to be due to aliasing effects. Until the procedure is modified in order to eliminate this unwelcome phenomenon, it is suggested to repeat an analysis involving multiple nonlinearities with various integration parameters.The influence of the frequency dependence of the foundation stiffness coefficients is less important in the nonlinear cases investigated than in the linear case. For all practical purposes, this frequency dependence may be neglected in the nonlinear cases. Because the data base is still limited, this conclusion should not be extended to soil-structure interacting systems whose characteristics depart significantly from the ones of the building studied here.6. ACKNOWLEDGMENTSThis study was conducted during a visiting stay at the University of California Berkeley. The financial support of the Swiss National Science Foundation under grant No. 82.598.0.88 and the computer time provided by the University of California are gratefully acknowledged. The hospitality of Prof. A.K.Chopra during the author’s stay at UC Berkeley is sincerely appreciated. 7. REFERENCES1. G.R.Darbre and J.P.Wolf, ‘Criterion of stability and implementation issues of hybrid frequency-time-domain procedure for nonlinear dynamic analysis’, Earthquake eng. struct. dyn. 16, 569–581 (1988).2. G.R.Darbre, ‘Seismic analysis of non-linearly base-isolated soil-structure interacting reactor buikding by way of the hybrid frequency-time-do-main procedure’, Earthquake eng. struct. dyn. 19, 725–738 (1989).3. G.R.Darbre, ‘On the application of the hybrid frequency-time-domain procedure to the seismic analysis of non-linear systems with frequency-dependent characteristics’, 9th european conf. earthquake eng., Moscow (1990).4. G.R.Darbre, ‘Nonlinear seismic analysis of base-isolated reactor building by the hybrid frequency-time-domain procedure’, Proc. 10th int. conf. struct. mech. reactor techn., Anaheim K, 661–666 (1989).5. A.S.Veletsos and B.Verbic, ‘Dynamics of elastic and yielding structure-foundation systems’, Proc. 5th world conf. earthquake eng., Rome 2, 1905–1908 (1972).



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< previous page page_454 next page >Page 4546. A.S.Veletsos and B.Verbic, ‘Basic response functions for elastic foundations’, J. eng. mech. div. ASCE 100, 189–202 (1974).7. R.Villaverde and R.C.Russell, ‘Scheme to improve numerical analysis of hysteretic dynamic systems’, J. struct. div. ASCE 115, 228–233 (1989).8. C.E.Ventura and A.S.Veletsos, ‘Steady-state and transient responses of non-classically damped linear systems’, Earthquake eng. struct. dyn. 14, 595–608 (1986).



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< previous page page_455 next page >Page 455Dynamic Soil-Structure-Interaction of Nonlinear Shells of Revolution in the Time Domain W.Wunderlich, B.Schäpertöns, H.Springer, C.TemmeTechnical University, Arcisstr. 21, Postfach 202 420, D-8000 Munich 2, GermanyABSTRACTThis paper deals with the influence of the soil on the nonlinear dynamic response of axisymmetric structures under arbitrary excitation. Nonsymmetrically loaded shells of revolution, modelled through ring elements in the axial direction, are coupled with isoparametric continuum ring elements for the soil. The numerical simulations are performed in the time domain using the finite-element-method.INTRODUCTIONThe realistic assessment of the nonlinear vibration characteristics of liquid storage tanks, i.e. under strong earthquake excitation, requires the inclusion of the liquid and the soil in the computational model. In the linear case, these procedures are well established, and they are normally performed in the frequency domain. However, in the more realistic nonlinear case, computation in the time domain must be used. Here, the soil region is subdivided into two parts: a near field, which is fully discretized and permits to treat the nonlinearities of the soil, and the far field, which enables the treatment of radiation of energy through infinite elements.To avoid the requirement of an expensive, fully three-dimensional discretization procedure, a semi-analytical approach for the nonlinear analysis of structures with rotational geometry was developed, which reduces the discretization effort to one dimension for the shell-structures and to two dimensions for the soil using Fourier series.The interior region of the soil is modelled by standard continuum ring elements. For the unbounded region a solution procedure is desirable, which do not require excessive numerical effort, and can easily coupled to the exist-



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< previous page page_456 next page >Page 456ing shell-structure. The exterior region of the soil may be considered as linear and is simulated by the—later described—infinite elements.

Fig. 1. Typical soil-structure systemIn comparison with the Boundary-Element-Method the use of infinite elements has, among other things, the advantage of decoupled boundary nodes resulting in a smaller bandwith of the system matrices. It also avoids the subdivision of the soil into a large number of elements if the usual approach is applied to unbounded domains.SHELLSAs described in some detail in [1–3] and the references given there, a semi-analytic spatial discretization is used to treat the governing nonlinear shell equations, which are valid for arbitrary shells of revolution undergoing large nonaxisymmetric deflections and moderate rotations, and the Newmark temporal operator is employed to perform a discretization in time. In contrast to standard two-dimensional displacement finite element approaches we start from the set of partial differential equations which govern the displacement variables and stress resultants appearing in the line integrals around the circumferential boundaries. Furthermore, the circumferential variation of all geometrical and field variables is approximated by Fourier series and the remaining first-order differential equations are integrated numerically along a finite meridional interval using an asymptotically exact numerical integration procedure. In this fashion we first obtain for each Fourier wave number n a linear stiffness matrix Kl, which connects the components of the associated displacement vectors at the nodal circles of a ring element as well as the various nodal vectors of the applied external



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< previous page page_457 next page >Page 457loads λp, the consistent mass matrices M, the damping matrices C and the pseudo-load vectors Ps. Due to the geometric nonlinearities the latter is a linear, quadratic and cubic function of all n displacement vectors.After assembling the matrices and vectors of the individual ring elements in the usual manner, one obtains n implicit sets of nonlinear (incremental form), algebraic, second-order ordinary equations in time. Considering the various Fourier harmonics to be collected into corresponding single matrices and vectors the equations may be written concisely in the form

(1)where the matrices M, C and K are block diagonal. The ‘secant‘-vector Ps is given by

ps(v0, v)=pt(v0, v)+pts(v0,v) (2)where the ‘tangent‘-vector Pt is linear in the increment v, while the remaining vector Pts is a quadratic and cubic function of v. The reference configuration are denoted by v0. Also, Pt, Kl, and the familiar global tangential stiffness matrix Kt are formally related by

Kt(v0)v=Klv+pt(v0, v) (3)FLUIDThe liquid filling may be taken into account by substructure methods in the numerical model. The fluid is assumed as inviscid and incompressible. In the most practical cases this description according to linear potential flow theory is sufficient [4, 5]. The incompressibility of the fluid allows a static condensation of the hydrodynamic pressure of the form of the equations of motion for a ’dry’ shell with a symmetric structure. The mass matrix contains additional terms which account for the interaction. In a formal sense, these additional terms may be viewed as ’added masses’ although, strictly speaking, they are ’added pressures’.MODELLING OF THE SOILThe soil region near the structure is fully discretized by isoparametric finite elements using Fourier decomposition. This procedure implies the reduction of the originally three-dimensional discretization problem to only two dimensions. At this stage the behavior of the soil is treated linear-elastic, but it is straightforward to consider the nonlinear effects of the soil i.e. [6].



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< previous page page_458 next page >Page 458Absorbing infinite elements for the time domainThe radiation of energy into the half space is simulated by a doubly asymptotic approximation. The infinite elements represent the behavior of the unbounded domain. The virtual work in the unbounded domain may be splitted into two parts:

(4)The first term of the right hand side represents the virtual work of the initial forces as a limit in the case of high frequencies and the second term represents the static case as a limit for low frequencies [7],For the high frequencies the static stiffness is negligible. To prevent outgoing radiating waves from reflecting at the edges of the discretized region an artificial absorbing layer is introduced. Following the approach of Lysmer/ Kuhlemeyer [9] viscous damping forces are arranged along the boundary to infinity. The virtual work of the inertial

forces in the infinite domain must be equivalent to the virtual work of the stresses on the surface between the discretized near field and the infinite far field. Thus, the first term of equ. (4) is replaced by

(5)By adding these two parts the doubly asymptotic approximation of the behavior of the surrounding half space is obtained:

(6)Static stiffness For the evaluation of the static stiffness matrix for one element of the surrounding elastic medium the formulation of ‘mapped’ infinite elements due to Zienkiewicz [8] is applied. With geometric shape functions the original infinite domain is mapped to a finite domain, the parameter space.The mapping into two dimensions is performed by using Lagrange-polynoms in the finite domain and special rational functions in the infinite direction. The shape functions for the unknowns are varying like inverse powers with the distance from the pole, and there the geometric shape functions approach infinity.This formulation is extended to three dimensions by the Fourier decomposition.



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< previous page page_459 next page >Page 459Absorbing layer In the formulation for the special boundary conditions the approach of Lysmer/Kuhlemeyer [9] is employed extending the relations to

Fig. 2. Stress componentsthree dimensions. The special conditions are assumptions which are verified by numerical results. For the one dimensional case this assumption is exact. The boundary conditions are

(7)with the two wave-velocities cs (shear) and cp (dilatational), the velocity , the density and indices n, t, φ for the normal, tangential and the circumferential direction.Transformation of the stresses and the velocities to their global components yields the specific damping matrix

(8)Here, T is the rotation matrix containing the direction cosines. By inserting the shape functions NA for the displacements at the surface, the damping matrix for the absorbing layer reads

(9)



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< previous page page_460 next page >Page 460EXCITATIONUsually the earthquake excitation of a system is defined as a rigid base excitation or as a basement rock excitation. Now the soil is treated as an infinite domain. So no boundaries exist, where the input motion could be defined. Furthermore, in most cases only the free field motions are known.It is possible, however, to express the effective earthquake forces also in terms of the free field motion. The system is divided into three parts: The structure, the soil and the interface between the structure and the soil. When the structure is superposed on the foundation, the response may be divided into the free field motions and the added response resulting from the soil-structure-interaction.The effective force vector results from the added structure acted upon by the free field motions. In fact, only the free field displacements of the interface are needed. This formulation makes more feasible to use frequency independent boundaries, since the source of excitation is not close to the boundary [10, 11].Here, the added motions of the structure are total displacements, but those of the interface are not. Assuming that the free field excitation at the interface is a rigid body motion, the corresponding parts of the structure may behave nonlinear. Defining the motions of the structure relative to those interface displacements simplifies the right hand side of the equations of motion as the stiffness terms drops out, and thus permits the nonlinear treatment of the total structure.NUMERICAL EXAMPLES1. Horizontal pulse loading on elastic half spaceTo show the absorbing properties of the infinite elements a test problem is considered. The elastic half space is loaded with a horizontal pulse as a generally three dimensional problem. Using Fourier decomposition the horizontal pulse corresponds to a loading in the first harmonic.

The numerical calculations are performed with the values cp=0.5m/s, cs=0.267m/s, µ=0.3 and . The half sine pulse has a duration of 2 seconds. It is applied as a surface load on the first element near the axis of revolution.The mesh consists of 9 times 9 elements of width 0.5m. The infinite elements with the absorbing layers are placed at the artificial boundary to infinity. With these values, the dilatational waves reach the boundary in 9 seconds and the shear waves in 17 seconds. No material damping is present in the system.



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Fig. 3. Discretization of the half spaceCompared to fixed boundary conditions, the numerical results show the good performance of the viscous boundary in eliminating wave reflections. In Fig. 4 the propagation of waves in the two main directions X and Y can be observed. The two kinds of waves -dilatational and shear- may be identified in the sampled plots. The dilatational waves, here combined with a surface waves, travel about two times faster than the shear waves.Some spurious reflections remain in the system with the infinite elements. On the one hand, these reflections are the result of the inaccurate formulation of the viscous boundary at the surface to infinity, on the second hand these are node to node reflections caused by the discretization with finite elements. But from an engineering point of view, the infinite elements with their absorbing layers work very satisfactory.2. Earthquake like base excitation of a cylindrical liquid storage tankAs a typical engineering problem, a water-filled liquid storage tank under horizontal base excitation is investigated. This system, Fig. 5, was also analyzed in [12] with fixed boundary conditions without soil. For the storage tank an elastic-ideally plastic material behavior is assumed (structural steel). The soil is treated linear and the properties are also given in Fig. 5. The time history of the ground accelerations is taken to be a periodic saw-tooth like function.The structural response was analyzed for three choices of finite models: (I) storage tank with rigid base; (II) tank with finite soil elements and rigid boundary conditions and (III) tank with finite and infinite elements. The models (II) and (III) were performed without any damping within the soil region. In this numerical example the fluid filling is replaced by added masses at the shell nodes, in that way that the eigenvalues of this system cor-



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Fig. 4. Radiation of waves in the elastic half space



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< previous page page_463 next page >Page 463respond to the eigenvalues which are computed including fluid elements as described above.In particular, we study the behavior of the top node of the structure and the stress component N22 at the bottom of the tank. In these proceedings first numerical results are given.

Fig. 5. Liquid storage tank under ground excitation



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< previous page page_464 next page >Page 464Varying the amplitudes of the base excitation of the three different models it turns out, that after several response cycles a localized bulge deformation near the bottom (“elephant’s foot”) occur at obviously different limits.Fig. 6 shows different time response curves of the stress component N22 of the element nearest to the bottom.

Fig. 6. Stress component N22, Element 1REFERENCES[1] Wunderlich, W., Cramer, H., Obrecht, H.: Application of ring elements in the nonlinear analysis of shells of revolution under nonaxisymmetric loading. Comp. Mech. Eng. 51 (1985), 259–275[2] Wunderlich, W., Cramer, H., Redanz, W.: Nonlinear analysis of shells of revolution including contact conditions. In ‘Finite Element Methods for Nonlinear Problems‘, Bergan, Bathe, Wunderlich, (eds.), Springer-Verlag, Berlin (1986), 697–717[3] Obrecht, H., Goebel, W., Wunderlich, W: Nonlinear Dynamic Analysis of Shells of revolution. In ‘Refined Dynamical Theories of Beams, Plates and Shells and There Applications‘, Elishakoff, Irretier, (eds.), Springer-Verlag, Berlin (1987), 402–419[4] Housner, G.W.: Dynamic Pressures on Accelerated Fluid Containers. Bull. Seism. Soc. Am., Vol. 47 (1957), 15–37[5] Veletsos, A.S.: Seismic Effects in Flexible Liquid Storage Tanks. Proc. 5th World Conf. Earthqu. Eng., Vol. 1 (1974), 630–639



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< previous page page_465 next page >Page 465[6] Cramer, H., Wunderlich, W.: Multiphase models in soil dynamics. Proc. European Conf. on Structural Dynamics, Eurodyn 90, Bochum (1990), 568–575[7] Haeggblad, B., Nordgren, G.: Modelling nonlinear soil-structure interaction using interface elements, elastic-plastic soil elements and absorbing infinite elements. Comp. & Struc., Vol. 26, No. 1/2 (1987)[8] Zienkiewicz, O.C., Emson, C., Bettes, R: A novel boundary infinite element. Int. Jour. for num. meth. in eng., Vol. 19, (1983), 393–404[9] Lysmer, J.. Kuhlmeyer, R.L.: Finite dynamic model for infinite media. J. of Eng. Mech. Div., EM4, (1969)[10] Clough, R.W., Penzien, J.: Dynamics of Structures. McGraw-Hill, Inc. (1975)[11] Bayo, E., Wilson, E.L.: Numerical Techniques for the Evaluation of Soil-Structure Interaction Effects in the Time Domain. College of engineering, Univ. of Calif., Report No. UCB/EERC 83/04 (1983)[12] Wunderlich, W., Springer, H., Goebel W.: Discretization and Solution Techniques for Liquid Filled Shells of Revolution under Dynamic Loading. In ‘Discretization Methods in Structural Mechanics’, Mang, H.A, Kuhn, G., (eds.), Springer-Verlag, Heidelberg (1989)



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< previous page page_467 next page >Page 467Dynamic Soil-Structure Interaction of Rigid and Flexible FoundationsL.AuerschBundesanstalt für Materialforschung und -prüfung, D 1000 Berlin 45, Germany1. SummaryDifferent flexible foundations such as plates, beams and railway tracks resting on the elastic halfspace are examined by use of a combined finite and boundary element method. The compliance functions for concentrated vertical harmonic loads are calculated for realistic structure and soil parameters and for a wide frequency range. The influence of soil and foundation stiffness is investigated, showing that the soil mainly affects the low frequency response whereas the structural properties are more important at higher frequencies. Two related models—rigid foundations on halfspace and flexible foundations on elastic support (Winkler soil)—are discussed briefly. By two-dimensional analysis (plate under lineload) the approach of finite plate to infinite plate behaviour is studied. High frequency asymptotes are observed for the plate and track foundations, which have also been measured in connection with railway research work. Therefore this contribution may be seen as a useful orientation about the dynamic behaviour of flexible foundations.2. Problem, system parameters and method of solutionIn the present study flexible foundation on a homogenous elastic soil medium are considered under concentrated dynamic loads. The study was motivated by research works on vibrations induced by railway traffic where the dynamic behaviour of track-soil or tunnel-soil systems were examined. To generalize some



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Figure 1Calculated foundation systems and their dimensions



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< previous page page_469 next page >Page 469observations simple foundations as elastic plates and beams are calculated for realistic soil and foundation parameters.The calculated systems are shown in figure 1: elastic plates, beams and a railway track under vertical harmonic point-loads p. For the plate also the two-dimensional problem of a semi-infinite plate under vertical line-load is solved. The geometric parameters can be read from figure 1. The material of the plates, the beams and the sleepers of the railway track is concrete withE=3·1010N/m2 modulus of elasticity,

ν=0, 15 Poisson’s ratio,

s=2, 5·103kg/m3 mass density.The soil is defined byvT=100…300m/s shear wave velocity,

ν=0, 33 Poisson’s ratio,

s=2·103kg/m3 mass density.The rail parameters are (UIC 60)EI=6, 4·106N m2 flexural stiffness,

m=60kg/m mass per length.We are interested in the vertical displacements u at the point of excitation as a function of frequency and compliance functions u/p or admittance functions v/p are calculated.The problem in solved numerically by use of a combined finite element and boundary element method which is described in /1/ and /3/. The foundation structures are calculated by finite beam and plate elements whereas the soil is calculated by the boundary element method using the half-space solution given in /5/.3. Rigid platesIn the literature of dynamic soil-structure interaction we find a lot of distributions about the behaviour of rigid foundations. When we show some more results about this topic, it is to facilitate the comparison of rigid and flexible foundations. The results for a rigid plate with dimension



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Figure 2 Compliance functions of rigid plates on different soils


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Figure 3 Admittance functions of rigid plates on different soils



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< previous page page_471 next page >Page 471a=b=5md=0, 5ms=2, 5·103kg/m3resting on different soils are given in figure 2. It shows the amplitude and the phase of the compliance (displacement divided by force) due to vertical excitation. One can see that there are extremely different static values whereas at high frequencies we have almost the same amplitudes for all soils under consideration. The vertical eigenfrequencies can be found in the phase diagram for φ=−90° lying between 20 and 60Hz, but no resonance amplifications compared to the static displacements occur for this light plate because of the high damping of the soil.The corresponding admittance (velocity divided by force) can be obtained by multiplying the results of figure 2 by the circular freguency i (figure 3) . The admittance function better shows the high frequency response whereas the static response cannot be seen. It is well suited for two-dimensional results (see section 5) and for experimental results where velocities are measured rather than displacements. In the admittance functions of figure 3 we find a region of maximum amplitudes around the eigenfrequency of the rigid foundation.4. Flexible platesNow the same foundation is considered as a flexible plate and the thickness d as well as the soil stiffness are varied. Figure 4 shows the influence of the soil stiffness for a plate with thickness d=0, 25m. As for the rigid foundation there are great differences of the static compliance and only little differences at high frequencies. It seems that—independent of soil stiffness—a common asymptote with constant phase φ=−50° and corresponding amplitude decay of A~ω−0,6 is reached at high frequencies.When the thickness and that means mainly the stiffness of the plate is varied (fig. 5) we have also differences in static displacements but even stronger differences at high frequencies. So it can be concluded that the static behaviour of the soil-foundation system in strongly affected by the soil whereas the dynamic behaviour at higher frequencies is ruled by the properties of the plate.



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Figure 4 Compliance functions of flexible plates (concrete, d=0, 25m) on different soils


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Figure 5 Compliance functions of concrete plates with different thickness (soil with vT=200m/s)



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< previous page page_473 next page >Page 473In figure 5 there are also shown the results of a rigid plate corresponding to the thickest plate with d=0, 5m, demonstrating that even a concrete plate of 0, 5m×5m×5m on a soft soil (vT=200m/s) cannot be regarded as rigid. Also by comparison of figures 2 and 4, the amplitudes of the flexible plates are considerably higher and the phase shift is smaller, not reaching the value of φ=−90° that would indicate an eigenfrequency.As can be read from figure 5, the static displacements of a flexible plate decrease with increasing thickness d and one might expect that a plate of d=1m would give the same static response as the rigid foundation. For larger thickness to length aspects d/a≥0, 2 the foundation (concrete on soft soil with vT=200m/s) may be regarded as rigid. But one should keep in mind that the differences between rigid and flexible plate response increase with increasing freguency and that the “rigidity” of a foundation also depends on the type of loading.5. Plates under line-load excitationMotivated by research work about tunnels of underground railways, plates under vertical line-loads are considered. The two-dimensional analysis allows to calculate larger systems and here especially the influence of the width b of the plate is examined. The vibration modes of different plates with thickness d= 0, 7m on soft soil (VT=200m/s) are shown for f=50Hz in figure 6. The deformations are very similar for all widths of the plate, only near the edges of the plate there is a smaller curvature (less bending).One may expect that the finite plates behave almost like the corresponding infinite plate. According to that, the admittances v/p′ of the midpoint of the plates with different widths (fig. 7) do not vary much in the given frequency range. The admittance reaches an almost constant real asymptote for frequencies higher than 20Hz, which means a phase of φ=−90° and an amplitude decay of A~ω−1 for the compliance function. In other words this is a damper like behaviour of the plates under line-load.



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Figure 6Concrete plates under line-load excitation: vibration modes for different widths of platea) b=5mb) b=9mc) b=15m d) b=25m(d=0, 7m, soil with vT=200m/s, f=50Hz


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Figure 7Admittance functions of concrete plates under line-load excitation for different widths



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< previous page page_475 next page >Page 4756. Beam foundationsFor elastic beams on halfspace again the compliance functions are calculated for different soil stiffnesses (fig. 8) and the same general tendencies can be observed: large differences in static stiffness, small differences for higher frequencies. However there are some differences compared with the given plate compliances (fig. 4). The phase shift reaches values greater than −90° and some small resonances occur around 100Hz. This is due to the fact that the contact area between beam and soil is small and therefore the damping is smaller than for the other cases considered here.For a beam foundation another model—the infinite beam on a Winkler soil—is often used (especially for railway tracks for which it had been introduced), because it has a simple explicit static solution:

with the characteristic length

and the stiffness k′ of the Winkler soil

This stiffness k′ can be chosen just to match the static compliance of the beam on halfspace system by taking

k'≈G or G/(1−v), but the displacements will be more concentrated and the soil stresses will be less concentrated around the loading point than for the beam on halfspace.7. Railway trackAt last a conventional railway track consisting of rails, sleepers and ballast as shown in fig. 1 is calculated for different soils (fig. 9) . As for the foundations treated before, the static compliance strongly depends on the soil stiffness, whereas the differences become small with increasing frequency. The phase reaches an asymptotic value of about 50°.



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Figure 8 Compliance function of a concrete beam (a=d=0, 5m) on different soils


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Figure 9 Compliance function of a conventional railway track on different soils



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< previous page page_477 next page >Page 477The calculated results are compared with measurements, here results measured by the university of Karlsruhe /4/, /6/ are used. Figure 10 shows the measured compliances of two ballasted tracks at different locations (and soil conditions) and of a slab track. Figure 11 again shows calculated results of ballasted tracks and of a slab track which was modelled by an additional concrete plate of thickness d=0, 2m and width a=2, 6m. The influence of the stiffer track structure is more evident at higher frequencies. The slab track has a phase asymptote of φ=−90º different from that of the ballasted track. These observations are in good agreement for calculation and measurement.References/1/ L.Auersch: Wechselwirkung starrer und flexibler Strukturen mit dem Baugrund insbesondere bei Anregung durch Bodenerschütterungen. BAM-Forschungsbericht 151, Berlin (Verlag für neue Wissenschaften, Bremerhaven), 1988/2/ L.Auersch: Zur Entstehung und Ausbreitung von Schienenverkehrserschutterungen—Theoretische Untersuchungen und Messungen am Hochgeschwindig-keitszug Intercity Experimental. BAM-Forschungsbericht 155, Berlin (Verlag für neue Wissenschaften, Bremerhaven), 1988/3/ L.Auersch: A simple boundary element method and its application to wavefield excited soil-structure interaction. Earthquake Engineering and Structural Dynamics, Vol. 19, 931–947, 1990/4/ G.Huber: Erschütterungsausbreitung beim Rad/Schiene-System. Veröffentlichungen des Instituts für Bodenmechanik und Felsmechanik, Universität Karlsruhe, Heft 115, 1988/5/ W.Rücker: Dynamical behaviour of rigid foundations of arbitrary shape on a halfspace. Earthquake Engineering and Structural Dynamic, Vol. 10, 675–690, 1982/6/ Ch.Vrettos, G.Huber, B.Prange: Identifikation des Gleisrostes für Erschütterungsausbreitung. 7. Technischer Bericht zum Forschungsvorhaben BMFT TV 82273B, Universität Karlsruhe, 1986



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Figure 10 Measured compliance functions of different railway tracks (after Huber, Prange et al. /4/, /6/)


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Figure 11 Calculated compliance functions of different railway tracks



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< previous page page_479 next page >Page 479Experimentally Determined Impedance Functions of Surface FoundationsB.Verbi•, S.MelerDepartment of Civil Engineering, University of Sarajevo, Hasana Brki•a 24, 71000 Sarajevo, YugoslaviaABSTRACTForced vibration tests on a series of four circular footings supported at the surface or relatively deep soil deposits with constant properties were performed. The 2.0m diameter model footings were subjected to a rotating-mass-type horizontal and vertical harmonic force in the frequency range from 5 to 40Hz. For each mode of vibration the independent displacements and phase angles were recorded, providing thus enough data to compute actual impedance functions of model footings. The experimentally determined impedances were then compared with the theoretical solutions for a rigid disc resting on the surface of an elastic halfspace.INTRODUCTIONThe last twenty years in the field of foundation vibrations has been characterized by an intensive development of theoretical methods for determination of dynamic stiffness of foundations based on numerical solution of the stress-wave propagation problems in elastic solids. The solutions were in most cases obtained as impedance functions, i.e. as complex functions of the frequency. Now when the theoretical and computational problems have been in general solved, the interest of researchers has shifted toward the experimental valuation of those solutions, which is understandable since the soil is a markedly nonlinear medium. In the past five to six years results of a number of studies were published, only a few of which are listed in this paper, in which impedance functions were determined directly from the results of forced vibration tests of foundations and compared with theory, or



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< previous page page_480 next page >Page 480in which the theoretical impedance functions were validated indirectly by comparing the measured responses of foundations with theoretical predictions. This paper presents results of a study which belongs to the first type of experiments, but the main purpose of it was not, however, to determine impedance functions of a particular type of foundation or site characteristics, but to try to provide a basis for a general evaluation of the theory based on elastic soil models, since in some of previous studies a significant disagreement between the theory and measured response of actual foundations was observed [1, 5, 7, 11].It was considered that this could be best done through tests with rigid circular footings resting at the surface of a deep homogeneous soil deposit for which the theoretical impedance functions are simple and depend on the least possible number of parameters. The experiments were performed with 4 circular concrete footings 2m in diameter and 0.75m in height. These were the largest foundations for which, on the basis of preliminary analysis, the maximum intensity of vibrations without lifting (separation from the soil) could be achieved with the available vibration generator in all vibration modes, and which could be at the same time considered as rigid. It is obvious that in experiments with foundations of such proportions supported on a real undisturbed soil the second part of the model could not be fully realized. The best that could be done was to construct the model footings on relatively deep soil deposits with relatively constant properties.SITE CHARACTERISTICSThe experiments were conducted in Skopje on two different sites with two model footings at each. Skopje was chosen as a place to carry out the tests because in the period following the catastrophic earthquake of 1963 a thorough investigation of geomechanical and seismical characteristics of soil deposits was done, making easier the search for sites with the desirable characteristics. The first testing site was located on a deposit about twenty meters deep of Quaternary sandy clays underlaid by marl, and farther on will be referred to as “clay”, while the second testing site was located on a deposit about ten meters deep of predominantly clayey and silty alluvial sands underlaid by gravels, which farther on will be referred to as “sand”. Both sites were in investigated in the 1980s by soil boring and the geomechanical properties of the soil were determined in the laboratory. Within the scope of this study the in-situ shear-wave velocities Vs were measured by seismic refraction and by steady-state-vibration technique during the forced vibration tests of footings. The second technique could be applied with the available equipment only to the depth of 6m. The site characteristics without the top weathered layer, which was removed prior to model



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< previous page page_481 next page >Page 481footing construction, are given in Tables 1 and 2.Table 1. Basic properties of site CLAYDepth m AC Clasif. VPm/sec Vsm/sec Poisson’s Ratio Unit Weight kN/m3

1.2–1.4 CI 425 150 0.429 16.5

10–11 CI, CL 720 315 0.382 19.7

21–24 CI/CH 1240 435 0.430 19.8

− Marl 2430 990 0.400 23.0Table 2. Basic properties of site SANDDepth m AC Clasif. VPm/sec Vsm/sec Poisson’s Ratio Unit Weight kN/m3

1.7–2.1 SW/SFc 375 185 0.339 14.6–15.7

8.5–9.5 SFs/SFc

SU, C 480 250 0.314 15.6

19–21 GW, ML 960 380 0.407 19.5Based on the measurements carried on within the scope of this study and on the results of a number of investigations of similar deposits in Skopje, the following distribution of shear-wave velocities with depth are proposed for the testing sites up to the depth of 10m:Site CLAY Vs=192H0.25 refraction measurements

Vs=124H0.5 steady-state vibration technique

Site SAND Vs=142H0.25 refraction measurements

Vs=114H0.5 steady-state vibration techniqueAs could be expected the shear-wave velocities obtained from the steady-state vibration measurements are smaller at the upper layers than the velocities obtained by seismic refraction due to higher strain level in the soil induced by the footing vibration.The ground water table was below 10m from the surface at both sites.EXPERIMENTSTwo types of forced vibration tests were performed on each model footing. In the first experiment the vertical harmonic force was applied in the direction passing through the common center of gravity of a footing and the vibrator and through the center of a footing-soil interface. To achieve that an additional weight was attached to the footings at the proper place. In the second experiment the horizontal harmonic force was applied in the direction centered over the center of gravity. The vibrations were excited



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< previous page page_482 next page >Page 482by a rotating-mass-type vibrator EX-50 made by Itoh Seiki, Japan, capable of producing unidirectional horizontal and vertical harmonic force in a frequency range from 2 to 40Hz with the maximum amplitude of 10kN.The induced vibrations were measured with three “Kistler” 305A accelerometers coupled with bridge amplifiers Honeywell Accudata 218 and recorded on a 906T Honeywell Visicorder UV recorder. The vertical response of each footing was measured with two accelerometers mounted on the upper surface of the footing in the direction in which the horizontal force was applied. The same accelerometers were used to measure the rocking response of a footing. The third accelerometer was used to measure horizontal translation of the footing and was mounted at the bottom of the footing in the direction of the horizontal exciting force. Parallel to that accelerometer one LVDT was mounted to measure eventual horizontal slippage of the footing. However, no slippage was registered at any test. Together with the footing response the peak of the exciting force was recorded for measuring the phase angle between the force and the footing response. The position of transducers on the footings and the direction of applied forces are shown in Fig.1.

Figure 1. Position of accelerometers and direction of applied forces on model footings.Each experiment was conducted with four different moments of eccentricity of the rotating masses of the vibrator: 0.1, 0.2, 0.3 and 0.4kgm in the frequency range from 5Hz up to a frequency for which the maximum allowable force of the vibrator of 10kN was produced. For the moment of eccentricity of 0.1kgm the upper limit was the limit of the vibrator frequency range of 40Hz. In Fig. 2 are shown steady-state translational responses (displacement amplitudes and phase angles) of the center of a footing-soil interface and rocking responses about the axis through the same center during the horizontal vibration of one footing on clay and one on sand and in Fig. 3 are shown vertical responses of one model footing on clay and one on sand.



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Figure 2. Translational (a) and rocking (b) responses of two model footings during the horizontal vibration tests.



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Figure 3. Responses of two model footings during the vertical vibration tests.EXPERIMENTAL IMPEDANCE FUNCTIONSExperimental impedance functions were obtained by solving the equations of motion for the forced vibration experiments at each frequency of excitation:

(Kvv−mω2+iωCvv)Z=F (1)for the case of vertical excitation, and

(−mω2+Khh+iωChh)X−mhcω2Φ=F−mhcω2X+(−Iω2+Kmm+iωCmm)Φ=Fhp

(2)

for the case of horizontal excitation, where Kii are the real parts and Cii imaginary parts of an impedance function, F is a force amplitude, Z, X and Φ the complex steady-state vertical, horizontal and rocking displacements. The overall mass m of a footing, vibrator and ballast, computed assuming a unit weight of concrete 24.0kN/m3, was 6200kg (410kg vibrator, 136 kg ballast). The mass-moment of inertia I about the axis passing through the center of the footing-soil interface was 3190kgm2. The height of the



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< previous page page_485 next page >Page 485center of gravity of the system is 0.43m from the footing-soil interface, and the horizontal force was applied 0.34m above the upper surface of each footing. It is obvious from the way the experiments were conducted that in the case of the horizontal excitation the off-diagonal terms in the impedance matrix corresponding to the coupled translational and rotational response were ignored in these computations. However, this has little effect on the computed values of the diagonal terms since the off-diagonal impedances are relatively small for surface foundations as theory [6, 7] as well as some experiments indicate [2]. The impedance functions were computed using as the point of reference the center of the footing-soil interface.The experimentally determined impedances for one footing on clay and one on sand are shown in Figs. 4 and 5 and compared with theoretical impedance functions computed using the corresponding solutions for a rigid disc supported at the surface of an elastic halfspace [6, 9, 10], and assuming for Vs average value determined by in-situ measurements to the depth of 1.5m, and zero material damping. The comparison shows that experimental impedances differ, in some cases significantly from the theoretical impedances obtained for an elastic halfspace model of the supporting soil. However, the differences are not of the same nature for all modes of vibration. For example, the horizontal experimental damping coefficients have much lower values than predicted by theory, while the rocking experimentally determined damping coefficients are larger than the theoretical. The disagreement between the theoretical and experimental impedances is not only in their values but also in the shape. In general, the experimental stiffness functions decrease faster with increasing frequency than the theoretical functions, while the experimental damping functions show higher variation with change in frequency than the theoretical functions. Similar kinds of disagreements between the theoretical and actual stiffness functions were predicted in one previous study [11] in which through a comparison of computed and measured responses of model footings it was concluded that the theory based on an elastic halfspace model of the soil overestimates the soil stiffness for foundations excited with both vertical and horizontal force, overestimates damping for vertically vibrating foundations, and underestimates damping in the case of a horizontal excitation. Results of a similar kind of study reported recently by Crouse et.al. [3] indicate that an improvement in agreement between the theory and the results of the presented experiments could be expected if the increase in shear-wave velocity with depth in soil is taken into account in theoretical impedances. Even better agreement between the theory and the results of experiments could be obtained by modifying the dynamic soil properties (shear-wave velocity and material damping) as was done in some studies [3, 11]. However, an argument against such an approach is that different modifications are needed for different impedance functions or modes of foundation vibration, and it



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Figure 4. Experimental vertical (Kvv, Cvv), horizontal (Khh, Chh) and rocking (Kmm, Cmm) impedance functions for one footing on clay.



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Figure 5. Experimental vertical (Kvv, Cvv), horizontal (Khh, Chh) and rocking (Kmm, Cmm) impedance for one footing on sand.also opens the problem of determination of the “actual” soil properties.



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< previous page page_488 next page >Page 488CONCLUSIONSImpedance functions of circular surface foundations were experimentally obtained and compared with the theoretical solutions based on an elastic halfspace model of the soil.The results of this study as well as the results of previous similar studies suggest that at least in the low frequency range in which the forced vibration tests were run (5 to 40Hz or a0=0.2 to 1.5), the theoretical solutions generally overestimate horizontal and rocking stiffnesses and underestimate the rocking damping of surface foundations. The best agreement between the theory and experimental results was obtained for vertical impedances.A difference is also found in the type of frequency dependence between the experimental and the theoretical impedances. The values of theoretical stiffness coefficients decrease generally slower with increasing frequency than the experimental coefficients, which itself may affect the accuracy of the computed response of a structure-soil system. However it is clear that more theoretical and experimental studies need to be performed for definite evaluation of the foundation-soil interaction theory based on an elastic soil model and in general for better understanding of interaction phenomena in real soil.ACKNOWLEDGMENTSThis study was supported by the National Science Foundation of the U.S.A. and the Science Foundation of the SR Bosnia and Hertzegovina, Yugoslavia, through the American-Yugoslav Board for Scientific and Technological Cooperation, grant No. JF844. The authors also wish to express their gratitude to the Institute for Earthquake Engineering and Engineering Seismology in Skopje for help in organizing the experiments and to its associates Dr. Dušan Aleksovski who performed the in-situ shear-wave velocity measurement and Dr. Mihael Garevski, Metodije Bojadžiev and Blagoje Keram•iev who assisted during the vibration testing.REFERENCES1. Barkan, D.D., and Shaevitch,V.M. Influence of Coupled Soil Mass and Its Nonlinear Characteristics on Vibration of Foundations, Osnovaniya fundamenti i mehanika gruntov, No. 5, pp. 11–14, 1976, (in Russian).2. Crouse, C.B., Liang,G.C., and Martin, G.R. Experimental Foundation Impedance Functions, J. Geotechnical Engrg. Div., ASCE, Vol. 111, No.6, pp. 819–822, 1985.



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< previous page page_489 next page >Page 4893. Crouse, C.B., Husmand, B., Luco, J.E., and Wong, H.L. Foundation Impedance Functions: Theory versus Experiment, J. Geotechnical Engrg. Div., ASCE, Vol. 116, No.3, pp. 432–449, 1990.4. Dobry, R., Gazetas, G., and Stokoe, K.H. Dynamic Response of Arbitrarily Shaped Foundations: Experimental verification, J. Geotechnical Engrg. Div., ASCE, Vol. 112. No.GT2, pp. 136–154, 1986.5. Ilichev, V.A. and Taranov, V.G. Experimental Study of Interaction Between a Vertically Vibrating Foundation and the Soil, Osnovaniya, fundamenti i mehanika gruntov, No.5, pp. 9–13, 1975, (in Russian).6. Luco, J.E., and Westman, R.A. Dynamic Response of Circular Footings, J. Engineering Mechanics Div., ASCE, Vol.97, No.EM5, pp. 1381–1395, 1971.7. Novak, M. Experiments with Shallow and Deep Foundations, pp. 1–26, Proc. ASCE Symp., Vibration problems in Geotech. Engrg., ASCE, New York, 1985.8. Novak, M. Discussion of “Dynamic Response of Arbitrarily Shaped Foundations: Experimental Verification”, [by Dobry et al.], J. Geotechnical Engrg. Div., ASCE, Vol. 113, No.11, pp. 1410–1412, 1987.9. Veletsos, A.S., and Wey, Y.T. Lateral and Rocking Vibration of Footing, J. Soil Mech. Found. Div., ASCE, Vol. 97, No.SM9, pp. 1227–1249, 1971.10. Veletsos, S.A., and Verbic, B. Vibration of Viscoelastic Foundations, Earthquake Engrg. Struct. Dynamics, Vol. 2, No.1, pp- 87–102, 1973.11. Verbic, B. Experimental and Analytical Analysis of Soil-Structure Interaction. Part One—Block Foundations, Research Report, Inst. for Materials and Structures, Faculty of Civil Engrg., Sarajevo, Yugoslavia 1985.12. Verbic, B. Application of Impedance Functions of Rigid Foundations for Analysis of Soil-Building Interaction, pp. 5.5/9–5.5/16, Proc. of the 8th European Conf. on Earthquake Engrg., Lisbon, 1986.13. Yan, R.J. In-Situ Measurements of Coupled Vibration Parameters, pp. 327–330, Vol. 3, Proc. of the 10th Int. Conf. on Soil Mechanics and Found. Engrg., Stockholm, A.A.Balkema, Rotterdam, 1981.



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< previous page page_491 next page >Page 491Stiffness and Damping of Closely Spaced Pile GroupsB.Boroomand, A.M.KayniaDepartment of Civil Engineering, Isfahan University of Technology, Isfahan, IranABSTRACTA general analytical solution is presented for the dynamic analysis of pile groups in a homogeneous soil stratum under steady-state vertical vibrations. The analysis is formulated for end-bearing piles and is extended to floating piles by treating the soil column beneath each pile as the pile’s extension. The special feature of this formulation is that the displacement compatibility is enforced on the entire pile-soil interface. This allows actual variation of pile-soil tractions to be taken into account. The presented analytical model thus makes it possible to analyze closely spaced pile groups, in which the widely used approximation of uniform pile-soil tractions is not justified.INTRODUCTIONPile foundations are usually used when the soil conditions or certain design constraints do not justify the use of the more conventional shallow foundations. Piles are usually used in a group in which case they are often connected to a rigid pile cap supporting the superstructure. The behavior of pile groups under static loads has been a subject of considerable research (eg. Poulos and Davis [1]). However, the first set of results obtained for the dynamic response of pile groups displayed a marked difference between the dynamic and static behaviors (Wolf and von Arx [2], Nogami [3]). This motivated more detailed analyses of pile groups through analytical or numerical models (Waas and Hartman [4], Kaynia [5], Kaynia and Kausel [6], Sheta and Novak [7], Kagawa [8]). These studies enlightened the mechanism of pile group dynamic behavior and paved the road for more research through boundary element and finite element formulations or approximate schemes (Roesset [9], Novak and EL-Sharnouby [10], Sen et al. [11], Kaynia [12], Dobry and Gazetas [13], Mamoon et al. [14], Kobori et al.[15]). A comprehensive survey of the achievements in the dynamics of pile groups has been recently presented by Novak [16].



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< previous page page_492 next page >Page 492Most of the existing numerical formulation, whether of the finite element or boundary element type, are based on the calculations of Green’s functions due to point forces or distributed forces stemming from the pile-soil interaction. An assumption inherent in most of these studies is that the pile-soil tractions are uniform on the pile’s circumference. Consequently, the pile-soil displacement compatibility can only be enforced at the centerline of the piles. Whereas this assumption is reasonable for pile groups, with a pile spacing larger than four times the pile diameter, its validity is questionable for closely spaced groups, as indicated through an approximate analysis by Sanchez-Salinero [17].The objective of this paper is to present a new formulation for the dynamic response of pile groups under steady-state vertical vibrations accounting for nonuniform pile-soil tractions. The results of this study also provide a ground for examining the assumption of uniform pile-soil tractions and its implications on the response of closely spaced pile groups.FORMULATIONIn the present study only the steady-state vertical vibration of pile groups is considered. The piles are assumed elastic rods with elasticity modulus Ep, mass density ρp, length LP and diameter d (or radius ro). These properties, however, do not need to be the same for the piles in a group. The soil medium is considered to be a uniform stratum, overlying a rigid bedrock, with depth H, shear and pressure wave velocities Vs and Vp, mass density ρs and hysteretic damping ratio βs. It is further assumed that under vertical vibrations the horizontal displacements are negligible. This approximation was made by other researchers [3, 18] and the results of a more rigorous model substantiated it [5].Consider the soil stratum in Figure 1 under vertical vibrations. For a steady-state axisymmetric vibration with frequency Ω the vertical displacement can be expressed as w(r, z)eiΩt. Then by solving the differential equation of the stratum, Nogami and Novak [18] showed that w (r, z) is given by the following expression

(1)where Ko is the modified Bessel function of order zero and

(2)

(3)

Figure 1. The soil stratum



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< previous page page_493 next page >Page 493Equation (1) expresses the displacement at any point as the superposition of infinite number of vertical modes with modal shaps cos (hnz). The quantity wn=AnKo(qnr) then reflects the radial variation of the amplitudes of the n th mode (radial mode shape).Equation (1) along with the associated expression for the shear stress [18] were used by Nogami to formulate the dynamic response of pile groups [3]. By this formulation, however, one is bound to assume a uniform variation of tractions on each pile’s perimeter and impose the pile-soil displacement compatibility only at the axis of the piles.To mend this model a new formulation is presented here which will make it possible to achieve the full compatibility between the piles and the soil at their interface. To this end the previous expression for w is replaced by another expression depending on θ, as well as on r and z, in the form

w=w(r, z, θ)eiΩt (4)The differential equation of the soil is then given by

(5)The solution of this equation satisfying the radiation condition as well as the stress and displacement boundary conditions at the top and bottom of the stratum can be written as

(6)where Km is the modified Bessel function of order m, Anm and Bnm are undetermined coefficients and hn and qn are given by equations (2) and (3).If now there are N vibrating piles in the stratum one can assume that the displacement field in the medium is the super-position of the displacements generated by the individual piles. Therefore, if the vertical displacement at a point due to the j th pile is denoted by wj, the total displacement at the point is

(7)where wj, given by Equation (6), will be expressed in the following form

(8)



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(9)in which rj and θj are the polar coordinates of the point defined with respect to a local coordinate system attached to pile j.At this stage it is necessary to enforce the displacement compatibility between the piles and the soil. This means one has to insure that, on the perimeter of each pile, the soil has a uniform vertical displacement. This can be achieved by requiring that for each pile, say pile i, the following conditions are satisfied

(10)

(11)where the angle α is shown in Figure (2) for the integrations being performed on pile i when it is within the displacement

field of pile j. Also note that in these integrations , which depends on rj and θj (as in Equation 9), can be expressed as a function of α.

Figure 2. Pile i within the displacement field of pile jUsing the notation defined by Equations (8) and (9) one can then calculate the displacement of pile i as

(12)Although Equations (10)–(12) indicate on infinite number of terms (integrations) to be calculated (corresponding to m=0, 1,…) usually only a few terms, M≤3, provide satisfactory



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< previous page page_495 next page >Page 495results. Therefore each of Equations (10) and (11) result in N×M equations.The equations resulting from the expressions (10) to (12) can be cast in a matrix form as

(13)where [Z] is a square matrix of order (2N×M+N).The next step is to calculate the shear stresses associated with the above displacements and to integrate them on the perimeter of each pile to obtain the force per unit length of the pile. One can show that this force can be expressed as (for pile i)

(14)where η is shown in Figure (2). Introduction of Equation (7), to express wi as Σwj, along with Equation (8) into Equation

(14) leads to the following expression for

(15)

Substituting Equation (9) for into Equation (15) and carrying out the integrations one can obtain as a function of Anm and Bnm which can be expressed in matrix form as

(16)The final step is to eliminate the vector containing the and Bnm values from Equations (13) and (16). This will result in an equation of the form

(17)

where is a square matrix, referred to as the n th modal soil stiffness matrix, relating forces per unit length of piles to the corresponding displacements. This matrix can then be used, in the same way as described by Nogami [3], to couple the soil stiffness with that of the piles to obtain the dynamic response of the pile group.



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< previous page page_496 next page >Page 496RESULTSA number of representative results are presented in this section to investigate the applicability of the presented formulation and the characteristics of the response of closely spaced pile groups.The quantities of interest are the dynamic stiffness of pile groups and the distribution of an applied force among the individual piles in a group. The dynamic stiffness of the pile group (relating the vertical force on the pile cap to the corresponding displacement) is a complex quantity which can be expressed as

K=k+iaoc (18)where ao= Ωd/Vs is the nondimensional frequency. In this form k and c are usually referred to as stiffness and damping of the foundation.The presented results correspond to a pile-soil system with νs=0.3, βs =0.05, νp=0.25, ρs/ρp=0.7, H/d=40 and Lp/d=20. Two types of piles are considered: stiff piles with Ep/Es=1000 and flexible piles with Ep/Es=100.Figure 3 shows the variations of stiffness and damping of a 2×2 stiff pile group with s/d=2 (s denotes the center-to-center spacing between the piles) as a function of ao. The stiffness and damping have been normalized with respect to the static stiffness of a single pile. The dashed curve in this figure corresponds to taking only one circumferential mode (M=0) and thus defines the solution associated with uniform pile-soil tractions. The solid curve, on the other hand, corresponds to M=2 (in obtaining the presented results it was observed that the accuracy did not change for M>2). Also shown in this figure are

Figure 3. Stiffness and damping of a 2×2 stiff pile group



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< previous page page_497 next page >Page 497the results obtained from Kaynia’s model [5]. As expected, the latter results match those calculated for M=0. Comparing these results one can conclude that for low to intermediate frequency ranges (ao<1.0) the accuracy of conventional solutions for pile groups is satisfactory. However, as frequency increases, the accuracy of the results from such models may deteriorate drastically. Figure 4 shows a comparison between the solutions associated with M=0 and M=2 for the same pile group over a larger frequency range. This figure clearly displays the difference between these two solutions.

Figure 4. Stiffness and damping of a 2×2 group for high frequencyFigure 5 shows the variations of stiffness and damping of a flexible 2×2 pile group with s/d=2 Similar observations apply to this figure.



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Figure 5. Stiffness and damping of a 2×2 flexible pile group with s/d=2As was stated in the introduction, the effect of assuming a uniform pile-soil traction on the solution accuracy is negligible in pile groups with moderate pile spacing. For example, in Figure 6 the stiffness and damping of a 3×3 stiff pile group

Figure 6. Stiffness and damping of a 3×3 stiff pile group with s/d=5with s/d=5 is plotted for M=0 and M=2. This figure and other similar results (not shown) indicate that if the piles are not too close then the assumption of uniform pile-soil traction is satisfactory.Finally, in order to investigate the effect of the above assumption on the distribution of an applied force among the individual piles, the distribution of forces for three piles in a 3×3 stiff group are plotted as a function of ao in Figure 7.



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Figure 7. Distribution of forces in a 3×3 stiff pile group with s/d=2

Figure 8. Distribution of forces in a 3×3 flexible pile group with s/d=2This figure corresponds to s/d=2 and shows the absolute values of the forces normalized by the average force in a pile. Besides displaying a noticeable difference between the two solutions the results suggest that the center pile is most affected by the assumption corresponding to M=0. Also, Figure 8 shows the distribution of forces in a 3×3 flexible pile group with s/d=2, which supports the previous observation.CONCLUSIONSA general analytical solution was presented for the steady-state vibration of pile groups. The solution takes into account the variation of traction on the perimeter of piles-soil interface. The results from this model suggest that for closely spaced pile groups the interaction between the piles, as displayed by the peaks in the variation of stiffness, is exaggerated by the more conventional pile group analyses. For low to intermediate frequency ranges, however, such analyses can provide satisfactory results even for close piles.REFERENCES1. Poulos, H.G. and Davis, E.H. Pile Foundation Analysis and Design, John Wiley and Sons, 1980.2. Wolf, J.P. and von Arx, G.A. Impedance Functions of a Group of Vertical Piles, Vol. 3, 1024–1041, Proc. ASCE Specialty Conf. on Earthq. Eng. and Soil Dyn., Pasadena, California, 1978.


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< previous page page_500 next page >Page 5003. Nogami, T. Dynamic Group Effect of Multiple Piles Under Vertical Vibration, pp 750–754 Proc. ASCE Eng. Mech. Specialty Conf., Austin, Texas, 1979.4. Waas, G. and Hartman, H.G. Analysis of Pile Foundations Under Dynamic Loads, Proc. SMIRT Conf., Paris, 1981.5. Kaynia, A.M. Dynamic Stiffness and Seismic Response of Pile Groups, Research Rep. R82–03, M.I.T., Cambridge, MA; 1982.6. Kaynia, A.M. and Kausel, E. Dynamic Behavior of Pile Groups, pp 509–532, Proc. 2nd Int. Conf. Numerical Meth. Offshore Piling, Austin, Texas, 1982.7. Sheta, M. and Novak, M. Vertical Vibration of Pile Groups, J. Geotech. Eng., ASCE, Vol. 108, No. 4, pp 570–590, 1982.8. Kagawa, T. Dynamic Lateral Pile Group Effects, J. Geotech. Eng., ASCE, Vol. 109, No. 10, pp 1267–1285, 1983.9. Roesset, J.m. Dynamic Stiffness of Pile Groups, Pile Foundations, ASCE, New York, 1984.10. Novak, M. and El-Sharnouby, B. Evaluation of Dynamic Experiments on Pile Groups, J. Geotech. Eng., ASCE, Vol. 110, No. 6, pp 738–756, 1984.11. Sen, R., Davies, T.G. and Banerjee, P.K. Dynamic Analysis of Piles and Pile Groups Embedded in Homogeneous Soil, Earthq. Eng. and Struct. Dyn., Vol. 13, pp 53–65, 1985.12. Kaynia, A.M. Characteristics of the Dynamic Response of Pile Groups in Homogeneous and Nonhomogeneous Media, Vol. 3, pp 575–580, Proc. 9th World Conf. on Earthq. Eng., Tokyo-Kyoto, Japan, 1988.13. Dobry, R. and Gazetas, G. Simple Method for Dynamic Stiffness and Damping of Floating Pile Groups, Geotechnique, Vol. 38, No. 4, pp 557–574, 1988.14. Mamoon, S.M., Kaynia, A.M. and Banerjee, P.K. On Frequency Domain Dynamic Analysis of Piles and Pile Groups, J. Eng. Mech., ASCE, Vol. 116, No. 10, pp 2237–2257, 1990.15. Kobori, T., Nokazawa, M., Hijikata, K., Kobayashi, Y., Miura, K., Miyamoto, Y. and Moroi, T. Study on Dynamic Characteristics of a Pile Group Foundation, Proc. 2nd Int. Conf. on Recent Adv. in Geot. Earthq. Eng., St. Luis, Missouri, 1991.16. Novak, M. Piles Under Dynamic Loads, State of the Art Report, Proc. 2nd Int. Conf. Recent Adv. in Geot. Earthq. Eng., St. Louis, Missouri, 1991.



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< previous page page_501 next page >Page 50117. Sanchez-Salinero, I. Dynamic Stiffness of Pile Groups: Approximate Solutions, Geotech. Eng., Report GR83-5, University of Texas at Austin, 1983.18. Nogami, T. and Novak, M. Soil-pile Interaction in Vertical Vibration, J. Earthq. Eng. and Struct. Dyn., Vol. 4 pp 277–293, 1976.



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< previous page page_503 next page >Page 503Chaotic Motions in Pile-DrivingM.StorzInstitut für Mechanik, Universität Karlsruhe, Kaiserstraβe 12, D-7500 Karlsruhe 1, GermanyABSTRACTVibratory pile-driving is modelled by unconstrained motions of a harmonically forced rigid body carrying an additional static load. The equation of motion for this single degree of freedom system can be easily given, if the characteristic of the restoring force is assumed to be known from experimental investigations. The restoring force is hysteretical. This leads to a strongly nonlinear equation of motion.For a certain class of motions, the characteristics are quasi-stationary during a finite number of forcing cycles. In the case of pure side resistance, the restoring force is described by a Coulomb type law. Tip resistance strongly increases the characteristic.Stationary solutions have the same frequency as the excitation. Dominant tip resistance gives rise to bifurcation phenomena as period doubling. Assuming the limiting case of a Coulomb side resistance and impact type tip resistance, subharmonic and even chaotic motions are observed. This is confirmed by experiments.SYSTEMWe consider a single degree of freedom system with harmonic excitation forces F0 sin Ωt (see Fig. 1), produced by rotating unbalanced masses. The pile is assumed to be rigid and the surrounding soil unmoveable. All parameters of the system are known, except for the restoring force. The restoring force depends upon the depth of penetration and the direction of velocity. This force

is divided into two parts, side resistance and tip resistance.



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Figure 1: Mechanical ModelFig. 2 shows a restoring force diagram, measured by Verspohl [6]. For , the restoring force is strongly influenced by tip resistance. For , there is only side resistance.

Figure 2: Restoring Force (Measured)If the characteristic is strongly influenced by tip resistance, we observe periodic multiplication of the system response [6] in experiments as well as in numerical calculations. For fundamental studies of the system behaviour,



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< previous page page_505 next page >Page 505we assume the limiting case of pure Coulomb side resistance and an inelastic impact (see Fig. 3).

Figure 3: Restoring Force with Coulomb Side Resistance and Inelastic ImpactIn the mechanical model, the inelastic impact corresponds to unmoveable ground. In this situation, the restoring force

depends only upon the direction of velocity and not on the depth of penetration x.The equation of motion is given by

z″+sgnz′+A=Bsinτ (1)with the dimensionless quantities

In the differential equation (1), the type of motion depends only upon the two parameters A and B.• A represents the sum of the pile weight N and an additional statical load G, normalized with the Coulomb frictional force R0.• B represents the amplitude of excitation F0, also normalized with the Coulomb frictional force R0.The switching conditions

impact: z(τ1)=0, reversal of velocity: z′(τ1)=0 (2)



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< previous page page_506 next page >Page 506will be required for the calculation of the switching time τ1 at an impact and at a reversal of velocity, respectively.If τ fulfills the condition

(3)then the pile is sticking after switching.The integration of (1) leads to

(4)which is valid for the motion in the interval between two impacts or until the following reversal of velocity. The constants of integration are given by

C1=B cosτ1,C2=B sinτ1,

It is worth to note, that the equation of motion is piecewise linear, whereas the nonlinearity of the total problem lies in the time-history dependent switching conditions.MOTIONS

Figure 4: A B Parameter DiagramThe possible motions depend on the two parameters A and B (see Fig. 4). To explain Fig. 4, it is assumed that the static load A remains constant and the amplitude of excitation B increases from the left to the right. In the hatched range, the pile sticks to the soil (τ∈S see (3)). In the case



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< previous page page_507 next page >Page 507of an increasing amplitude of excitation, simple one impact motions appear, which are denotated by (n, 1). The notation represents:first number: period of response as a multiple of the period of excitation.second number: number of impacts per period of response.In the shaded parameter ranges complex motions are observed.

Figure 5: Series of Motions for a Constant AFig. 5 shows in detail the scheme for the series of motions when the amplitude of excitation B increases. The simple one impact motions of Fig. 4 are found in the nonshaded boxes. The motions in the shaded boxes follow the following rules:Starting with a one impact motion, the period of response and the number of impacts are doubled. After this period doubling complex motions appear, which have a variety of periods and number of impacts. In the next box, the period is reduced by one, but the number of impacts remains the same. There is always a parameter range of B with complex motions in the interval between two impact motions (see CHAOTIC MOTIONS). Finally, the next one impact motion appears with the period increased by one.The series of motions always follows these rules. Only the number of motions within the shaded ranges increases by two.It is possible to determine the rules of these bifurcation phenomena by means of a point mapping method [1] [4]. This has been described in a former paper [5].EXPERIMENTS AND VERIFICATIONFig. 6 shows the experimental arrangement for the verification of the calculated results.



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Figure 6: Laboratory Model with Measuring DevicesThe pile is modelled by a hollow steel tube. The exciter at the top possesses two gear wheels with unbalanced masses. To minimize system weight, the driving motor is not mounted on the exciter. The Coulomb frictional force is produced by means of three steel plates with cork layers. The amount of friction is controlled by springs. Otherwise, the pile moves frictionless and impacts inelastically into a barrel filled with sand. By separating the force components, Coulomb friction and impact, it is guaranteed that the restoring force (Fig. 3) is well reproduced.The frequency of excitation and the pile acceleration are measured. The velocity and the displacement are integrated from the acceleration by an analog computer. It is therefore possible to produce phase curves on the oscilloscope.Figures 7 to 12 show a comparison of calculated phase curves and restoring forces with measured curves, up to and including the (2, 1)-motion. To produce these curves, A was held constant and B was gradually increased. The motions in this comparison are steady-state motions. The experimental results concur completely with the calculated motions. The resulting diagrams agree in both the shape of the curves and in the series of motions (see Fig. 5).Figures 7 and 8 show a (1,1)-motion with one impact per period. The response has the same frequency as the excitation. The left hand diagram of Fig. 7 shows the calculated phase curve. The hatching represents the unmoveable ground for z=0. The pile impacts inelastically for z=0 and the trajectory starts again for z, z′=0. In the interval between the impact and the starting point, sticking is possible. This can not be seen in the phase curves, because they are the parametric representations of the two time dependent functions z(t) and z'(t), The right hand diagram of Fig. 7



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Figure 7: (1, 1)-Motion (Calculated)

Figure 8: (1, 1)-Motion (Measured)




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< previous page page_511 next page >Page 511shows the corresponding restoring force. The measured results for the phase curve and the restoring force (Fig. 8) have the same shape as those calculated, except for the fact, that the impact is not completely inelastic. A (2, 2)-motion (see Fig. 9 and Fig. 10) appears for an increasing frequency of excitation f, which corresponds to an increasing amplitude of excitation B~f2. Again, the phase curves and the restoring forces concur with the calculated results. The period of response of this motion is double the period of excitation with response having two impacts per period. This can be proved by counting the cycles of response. The last two figures show a (2, 1)-motion (Fig. 11 and Fig. 12) with the same period as the (2, 2)-motion, but having only one impact.For any additional increasing of the amplitude of excitation B, the series of motions follow the calculated predictions.The two phase curves in Figures 13 and 14 show complex motions.

Figure 13: Phase Curve of a Multi-Periodic Motion (Calculated)

Figure 14: Phase Curve of a Multi-Periodic Motion (Measured)This occurs in ranges of B where it is impossible to determine the periodic



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< previous page page_512 next page >Page 512multiplication and the number of impacts.It is also possible to calculate or to measure much more complex phase curves. But their diagrams are very complicated, so they were not included for the sake of clarity.CHAOTIC MOTIONSThe phenomena of complex motions in the interval between two impact motions has been investigated in [5] by means of a point mapping method. The traces of the point mappings leads to the conclusion, that there is the unique possibility of chaotic motions in these parameter ranges. It is interesting to note, that there are no critical values for the parameters A and B. Every parameter range with chaotic motions is followed by a range with steady-state motions.It is not possible to prove chaotic behaviour by numerically calculated point mappings. There is always a limiting cycle because of the finite number of mapping points. ”The numeric determines the chaos”.Acknowledgement: The author would like to thank Prof. Dr.-Ing. P.Vielsack and Dr.-Ing. H.Schmieg for the interesting discussions and help with the experiments. Many thanks to the Deutsche Forschungs Gemeinschaft for their financial support.REFERENCES1. Guckenheimer, J., Holmes, P. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Applied Mathematical Sciences, Vol.42, Springer-Verlag, Berlin and New York, 1983.2. Moon, F.C. Chaotic Vibrations, John Wiley & Sons, New York, 1987.3. Schmieg, H., Vielsack, P. Vibratory Driving of Tubes into Dry Granular Soil, Structural Dynamics, Ed. Krätzig et al., pp. 779–785, 1990.4. Shaw, S.W., Holmes, P.J. A Periodically Forced Impact Oscillator with Large Dissipation, Journal of Applied Mechanics, Vol.50, pp. 849–857, 1983.5. Storz, M. Ein ungefesseltes, harmonisch erregtes System mit Coulombscher Reibung und Stoß als Modell für die Vibrationsrammung, to appear in Proceedings of GAMM Annual Scientific Conference, Cracow, Poland, 1991.6. Verspohl, J. Ungefesselte hysteretische Systeme unter besonderer Berüksichtigung der Vibrationsrammung, Thesis, University of Karlsruhe, 1990.



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< previous page page_513 next page >Page 513SECTION 8: EARTHQUAKE ENGINEERING OF STRUCTURES



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< previous page page_515 next page >Page 515Seismic Damage Assessment for Reinforced Concrete StructuresA.S.Cakmak (*), S.Rodriguez-Gomez (**), E.DiPasquale (***)(*) Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ 08544, U.S.A.(**) T.Y. Lin International, San Francisco, CA 94133, U.S.A.(***) Engineering Systems International, 20 rue Saarinen-Silic 270–94578, Rungis-Cedex, FranceABSTRACTThe aim of this paper is to show that structural damage can be detected through the analysis of strong motion records. without any immediate need for inspection after earthquakes.For the purpose of post-earthquake reliability assessment, a damage model based on the evolution of the natural period of a time-varying linear system equivalent to the actual nonlinear system for a series of non-overlapping time windows has been developed. A maximum likelihood criteria for the identification of the time-varying equivalent linear system from the acceleration records at the top and at the base of the structure has been used. The functional form of the damage indicies has been derived from the experimental and theoretical analysis. The analysis and the calibration of the model have been presented in detail. The model has been validated by other experimental data and real examples have been studied. The model has also been compared to the traditional measures of damage and some of the more recently developed indicies.In addition to the post-earthquake damage assessment, this analysis of strong motion records can be used in performance prediction of a structure in the design stage and reliability studies of existing facilities.



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< previous page page_516 next page >Page 516INTRODUCTIONLocal and Global Damage IndiciesStructural mechanics and previously set standards are the main tools in the process of structural design. In structural design, the earthquake loads or so called seismic loads, which are stochastic in nature, have been simplified so that the structural engineer can easily analyze the structure under design for seismic loading which may act on the structure throughout its useful life.In current practice, structures (especially reinforced concrete ones) are designed in such a way that they can withstand only minor to moderate earthquakes within the elastic range. The safeguard against large earthquakes is the inelastic response of the structural elements which provide a mechanism for the dissipation of the earthquake energy. When the inelastic response takes place, the seismic forces are reduced. Thus, it is important to know the degree of damage for structures at which they undergo inelastic deformations and dissipate the earthquake energy.In the classical structural design for static loads, the structural safety is attained by keeping the stresses well below the material yield limit. This simple definition of safety is too conservative for the evaluation of the state of a structural system after earthquakes. Even moderate earthquakes may produce yielding in some of the structural members without producing a dangerous situation. Since inception of the earthquake engineering practice both the expected performance of structures subjected to earthquakes and the state of damage after actual seismic events have been characterized using indicators other than the stress level. The width and distribution of cracks. the interstory or global drift and ductility ratio have been used as traditional or engineering measures of seismic damage. Similar to these measures, many local and global damage indices have been proposed to indicate the remaining capacity after earthquakes. Most of the research in the field of damage assessment has dealt with the analysis of simple structural elements. The application of these techniques to the post earthquake analysis of existing structures is limited by the scarcity of data available in the field regarding the history of stress and strain for single elements. In order to obtain such detailed information a great number of instruments must be installed on the structure. Furthermore, when a certain degree of redundancy is present in the structure, local features of damage may not be very important in determining what, if any, repair actions should be carried out. Some average or global measures of damage may, in this case, be as useful to engineers as detailed information.



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< previous page page_517 next page >Page 517Global damage indices that give information about the state of complex structual systems are generally defined as a weighted average of the local damage in the simple structural elements that form the system. A different approach for the definition of global damage indices is proposed in this paper. The new proposed global damage indice is based on the changes in vibrational parameters of the structure.Definition of Global Damage StateEarly in this paper, it was pointed out that for the purpose of postearthquake reliability assessment, it is necessary to define a global damage state for a structure that has experienced an earthquake. An intuitive approach is to define global damage as a combination of the damage at each point of the structure. The definition of “point” depends on the structural model that is chosen for the analysis. The damage state can also be defined in several ways. One can think of a binary damage state (failure/no failure), or of a discrete-valued damage state using qualitative indicators such as safety, light damage, damage and critical damage (Stephens and Yao, 1987), or of continuous-damage indices, as considered in continuum-damage mechanics. The damage state may or may not be dependent on the history of loads. depending on the particular model that has been chosen to describe the structural behavior (Kachanov, 1958, Krajcinovic and Fonesca, 1981, Lemaitre, 1984).In general. therefore, a damage event that the structure undergoes can be described as a function, f: Rn→Rn. The function f, which customarily takes values in the interval (0, 1), is defined on the volume (Ω) occupied by the structure. The function f describes the loss of resistance experienced in the neighborhood of a given point. The global damage state (D) can thus be definded as a functional of f.

D=∫Ωw(x)f(x)dx (1)where w(x)=an appropriate weighting functionThis formulation of the damage problem, based on an infinite dimensional damage state, is theoretically correct but very impractical. The damage state must be reduced to a finite number of dimensions in order to solve the problem of damage assessment for a real structure. The need for this reduction was first pointed out by Yao (1982) It was assumed that a finite number of quantitative indicators is sufficient to determine the damage state of the whole structure. Such indicators are usually referred to as “damage indices.” Once the number of dimensions has been reduced,



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< previous page page_518 next page >Page 518the damage state of the structure can be inferred from the history of a finite number of structural parameters. The analysis of these histories yields numerical values for the corresponding damage indices.The space of the damage indices can be termed “damage space.” The expression (Eq.1) for the structure’s damage can thus be simplified. If δ1,…., δm are the damage indices considered, the global damage (D) is given by function (f) defined on the damage space:

D=f(δ,….,δm) (2)Similarly, a state is defined as a surface g in the damage space:

g(δ1,….,δm)=0 (3)Examples of limit states are the serviceability, or elastic, limit state, and the collapse limit state.The reduction to a finite number of dimensions can be obtained by lumping procedures. The structure is modeled as an assemblage of elements and joints, for each of which a damage index is computed from the history of loading during the earthquake. The global damage indices are for the single elements (Stephens and Yao, 1987, Park et al., 1985). If nel elements and joints are considered, for each of them a damage index (δi) and a weight (βi) can be defined, so that the global damage is measured as:

(4)Reduction by lumping procedures requires that generalized displacements and restoring forces are available for a large number of nodes. This is possible in the analysis of shaking-table experiments, in simulation studies, or for full-scale structures that are extensively instrumented. In the practical case of a structure where only two accelerometer arrays are installed, reduction by lumping is not directly possible.After considering the above mentioned facts and previous observations in the damage assessment field. DiPasquale and Cakmak (1987, 1988) defined a global damage indice based on the vibrational frequencies of the structure in the linear phase, and of an equivalent linear structure in the nonlinear phase. The functional form of the damage indices is derived from the experimental and theoretical analysis of damage of simple struc tural elements and of complex structures.Effect of Structural Damage on the Modal ParametersA linear structural system described in terms of the modal parameters was proved to be indentifiable by Beck (1979). The modal parameters of a structure are the damping factor, the natural frequencies and the effective participation factor for each natural mode. The modal parameters are functions of the vibrational characteristics, mass, damping and stiffness, of a structure.



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< previous page page_519 next page >Page 519When an earthquake hits a structure, the local effect of structural damage will be cracking. buckling and plastic deformation, in any case degradation of the resistance properties of structural elements. For the case of damage in reinforced concrete structures, due to the microcracking of concrete and due to plastic deformation related to yielding of reinforcement bars, stiffness characteristics of an element or of a joint will degrade. From this local stiffness degradation, a general shift of the natural frequencies towards lower values will result. (Rayleigh, 1945, Dowell, 1979).Stiffness degradation of both full scale structures and of small scale models, consequent to seismic damage, has been observed by several authors (Chen et al., 1977, Foutch and Housner, 1977, Meyer and Roufaiel, 1984, Mihai et al., 1980, Vasilescul et al., 1980). Ogawa and Abe (1980) and Carydis and Mouzakis (1986) attempted to correlate stiffness degradation, as a function of the variation in the fundamental period, and the severity of damage.When time variant linear models are fitted to strong motion records, it is confirmed that the natural frequencies of the structure tend to shift towards lower values (Beck, 1983). This can be due to nonlinearities in the mechanical behavior of the structure, due to the soil-structure interaction, as well as due mostly to the stiffness decrease subsequent to structural damage. After considering the above mentioned facts and previous observations in the damage assessment field, a global damage indice based on the vibrational frequencies of the structure in the linear phase, and of an equivalent linear structure in the nonlinear phase has been defined. the functional form of the damage indicies is derived from the experimental and theoretical analysis of damage of simple structural elements and of complex structures.While studying the decrease in the natural-frequencies or the lengthening in the natural periods, the nonlinear mechanical behavior was taken into account by evaluating the natural period of a time-varying linear system equivalent to the actual nonlinear system for a series of non-overlapping time windows. Throughout this work, it has been asssumed, as a working hypothesis, that the acceleration, recorded at the basement of the structure, was in fact the input motion. Any interaction between foundation and soil was thus neglected. This has been so far assumed, by all the researchers in the field, but it is in general not true. Unfortunately, the importance of soil-structure interaction increases with the amplitude of the motion, and therefore with the expected severity of the damage. This is especially true for low-rise buildings, as well as nuclear facilities, whereas soil-structure interactions would definitely be less important in the analysis of tall, slender structures.DEFINITION OF THE PROPOSED GLOBAL DAMAGE INDEXKnowing that a linear structural system described in terms of the modal



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< previous page page_520 next page >Page 520parameters can be identifiable, one can think of using equivalent linearization techniques to consider nonlinear mechanical behavior so that when the structural behavior is nonlinear, the system identification algorithm based on linear models will yeld estimates of equivalent linear parameters. The nature of such an equivalence will depend on the criteria that the analyst has chosen for the purpose of indentification. Traditional equivalent linearization techniques (Caughey, Valdimarsson et al.) seek equivalent linear models using analytical methods such as the error-in-the-equation criterion. Beck and Jennings introduced error in the output criteria instructural analysis and DiPasquale and Cakmak treated “maximum likelihood” criteria which yields statistically optimal estimates of the time varying modal parameters.

FIGURE 1. Evolution of Equivalent Fundamental Period for Milikan Library (San Fernando Earthquake, 1971)The equivalent linear model that fits strong motion records coincides with the actual structure as long as the structure’s behavior is linear. When the structure enters a nonlinear phase, the equivalent linear model will change. reflecting the nonlinearities that take place during the strong motion. In particular, the structure will experience an apparent softening



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< previous page page_521 next page >Page 521as the amplitude of the oscillation increases, and the equivalent natural frequencies will decrease. By fitting a time variant linear model to the records, a history of the equivalent parameters is obtained. The algorithm for Maximum Likelihood estimation of the equivalent modal parameters and the details of the fortron code, MUMOID which was developed to implement the procedure were presented in [12].TABLE 1. Damage Indices Based on Equivalent Linear Models

LOCAL PHENOMENON MACROSCOPIC FEATURE GLOBAL INDEX

Stiffness degradation Cracks Final softening

(5a)

Plastic deformation Yielding of reinforcement bars Plastic softening

(5b)

Combined effect of plastic deformation and stiffness degradation

Onset of structural damage (service-ability limit state)

Maximum softening

(5c)First, the goal is to extract information concerning damage from the history of the modal parameters. It is clear from the start that only the natural frequencies will provide valuable information. Damping factors are entities of uncertain physical meaning, and their estimation, when the structure is in the nonlinear phase, yields results of questionable reliability. Furthermore, the estimates of damping factors and of effective participation factors are statistically correlated. [Beck, 1978] This correlation is reduced but not eliminated when constraints are imposed on the effective participation factors. In this research, only the first (fundamental) natural frequency is considered. All the computations are actually carried out on its inverse, the fundamental period of vibration. Engineers



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< previous page page_522 next page >Page 522commonly use only the fundamental period. Nevertheless it should be noted that consideration of other modes will yield better results after a more complicated analysis.The interval (0, s) of duration of the earthquake is divided into nwind nonoverlapping windows of width si seconds. For each of these windows, an equivalent fundamental period (T0)i is computed. The first window can be made small enough, so that it can be assumed that the structure is still vibrating in the linear regime and that (T0)1 is equal to the fundamental period of the linear oscillation of the building before the earthquake, (T0)initial. When the record is long enough, so that the vibrations due to the strong motion have abated by the end of the record, and the behavior of the structure can be considered linear, the estimate of (T0) corresponding to the last window, (T0)nwind, can be assumed to be equal to the fundamental period of the linear oscillation after the earthquake (T0)final. When the final portion of the record still presents apparent nonlinearities, (T0)final can sometimes be obtained from postearthquake tests.Fig. 1 shows the evolution of the equivalent fundamental period estimated from the response of the Millikan Library in Pasadena, California, during the San Fernando earthquake, 1971. Three parameters can thus be typically observed in the evolution of the equivalent fundamental period during an earthquake: (1) An intial value, (T0)initial, which is assumed to correspond to the linear behavior of the undamaged structure; (2) a final value, (T0)final, corresponding to the linear behavior of the damaged structure; and (3) the maximum value of the estimate of the fundamental period, (T0)max, where the effect of nonlinear behavior and of soil-structure interactions is superimposed onto stiffness degradation.Several indices can be proposed as measures of global structural damage. They are functions of the fundamental periods (T0)n estimated during an earthquake, as well as of the initial fundamental period (T0)initial and the final one (T0)final. The functional form of the index may depend upon phenomenological aspects of damage at the local level, upon analytical considerations and upon the analysis of data recorded from damaged structures. Table 1 describes some parameter-based global damage indices that have been proposed by Cakmak and DiPasquale and their correlation with local damage variables and with macroscopic features of damage for reinforced concrete structures.A procedure for the detection of seismic structural damage will be introduced and validated below. Analysis of empirical results shows that there exists a correlation between the damage state of a reinforced concrete structure that has experienced an earthquake and the maximum softening δM, defined as:

(6)It is commonly believed (see for instance Sozen) that seismic damage to



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< previous page page_523 next page >Page 523reinforced concrete structures depends mostly on the maximum strain that is observed during an earthquake, while the particular sequence (or path) of loading is not very important in determining damage. It is therefore intuitive that the maximum softening δM. which depends on the combined effects of stiffness degradation and nonlinearities, be used as damage index for reinforced concrete structures.THEORETICAL VALIDITY OF THE PROPOSED INDICES FROM THE CONTINUUM-DAMAGE MECHANICS VIEWIn order to strengthen the validity of the proposed indices, DiPasquale, Ju and Cakmak (1989, 1990) studied the theoretical basis to the proposed indices, namely final stage softening, δf, plastic softening, δp and maximum softening, δM. It has been shown that for both cases of elastic and plastic damage, continuum-damage mechanics provide a theoretical foundation to global indices and that the proposed indices carry the necessary information of elastic and plastic damage. Repeating Table 1, final softening, δf, is defined for the stiffness degredation related to microcracking of concrete and plastic softening, δp, is defined for the plastic deformation related to yielding of reinforcement bars, it is obvious that maximum softening. δM, defined for reinforced concrete structures, is a theoretically grounded global damage indice, which carries the necessary elastic and plastic damage independent contributions [16].THE ANALYSIS AND THE CALIBRATION OF THE MODELPost earthquake damage assessment involves decisions regarding the future of the building, such as whether the building is safe or needs some actions, ranging from cosmetic repairs to demolition. In a model for the analysis of seismic damage these decisions will correspond to different limit states. The decision to undertake a particular action is then equivalent to assessing whether the corresponding limit state has been trespassed or not. So, the structural serviceability limit state, corresponding to the onset of seismic structural damage, should be defined.Up to here a parameter based global damage index has been introduced as a measure of seismic structural damage. It has been shown that the maximum softening, as defined in equation (6), depends on the combined effect of stiffness degradation and plastic deformations.Analysis of Strong Motion RecordsIn order to use the maximum softening to measure global damage, a quantitative relationship must be established between the numerical value of the index, as obtained from strong motion records, and engineering features of damage, as reported by post earthquake inspection and analysis. Seismic response data from damaged structures must be analyzed. Unfortunately, there are very few records from buildings that have been damaged during an earthquake. In order to find enough data to support



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< previous page page_524 next page >Page 524MODEL AUTHORS TYPE OF

STRUCTUREEARTHQUAK EEXCITATION

UNDAMAGED FIRST MODE

FREQUENCY (Hz)

TOTAL NUMBER OF

RUNS

MF1 Healey and Sozen

10-Storey, 3-bay double frame tall

first storey

EI Centro (1940) 3.7 3

FW1 Abrams and Sozen

10-Storey, 3-bay double frame

+heavily reinforced wall

EI Centro (1940) 4.3 3


10-storey 3-bay double frame+lightly

reinforced wall

EI Centro (1940) 4.5 3


10-Story 3-bay double frame+lightly

reinforced wall

Taft (1952) 4.0 3


10-storey, 3-bay double frame

+heavily reinforced wall

Taft (1952) 5.2 3

H1 Cecen Sozen 10-storey, 3-bay double frame weak

beams

EI Centro (1940) 2.2 3

H2 Cecen and Sozen 10-storey, 3-bay double frame weak

beams design

EI Centro (1940) 2.7 7

TABLE 2. Shaking Table experiments at UIUC



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< previous page page_525 next page >Page 525a statistical analysis, it is necessary to resort to seismic simulations on shaking tables. Such experiments are very useful for model validation. A particularly interesting program of shaking table experiments has been performed at the University of Illinois at Urbana-Champagne (UIUC) by Sozen and his associates (Healey and Sozen (1978), Abrams and Sozen (1979), Cecen (1979), Sozen (1981). The experiments analyzed here come from a population of seven ten-storey, 1/10th -scale structures (Table 2). These structures were representative of a variety of options in the design of structures against lateral (seismic) loads.Each structure was tested at the University of illinois Earthquake Simulator. Test runs of a given structure included repetitions of the following sequence:(1) Free vibration test to determine low-amplitude natural frequencies.(2) Earthquake simulation(3) Recording of any observable signs of damage.This sequence was repeated with the intensity of the earthquake simulation being increased successively. For all but one structure (H2), the first earthquake simulation represented the ‘design’ level. The design earthquake for each structure was the earthquake for which the interstorey drift reached the limit level of 1%.The natural frequency of a structure’s linear oscillations can be estimated from the strong motion records. For the analysis of real world buildings, it must be assumed that no prior information about the structure is available and that the undamaged natural frequency must be estimated from the strong motion records.In order to estimate the equivalent linear parameters of a structure, records from the basement and from some upper level are needed. The basement record is used as input to a numerical model that is equivalent to the original structure in the Maximum Likelihood sense. As the actual structure is nonlinear, the equivalent linear model must be time variant. Therefore, the interval of duration of the earthquake is divided into segments (time windows) of appropriate length, and the parameter estimation is performed separately for each of these windows. Fig. 2 shows the accelerations records for the structure FW1, run 1 (Abrams and Sozen, 1979) and the corresponding evolution of the structure, as defined as

(7)where the subscript n indicates the ith window considered, and (T0)n the respective equivalent fundamental period estimated.The analysis of each window can be divided into two phases. First the order of model, i.e., the number of modes whose parameters are estimated, is determined (model identification). Then the parameters are estimated using maximum likelihood techniques. If the effect of input



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< previous page page_526 next page >Page 526noise is neglected, the estimation of the modal parameters using the Maximum Likelihood criterion reduces to matching the recorded acceleration of the upper floor with the output of the model considered. Experience shows that one-mode models may sometimes match the observed mation very poorly, although the estimates of the fundamental frequency are very close to those obtained using higher order models. Two-mode models usually can be fitted to the output with very good results, while three-mode models are very difficult to treat, due to the large number of parameters involved.

FIGURE 2. Acceleration Records and Evolution of Softening for the Structure FW1, Run 1Calibration of the ModelThe model has been calibrated according to the acceleration records from seismic simulation experiments performed at the Univeristy of Illinois at Urbana-Champaign [21]. The seismic behavior of seven reinforced concrete structures has been analyzed, for a total of 25 tests. Figure 3 shows



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< previous page page_527 next page >Page 527a plot of the normalized intensity versus the maximum softening of the earthquake for each of the seven structures analyzed. The normalized intensity is the ratio of the of the intensity of the earthquake to which the structure is subjected to the intensity of the design earthquake. As customary, the intensity of an earthquake is defined as the peak acceleration recorded during the strong motion. All the structures considered were designed so that they would undergo moderate damage (corresponding to a maximum storey drift of 1%) when subjected to an earthquake of the design intensity (typically 0.4g). Models of structures for shaking table tests were designed and built with much greater care than standard buildings. Their actual behavior was thus very close to analytical or numerical predictions. In particular, they were stiffened against out-of-plane (lateral and torsional) motion. Furthermore, the structure was subjected to a series of amplified replicas of the same earthquake ground motion (in most cases, El Centro, NS, 1940). For these reasons the ‘design’ earthquake intensity relative to the models considered is a very meaningful quantity. The structures were expected to undergo moderate damage when subjected to an earthquake of the design intensity, while no damage was expected for weaker earthquakes. For this set of structures, therefore, the design intensity ades and the lowest damaging intensity adam can be assumed to coincide. The normalized intensity can thus be used to detect the insurgence of structural damage. When this ratio is less than one, the structure may be considered safe, otherwise some damage must be expected. The functional form of the relationship between amax/ades and δM is certainly complicated. Nonetheless, as most data lie in the intermediate damage class, it can be expected that a linear regression would yield results valid in the neighborhood of the damage theshold.A linear model:

(8)has been fitted to the data of Table 2. Due to the uncertainties present in the problem. the parameters α and β must be treated as random variables. The leastsquare estimates of the regression parameters are:

α=−1.081 (9a)

(9b)with standard deviations:

σα=0.260 (10a)σβ=0.446 (10b)

For simplicity the covariance σαβ of the regression parameters has been neglected. α and β are therefore treated as

independent random variables. For the purpose of damage detection, the maximum softening is estimated from the analysis of strong motion records. An expected value E(amax/ades) for the normalized intensity is easily computed:



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(11)The uncertainties present in the problem can be easily taken into account if it is assumed that α, β and δM are

independent random variables with means , and and standard deviation σα, σβ, and and σδ. The variance of the normalized intensity will then be:

FIGURE 3. Intensity Ratio vs. Maximum Softening for UIUC Models (Dotted Line: Linear Regression)

(12)A conservative estimate of the standard deviation σδ of the maximum softening is:

(13)As a first approximation, the normalized intensity can be treated as a Gaussian random variable. Under this assumption it is easy to compute the probability that the damaging earthquake intenstiy has been exceeded:



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FIGURE 4. Fragility Curve, Structural Serviceability Limit State



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(14)From the analysis of a set of data, for which the onset of damage could be identified using the normalized intensity of the earthquake, a fragility curve has been defined, giving the probability of the onset of damage (limit state probability) as a function of the maximum softening δM (Fig. 4) estimated from the strong motion records. Such probability of damage must be considereed as the fraction of damaged structures, over a large number of structures for which the same δM has been estimated. In particular, when δM=0.43, the model predicts that 50% of the structures considered will present damage. A set of fragility curves, relative to a sequence of limit states, can be defined similarly [13].These results are valid for moderately slender reinforced concrete structures, whose resistance to horizontal load is provided by moment resistant frames or shear walls. The applications described in the next section suggest that this method of the detection of seismic structural damage can be applied to full-scale structures. It should be pointed out that structures analyzed in this project represented the widest database available in the USA. regarding seismic structural damage to reinforced concrete structures.VALIDATION OF THE DAMAGE DETECTION MODELAnalysis of Shaking Table ExperimentsA series of seismic simulations on shaking tables constitute an excellent test for damage assessment procedures. The experimental program considered here was carried out at UCB by Bertero and his associates within the US-Japan joint research project [5]. The structure tested was a 1/5th scale model of a seven-storeyed reinforced concrete building designed according to the Uniform Building Code. Resistance to lateral loads was provided by a moment resistant frame acting in parallel with a shear wall. After a series of low amplitude tests. the structure was subjected to simulated seismic excitation of increasing intensity until collapse occurred at the first floor. The structure was then repaired and tested again, until a second collapse occurred.The experimenters have reported information on the damage state subsequent to each of seven tests. Records from six of these tests were available and have been analyzed here. The procedure described above has been used for the detection of seismic damage. From the analysis of the acceleration records of each test, the maximum softening and its standard deviation were estimated. Hence the probability of the onset of seismic structural damage was computed. Figure 5 has been derived from the experimenters’ report (Bertero et al.). The points marked by circles identify the test runs in the maximum-storey-drift versus maximum-base-shear plane. The connecting lines represent an envelope of the hysteretic cycles



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< previous page page_531 next page >Page 531of the structure, and provide information on the progression of nonlinearities and the eventual collapse. Corresponding to each test run, amax, δM and P[damage] are reported. It should be noted that the method is successful in identifying the onset of damage. The simulated earthquakes that damage the structure are clearly identified by a very large probability of damage. Very low probabilities of damage correspond to those tests in which the structure remains safe.

FIGURE 5. Detection of Structural Damage for the UCB ModelsFor the purpose of damage detection, the maximum softening contains the same information as the envelope curve shown in Fig. 5. The advantage of using the maximum softening, as well as the other two global damage indices described in Table 1, is that only two acceleration records are needed for its computation, while extensive instrumentation is necessary to obtain the curve of Fig. 5 Criterion (14) provides a quantitative indication of the degree of damage, thus simplifying the decision process.Analysis of Actual Strong Motion RecordsFive strong motion records of the San Fernando earthquake were analyzed. The structures considered sustained little or no damage (Jen-



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< previous page page_532 next page >Page 532nings et al.). They are therefore a very good test for the sensitivity of the damage detection procedure. Strong motion records of the Imperial Valley eathquake, 1979, relative to the Imperial County service building, have also been analyzed. This building was severely damaged during the earthquake. Its repair was considered not advisable and it was decided to demolish it (Wosser et al.). Some similar analysis has been done for the double deck structure on 1–880 which collapsed in the 1989 Loma-Prieta earthquake [27]. The results of the serviceability analysis are shown in Table 3. The table reports the estimated value of the maximum softening δM and the probability that the structure is damaged (limit state probability). It can be seen that the model performs very well in detecting damage even at very low levels. TABLE 3. Damage Detection for Actual BuildingsSAN FERNANDO EARTHQUAKE (1971)

STRUCTURE DAMAGE δM P[damage]

611 6th Street None 0.13 0

Sheraton Hotel None 0.29 0

Milikan Library None 0.32 0.03

Holiday Inn Repaired 0.52 0.88

Bank of California Repaired 0.81 1IMPERIAL VALLEY EARTHQUAKE (1979)STRUCTURE DAMAGE δM p[damage]

Imperial Countv (EW) Demolition 0.6 0.97

Imperial County (NS) Demolition 0.59 0.97LOMA PRIETA EARTHOUAKE (1989)

STRUCTURE DAMAGE δM p[damage]

Double-deck structure on 1–880 Demolition 0.62* 0.97



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< previous page page_533 next page >Page 533Evaluation of Maximum Softening as a Damage Indicator for Reinforced Concrete Under Seismic ExcitationThe properties of the maximum softening as a global damage indicator for reinforced concrete structures subjected to seismic loads have been evaluated [23]. Especially the Markov property of the damage indicator is investigated, which is mandatory if statements on the post earthquake structual reliability is intended solely from knowledge of the latest recorded damage value. The transition probability density frunction (tpdf) of the damage indicator during sequential earthquakes has been determined by Monte-Carlo simulation for a planar 3 storey 2 bay reinforced concrete frame designed according to the Uniform Building Code (UBC) zone 4. The applied earthquake excitation model is the non-stationary single earthquake ARMA-model of Ellis and Cakmak (1987), which depends on the magnitude of the earthquake, the epicentral distance, the duration of strong shaking and local soil quality parameter. The dynamic analysis of the structure is performed by the SARCF-II program of Chung et al., (1987). For each realization of the earthquake process a sample of the damage indicator during the earthquake is obtained by numerical integration of the structural equations of motion. From the observed data sample, the Markov assumption of the damage indicator sequence is justified for the mean value of the tpdf, whereas some deviation is observed for the variance and higher order statistical moments. In addition a regression analysis is performed for the increment of the damage indicator versus the sample peak acceleration and the sample peak frequency of the causing realization of the earthquake process, from which, it is concluded that the maximum softening is completely uncorrelated to these quantities [23].COMPARISONSRodriguez-Gomez and Cakmak (1990) compared maximum softening (the global damage index defined in equation 6) to weighted averages of some recently developed damage indices and to traditional measures of damage such as the maximum interstorey drift, the permanent interstorey drift and the maximum ductility ratio for beams and columns.Due to the scarcity of experimental data and because this was the only option available for the computation of some of the damage indices that had been considered. comparisons were obtained by means of numerical simulations. In order to compute the maximum softening DiPasquale and Cakmak (1987–90) used the ground acceleration and the acceleration at another location such as the top of the structure. On the other hand in the numerical simulations, besides these acceleration records, information about the instantaneous natural period was available. So, Rodriguez-Gomez and Cakmak (1990) developed a new procedure for the computation of the damage indices based on vibrational parameters from the instantaneous natural period [26]. An averaging window with a length



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< previous page page_534 next page >Page 534equal to two or two and a half times the initial natural period has been found to yield results consistent with the original way of computing the global damage indices of DiPasquale and Cakmak.The seismic response of several reinforced concrete frames subjected to a set of artificially generated and recorded earthquakes was computed by using an improved version of the computer code SARCF. A general description of the initial version of SARCF (Seismic Analysis of Reinforced Concrete Frames) and the modifications done by Rodriguez-Gomez can be found in [8] and [26] respectively. This new version was first tested by comparing computed results to experimental results obtained from a reinforced concrete model tested at the University of Illinois at Urbana-Champaigne. The conclusion that was extracted from the comparison of the recorded response to the computed response was that, given the uncertainties that exist in the description of the nonlinear behavior of reinforced concrete structures. a good approximation was obtained using the program SARCF-II [26].The numerical simulations have been carried out on three low rise buildings designed according to the existing codes. First, one two bay three storey building frame, one three bay four storey and another three bay four storey building frames are designed according to the ACI 318–83 [2] code , to resist the equivalent static lateral loads specified in the Uniform Building Code [32] for seismic zone 4. The details of the structures can be found in [26]. Then a total of 29 different earthquakes have been applied to the three structural models. Two types of input acceleration have been used: scaled versions of the N-S component of the acceleration history recorded at El Centro during the Imperial Valley Earthquake (1940), and artificially generated earthquakes using an ARMA model according to the method proposed by Ellis and Cakmak [19, 20]. The parametric relationships between modeling parameters and physical variables were obtained from a set of strong motion accelerograms recorded from the following Japaneses earthquakes: Tokachi-Ochi (1968), Miyagiken-Oki (1978), Nihonkai-Chubu (1983) and Michoacan (1985). Single event earthquakes have been generated using this capability of the program SARCF-II. The input parameters and their values used in this case are: an initial peak time of two seconds, a duration of the earthquake of 20 seconds, a distance from the epicenter of 10 or 100 kilometers, different values of the earthquake magnitude, and a soil factor, γf, with a value of 0.10.For each of the nonlinear dynamic analysis that have been carried out, information about traditional measures of damage, local damage indices, global damage indices and parameters that characterize the response of the damaged structure to a second earthquake were obtained by using the SARCF-II.Among the traditional means of damage the maximum interstorey



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< previous page page_535 next page >Page 535drift, the permanent interstorey drift and the maximum ductility ratio for beams and columns, have been computed.The interstorey drift is defined as the tangent of the angle between the original position and the deformed position of the columns. The interstorey drift has been used as a measure of damage by M.A.Sozen [28].The permanent drift is the interstorey drift after the earthquake. The permanent drift has been used as an indication of damage by J.E.Stephens and J.P.T.Yao [30 and 31].The ductility ratio which has been computed is a curvature ductility ratio, defined as the ratio between the curvature and the yield curvature.Chung, Meyer and Shinozuka’s damage index and the modified Park and Ang’s damage index have been computed at both ends of each beam or column.Chung, Meyer and Shinozuka’s local damage index combines a modified version of Miner’s Hypothesis with damage acceleration factors that reflect the effect of the loading history [8].Park and Ang’s local damage index includes two terms that reflect the influence of the maximum deformation and the absorbed hysteresis energy [24].The global damage indices that were studied are: 1. Energy average of Chung, Meyer and Shinozuka’s Local Damage Index [26].2. Weighted average of Chung, Meyer and Shinozuka’s Local Damage Index, using the triangular weighting function [26].3. Energy average of Park and Ang’s Damage Index [26].4. Maximum and Final Softening defined by Equations (5c) and (5a). Both indices have been computed in two different ways, using the Systems Identification Program, MUMOID, and computing a moving average of the instantaneous natural period.First, the correlation between system identification and moving averages methods for the computation of the maximum and final softening were obtained and it is concluded that two methods give equivalent results for the structures of study. Then maximum softening is compared to other damage indices. The comparisons have been done by means of a series of plots that show the value of the maximum softening as a function of the other damage indicator. Each point in the graphs has been obtained from the nonlinear dynamic analysis of any of the building frames described in [26]. The filled circles represent the cases for which the second occurrence of the same earthquake produced collapse. However, the filled circles correspond to the damage indices that were computed for the first earthquake.Whether or not the structure collapses in the second earthquake is considered to be an objective measure of the level of damage in the first



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FIGURE 6. Maximum Softening vs. Maximum Interstorey Drift


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FIGURE 7. Maximum Softening vs. Maximum Final Drift



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FIGURE 8. Maximum Softening vs. Maximum Ductility Ratio for Columns

FIGURE 9. Maximum Softening vs. Maximum Ratio for Beams



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FIGURE 10. Maximum Softening vs. Energy Average of Park’s Index

FIGURE 11. Maximum Softening vs. Energy Average of Chung’s Index



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< previous page page_539 next page >Page 539earthquake, the best global damage index would be the one that can predict the collapse of the structure in the second earthquake. In that case, there should be a cutoff value of that damage index so that a collapse resulting from identical earthquake only occurs for values greater than that cutoff value.Since it is possible to draw a horizontal line that seperates the squares and filled circles in all figures 6, 7, 8, 9, 10, 11 and 12, recalling that the serviceability limit state was related to an average value of maximum softening of 0.43, one concludes from figures 6, 7, 8, 9 ,10, 11 and 12 that for all cases, a value of the maximum softening larger than 0.43 implies a collapse in the second identical earthquake and that there is not such a value for the other damage indicators that seperates the structures that collapse in the second earthquake. The reader is referred to reference [26] for detailed information about the calculation of other damage indicators.

FIGURE 12. Maximum Softening vs. Chung’s Weighted AvergageCONCLUSIONSIn this paper, for the detection of seismic structural damage, global dam-



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< previous page page_540 next page >Page 540age indices based on the lengthening in the fundamental period has been first introduced and then validated.For post-earthquake reliability assessment, a damage model based on the evolution of the natural period of a time-varying linear system equivalent to the actual nonlinear system for a series of non-overlapping time windows has been developed. For the identification of the time varying equivalent system from the acceleration records at the top and at the base of the structure a maximum likelihood criteria has been used.Noted that previously in [38, 40] the theoretical basis to the intuitvely defined global damage indices (final softening, plastic softening and maximum softening) had been studied and it was concluded that continuum mechanics provide a theoretical foundation to the defined global indices and the proposed indices carry the necessary information for both cases of elastic and plastic damage. The details of the theoretical study is not presented in this paper.In order to calibrate the model, acceleration records from seismic simulation experiments performed at the University of Illinois at Urbana-Champaigne have been used. The seismic behavior of seven reinforced concrete structures has been analyzed, for a total of 25 tests. The damage detection criterion proposed yields the probability that the structure has undergone damage after the seismic event or not. So, a fragility curve, relative to the structural serviceability limit state has been derived. The probability of the onset of seismic structural damage is computed as a function of a global damage index (maximum softening) which is based on equivalent modal parameters,This criterion has been used for the damage analysis of shaking table experiments performed at the University of California at Berkeley and strong motion records from the San Fernando (1971), Imperial Valley (1979) and Loma Prieta (1989) earthquakes. In all cases considered, an excellent correlation has been found between the values of the parameterbased damage indices estimated from strong motion records and the onset of seismic structural damage.Since the post earthquake structural reliability is intended solely from the latest recorded damage value, the Markov assuption of maximum softening has been studied and it is justified for the maximum value if transition probability density fuction where as some deviation is observed for variance and higher order moments.In order to compare maximum softening with other damage indicators, the response of three code-designed buildings frames subjected to artificially generated and recorded earthquakes has been obtained by using the computer code SARCF-II In this numerical study, several damage indices and traditional measures of damage have been computed and compared to maximum softening. A new way of computing maximum softening by moving averages of the instantaneous natural period has been



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< previous page page_541 next page >Page 541derived and it is oberved that system identification and moving averages methods are equivalent from practical point of view. Before graphic comparisons, simulation results are compared to experimental results of the tests done at UIUC by Sozen et al. Considering the uncertainities that exist in the description of the nonlinear behavior of reinforced concrete structures, a good approximation to the experimental values was obtained by using SARCF-II. For the studied case, it is concluded that maximum softening predicts the structural capacity for future earthquakes more consistently than any of the traditional measures of damage and that maximum softening is a more robust global damage index than the weighted averages of the local damage indices. It should be noted that local damage indices give important information at the local level, but for obtaining global damage, an appropriate weighting function has to be defined. The maximum softening avoids the definition of a weighting function.The maximum softening and the limit state probability defined can therefore be used in the detection of seismic structural damage.ACKNOWLEDGEMENTIn carrying out this work, the authors were supported by the National Center for Earthquake Engineering Research under contract number 89–1104. The authors also acknowledge the support of Princeton-Kajima Joint Research Program.REFERENCES1. ABRAMS, D.P., SOZEN, M.T., (1979), “Experimental Study of Frame-Wall Interaction in Reinforced Concrete Structures Subjected to Strong Earthquake Motions”, Report No. SRS 460, UILU-ENG-79–2002, University of Illinois at Urbana-Champaigne.2. AMERICAN CONCRETE INSTITUTE, (1984), “Building Code Requirements for Reinforced Concrete”, ACI 318–83, Detroit, MI.3. BECK, J.L., (1979), “Deteriming Models of Structure from Earthquake Records”, Ph.D Dissertation, California Institute of Technology.4. BECK, J.L., JENNINGS, P.C., (1980), “Structural Identification Using Linear Models and Earthquake Records”, Earth. Eng. Struc. Dyn., Vol. 8 pp. 145–160.5. BERTERO, V.V., AKTAN, A.E., CHARNEY, F.A., SAUSE, R., (1984), “U.S.-Japan Cooperative Research Program: Earthquake Simulation Tests and Associated Studies of a 1/5th Scale Model



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< previous page page_542 next page >Page 542 of a 7-Story Reinforced Concrete Test Structure”, Report No. UCB/EERC-84/05, University of California at Berkeley.6. CAUGHEY, T.K., (1963), “Equivalent Linearization Technique”, J. Ac. Soc. Am., Vol. 27, No. 4, pp. 1706–1711.7. CECEN, H., (1979), “Response of Ten Story, Reinforced Concrete Models Frames to Simulated Earthquakes”, Ph.D. Dissertation, University of Illinois at Urbana-Champaigne.8. CHUNG, Y.S., MEYER, C. AND SHINOZUKA, M., (1987), “Seismic Damage Assessment of Reinforced Concrete Buildings”, NCEER-87–0022, National Center for Earthquake Engineering Research, State University of New York at Buffalo.9. CHUNG, Y.S., MEYER, C. AND SHINOZUKA, M., (1988), “Automated Seismic Design of Reinforced Concrete Buildings”, NCEER-88–0024, National Center for Earthquake Engineering Research, State University of New York at Buffalo.10. CHUNG, Y.S., MEYER, C. AND SHINOZUKA, M., (1988), “SARCF User’s Guide, Seismic Analysis of Reinforced Concrete Frames”, NCEER-88–0044, National Center for Earthquake Engineering Research, State University of New York at Buffalo.11. DIPASQUALE, E., CAKMAK, A.S., (1987), “Damage Assessment from Earthquake Records”, Proc. 3rd Int. Conf. Soil Dyn. Earth. Eng., Princeton, NJ.12. DIPASQUALE, E., CAKMAK, A.S., (1987), “Detection and Assessment of Seismic Structural Damage”, Report NCEER-87–0015, National Center for Earthquake Engineering Research, State University of New York at Buffalo.13. DIPASQUALE, E., CAKMAK, A.S., (1988), “Identification of the Serviceability Limit State and Detection of Seismic Structural Damage”, Report NCEER-88–0022, National Center for Earthquake Engineering Research, State University of New York at Buffalo.14. DIPASQUALE, E., CAKMAK, A.S., (1989), “On the Relation Between Local and Global Damage Indices”, Technical Report NCCEER-89–0034, National Center for Earthquake Engineering Research, State University of New York at Buffalo.15. DIPASQUALE, E., CAKMAK, A.S., (1990), “Detection of Seismic Structural Damage Using Parameter-Based Global Damage Indices”, Probabilistic Engineering Mechanics, Vol. 5, No. 2, pp. 60–65.16. DIPASQUALE, E., JU, J.W., ASKAR, A., AND CAKMAK, A.S., (1990), “Relation Between Global Damage Indices and Local Stiffness Degradation”, J. of Struc. Eng., Vol. 116, No. 5, pp. 1440–1456.17. ELLIS, G.W., (1987), “Modeling Earthquake Ground Motions in Seismically Active Regions Using Parametric Time Series Methods”, Ph.D. Dissertation, Princeton University, Princeton, NJ.



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< previous page page_543 next page >Page 54318. ELLIS, G.W., CAKMAK, A.S., LEDOLTER, J., (1987), “Modelling Earthquake Ground Motion in Seismically Active Regions Using Parametric Times Series Methods”, Proc. 3rd Int. Conf. Soil Dyn. Earth. Eng., Princeton, NJ.19. ELLIS, G.W., CAKMAK, A.S., (1987), “Modeling Earthquakes in Seismically Active Regions Using Parametric Time Series Methods”, Technical Report NCEER-87–0014, National Center for Earthquake Engineering Research, State University of New York at Buffalo.20. ELLIS, G.W., CAKMAK, A.S., (1988), “Modelling Strong Ground Motion from Multiple Event Earthquakes”, Technical Report NCEER-88–0042, National Center for Earthquake Engineering Research, State University of New York at Buffalo.21. HEALEY, T.J., SOZEN, M.A., (1978), “Experimental Study of the Dynamic Response of a Ten Story Reinforced Concrete Frame with a Tall First Story”, Rep. No. UILU-ENG-78–2012, SRS 450, University of Illinois at Urbana-Champaigne.22. Ju, J.-W., (1989), “On Energy-Based Coupled Elasto-Plastic Damage Theories: Constitutive Modeling and Computational Aspects”, Intl J. of Solids and Struc., Vol. 25, No. 7, pp. 803–833.23. NIELSEN, S.R.K., CAKMAK, A.S., (1991), “Evaluation of Maximum Softening as a Damage Indicator for Reinforced Concrete Under Seismic Excitation”, Proc. of 1st Inter. Confrence on Computational Stochastic Mechanics (to appear September 1991).24. PARK, Y.-J., ANG, A.H.S., (1985), “Mechanistic Seismic Damage Model for Reinforced Concrete” ,ASCE J. Struc. Eng., Vol. 111, No. 4, pp. 722–739.25. PARK, Y.-J., ANG, A.H.S. AND WEN, Y.-K., (1985), “Seismic Damage Analysis of Reinforced Concrete Buildings”, ASCE J. Struc. Eng., Vol. 111, No.4, pp. 740–757.26. RODRIGUEZ-GOMEZ, S., (1990), “Evaluation of Seismic Damage Indices for Reinforced Concrete Structures”, M.Sc. Eng. Dissertation. Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ.27. RODRIGUEZ-GOMEZ, S., CAKMAK, A.S., AND SHINOZUKA, M., “Damage Analysis of Simulated 1–880 Structures Under the Loma Prieta Earthquake”, Soil Dyn. and Earth. Eng. (to appear).28. SOZEN, M.A., (1981), “Review of Earthquake Response of Reinforced Concrete Buildings with a View to Drift Control”, State of the Art in Earthquake Engineering, Turkish National Committee on Earthquake Engineering, Istanbul, Turkey.



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< previous page page_544 next page >Page 54429. STEPHENS, J.S., (1985), “Structural Damage Assessment Using Response Measurement”, Ph.D. Dissertation, Purdue University, Lafayette, IN.30. STEPHENS, J.S., YAO, J.P.T., (1987), “Damage Assessment Using Response Measurements”, ASCE J. Struc. Eng., Vol. 113, No. 4, pp. 787–801, “Theory of Evidence”, Structural Safety, Vol. 1, pp. 107–121.31. Toussi, S., YAO, J.P.T., (1983), “Hysteresis Identification of Existing Structures”, ASCE J. Eng. Mech., Vol. 109, No. 5, pp. 1189–1203.32. UNIFORM BUILDING CODE, (1988), “Earthquake Regulations”.33. VALDIMARSSON, H., SHAH, A.H., MCNIVEN, H.D., (1981), “Linear Models to Predict the Non Linear Seismic Behavior of a One-Story Steel Frame”, UCB/EERC 81/13, Berkeley, CA.34. YAO, J.P.T., (1982), “Probabilistic Method for the Evaluation of Seismic Damage of Existing Structures”, Soil Dyn. Earth. Eng., Vol. 1, No. 3, 1982, pp. 130–135.



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< previous page page_545 next page >Page 545Reduction of Linear Elastic Response Spectra due to Elastoplastic Behaviour of SystemsS.E.Ruiz, O.DíazInstituto de Ingenieria, UNAM, Apdo Postal 70–472, Coyoacan, 04510, Mexico, D.F., MexicoABSTRACTUsing the responses obtained from five groups of accelerograms recorded in soft and hard soils in Mexico City, the well known rules proposed by Newmark (1970) for estimating nonlinear response spectra on the basis of the elastic ones are analyzed for different types of ground motion. This study indicates that the mentioned rules (applied to the mexican seismic code of 1987), give place to different results with respect to those obtained directly from records, depending on: a) ductility reduction factors, b) frequency-content characteristics of motions, and c) structural vibration periods. A critical analysis of the results is made. A set of simple rules to reduce the elastic spectra due to elastoplastic behavior is given. Each rule corresponds to different soil conditions.INTRODUCTIONRules for reducing the ordinates of linear acceleration spectra used in seismic design to account for inelastic response of structures are given in some seismic design codes, i.e. Federal District Seismic Code of Mexico City (FDR-87), ref 1; UBC Code, ref 2, etc. A well known set of rules was proposed by Newmark, in 1970 where: for long and moderate periods a hypothesis of similar displacements of linear-elastic and elastoplastic systems is adopted, whereas for the short period range, similar energies are supposed. For long and moderate periods the reduced spectra are obtained simply by dividing the linear elastic spectral responses by a reduction factor (this is called Q in ref 1, and K in ref 2). For short

periods the spectral ordinates are divided by while the ordinates of nonlinear spectra tend to the peak ground acceleration, regardless of the ductility value assumed.



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< previous page page_546 next page >Page 546In this paper we analyzed the above mentioned rules for motions recorded on different types of soil in Mexico City. Five groups of intense ground motions are studied. Their elastoplastic spectra are obtained for ductility factors Q of 1.5, 2, 3 and 4. In order to evaluate the relations among the spectra coresponding to different soil conditions, as well as the spectra reduced by inelastic behavior, the mean values of the ratios between inelastic (Ai) and elastic (Ae) acceleration spectra were compared with those obtained by Newmark,s rules.EARTHQUAKE MOTIONSThe ground motions analyzed were originated at the subduction zone in southwestern Mexico with magnitudes equal or greater than 6.3.Group A. Ground motions recorded on soft soil. with peak period of 2s.This group consists of 11 accelerograms recorded on free field at the surface of soft clay deposits. Their magnitudes were in the range of 6.8 to 7.6. The mean acceleration spectrum—plus and minus one standard deviation—of the normalized accelerograms, for 5% of critical damping, is shown in Fig 1a.For normalizing each record its ordinates were scaled so that each motion had an energy content equal to the maximum energy presented in any accelerogram within each group.Group B. Ground motions affected by soil-structure interaction on soft soil, with peak period of 2s.These were recorded by instruments placed at the base of structures located on soft clay layers of 30 to 40m deep. This group with 6 accelerograms, is associated with earthquakes having magnitudes between 6.3 and 6.5. Figure 1b presents the mean spectrum—plus and minus one standard deviation—for 5% of critical damping. In view of the soil-structure interaction phenomenom, the bandwidth is wider than that associated with group A.Group C. Ground motions recorded on soft soil with peak period of 3s.This group consists of 7 records obtained in the Texcoco Lake area. The soil in this zone includes a superficial layer of alluvial deposits followed by a thick layer of soft clay. Underlying this there is a hard deposit of silty weakly cemented sand about 50 to 70m deep (Romo et al, 1988). The



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< previous page page_547 next page >Page 547mean spectrum—plus and minus one standard deviation—for 5% of damping is shown in Fig 1c.Group D.1. Ground motions on stiff soil with spectral peak periods between 0.9 and 2s.Ground motions of this group were recorded within the National University campus, located at the south-west of the city. This zone has a layer of fractured lava, about 12m deep, overlying soft rock at depths between 12 and 21m (Seed et al, 1988). The events’ magnitudes range from 6.4 to 7.8. The associated mean spectrum is shown in Fig 1 d.1. As the September 19, 1985 records obtained at this zone were considered seismologically uncommon (Singh et al, ref 6) they were analyzed separately.Group D. 2. Ground motions recorded on the stiff soil zone of Mexico City during the September 19, 1985 earthquake.This group consist of 6 records obtained at 3 sites at the National University campus during the 8.1 magnitude earthquake of 1985. The mean spectrum of Fig 1 d.2 shows a bandwidth between 1 and 2s.SPECTRAL RATIOSThe elastoplastic acceleration spectra were calculated for each accelerogram, assuming four different ductility levels: Q= 1.5, 2, 3 and 4. The ratios between these spectra (Ai) and the linear elastic ones (Ae) were obtained for different

vibration periods. Then, the mean ratios and the standard deviations for each frequency were calculated.REGRESSION CURVESA regression curve having the following form was fitted to Ai/Ae data:

(1)where

Q=ductility demand



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T=vibration period a, b, c, d=parameters to be fitted based on data

Powell’s method (ref 7) was applied to curve-fitting. Values of the parameters a, b, c and d for each case are presented in Table I. As an example, the regression curve r—plus and minus one standard deviation, σr—of group A, Q=2, is shown

in Fig. 2. In the same figure the mean values curve is presented.The regression analysis of groups D.1 and D.2 suggested that they behaved similarly, so they were grouped in a single group denoted as group D. Regression parameters for this case are also given in Table I. Figs 3a–d show regression

curves corresponding to the studied groups. They also show the ratios associated to the inelastic design

spectra divided by the elastic one , both obtained from the FDR-87 (ref 1). This code uses the Newmark’s reduction rules.RESULTSInfluence of the vibration period

Results in Figs 3a–d indicate that ratios are higher than those of the regression curve r, for periods between 1.5s or 2.5s and 3s or 4s, depending on the group. For short periods, 0<T<1.5s or 2.5s, those ratios (C) are always smaller. For T values greater than 3s or 5s the mentioned ratios tend to the value 1/Q.Peak inelastic spectra ordinatesFrom the regression curves (ec. 1 and Table I) it is observed that the minimum value of each curve occurs at periods which are functions of the ductility factor and the type of motion. For example, Fig 3a illustrates the fitted curves for Q=1.5, 2, 3 and 4 corresponding to group A. The minimum value of each curve occurs at larger periods for higher values of Q. This shows that for high values of Q the peak values of the elastoplastic acceleration spectra occur at shorter periods than that associated to the peak elastic spectra. This indicates that for high Q values the inelastic spectra ordinates are higher for short structural periods than for large ones, which is reasonable.



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< previous page page_549 next page >Page 549Comparison among acceleration ratios of the same group, associated to different Q valuesPeak values of ratios C/M versus Q are plotted in Fig 4. From this figure it can be seen that for groups A, C and D such ratios are higher for Q=2 than for ductility factors of Q=3 or 4.Comparison between groupsFigure 4 also shows that in general the maximum values of ratios C/M correspond to group A and the minimum ones to group D. This is an indication that dynamic response to narrow band excitations undergoes more important reductions due to inelastic behavior than that corresponding to wide-band motions.CONCLUSIONS1.The elastoplastic spectral accelerations corresponding to ground motions recorded on soft soils in Mexico City have

higher ratios C/M than those associated to motions recorded on the stiff soil zone (fig 4). ,

, subindex i and e correspond to inelastic and elastic responses respectively.This means that for systems subjected to narrow band motions like those of group A (soft soil) response reductions associated to elastoplastic behavior are higher than those associated with wider band motions like those of group D (stiff soils). This conclusion may not apply to stiffness degrading systems, where more pronounced period changes may occur, leading to different response patterns.The structural engineer should provide designs capable of absorbing the energy of structures without suffering damage, specially when they are subjected to narrow band motions.

2. Figures 3a–d show the behavior of the ratios between (related to Newmark,s rules, applied to the 1987 mexican seismic Regulations, ref 1), and those calculated from motions recorded on soft and stiff soils of Mexico City (ec 1). From these figures the following conclusions can be made:a)For intermediate periods (1.5 to 2.5s<T< 3.5 to 5s, depending on the group) C ratios based on Newmark,s rules are higher than those corresponding to results of real motions (ec 1).b) For short periods (0<T<1.5 to 2.5s) the ratios C are



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< previous page page_550 next page >Page 550 always smaller than r. Nontheless it must be noted that for short periods the structures are generally designed for vertical loads and the seismic effect is unimportant.c) For long periods the ratios C and r tend to the same value 1/Q. It should be noted that for very large T periods ec 1 is

equal to 1/Q. This condition is established by Newmark’s rules. Nonthelees some of the values obtained in this study from the accelerograms groups are greater than 1/Q for periods longer than 4s.3. From Fig 4 it can be appreciated that the peak ratios C/M are greater for Q values of 2 and they decrease for higher ductility factors.4. General expressions, including band-width characteristics, besides the parameters studied in this paper, are being explored in a current research project at the Institute of Engineering, UNAM. Such project includes the analysis of near field motions like those registered in stiff soil of Acapulco, Gro, Mexico.ACKNOWLEDGMENTSThanks are given to L Esteva, R Gómez and M Ordaz for their valuable suggestions. The enthusiastic participation of Hector Rosas and Omar Valladares, is greatly appreciated.REFERENCES1. “Mexico Code 1987” Reglamento de construcciones para el Distrito Federal, Diario oficial de 1a Federación, Mexico, D.F., Mexico, July , 19872. Hadjian, A.H., “A calibration of the lateral force requirements of the UBC”, Proc. 8th world conference on earthquake engineering, San Francisco, 19843. Newmark, N.M., “Current trends in the seismic analysis and design of high-rise structures”, in Earthquake. Engineering, edited by R.Wiegel, Prentice-Hall, Englewood Cliffs, N.J., 19704. Seed, H.B. et al, “Relationships between soil conditions and earthquake ground motions” Earthquake spectra, Vol 4, No. 4, pp 687–730, nov 1988



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< previous page page_551 next page >Page 551 5. Romo, M.P., Jaime, A and Resendiz, D, “General conditions and clay properties in the valley of Mexico”, Earthquake spectra, pp 731–752, nov 19886. Singh, S.K., Mena, E and Castro, R., “Some aspects of source characteristics of th 19 September 1985 Michoacan earthquake and ground motion amplification in and near Mexico City from strong motion data, Bull seism soc Am., Vol 78, No. 2 pp 451–477, April, 19887. Press, W.H. et al, Numerical recipes, Cambridge University Press, London, pp 294–307, 1986TABLE IParameters of the regression curves Parameters

Group a b c d

A 3(Q)−1/2 0.4158+3.33Q 8.0 0.9Q2

B 1.559 0.506+0.772Q 1.846 0.431

C 1.40 0.298–0.77Q 2.10 0.02

D.1 1.502 0.165+0.133Q 0.068 0.147

D.2 1.926 0.749+0.795Q 1.513 0.662

D 1.70 0.2917+0.711Q 1.310 0.428



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Fig 1 Mean acceleration spectra, plus and minus one standard deviation of the normalized records



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Fig 2 Data Ai/Ae and regression (r) curves, plus and minus one standard deviation (σr) of group A, Q=2

Fig 3 a, b Regression curves



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< previous page page_555 next page >Page 555Problems in the Determination of Input Data for the Seismic Design of Structures in Regions of Low SeismicityJ.Eibl, E.KeintzelInstitut für Massivbau und Baustofftechnologie, Universität Karlsruhe, W-7500 Karlsruhe, GermanyABSTRACTPossibilities are analysed in the paper to consider in the design of structures in regions of low seismicity the favourable effect of short earthquake duration. A reduction of the design peak ground acceleration, used in traditional codes, is not applicable in modern codes, based on ductility considerations, because it would prevent a realistic evaluation of ductility demands. As an alternative a detailed analysis of the design provisions is proposed, aiming to adapt them to the reduced requirements regarding the behaviour under cyclic loading of structures in regions of low seismicity. As an example the elaboration of design provisions for reinforced concrete structures of ductility class L in Eurocode 8 in examined. Suggestions concerning the design of masonry structures in regions of low seismicity are also given.INTRODUCTIONIn the design of structures in seismic regions three parameters of a presumed future seismic event must be considered, on which depends the effect of earthquakes on structural elements: the intensity of the seismic ground motion, its frequency content and its duration. The first two characteristics—intensity and frequency content—enter directly into the input data for seismic design by the response spectrum method. So the intensity of the ground shaking is characterized usually by a conventional value of the peak ground acceleration and the frequency content is reflected by the shape of the spectrum.In what concerns the strong motion duration, it is recommended frequently—e.g. in the Workshop on Seismic Input Data in Lisbon [1]



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< previous page page_556 next page >Page 556—to adjust the value of the intensity parameter—that is the value of the effective peak ground acceleration—with this characteristic of the ground motion. Code provisions for seismic design generally are oriented by the needs of structures in regions of high seismicity. So they are aimed to ensure the integrity of structures for relative long durations of the ground shaking, occuring at earthquakes of high magnitudes. Correspondingly for regions of low seismicity, where only seismic events of low magnitudes are considered, a double revision of code provisions is possible, reflecting as well the effect of lower ground accelerations as that of shorter strong motion durations. In this situation the favourable effect of shorter strong motion durations is introduced in codes frequently by reducing the design peak ground acceleration. As an example may be considered the actual German Seismic Code DIN 4149 [2], stating in Clause 7.1 that “the assumed accelerations….are not identical with the peak soil accelerations, which are higher, but which cannot exert their full effect because of the very short duration of action occuring in German earthquake regions….”. In Fig. 1 a comparison is shown between elastic response spectra, corresponding to the assumptions of DIN 4149 [2] and to a proposal for seismic input data by Hosser et al. [3], to be used in a design according to Eurocode 8 [4]. The realistic spectral values, proposed by Hosser et al. [3], exceed the values of DIN 4149, reduced by considering the influence of short earthquake duration, by more than two times. The question is raised how to consider this influence also in a design according to Eurocode 8.

Fig. 1 Elastic Response Spectra for German sites on rock or firm soil, according to DIN 4149 and to a proposal for Eurocode 8



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< previous page page_557 next page >Page 557DUCTILITY BASED DESIGN IN EUROCODE 8One of the most important differences between Eurocode 8 and traditional seismic codes, as DIN 4149, consists in the fact that in the traditional codes the prescriptions for the design of ductile structures are more qualitative than quantitative, whereas in Eurocode 8 ductility demand and ductility supply are controlled numerically in a high degree. So in Eurocode 8 the favourable effect of inelastic deformations is introduced by behaviour factors q, by which the ordinates of elastic response spectra are devided. Subsequently these behaviour factors are converted into quantitative ductility requirements and further into quantitative prescriptions for detailing.The implementation of a given behaviour factor q, that is the design of a structure, complying with q-dependent ductility demands, is shown schematically in Fig. 2 for reinforced concrete structures. In a first step displacement ductility factors for single degree of freedom systems are expressed as a function of behaviour factors. They are calculated as the mean values of two expressions, the one being derived by equating the deformation energies of the elastic and the inelastic system, but the other by equating their displacements. In a second step curvature ductility factors in the critical sections of real, multi degree of freedom structures are derived from displacement ductility factors of associate single degree of freedom systems and then expressed approximately as a function of behaviour factors. They are proved to increase proportionally with the squares of behaviour factors. Finally, in a third step, constructional details of the confining reinforcement are adapted to the required curvature ductility factors, by using an appropriate constitutive law for confined concrete and by inserting a verification of the section after the spalling of the concrete cover. In this way the whole process of designing and detailing is based on a quantitative evaluation of ductility requirements, corresponding to a realistic assessment of behaviour factors as ratio between elastic seismic loads and design loads.A similar situation is encountered in the case of steel structures, where the width-thickness ratio b/t is restricted for compressed zones of sections, its limit values depending on the behaviour factor q as a measure for the required rotation capacity of the plastic zones.The admittance in Eurocode 8 of non-dissipative steel structures, for example of K-shaped truss bracings, which must remain in the eleastic range, equally implies the evaluation of realistic elastic seismic loads.As a conclusion it is stated that the quantitative definition of ductility requirements in Eurocode 8 implies the use of realistic spectral values. The consideration of the influence of short earthquake duration



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Fig. 2 Implementation of a given behaviour factor q for RC structures according to Eurocode 8by reducing the spectral values, as in DIN 4149, would falsify the design calculations and is therefore unsuitable. So the consideration of this effect in a design according to Eurocode 8 or to an other similar modern code must be based on a detailed analysis of the behaviour of structural members in the nonlinear domain, as shown in the following for reinforced concrete members.



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< previous page page_559 next page >Page 559RC STRUCTURAL MEMBERS UNDER CYCLIC LOADING IN THE NONLINEAR DOMAINIn Fig. 3 typical hysteresis loops for RC structural members under cyclic loading in the nonlinear domain are represented. Curve 1 shows a perfectly elastic-plastic load-displacement relationship, unattainable in real RC plastic hinges. Curve 2, showing a load-displacement relationship characterised by Bauschinger effect, corresponds to a RC member dominated by flexure, in which the internal forces are transferred predominantly by reinforcing steel. Curves 3 and 4 correspond to a RC member, dominated by shear. In this case both important strength and stiffness degradation occurs; the hysteresis loops are of pinched shape, and the area eclosed by a hysteresis loop, which represents the energy dissipation capacity, is small. So the earthquake-resistant capacity is poor in comparison with that of a member dominated by flexure.In Fig. 4 examples of experimentally obtained hysteresis loops are represented, belonging to RC members with flexural, respectively with shear failure. Whereas for the member with flexural failure a large number of hysteresis cycles can be supported without strength degradation, for the member with shear failure an important strength degradation occurs after few cycles.

Fig. 3 Idealized, optimal and degrading displacement response of a RC member during an inelastic pulse, from Paulay et al. [5]



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Fig. 4 Hysteresis loops for RC structural members under cyclic loading.a) Member with flexural failure, from Park et al. [6],b) member with shear failure, from Wight et al. [7].

Fig. 5 Load and energy degradation under cyclic loading in beams with a given displacement ductility factor, from Brown et al. [8].a) Beam with flexural failure,b) beam with shear failure.


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In Fig. 5 load and energy degradation (degradation of the area enclosed by a hysteresis loop) are represented for two cantilever beams, the one with flexural and the other with shear failure, as a function of the number of inelastic deformation cycles, corresponding to a displacement ductility factor µ=5. Likewise as in Fig. 4 it is shown that the supportable number of inelastic deformation cycles is much lower at members with shear failure than at members with flexural failure. The high sensibility of members with shear failure to the number of load cycles denotes an increased importance for these elements of the duration of ground shaking.



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< previous page page_561 next page >Page 561THE CONSIDERATION OF EARTHQUAKE DURATIONS IN THE DESIGN OF RC STRUCUTRES ACCORDING TO EUROCODE 8The design and dimensioning of RC structural members under seismic loading for flexural strength and ductility is based in Eurocode 8 on the consideration of the moment-curvature-diagram under static loading in the nonlinear range, as shown in Fig. 6. Depending on the chosen ductility class—high (H), mean (M) or low (L) ductility—a higher or a lower behaviour factor is introduced, leading to a higher or to a lower curvature ductility factor, as represented in Fig. 2. The duration of the expected earthquake does not occur directly in the design calculations.The procedure outlined above, neglecting the influence of the earthquake duration, is based on the assumption that even in an earthquake of relative long duration, leading to a large number of load reversals, no important strength and stiffness degradation occurs, that is that the behaviour of the structure is not similar to that represented in Fig. 7a, but to that represented in Fig. 7b. Thus it is supposed, that the behaviour of the structure is characterized by flexural failure, and not by shear failure.In order to ensure stable hysteresis loops, that is in order to avoid shear failure under a large number of load and deformation cycles in the nonlinear range, in Eurocode 8 a series of special measures are imposed on the shear dimensioning of RC structures, belonging to the ductility classes H and M. So in beams and in columns, where due to shear reversals bidiagonal cracking is probable, bidiagonal shear reinforcement is provided. Similarly in shear walls, in order to prevent shear sliding

Fig. 6 Ultimate curvature and curvature at yield for a RC member, according to Eurocode 8



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< previous page page_562 next page >Page 562failure, also bidiagonal reinforcement is suggested. Further within the critical regions of potential plastic hinge formation in beams, columns and shear walls, where the normalised design axial force is lower than 0.1, the contribution of concrete to the shear strength of the element (the term Vcd in the dimensioning for shear according to Eurocode 2) is not taken into account.The third ductility class (L) has been introduced in Eurocode 8, according to a proposal from Germany at a workshop in Pavia [10], as lowest ductility class for structures in regions of low seismicity. So this ductility class has been conceived for structures that have to resist to earthquake of low magnitude, leading to a short duration of the strong motion phase and accordingly only to a reduced number of load reversals. Thus it was possible to introduce the influence of short earthquake duration in the design of RC structures of ductility class L by renouncing for this ductility class to the special measures, ensuring for structures of the ductility classes H and M a satisfactory behaviour under a large number of load reversals. The few load reversals, considered in the case of ductility class L, corresponding, due to the low values of the behaviour factors, to rather insignificant incursions into yielding state, one could renounce likewise to a series of supplementary provisions for detailing. In this way important simplifications could be admitted in the design of structures in regions of low seismicity.

Fig. 7 Behaviour of structures under cyclic loading, from Wakabayashi [9].a) Poor behaviour with strength and stiffness degradation,b) good behaviour with stable hysteresis loops.



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< previous page page_563 next page >Page 563CONCLUSIONSIn the paper possibilities are analysed to introduce in the design of structures in regions of low seismicity the favourable effect of short earthquake duration. It is shown that in modern codes, as in Eurocode 8, where ductility demand and ductility supply are controlled numerically in a high degree, this effect cannot be introduced by reducing the design peak ground acceleration as in traditional codes, but only by taking into account the real behaviour of structural members under cyclic loading in the nonlinear domain. Considerations of this kind in the field of RC structures are translated in Eurocode 8 into more favourable conditions in the shear design of structures belonging to the ductility class L—conceived especially for structures in regions of low seismicity.The behaviour of masonry members under cyclic shear loading, characterized by an important strength and stiffness degradation, leads to the idea to apply similar considerations also to masonry structures, dominated in their seismic behaviour by shear. As a final result of research efforts in this domain, higher behaviour factors for the seismic design of masonry structures in regions of low seismicity may be expected. First investigations on this problem are presented by Vratsanou [11].REFERENCES1. Workshop on Seismic Input Data held in Lisbon, at LNEC, on July 2nd and 3rd, 1990. Synthesis Report on the agreed decisions concerning the lines to be pursued in order to arrive at an harmonized definition of a model for the seismic action and of the design input data to be introduced in Eurocode 82. DIN 4149, Teil 1. Bauten in deutschen Erdbebengebieten, Lastannahmen, Bemessung und Ausführung üblicher Hochbauten, Ausgabe April 19813. Hosser, D., Keintzel, E. and Schneider, G. (Coordination J. Eibl and E.Keintzel): Proposal for Harmonized Rules for the Determination of Seismic Input Data. Preliminary Report. In: Background Documents for Eurocode 8, Part 1 (May 1988). Vol 1—Seismic Input Data. Commission of the European Communities, 19894. Eurocode 8. Strucutres in seismic regions. Design. Part 1. General and building. May 1988 edition. Commission of the European Communities, 19895. Paulay, Th. and Bull, J.N.: Shear Effects on Plastic Hinges of Earthquake Resisting Reinforced Concrete Frames. AICAP—CEB Symposium Structural Concrete under Seismic Actions, Rome, 1979. Bulletin d’Information CEB No. 132, Paris, 1979



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< previous page page_564 next page >Page 5646. Park, R., Kent, D.C. and Sampson, R.A.: Reinforced Concrete Members with Cyclic Loading. ASCE, Journal of the Structural Division, Vol. 98, pp. 1341–1360, 19727. Wight, J.K. and Sozen, M.A.: Shear Strength Decay in Reinforced Concrete Columns Subjeted to Large Deflection Reversals. ASCE, Journal of the Structural Division, Vol. 101, pp. 1053–1065, 19758. Brown, R.H. and Jirsa, J.U.: Reinforced Concrete Beams Under Load Reversals. Journal of the American Concrete Institute, Vol 68, pp. 380–390, 19719. Wakabayashi, M.: Design of Earthquake Resistant Buildings. McGraw-Hill, New York, 198610. Eurocode 8—Structures in seismic regions, Specific rules for concrete structures—workshop in Pavia, February 1988. European Earthquake Engineering. Vol 2, No. 2, pp. 43–44, 198811. Vratsanou, V.: Determination of the Behaviour Factors for Brick Masonry Panels, Subjected to Earthquake Actions. Proceedings of the 5th International Conference on Soil Dynamics and Earthquake Engineering, Karlsruhe 1991. Computational Mechanics Publications, Southampton, 1991



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< previous page page_565 next page >Page 565Determination of the Behaviour Factors for Brick Masonry Panels Subjected to Earthquake ActionsV.VratsanouInstitut für Massivbau und Baustofftechnologie, Abteilung Massivbau, D7500 Karlsruhe 1, GermanyABSTRACTAn analytical constitutive model for the analysis of the response of brick masonry panels to in-plane monotonic and cyclic actions is presented. Masonry is treated as an ideal nonlinear homogeneous material. The model is based on the concept of the “equivalent uniaxial strain” first introduced for concrete by Darwin/Pecknold. For the determination of the stress-strain relations under monotonic and cyclic actions the experimental findings of Naraine/Sinha are used. A failure criterion has been elaborated, which is based on the experimental results of Page. The proposed constitutive model can predict (i) failure by splitting parallel to the free surface of the panel and (ii) failure by cracking either in the joints alone or in a combined mechanism involving both bricks and joints. The results from the analysis of masonry panels are compared with experimental findings from the literature and a very good agreement is found.The proposed material model is used for the determination of the behaviour factors for brick masonry panels in seismic regions.INTRODUCTIONMasonry is a material which exhibits distinct directional properties due to the mortar joints which act as plane of weakness. The great number of the influence factors, such as anisotropy of bricks, dimension of the bricks, joint width, material properties of brick and mortar, arrangement of bed and head joints and quality of workmanship, make the simulation of plain brick masonry extremely difficult. For convenience, masonry is often modelled as a linear, isotropic, elastic or elastic-plastic material.In the recent years several investigations have dealt with the structural behaviour of masonry, using the finite element method. Remarkable efforts in this field have been made i.a. by Page [5], Bernardini et al. [6],



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< previous page page_566 next page >Page 566Table 1: Comparison between one- and two-phase material models for masonryONE-PHASE MODELS TWO-PHASE MODELS

Relatively simple use Relatively costly use due to the great number of the degrees of freedom

Less input data More input data

The failure criterion has normally a simple form The failure criterion has a complicated form due to the brick-mortar interaction

The constitutive equations for the material “masonry” are relatively complicated

The constitutive equations of the components have normally a simple form

Unloading and reloading of the material can be simulated The simulation of unloading and reloading is extremelly difficult

At best suitable for the simulation of structures or greater structural elements

Suitable for the simulation of small structural elements (e.g. small test specimens), and only if the contact regions are clearly definable

At best suitable for the study of the global behaviour of masonry

Suitable for the study of the local behaviour of masonry

Motta/D’ Amore [9], Mengi/McNiven [12], Calvi [7], Ignatakis et al. [11]. The proposed constitutive models can be classified in two categories: (i) the “one-phase” material models, where masonry is treated as an ideal homogeneous material, whose constitutive equations differ from those of the components (bricks, mortar), and (ii) the “two-phase” material models, where the components are treated separately and the interaction between them is taken into account. The advantages and the disadvantages of the two categories are shown in Tab. (1).The present paper briefly describes an one-phase constitutive model for plain brick masonry subjected to in-plane monotonic and cyclic actions, which is based on the principle of the “equivalent uniaxial strain” first introduced for concrete by Darwin/Pecknold [2]. For the description of the model the following material parameters are required: (i) the

initial tangent modulus, Eo, (ii) the Poisson’s ration, ν, (iii) the uniaxial compressive strength, , (iv) the strain corresponding to the uniaxial compressive strength, ,



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Figure 1: Typical stress-strain curve for masonry in uniaxial loading

(v) the uniaxial tensile strength, and (vi) the parameters Cσ and which define the point of material disintegration.The above material properties, which are indicatcd in Fig. (1), can be determined by conventional tests on brick masonry panels and/or by emprirical formulas available in literature.MATERIAL IDEALIZATIONForm of the incremental constitutive relationsThe biaxial stress-strain relations for plain masonry are idealized as incrementally linear orthotropic, with the axes of orthotropy coinciding with the current principal stress axes. Thus:

(1)in which the axes 1, 2 are the current principal stress axes. The shear modulus term is obtained by requiring it to remain independent of direction. The tangent modulus Ei, i=1, 2, in the current principal stress direction i, is determined from a

family of —curves (σi is the principal stress and the equivalent uniaxial strain in the i-direction).



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Figure 2: Proposed stress-strain curve for masonry in biaxial stress stateEquivalent uniaxial strainThe introduction of the equivalent uniaxial strain is a clever method to “remove” the Poisson’s effect from Eq. (1). The equivalent uniaxial strain is always associated with the current principal stress axis i and depends on the current stress ratio α=σ1/σ2. It is defined as the strain corresponding to the stress σi on the uniaxial loading curve and is determined by:

(2)in which dσi is the incremental change in stress. The equivalent uniaxial strain does not transform in the same manner as stress; it is only a “fictitious” measure on which to base the variation of the material parameters.Monotonic stress-strain relationsOnce the various expressions describing the stress-strain curve under uniaxial loading are known, these can be directly transformed into biaxial loading conditions in terms of the biaxial stresses, σi, and equivalent uniaxial strains, in

place of the uniaxial stress, σ, and strain, respectively. The parameters and

(Fig. 1) are now replaced by the biaxial parameters and respectively (Fig. 2), where σic is the biaxial compressive strength and the corre-



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< previous page page_569 next page >Page 569sponding strain in i-direction.

To describe the nonlinear compressive loading portion of the —curve the exponential equation suggested by Naraine/Sinha [3] is used. This expression has been slightly modified so as to account for the fact that the stress-strain curve at point C (attainment of compressive strength) can be very pointed. Thus:

(3)

This curve has initial slope Eo and passes through the point of max. stress, which is a function of the

continuously changing principal stress ratio α, the strengths and and the strain . At point the

tangent modulus becomes equal to null. The coordinates of point are determined by in

which 0<ρσ≤1 and are functions of Eo, and .The downward portion of the envelope curve drops linearly from the point of max. stress (a function of α) until a max. compressive strain is reached and the material crushes. The stress and strain at the lower end of the line are independent

of α, and the parameters Cσ and , which define them, reflect the ductility of masonry. In the tension region a linear stress-strain relation is assumed with a brittle failure as the tensile strength σit is reached.Cyclic stress-strain relationsSince no experimental data on the behaviour of plain masonry under biaxial cyclic loading are available, the model is based on the experimental work of Naraine/Sinha [3, 4], who investigated the behaviour of plain brick masonry under uniaxial compressive loading perpendicular and parallel to the bed joint. Using the exponential functions proposed by the same authors the unloading and reloading curves for both load cases (perpendicular and parallel to the bed joints) were compared and it was found out that the effect of the bed joint orientation is negligible.A good match with the experimental results is obtained by substituting the proposed exponential through linear functions (Fig. 3). The reloading curve is now represented by a straight line from the “plastic strain” point P through the common point K. The unloading curve is approximated by two straight lines. Load reversals follow the line with slope EAS between the unloading and reloading lines. For low values of the equivalent uniaxial strain the unloading and reloading take place on a single line with slope Eo.Failure criterionThe failure criterion adopted is depicted in Fig. 4. It is based on the experimental results of Page et al. [10], who tested 180 masonry panels with five different bed joint orientations for a range of principal stress ratios.In the compression-compression region the bed joint orientation exerts little



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Figure 3: Typical hysteresis curve for the proposed model

Figure 4: Proposed failure criterion for brick masonry in plane stress state



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< previous page page_571 next page >Page 571influence at the strength and failure occurs by splitting in a plane parallel to the free edges of the panel. It was found out that the biaxial strengths σ2c and σ1c can be satisfactorily determined using the following expressions (for σ1≥σ2):

(4)According to the experimental results of Page, when one principal stress dominates, the bed joint orientation becomes significant, since failure occurs in a plane normal to the panel in all cases. Thus it was assumed that Eq. (4) is valid only for α>0.10. For lower values of α (−∞<α<0.10) failure occurs by cracking and sliding in the joints and/or bricks. In view of the need to distinguish between tensile and sliding failures in this region, it was assumed that sliding occurs if the

normal compressive stress σyy is relatively low, that is if . The experimental results of Mann [8] are also consistent with this assumption. In the tension-compression region the assumption of a straight line reduction in compressive strength (abs.) with increased tensile stress worked satisfactorily in the investigated problems. In the tension-tension region failure occurs by joint sliding; the simple criterium of constant tensile strength, equal to the

uniaxial tensile strength of masonry , was assumed.Expressions for the corresponding strains , i=1, 2, in the compression range were obtained from curve-fitting of the biaxial test results. It was found out that the variation of with σic can be simply expressed by

(5)CrackingA smeared set of cracks occurs when the tensile stress in a principal stress direction exceeds the tensile strength of masonry. In this case, it is assumed that a plane of failure develops perpendicular to the principal stress sirection. The effect of this material failure is that the normal and shear stiffness across the plane of failure are reduced, and the corresponding normal stress is released. However, a strain softening behaviour in the tension region was adopted for the purpose of the numerical solution. The tangent modulus of the corresponding direction on the downward portion of the stress-strain curve is set to zero and the stresses are corrected to their proper value at the end of each iteration.The model includes the following crack configurations: one crack open; first crack closed; first crack closed, second crack open; both cracks closed; both cracks open. The second crack may form while the first crack is either open or closed, and forms like the first when the tensile strength of masonry is exceeded. If the first crack is open, the second crack forms perpendicularly to the first; if the first crack is closed, then the orientation of the second



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Figure 5: Comparison of experimental and numerical resultscrack is linked to the orientation of the principal stress axes.NUMERICAL APPLICATIONSThe proposed material model was implemented to the F.E. Program ADINA and several parametric studies were conducted to establish the sensitivity of the material model to variations in mesh size and load step in the analysis of masonry panels.As a first example the simulation of the experimental results of Bernardini/Modena/Vescovi [1] is shown here, who performed uniform compression tests in masonry panels in order to determine the uniaxial compressive strength normal to the bed joints. The dimensions of the masonry panels were 0.77m·0.83m·0.12m. They were made of normal ceramic hollow bricks with a mean compressive strength of 24.7M Pa and M3 mortar with a minimum compressive strength greater than 8M Pa. The material properties that were given as input to the program were: Eo=7270M Pa, ν=0.20,

, , , Cσ=0.85 and .The experimental results were compared with the numerical solution and a very good agreement was found (Fig. 5).As a second example the experimental results of Naraine/Sinha [3] were simulated. The test specimens were of dimensions 0.70m·0.70m·0.23m



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Figure 6: Comparison of experimental and numerical resultsand were constructed with frogged clay bricks and 1:5 (cement: sand by volume) mortar mix (mean compressive strength of the bricks fbc=13.1M Pa, mean compressive strength of the mortar cubes fmc=6.1M Pa). The case of compressive loading perpendicular to the bed joints was investigated. The material properties that were given as input to the program

were: Eo=2400M Pa, ν=0.20, , , , Cσ=0.85 and

. The experimental results were compared with the numerical solution and again a very good agreement was found (Fig. 6).EVALUATION OF THE BEHAVIOUR FACTORThe behaviour factor q, which is an essential term in Eurocode 8 (“Structures in Seismic Regions”) is an oversimplified means in order to take into account the capability of the structural system to resist seismic loads in the post-elastic region. It is defined as the ratio between the earthquake intensity (in the sense of acceleration) resulting to a collapse of the structure and that leading to the attainment of the elastic limit.In the CIB-Recommendations [13] the following values for the q-factor are suggested:



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Figure 7: Indirect method for the numerical evaluation of the behaviour factorUnreinforced masonry: q=1.50

Confined masonry: q=2.00

Reinforced masonry: q=2.00÷3.00However, no systematical studies have been carried out for the evaluation of the behaviour factors. The above values are mainly based on the available experience from the behaviour of masonry buildings during earthquakes and on a critical review of the available experimental and theoretical data.The behaviour factor can be defined experimentally or numerically. The ideal path would be, given a structure and a loading history, to define the strength needed to reach a 30% deterioration (according to the definition of “failure” suggested by the CIB-Recommendations [13]) and to divide this value by the elastic strength. The randomness of the seismic excitation and the variation in the geometry of the structure make this method very costly and time-consuming.In this work an “indirect” numerical method was chosen, whose basic steps are depicted in Fig. 7. The local ductility of the basic masonry element can be defined using the above described material model. The passage from the local to the global ductility and from the global ductility to the behaviour factor can be achieved by means of semi-empirical formulas available in literature. The advantage of this method is the considerable reduction of the influence parameters.Particular consideration is given on the relation between the duration of the seismic excitation and the q-value. The research is still in progress, and the first results will be presented at the Conference.



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< previous page page_575 next page >Page 575REFERENCESPaper in a journal1. Bernardini, A., Modena, C., Vescovi, U., “An anisotropic biaxial failure criterion for hollow clay brick masonry”, The International Journal of Masonry Construction, Vol. 2, No. 4, 19822. Darwin, D., Pecknold, D.A., “Analysis of r.c. shear panels under cyclic loading”, ASCE Journal of the Structural Division, Vol. 102, No. ST2, Feb. 19763. Naraine, K., Sinha, S., “Behaviour of brick masonry under cyclic compressive loading”, ASCE Journal of the Structural Engineering, Vol. 115, No. 6, June 19894. Naraine, K., Sinha, S., “Loading and unloading stress-strain curves for brick masonry”, ASCE Journal of the Structural Engineering, Vol. 115, No. 10, Oct. 19895. Page, A.W., “Finite element model for masonry”, ASCE Journal of the Structural Division, Vol. 104, No. ST8, Aug. 1978Paper in Conference Proceedings6. Bernardini, A., Rossetto, P., Sproccati, A., Vitaliani, R., “Soluzione numerica dello stato fessurativo in lastre piane di muratura ordinaria o armata”, 6th I.B.Ma.C., Rome, May 19827. Calvi, G.M., Cantù, E., “A finite element analysis of masonry walls under cyclic actions”, CIB Symposium on Wall Structures, Warsaw, Jun.19848. Mann, W., “Failure of shear-stressed masonry. An enlarged theory, tests and application to shear walls”, 7th International Symposium on Load-Bearing Brickwork, London, Nov. 19809. Motta, F., D’ Amore, E., “Numerical modelling of the structural behaviour of masonry buildings”, 7th I.B.Ma.C., Vol. I, Melbourne, Feb. 198510. Page, A.W., Samarasinghe, W., Hendry, A.W., “The in-plane failure of masonry—A revue”, 7th International Symposium on Load-Bearing Brickwork, London, Nov. 1980



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< previous page page_576 next page >Page 576Chapter in a Book11. Ignatakis, C., Stavrakakis, E., Penelis, G, “Analytical model for masonry using the finite element method”, in “Structural repair and maintenance of historical buildings”, Computational Mechanics Publications, 1989Report12. Mengi, Y., McNiven, H.D., “A mathematical model for predicting the nonlinear response of unreinforced masonry walls to in-plane earthquake excitations”, Report No. UCB/EERC-86/07, Earthquake Engineering Research Center, University of California, Berkeley/California, May 198613. “International Recommendations for Design and Erection of Un-reinforced and Reinforced Masonry Structures” with an Appendix on “Recommendations for Seismic Design of Unreinforced, Confined and Reinforced Masonry Structures”, CIB, Publication 94, 1987



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< previous page page_577 next page >Page 577Statistical Study of Nonlinear Response Spectra for Aseismic Design of StructuresE.MirandaEarthquake Engineering Research Center, University of California at Berkeley, Berkeley, CA 94720, U.S.A.ABSTRACTResults from a statistical analysis of the nonlinear response of single-degree-of-freedom systems subjected to earthquake ground motions are presented. The study was conducted to develop practical means of estimating strength demands on earthquakes that are more rational than those presently specified in various seismic code requirements. The study was based on inelastic response spectra of 124 recorded ground motions. The records were selected giving emphasis to those recorded in California and to those recorded in the last six years. Special attention was devoted to the influence of local site conditions on the inelastic strength demands. Mean inelastic strength demand spectra are presented for six levels of displacement ductility. The results indicate that the use of a period- and soil-independent strength reduction factor may lead to unconservative designs.INTRODUCTIONThere is a general consensus that the greatest source of uncertainty in the determination of the response of structures to earthquake ground motions is that associated with the prediction of the intensity and characteristics of the seismic input. Since the concept of response spectrum was developed in the late 30’s, response spectra have been widely used to estimate strength demands on structures imposed by earthquake ground motions.A number of statistical studies have been conducted over the years with the purpose of improving the knowledge on design spectra. These studies have been improved in time as more earthquake ground motions have been recorded. Linear elastic response spectra provide a reliable tool to estimate the level of forces and deformations developed in structures responding elastically during earthquake. There has been a good number of statistical studies



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< previous page page_578 next page >Page 578that, by considering a large number of recorded ground motions, have investigated the characteristics of linear elastic response spectra including the influence of earthquake magnitude, epicentral distance, frequency content, damping ratio and, local site conditions (Newmark et al. [1], Seed et al. [2], Mohraz et al. [3], Katayama et al. [4], Kiremidjian et al. [5]). During strong earthquakes, however, present seismic design philosophy accepts structural and non-structural damage. Thus buildings designed according to this philosophy are likely to experience significant inelastic excursions which produce produce reductions in seismic forces which cannot be predicted with the use of linear elastic models. Typically, statistical studies that have included non-linear behavior have only considered a small number of recorded ground motions (Veletsos [6], Riddell et al. [7]). Recently, Krawinkler [8] studied the strength reductions due to nonlinear behavior by using 33 horizontal ground motions recorded during the 1987 Whittier Narrows, California earthquake. However, the effect of soil conditions was not taken into account.The objective of this paper is to present the result of a statistical study of inelastic strength demands on single-degree-of-freedom (SDOF) systems in which a large number of recorded ground motions was considered.RESPONSE OF NONLINEAR SDOF SYSTEMSIn the present study, constant ductility nonlinear spectra are obtained by computing the response of a family of SDOF systems with the use of the computer program NLSPECTRA which was developed specifically for this purpose. The equation of motion of a SDOF system under earthquake excitation is given by

(1)where u is the relative displacement of the system, m is the mass, c is the damping coefficient, üg is the ground acceleration, and R(t) is the restoring force. Equation 1 is frequently expressed as a function of the natural circular frequency, ω, and damping ratio, ξ

(2)where the natural circular frequency and damping ratio are given by

(3)and k is the initial stiffness of the system.The displacement ductility ratio is defined as the ratio of the maximum absolute value of the displacement response divided by the yield displacement of the system,

(4)



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< previous page page_579 next page >Page 579The ductility ratio gives a measure of the severity of the peak displacement relative to the displacement necessary to initiate yielding. A displacement ductility ratio less than unity represents elastic response, while a value greater than unity indicates inelastic response. Computation of constant ductility response spectra involves iteration on Equation 2 with different values of yielding strength. The iteration is successful when the computed ductility reaches the specified (target) ductility within a certain tolerance that can be specified by the user. In this study, ductilities were considered satisfactory if they were within 1% of the target ductility.In NLSPECTRA iteration is done with the secant method. Equation 2 is solved by numerical step-by-step integration using the linear acceleration method. Equilibrium violations due to stiffness changes within a step are minimized by using a variable integration time-step and by imposing equilibrium by modifying the acceleration at the end of the step.STATISTICAL STUDY OF INELASTIC STRENGTH DEMANDSIn order to improve current methods to estimate inelastic strength demands on structures a statistical study of nonlinear response spectra was conducted. Constant ductility nonlinear spectra were computed for 124 earthquake ground motions recorded on various soil conditions ranging from rock to very soft soils. The following values of ductility were selected for this study: 1 (elastic), 2, 3, 4, 5 and 6. Due to the large number of records and the computational effort involved in calculating constant ductility nonlinear spectra, the study was limited to bilinear systems with a post-elastic stiffness of 3% of the elastic stiffness and with a damping ratio of 5% of critical.Selected Ground MotionsIn the last six years an extensive number of earthquake ground motions has been recorded in different parts of the world. These ground motions have more than doubled the number of records previously collected. For this study 124 records were selected, with emphasis on those recorded in California and on those recorded during the last six years. Of the total number of records, 96 (77%) were recorded in the last six years and 90 (73%) were recorded in California.The ground motions were classified into three groups according to the geologic conditions at the recording station. These groups were rock, alluvium and very soft soil. Tables 1, 2, and 3 list the selected ground motions recorded on rock, alluvium, and soft soil sites, respectively.Results from the Statistical StudyConstant ductility nonlinear spectra were computed for all records in each soil group. Strength demands for each record were then normalized using peak ground acceleration (PGA). For ground motions recorded on rock or



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< previous page page_580 next page >Page 580alluvium sites, nonlinear spectra were computed for a fixed set of 50 periods between 0.05 and 3.0 seconds. In the case of ground motions recorded on very soft soil, spectra were computed for a fixed set of 50 ratios of T/Tg, where Tg is the predominant period of the ground. The reason for using T/Tg instead of T is that Tg can have large variations depending on the shear wave velocity of the soil and the depth of the soft deposits. For instance, in Mexico City where the soft clay deposits have approximately the same characteristics throughout the city, the predominant period of the motion can vary anywhere from 0.6 second to more than 3.8 seconds depending on the depth of these deposits. For Statistical analyses of spectra it makes no sense to average spectral ordinates at a certain period for ground motions with significantly different predominant periods. For structural design purposes, it is important to characterize the demands on structures with periods shorter, longer or near the predominant period.In this study the predominant period was computed as the period corresponding to the maximum spectral velocity ordinate. It can be shown that essentially the same predominant period would be obtained if the Fourier amplitude spectrum or the input energy spectrum are used instead of the velocity spectrum because of the relationship between these three spectra.Mean and mean plus one standard deviation inelastic strength demand spectra of 38 ground motions recorded on rock are shown in Figures 1 and 2. The spectra are plotted for displacement ductilities of 1, 2, 3, 4, 5, and 6 (from top to bottom). Mean and plus one standard deviation inelastic strength demand spectra of 62 ground motions recorded on alluvium are shown are shown in Figures 3 and 4. The maximum amplification for alluvium sites is larger than that observed at rock sites. Mean and mean plus one standard deviation inelastic strength demand spectra of 24 ground motions recorded on soft soil sites are shown in Figures 5 and 6.By comparing the average spectra of the three different soil conditions it can be seen that the largest dynamic amplification for elastic response (μ=1) is produced for soft soil sites. These results are different to those reported previously by Seed et al. [2] who computed larger amplifications for rock and alluvium sites than for soft soil sites. Moreover, the maximum amplification (with respect to PGA) computed in that study is nearly 30% smaller than the maximum amplification computed here. For rock and alluvium sites the maximum amplifications computed in this study are practically the same as those found by Seed et al. with a smaller set of ground motions.The shape of inelastic response spectra is significantly different to that of linear elastic spectra. Hence, scaling of the elastic spectra using a period-independent factor to estimate inelastic strength demands is not rational. Moreover reductions are significantly affected by the soil conditions. For buildings on soft soil and with fundamental periods smaller than the predominant site period, strength reductions are very small, while for fundamental



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< previous page page_581 next page >Page 581period close to the predominant site period the reduction factors are larger than µ. A comprehensive study of the reduction factors and their dispersion can be found in Ref. [9] (Miranda).CONCLUSIONSElastic and inelastic response spectra were computed for 124 ground motions recorded on rock, alluvium and soft soil sites. The results indicate that the shape of the inelastic strength demands differs from the shape of elastic strength demands. It is concluded that strength reductions produced in non-linear systems are strongly affected by the natural period of vibration, the level of inelastic deformation, and the local site conditions. For soft soil sites, the estimation of the predominant period of the site is particularly important on the estimation of strength and deformation demands. The use of period-independent strength reduction factors, as currently specified in many seismic design recommendation may lead to unconservative designs.REFERENCES1. Newmark, N.M., Blume, J.A., and Kapur, K.K., Seismic Design Spectra for Nuclear Power Plants, Journal of the Power Division, Proceedings of the American Society of Civil Engineers, Vol. 99, No. P02, pp. 287–303, November, 1973.2. Seed, H.B., Ugas, C., Lysmer, J., Site-Dependent Spectra for Earthquake-Resistant Design, Report No. EERC 74–12, Earthquake Engineering Research Center, University of California, Berkeley, November, 1974.3. Mohraz, B., A Study of Earthquake Response Spectra for Different Soil Conditions, Civil and Mechanical Engineering Department, Southern Methodist University, Dallas, Texas, August, 1975.4. Katayama, T., Iwasaki, T., and Saeki, M., Statistical Analysis of Earthquake Acceleration Response Spectra, Transactions of the Japan Society of Civil Engineering, Vol. 10, pp. 311–313, 1978.5. Kiremidjian, A.S., and Shah, H.C., Probabilistic Site-Dependent Response Spectra, Journal of the Structural Division, Vol. 106, No. ST1, pp. 69–86, January, 1980.6. Veletsos, A.S., Response of Ground-Excited Elastoplastic Systems, Report No. 6, Research at Rice, Rice University, Houston, Texas, December, 1969.7. Riddell, R., and Newmark, N.M., Statistical Analysis of the Response of Non-linear Systems Subjected to Earthquakes, Structural Research Series No. 468, Department of Civil Engineering, University of Illinois,Urbana-Champaign, August, 1979.8. Krawinkler, H., Nassar, A., Strength and Ductility Demands for SDOF and MDOF Systems Subjected to Whittier Narrows Earthquake Ground Motions, SMIP-1990 Proceedings of the Seminar on Seismological and Engineering Implications of Recent Strong-Motion Data, California Department of Conservation, Sacramento, California, June, 1990.9. Miranda, E., Seismic Evaluation and Upgrading of Existing Buildings, Ph.D. Thesis, University of California at Berkeley, Berkeley, California, May 1991.



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STATION NAME GEOLOGY EARTHQUAKE DATE MAGN. EPICTR. DIST.[km]

DIRECTION PGA [g’s]

SAN FRANCISCOGolden Gate Park

Slliceoussandstone

San FranciscoMarch 22, 1957

5.3(ML) 11 N10ES80E

0.080.11

PARKFIELDCholame Shandon No.2

Rock ParkfiledJune 27, 1966

5.6(ML) 7 N65E 0.48

CASTAICold Ridge Road

Sandstone San FernandoFebruary 9, 1971

6.5(ML) 29 N21EN69W

0.320.27

LLOLLEO Sandstone &volcanic rock

Central ChileMarch 3, 1985

7.8(MS) 45 N10ES80E

0.670.43

VALPARAISO Volcanicrock

Central ChileMarch 3. 1985

7.8(MS) 84 N70ES20E

0.180.16

LA UNION MetavolcanicRock

MichoacanSept. 19, 1985

8.1(MS) 84 NOOEN90E

0.170.15

LAVILLITA GabbroRock


8.1 (MS) 44 NOOEN90E

0.130.12

ZIHUATANEJO TunaliteRock


8.1(MS) 135 N90WSOOE

0.100.16

NATL GEOGR.INSTITUTE

BalsamoFormation

San SalvadorOctober 10,1986

5.4(MS) 5.7 270180

0.530.39

INST. URBANCONSTRUCTION

FluviatePumice rock


5.4(MS) 5.3 90180

0.380.67

GEOTECH. INVEST.CENTER

FluviatePumice rock


5.4(MS) 4.3 18090

0.420.68

MT WILSONCaltech Seismic Station

Quartzdiorite

Whittier-NarrowsOctober 1, 1987

6.1(ML) 19 90360

0.190.13

CORRALITOSEureka Canyon Road

Landslidedeposits

Loma PrietaOctober 17, 1989

7.1(MS) 7 90360

0.470.62

SANTA CRUZUCSC

Limestone Loma PrietaOctober 17, 1989

7.1(MS) 16 90360

0.410.43

SAN FRANCISCOCliff House

Franciscansandstone


7.1(MS) 99 900

0.110.07

SAN FRANCISCOPacific Heights

Franciscansandstone


7.1(MS) 97 360270

0.050.06

SAN FRANCISCOPresidio

Serpentine Loma PrietaOctober 17, 1989

7.1(MS) 98 900

0.200.10

SAN FRANCISCORincon Hill

Franciscansandstone


7.1(MS) 95 90360

0.090.08

YERBA BUENAISLAND

Franciscansandstone


7.1(MS) 95 90360

0.060.03

Table 1. Selected ground motions recorded at rock sites



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STATION NAME GEOLOGY EARTHQUAKEDATE

MAGN. EPICTR.DIST.[km]

DIRECTION PGA[g’s]

EL CENTROIrrigation District

Alluvium Imperial ValleyMay 18, 1940

63 (ML) 8 S90WSOOE

0.210.34

TAFTLincoln School Tunnel

Aluvium Kern CountyJuly 21, 1952

7.7 (MS) 56 N21ES69E

0.150.17

FIGUEROA445 Figueroa St.

Alluvium San FernandoFebruary 9, 1971

6.5 (ML) 41 N52ES38W

0.15 0.12

HOLLYWOODFree Field

Alluvium San FernandoFebruary 9, 1971

65 (ML) 35 N90ESOOW

0.21 0.17

AVE.STARS1901 Ave. of the Stars

Silt & sandlayers

San FernandoFebruary 9, 1971

6.5 (ML) 38 N46WS44W

0.140.15

SENDAI CITYKokutetsu Bldg.

Alluvium Miyagi-Ken-OklJune 12, 1978

7.4 (MS) 110 N90WNOOE

0.440.24

MELOLANDInterstate 8 Overpass

Alluvium Imperial ValleyOctober 15, 1979

6.6 (ML) 21 360270

0.310.30

BONDS CORNERHighways 98 & 115


6.6 (ML) 3 S40ES50W

0.580.77

JAMES ROADEl Centro Array # 5


6.6 (ML) 22 S40ES50W

0.520.37

IMPERIAL V. COLLEGEEl Centro Array # 7


6.6 (ML) 21 S40ES50W

0.330.45

EL ALMENDRAL Compactedfill


7.8 (MS) 84 N50ES40E

0.290.16

VINA DEL MAR Alluvialsand


7.8 (MS) 88 N70WS20W

0.230.36

ZACATULA Alluvium MichoacanSept. 19, 1985

8.1 (MS) 49 SOOEN90W

0.260.18

ALHAMBRAFreemont School

Alluvium Whittier-NarrowsOctober 1, 1987

6.1 (ML) 7 270180

0.400.30

ALTADENAEaton Canyon Park


6.1 (ML) 13 90360

0.160.31

BURBANKCal. Fed. Savings Bldg.


6.1 (ML) 26 13040

0.220.17

Table 2. Selected ground motions recorded at alluvium sites



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DOWNEYCounty Maint. Bidg.

Deep alluvium Whittier-NarrowsOctober 1, 1987

6.1 (ML) 17 270180

0.160.20

INGLEWOODUnion Oil Yard

Terracedeposits


6.1 (ML) 25 90360

0.230.27

LOS ANGELES116th School

Terracedeposits


6.1 (ML) 22 360270

0.400.29

LOS ANGELESBaldwin Hills

Alluviumover shale


6.1 (ML) 27 90360

0.17 0.15

LOS ANGELESHollywood Storage FF


6.1 (ML) 25 90360

0.120.21

LOS ANGELESObregon Park


6.1 (ML) 10 360270

0.440.45

LONG BEACHRancho Los Cerritos


6.1 (ML) 27 90360

0.250.15

SAN MARINOSouthwestern Academy


6.1 (ML) 8 360270

0.200.15

TARZANACedar Hill Nursery


6.1 (ML) 44 90 360 0.630.46

WHITTIER7215 Bright Tower


6.1 (ML) 10 90360

0.630.43

ALBA900 S.Fremont


6.1 (ML) 8 90360

0.29 0.25

CAPITOLAFire Station

Alluvium Loma PrietaOctober 17, 1989

7.1 (MS) 9 90360

0.390.46

HOLLISTERSouth & Pine


7.1 (MS) 48 90360

0.170.36

OAKLAND2-Story Office Bldg.


7.1 (MS) 92 290200

0.240.19

STANFORDParking Garage


7.1 (MS) 51 36090

0.260.22



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BUCHARESTBuilding Research Inst.

Soft RomaniaMarch 4, 1977

7.1 (MS) 174 EWSN

0.170.20

SCT Sria. deComunic. y Transport.

Softday

MichoacanSept 19, 1985

8.1 (MS) 385 N90WSOOE

0.170.10

CENTRAL DE ABASTOSFrigorifico

Softday


8.1 (MS) 389 99.5377.52

0.100.08

CENTRAL DE ABASTOSOficina

Softday


8.1 (MS) 389 76.5667.95

0.080.07

COLONIA ROMA Softday

AcapulcoApril 25, 1989

6.9 (MS) – N90WSOOE

0.060.05

EMERYVILLEFree Field South

Bay mud Loma PrietaOctober 17, 1989

7.1 (MS) 97 350260

0.210.26

EMERYVILLEFree Field North


7.1 (MS) 97 350260

0.200.22

OAKLANDOuter Harbor Wharf


7.1 (MS) 95 305125

0.270.29

TREASURE ISLANDNaval Base

Fill Loma PrietaOctober 17, 1989

7.1 (MS) 98 90 360 0.160.10

SAN FRANCISCOInternational Airport


7.1 (MS) 79 90360

0.330.23

SAN FRANCISCO18-Story Comercial Bldg.

Fill overbay mud


7.1 (MS) 95 980350

0.130.16

FOSTER CITYRedwood Shores


7.1 (MS) 63 900

0.280.26

Table 3. Selected ground motions recorded at soft soil sites



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Figure 1. Mean strength demands of ground motions recorded on rock when normalized using PGA (μ=1, 2, 3, 4, 5, 6).

Figure 2. Mean plus one standard deviation strength demands of ground motions recorded on rock when normalized using PGA (µ=1, 2, 3, 4, 5, 6).


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Figure 3. Mean strength demands of ground motions recorded on alluvium when normalized using PGA (µ=l, 2, 3, 4, 5, 6).

Figure 4. Mean plus one standard deviation strength demands of ground motions recorded on alluvium when normalized using PGA (µ=l, 2, 3, 4, 5, 6).


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Figure 5. Mean strength demands of ground motions recorded on soft soil when normalized using PGA (µ=1, 2, 3, 4, 5, 6).

Figure 6. Mean plus one standard deviation strength demands of ground motions recorded on soft soil when normalized using PGA (μ=1, 2, 3, 4, 5, 6).


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< previous page page_589 next page >Page 589Shear Transfer and Friction across Cracks in Concrete under Monotonic and Alternate LoadsC.KarakoçDepartment of Civil Engineering, B•oazçi University, 80815 Istanbul, TurkeyABSTRACTThe object of this paper is to introduce the constitutive relations for the crack interface, which express the stresses in terms of the related displacements for alternate loading in addition to the monotonic loading case. Global significance and various new aspects of such formulations, and coefficient of friction will be specially emphasized, assuming that the aggregate interlock is of frictional nature. Such considerations and formulations for the alternate loading case will be confronted with experimental results, also considering the reversed loading case.INTRODUCTIONIn a cracked plain or reinforced concrete element, substantial shear forces can be transmitted across the cracks even though there is a significant reduction in the shear strength. This is due to the “Aggregate Interlock” which is caused by the interlocking of the aggregate particles protruding from the opposite rough crack faces, and is one of the most important mechanisms for the transfer of the shear force.In recent years, extensive research work has been carried out to disclose experimental results related to the stresses transmitted across such typical crack faces in terms of the related displacements (Fig. 1), and also to develop anaytical models. References [1–3] are some of the important works disclosing such results for monotonic loading, while references [2–9, 12] are some of the works which further analyze the aggregate interlock phenomenon and its various aspects. In this paper, the stress formulations used are based on the experimen-



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< previous page page_590 next page >Page 590tal results of the first two references for monotonic loading. These stress formulations are represented by the “Gambarova-Karakoç Model” which is one of the rather few successful models which are widely recognized and used, Refs. [10, 11].

Figure 1. Stresses and related displacements across a typical crack.EXPERIMENTAL ASPECTS AND FIELDS OF APPLICATIONThe typical experimental set-ups which are employed in the tests to obtain results and information about the behaviour of the cracks and especially the relation between the stresses and displacements across the crack are shown in Fig. 2. In types 2.a) and

Figure 2. The types of set-ups.b) the confinement is provided by external and internal (embedded) bars to simulate and study the effects of reinforcement crossing a crack. In c), the crack opening is kept constant throughout the experiment while in d) the parameter kept constant during the test is the confinement stress. Gambarova and Karakoç [5] have presented a short survey on the researchers and their works related to such tests.As to the fields of application and design, it should be kept in mind that the aim of all such experiments is basically oriented towards a more realistic establishment of the constitutive laws of cracked concrete. Bazant and Gambarova [4] and then Walraven [2] have already shown based on smeared approach, for a media characterized by parallel, regular, and closely spaced cracks, that such constitutive laws can be obtained from the stress-displacement relations for the crack. It naturally follows that in numerical methods such as finite element method, it is possible to express and update the constitutive laws of concrete at each increment of load by making use of such relations. Kupfer et al. [7], using such relations proposed by Walraven [2], and Dei Poli et al. [8] using the for-



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< previous page page_591 next page >Page 591mulations of Gambarova and Karakoç [5], have analyzed the contributions of aggregate interlock to the shear capacity of thin webbed reinforced and prestressed I beams.FORMULATION OF THE STRESSES AND COEFFICIENT OF FRICTIONAs for the formulations for the relation between stresses and displacements related to a crack in concrete proposed by Gambarova and Karakoç [5], the following expression was introduced for the evaluation of confinement stress for the case of varying displacement and stress:

(1)This semi-empirical expression checks the basic physical criteria expected for the behaviour of the crack, such criteria having been summarized by Bazant and Gambarova [4]. Gambarova and Karakoç [5, 16] have shown that the predictions of Eq. (1) were in very good agreement with the test results given by Walraven [2] and those by Millard and Johnson in the papers referred in [16].For the constant crack opening case, Karakoç [9] has proposed the following formulation :

(2)This formulation which is quite handy because of its simplicity, can be seen to be considerably different from Eq. (1), even though they are similar in basic form. This is expected; Walraven and Keuser [15] cites the words of Millard and Johnson [16]: “If a cracked specimen undergoes shear displacement without crack widening occurring there must be a great deal of irreversible crushing damage caused. If the crack is then widened, it might be expected that the shear and normal stresses would be quite different from those that would result if the crack was first widened and then caused to shear.” Furthermore, Walraven and Keuser [15] also refer to Nissen [14] and present his results for the path dependency. Karakoç [9], has already shown that the predictions of Eq. (2) were in a very good agreement with all the results of Daschner [1], for the tests he has carried out at constant crack opening in which all basic variables (two stresses and related displacements) were disclosed. In Fig.3 and Fig.4, predictions of Eq. (2)



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Figure 3. Test results of Daschner and Kupfer [6] and predictions of Eq. (2).have been compared with the results given by Daschner and Kupfer [6] for the constant crack opening case of δn=0.2mm for normal and lightweight concrete, both having a cube strength of 55Mpa, Da being the max. aggregate size.

Figure 4. Test results of Daschner and Kupfer [6] and predictions of Eq. (2).Making use of Eq. (1), the coefficient of friction (μ=σnt/σnn) can be given as:



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(3)

Figure 5. Test results and predictions of Eq. (3).Using the same data as for Figs. 3–4, Eq. 3) is checked as shown in Figs. 5 and 6.

Figure 6. Test results and predictions of Eq. (3).



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< previous page page_594 next page >Page 594FORMULATION OF THE STRESSES FOR THE ALTERNATE LOADINGThe same formulation as given in Eq. (2) is adopted for the alternate loading case. This gives remarkably close predictions for the experimental results which are unfortunately quite scarce. The only new concept introduced is the effective slip (δt eff) which should take care of the previous displacement history. This effective slip will be based on the damage experienced by the protruding particles during the previous load cycles. Walraven [13] observes that further damage occurs even after 55 loading cycles, drawing attention to also the significance of the damage in previous loading cycles, and finally concludes that the load history effect is due to the crushing of the cement matrix between the aggregate particles which may be considered as perfectly brittle. To account for this irreversible and continuous damage, it is quite natural that the slip value to be substituted in Eq. (2) should be equal to the slip measured for the new cycle added to the summation of the final slip values (δtf) attained in the previous cycles:

(4)where r stands for reversed loading if there is any and C is a constant less than 1 to take into consideration the contribution of the reversed loading.

Figure 7. Test results given by Maekawa and Li [12] for the first loading cycle and predictions of Eq. (2).In Figs. 7, 8 and 9, predictions of Eq. (3) are compared to all the test results of Maekawa and Li [12] for alternate loading including the reversed loading case. It is to be noted that δt eff in Eq. (4) is identical to δt in Eq. (2) for the first cycle, Fig. 7. It is also important to note again that the slip values indicated on the abscissa in Figs.8 and 9 are the test results (δt), while for the predictions of Eq. (3) effective slip values (δteff) are used. Finally, it is worth noting that assuming values of (0.6) or (0) for C are



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< previous page page_595 next page >Page 595both fairly satisfactory for the reversed loading. The value for C can be fixed according to the -hopefully- more abundant future test results.

Figure 8. Test results given by Maekawa and Li [12] for the reversed loading cycle and predictions of Eq.(2).

Figure 9. Test results given by Maekawa and Li [12] for the second loading cycle and predictions of Eq.(2).



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< previous page page_596 next page >Page 596CONCLUDING REMARKSImportant conclusions can be drawn in the light of Eq. (2). The obvious feature and advantage of this formulation is that either the confinement or the shear stress can be computed in terms of the other, the only other variable necessary being the slip between the crack faces. A very important consequence that follows is that the coefficient of friction is a variable being a function of only the slip. Paulay and Loeber [3], had indicated a constant value for this coefficient (1.7) which was a medium for their experiments and which was close to the value recommended in the recent ACI Building Code requirements (1.4) for the shearfriction design method.Therefore, this new introduction paves way to a new and more realistic assessment of this coefficient both as a concept on theoretical level and as a variable for design basis.It may finally be concluded that, even though based on the few available test results, the formulation for the alternate loading is quite satisfactory and yet its reliability is to be confirmed with future experimental results.REFERENCES:1. Daschner, F., Shubkraftübertragung in Rissen von Normal-und Leichtbeton, Bericht Erstattet, Institut für Bauingenieurwesen III, Lehrstuhl für Massivbau, Technische Universität München, München, 1980.2. Walraven, J.C. and Reinhardt, H.W., Theory and Experiments on the Mechanical Behaviour of Cracks in Plain and Reinforced Concrete Subjected to Shear Loading, Concrete Mechanics -part A, Heron, 26, no.1A, 1981.3. Paulay, T. and Loeber, P.J., Shear Transfer by Aggregate Interlock, Paper 42–1, vol. 1, ACI Special Publication, pp. 1–15, 1974.4. Bazant, Z.P. and Gambarova, P.G., Rough Cracks in Reinforced Concrete, Journal of the Structural Division, ASCE, vol. 106, No. St4, pp. 819–842, 1980.5. Gambarova, P.G. and Karakoç, C, A New Approach to the Analysis of the Confinement Role in Regularly Cracked Concrete Elements, Trans. of the 7th Int. Conf. on St. Mech. in Reac. Tech., Paper H5/7, Chicago, U.S.A., pp. 251–261, 1983.



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< previous page page_597 next page >Page 5976. Daschner, F. and Kupfer, H., Versuche zur Schubkraftübertragung in Rissen von Normal-und Leichtbeton, Bauingenieur 57, pp. 51–55, 1982.7. Kupfer, H.,Mang, R.and Karavesyroglou, M., Bruchzustand der Schubzone von Stahlbeton-und Spannbetonträgern-Eine Analyse unter Berücksichtigung der Rißverzahnung, Bauingenieur 58, pp. 143–149, 1983.8. Dei Poli, S., Gambarova, P.G. and Karakoç, C., Aggregate Interlock Role in Reinforced Concrete Thin Webbed Beams in Shear, Journal of the Structural Division, ASCE, No.1, pp. 1–19, 1987.9. Karakoç, C., On Aggregate Interlock, Bulletin of the Technical University of Istanbul, Vol. 40, pp. 705–722, 1987.10. Feenstra, P.H., de Borst, R. and Rots, J.G., Stability Analysis and Numerical Evaluation of Crack-Dilatancy Models, Proceedings of SCI-C 1990, Second International Conference, pp. 987–999, 1990.11. Bangash, M.Y.H., Concrete and Concrete Structures : Numerical Modelling and Applications, Elsevier Applied Science, London and New York, 1989.12. Maekawa, K. and Li, B., Contact Density Model for Cracks in Concrete, Report, Iabse Colloquium, Vol. 54, Delft, pp. 51–62, 1987.13. Walraven, J.C., Aggregate Interlock Under Dynamic Loading, Darmstadt Concrete, Ann. Jour. on Conc. and Conc. St., Vol. 1, pp. 143–156, 1986.14. Nissen, I., Rißverzahnung des Betons—Gegenseitige Rißuferverschiebungen und übertragene Kräfte, PhD-Thesis, Technical University of München, 1987.15. Walraven, J.C. and Keuser, W., The Shear Retension Factor as a Compromise between Numerical Simplicity and Realistic Material Behaviour, Darmstadt Concrete, Ann. Jour. on Conc. and Conc. St., Vol. 2, pp. 221–234, 1987.16. Dei Poli, S., Gambarova, P.G. and Karakoç, C., -with reply by Millard S.G., Johnson R.P.- Discussion of the papers MCR 126 (1984, pp. 9–21) and MCR 130 (1985, pp. 3–15), Magazine of Concrete Research, Vol. 38, No. 134, pp. 47–51, 1986.



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< previous page page_599 next page >Page 599Tests on Upgrading Dynamic Properties of Existing Damaged Structures for a Better Seismic PerformanceO.YuzugulluDepartment of Earthquake Engineering, B•oazçi University, Kandilli Observatory, 81220 Cengelkoy, Istanbul, TurkeyABSTRACTThe results of three sets of tests are presented and discussed. All the tests had the objective of upgrading the dynamic properties of existing structures for a better seismic performance.INTRODUCTIONDamage to structures can occur due to various reasons, among which earthquake generated ground motions should seriuosly be accounted for.The available seismic codes and eartquake resistant design principles are generally sufficient to design and construct earthquake resistant new structures. On the other hand frequently need arise to upgrade existing older structures which are generally weaker in earthquake resistance in terms of new developments and standards or which might have been actually damaged during an earthquake. It is not easy to state general rules to strengthen (retrofit) existing insufficient or damaged structures since there exists a large variety of structures and each strengthening problem has its own merits. However it is possible to approach the strengthening problem of existing structures by varying some or all of the dynamic parameters, such as by increasing or decreasing the mass, stiffness and damping properties either by repairing the existing damage or by introducing new elements to the structure. In any case the efficiency and applicabi1ity of each method should be verified by means of tests and be supported by a sound theory.Three sets of tests were carried out to partially fulfill some of the above mentioned goals. The first two sets of tests [1, 2, 3] were involved in improving



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< previous page page_600 next page >Page 600the stiffness and damping properties of existing structures whereas the last set [5] was aimed in reducing the roof mass.TEST RESULTSTest Set No.1:The stiffness and damping properties of reinforced concrete frames were upgraded by adding new structural elements into the frame in the form of precast r.c. panels. Due to minimum construction time and obstruction involved in the application, the above method of strengthening seems to have a promising future especially for the case of continous functioning of the structure which is mandatory immediately after an earthquake, such as telecomminication buildings, hospitals, fire stations etc.Two types of connection detailing were considered; Welded and Bolted.Welded Connections [1] Ten reinforced concrete one story one bay frame models (Fig. 1) which measured 79.5cm by 138.0cm were tested under reversed cyclic loading either as bare frames (M1, M3) or frames strengthened by means of two or more reinforced concrete precast panels.The frame had 15×15cm beams and 15×7.5cm columns. The panels were 3cm thick and connected to the frames and among themselves by means of dicrete or continous welded type of connections; the details of which are given in Fig. 2. An example for a 1oad-deflection diagram is given in Fig. 3. The test results are summarized in Table 1.Bolted Connections [2] Two reinforced concrete beam elements measuring 15×15×75cm and two precast reinforced concrete panel elements measuring 30×30×3cm were assembled by means of standard steel angles, metal pieces and bolts. The models formed as such were tested under reversed cyclic loading. Out of ten models two were cast monolitically (Models M1 and M6). Fig. 4 shows the model designation and connection types. Connection details are given in Fig. 5. An example for 1oad-def1ection diagram is given in Fig. 6. The test results are summarized in Table 2.Test Set No. 2 [3]The objective of this set of tests was to upgrade the dynamic properties of existing timber framed building structures. This time ferrocement plastering both from inside and outside was considered for improving the lateral stiffness and possibly damping. If the joint



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< previous page page_601 next page >Page 601TABLE 1. BEHAVIOUR OF WELDED COONECTIONS [1]Model Desig. Stiffness k (kg/cm) Failure Mode Ult. Load P (kg) Energy E (kg-cm)

M1 5000 B-Frame 1800 4800

M2 12500 P-Panel 6000 6350

M3 5076 B-Frame 1250 6232

M4 12500 P-Panel 7000 14743

M5 11111 S-Frame 12750 19550

M6 12500 S-Frame 12000 29363

M7 7895 P-Panel 7500 13404

M8 14286 S-Panel 11000 18058

M9 7142 S-Panel 8000 22500

M10 Weld Fail. 5000 8390

Legend: B: Flex.Fail./P: Punching/S: Shear Fail.TABLE 2. BEHAVIOUR OF BOLTED CONNECTIONS [2]Model Desig. Stiffness k (kg/cm) Failure Mode Ult. Load P (kg) Energy E (kg-cm)

M1 4000 SC Sudden 4800 3646

M2 769 R-Cr 2500 4375

M3 2222 SC Delayed 5300 9884

M4 2500 SC Sudden 5450 7364

M5 1818 R-Cr 3400 8046

M6 4000 SC Sudden 4800 9045

M7 1111 R-Cr 6400 26545

M8 2000 SC Delayed 7500 16932

M9 1428 SC Sudden 7300 7839

M10 1818 R-Cr 2800 3943

Legend: SC: Shear-Corapr. Fail. /R-Cr: Rotation-Crushing



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< previous page page_602 next page >Page 602connections are weak, timber framed buildings have been reported [4] to be poor in behavior against lateral 1oads. Timber framed buildings are commonly used for rural housing in Middle East, West Europe, Mediterranean and South America.In the actual application it was expected that by replacing the existing mud plaster with ferrocement and, partially or totally avoiding the the existing heavy nonstructura1 fillers, considerable upgrading in the dynamic properties would be achieved.The models for these tests were selected as 30 cm square wooden frames which were reinforced either by means of one layer of expanded mesh (models M2, M3) or two layers of hexagonal mesh (models M5, M6). Model M1 was a bare frame for further comparisons. Fig. 7 shows the model designation and the reinforcement details. To study the effect of interconnection between the ferrocement faces, internal ties were used in two of the models (models M3 and M6). The mortar for ferrocement layers consisted of cement and natural sand which was applied manually on each face upto a thickness of about 0.5cm.The models were subjected to monotonically increasing diagonal compression .The loading and the load-def1ection diagrams corresponding to in-plane and out-of-plane deformations are shown in Figs.8 and 9 respectively.Test Set No. 3 [5]In this set of tests precast ferrocement hollow box roof elements were proposed to replace heavy roofs of existing single story masonary houses.The improvement in dynamic property was expected to be achieved by reducing the mass.It was anticipated that high death risk caused by heavy traditional earth roofing which is commonly used in rural houses of Middle East countries,can considerably be reduced if such roofs are totally avoided and replaced by another cost competetive system with improved dynamic performance. In this respect ferrocement roofing has advanced as a potential solution for such applications, due to its low-cost, durability and light-weight [6].Structural behaviour of six open-end box shaped precast ferrocement roof elements were studied under monotonically icreasing flexural loads.Three types of cross-sections and two types of mesh were combined in



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< previous page page_603 next page >Page 603six models.The designaton and the details of the models are given in Fig. 10. Experimental and theoretical results are summarized in Table. 3. The 1oad-def1ection behaviour for the two types of mesh used are compared in Fig. 11. TABLE.3 SUMMARY OF HOLLOW BOX ELEMENT TEST RESULTS [5]Element Desig. Exp. Ult (kg) Theo. Ult (kg) Exp/Theo

M1-B 700 639 1.095

M2-B 750 639 1.174

M3-B 750 639 1.174

M1′-B 500 496 1.008

M2′-B 500 496 1.008

M3′-B 500 496 1.008DISCUSSION OF RESULTSTest Set No. 1Welded connections behaved satisfactorily for both in increasing the lateral stiffness and damping properties; it was possible to increase the load carrying capacity in the lateral direction as much as 7 to 9 times and the new stiffness reached upto 3 times of the stiffness possesed before strengthening Energy dissipation which was used to indicate damping, increased considerably (upto 5 times) in direct proportion to the number of panels (2 or 4) and the connection type (discrete or continous). Due to lack of columns in the models used with bolted connection detaling, almost all the stiffness and damping was attributed to the panels alone. Monolitically cast models provided the basis for comparison of initial stiffness strength.Accordingly when precast panels were used, the lateral load carrying capacity and the failure modes were not effected,however the stiffness dropped as much as 50% ,energy dissipation capacity either remained at the same level of the



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< previous page page_604 next page >Page 604monolitic counterpart or increased upto 3 times as a function of the connection pattern used. Poor behaviour, in terms of stiffness and energy dissipation was observed in the models which had only end and side stoppers to hold the panels (Models M5, M10). Test Set No. 2In comparison with the negligible amount of the stiffness of the bare timber frame, almost 100% of the lateral stiffness was provided by the two ferrocement 1ayers. Fai1ure mode of all the strengthened models was local crushing of ferrocement at the loaded corners. This failure mode did not change with the presence of internal ties and was independent of the type of the mesh used. The internal ties helped to limit the out-of-plane deformations and delayed the failure at ultimate load, accordingly increased the amount of dissipated energy to some extend.Test Set No. 3The bending stiffness and load carrying capacity of models with hexagonal mesh were almost half as much of models with hexagonal mesh, although the percentage of both types of reinforcement were similar (Expanded mesh: one layer at top and bottom flanges and two layers at each side web; Hexagona1 mesh: three layers at the top and bottom flanges and six layers at each side web). Theoretical ultimate load carrying capacities were calculated by means of simple flexure theory, idealizing the box shapes as equivalent I-beams.CONCLUSIONSSince the number of tests were limited and scale effects were present, the test results obtained from all of the three sets cannot be used directly in actual construction; in fact they were not intended as such. Their common objective was explore and compare an appropriate model for a full scale detailed testing in the future. Keeping the above remark in mind following could be concluded:Precast panel strengthening:There was an obvious increase in both stiffness and damping properties when precast panels were used inside the frames. The initial stiffness remained almost same however more energy was dissipated as



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< previous page page_605 next page >Page 605the number of panels to fill the space inside the frame increased. In spite of the added cost of increasing the number of connections, more panels inside a frame had the advantage of ease of handling in full scale application since smaller size panels would be involved. Bolted connection detailing required more effort and precision than welded connection detailing for fixing the panels inside the frames.Ferrocement Plastering:The ferrocement plastering of timber framed structures resulted with a direct increase in lateral stiffness. Internal ties helped to limit the out-of-plane deformations and resulted in a better distribution of the applied loads between the layers.Hollow Box Elements:Overall flexural behaviour of elements with expanded mesh and/or flat top or concave top gemetry were much better when compared to the behaviour of elements with hexagonal mesh and/or convex top geometry.In summary:1-Welded type of connection detailing should be preferred to fix the precast reinfoced concrete panels inside the frames. In a further study [7] it was shown that the precast panels could be idealized as two Inclined parallel equivalent struts to calculate their contribution to stiffness. For immediate application of strengthening, panels which can be manufactured and cured in a shorter time, such as sulfur concrete reinforced with steel or glass fiber should be tried as an alternate to the reinforced concrete panels.2-Ferrocement plastering has an outstanding advantage in upgrading the lateral stiffness of timber structural frames of rural houses. If the heavy stone or adobe fillers are partially or totally avoided.there is an added advantage of reduction in mass.3-Expanded mesh reinforcement and either flat-top or concave top cross-sectiona1 geometry should be preferred to manufacture box-shaped precast ferrocement roof elements. To ease the manufacturing process and further improve the insulation properties a light-weight infill, such as styrofoam is recommended.



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< previous page page_606 next page >Page 606REFERENCES1. Yuzugullu.O., Multiple Precast Reinforced Concrete Panels for Aseismic Strengthening of R.C.Frames. Vol. 6, VII World Conference on Earthquake Engineering, Sept. 1980, Istanbul, Turkiye2. Yuzugullu.O., Bolted Connections for Precast R.C. Panels used for Repair and/or Strengthening, Proceedings VIII World Conference on Earthquake Engineering. Vol. I, July 21–28, 1984 San Francisco, California, USA3. Yuzugullu, O., Ferrocement to Increase the Lateral resistance of Timber Framed Rural Houses, Vol. 18, No. 1, Journal of Ferrocement, Jan. 19884. Earthquake Report, Caldiran Earthquake of 24 Nov., 1976, published by the Turkish Ministry of Reconstruction and Resettlement, Earthquake Research Institute, June 1977 (in Turkish)5. Yuzugullu, O., Precast Ferrocement Roof and Wall Elements for Low-Cost Housing in Earthquake Prone Areas, Jan. 1991, paper accepted for publication in the Journal of Ferrocement6. Naaman, A.E., Shah.S.P., Proceedings, IAHS International Symposium on Housing Problems, 1976, Atlanta, Georgia, USA7. Yuzugullu,O., Initial Stiffness Computation of Precast Concrete Panels, Proceedings of the Symposium on Concrete and Concrete Structures in the Middle East. Vol. II, April 25–29, 1987 Riyadh, Saudi Arabia



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Figure 1. Model Designation and Connection Types (1)

Figure 2. Welded Connection Details (1)



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Figure 3. Load-Deflection Diagram (1) Model M8


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Figure 4. Model Designation and Connection Types (2)



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Figure 5. Bolted Connection Details (2)


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Figure 6. Load-Deflectior Diagram (2)



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Figure 7. Model Designation and Reinforcement Details (3)


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Figure 8. In-plane Deformations (3)



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Figure 9. Out-of-Plane Deformations (3)


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Figure 10. Model Designation and Details (5)



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Figure 11.a Models with Hexagonal Mesh (5)

Figure 11.b Models with Expanded Mesh (5)



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< previous page page_613 next page >Page 613Helical Springs in Base Isolation SystemsG.K.HueffmannGERB Gesellschaft für Isolierung mbH & Co. KG, Sylviastraβe 21, D-4300 Essen 1, GermanyABSTRACTHelical springs are standard elements in general vibration control systems, but they may also be used to protect components or complete buildings against seismic excitation.Advantages and disadvantages of three-dimensionally elastic helical springs and only horizontally elastic elastomer mounts have been discussed at many earthquake conferences.In Los Angeles for the first time two residential buildings have been erected on helical springs combined with VISCODAMPERS. Details will be reported.Helical springs can also be applied in sliding systems that were developed in the United States and Japan parallel to elastic systems.A building on sliders will move when dynamic forces exceed static friction. To limit travelling of the building restoring forces are necessary.Tests at the National Center for Earthquake Engineering Research at the State University of New York at Buffalo with a slider system using helical springs to provide the restoring forces have been very successful. Test results will be discussed.INTRODUCTIONBase isolation of buildings and other structures has become a key word worldwide at all earthquake



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< previous page page_614 next page >Page 614conferences for now more than a decade. A variety of systems and elements have been developed over this period, based mainly on elastic deformation of the support elements, but also on sliding. Laminated elastomer elements have been favored in a number of actual projects in Japan, the western USA and in Italy providing where necessary very low horizontal natural frequencies of down to 0.5Hz and permitting horizontal response deflections of up to 500 mm. In vertical direction they are very stiff, nearly rigid, leading to vertical natural frequencies of the supported structures of 15–25Hz and thus providing no vibration isolation effect in this direction for typical earthquake spectra with a major low freguency content.While solely horizontal base isolation may be sufficient protection for a wide range of buildings, it seems too ignorant of reality to define in general an isolator unit as “a horizontally-flexible and vertically-rigid structural element…which permits large lateral deformations” (SEAOC Blue Book 1990 [4], UBC Code 1990 [5]).There should be no doubt that a three-dimensional protection against earthquakes must be better than a two-dimensional even if not always necessary.From the early days of base isolation a three-dimensional system based on helical spring elements have been in the discussion, especially where not only a building, but also equipment inside have to be protected, for which there is usually no preference for the vertical or horizontal direction. But although even a nuclear power plant was planned with such a three-dimensional system, in no project was it actually applied until 1990 when two residential houses in Los Angeles were built on a helical spring system combined with high damping VISCODAMPERS. This project will be discussed later in more detail.Parallel to the two-dimensional or three-dimensional elastic support systems, various slider systems have been developed allowing a building to slide into any horizontal direction when accelerations exceed friction. The problem is not so much where and how the building slides, but how to get it back into the original position, e.g. where to get the necessary restoring force. The National Center for Earthquake Engineering Research at the State University of New York at Buffalo has worked on sliding systems for a number of years. In one of their latest research projects they have used helical springs, supporting the structure from below, but without being vertically loaded. All the static load is taken by the friction elements. It is the horizontal deflection of the helical springs that gives the restoring force. The three-dimensionally elastic springs offer a number of advantages over other types of elements that can also provide in general restoring forces.



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< previous page page_615 next page >Page 615BASE ISOLATION WITH A HELICAL SPRING SYSTEMGeneral InformationTwo residential houses in West Los Angeles, California, were erected in 1990 on a helical spring base isolation system. Figure 1 shows a picture of these houses. The buildings have a partial basement with concrete walls extending over that basement area, so that the entire base isolation system could be placed on top of these walls.The base isolation system consists of the load carrying helical springs that provide three-dimensional elasticity at zero material damping and separate no-static load carrying, energy dissipating viscoelastic damping elements, so-called VISCODAMPERS, providing in general velocity proportional damping with a dynamic elasticity component being usually negligible compared to the spring stiffness of the helical springs.Each of the 17 building columns is supported by one element. Figure 2 shows a general plan view with the distribution of the elements. Eight of the elements are equipped with four springs, the other nine with two springs only. Six of the eight elements with four springs have integrated VISCODAMPERS. Figures 3 and 4 show the different elements.The three-story superstructure of these buildings is a light weight steel structure. The total spring supported weight, dead load plus actual live load, is only 630kN. The first bending natural frequency of the superstructure with columns assumed to have fixed ends, e.g. no spring support, was calculated to be at 7.75Hz.



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< previous page page_616 next page >Page 616The basic design idea for the base isolation system was not so much to reduce input accelerations of the design basis earthquake, but to avoid resonance amplified responses in the structure in all three dimensions.By choosing slightly higher natural frequencies than in typical two-dimensional elastomer systems, response amplitudes are limited to lower values, which is obviously an advantage for the survival of the life lines etc., and high damping guarantees that system response is low even when the natural frequencies are excited.System design refers in general to the American UBC Code 1988 and the SEAOC Blue Book 1988 with the so-called Tentative Seismic Isolation Design Requirements, but certain demands do not apply as these requirements are mainly referring to two-dimensional base isolation systems being rigid in vertical direction.While the two-dimensional system moves with all parts parallel in any horizontal direction (Figure 5), the three-dimensional system tends to rock (Figure 6). This leads to smaller horizontal displacements in the base isolation interface of the three-dimensional system, but additional vertical displacements have to be taken into account. The actual design displacements are evaluated from a response spectra analysis.Torsion can completely be neglected in this system. The theoretical center-of-gravity (COG) coincides nearly exactly with the center-of-rigidity (COR) in plan view. This can easily be checked by measuring the spring height. Equal spring compression is the reference for the coincidence of COG and COR. Major eccentricities would result in a tilting of the building. Even accidental eccentricities of COG and COR would not lead to major torsional motion, as modal damping for this mode is so high that this mode is completely suppressed. This special behavior of the helical spring system in comparison to elastomer systems was already shown in earlier corresponding shaking table tests.Irregular load distribution has also not to be taken into account. The compression of the linear spring gives the exact load for each element. Where



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< previous page page_617 next page >Page 617necessary the regular design load distribution can easily be arranged through spring adjustment by shimming.The elements are designed to transmit lateral forces on their top sides by a centering middle shear pin, but in addition all elements are fixed with four bolts.On the bottom side the elements are resting on adhesive pads, which are resistant to lateral accelerations up to 2g. The adhesivity grows with time. These pads were tested in the past for many applications in nuclear power plants, but for additional safety reasons and taking an unlikely deterioration of the pads into account, elements are additionally fixed with four anchor bolts each.When installed and for the entire erection time of the building the springs were prestressed up to design load and locked in this position so that the building was erected on a rigid subsystem.When all major loads including live loads were applied, the system was activated by removal of the prestressing bolts in the elements. A check of spring heights proved that earlier load assumptions, especially of live loads, were quite accurate.Spring DataThe springs in the base isolation system are made of chrome silicon alloy steel 60SiCr7.They have a definite vertical and horizontal spring stiffness and the following dimensions and properties:Shear modulus G=78, 500N/mm2

Wire diameter d=24mm

Mean diameter of windings D=110mm

Number of windings n=7

Height, when not loaded LO=285mm

Vertical spring stiffness kV=0.350kN/mm

Horizontal spring stiffness kH=0.14kN/mm

Maximum spring compression (down to solid height) fB=88mm



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< previous page page_618 next page >Page 618VISCODAMPERSThe viscoelastic VISCODAMPER is an energy absorbing element. It does not carry any static load. It consists of the bottom part filled with highly viscous liquid and a piston-type top part that dips into the viscous liquid with the possibility of moving inside the bottom part in all degrees of freedom (Figure 7).Motion of the piston means shearing of the viscous liquid and consequently energy dissipation. The force necessary to move the piston is in general velocity proportional:

FD=W·V. W is the so-called damping resistance (kN·s/m), a frequency dependent property of the VISCODAMPER.The following damping resistances were used in the dynamic analysis.

Frequency (Hz) W (kN·s/m)

Vertical Horizontal

1 150 50

2.5 90 – –The motion of the piston inside the VISCODAMPER is limited to 55mm. It then hits its wall or bottom, thus providing an ultimate restraint system.System AnalysisWith the first bending natural frequency of the steel structure ≥7.5Hz, it is permissible to neglect the elasticity of the building superstructure in the calculation of the basic natural frequencies. They are therefore calculated for a rigid body with spatially equal mass distribution over a rectangular plan view (Figure 2), which has the same area as the actual plan view of the building supported by the actual spring and VISCODAMPER distribution. (For a given total mass, mass distribution has only a small influence on the main natural frequencies of a spring supported rigid body system.)The main natural frequencies and damping values are:



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< previous page page_619 next page >Page 619Direction Natural Frequency Damping

Vertical 2.5Hz 28% of critical

Horizontal/rocking (both directions) 1.4Hz 25% of criticalThe response analysis was based on the normalized response spectra given in the UBC Code for soil type S1. Vertical spectral accelerations were choses at 2/3 of the horizontal values.Maximum response acceleration and displacements in the springs are:Vertical from vertical excitation 0.4g ±15mm

Vertical from rocking (spring unit 5) 0.15g ±21mm

Vertical RMS 0.43g ±26mm

Horizontal at spring level 0.33g ±30mmSpring shear stresses were calculated for static plus maximum vertical RMS compression from the vertical mode and the horizontal mode (65mm) and maximum horizontal displacement (30mm).Shear stress at 65mm vertical spring compression (zero horizontal displacement) γ=457N/mm2

Total shear stress for additional 30mm horizontal displacement γ=708n/mm2So even at maximum design displacements the springs neither bottom-out nor is there a lift-off, and stresses are still below permissible working stress level (γ=820N/mm2).



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< previous page page_620 next page >Page 620HELICAL SPRINGS IN SLIDER SYSTEMSSliding systems are an alternative in base isolation to elastomer systems. Motion of the slider supported structure will only occur when response accelerations exceed friction. The influence of the frequency content in the input signal on the response is negligible.If friction is exceeded displacements may become large and permanent, especially on not perfectly horizontal surfaces. To avoid this restoring forces have to be added to the sliders.Mainly three sliding systems have been investigated in analysis and tests, the friction pendulum system (FPS) [1], the Japanese TASS system [2] and the resilient friction base isolators (R-FBI) [3]. The restoring force is provided in different ways. The FPS element has a spherical shape (Figure 8), where the building slides back to the lowest point. TASS uses rubber springs and R-FBI elements (Figure 9) have a rubber core. In all cases the restoring forces are higher than friction, and system response is not independent of frequency. Especially long-period earthquakes, such as Mexico City 1985, cause major amplified displacements. In general, response displacements would be smaller than in a system with pure elastomeric support; response accelerations, however, are usually higher.In a new test series at the National Center for Earthquake Engineering Research at the State University of New York at Buffalo, sliding Teflon disc bearings (Figure 10) were used in combination with helical springs. These springs were mounted below the test base frame without being vertically loaded providing the intended restoring forces in shearing direction (Figure 11).The end windings of the springs are fixed in an epoxy ring to prevent their partial lift-off during high horizontal deflections. The also vertically very flexible spring adapt easily to height differences between support level and test frame without taking major load away from the disc bearings.Frictional forces were in this case two times higher than peak restoring forces developed in the springs.



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< previous page page_621 next page >Page 621The tests showed that these weak springs did not influence the response signal and therefore not the base isolation effect, but they were able to limit permanent displacements to only 6% of the design displacements in the disc bearings.Peak deflections in the springs reached 54mm for a 120% Mexico test earthquake where shear stresses in the springs were still far below the permissible working stress level.CONCLUSIONSHelical springs may be used in base isolation systems in different ways:– – as a three-dimensional support system for structures; or

– – as elements to provide restoring forces in slider systems.In both cases it is mainly the horizontal deformation of the spring that is used and not so much the vertical compression.There are other base isolation systems and elements for the recentering of sliding structures, but there will be in the future applications where helical springs not only offer technical advantages, but can compete even economically.



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< previous page page_622 next page >Page 622REFERENCES1. Mokha, A. et al. Experimental Study of Friction-Pendulum Isolation System, Journal of Structural Engineering, Vol. 117, No. 4, April 1991.2. Kawamura, S. et al. Study of a sliding type base isolation system, Proceedings of the 9th WCEE, Vol. 5,3. Mostaghel, N. Response of Structures Supplied on R-FBI Bearings, Proceedings of the 9th WCEE,4. Structural Engineers Association of California (SEAOC). Recommended Lateral Force Requirements (Blue Book), 1988, 1990.5. United Builders Code (UBC) 1988, 1990.6. Constantinou, M.C., Study of Sliding Bearing and Helical Steel Spring Isolation System, Journal of Structural Engineering, Vol. 117, No. 4, April 1991.



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Fig. 1 Steel structure of base isolated building

Fig. 2 Layout of base isolated buiding



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Fig. 3 Spring Unit

Fig. 4 Spring Unit with integrated VISCODAMPER

Fig. 5 Horizontal motion


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Fig. 6 Rocking

Fig. 7 VISCODAMPER



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Fig. 8 FPS element

Fig. 9 R-FBI element

Fig. 10 Teflon disc bearing

Fig. 11 Horizontally deflected helical spring



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< previous page page_627 next page >Page 627Damage Reduction with Controlled Seismic PoundingS.Govil, A.SinghalDepartment of Civil Engineering, Arizona State University, Tempe, AZ 85287, U.S.A.ABSTRACTSeismic pounding of neighboring buildings can cause serious damage especially in congested areas of metropolitan cities as reported in the Loma Prieta earthquake of 1989 and Mexico City earthquake of 1985. The primary reason for pounding is insufficient separation and clearance between adjacent buildings. Although all tall buildings are designed for existing seismic codes, however current seismic codes do not adequately account for pounding between adjacent buildings. Large spacing can result from codes, but that does not guarantee the prevention of pounding due to the relatively low code lateral yielding strength requirements. Moreover little consideration for pounding is given to buildings separated by expansion joints. Thus the effects of pounding are devastating leading to substantial building damage, collapse, and increased public hazard.In this paper pounding of buildings has been studied numerically. Numerical studies are performed to study the effect of pounding on buildings with various structural properties subjected to strong ground motion using the DUHAMEL integration technique. The response of adjacent building is studied for (1) no impact, (2) single impact, and (3) multiple impact. A parametric study has been performed to study the effect of (a) relative masses of the buildings, (b) relative stiffness of the buildings, (c) relative damping, and (d) coefficient of restitution and energy losses during impact. It is also found that for larger mass ratio the effect of pounding is more pronounced than the structure with a relative smaller mass ratio. Damping is found to be an important parameter. Several figures summarize interesting parametric studies.INTRODUCTIONPounding of buildings could be a severe problem during earthquakes. These days tall buildings are constructed with very small gap between them. Because of this small gap, there is always a risk of collision of adjacent building during a strong earthquake. It could also occur by severe wind



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< previous page page_628 next page >Page 628forces. Several examples could be cited from past earthquakes, eg., in Alaska Earthquake of 1964, Anchorage-Westward Hotel and the adjoining three story building hammered each other [1]; in the San Fernando earthquake the building of Olive View hospital hit against the neighboring stair tower [2, 3]. Other pounding damages are reported from 1976 Friuli Earthquake [4], 1977 Romanian Earthquake [5] and 1985 Mexico City Earthquake [6]. In the recent 1989 Loma Priata Earthquake, extensive structural damage due to pounding was observed [7].It should be noted that when ever there is an impact between two adjacent buildings, there is a degradation of the structural stiffness. This in turn leads to the degradation of the dynamic properties. This effect will be more pronounced when the building undergoes multiple impacts. In the current building codes there is a provision for the building separations to avoid pounding. This is considered to be highly conservative. Thus we can see that pounding can be a serious seismic hazard. It is very important to study the response of buildings that are subjected to multiple pounding as well as to provide guidance in future building design. This paper contributes to the above mentioned goals by presenting a study of the dynamics of structural pounding.Computer analysis results of controlled pounding of structural systems subjected to strong ground motion are presented. Controlled pounding reduces the overall seismic response of the building. Consideration of multiple impact is important in the analysis.THEORYAssuming that both buildings are subject to same ground motion, there are two basic approaches to study the response of buildings subjected to pounding.(1) The pounding between the buildings can be simulated by using a spring and a dashpot between the masses of the buildings. This spring and dashpot are active only when the two buildings are in contact. In this kind of analysis it has been observed that the spring should have a sufficiently large stiffness; approximately 20 times that of the building [8, 9]. The damping used is a function of the coefficient of restitution. The smaller the coefficient of restitution the larger is the damping. This approach will require that the spring connecting the two masses behave as a non-linear material. In this mentioned the system is modelled as a two degree of freedom system. The two degrees of freedom system are uncoupled into two single degree of freedom system when they are not in contact. The equation of motion for such a model can be solved by the linear acceleration method [8, 9]. One of the main sources of error in the above method is that it is very difficult to asses the stiffness and damping of the pounding system. This method of analysis requires nonlinear analysis. Moreover the idea behind this study was to come up with a simple approach which models the problem with sufficient accuracy.(2) The other approach, which is used in the current analysis requires that the two buildings be modelled as a single degree of freedom system and their response be studied simultaneously. Same ground motion is applied to both



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< previous page page_629 next page >Page 629structures. They are separated by a distance ‘d’. The masses of these buildings are m1 and m2 with the same stiffness and damping, ‘k’, and ‘c’.There are two extreme cases of impact. One is the elastic impact and the other plastic. In the elastic impact the two rigid bodies impacting are in contact for a very short interval of time. The impact is considered to be instantaneous. Where as in a plastic impact the two rigid bodies are permanently in contact after the impact Thus we can say that the duration of impact can be represented as a function of, how elasto-plastic the impact is. This is taken into consideration by means of the coefficient of restitution, ‘e’. e=0 corresponds to the plastic impact and e=1 corresponds to an elastic impact.The various steps involved in this analysis are as follows:(a) The response of the two single degree of system is obtained simultaneously using the Duhamel Integral [10, 11]. To start with the initial displacement yo and the initial velocity vo are zero for both the systems. That is both the systems are starting at rest.(b) During the response the relative displacement of the two masses is closely observed. As soon as the relative displacement of the two masses is greater than the distance between the buildings, d, a collision between the two masses has occurred. When there is a collision there is a change in the velocity of both the masses depending on their approach velocities, mass and the coefficient of restitution. This change in velocities can be obtained by applying the conservation of momentum and the definition of coefficient of restitution as given by the equations below. Moreover during this process some energy is lost during the impact depending on the coefficient of restitution.

maVa0+mbVb0=maVaf+mbVbf (1)

(2)From the above two equations we can obtain the velocity of both the masses after the impact. The velocities after an impact are given by

(3)Vaf=−e (Va0−Vb0)+Vbf (4)

The displacement of both the masses at the time of impact is obtained as part of the response of the masses obtained by the numerical solution of the Duhamel Integral.



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< previous page page_630 next page >Page 630(c) The third and the final step is to obtain the response of the above two masses by taking the initial displacement and velocity of the masses as their displacement at the time of impact and the velocity after the impact.A program called IMPACT is developed to obtain the response of adjacent building subjected to multiple pounding. This program considers the buildings as single degree of freedom systems with generalized mass, stiffness and damping.A nonlinear dynamic response of multilayered complex structural system with multiple masses and multiple impact between various masses was previously studied by Singhal [12], which also had oblique impact. The present study however has more direct application to two impacting masses.PARAMETRIC STUDYIn order to study the general nature of the response, a large number of analysis are carried out. The first 10 sec. of the N-S component of El Centro Earthquake was used for the analysis [10]. The various parameters that have been studied are(i) Effect of damping(ii) Ratio of the masses of the two buildings(iii) Coefficient of restitutionAnalysis has been carried out for two different values of damping ie 0.0% of critical and 10% of critical. This values are assumed to be the same for both the structures. Coefficient of restitution of 0.0 and 1.0 has been used in the analysis. In order to study the effect of relative mass, mass ratios of 1:0 to 1:4 has been used in the analysis. The maximum velocity in inches/sec of both the masses for all the possible combinations of the above parameters with the frequency of masses varying from 0.1cps to 20.0cps approximately.RESPONSE OF SDOF UNDER CONFINED CONDITIONSA study was carried out to study the response of a SDOF system subjected to an excitation under confined conditions. The system is enclosed by fixed walls on either side such that the displacement of the SDOF system is restricted, and if the displacement exceeds the given separation, there is an impact. As a test case, numerical computations were made for a single story frame whose mass was taken as 200,000 lbs, equivalent stiffness of 50,000 lbs/in and damping of 20% of critical. The coefficient of restitution was taken as 0.75. The separation was assumed as 2 in on either side. The response was studied for (a) Free vibration with initial velocity of 100 in/sec and initial displacement of -2 in, and (b) A sinusiodal forced excitation with a peak acceleration of 0.75g. The displacement time history for case (a) and (b) for confined and unconfined response are given in figure 1. It is observed that the response is reduced significantly by controlled pounding.



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< previous page page_631 next page >Page 631ENERGY ABSORBERSIn the previous section we have discussed the reduction in response due to controlled pounding. It is also important to have some kind of energy absorbers in order to absorb the energy released during impact. Shear pins are found to be very effective in absorbing energy, Salvadori and Singhal [13].RESULTS AND SUMMARYFirst the results are summarized based on the parametric study and then a general discussion is made on the pounding of buildings.Response SpectraResponse spectrums have been developed in order to study the effects of mass ratios, damping and the coefficient of restitution.Effect of Mass RatioResponse spectrums have been plotted for mass 1 for mass ratios of 1:1, 1:2, and 1:4 in figure 2 for ξ=0% and 10% and e=0 and 1. For example in Figure 2(a), m11-c0 means a mass ration of 1:1 and 0% damping. m12-c0-e0 means a mass ratio of 1:2, damping 0% and coefficient of restitution e=0. Consequently, m11-c0-e0 is the baseline spectrum for a SDOF system. From figure 2, it is noted that if the system has significant damping than the response for various mass ratios remains unaffected for all values of coefficient of restitution as seen in figure 2(b) and 2(d). For 0% damping the response is reduced as the mass ratio reduces. It is interesting to note that for 0% damping the response is very much a function of the coefficient of restitution, figure 2(a) and 2(c). For e=0, as the mass ratio increases the response increases and approaches to that of a single degree of freedom system without impact where as for e=1.0 as the mass ratio reduces it approaches to that of the single degree of freedom response without impact.Effect of DampingComparison of figure 2(a) and 2(b) shows that damping reduces the response significantly. This is also observed in figures 3 and 4. Damping ratios of 0% and 10% are used in the analysis. Figures 3(a) and 3(b) are for mass 1 and 2 for a mass ratio of 1:4 and e=0 where as figures 4(a) and 4(b) are for the same mass ratio but for e=1.0.Effect of Coefficient of RestitutionThe response is reduced as the coefficient of restitution increases as can be seen from figures 5(a) and 5(b) for mass ratios of 1:4 and damping of 0% and 10% respectively.Another test case treated, mass ratios of 1:2 and 1:4, stiffness= 100 units and gap d=6 units. Figures 6 and 7 give the relative displacement time history for both mass ratios and various coefficient of restitution and damping.



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< previous page page_632 next page >Page 632In summary, for very flexible structures an increase in response occurs, where for stiff structures a slight decrease is observed. It is also seen that for larger mass ratios the effect of pounding is more pronounced than the structure with a smaller mass. In designing buildings which are safe against pounding the maximum distance between the buildings can be kept equal to the sum of the maximum displacement of the two adjacent buildings although this is very conservative because the maximum of both the buildings will not occur at the same time.CONCLUSIONSBased on the above analysis following conclusions have been arrived:(a) Significant reduction in the response of the building can be achieved by means of controlled pounding between various elements of the building which in turn reduces the damage due to strong ground motion.(b) It is observed that this reduction in response is very much a function of the relative mass and stiffness of the structural elements.(c) Shear pins can be effectively used at each floor level in order to absorb the impact energy.REFERENCES1. The Great Alaskan Earthquake of 1964, Engrg, NAS, Pub. 1606, National Academy of Science, Washington, D.C., 1973.2. Mahin, S.A., V.V.Bertero, A.K.Chopra and R.G.Collins, EERC 76–22, University of California, Berkeley, 1973.3. Bertero, V.V. and R.G.Collins, EERC 73–26, University of California, Berkeley, 1973.4. Glauser, E.C., Proc. Sixth World Conference, Earthquake Engineering, New Delhi, Vol. 1, 1977, pp. 279–288.5. Tezcan, S.S., V.Yerlici and H.T.Durgunoglu, International Journal of Earthquake Engineering and Structural Dynamics, Vol. 6, 1978, pp. 397–421.6. Rosenblueth, E. and R.Meli, “The 1985 Earthquake: Causes and Effects in Mexico City”, Concrete International, 8 (5), pp. 23–24.7. Lew, H.S. (Editor), “Performance of Structures During The Loma Prieta Earthquake of October 17, 1989”, Publication No. 778, ICCSSCTR11, National Institute of Standards and Technology, pp. 1–201, United States Department of Commerce, Gaithersburg, M.D.8. Anagnostopoulos, S.A., “Pounding of Buildings in Series During Earthquakes”, Earthquake Engineering and Structural Dynamics, Vol. 16, 1988, pp. 443–456.9. Wolf, J.P. and P.E.Skrikerud, “Mutual Pounding of Adjacent Structures during Earthquakes”, Journal of Nuclear Engineering Design, Vol. 57, 1990, pp. 253–275.10. Paz, Mario, Structural Dynamics, 3rd edition, Van Nostrand Reinhold Co., 1985.11. Clough, R.W. and J.Penzien, Dynamics of Structures, McGraw-Hill Book Company, New York, 1975.12. Singhal, A.C., “Nonlinear Dynamic Response of Bonded and Unbonded Rings Under High Intensity Blast”, Structural Group, Vol. 50, 1969.



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< previous page page_633 next page >Page 63313. Salvadori, M.G. and A.C.Singhal, “Seismic Analysis of the Suspended Steam Generator Supporting Structure”, Foster Wheeler Unit #2-85-610, Socal-Ormond Beach Power Station, California, Weidlinger Associates, New York, 1975.APPENDIX I—Notationsk=generalized stiffnessm=generalized massξ=% of critical dampingd=distance between buildingse=coefficient of restitutionag=ground accelerationm1=mass of the first buildingm2=mass of the second buildingωD=damped natural frequencyω=natural frequencyVo=initial velocityYo=initial displacementy(t)=displacement at time tt, τ=timema=mass of body amb=mass of body bVaf=velocity of ‘a’ after impactVbf=velocity of ‘b’ after impactVa0=velocity of ‘a’ before impactVb0=velocity of ‘b’ before impactmij=mass i with a mass ratio of i:jmj=mass j with a mass ratio of 1:je1=coefficient of restitution of 1e0=coefficient of restitution of 0c0 =0% of critical dampingc1=10% of critical damping



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Figure 1. Displacement time history for SDOF system under unconfined and confined condition. (a) Free vibration with


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Vo=100 in/sec, Yo=−2 in (b) Forced vibration with sinusiodal excitation of 0.75g.



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Figure 2. Response spectra for mass 1 for mass ratios of 1:1, 1:2, 1:4. Coefficient of restitution e=0 and 1. Damping is 0 and 10 % of critical.



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Figure 3. Response spectra for mass 1&2 for mass ratio of 1:4. Coefficient of restitution e=0. Damping is 0 and 10 % of critical.

Figure 4. Response spectra for mass 1&2 for mass ratio of 1:4. Coefficient of restitution e=1. Damping is 0 and 10 % of critical.



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Figure 5. Response spectra for mass 1 for mass ratio of 1:4. Coefficient of restitution e=0 and 1. Damping is 0 and 10 % of critical.

Figure 6. Relative displacement time history for mass 1 for mass ratio of 1:2. Coefficient of restitution e=1. Damping is 0 and 10 % of critical.



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Figure 7. Relative displacement time history for mass 1 for mass ratio of 1:4. Coefficient of restitution e=0 and 1.


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Damping is 0 and 10 % of critical.



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< previous page page_639 next page >Page 639On-Line Hydraulic Servodrives to Protect Serviceability of Antiseismic Structures—Pre-Design CriteriaA.CarottiDepartment of Structural Engineering, Politecnico di Milano, 20133 Milan, ItalyABSTRACTThe active alternative offers a number of distinct advantages for protection against brief and intense disturbances occurring at random during the life of a construction.One major aspect of pre-designing the electrohydraulic servosystem which controls, by means of activated bracings, the lateral sway of a multistorey buildings under horizontal loads, is that of determining the analytical structure of the central control processor able to drive the on-line redistribution of the control actions when small relaxations in tracking the dynamical specifications could intervene at certain floors and influence the corrective actions at other floors where the implementation of the control objectives is strictly required. General criteria for math-modelling and answering such a problem are presented, establishing simple relations between certain known building parameters and the main design variables of the active servodrive.OUTLINE. PROBLEM STATEMENTThe active-structural-control technology for protecting civil structures under extraordinary horizontal loads (seismic or wind) has been widely discussed in the literature during the last twenty years (see References).A representation of an active on-line system for use on civil structures is given in fig. 1, where the external power supply and the various stages of the closed loop are shown. Traditional antiseismic protection usually involves elastic design in compliance with serviceability limit states in the expectation of moderate earthquakes (ground acceleration 4–6 times weaker than the maximum foreseeable). Under severe earthquakes energy is dissipated by plastic deformation with inevitable, possibly serious, structural damage. The active



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Fig. 1system can be viewed as an alternative approach to building safety for cases where continued functioning is all-important, for example in “1st-class” buildings, which must remain usable in a post-earthquake emergency.In the event of a relatively brief but intense horizontal disturbance, an on-line servodrive which intervenes, adapting its control action to the level of the disturbance, could constitute an economically worthwhile, high performance mechanical alternative to traditional protection. Costs has been addressed specifically in Carotti [13].There has undoubtedly been a widespread increase in interest in the active approach over the past five years and full-scale projects could be imminent. An attempt must be made, in particular, to summarise standard criteria for pre-sizing components and subsystems, stating simple relations between known global structural parameters and the characteristic variables of an active plant.A major problem which arises in the management of active subsystems for the control of the lateral response yi(t) (i=1, n) of n-storey framed buildings (e.g. Carotti [2, 6, 13, 14], Masri [8, 10], Meirovitch [7, 21]) is to prevent situations in

which the effects of small relaxations in tracking the dynamical specifications at certain floors—and the correspondent actual displacements yi(t)- could influence the response of the active bracings at other floors, i.e. those for which the implementation of the control objectives is strictly required. Such relaxations could be accidental (due to natural decay or failure of equipments) or due to a design strategy for controlling the external energy consumption when active levels exceed certain predetermined thresholds.In the case of active control of the lateral response of tall



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< previous page page_641 next page >Page 641buildings with the external horizontal disturbance exceeding predetermined magnitude threshold, it is economically advisable and worthwhile to adopt a control hierarchy with different corrective actions at the floors (e.g. protection of the serviceability-limit-state of the structure at certain floors only, or, in other words, privilege to the control of the relative sways at high floors when, for example, high flexibility and inertia could induce second order effects).From a mechanical point of view, a small local relaxation of the control specifications at a number of floors, turns out in a re-distribution of the inertia forces effects which must be promptly counteracted by a correspondent re-distribution of the auxiliary control actions if a severe observation of the response targets is required at the other floors.At a final analysis the problem arises of determining the analytical structure of the central control processor, whose manipulated signals, when distributed to the electrohydraulic servovalves drives the bracings, could be able to implement the before mentioned control re-distribution. This is a major aspect of predesigning the hydraulic subsystem (“independence” of the activated bracings): in this paper, general criteria for math-modelling and answering such a problem are presented, establishing simple relations between certain known building parameters and the main design variables of the active servodrive and of the central controller, for the case of a two-storey linear elastic frame.MATH-MODELLINGLet us consider fig. 1 taking into account the block diagram in fig. 2; a simple rigid-elastic scheme is considered with rigid members and elastic damped nodes. Structural responses are:

(1)

Fig. 2



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< previous page page_642 next page >Page 642Let vi(t): R→R, be the active horizontal control action transmitted from the activated bracing system to the i-th floor (fig. 1).

Let T be the time-interval of the observed dynamics, Tc=[t1, tf] that of the structure/active-bracing interaction and the control vector

(2)The state-space state vector of the frame dynamics is

(3)qi(t) being the angular displacement (from the vertical) of the i-th column pair. From the set of the lagrangian equations, taking into account (1), (2) and (3), the state motion and the output transformation become:

(4)

with WεR4x4 natural matrix of the frame (w21, w23, w41, w43 derived from the geometric, inertia and elastic characteristic; w12=w34=1) and vεR4×2 matrix of the controls distribution (v21, v22, v41, v42 depending on the geometrical and physical characteristics)

(4)′

being , , matrix of the response transformation, with non-zero u11 and u23.Formal properties of (4), (4)′:(i) asymptotic stability of W,(ii) complete controllability of the pair (W V),

(iii) , (i=1, 2),

(iv) linear independence of , , , .The following can be stated:Problem. Pre-design of a negative-feedback controller such that:(a) the two pair of activated diagonal bracings are made independent: i.e. when changing from one dynamical

specification to a different one b*(t))T with b(t)≠b*(t), the frame

dynamics passes from to , for each initial state of the frame ,(b) a predetermined eigenvalue geography in the left gaussian half-plane could be implemented.The following are assumed:Hyp.1: on-line availability of the structure state (fig. 2, continuous-line feedback signal),



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< previous page page_643 next page >Page 643Hyp. 2: implementation of a state feedback (e.g. Carotti [1, 14, 16], Gabosov [19], Abgarian [20], Meirovitch [21]) with characteristic:

(5)with E εR2×4 and AεR2×2 matrices to be determined, of feedback gains and of feedforward gains respectively.

Hyp. 3: each floor specification could influence the structural dynamics (the trivial solution A=0 is excluded). Due to Hyp. 2 (5) the closed loop frame dynamics is :

(6)

(6)′with TεR2×2 matrix of the impulsive responses (i/o transfer matrix) which must be diagonal:

The Hypothesis 3, together with the requested independence of the bracings (diagonality of T(t)) are satisfied by virtue of the linear independence of the columns of V·A, from which the linear independence of the i/o transfer matrix directly follows (rank (V·A)=2). From such condition and from rank(V)=2, the non-singularity of the a-priori unknown matrix A of the feedforward gains can be deduced.The solution of the stated problem has been obtained by adjustements of the closed solution of the before mentioned Problem (a), in order to comply with Problem (b).ANALYSISThe following auxiliary augmented state-space is introduced:

(7)with M a-priori satisfying the linear independence of the rows of the two upper blocks, the lower block being available to comply with the non-singularity of the transformation. In the augmented space:

(8)(8)′



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< previous page page_644 next page >Page 644in which

(9)

the unknown E and A appearing in and respectively. When is partitioned into , into ,

into and into , from (7) (9) the following can immediately be stated:

(10)The independence of the floor bracings can thus be represented by means of the block structure in fig 3 which turns out in the independence of the two subsystems τi.The state space dynamics of each τi can be expressed in the following way:

(11)and that of the auxiliary subsystem τ2+l is

(12)It follows that the auxiliary matrices must have structure

Fig. 3



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In order to comply with the non-singularity of (7) and rearrange the results in term of the original dynamical system, the preceding condition (iv) has been used, with the following assumption:

(14)from which the first member of (10) takes the form:

(15)From the matrix structure in (13), taken into account (15), it follows that (10) is satisfied if-and-only-if the first row of Mi(W+V E) (i=1, 2) has non-zero elements and coincides with the last row of the Mi given in (14): the following matrix of the feedback gains is obtained

(16)In the same way, taken into account (9) (b), (14) and the 2nd and 3rd of (13)—which are a consequence of the formal structure of matrix T—the matrix A can be obtained:

(17)The control law (5) with (16) and (17) are now adjusted in order to shift the predetermined eigenvalues deeper inside an admissible region (depending on the design specifications) of the gaussian left half plane. As for the augmented auxiliary structure given in fig. 3, a modified design specification has been assumed for each subsystem τi; this is of the type

, the two pairs of the desired eigenvalues being taken into account in the

factors. When the preceding condition is put into the first (11), a characteristic polynomial is obtained whose coefficients, which coincide with the elements of , can be arbitrarily assigned.We finally return to the original space state by means of the (7) and of the modified control

,



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which must be put into the of (5) together with (16) and (17).RESULTSAs a consequence of the non-singularity of

for the dynamics (4) we have:

and

When the following reference eigenvalues are introduced:

the control forcing function at frame nodes (with columns of unitary height) must be the following:

The preceding expression defines the central controller which implements both the requested “independence” of the bracing subsystems and the eigenvalues shifting.



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For the practical realization of the preceding control strategy a set [q1(t) q2(t) ] of displacement and velocity signals must be available from the on-line measurement systems at the building floors. From these signals and

from the reference signals and , the central supercontroller, which interacts with the standard microprocessor board of the electrohydraulic servovalve, processes on-line the algorythm and sends the manipulated signals to the servovalves. These drive the pressure differentials in the two chambers of the hydraulic cylinders and thus the corrective mechanical actions v1(t) and v2(t) to the frame nodes, via the interposed activated bracings.REFERENCES1. Carotti A., Electrohydraulic subsystem control-law design for active structural protection under extraordinary loads. Proc. Struct. Mech. in Reactor Technology SMiRT11 Intern. Conf., Tokyo (Japan) August 18–23, 1991 (to appear).2. Carotti A., Chiappulini R. Active protection of large structures under seismic loads: artificial damping and stiffness supplied -by a hydromechanical servodrive. Soil Dynamics and Earthquake Engineering International Journal, Vol. 10, No. 2, pp. 110–126, 1991.3. Carotti A., Chiappulini R. Assessment of active alternatives for the control of long-span bridge dynamics. Implications for train runnability. Proceedings Eurodyn 90, European Conf. on Structural Dynamics, Ruhr Universität-Bochum (FRG), June 5–7, 1990; W.B.Kratzig et al. (Editor), Rotterdam Vol.1, pp. 385–392, 1991.4. Natke H.G. .Recent trends in System Identific., Proceedings Eurodyn 90, European Conf. on Structural Dynamics, Ruhr Universität-Bochum (FRG), June 5–7, 1990; W.B.Kratzig et al. (Editor), Rotterdam 1991, Vol.15. Carotti A., Lio G. Experimental active control: bench tests on controller units. Engineering Structures International Journal. To appear July ’91.6. Carotti A., Chiappulini R., Reconsidering active control in civil engineering: criteria for the design of structural and hydraulic auxiliary subsystems. Proceedings 9th ECEE, European Conf. on Earthquake Engineering, Moscow, USSR, Vol. 10-A, pp. 23–32, 1990.7. Meirovitch L., Control of structures subjected to seismic excitations, ASCE, J. Eng. Mech., 109, pp. 604–618, 1983.8. Masri S.F., Session Report: Seismic Response Control of Structural Systems: Closure. Proceedings of 9th World Conf. on Earthquake Eng., Tokyo-Kyoto, Japan(Vol. VIII), pp. 497–502, 1988.9. Carotti A., de Miranda F.G. Passive antiseismic protection of multistorey buildings by controlled buckling of a base



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< previous page page_648 next page >Page 648 isolation steel system. Mechanics Research Communications. International Journal, Vol. 17, 3, pp. 149–155, 1990.10. Miller R.K., Masri, S.F., Dehghanyar, T.J. and Caughey, T.J., Active Vibration control of large civil structures. The Journal of the Engineering Mechanics, Division ASCE, Vol. 114, No. 9, pp. 1542–1570, 1988.11. Meirovitch L., and D. Ghosh, Control of flutter in bridges, ASCE J. of Eng. Mech., 113, 5, pp. 720–726, 1987.12. Soong T.T., State-of-the-Art Review, Active structural control in civil engineering. Eng. Strct. 10, pp. 74–84, 1988.13. Carotti A., de Miranda F.G., “Active walls” for the antiseismic protection of multistorey r.c. building. Design criteria and feasibility analysis. Proceedings 7th Symp. on Dynamics and Control of Large Structures, Virginia Polytechnic Inst. & State Univ., Blacksburg, USA, pp. 83–98, 1989.14. Carotti A., Active Control of Stress in torsional dynamics of structures under seismic disturbance. Proc. 2nd Int. Conf. Fatigue & Stress, Imperial College, London, UK, pp. 164–179, 1988.15. Carotti A. , de Miranda M. and Turci E. , An active protection system for wind induced vibrations of pipeline suspension bridges. -Proc. 2nd Int. Symp. on Structural Control, Waterloo, Ontario, Canada (1985), Martinus Nijhoff, Amsterdam; pp. 76–104, 1987.16. Carotti A., Automatic control of drift vibrations in steel stacks subject to Benard-Karman vortex discharges. Proc. 5 ICC Int. Conf. Essen, West Germany, pp. 169–174, 1984–17. Guillon N.M., Hydraulic and Electrohydraulic Servo-systems, Vol. I, II, Paris, 1972.18. Morse, Electrohydraulic Servomechanisms, N.Y., 1964.19. Gabosov R., Theory of controllability of dynamical systems, Diff. Uraunemia, 4, 3A, pp. 1575–1583, 1968 (in russian).20. Abgarian K.A., Matrix and Symptotic methods in the theory of linear systems. Nauka Moscow, 1973 (in russian).21. Meirovitch L., Dynamics and Control of Structures, New York, 1990.



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< previous page page_649 next page >Page 649SECTION 9: VIBRATIONS



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< previous page page_651 next page >Page 651Shielding of Structures from Soil VibrationsG.Schmid, N.Chouw, R.LeDepartment of Civil Engineering, Ruhr- University, D-4630 Bochum 1, GermanyABSTRACTWe are proposing a new approach to shield structures from soil vibration. It is based on the dynamic behaviour of a soil layer over a bedrock, not to transmit soil vibration. This can be achieved artificially by a stiff obstacle at a certain depth under the structure’s foundation.A comparison to the shielding effect of a stiff wall constructed as a wave barrier shows the effectiveness of our approach.INTRODUCTIONExcitation of buildings due to soil vibration is occurring more frequently nowadays and is getting more and more attention. One example, is road and rail traffic in densely populated urban areas. To lessen the effect of soil vibration on sensitive equipment and on the well-being of humans, it is necessary to consider these effects in the design of structures.Arrangements like for e.g., rubber bearings that are attached to the foundation could be used to reduce the effects of incoming waves. Another possibility would be to install a wave barrier i.e. a solid



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< previous page page_652 next page >Page 652wall at the surface. Then, the vibration in a certain area behind the barrier will be reduced. Recently air cushions have been used to increase the screening effect, e.g. Massarch [1].Our approach to impede the development and transmission of soil vibration takes advantage of the vibration transmitting behaviour of a soil layer over a bedrock. The numerical calculations are performed using boundary element method [2].SOIL TRANSMITTING BEHAVIOURThe soil considered, has a density of 1800 kg/m3, a Poisson’s ratio ν1 of 0.33, a shear modulus G1 of 53·106Pa and has no material damping. The soil vibration is caused by a horizontal, harmonic unit excitation with a frequency of 16Hz . It acts on a strip surface foundation which is assumed to be rigid and massless. The foundation has a width b of 3m. The vibration transmitting behaviour is represented by the dimensionless amplitude A[−]=u·G1 of the steady-state vibration of the soil surface. u is the displacement related to a harmonic line-load.If the amplitude of the vibration of the soil layer is compared to that of the half-space, it becomes clear that the vibration transmitting behaviour of the soil layer is dependent on its thickness. This is due to the fact that the transmitting behaviour of a soil layer on a bedrock is determined by its lowest eigenfrequency or its corresponding critical thickness Hcrit=C/4f. In the case of horizontal excitation, C is the propagation velocity of the shear wave and Hcrit has a value of 0.268LR. LR=10 m is the Rayleigh wave length.If H<Hcrit, then there is no wave spreading and the soil behaves like a finite system. Only when H> Hcrit, can the wave propagate laterally. H=0.25LR



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< previous page page_653 next page >Page 653and H=0.3LR are slightly below and above the critical thickness and the behaviour is similar to resonance (Fig. 1).If H=0.25LR, then there is no lateral wave spreading but due to the resonance effect, a much larger area of the soil layer vibrates than in the case of H=0.1LR.With increasing thickness of the soil layer it approaches the behaviour of a half-space (see e.g. H =1.0LR in Fig. 1). More details of vibration transmitting behaviour of a soil layer over a bedrock is described by Chouw, Le and Schmid [2].

Figure 1. Influence of the layer thickness on the amplitude of the horizontal vibration at the soil surface



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< previous page page_654 next page >Page 654WAVE IMPEDIMENTUsually the ground is not a soil layer over a bedrock. Even when there is a stiff subsoil, it is seldom located at the desired depth. Nevertheless, impediment of spreading of waves can be achieved artificially by installing a stiff obstacle at a certain depth below the source. Thus we have created an artificial bedrock.Fig. 2 shows the application of the approach in the case of a half-space. The soil, the foundation and the excitation are the same as at the previous section. The artificial bedrock lies at a depth of 0.05LR. It is 1.0LR wide and has a thickness

of 0.1 LR. Its density is 2400kg/m3, its Poisson’s ratio ν2 is 0.2 and has no material damping. The ratio of the shear wave velocity of the artificial bedrock CS2 to that of the surrounding soil CS1, is 12.If there is a bedrock below the foundation (as indicated in Fig. 2 by CS2=∞) the amplitude of its horizontal vibration is reduced significantly. The amplitude of the vertical vibration nearly vanishes. A bedrock of limited size can also entirely screen the surrounding from spreading waves. The soil, only close to the foundation, vibrates.If an artificial bedrock is used, the foundation and soil vibration can be reduced but wave propagation into the surrounding area cannot be totally impeded because the artificial bedrock vibrates. The effectiveness of the artificial bedrock can be improved by increasing its stiffness. Results are presented by Chouw, Le and Schmid [3].



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Figure 2. Impediment of spreading waves by a bedrock and an artificial bedrock



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< previous page page_656 next page >Page 656SHIELDING OF STRUCTURESShielding of buildings from soil vibration can also be achieved by using the method described before. An artificial bedrock has to be installed below the building’s foundations.In Fig. 3 a half-space with an artificial bedrock below the building’s foundation (footing 2) can be seen. The source of the soil vibration is the neighbouring foundation (footing 1) which is loaded by a horizontal harmonic unit excitation with a frequency of 16Hz. The material of soil and foundations and the assumptions are the same as in the second section. The artificial bedrock is the same as in the third section.

Figure 3. Foundation on a half-space with an artificial bedrockThe amplitudes of the foundation and surface vibration with shielding are compared in Figs. 4 and 5 with those of the half-space without artificial bedrock.



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Figure 4. Impediment of the incoming waves by a bedrock



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Figure 5. Impediment of incoming waves by an artifi cial bedrock



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< previous page page_659 next page >Page 659If a real bedrock does exist at a depth of 0.05 LR below the footing , the incoming of waves is entirely impeded. The footing 2 is not excited by the soil. The bedrock is numerically described by a fixed interface, therefore the soil waves can pass underneath it and cause vibration behind footing 2.If an artificial bedrock is installed at the same position, soil vibration can largely be held off.COMPARISON OF THE SHIELDING EFFECTIVENESSUsually, to shield a building from vibration, a wave barrier has to be installed close to it, because the shielding effect is a maximum in the area closely behind the barrier. To compare the shielding effectiveness of an artificial bedrock to that of a wave barrier, a wall consisting of the material of the artificial bedrock is built at a distance d of 0.4LR from the building’s foundation (footing 2). The wall has a width T of 0.1LR and a depth w of 1LR (see Fig. 6).

Figure 6. Foundation on a half-space with a wave barrier



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Figure 7. Influence of the depth a of excitation on the shielding effectiveness of a solid wallIn all investigated cases of varying source depth it was shown that the artificial bedrock is more effective than the solid wall. This can be seen from Figs. 7 and 8. If the excitation occurs horizontally at the soil surface, it is difficult to shield a building using a solid wall. This is clearly indicated in Fig. 9 by comparing the amplitude of the horizontal vibration of the footing 2. The artificial bedrock is equally effective in reducing the vertical and horizontal vibration (heavy dark line) and more effective than the solid wall (dotted line).



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Figure 8. Influence of the depth a of excitation on the shielding effectiveness of an artificial bedrockREFERENCES1. Massarsch, K.R. Ground vibration isolation using gas cushions, Proc. of Second Int. Conf. on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, St. Louis, Missouri, 1991.2. Chouw, N., Le, R. and Schmid, G. Propagation of vibration in a soil layer over bedrock, Engineering Analysis with Boundary Elements, 1991.3. Chouw, N., Le, R. and Schmid, G. An approach to reduce foundation vibrations and soil waves using dynamic transmitting behaviour of a soil layer, Bauingenieur, 1991.



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Figure 9. Shielding effectivness of an artificial bedrock and a solid wall in comparison



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< previous page page_663 next page >Page 663Vertical Vibration of a Rigid Plate on a Continuously Nonhomogeneous SoilS.Savidis, C.Vrettos, B.FaustTechnical University of Berlin, Geotechnical Engineering Institute, 1000 Berlin 12, GermanyABSTRACTThe vertical response of a rectangular rigid foundation placed on a linear-elastic, nonhomogeneous half-space and subjected to time-harmonic vertical excitation is studied. The mixed boundary value problem is solved by means of the method of subdivision of the contact area. The influence functions for the sub-regions are determined from the time-harmonic Green’s functions of the particular nonhomogeneous soil model. The frequency dependent stiffness functions are presented for selected geometries and are compared with the corresponding solutions of the homogeneous half-space. A numerical example illustrates the effect of the soil nonhomogeneity.INTRODUCTIONThe analysis of foundation vibration has experienced a tremendous growth over the last decades through the introduction of the lumped-parameter model and the development of efficient numerical and analytical techniques, e.g. Gazetas.1 In engineering applications, however, some uncertainities in the characterization of the free-field motion and the soil properties still remain. One factor which requires particular attention is the variation of the soil stiffness with depth which occurs even in structurally homogeneous deposits of sand or clay, Richart et al.2 One way to circumvent the soil nonhomogeneity consists in choosing an equivalent value of the shear modulus assigned to the soil conditions at a particular depth. While for free-field surface wave propagation this approximation leads to simple expressions for the equivalent shear modulus, difficulties arise when considering foundation vibration: The equivalent shear modulus is expected to depend on vibration mode and be different for the stiff-



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< previous page page_664 next page >Page 664ness and the damping constant of the foundation-soil system, Holzlöhner,3 Waas & Werkle.4The dynamic response of arbitrarily shaped rigid foundations can be determined by standard numerical methods, such as the method of sub-division of the contact area, Savidis &; Richter,5 Wong &: Luco,6 or the boundary element method, Kobayashi.7 The key to the solution when considering nonhomogeneous soils is the derivation of the Green’s functions for the particular nonhomogeneous soil medium. This can be accomplished by using either the solutions for a medium consisting of a stack of thin layers developed by Waas8 and Kausel9 or the solution for a prescribed continuous shear modulus depth-variation recently developed by Vrettos.10 Although the first is more flexible in modelling an arbitrarily layered soil stratum, the latter is more appropriate for continuously nonhomogeneous soils and has some advantage with respect to nondimensional parametric studies.In the sequel the dynamic stiffness function for vertical vibration of a rigid massless plate will be derived for a nonhomogeneous half-space of constant density and Poisson’s ratio and shear modulus varying sublinear with depth (z) according to 1−e−βz. This model is chosen so as to describe uniformly deposited cohesionless sands.ANALYSIS PROCEDUREConsider a rigid massless rectangular plate of contact area S=2a×2b, a<b, resting on the surface of a linear-elastic, isotropic half-space of constant mass density and Poisson’s ratio ν and shear modulus G(z), as shown in Figure 1.

Figure 1: Plate on nonhomogeneous soil



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< previous page page_665 next page >Page 665The depth-variation of the shear modulus is given by

G(z)=G0+(G∞−G0)(1−e−βz), 0<G0≤G∞ (1)where G0 and G∞ are the shear moduli at the surface and at infinite depth, respectively, β is a constant of the dimension of inverse length. This plate is excited by a time-harmonic vertical force P=P0eiωt, where ω is the circular frequency, t

is time and . Making use of the lumped parameter concept it is possible to define an equivalent effective dynamic spring kz and a radiation (wave propagation) dashpot coefficient cz so that

(2)where ω is the complex vertical displacement of the plate.

and cz are functions of the frequency ω as well as of the foundation-soil contact area and the elastic properties of the half-space. They are determined herein by using the computer program DYBAST, Sarfeld et al.10 The method of analysis is reviewed elsewhere, cf. Savidis & Sarfeld,11 and only a brief description will be given here: The contact area S is divided into a finite number of rectangular subregions with uniformly distributed pressure over each subregion. Further, it is assumed that the displacement of each subregion is represented by the one at the center of the subregion. Then, the unknown contact pressure under the rigid foundation is determined by the equation

(3)where N is the number of subregions, ωi denotes the displacement of the subregion i, pk is the magnitude of the uniformly distributed pressure on the subregion k and (xi, yi) and (xk,yk) are the coordinates of the centers of the

subregions i and k, respectively. The influence function is derived by integrating the surface Green’s function Uzz over the area Sk of the subregion k. The surface Green’s function Uzz(x, y) relates the vertical displacement field at a distance (x, y) to a unit vertical point load at the origin (0,0). For the nonhomogeneous soil medium treated herein Uzz is derived by Vrettos10 by analytical techniques. The limiting case of a homogeneous half-space is recovered by setting G0=G∞ in equation (1). Imposing the displacement boundary condition for rigid foundation

ω(x,y,0)=∆z=const. (4)



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< previous page page_666 next page >Page 666at the N points on the contact area S leads to a system of linear simultaneous equations for the unknown contact pressures pk which can be solved in a straightforward manner. Once the contact pressures are obtained the desired complex-valued force-displacement relationship for the rigid foundation is calculated by

(5)It should be noticed that the approximate approach outlined above i) does not consider the influence of shear tractions acting on the contact area and ii) may not describe in detail the singularities of the surface tractions along the boundary of the rigid plate. However, the accuracy of the results should be sufficient for most practical purposes.NUMERICAL RESULTS AND DISCUSSIONSolutions were obtained for two different geometrical configurations b/a= 1 and b/a=2 for Poisson’s ratio ν=1/3 and for two shear modulus depth-profiles. The two profiles have the same degree of nonhomogeneity

(6)but different values of the nonhomogeneity gradient parameter θ which is defined by, Vrettos10

(7)

where

The two soil profiles are given by and θ=7.65 and θ= 19.13, respectively. For a/b=1 the contact area has been discretized by means of 64 equal square subregions while 128 subregions have been chosen for b/a=2. The dynamic

stiffness functions have been normalized with respect to the static stiffness of a rectangular plate resting on a

homogeneous half-space of shear modulus G0. The static stiffness is determined numerically for the limiting case

ω=0 using the program DYBAST: for b/a=1 and for b/a=2. The normalized dynamic stiffness functions

(8)are depicted in Figures 2 and 3 versus the nondimensional frequency

(9)



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Figure 2: Nondimensional dynamic spring Kz versus the nondimensional frequency a0 for two plate geometries. The

solid lines are for the shear modulus depth-profile equation (1) with and θ=7.65 (I) and θ=19.13 (II), resp. The dashed lines are for a depth-profile with constant shear modulus G0. The Poisson’s ratio v=1/3.


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Figure 3: Nondimensional radiation dashpot Cz versus the nondimensional frequency a0 for two plate geometries. The

solid lines are for the shear modulus depth-profile equation (1) with and θ=7.65 (I) and θ=19.13 (II), resp. The dashed lines are for a depth-profile with constant shear modulus G0. The Poisson’s ratio ν=1/3.


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< previous page page_669 next page >Page 669It can be seen that as the nonhomogeneity gradient parameter θ increases the solution approaches the corresponding solution of the homogeneous half-space with shear modulus G0.A numerical example should demonstrate the influence of soil nonhomogeneity on the dynamic response of a rigid rectangular foundation forced to vibrate in the z—axis:Frequency: f=50HzGeometry: a=1m, b=2m, b/a=2

Soil: ν=1/3, , G0=25MN/m2 and β=0.35 m−1 (profile I) and β=0.14m−1 (profile II). The two shear modulus depth profiles correspond to θ=7.65 (profile I) and θ=19.13 (profile II), respectively, and are plotted in Figure 4.

Figure 4: Shear modulus depth-profiles given by equation (1) with and β=0.35m−1 (I) and β=0.14m−1 (II), resp.

From equation (9) we find a0=2.66. From Figures 2 and 3 and equations (8) we finally obtain

and cz=3.13MN s/m for the profile I and cz=3.19MN s/m for the profile II, respectively.CONCLUSIONSThe dynamic stiffness functions for the vertical vibration of a rectangular rigid plate on a continuously nonhomogeneous soil deposit have been presented. The solution procedure followed can be easily extended to treat all possible vibration modes as well as to solve problems of dynamic interaction between rigid or flexible plates on nonhomogeneous soils. The soil model proposed allows an immediate engineering application. However, a more refined study should also incorporate the horizontal and vertical variation of the soil stiffness resulting from the pressure distribution due to the foundation static load.



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< previous page page_670 next page >Page 670REFERENCES1. Gazetas, G. Foundation Vibration. Chapter 15, Foundation Engineering Handbook, (Ed. Fang, H.Y.), pp. 553–593, Van Nostrand Reinhold, 1991.2. Richart, F.E.,Jr., Hall, J.R.,Jr. and Woods, R.D. Vibrations of Soils and Foundations, Prentice-Hall, Englewood Cliffs, N.J., 1970.3. Holzlöhner, U. The Use of an Equivalent Homogeneous Half-Space in Soil-Structure Interaction Analysis, SMiRT 5, M 10/3, Berlin 1979.4. Waas, G. and Werkle, H. Schwingungen von Fundamenten auf inhomogenem Baugrund, VDI-Berichte 536, pp. 349–366, 1984.5. Savidis, S. and Richter, T. Dynamic Interaction of Rigid Foundations, Vol. 2, pp. 369–374, Proc. 9th Int. Conf. Soil Mech. Found. Eng., Tokyo, 1977.6. Wong, H.L. and Luco, J.E. Dynamic Response of Rigid Foundations of arbitrary shape, Earthquake Eng. Struct. Dyn, Vol. 4, pp. 579–587, 1976.7. Kobayashi, S. Elastodynamics. Chapter 4, Boundary Element Methods in Mechanics (Ed. Beskos, D.E.), pp. 192–255, North-Holland, Amsterdam, 1987.8. Waas, G. Linear Two-Dimensional Analysis of Soil Dynamics Problems in Semi-Infinite Layered Media, Ph.D. Thesis, University of California, Berkeley, 1972.9. Kausel, E. An Explicit Solution for the Green’s Functions for Dynamic Loads in Layered Media, Research Report R81–13, Publ. No. 699, Dept. of Civil Eng., M.I.T., Cambridge, Massachusetts, 1981.10. Vrettos, C. Time-Harmonic Boussinesq Problem for a Continuously Nonhomogeneous Soil, Earthquake Eng. Struct. Dyn., Vol. 20, 1991.



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< previous page page_671 next page >Page 67111. Sarfeld, W., Savidis, S. and Faust, B. DYBAST-A Computer Program for the Dynamic Calculation of Bases and Structures, Internal Report, Technical University of Berlin, 1990.12. Savidis, S. and Sarfeld, W. Verfahren und Anwendung der dreidimensionalen dynamischen Wechselwirkung, pp. 47–78, Vorträge der Baugrundtagung, Mainz, 1980.



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< previous page page_673 next page >Page 673The Influence of Thickness Variation of Subway Walls on the Vibration Emission Generated by Subway TrafficR.Thiede, H.G.NatkeCurt-Risch-Institut for Dynamics, Acoustics, and Measurement Technique, University of Hannover, GermanyABSTRACTThe wall thickness of a tunnel is an essential parameter for the dynamic response in the vicinity of traffic and subway tunnels. The results from numerical parametrical investigations are presented to clarify the effect of this parameter. In addition to the variation of the wall thickness, a variation of the excitation frequency is also carried out.INTRODUCTIONIn the vicinity of traffic and subway tunnels, the inhabitants of nearby houses are often disturbed by the vibrations and the indirect airborne noise (radiated from vibrating structures due to structure-borne noise). The cause of this annoyance is to be found in vibrations of the tunnel structure which are excited by the traffic in the tunnel. These dynamic interaction forces generate waves which propagate through the surrounding soil to the nearby houses, where they excite the disturbing vibrations of parts of the buildings.The vibrations of the tunnel as well as the excitation of waves in the surrounding soil are affected by a number of parameters of the excitation, of the tunnel, and of the soil. Thus the dynamic responses of the tunnel and of the surrounding soil can be influenced by the modification of these parameters. Such parameters are, for example,– frequency of “excitation”,– shape of the tunnel cross section,–wall thickness of the tunnel,–inside dimensions of the tunnel,–material of the tunnel,



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< previous page page_674 next page >Page 674 – depth position of the tunnel,– dynamic soil parameters.Extensive comparative numerical investigations revealed the influences of the various parameters. The results are presented in the papers of Tahadjodi et al. [1] and [2]. They show that the wall thickness of the tunnel essentially affects the dynamic response of the tunnel and of its vicinity. For this reason, information on the investigations concerning this parameter is to be provided here.HALFSPACE MODELThe tunnel represents a linear structure in the soil, that is excited in the first approximation by a line excitation from the traffic (instead of looking for the vehicle-road interaction). Additionally, the dynamic responses will be determined only in the short-range field of the tunnel. Thus the actual spatial 3-dimensional problem can be reduced to a spatial 2-dimensional one, i.e. to a slice orthogonal to the longitudinal axis of the tunnel. The idealization can be carried out extensively, because no quantitative results for given real cases will be obtained. Within the scope of the parameter investigations, it is sufficient to work out the influence of the parameter (thickness of the tunnel wall) on the dynamic responses of the tunnel structure and at the soil surface in a relative ratio. That is why tunnel fittings (roadway plates, railroad track structures, and so on) are neglected.It is assumed that the soil will behave linearly elastically and that it is homogeneous and isotropic. Because all the boundaries of the soil slice, except the free surface, lie at infinity, propagating waves have to be modelled. Finite element models of the finite model section of the infinite soil slice, which are treated here for numerical investigations with the FEM, must be modelled as being nonreflecting at their artificial boundaries, thereby taking the geometrical damping into account.In traffic tunnels, the vehicle-tunnel interactions are the cause of the tunnel vibrations. Investigations show that essential frequencies of this interaction occur in subway tunnels up to about 80Hz and in traffic tunnels up to about 60Hz (see Rücker [3], Natke [4]).



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< previous page page_675 next page >Page 675On the basis of the details given above, a model is chosen for the numerical parametrical investigations, as represented in Fig. 1.Horizontal and vertical active dampers are modelled at the artificial inserted lateral and lower boundaries of the model (see Lysmer et al. [5], Elmer et al. [6]) while in the nodes in the axis of symmetry only static equivalent supports (Fig. 2) are permitted to be included (Tahadjodi [1]).In the area of the tunnel cross-section, the element net is shaped in such a way that all the calculations of the parameter variation of the thickness of the tunnel wall can be performed without net modification (the internal diameter of the tunnel is always constant), and only the appropriate material data can be assigned case by case to the particular element rings. In order to have at least 5 elements for each wave length [6], only frequencies of a maximum of 40Hz can be determined with the given material data from the soil and element dimensions of 0.75 m. For higher frequencies (up to 80Hz) the element dimensions must be reduced to 0.375m.The tunnel is excited at its base with frequencies from 10 to 80Hz in steps of 10Hz.RESULTSAll the investigations with the FE-model of Fig. 1 are performed with the help of the FEM-program ADINA.In order to obtain an insight into the vibration velocity distribution at the contact surface between the tunnel structure and the surrounding soil and at the free surface of the soil, the maximum vibration velocities for various thicknesses of the tunnel walls are calculated in preliminary investigations with an excitation frequency of f=20Hz and the soil and tunnel data specified in Fig 1. The results are plotted in Figs. 3 and 4.Fig. 3 shows clearly that in the case of small wall thicknesses, much larger vibration velocities occur in the area of the tunnel base than in the area of the tunnel head, i.e. as a result of the comparatively small flexural stiffness of the tunnel wall the excitation energy essentially passes into the soil near the points of excitation, and it is not able to excite remarkable vibrations in parts of the tunnel which are at some distance. With increasing thickness of the tunnel wall the vibration velocity



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Figure 1. FE-model for the numerical parameter investigations


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Figure 2. Static equivalent supports in the axis of symmetry



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Figure 3. Vertical vibration velocities in the contact surface between the tunnel structure and the soil (circular tunnel)


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Figure 4. Vertical vibration velocities at the free surface of the soil (circular tunnel)



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< previous page page_678 next page >Page 678decreases continuously at the tunnel base and, on the other hand, it increases at the tunnel head as a consequence of the increasing flexural stiffness. In that way the excitation energy passes into the surrounding soil after being distributed more and more over the tunnel circumference. The increase of the vibration velocity at the tunnel head does not take place continuously, but it decreases from a definite thickness of the tunnel wall onwards. The reason for this behaviour is that not only the flexural stiffness but also the mass of the tunnel wall increases with the increasing thickness of the wall. At the tunnel head the vibration velocity increases as long as the effect of the increasing flexural stiffness, which increases the vibration velocity, preponderates over the effect of the increasing mass, which decreases the vibration velocity. From a particular wall thickness onwards, however, the effect of the mass preponderates, so that now the vibration velocity also decreases continuously at the tunnel head, just as in the other tunnel areas. In the course of this the amplitudes of the vibration velocity equate more and more over the whole circumference of the tunnel, till at very large wall thickness the entire tunnel cross-section oscillates as a rigid body.At the free surface of the soil Fig. 4 shows very different dynamic responses directly above the tunnel head and in the area beside the tunnel. Above the tunnel head the increasing thickness of the tunnel wall causes an increase of the vibration velocity, as was expected in consequence of the increasing vibration velocity at the tunnel head. But at the free surface the same effect cannot be identified as at the tunnel head: there is no decrease of the vibration velocity from a particular wall thickness onwards. But the rate of increase of the vibration velocity at the free surface decreases with the increasing thickness of the tunnel wall, so it can be expected that with even thicker walls a decrease of the vibration velocity will occur above the tunnel head. In Fig. 5 an effect of this kind appears clearly above a rectangular tunnel. At the free surface the situation inverts very quickly in the lateral zone of the tunnel when compared with the zone above the tunnel head, i.e. with increasing thickness of the tunnel wall the vibration velocity decreases. Apart from some irregularities, this holds true the more the lateral distance to the tunnel grows (see Fig. 5). The nonuniform decrease of the vibration velocity at the free surface of the ground may be mainly attributed to the fact that the waves which are excited in the soil at the tunnel base are screened by the tunnel



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Figure 5. Vertical vibration velocities at the free surface of the soil (rectangular tunnel)


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Figure 6. Influence of the thickness of the tunnel wall and of the excitation frequency on the vibration velocities at the soil surface (zone 1)



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< previous page page_680 next page >Page 680during their direct propagation toward the free surface of the soil. Thus they strike this surface diagonally at some lateral distance, and there they superimpose with the already partly decayed vibration velocity of the waves, which propagates at the free surface from the area above the tunnel head. This means that the amplitudes of the waves become enlarged or reduced due to interferences.The influence of various thickness of the tunnel walls on the vibration velocity at the free surface of the soil is also of interest when an excitation is chosen that depends on frequency. In Figs. 6 to 8 the calculated maximum amplitudes of the vibration velocities are plotted in three zones at the free surface of the soil above and beside the tunnel. The partition into zones is marked in Fig. 1. Fig. 6 shows that here, too, in zone 1 (above the tunnel) the vibration velocities increase with increasing thickness of the tunnel wall at nearly all the studied excitation frequencies. But at least for frequencies

less than 40 Hz the results indicate that above a certain wall thickness the vibration velocities decrease again. In addition, the vibration velocities decrease on average with the increasing excitation frequency. In zones 2 and 3 (Figs. 7 and 8) for nearly all the excitation frequencies it holds true that the vibration velocity at the free surface of the soil decreases with increasing thickness of the tunnel wall. Moreover, in zone 3 there is a tendency for the dynamic responses at the free surface to decrease with the increasing excitation frequency. In zone 2 an effect of this kind cannot be observed to the same extend.CONCLUSIONSThe thickness of the tunnel wall is an essential influencing factor for the dynamic response at the free surface of the soil. The results presented from numerical parametrical investigations show that for the zone directly above the tunnel the statements are different from those in the zones beside the tunnel. The influence of a variation of the wall thickness with various excitation frequencies is also shown.ACKNOWLEDGMENTAll the investigations have been carried out mainly by Dr.-Ing. A.Tahadjodi at the Curt-Risch-Institut for Dynamics, Acoustics, and Measurement Technique, University of Hannover, within the scope of a research program funded by the Deutsche Forschungsge-



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Figure 7. Influence of the thickness of the tunnel wall and of the excitation frequency on the vibration velocities at the soil surface (zone 2)


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Figure 8. Influence of the thickness of the tunnel wall and of the excitation frequency on the vibration velocities at the soil surface (zone 3)meinschaft (DFG). The computations have been carried out with the help of the FEM-program ADINA at the Regionales Rechenzentrum für Niedersachsen (RRZN) at the University of Hannover.



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< previous page page_682 next page >Page 682REFERENCES1. Tahadjodi, A. Beitrag zur Optimierung von Tunnelbauwerken in dynamischer Hinsicht—Bemessungsdiagramme—, Diss. Univ. Hannover, 1989.2. Tahadjodi, A. and Thiede, R. Schwingungsuntersuchungen von Tunnelbauwerken und Auswirkungen in der Umgebung, Final Report for the Deutsche Forschungsgemeinschaft (DFG), Institute Report of the Curt-Risch-Institute, Univ. Hannover, CRI-F-1/1990.3. Rücker, W. Ermittlung der Schwingungserregung beim Betrieb schienengebundener Fahrzeuge in Tunneln sowie Untersuchung des Einflusses einzelner Parameter auf die Ausbreitung von Erschütterungen im Tunnel und dessen Umgebung, Diss. D 83, TU Berlin, 1979.4. Natke, H.G. Ursachenfindung zur Lärm—und Erschütterungsentstehung in der Nachbarschaft des Elbtunnels, STUVA Forschung+Praxis, No.23, pp. 166–172, 1980.5. Lysmer, J. and Kuhlemeyer, R.L. Finite Dynamic Model for Infinite Media, Journ. Eng. Mech. Div., ASCE, Vol. 95, pp. 859–877, 1969.6. Elmer, K.-H., Natke, H.G. and Thiede, R. Modelling in Soil Dynamics by a Finite Domain with Respect to Transient Excitation, in Ground Motion and Engineering Seismology (Ed. Cakmak, A.S.), pp. 347–364, Proceedings of the 3rd Int. Conf. on Soil Dynamics and Earthquake Engineering, Princeton, N.J., USA, 1987. Elsevier, Amsterdam, Computational Mechanics Publications, Southampton, 1987.



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< previous page page_683 next page >Page 683Vibration Isolation by an Array of PilesB.Boroomand, A.M.KayniaDepartment of Civil Engineering, Isfahan University of Technology, Isfahan, IranABSTRACTA general analytical solution is presented for the dynamic analysis of pile-soil-pile interaction for vertical piles in a homogeneous soil stratum. The important characteristic of this solution is that by using a Fourier expansion of variables on the circumference of each pile the nonuniform variation of pile-soil tractions are accounted for. The application of this model to the problem of vibration isolation by a row of closely-spaced piles is then demonstrated. A number of results are presented to highlight the influence of certain parameters on the effectiveness of isolation.INTRODUCTIONIsolating a structure from disturbing vibrations generated by an external source and transmitted through the ground has been facing foundation engineers for many years. It is certainly most desirable to inhibit the vibrations at the source. This can be accomplished in certain cases, such as in machine foundations, by installing mechanical isolation devices. However, for other vibration problems, where this scheme is not adequate or not possible, such as the traffic-induced vibrations, foundation isolation might be a viable alternative. An obvious solution for such problems is the use of an open trench in the path of wave propagation. Indeed, experimental studies have supported this idea [1,2]. Also numerical formulations, such as the boundary element method, have been applied to this problem resulting in a better understanding of the effects that different parameters have on the performance of open trenches (eg. Dasgupta et al. [3]).Another solution to the vibration isolation problem is the use of a row of closely installed piles. In fact, for difficult soil conditions piles provide a better alternative to open trenches. The idea of using a row of piles as wave barriers was



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< previous page page_684 next page >Page 684seemingly proposed by Richart et al. [2], Subsequently, Woods et al. [4] undertook an experimental study using holography to investigate the effectiveness of an array of cylindrical cavities, as well as a row of rigid inclusions, as vibration barriers. They showed that for wavelengths less than six times the diameter of the cavities and cavity separations less than a quarter of the wavelength a satisfactory isolation could be obtained. Also Liao and Sangrey [5] conducted experiments on acoustic waves and confirmed the earlier observations. A few analytical models have also been proposed to solve the problem. Aviles and Sanchez-Sesma [6, 7] studied the scattering problem by a row of infinitely long rigid piles in an unbounded space under incident SH, SV and P waves and showed that satisfactory wave screening is obtained if the wavelength of the incidence field is between one to four times the diameter of the piles. They also investigated the scattering of incident Rayleigh waves for a row of piles in an elastic half space.The objective of the present paper is to propose an analytical model for the dynamic analysis of closely spaced piles under steady-state vertical vibrations. This formulation, which properly accounts for the scattering of waves by piles, is used to solve a few vibration isolation problems by piles and high-light the influence of certain parameters on the effectiveness of isolation.ANALYTICAL MODELIn the present study it is assumed that under vertical vibrations the horizontal displacements are negligible. This approximation has been used for the analysis of pile groups [8] and has been shown to provide satisfactory results. The piles are assumed to be elastic rods with elasticity modulus Ep, mass density ρp, length Lp and diameter d. The soil medium is considered to be a homogeneous stratum with depth H, shear and pressure wave velocities Vs and Poisson’s ratio νs mass density ρs and hysteretic damping ratio βs.

Figure 1. The soil stratumConsider the soil stratum in Figure 1. For a steady-state vertical vibration with frequency Ω the vertical displacement can be expressed as w(r,z,θ)eiΩt. The differential equations of the soil, considering zero horizontal displacements, then reduce to the following equation

(1)



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< previous page page_685 next page >Page 685The solution of this equation satisfying the radiation condition as well as the stress and displacement boundary conditions at z=0 and z=H can be written as

(2)where Km is the modified Bessel function of order m, Anm and Bnm are the coefficients (to be determined for a given condition), and hn and qn are given by

(3)

(4)If there are N vibrating piles in the medium one can assume that the displacement field is the superposition of the displacements generated by the individual piles. Therefore, Equation (2) is used to obtain the displacements caused by one N piles at all points on the piles-soil interface. In order to ensure pile-soil displacement compatibility; that is, a uniform soil displacement around each pile, the condition that the circumferential variation of displacements should be normal to all circumferential modes, I.e. cos (m′θ) and sin (m′θ), except for m′=0, is imposed on the displacements [9]. Also, Equation (2) is used to obtain the shear stresses on the circumference of the piles. The final stage of the work is to consider the equilibrium of the piles, under the pile-soil tractions, by solving the differential equations of the piles. More details of this formulation are given in reference [9].RESULTSA number of results are presented in this section to investigate the performance of an array of piles as wave barriers. As shown in Figure 2 the soil stratum is excited by a harmonically

Figure 2. The soil stratum and the row of piles



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< previous page page_686 next page >Page 686vibrating vertical force at a distance 1 from the piles. The distance between the center of adjacent piles is denoted by s and the frequency of vibration is defined by the nondimensional factor ao=Ωd/Vs.The quantity of interest in this study is the ratio between the displacements behind the barrier, after and before pile installation. The displacements are complex quantities which will be represented by their absolute values. Therefore, if the two displacements at a point, before and after pile installation are denoted by wo and w, then displacement ratio is

. Then the effectiveness of the isolation could be taken as (1−R).The presented results correspond to a pile-soil system with νs=0.3, βs=0.05, ρs/ρp=0.7, H/d=40, Lp/d=20 and 1/d=10. Two types of piles are considered: stiff piles with Ep/Es=1000 and flexible piles with Ep/Es=100. In all cases there are N=8 piles in the array.Figure 3 shows the variation of R along different lines parallel to the x axis for ao=1.0 and s/d=2 (see Figure 2 for the orientation of the x and y axes). The lines correspond to y=0 (i.e. the x-axis), y=2d, 4d and 6d. The piles in this case are stiff.

Figure 3. Variations of R for stiff piles, s/d=2 and ao=1.0The figure clearly reveals a considerable reduction of vibration in a wide area behind the piles (about 90 percent reduction on the average). Figures 4 and 5 show similar results for s/d=2.5 and s/d=3.0, respectively. The frequency and pile stiffness in these figures are the same as those in Figure 3. These figures indicate a remarkable reductions in vibrations, too. As expected, increasing the pile spacing reduces the effectiveness of isolation. This can be verified by comparing the results in Figures 3, 4 and 5. In fact, as previous studies have concluded, the effectiveness of isolation by piles is dictated by the free space between the piles, to the extent that if the piles are too



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Figure 4. Variations of R for s/d=2.5 and ao=1.0

Figure 5. Variations of R for s/d=3.0 and ao=1.0wide apart almost no isolation is gained.To show the influence of frequency on the effectiveness of isolation, variations of R for three different values of ao(0.5, 1.0 and 1.5) for an array of stiff piles with s/d=2.5 and 1/d=20 are plotted in Figure 6. This figure suggests that, for this arrangement of piles and parameters, as frequency increases the isolation effectiveness reduces; as if more energy can pass through the spaces between the piles.To show the influence of pile stiffness on the isolation effectiveness Figure 7 displays the variations of R for flexible piles as well as for stiff piles for ao=1.5. This figure clearly shows that flexible piles are not as effective as stiff piles. In fact with flexible piles the vibration isolation is practically restricted to a close neighbourhood of the piles.Finally to give a more clear picture of the isolation phe-



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Figure 6. The influence of frequency on vibration isolationnomenon Figure 8 shows a contour diagram of the displacement ratio behind a row of piles. This figure corresponds to


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ao=1.5, s/d=2 and Ep/Es=1000. The figure depicts the plan diagram of the isolated area and shows that the isolation effectiveness diminishes from the centerline (x-axis) in the circumferential direction. It is interesting to note that the displacements in front



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Figure 7. Variation of R for stiff and flexible piles for ao=1.5

Figure 8. contour diagram for stiff piles, ao=1.5 and s/d=2.0



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Figure 9. contour diagram for flexible piles, ao=l.5 and s/d=2of the barrier are magnified after pile installation. Figure 9 presents a similar contour diagram for the case where Ep/Es=100. Other parameters are the same as those in Figure 8. This figure now more clearly shows the previously stated fact that flexible piles are not as effective as stiff piles for vibration isolation.REFERENCES1. Woods, R.D. Screening of Surface Waves in Soils, J. Soil Mech. and Found. Eng., ASCE, Vol. 94, No. 4, pp 951–979,1968.2. Richart, F.E., Hall, J.R. and Woods, R.D. Vibrations of Soils and Foundations, Prentice-Hall, Englewood Cliffs, NJ, 1970.3. Dasgupta, G., Beskos, D.E. and Vardoulakis, I.G. 3-D Analysis of Vibration Isolation of Machine Foundation, Vol.4, pp 59–73, Proc. 10th Boundary Elements Conf., Berlin, Germany,1988.4. Woods, R.D., Bornett N.E., Sagesser R. HOLOGRAPHY-A New Tool for Soil Dynamic, J. Geotech. Eng., ASCE, Vol. 100, No. GT 11, pp 1231–1974.5. Liao, S. and Sangrey, D.A. Use of Piles as Isolation Barriers, J. Geotech. Eng., ASCE, Vol. 104, No. 9, pp 1139–1152, 1978.



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< previous page page_691 next page >Page 6916. Aviles, J. and Sanchez-Sesma, F.J. Foundation Isolation From Vibrations Using Piles as Barriers, J. Eng. Mech. ASCE, Vol. 114, No. 11, pp 1854–1870, 1988.7. Aviles, J. and Sanchez-Sesma, F.J. Piles as Barriers for Elastic Waves, J. Eng. Mech. ASCE, Vol. 109, No. 9, pp 1133–1146, 1983.8. Nogami, T. Dynamic Group Effect of Multiple Piles Under Vertical Vibration, Proc. Third ASCE Eng. Mech. Specialty Conf., Austin, Texas, 1979.9. Boroomand, B. and Kaynia, A.M. Stiffness and Damping of Closely Spaced Pile Groups, Proc. Soil Dyn. Earthq. Eng. 91, Karlsruhe, Germany 1991.



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< previous page page_693 next page >Page 693Numerical Modelling of Stability Cases for Caisson-Type Breakwaters without Through-flowE.Stein, M.LengnickInstitute for Structural and Numerical Mechanics, University of Hannover, D-3000 Hannover, GermanySUMMARYCaisson-type breakwaters represent essential forms of constructions in coastal protection. They have especially proved themselves in deep water applications, in cases where short construction times are aimed at, and where the supply and transport of crushed rock is difficult. The dynamic stability behaviour of these special coastal protection structures is being investigated both experimentally and numerically in the Special Research Field 205, “Coastal Engineering”, at the University of Hannover, and at the Technical University of Braunschweig.INTRODUCTIONIn this contribution, results are presented which were obtained during 1989 and 1990 from the numerical modelling of a caisson-type breakwater. The first questions to be answered concerned the physical influencing factors which contribute most to the failure of caisson-type breakwater. These fundamental questions could only be clarified in collaboration with researchers involved in experimental investigations of the caisson-type breakwater using the Large Wave Channel. In this context, clarification of the following points was sought in the first instance:1. What is to be understood by “failure” of the caisson- type breakwater?2. How can the complex system of the “caisson-type breakwater” be simplified for the mathematical-numerical treatment so that the processing reduces to well-defined initial boundary value problems without neglecting essential physical influences?3. What results can be sensibly expected from the numerical breakwater model?4. Questions concerning the modelling of the rubble bed and the sand.5. Should the numerical breakwater model represent a purely deterministic model or should an attempt be made to account for probabilistic influences in the model, such as, e.g. the wide range of stochastic fluctuations in loading as a result of air absorption?



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< previous page page_694 next page >Page 694Caisson-type breakwaters without throughflow exhibit three principle modes of global failure: sliding of the breakwater on the coarse-grained rubble bed, tilting of the breakwater, and the formation of shearing surfaces in the subgrade which is influenced by the wave-induced flow field within the rubble layer and the basic foundation comprised of the water-saturated sand half-space. Field observations indicate that there is generally a high degree of interaction between these stability cases.THE BREAKWATER MODEL FOR NUMERICAL CALCULATIONSThe numerical model of the caisson-type breakwater is illustrated in Fig. (1). The calculations are carried out for the plane deformation condition. The caisson is initially assumed to be a rigid body. At a later stage, more realistic material laws for concrete under severe impacts should lead to an improvement of the model. The rubble layer between the caisson and the water-saturated sand is neglected at first in setting up the model. A homogeneous and isotropic half-plane is assumed. The Cambridge-Clay plasticity model was employed as the material law for the water-saturated sand. It is intended at a later stage to apply a formulation in terms of multi-surface plasticity in order to describe cyclic loading and unloading processes. The horizontal loading of the caisson caused by waves breaking on the structure, as well as the distribution of pore water pressure at the base joint of the breakwater are based upon measurements made by Professor Partenscky and colleagues (Franzius-Institut, University of Hannover) in the Large Wave Channel in Hannover-Marienwerder. Interaction between the caisson, the breaking waves and the pore water pressure field in the subgrade is not considered at this stage.

Figure 1. Numerical model of the caisson-type breakwater.



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< previous page page_695 next page >Page 695SLIDING OF THE CAISSON-TYPE BREAKWATERIn contrast to many previous numerical models, in which a rigid connection between the caisson-type breakwater and the rubble bed was assumed, the present model includes effective pressure and frictional forces in the contact zone dependent on the distance and relative tangential displacement of the contact surfaces. The friction law developed in [10] is based on the following assumptions:1. there is a non-linear relationship between the contact surfaces and the pressure load,2. the shear stress over the contact surfaces is dependent on the contact pressure.TILTING OSCILLATIONS OF THE CAISSON-TYPE BREAKWATERTilting oscillations of the caisson-type breakwater were first investigated on the basis of simplifying assumptions . On the one hand, the caisson was assumed to be a rigid body and on the other, the sand filling of the caisson was only treated as an additional mass of concrete. Consequently, the interaction between the sand filling and the caisson-type breakwater was neglected. Furthermore, simplifications were made concerning the spatial and temporal distributions of the horizontal and vertical pressures. Accordingly, a simplified spatial line load is applied to the system in the form of a sine wave impulse combined with a time-displaced, linear pore water pressure distribution, likewise in the form of a sine wave impulse.Sinking on the seaward side of the caisson-type breakwater was observed under certain boundary conditions. Oumeraci et al. report on this in [9]. This effect can be explained by material transport and/or soil liquefaction. Up to now, it has not been possible to take account of this effect in the numerical breakwater model.EXCESS PORE WATER PRESSURE DISTRIBUTION IN THE SOILThe calculation of excess pore water pressures resulting from the sudden application of unit area loads can be regarded as a first step towards coupling the caisson-type breakwater and the pore water pressure distribution in the soil. These calculations were carried out using the finite element program PLASCON, details of which are described in [11]. The program is suitable for calculating elasto-plastic consolidation processes in fine-grained soils. For the fluid flow, a linear relationship was assumed between the gradients of the pore water pressure and the flow velocity in accordance with Darcy’s filter law. Since the flow velocities of the fluid in the rubble layer deviate considerably from those in the sand, which in turn are highly dependent upon the depth of the sampling point beneath the base of the breakwater, the assumption of a “slow” groundwater flow, as is presupposed by adopting the Darcy filter law, can only be considered as an approximation for large areas of the sand half-plane. The actual flow velocities—particularly in front of the breakwater—will necessitate the inclusion of higher flow velocities in the numerical model. A further problem at the present time is how to describe cyclic pore water pressure variations in the sand. An increase in the pore water pressure resulting from cyclic loading of the soil due to waves and movement of the caisson itself cannot be reproduced at present. The inclusion of these physical effects in the numerical breakwater model is a task to be dealt with during a further research period.



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< previous page page_696 next page >Page 696CAMBRIDGE-CLAY PLASTICITYIn the following, the terminology of the Cambridge-Clay plasticity model will be briefly explained insofar as is necessary for the further investigations in this contribution. A detailed account of “Critical State Soil Mechanics” can be found in Schofield and Wroth [6].The Cambridge-Clay plasticity model is based upon a functional relationship between the preloading of the soil, the pore number, the effective stresses and plastic strains. In relation to plasticity theory, this plasticity model can be categorized under the group of cap models. In order to define the position and shape of the yield surface, a hardening law is applied which is formulated as a function of the plastic volume changes.The consolidation stress pc represents the largest possible average compressive stress in the soil. In this respect, it is presupposed that the excess pore water pressure is completely dissipated due to external influences. If the existing average compressive stress in the soil is smaller than the consolidation stress, it is referred to as overconsolidated. If the existing average compressive stress is equal to the consolidation stress, the soil is referred to as normally consolidated. In the Cambridge-Clay plasticity model, a fixed relationship between cohesion and the initial consolidation stress is assumed. The compression curve represents the relationship between a hydrostatic state of stress and the pore number e. The critical state line defines the points in the (p, g, e) space which do not alter under further shearing load. A further concept is that of ideal critical states. The state boundary surface is the geometrical location of those stress paths in the (p, q,e) space which have been determined from undrained triaxial tests (normally consolidated samples). The swell curve represents the hydrostatic unloading curve.The modified Cambridge-Clay plasticity model has a yield surface of

(1)and a plastic potential of

(2)



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< previous page page_697 next page >Page 697where

(3)represents the volumetric stress and

(4)

the deviatoric stress. In addition, denotes the (euclidian) norm of the stress deviator

(5)

(6)The quantities Mf and Mq are material parameters and must be determined from tests. The generalized formulation of Hooke’s law is applied as the material operator for the elastic range of the stress space. It should be noted that Ke and Ge are not constants but, in the form Ke(p) and Ge(p), are dependent on the stress.

(7)where

(8)For the plastic strain increment, the evolution equation

(9)should be used. The loading/unloading conditions can be interpreted in terms of a convex mathematical optimization as an associated Kuhn-Tucker condition of the optimization problem

(10)The consistency condition is

(11)In this contribution, the hardening of the material is assumed to be purely isotropic. The inclusion of a kinematic hardening law, e.g. in the form of a generalized Ziegler formulation, is possible. This form of hardening has been dealt with inter alia by Duszek in [2]. A derivation of the hardening law for the Cambridge-Clay plasticity model from experimental values can be found, for example, in [6]. The isotropic hardening law (evolution equation



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< previous page page_698 next page >Page 698for the hardening parameter pc) is applied in the following form in the plasticity model considered

(12)

in which denotes the rate of equivalent plastic strain,

(13)pc the average consolidation stress, e0 the pore number at the start of the loading procedure, λ the bulk modulus, and k the swell factor. Specific to the Cambridge-Clay plasticity, the gradient of the yield surface in the stress space is given by

(14)and the gradient of the plastic potential by

(15)in which the following is defined

(16)

With the equation for the yield parameter is obtained from the consistency condition (11)

(17)

(18)in which it is presupposed that

(19)is valid. This leads to the following relationship for the elasto-plastic material operator of the Cambridge-Clay plasticity model [8]

(20)



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< previous page page_699 next page >Page 699Hence, the material law has the general form

(21)Equation (21) represents a system of first order differential equations for the stress tensor S. Since S must lie within the elastic range of the stress space, this also indicates an initial value problem with constraint conditions. An implicit integration method is given by Borja and Lee [8] specifically for the Cambridge-Clay plasticity model.CONDITION FOR LOCALIZING DEFORMATIONSIn the following, the bifurcation problem for the elasto-plastic material operator is formulated specifically for the Cambridge-Clay plasticity model. A general account of bifurcation problems for a large class of material laws is given by Ottosen and Runesson in [1]. The elasto-plastic acoustic tensor for the Cambridge-Clay plasticity is

Qep:=n·Cep·n. (22)The necessary condition for localizing plastic deformations in shear bands leads to an equation for the hardening modulus H, Ottosen and Runesson [1]. In the following, this equation is applied to a non-associated model of the Cambridge-Clay plasticity. Cambridge-Clay plasticity models are frequently used to describe the material behaviour of cohesive soils under three-dimensional stress conditions. The hardening modulus may be represented in the following form [1]

(23)

(24)

(25)

(26)In compliance with the procedure of Ottosen and Runesson in [1], the gradients of the yield surface Nf and the plastic potential Nq are subdivided into their deviatoric and spherical tensor components. Using the definitions

(27)

(28)



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(29)and

(30)the following representation is obtained for the gradients

(31)

(32)The relationship for the hardening modulus H thus has the following form

(33)

(34)

(35)

(36)in which the variables are defined by

(37)

(38)

The task of seeking the maximum value of H under consideration of the constraint condition can be solved with the aid of the Lagrangian multiplier method. This leads to the formulation

(39)Ottosen and Runesson [1] have specified analytical solutions for the critical hardening modulus as well as the associated localization directions for a large class of plasticity models. For these solutions, the following functional forms are permitted for the yield surface and the plastic potential

(40)

(41)in which A denotes the back stress tensor and qα, α=1,2,... denotes the scalar hardening variables. If F and Q are chosen

specifically as isotropic functions of the tensor S—A, then Nf and Nq, and hence and



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, have the same principal axes. The plastic models considered by Ottosen and Runesson thus include both isotropic as well as kinematic and isotropic-kinematic hardening.NUMERICAL ALGORITHMS AND RESULTSPlain 4 node isoparametric finite elements were used within the concept of consistent tangent tensors for describing the projection of the elastic predictor the actual yield surface. The whole algorithm was published by Borja and Lee [8].Numerous parameter studies for the consolidation process were performed using an APOLLO DN 10000. Comparisons of results for original Cambridge-Clay plasticity and sophisticated plastic potentials will be given separately. The localization process for inhomogeneous deformation states is being implemented into PLASCON in that time. First tests using the mesh generator Patran for refining the mesh at the localization zone are shown in figure 2. The main features of our research project will be reliable computations for the possible failure modes and the etection of the governing ones. A main problem will be the wave induced change of excess pore water pressure in the soil layer.

Figure 2. Finite element mesh and first approximation of shearing surface



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< previous page page_702 next page >Page 702CONCLUDING REMARKS AND OPEN PROBLEMSOttosen and Runesson [1] have investigated the characteristics of discontinuous bifurcation solutions for a large class of plasticity models. With the aid of a spectral analysis of the elasto-plastic material operator, analytical expressions for the critical hardening modulus and localizing directions can be specified for associated and non-associated plasticity models.The contribution presented here can be interpreted as a first step towards determining localized failure by means of finite element computations. With regard to J2 plasticity, various finite element models have been investigated in [4] to assess their ability to reproduce localized deformations. Strategies for the numerical treatment of localization of the equations of field using the finite element method with displacement formulations are (a) partial refinement of the computational mesh (h-adaption), (b) use of higher order polynomials for the shape functions (p-adaption), (c) regional optimization of the position of the element nodes (r-adaption) [4].In water-saturated soils, localizations of deformations occur which are affected by the pore water pressure field in the soil. Criteria to determine the onset of localizations should be further developed for this problem area. In addition, dynamically-loaded sand should be included in the formulations. Also, the material equations described must be better matched to natural conditions. The influence of cyclic loads represents a further possible extension.LITERATURE1. Ottosen, N.S. and Runesson, K. Properties of Discontinous Bifurcation Solutions in Elasto-Plasticity, Int. J. Solids Structures, Vol. 27, pp. 401–421, 1991.2. Duszek, M.K. Finite Plastic Flow and Adiabatic Shear Band Localization in Geotechnical Materials, in Inelastic Solids and Structures (Eds. Kleiber, M. and König, J.A.), pp. 95–109, Antoni Sawczuk Memorial Volume, Pineridge Press, Swansea, 1990.3. Loret, B., Prévost, J.H. and Harireche, O. Loss of hyperbolicity in elastic-plastic solids with deviatoric associativity, Eur. J. Mech., A/Solids, n° 3, 225–231, 1990.4. Steinmann, P. Strategien zur Erfassung lokalisierten Versagens in FE-Berechnungen, GAMM-Tagung, Krakau, 1991.5. Stein, E., Lammering, R. and Wagner, W. Stability problems in continuum mechanics and their numerical computation, Ingenieur-Archiv, Vol. 59, pp. 89–105, 1989.



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< previous page page_703 next page >Page 7036. Schofield, A. and Wroth, P. Critical State Soil Mechanics, McGraw-Hill, London, 1968.7. Simo, J.C. and Taylor, R.L. Consistent Tangent Operators for Rate-independent Elastoplasticity, Computer Methods in Applied Mechanics and Engineering, Vol. 48, pp. 101–118, 1985.8. Borja, R.I. and Lee, S.R. Cam-Clay Plasticity, Part I: Implicit Integration of Elasto-Plastic Constitutive Relations, Computer Methods in Applied Mechanics and Engineering, Vol. 78, pp. 49–72, 1990.9. Oumeraci, H., Partenscky, H.W. and Tautenhain, E. Large-scale model investigations: A contribution to the revival of vertical breakwaters, Proceedings of the Conference on Coastal Structures and Breakwaters, London, Great Britain, 1991, Paper accepted.10. Stein, E., Wriggers, P. and Vu Van, T. Models of Friction, Finite-Element-Implementation and Application to Large Deformation Impact-Contact-Problems, in Computational Plasticity Models, Software and Applications (Eds. Owen, D.R.J., Hinton, E. and Onate, E.), pp. 1015–1041, Proceedings of the 2nd International Conference on Computational Plasticity, Barcelona, Spain, 1989. Pineridge Press, Swansea, 1989.11. Lewis, R.W. and Schrefler, B.A. The Finite Element Method in the Deformation and Consolidation of Porous Media, John Wiles & Sons, Chichester, 1987.12. Bigoni, D. and Hueckel, T. A note on strain localization for a class of non-associative plasticity rules, Ingenieurarchiv, Vol. 60, pp. 491–499, 1990.



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< previous page page_705 next page >Page 705SECTION 10: ROCK DYNAMICS



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< previous page page_707 next page >Page 707Explosion Effects in Jointed Rocks—New InsightsF.E.Heuzé, T.R.Butkovich, O.R.Walton, D.M.MaddixLawrence Livermore National Laboratory, Livermore, CA 94551–0808, U.S.A.ABSTRACTThis article deals with the effects of high-explosive and nuclear explosions in rock masses. We first highlight the strong influence of geological discontinuities, such as joints and faults, on ground motion characteristics. Then, we briefly introduce the Discrete Element method, as a new technique for numerical simulations. It is shown to be far superior to continuum-based approaches, when dealing with the dynamics of discontinuous media such as jointed rocks. Finally, we simulate the ground effects from a generic contained nuclear explosion, which bears some similarity to the SHOAL event in granite. This calculation is intended to emphasize the influence of the near-source geology on the distribution of energy, and on the motion at various azimuths in the medium.INTRODUCTIONThe phenomenology of explosive effects in hard rocks is of interest to workers in the energy field [1–2], in the defense field [3– 5], and in planetary science [6].The salient characteristic of hard rock masses is that they are seldom massive monolithic formations, but rather are penetrated by numerous geologic discontinuities such as joint sets, faults, shears, contacts, etc.. (often simply referred to as “joints” or “faults”). These discontinuities control the propagation of ground shock, and the kinematics of the resulting motions, as observed around many nuclear and high explosive (HE) tests [4].GROUND MOTION EFFECTS OF GEOLOGICAL DISCONTNUITIESFigure 1 shows discontinuity-controlled roof failure in an underground chamber loaded dynamically. It also shows the blocky nature of the roof fall. The term “block motion” has been coined to



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Figure 1: Roof Failure in a Underground Chamber Near a Nuclear Explosion in Tuff

Figure 2: Ground Motion Records in the STARMET High-Explosives Test, After [3]



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< previous page page_709 next page >Page 709characterize the discrete displacements taking place along “joints” and “faults” under dynamic loading [3, 4]. Two examples, one from the field and the other from the laboratory, will illustrate the large effect that even a single discontinuity has on dynamic motion.Figure 2 shows the contrast in ground motion histories for gages located on either side of a shear plane, under HE loading in the STARMET event [3]. The duration of the motion of the near-gage is strikingly longer than that of the farther gage; it reflects the upthrust of a large rock block, with high inertia, in which the near-gage was located. Laboratory-scale observations also can be very revealing with regard to the effect of discontinuities. For example, Figure 3 shows a contrast in velocity histories for gages on either side of a well-lubricated interface in a sample of 2C4 grout under the action of a small HE charge [7]. The durations are about the same, but the magnitude of the peak velocity varies by more than one order of magnitude within a very short distance; in this case, the low-friction joint has decoupled the motion of the lower half of the sample. Such a dramatic influence on ground motion can also be expected in the field when very low friction fault surfaces are present.

Figure 3: A Laboratory-Scale Explosive Test in a Grout Sample with a Lubricated Joint. After [7].



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< previous page page_710 next page >Page 710THE NEED FOR NEW ANALYSIS CAPABILITIESClearly then, a realistic analysis of explosions in hard rocks must include the effect of these discrete discontinuities. If few are present, it may be possible to predict their individual motion with simple analytical methods [4]. But, typically, their large number requires the use of numerical models. Traditionally, analyses of explosive events have been done with continuum codes, mostly based on finite difference or finite element methods. Whereas these methods are mature today and have been quite useful in simulating some ground shock effects, they are not adequate for representing multiple dynamic block motion processes. Figure 4 is a case in point. This continuum simulation of a particular underground nuclear explosion in a hard rock shows a very large (over 60-metre) mounding of the surface, and sharp displacement discontinuities over a zone extending from the cavity to the surface. It is clear that extensive rock breakage would take place in the medium, in the mound, and along the displacement discontinuity surfaces, prior to material being ejected. Also, a large amount of relative block motion will take place in the ejecta, with attendant dilation. The above physics is beyond the reach of continuum codes, even if they contain so-called “damage” models for tensile and shear failure, because, after the medium fragments, the laws of continuum mechanics do not apply any more. These models also may have continuum-based parameters such as bulking factor or bulk dilatancy, which are inadequate to represent the shear dilatancy along discrete surfaces through a system of blocks. Empirical ballistic ejecta throwout and crater-lip slumping models have occasionally been coupled to continuum models, in order to approximately predict final crater dimensions. Typically, they do not provide for the momentum exchange during collisions of individual blocks, and do not have geologically-based laws for energy loss during collision.A representative analysis would require addition of realistic fragmentation models to continuum codes, and the interfacing of these with other codes which can track discrete particles and correctly account for their dynamic interaction. In contrast to Figure 4, Figure 5 shows examples of the discrete element kinematics resulting from a large displacement discontinuity or mound break up. These results were obtained with the LLNL discrete element code DIBS [8]. The remainder of this paper briefly introduces discrete element models, summarizes the essential features of DIBS, shows graphical results from recent DIBS calculations of a deep underground explosion, discusses current modeling limitations, and suggests future work to be done to further enhance the value of discrete element models.DISCRETE ELEMENT (DE) MODELINGDiscrete Particle ModelsSimulation of discrete particle motion has its origins in the field



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Figure 4: Displacement. Field in a Continuum Calculation of an Underground Nuclear Explosion


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Figure 5: Examples of DIBS Calculations Showing a) Shear Surface Development b) Break up of Mound



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< previous page page_712 next page >Page 712of molecular-dynamics, where the bulk properties of systems of particles on a molecular scale are obtained by space and time averaging the velocities and forces acting on the individual particles. Such molecular scale models, however, do not handle properties specific to rock masses such as inelastic normal forces, irregular shapes, and tangential friction at contacts. Over the past several years, a few numerical models have been developed specifically aimed at simulating the motion of macroscopic, inelastic, frictional particles. An excellent overview of the state-of-the-art is afforded by the Proceedings of the 1st U.S. Conference on Discrete Element Modeling [9].The LLNL DIBS CodeBecause of space limitations only a brief description is given here. Detailed information can be obtained in a recent report [10]. The DIBS (Discrete Interacting Block System) model is a two-dimensional polygonal-particle model that was originally applied to the flow behavior of granular solids. Basically, the model tracks the motion of each individual particle (or element) in a system of many, as it interacts with other particles and boundaries, under applied loads and gravity. Several simplifications are made concerning the interactions between elements and the properties of the elements themselves, in order to efficiently calculate the forces and motions of large numbers of distinct blocks. The major assumptions are:• elements consist of arbitrary (nearly convex) polygonal shapes.• both the interface stiffness and the block stiffnesses are included in the contact law, while the blocks themselves remain rigid during the simulations.• all contacts are modeled as corner-on-side or corner-on-corner.• small but finite “overlaps” can occur as normal forces are developed. Similarly, a finite, partially recoverable shear-strain develops in the joint-elements before frictional sliding is initiated.Recent enhancements to DIBS have included:• a capability for multi-material domains• new hysteretic normal contact force laws• rounded, block corners• “silent” (non-reflecting) boundaries• infinite-medium boundaries• a new algorithm to speed convergence of settling under gravity• a modified algorithm allowing small initial block overlaps• specification of load- or velocity-time histories for any block• time-history files of block positions and velocities for post-processing• restart capability in the course of an analysis



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< previous page page_713 next page >Page 713SIMULATION OF A CONTAINED NUCLEAR EXPLOSIONWe simulated a large explosion comparable to those of interest in the seismic range. A 22m radius cavity, with a rock cover of 340m, was loaded to simulate a 12.5 kiloton nuclear cxplosion in granitic rock. The total duration of the loading was 150ms and the peak surface velocity was 5.66m/s. The P-wave speed in the model was about 4,300m/s. All these numbers are somewhat similar to those of the SHOAL event [11]. However, the DIBS mesh structure (Figure 6) was different from the rather complex geology of the SHOAL site; in particular, the 2-D DIBS representation is not appropriate for replicating the various faults and dykes at SHOAL. So, these calculations were essentially generic. This study was designed to illustrate the effect of the near-source geology on the radiated shock energy, and on the rock mass motion in both the vertical (spall) and the horizontal direction. The total impulse was applied to the blocks immediately surrounding the cavity.As regards spall behavior, the vertical velocity history of the surface point right above the cavity (also called surface-ground-zero, or SGZ) is shown in Figure 7. The mean return acceleration of about-1.46g, is significantly greater than that of gravity. This value is close to the -1.40g for the SHOAL SGZ. This result is also consistent with those cited for PILEDRIVER, in which fall-back acceleration of up to -1.75g was reported, with a mean for surface records of about -1.25g [12]. It has been suggested [13] that the greater-than-gravity acceleration could be explained by a vacuum developing on the underside of a spall layer. Based on the DIBS simulations, it is also explainable by the storing of strain energy in the upward deforming layer and its partial recovery upon return, provided that the top layer has not been dislocatcd. The vertical displacement histories of the four near-surface blocks shaded in Figure 6, are compared in Figure 8. As a perspective on SHOAL, we also show in Figure 8, the SHOAL surface-ground zero displacement. Both the lower peak motion and the residual vertical displacement could be explained by the SHOAL near-surface region being under much more horizontal constraint. The grid had a horizontal to vertical stress ratio less than 0.5, whereas the Nevada granite is known to be under high horizontal stresses. This is another reason why this study is not a direct modeling of SHOAL.The distribution of energy is illustrated as a function of time in Figure 9, in terms of particle velocity vectors. The brick-like structure of the rock mass model, with predominantly horizontal and near-vertical interfaces, creates a strong directionality in the way the energy emanates from the near-cavity region. The effect is particularly pronounced in the horizontal direction. These results are relevant in terms of seismic wave patterns and seismic monitoring of such explosions. We note that the SHOAL explosion was reported as having radiated non-symmetrical surface waves; the asymmetries were attributed to relative motions of large blocks of rocks [11]. The HARD HAT and PILEDRIVER nuclear explosions which were in granite, like SHOAL, also were reported as giving asymmetrical azimuthal ground motion [14–17].



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Figure 6: Grid for SHOAL-Like Simulation

Figure 7: SGZ Vertical Velocity History


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Figure 8: Vertical Displacement History of Selected Near-Surface Blocks



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Figure 9: Velocity Field in the Near-Source Region Showing the Influence of the Layered Horizontal Geology



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< previous page page_716 next page >Page 716DISCUSSION AND CONCLUSIONSThe power of the Discrete Element (DE) Method is clear. It is currently the only way to describe and simulate the interaction of a large number of rock blocks and their dynamics through very large motions and dislocations. The DIBS simulation of the SHOAL-like explosion confirmed field observations of spall fallbacks with acceleration largely exceeding gravity. It also showed the large effect that the geologic structure in the near-source region has upon the azimuthal distribution of energy emanating from the explosion. It is gratifying that DE analyses can capture very important aspects of the kinematics of explosion, and provide results which are quantitatively similar to those observed in the field.However, there should be no delusion that the DE analysis is mature. Development of the method stands today where finite elements stood perhaps 15 years ago. Much work lays ahead to enhance the approach as a practical tool of analysis. Some of it concerns the numerical developments themselves, and some consists of the physical experiments which should be done in support of this technique. For example:• the fundamental interaction of blocks is through their colliding and rebounding; but very little is known of the loading and unloading mechanics of such collisions. Today, we do not know of data which can be used to guide the choice of input quantities such as a realistic coefficient of restitution, for example. Also, the damage or fracturing induced in the colliding blocks remains to be quantified and modeled. It is yet another source of energy loss, and it contributes to dilatancy.• for a complete simulation of explosion effects, the discrete element analysis can be interfaced with the results of analyses in the very-near source region of extreme pressures (several tens of kilobars and up), which would be modeled with continuum-based hydrocodes. In turn, these continuum codes must evolve to provide a representation of the fragmentation of the rock mass, if required. Only the crudest comminution algorithms exist today in such codes.• the influence of block size on the results of DE simulations is an unresolved question. Systematic studies are desirable to determine the degree of discretization which is needed to capture all the important features of ground motion.• except for rare geologies, the block motion truly has a 3-dimensional character. Rock reinforcement such as bolts is also 3-D in nature. But there are precious few 3-D discrete polyhedral models, see [10]. And even so, their usage is enormously cumbersome and computationally expensive. To achieve practical simulations of 105 or 106 blocks, as might be needed for adequate 3-D representation of a geologic region of interest, will require that the power of parallel processing be applied to the DE models. This is a very challenging task. On the other hand, much insight still can be gained by pursuing the



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< previous page page_717 next page >Page 717 enhancement and application of 2-D models, as demonstrated in this writing.Certainly the field of dynamics of jointed rocks is rich in opportunities for significant experimental and numerical developments, and the discrete element method is by far the most promising numerical approach in that field, today. Because geological discontinuities have such an influence on ground shock, it is also clear that the usefulness and credibility of numerical simulations is intimately tied to the quality of the geological site characterization.REFERENCES1. Grady, D.E., and Kipp, M.E. “Dynamic Rock Fragmentation”, Chapter 10 in Fracture Mechanics of Rock, (Academic Press, London), pp 429–475, 1987.2. Fourney, W.L., and Dick, R.D., Eds. (1987) “Rock Fragmentation by Blasting”. Proc. 2nd Int. Symp., Keystone, Co., (Soc. Exp. Mech. Bethel, CT), 676 p., 1987.3. Blouin, S., “Block Motion from Detonation of Buried Near-Surface Explosive Arrays”, U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, NH, Report CRREL 80–26, 62 p., Dec. 1980.4. Bedsun, D.A., Ristvet, B.L., and Tremba, E.L. “Summary and Evaluation of Techniques to Predict Driven and Triggered Block Motion: A State-of-the-Art Assessment”, S-Cubed, Albuquerque, NM, Report to Defense Nuclear Agency, DNA-TR-85–249, 122 p., June 19855. Gold, K.E. “Status of Nuclear Crater Prediction Methodologies”, Kaman Tempo, Santa Barbara, CA, for Defense Nuclear Agency, Report DASIAC-TN-85–5.. 1985.6. Roddy, D.I., Peppin, R.O. and Merrill, R.E.Eds. Impact and Explosion Cratering. (Pergamon Press, New York, NY) 1977.7. Fogel, M.B., Kilb, D.L., Nagy, G., and Florence, A.L., “Experimental and Calculational Study of Wave/Fault Interaction”, Proc. 4th Symp. Containment Underground Nuclear Explosions, Colorado Springs, CO, pp 81–89, CONF-870961. (NTIS, Springfield, VA), Sept. 1987.8. Walton, O.R. “Explicit Particle Dynamics Model for Granular Materials”, Numerical Methods in Geomechanics, Edmonton, Canada, Z.Eisenstein, ed., (A.A.Balkema, Brookfield, VT), pp 1261–1268, 1982.



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< previous page page_718 next page >Page 7189. Mustoe, G. Ed. Proceedings 1st U.S. Conference on Discrete Element Methods, Golden, Colorado (Colorado School of Mines Press, Golden), 1989.10. Heuze, F.E., Walton, E.R., Maddix, D.M., Shaffer, R.J., and Butkovich, T.R., “Analysis of Explosions in Hard Rocks: The Power of Discrete Element Modeling”, Lawrence Livermore National Laboratory, UCRL-JC-103498, 68 p, March 1990.11. Weart, W. “Vela Uniform. Project SHOAL. Project 1.1: Free Field Earth Motion and Spalling Measurements in Granite”, Sandia Corporation, Report VUF-2001, February, 99 p 1965.12. Hoffman, M.V., and Sauer, F.M. “Operation Flint Lock, Shot PILEDRIVER. Project Officers Report. Project 1.1: Free-Field and Surface Motions” Stanford Research Institute, CA, Report POR-4000, June, 139 p 1969.13. Perret, W.R. Sandia Corporation, Albuquerque, oral communication to Hoffman, H.V. 1969.14. Perret, W.R. “Shot HARD HAT: Free-Field Ground Motion Studies in Granite, Sandia Corporation Report Albuquerque, NM, POR-1803, 1963.15. Swift, L.M., and Eisler, J.D. “Vela Uniform. Nougat-Series, ANTLER and HARD HAT Events. Project 1.2: Measurements of Close-In Earth Motion”, Report VUF-2100 from Stanford Research Institute to DASA May, 131 p, 1965.16. Perret, W.R. “Operation Flintlock. Shot PILEDRIVER. Project Officers Report. Project 1.2a. Free-Field Ground Motion in Granite”, Sandia Laboratory Report, Albuquerque, NM, POR-4001. 1969.17. Murphy, J.R. “A Review of Available Free-Field Seismic Data from Underground Nuclear Explosion in Salt and Granite”, Computer Sciences Corp., Falls Church, VA, AD-A06630113 (NTIS, Springfield, VA), 61 p., Sept. 1978.ACKNOWLEDGEMENTSThis work was performed with funding from the Treaty Verification Program, of Lawrence Livermore National Laboratory, under contract W-7405-ENG-48 with the U.S. Department of Energy.Mrs. K.Kirk skillfully typed the manuscript, and we are grateful to P.Proctor for her fine graphics artwork. We also appreciate the comments from J.Hannon and S.Taylor, at LLNL.



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< previous page page_719 next page >Page 719Numerical Analysis and Measurements of the Seismic Response of GalleriesH.-J.Alheid, K.-G.HinzenBundesanstalt fuer Geowissenschaften und Rohstoffe, D-3000 Hannover, GermanyABSTRACTIn-situ dynamic load experiments and corresponding numerical dynamic load experiments were performed to measure and simulate the dynamic response of a gallery in rock salt and compare the results to a former test in dolomite. The kinematic analysis of the generated seismic wavefields allows comparison of the effects of different geometries of the underground openings. Comparison of measured and calculated data constrains material inhomogeneities in the vicinity of the gallery.INTRODUCTIONSite investigations for the proposed permanent repository for radioactive wastes in rock salt in Germany include the determination of the in-situ dynamic properties of the host rock. Knowledge of these properties is essential in earthquake safety analysis. For this reason a research program was initiated to develop equipment and experience to perform in-situ dynamic load experiments (IDLEs) for determination of the dynamic material properties in the vicinity of underground openings in rock salt.Corresponding numerical dynamic load experiments (NUDLEs) were performed using the Finite-Element-Code ANSALT. This code was developed for nonlinear seismic analysis of underground openings for permanent storage of hazardous wastes. Reliable earthquake safety-analysis of such structures requires an extensively verified and validated computer code.In the present stage of the research the results from the IDLEs and NUDLEs are compared to validate the computer code as well as to interpret the measured data.



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< previous page page_720 next page >Page 720The comparative interpretation of measured and calculated data helps to discriminate effects on the seismic wavefield caused by either geometry of the opening or material inhomogeneities in the surrounding rock mass.In 1989 a first set of in-situ experiments was carried out in order to test the

Figure 1. Residuals of measured and calculated first arrival times for a dynamic test in dolomite (from Alheid and Hinzen [1]).equipment and processing software in a gallery 3.5m high, 3.0m wide and with a maximum overburden of about 55m. The gallery was situated in a rather homogeneous dolomite with an average P-wave velocity of 3800m/s. For construction of the gallery, a drilling and blasting technique was used. The experimental procedures and detailed results are given in Alheid and Hinzen [1]. The in-situ load was realized by the detonation of chemical explosives 20.5m above the center of the gallery. The residuals of measured and calculated travel times are shown in Figure 1. Positive values indicate a delay in measured travel–times. If one assumes that geometrical effects are modeled properly by the numerical model, gradients in the isochron plan of the travel-time residuals indicate deviations in the in-situ compressional wave velocities. The highest gradient, i.e. highest density in isochrones, is observed in the vicinity of the gallery. The gradient decreases with increasing distance from the gallery. This



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< previous page page_721 next page >Page 721effect is interpreted as a decrease of the P-wave velocity due to weakened material in this zone, which extends up to the dimension of the gallery. The maximum observed time-delay is 1.05ms.A second set of experiments was carried out in 1990 in a deep underground gallery in the Asse salt mine in Germany. Since the opening was excavated using a cutting-machine, the material surrounding the gallery was expected to be more homogeneous than the near-surface experiments. Therefore, we felt this area provides a better test for modeling the geometrical effects with the numerical calculations.THE IN-SITU CONCEPTThe in-situ dynamic load experiments (IDLEs) were carried out in an underground gallery in rock salt. The gallery of 4m height, 5m width and 90m length was constructed at the -800m level in the Asse salt mine, Germany. For the observation of the response of the rock in the vicinity of the gallery a measuring field of a total of 22 15m long boreholes was constructed. The vibration time history during the dynamic experiments was observed with 20 three-component borehole geophone stations. The geophones have a flat response between frequencies of 30Hz and 1200Hz. The seismic signals are digitized at 31,250 samples per second for each channel. Details of the recording system are given in BGR report [2],For this study two main dynamic experiments, named A and B, were carried out in the dynamic test gallery. The 2D measuring array for the experiments consists of eight radial boreholes. Figure 2 shows a crosscut through the left half of this measuring field. The right hand side of the measuring field was symmetric to the left side. The dynamic load was performed by the detonation of chemical explosives in boreholes. The charge weight was 70g including the detonator. The charges were fired at the bottom of two 15m deep holes, which were drilled vertically down from the middle of the gallery floor. The hole for experiment B was 6m away from the measuring array, while the shot hole for experiment A was part of the array. The charges were were put into the center of cylinders made from salt-concrete and then lowered into the shot holes. The cylinders gave a reproducible coupling for each individual shot. The right half of the figure shows the corresponding finite element model of the measuring field. In experiment A a total of three individual shots were fired in the vertical bottom hole. After each shot the position of the geophone stations was changed. In this way the number of observation points for the travel times of the seismic waves could be larger than the number of stations available. A total of 45 P-wave travel times was measured. The first arrival times were picked interactively from the digital seismic



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Figure 2. Cross-cut of the gallery and measuring field for the in-situ dynamic load experiment (left) andcross-sectional discretization for numerical calculations.recordings. The exact moment of firing was recordedonanextrachannel. In this way the exact travel times of the seismic waves could be determined.THE NUMERICAL CONCEPTThe finite element code ANSALT was used for the calculations. The rock salt is modeled by 2794 4-node isoparametric solid elements (Fig. 2, right hand side) with 2910 nodes and 5765 degrees of freedom. The maximum node distance is 0.5 m and the time-step is 0.13333ms. The overall dimensions of the discretization are 24m×29m. A linear elastic material was assumed with a Young’s modulus of 35,480MPa, a density of 2100kg/m3 and a Poisson ratio of 0.25.These properties correspond to a P-wave velocity of 4500m/s and an S-wave velocity of 2600m/s. Due to maximum node distance the finite element discretization yields an upper frequency limit of about 1kHz with these material properties.The center line of the gallery is used as a symmetry axis. The fullspace surrounding the center is simulated by viscous boundary elements at the top and



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< previous page page_723 next page >Page 723bottom and transmitting boundary elements in time domaine (Alheid et al. [3]) for horizontal energy absorption.The calculation was performed assuming a seismic point source 15m below the gallery. The load time-history used was lowpass-filtered with a cutoff frequency of 400Hz to insure that the results would be below the upper frequency limit of the discretization. The excitation was only applied in the vertical direction.Two different cross-sections were chosen (Fig. 3) to model the geometry of the gallery. Geometry 1 (left) was used for precalculations before the actual geometry of the gallery could be measured whereas geometry 2 (right) represents the actual geometry of the gallery. With geometry 2 the gallery is 0.1m higher than in geometry 1, and the sidewalls are more smoothly curved.

Figure 3. Cross-sectional finite element discretization for two geometries of a gallery.Isochron plans were assembled from the results of these two calculations. The important clippings are shown side by side in figure 4. The interval of the isochrones is 0.1ms. Up to a value of 3.5ms no differences between the two geometries are observed. Close to the sidewalls only minor differences occur. At the top of the galleries geometry 1 apparently leads to larger travel times than geometry 2. This effect can be seen from the offset of corresponding isochrones. The difference in travel times, however, is only 0.05ms and thus smaller than the time-step in the calculations. Thus the differences in the isochrones are only



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Figure 4. Isochrone plan of the calculated first arrival times for the two geometries of Figure 3.caused by slightly different node coordinates in the two discretizations close to the gallery. Nevertheless the results of the calculations using geometry 2 clearly yield a better fit to the results of the in-situ dynamic load experiments. Therefore in the following only the results of calculations using geometry 2 are presented.RESULTSKinematic analysisNumerical travel times From the results of the FE calculations the velocity-time histories of 177 nodes were constructed. These synthetic seismograms were processed the same way as measured data. The first breaks were picked and an isochron plan was assembled (Fig. 5). The interval of the isochrones is 0.2ms. Below and beside the gallery the isochrones are equally spaced, indicating undisturbed wave propagation. Above the gallery delayed arrival times are obvious. As these calculations do not include material inhomogeneities, the time delay above the gallery is due to the shadowing effect of the geometry. The shadowing zone extends out to about one dimension of the gallery.In-situ travel times Figure 6 shows an isoline plan of the position of the P-wave front, which was calculated on the basis of the measured travel times from



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Figure 5. Isochrone plan of first arrival times from the finite element calculations for geometry 2.(Distances are given in meters)


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Figure 6. Isochrone plan of the measured first arrival times. (Distances are given in meters)



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< previous page page_726 next page >Page 726experiment A. The plan shows the crosscut through the measuring array, expanding 16m to the left and the right from the gallery center and 10m below the gallery floor to about 18m above it. The crosses indicate the locations of the geophone stations in the boreholes. The positions of the wave front are given in steps representing a time span of 0.2ms. This corresponds to 6.5 samples of the recording. The first wavefront position at the bottom of the figure is that for 1.4 ms after the shot and the last position (top) is for 7.4ms. Besides the deformation of the wavefront due to the shadowing effect of the gallery, only a few minor disturbances in the kinematic wavefield are seen in the lower right quadrant of the measuring array if compared to the calculations. This proves the high degree of homogeneity of the rock salt in the vicinity of the gallery.Comparison of travel times Figure 7 shows the residuals of measured and

Figure 7. Residuals of measured and calculated first arrival times.calculated travel−times. As with Figure 1 positive values indicate a delay in measured travel-times. The interval of the isochrones is 0.05ms. The maximum delays are located at the lower corners of the measuring field. This effect is due to differences in the radiation patterns of IDLE( explosive ) and NUDLE ( single vertical excitation ). In the other region the delay-times do not exceed 0.2ms, i.e. less than 2 samples, and seems to be uncorrelated to the gallery opening. This result demonstrates that within the resolution of these calculations no weakened zone is detectable in the vicinity of the gallery. Furthermore, the effects of wave diffraction into the shadow zone are modeled properly by the FE calculations.



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< previous page page_727 next page >Page 727Dynamic interpretationComparison of seismograms Figure 8 shows a comparison of measured (left) and

Figure 8. Measured and calculated vertical component velocity seismograms for a vertical profile through the middle axis of the gallery.calculated (right) vertical component velocity seismograms. The time window displayed in the figure is 20ms. All seismograms are normalized to their maximum amplitudes. The measured seismograms were recorded during experiment B, where the shots were fired in the borehole which is located 6m away from the measuring array. The small insert on the left side of Figure 8 gives



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< previous page page_728 next page >Page 728the location of the borehole stations at which the seismograms were recorded. The lower trace A gives the movement 7.5m below the center of the gallery floor. The traces B and C were measured 0.5m below the gallery floor and 0.5m above the gallery roof, respectively. The upper trace D was measured 7.5m above the gallery roof. The measured seismograms of this figure were lowpass-filtered with a comer frequency of 400Hz in order to match the bandwidth of the calculated seismograms. The travel time delay with increasing distance from the source is obvious. All traces show clear first arrivals which are followed by a

Figure 9. Spectral ratios of the traces B, C and D to traces A from Figure 8. Bold lines are for measured data.negative half wave which represents an upward movement. The measured bottom trace shows a more or less smooth decay of the amplitude after a positive half wave which followed the first negative arrival. In the trace from the gallery floor, a positive onset is observed after the second half wave. This onset is clearer in the



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< previous page page_729 next page >Page 729traces measured above the gallery. The duration of the second pulse is close to that of the first one. The overall pulse shape and frequency content of the calculated seismograms matches the measured data fairly well, but some differences are obvious. The amplitude of the first positive half wave in the upper two traces is reduced by a negative onset which occurs at the end of the first negative half wave. This onset does not exist in the measured data.Comparison of frequency content Figure 9 shows ratios of amplitude spectra. The upper middle and lowerparts of the figure give the ratios of the signals from 7.5m, 0.5m above and 0.5m below the gallery to the signal 7.5m below the gallery, respectively. The bold traces represent the measured data, the thin lines the calculated data. The ratios are given in log-log scale for the frequency range from 30Hz to 1kHz. The overall shape of the ratios fits very well. At the low frequency end of the spectra between 30Hz and 200Hz the ratios of the measured data show values lower than those derived from the calculations. This effect is due to the lack of inelastic absorption in the calculations and the two-dimensional numerical model. While the values for the stations below the drift are close to unity the ratios of the upper stations are significantly smaller. This is mainly due to the shadowing effect of the tunnel and only to a minor extent due to geometrical spreading.CONCLUSIONSA significant difference in the results can be seen between the two test series: whereas the kinematic analysis of the data from former experiments in dolomite clearly show the presence of a weak zone close to the opening, the recent experiments in rock salt show no comparable effects. This is due to the more homogeneous material and the construction by cutting-machine instead of drilling and blasting. The high degree of homogeneity of the rock salt in vicinity of the dynamic test drift allows a detailed validation study between IDLEs and NUDLEs. Even small changes in the geometry of the crosscut of the underground opening in the FE model become obvious in the kinematic interpretation of the seismic wavefield. In the tests in dolomite, these small geometry effects are hidden in the bigger effects due to material inhomogeneities.ACKNOWLEDGEMENTSThe authors would like to express sincere thanks to Renate Pfeiffer for conducting the numerical calculations. The contribution of the BGR B2.12 seismic field crew is also very much appreciated. The authors also acknowledge the valuable discussions with Sharon K.Reamer.



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< previous page page_730 next page >Page 730REFERENCES1. Alheid, H.-J. and Hinzen K.-G. Dynamic Response of a Gallery—Calculations and Measurements, in Structural Dynamics and Soil-Structure Interaction (Ed. Cakmak, A.S. and Herrera, I.), pp. 355–370, Proceedings of the fourth International Conference on Soil Dynamics and Earthquake Engineering, Mexico City, Mexico, 1989. Computational Mechanics Publications, Southampton, 1989.2. BGR Report. Stoffverhalten von Salz bei kurzzeitigen Wechselbelastungen. Final Report for Projekt KWA 5502 8, BGR, Hannover, 1988.3. Alheid,H.-J., Honecker, A., Sarfeld, W. and Zimmer, H. Transmitting boundaries in time domain for 2-D nonlinear analysis of deep underground structures, in NUMETA85 (Ed. Middelton J. and Pande G.N.) pp. 117–127, Proc. Int. Conf. on Numerical Methods in Engineering: Theory and Applications, Swansea, 1985. Balkema, Rotterdam, 1985.



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< previous page page_731 next page >Page 731Dynamic Solution of Poroelastic Column and Borehole Problems of Soil and Rock MechanicsD.E.Beskos (*), I.Vgenopoulou (**)(*) Department of Civil Engineering, University of Patras, GR-261 10, Patras, Greece(**) School of Applied Technology, Technical and Educational Institute of Patras, GR-263 34 Patras, GreeceABSTRACTThe one-dimensional dynamic column and borehole problems of soil and rock mechanics are solved analytically-numerically. The poroelastic soil medium obeys the Vardoulakis-Beskos theory, while the poroelastic, fissured rock medium the Aifant is-Beskos theory. Use of Laplace transform with respect to time reduces the column and borehole problems to ordinary differential equations with constant and variable coefficients, respectively. The transformed solution of these problems is obtained analytically for the column and by finite differences for the borehole problem and after a numerical Laplace transform inversion, produces the time domain response. Numerical results are presented in order to access the significance of various dynamic and material parameters on the response.INTRODUCTIONThe dynamic behavior of nearly-and fully-saturated, isotropic, poroelastic soil has recently been studied by the Vardoulakis-Beskos model in Vardoulakis and Beskos [1] and Vgenopoulou et al [2], where model development and body wave propagation studies are reported. Also, the dynamic behavior of fully-saturated, fissured, isotropic, poroelastic rocks has recently been studied by the Aifant is-Beskos model in Beskos [3], where the model development is described and in Beskos et al [4, 5], where body and Rayleigh wave propagation is studied.In this paper the one-dimensional dynamic column and borehole problems of soil and rock mechanics, formulated on the basis of the above two models, are solved analytically-numerically. Use of Laplace transform with respect to time reduces the column and borehole problems to ordinary differential equations with constant and variable coefficients, respectively. The transformed solution of these problems is obtained analytically for the column and by finite differences for the borehole problem and after a numerical Laplace transformed inversion, produces the time domain response. Numerical results are presented in order to access the importance of the various dynamic and material parameters on the response.



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< previous page page_732 next page >Page 732THE DYNAMIC SOIL COLUMN PROBLEMVardoulakis and Beskos [1] have recently proposed a more refined model thanBiot’s [6] for the description of the dynamic behavior of fully saturated poroelastic soil media. This model reduces to the following system of equations for the one-dimensional dynamic soil column problem:

(1)

(2)

(3)In the above, commas indicate differentiation with respect to z along the column, dots differentiation with respect to time

t, u is the solid phase displacement, p is the fluid pore pressure, and are the relative solid and fluid densitites given

in terms of the actual ones by the relations where n is the porosity, λ and µ, are the Lame’ elastic constants, γ, α1 and α2 are material constants depending on the porosity and the compressibilities of the constituents, b=vf/K with vf being the dynamic viscosity of the fluid and K the Muskat permeabillity and Q=n(u−u) , with u being the fluid displacement.The soil column with a height h is laterally confined, has a rigid and impervious bottom at z=h and is subjected at its top surface z=0 to a suddenly applied compressive load of magnitude σ0 through a porous slab so that water can escape from the suface. The boundary and initial conditions of the problem read (Vgenopoulou and Beskos [7])

(4)and



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(5)respectively. Application of Laplace transform with respect to time to Eqs (1)—(3) under the initial conditions (5) and

after elimination of yields

(6)

(7)where overbars denote transformed quantities and A11, A12,…., F are constant coefficients, functions of various material constants. Equations (6) and (7) can be solved analytically and in conjunction with the boundary conditions (4)

provide the transformed solution and . A numerical Laplace transform inversion of this solution with the algorithm of Durbin [8] can finally produce the time domain response u(z, t) and p(z, t). The quasi-static (consolidation) behavior of the column can be easily obtained as a special case.THE DYNAMIC SOIL BOREHOLE PROBLEM.The Vardoulakis-Beskos soil model reduces to the following system of equations for the one-dimensional dynamic borehole problem:

(8)

(9)

(10)In the above, commas indicate differentiation with respect to the radial coordinate r, which is the only spatial coordinate of the problem due to plane strain and axial symmetry conditions satisfied herein. The borehole has a radius r0 and is subjected to a suddenly applied uniform pressure p0.The boundary and initial conditions of the problem read (Vgenopoulou and Beskos [7]).

P(r0, t)=p0, σr(r0, t)=−p0, p(∞, t)=0, σr(∞, t)=0 (11)and



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(12)respectively. Application of Laplace transform with respect to time to Eqs. (8)–(10) under the initial conditions (12) and

after elimination of yields

(13)

(14)where overbars denote transformed quantities and B11, B12,…, B22 are constant coefficients, functions of various material constants. Equations (13) and (14) have variable coefficients and are solved numerically by central finite

differences (Von Rosenberg [9]) to provide and . A numerical Laplace transform inversion of this solution with the algorithm of Durbin [8] can finally produce the time domain response u(r, t) and p(r, t). The quasi-static (consolidation) behavior of the borehole problem can be easily obtained as a special case.THE DYNAMIC ROCK COLUMN PROBLEMBeskos [3] has recently proposed a rock model, on the basis of Aifantis’ [10] theory, for the description of the dynamic behavior of fully saturated, fissured, poroelastic rock media, i.e., media with two kinds of porosities-one due to the fissures separating the rock into blocks and the other due to the pores of those blocks. This model reduces to the following system of equations for the one-dimensional dynamic rock column problem:

(15)

(16)

(17)In the above, a receives the values 1 (for fissures) and 2 (for pores), βα, γα, βf, and K are material constants, va=βαVf/Kα and nα and Kα denote the two most important parameters of the model, i.e., porosities and Muskat permeabilities.The boundary and initial conditions of this column problem read (Vgenopoulou and Beskos [11])

(18)



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< previous page page_735 next page >Page 735and

(19)respectively. The solution procedure is exactly the same as in the case of the soil column problem.THE DYNAMIC ROCK BOREHOLE PROBLEM.The Aifant is-Beskos rock model (Beskos [3]) reduces to the following system of equations for the one-dimensional dynamic borehole problem:

(20)

(21)

(22)The boundary and initial conditions of the above problem read (Vgenopoulou and Beskos [11])

(23)and

(24)respectively. The solution procedure is exactly the same as in the case of the soil borehole problem.NUMERICAL RESULTSNumerical results were obtained on the basis of soil and rock data described in Vgenopoulou and Beskos [7, 11]. Figures 1 and 2 show the displacement and pressure histories at soil column height h/2=10m. for two degrees of saturation S for the case of the suddenly applied load under both dynamic and quasi-static



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< previous page page_736 next page >Page 736conditions, which produce almost identical results. However, if the applied load is harmonically varying with time, then the dynamic response is higher than the quasi-static one as Fig. 3 clearly indicates. Figures 4–6 depict analogous things for the case of the soil borehole problem under dynamic and quasi-static conditions.Figures 7 and 8 portray the displacement and pore pressure histories at rock column height h/2=10m for various porosity ratios for the case of the suddenly applied “load under both dynamic and quasi-static conditions, which produce almost identical results. However, if the applied “load is harmonic, then the dynamic response is higher than the quasi-static one as Fig. 9 clearly indicates. Figures 10 and 11 depict the displacement and fissure pressure versus distance for various times for the rock borehole problem under dynamic and quasi-static conditions, producing almost identical results. However, if the applied pressure is harmonic, the dynamic response is higher than the quasi-static one as Fig. 12 clearly indicates.CONCLUSIONSThe major conclusions of this work are as follows:1) The proposed methodology is highly accurate and can easily handle “loads of any time variation and soils and rocks exhibiting viscoelastic behavior.2) The significance of inertial effects depends on the kind of dynamic loading. Thus inertial effects are negligible for suddenly applied loads and significant for harmonic loads.3) The effect of the degree of saturation was found to be significant when the soil model takes it into account.4) The effect of porosities and permeabilities in the rock model was found to be rather small and so was found to be the difference between the two degrees and the one degree of porosity rock models.5) Solution of these one-dimensional column and borehole problems provides valuable insight into the behavior of soils and rocks and a firm basis for checking the accuracy of general numerical methods, such as finite elements or boundary elements.



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Fig. 1: u versus t for dynamic soil column problem

Fig. 2: p versus t for dynamic soil column problem


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Fig. 3: u versus t for harmonic soil column problem



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Fig. 4: u versus r for dynamic soil borehole problem

Fig. 5: p versus r for dynamic soil borehole problem


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Fig. 6: u versus t for harmonic soil borehole problem



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Fig. 7: u versus t for dynamic rock column problem

Fig. 8: p2 versus t for dynamic rock column problem

Fig. 9: u versus t for harmonic rock column problem



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Fig. 10: u versus r for dynamic rock borehole problem

Fig. 11: p1 versus r for dynamic rock borehole problem


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Fig. 12: u versus t for harmonic rock borehole problem



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< previous page page_741 next page >Page 741REFERENCES1. Vardoulakis I. and Beskos, D.E. Dynamic Behavior of Nearly Saturated Porous Media, Mechanics of Materials, Vol. 5, pp. 87–108, 1986.2. Vgenopoulou, I., Beskos, D.E. and Vardoulakis, I. High Frequency Wave Propagation in Nearly Saturated Porous Media, Acta Mechanica, Vol. 85, pp.115–123, 1990.3. Beskos, D.E. Dynamics of Saturated Rocks I : Equations of Motion, Journal of Engineering Mechanics of the ASCE, Vol. 115, pp. 982–995, 1989.4. Beskos, D.E., Vgenopoulou, I. and Providakis, C.P. Dynamics of Saturated Rocks II: Body Waves, Journal of Engineering Mechanics of the ASCE, Vol. 115, pp. 996–1016, 1989.5. Beskos, D.E., Papadakis, C.N. and Woo, H.S. Dynamics of Saturated Rocks III: Rayleigh Waves, Journal of Engineering Mechanics of the ASCE, Vol. 115, pp. 1017–1034, 1989.6. Biot, M.A. Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid I. Low-Frequency Range, Journal of the Acoustical Society of America, Vol. 28, pp. 168–178, 1956.7. Vgenopoulou, I. and Beskos, D.E. Dynamic Poroelastic Soil Column and Borehole Problem Analysis, Acta Mechanica, submitted.8. Durbin, F. Numerical Inversion of Laplace Transform: An Efficient Improvement to Dubner and Abate’s Method, The Computer Journal, Vol. 17, pp.371–376, 1974.9. Von Rosenberg, D.U. Methods for the Numerical Solution of Partial Differential Equations, American Elsevier, New York, N.Y., 1969.10. Aifantis, E.C. On the Problem of Diffusion in Solids, Acta Mechanica, Vol. 37, pp. 265–296, 1980.11. Vgenopoulou, I. and Beskos, D.E. Dynamics of Saturated Rocks IV: Column and Borehole Problems, Journal of Engineering Mechanics of the ASCE, submitted.



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< previous page page_743 next page >Page 743Fundamentals of a Practical Classification of Mining Induced Seismicity (Rock Bursts)P.KnollCentral Institute for Physics of the Earth, Telegraphenberg, D-O-1501 Potsdam, GermanyABSTRACTThe most important prerequisite for the effective investigation and control of the seismicity in and around mines is the knowledge of the mechanics of the complete rock burst processes. The literature and case studies of rock bursts show that there is no accordance in relation to the fundamental source mechanism and to the definition of the rock burst phenomena in general. So, it is difficult to generalize the results of rock burst mechanism investigations from different mining areas.Therefore, the Economic Commission for Europe (Coal Committee) of UNO is going to develope a new definition of the phenomena “rock burst” and “mining-induced seismic event” as well as an international classification of rock bursts (1994). The new criteria for the classification should be the features of the fracture mechanism within the source and the mechanism of energy emission in opposite to the former classification which has been related to the effects and damages in mining openings, mainly. The new classification would not more exclude e.g. the so called “bumps”, i.e. the seismic events in mining areas without damages in the underground workings itself. Besides other advantages the classification should be in clother relation to the seismicity evaluation at a site, for earthquake engineering purposes, too.A proposal for the fundamentals of this classification will be presented and discussed. This is also a call to scientists and engineers in earthquake engineering and mining sciences to share by own proposals and experiences to a practicable classification assisting the seismicity investigations in mining areas as well as the mining safety, too.



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< previous page page_744 next page >Page 744INTRODUCTIONIn October 2–6, 1989 in Ostrava/Czechoslovakia took place an international “Symposium on Forecasting and Prevention of Rock Bursts and Sudden Outbursts of Coal, Rock and Gas” initiated by the Coal Committee of the UNO-Economic Commission for Europe (ECE). In the overall conclusions of the official report of the meeting /1/ was established:“Participants identified the following factors as impeding a fuller control of rock bursts and sudden outbursts:…(c) intentionally: a certain confusion as to the exact meaning/definitions of terms, classifications, parameters, etc. used in the various countries. This lack of accuracy impeded the international exchange of data, the common analysis of bursts/outbursts (case histories), …” (/1/, point 40, page 10)and“with a view to enhancing international co-operation,participants appealed to the ECE Coal Committee:…(b) to establish an ad hoc meeting of experts with the mandate to develop an ECE classification of rock bursts/sudden outbursts in underground mines, including related issues such as terminology, measuring devices and standards. Such a classification would be a first step in overcoming present obstacles in the international exchange of data, analysis of incidents, comparison of performance of devices/ appliances, and national regulations and norms. The classification could become a contribution of the Coal Committee to the United Nations International Decade for the Prevention of Natural Disasters (1990s). International governmental and non-governmental organizations dealing with mining, geomechanics, seismology, geology should be invited to contribute, so as to secure a comparable data flow between disciplines.” (/1/, point 44, page 11)As a consequence of this report and as a resolution of the 86th session of the ECE-Coal Committee /2/ and the 20th session of the Meeting of Experts on Productivity and Management Problems in the Coal Industry /3/ in January 15–17, 1991 took place the pronounced ad hoc Meeting on Classification of Rock Bursts and Sudden Outbursts in Underground Mines” in Geneva and– decided to set up two Working Groups for classifi-



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< previous page page_745 next page >Page 745cation of rock bursts or sudden outbursts, respectively, under the chairmenship of P.Knoll (Germany) for rock bursts and I.M.Petuchov (Soviet Union) for sudden outbursts and– adopted a working programme aimed on a new rock burst classification (and definition). The classification is to be presented and finally discussed at the next international ECE-Symposium on rock bursts and sudden outbursts /4/ in Leningrad (USSR), 1994.The following paper will give an impression on the existing problems and discuss the first proposals for a classification and definitions of rock bursts. To open the public discussion about this topic and to get a broad accepted classification it is of great importance for seismicity evaluations and engineering activities in mining areas taking into account that the maximum mining-induced seismic events had worldwide magnitudes up to 5 and more and epicentral intensities (MSK-scale) of about Io= 9, 5° /12/.STATE OF THE ART IN ROCK BURST CLASSIFICATIONRock bursts always cause very severe injuries in mines and the first and most common attempts to define and classify this phenomenon started therefore from the damaging effects within the mines. As “rock bursts” often only those events were classified which gave rise to damages in the mine workings. In this sense e.g. Bräuner /5/ defined the rock burst as “a fracture by which the coal breaks into the mine working with an explosion-like force” or later in an other publication/6/ he defined the “rock burst in a more narrow sense as a shock-like fracture the energy of which comes from the elasticity of the rock masses”. He separated the rock bursts from the “mining earth tremors” caused by mining and shaking the mine and the surface but not damaging the mine workings /6/.This separation, supported by a lot of other mining engineers, expresses the most serious problem of the classical rock bursts definitions and raises the questions: Is the emission of energy by brittle fracturing of rocks and sources situated far from an mining opening and connected or not connected with damages in the mine working also a rock burst or not? What is the limit in relation to the distance of the primary sources from the mine working? What about the brittle fracture of a residual pillar in the goaf of a mine in several hundred meters distance from the active stope connected with a big energy



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< previous page page_746 next page >Page 746emission but without damages in the active workings? Is that a rock burst or not?Gibowicz /7/ divided between tremors and rock bursts and said “there are tremors causing rock bursts and tremors causing not rock bursts” but tremors are mining-induced seismic events.” Rock bursts are a small subset within a large set of mininginduced seismic events”/7/. That means that a definition and classification of rock bursts would deal only with one small part of effects of brittle fracture processes and these effects are only partially determinated by the fracture process itself but also (and in some cases very clearly) by the technical conditions in the mine (support, bolting etc.), by technological conditions (absence of men in the stope, room and pillar mining methods etc.) or by the technological “history” of the opening (e.g. state of the contour strength) and so on.If we include only a small “subset” of all mining-induced brittle fracture processes in and around the mine in the investigation of rock bursts we are unable to discover the whole process “rock burst”, its preconditions, the interaction of technological and natural conditions and last but not least the real mechanism of the origin of rock bursts damages in the mine.Starting with the papers of Knoll /8, 9/ and the following papers /10, 11/ the weight has been more concentrated not to rock burst in the former narrow sense but to the complex phenomenon “brittle fracturing of rocks in and around mines and caused by mining” as a complex induced fracture process including the effects underground and at the surface.On this base further proposals for a more complex definition of rock burst were given by Konecny /13/, Ryder /14/, Semjakin /15/, Johnston /16/. The authors understand the rock burst as a mining -induced seismic event and discriminate two main types, one having the source near the mining openings and the other one far from it.The two main types can occur in different variations depending on natural (geological, tectonical, rock-mechanical) conditions. And the damages in the mine are consequences of the rock bursts determined by technical, technological and, of course, geological conditions.Under “induced seismic event” they understand (like earthquakes as natural, tectonic seismic events) the source process (focal fracture mechanism), the radiation of energy (seismic effect), the effect of seismic waves (dynamic loading) on a site (underground or at the surface, respectively).



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< previous page page_747 next page >Page 747FUNDAMENTALS FOR A NEW DEFINITION AND CLASSIFICATIONThe first question to be answered is the question for the aim of a new definition and classification of rock bursts in the general sense. In mines there are two main fields of interest: i) The interest to have a definition and a classification to describe and classify the effects in the mine, i.e. a tool for safety statistics and for communications with mining safety authorities. This is important for the evaluation of the development of the safety in the mine and to decide appropriate measures for fighting against damages.ii) The interest to have a definition and classification as a tool to be able to evaluate the interaction of mining technology and natural geological and tectonical conditions with respect to the origin of mining-induced seismic events and the developement of a potential for damaging effects within the mining workings and/or the surface above the mine.The first point contains a definition and classification more for the daily mining practice in certain districts. The regulations depend on the local conditions and the specific mining technology and agree—as a rule—with other regulations and definitions in the mine. The miners does not want to change this system of regulations and definitions. But the definitions and classifications on this base are local ones and not compatible with the corresponding definitions in other mining districts.Therefore most of results of statistical and technical investigations, some measures for forecasting, combatting and fighting against rock bursts are not comparable with corresponding topics in other mining regions.Another point is the growing knowledge about the mechanism of rock bursts. From case studies in a lot of mines is well known, that for several “rock bursts” the focal zone of the primary brittle fracture process and the site of damages within the mine are more or less distant. Several hundred meters or single kilometers has been observed as distances, particularly in the very well investigated deep gold mines of South Africa /14, 17/, in Ostrava-Karvina district /18/, in the Tkibuli-Shaor deposite /19/ and in other regions. It is impossible, especially in



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< previous page page_748 next page >Page 748the above named cases, to understand the whole process of preparation, fracture, energy exchange as well as place and extension of damage of the “rock burst phenomenon” if as a “rock burst” are considered only fractures of the immediate contour zones of mining openings causing damages.In the same sense, if one only considered the site of damages, even prevention measures cannot were directed to the primary reasons of the burst and so they cannot be effective from the very start in some cases. Particularly, one cannot fight successfully against even the very powerful rock bursts showing a complex and more-component fracture mechanism if one is going out form a single opening or mine working but not from the large scale interaction of the mine with the “geological environment” in a region extended in the order of several times of the extension of the whole mine.In the conclusions of the last ECE-syposium /1/ one can find:“20. It was clear from the discussion that while recognizing the achieved progress in understanding the circumstances and causes of rock bursts as well as of their forecasting and prevention, further research in this field should continue on a larger scale on national and international levels. This related particularly to the research on: measurement methods and techniques, modelling techniques, and simulation methods applied on local and regional scale.21. For the common benefit of the mining industrie of the ECE member countries, the basic research on these phenomena could be co-ordinated so as to direct the scientific efforts on subjects of priority to the safety of mines. Among others, such activities should focus on the following problems:– basic theoretical research on the rock burst occurrence,– uniform terminology and classification of rock bursts occurrence and rock burst mechanics,– rock burst occurrence in seismic regions as well as in regions with induced seismicity and its relationship,– evaluation of rock burst energy balance and its classification…– setting up regional measures for perventing and forecasting rock bursts…during the design, con-



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< previous page page_749 next page >Page 749struction and exploitation of mines,– geodynamic surveying of coal fields”.To fulfill these clear tasks, it is to develop even a rock burst definition and classification in the second above mentioned field of interest, containing– the whole and complex rock burst process,– in the first order not local but regional and general features,– generalisations, permitting the use of the results to all local mining conditions.The definition and classification must therefore have a “generalized model character” and must give the joint base for local site-related measures and regulations. So, the definition and classification can help the researchers to develope the modelling of the rock burst process, the engineers to carry out and compare case studies and methods for forecasting, prevention and limitation as well as can help to develop countermeasures really directed to the primary reasons of the fracture processes and to find the possible places of effecting the mine (and the surface).DRAFT PROPOSALS FOR A NEW DEFINITION AND CLASSIFICATIONThe following proposals are given for a new definition and classification corresponding with the above summarized fundamentals and with the task of the ECE-symposium /1/:Definition:1. Rock bursts are mining-induced seismic events.2. Mining-induced seismic events are sudden, brittle fractures of parts of the rock mass within the range of the mining-induced stress redistribution and connected with emission of deformation energy stored in the rock mass, which is influenced by mining stress redistribution.3. The sources of stored and emitted energy are– the tectonic stress field and– the gravitational stress field and– the immediately mining-induced stress field and/or– the mediately mining-induced stress field.



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< previous page page_750 next page >Page 750The immediately mining-induced stress field is the stress field induced by mineral extraction and changing of geometrical structure in a mine (underground and/or at the surface: e.g. by shafts, openings, stopes, roadways, open-pits, bunkers, pillars, insulas within the goaf etc.). The immediately mining-induced stress field occure, as a rule, in different scales: in the contour zones of openings (roof, bottom, seam edge etc.) and in bigger dimensions and distances from the openings within the rock mass.The mediately mining-induced stress field are the stresses caused by technogenous sinking or rising of ground water level, weight changes by reditributions of large masses, destressing of water, gas or oil accumulations, injection or extractions of liquids within the influence region of the mine, inducing of large scale deformation fields around mine etc.4. Depending on the position of the primary fracture zone (primary focal zone) and the real natural and technical conditions the mining-induced seismic events can result in local destructions within the mining openings or at the surface.Rock burst process:Corresponding to the definition given above a mining-induced seismic event (rock burst in the extended sense) is a complex process containing the following essential parts:1. Brittle fracture of primary source volume2. Emmission of energy (seismic waves, deformation steps, shocks)3. Fractures induced by dynamic loading of highly stressed underground structures and highly loaded parts of the rock mass outside the primary source4. Shaking the surface above and near the mine including all structures and buildings on the surface5. Development of a new equilibrium state in and around the underground openings and in the over- and underlying rock strata6. Local fractures (incidental also brittle ones) as a result of the development of the new equilibrium state in the rock mass including the surroundings of underground openings.



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< previous page page_751 next page >Page 751In general all parts of a mining-induced seismic event (rock burst) will occure but depending on the real local natural and technical conditions the weights of the different parts can be very different.Some examples should demonstrate that:The classical rock burst in the narrow sense is e.g. an outburst of the edge of a coal seam. Part 1 is the dominating one and occurs very near to the mi ne working; parts 1, 2 and 3 take place nearly at the same site; part 4 is very small, mostly only to point out by instrumental registrations; part 5 is limited to the immediately focal zone, too.The case of “sudden failure of a residual pillar within the goaf” shows a clear part 1 very distant from the active mining workings; also part 2 will be, of course, and part 3 can be, depending on the value of emitted energy and on the strength conditions of the contour zones of the nearest active openings. Often, the parts 4 and 5 are perceiveable and part 6 can occur, occasionally, in openings .it the edge of the goaf.The mining-induced seismic events in the Saskatchewan potash mines /20/ show the parts 1, 2, 4 and 5 but not part 3 because of the lack of sensitivity to dynamic loading of sylvinitic salt rocks in the mine.In the opposite the very powerful mining-induced seismic events in the German potash mines /12/ show the same parts of the burst process as those of Saskatchewan but additional very clear parts 3, 5, and 6. The part 3 is here dominating due to the sensitivity to dynamic loading of the high stressed carnallitic pillars.The “large area rock bursts” in the mines of Kolar Gold Fields in India /21/ are typical examples for dominating parts 1, 2, 5 and, particularly, 6. One big “bump” (parts 1 and 2) is accompanied by a number of small single rock bursts in a relatively narrow time window at different places of the mine (parts 5 and 6).On the base of a characterisation of the rock burst process by the 6 parts, one can use and compare the local technology—and geology—dependent classifications. It is important to point out the relations of the local events to the 6 parts of a generalized rock burst (mining-induced seismic event) and not longer to exclude some events from the phenomenon “rock burst” if one or more of the 6 parts



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< previous page page_752 next page >Page 752are not so clear recognizable or not present.Some examples should be given for the correspondence of classical terms of rock bursts and the 6 parts of the rock burst process:– “rock shooting” /e.g. as defined in / /: It is a micro-rock burst, consisting only of the part 1 and a very small part 2.– “bump” (e.g. as defined in /6/): The terminus bump stands in the literature for a rock burst with dominating part 1 distant from the workings and, partially, no part 3.– “rock fall” (e.g. as defined in /6/): Rock fall is excluded from the mining-induced seismic events by the definition; it is no rock burst because the source of emitted energy is only the gravity field; the other sources are not active immediately, and the primary fracture process is not a brittle one.– “tectonic rock burst” (e.g. as defined in /11/): In tectonic rock bursts, as a rule, all 6 parts of the rock burst process are clear developed and the tectonic rock bursts are therefore the most powerfull and damaging ones.– “pillar burst” (e.g. as defined in /22/): A pillar burst is a clear local fracture with dominating part 1 and the occurrence of parts 1, 2 and 5 at the same place (pillar).CONCLUSIONSThe very complex phenomenon “rock burst” should be understud as a “mining-induced seismic event” presenting as a complex fracture process. The process consists of six parts. The parts can occur more or less simultaneously but they can have very different weights or single parts can be absent, respectively, if the local conditions are not given for one or the other part. That means on the other side, if the conditions for the single parts are changing in time or space the features of rock bursts can change, too. So, a “bump” (part 1 is dominating far from the openings) can develop very quickly to a “tectonic rock burst” if the mining operations reach geological zones, which rocks are high stressed and sensitive for dynamic loading. Otherwise, one can change the very dangerous “tectonic rock burst” into “bumps”, not dangerous for the underground openings in the mines if one can change the sensitivity for dynamic loading of the adjacent rocks of the ope-



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< previous page page_753 next page >Page 753nings into a state of unsensitivity. This conception will be used e.g. in the Tkibuli-Shaorsk-coal deposite /19/.The formulation of the complex rock burst process is therefore a usefull tool– to recognize the full interaction of the single parts of the process,– to direct the measures for prevention, prediction, and combating to the real primary reasons of the burst,– to evaluate better a possible change of rock bursts in the course of the development of the mining extractions in time and space and– to give the very different local definition and classification systems a common base and to make possible the comparison of case studies in different mining districts.The proposals allows to integrate the different termini, data and measuring results and make they so comparable and exchangeable. The publication of the proposal is also a call to the miners, seismologists, geologists and rock-mechanicians engaged in the field of rock bursts to a critical discussion and to share our contributions and proposals to the common aim: create an international classification applicable in all mining regions and promoting the activities for further clearing up the mechanics of rock bursts.As the chairman of the corresponding ECE-Working Group the author will collect all remarks and proposals for evaluation in the group.Literature:1 – UNO/ECE: Official Document COAL/SEM.10/2; 23. October 19892 – UNO/ECE: Official Document ECE/COAL/121; 15. October 19903 – UNO/ECE: Official Dokument COAL/GE.1/34; 7. February 19904 – UNO/ECE: Official Document COAL/AC.6/CRP.1; 15 January 1991



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< previous page page_754 next page >Page 7545 – Bräuner, G.: Rock bursts and their prevention in the Ruhr mining district (in German) Verlag Glückauf Essen, 19896 – Bräuner, G.: Rock pressure and rock bursts (in German) Verlag Glückauf Essen, 2., neubearb u. erg. Aufl. 19917 – Gibowicz, S.J.: The mechanism of seismic events induced by mining: a review Rock bursts and Seismicity in Mines, Ch. Fairhurst (ed.), A.A.Balkema, Rotterdam, 19908 – Knoll, P.: About the mechanism of brittle fractures in mining areas (in German) Diss. B, Bergakademie Freiberg, 19819 – Knoll, P.; Thoma, K.; Hurtig, E.: Rock bursts and seismic events in mining areas (in German) Rock Mech., Suppl. 10, 85–102 (1980)10 – Knoll, P.: Investigations of the geomechanical mechanism of rock bursts by seismological methods (in German), Berg-und Hüttenmänn. Monatsh., Wien, New York (1987) 4, 97–10311 – Knoll, P.; Kuhnt, W.: Seismological and technical investigations of the mechanics of rock bursts Rockbursts and Seismicity in Mines, Ch. Fairhurst (ed.), A.A.Balkema, Rotterdam, 1990, 129–13812 – Knoll,P.: Discussion on the 4th. Int. Congr. Rock Mech., Montreux; Proc.Vol. 4, A.A.Balkema, Rotterdam, 1979, 406–40813 – Konecny, P.; Knezlik, J.; Kozak, J.; Vesely, M.: The development of mining-induced seismisity ISRM, Proc. Int. Congr. Rock Mech., G.Herget, S. and S.Vongpaisal (ed.), Vol. 2, 1017–1021, 198714 – Ryder, J.A.: Excess shear stress (ESS): an engineering criterion for assessing unstable slip and assocated rock burst hazards ISRM, Proc. Int. Congr. Rock Mech., G.Herget and S.Vongpaisal (ed.), Vol. 2, 1211–1215, 198715 – Shemjakin, E.I.; Kurlenja, M.B.; Kulakov, G.I.: About classification of rock bursts (in



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< previous page page_755 next page >Page 755 Russian) Fiz. tech. prob. raz. pol. iskop., Novosibirsk, 1986, 5, 3–1216 – Johnston, J.C.: Rockbursts from a global perspectivein: Special Issue “Induced Seismicity”, P. Knoll (ed.) Gerl. Beitr. Geophys., 98 (1989)6, 474–49017 – Mc Garr, A.; Bicknell, J.; Sembera, E.; Green, R.W.E.: Analysis of exceptionally large tremors in two gold mining districts of South Africa Seismicity in Mines, S.J.Gibowicz (ed.), Repr.from PAGEOPH, Vol. 29 (1989) No 3/4, 295–30818 – Konecny, P.: Mining-induced seismicity (rock bursts) in the Ostrava-Karvina coal basin, Czechoslovakia in: Special Issue “Induced Seismicity”, P.Knoll (ed.), Gerl. Beitr. Geophys., 98 (1989)6, 525–54719 – Yufin, S.A.; Shvachko, I.R.; Morozov, A.S.; Berdze- nishvili, T.L.; Gelashvili, G.M.; Gordeziani, Z.A.: Implementation of finite element model of heterogeneous anisotropic rock mass for the Tkibuli-Shaor coal deposite conditions ISRM, Proc. Int. Congr. Rock Mech., G.Herget and S.Vongpaisal (ed.), Vol.2, 1345–1348, 198720 – Gendzwill, D.J.: Induced seismicity in Saskatchewan potash mines Rockburst and Seismicity in Mines, N.C.Gay and E.C.Wainwright (ed.), Johannesburg, 1984, 131–14621 – Srinivasan, C.; Sringarputale, S.B.: Mine-induced seismicity in the Kolar Gold Fields in: Special Issue “Induced Seismicity”, P.Knoll (ed.), Gerl. Beitr. Geophys., 99 (1990)1, 10–2022 – Petuchov, I.M.: Forecasting and combating rock bursts: recent developments ISRM, Proc. Int. Congr. Rock Mech., G.Herget ans S.Vongpaisal (ed.), Vol.2, 1207–1210, 1987



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< previous page page_757 next page >Page 757AUTHORS’ INDEX Adachi T 403Alheid H.-J 719Andrus R.D 251Ansal A.M 49, 303Auersch L 467 Beresnev I.A 99Beskos D.E 731Boroomand B 491, 683Butkovich T.R 707 Cakmak A.S 515Carotti A 639Çelebi M 35Chang D.-W 111Chouw N 651Chu J 277 Darbre G.R 441Desai C.S 223Díaz O 545DiPasquale E 515 Eibl J 555Erdik M 3 Faust B 663Fei H.-C 201, 293Fujiwara A 139 Genske D 403Govil S 627Griffiths D.V 377Gucunski N 127Gurevich B 235 Haeri S.M 325Hataf N 215Haupt W 151Hayashi K 263Henseleit O 73Heuzé F.E 707Hinzen K.-G 719Hueffmann G.K 613 Ishihara K 365 Jagannath S.V 223Jiang T 23 Karakoç C 589Kastner R 415Kaynia A.M 491, 683Keane C.M 263Keintzel E 555


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Kiku H 341Kim D.-S 189Klapperich H 403Knoll P 743Kobayashi K 351Kokusho T 177Kostov M 73Kottnauer P 15Kundu T 223Kuribayashi E 23Kuroiwa S 23 Lav A.M 49Le R 651Lengnick M 693Little J.A 215Lopatnikov S 235 Madabhushi S.P.G 163Maddix D.M 707McGarr A 35Meler S 479Miranda E 577 Nagasaka H 23Nagase H 341Nakamura S 351Natke H.G 673Niiro T 23Nishioka S 23 Ohbo N 263



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< previous page page_758Page 758 Paronesso A 389Prevost J.H 263 Richart, Jr. F.E 201Rodriguez-Gomez S 515Roësset J.M 111, 189, 251Ruiz S.E 545 Sato K 351Savidis S 663Schäpertöns B 455Schenk V 15Schmid G 651Shen X 61Singhal A 627Song C 429Soubra A.H 415Springer H 455Steedman R.S 163Stein E 693Stokoe II K.H 111, 189, 251Storz M 503Sugito M 403 Takemiya H 139Takeuchi M 365Talaganov K 313Tanaka Y 177Temme C 455Thiede R 673 Uchida Y 341 Verbi• B 479Vgenopoulou I 731Vratsanou V 565Vrettos C 663 Walton O.R 707Wang C.Y 139Wolf J.P 389, 429Woods R.D 127Woodward P.K 377Wunderlich W 455 Yao S 351Yanagihara S 365Yasuda S 341Yoshida N 351Yoshida Y 177Yu J 61Yuzugullu O 599

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