Rüdiger Höffer Structural Dynamics with Applications in

Konstantin Meskouris Christoph Butenweg · Klaus-G. Hinzen Rüdiger Höffer

Structural Dynamics with Applications in Earthquake and Wind Engineering Second Edition

Structural Dynamics with Applicationsin Earthquake and Wind Engineering

Konstantin Meskouris • Christoph ButenwegKlaus-G. Hinzen • Rüdiger Höffer

Structural Dynamicswith Applicationsin Earthquake and WindEngineeringSecond Edition

123

With contributions from Ana Cvetkovic, Linda Giresini,Britta Holtschoppen, Francesca Lupi, Hans‐Jürgen Niemann

Konstantin MeskourisLehrstuhl für Baustatik und BaudynamikRWTH Aachen UniversityAachen, Germany

Christoph ButenwegFH Aachen—Universityof Applied Sciences

Aachen, Germany

Klaus-G. HinzenErdbebenstation Bensberg, Institutfür Geologie und Mineralogie

Universität zu KölnBergisch Gladbach, Nordrhein-WestfalenGermany

Rüdiger HöfferFakultät für Bau- undUmweltingenieurwissenschaften

Ruhr-Universität BochumBochum, Nordrhein-WestfalenGermany

ISBN 978-3-662-57548-2 ISBN 978-3-662-57550-5 (eBook)https://doi.org/10.1007/978-3-662-57550-5

Library of Congress Control Number: 2018949059

Originally published by Ernst & Sohn Verlag, Berlin, 20001st edition: © Ernst & Sohn Verlag, Berlin 20002nd edition: © Springer-Verlag GmbH Germany, part of Springer Nature 2019This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, expressed or implied, with respect to the material containedherein or for any errors or omissions that may have been made. The publisher remains neutral with regardto jurisdictional claims in published maps and institutional affiliations.

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Preface

The traditional design philosophy for buildings and structures, which takesalmost exclusively only statics considerations into account, is being steadily sup-plemented by the need for carrying out additional verifications concerning theirresponse and safety under dynamic loads. A reason for this may lie in the prolif-eration of modern bold architectural design forms favouring unorthodox and/orvery slender structures, which are often susceptible to vibration under dynamicexcitation. In addition, satisfying higher safety demands is increasingly required forbuildings serving important societal needs (e.g. hospitals), or structures with highintrinsic risk potential (e.g. large industrial units). A prerequisite for carrying outcomplex dynamic analyses is a familiarity with the theoretical foundations andnumerical methods of structural dynamics together with experience in the appli-cation of the latter and an insight into the nature of dynamic loads. The present bookaddresses both students and practising civil engineers offering an overview of thetheoretical basics of structural dynamics complete with the relevant software foranalysing the response of structures subject to earthquake and wind loadsand illustrating its use by means of many examples worked out in detail, with inputfiles for the programmes included. In the spirit of “learning by doing”, it thusencourages readers to apply the tools described to their own problems, allowingthem to become familiar with the broad field of structural dynamics in the process.Chapter 1 deals with the basic theory of structural dynamics followed by chapterson wind and earthquake loads. Chapters 4 and 5 deal with the behaviour ofbuildings and industrial units under seismic loading, respectively, while the finalchapter is devoted to the application of wind engineering methods to slendertower-like structures. May this book contribute to a deeper understanding and

v

familiarity of civil engineering students and practising engineers with the standardstructural dynamics methods, enabling them to confidently carry out all necessarycalculations for evaluating and verifying the safety of buildings and structures!

Aachen, Germany Konstantin MeskourisAachen, Germany Christoph ButenwegBergisch Gladbach, Germany Klaus-G. HinzenBochum, Germany Rüdiger Höffer

vi Preface

Contents

1 Basic Theory and Numerical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . 1Konstantin Meskouris1.1 Single-Degree-of-Freedom Systems . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Linear SDOF Systems in the Time Domain . . . . . . . . . . . 21.1.2 Linear SDOF Systems in the Frequency Domain . . . . . . . 151.1.3 Nonlinear SDOF Systems in the Time Domain . . . . . . . . 271.1.4 Applications of the Theory of SDOF Systems:

Response Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.2 Discrete Multi-degree of Freedom Systems . . . . . . . . . . . . . . . . . 41

1.2.1 Condensation Techniques . . . . . . . . . . . . . . . . . . . . . . . . 451.2.2 Lumped-Mass Models of MDOF Systems . . . . . . . . . . . . 491.2.3 Modal Analysis for Lumped-Mass Systems . . . . . . . . . . . 501.2.4 The Linear Viscous Damping Model . . . . . . . . . . . . . . . . 561.2.5 Direct Integration for Lumped-Mass Systems . . . . . . . . . . 601.2.6 Application to Base Excitation . . . . . . . . . . . . . . . . . . . . 61

Appendix: Descriptions of the Programs of Chapter 1, in AlphabeticalOrder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2 Seismic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Klaus-G. Hinzen2.1 The Earthquake Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

2.1.1 Earthquake Source Model . . . . . . . . . . . . . . . . . . . . . . . . 992.1.2 Seismic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

2.2 Strong Ground Motion Characteristics . . . . . . . . . . . . . . . . . . . . . 1062.2.1 Prediction of Ground Motion Parameters . . . . . . . . . . . . . 1082.2.2 Site Specific Ground Motion Parameters . . . . . . . . . . . . . 111

2.3 Source Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1152.3.1 Point Source Approximation and Equivalent Forces . . . . . 1152.3.2 Moment Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1212.3.3 Finite Source Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

vii

2.3.4 Seismic Source Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 1272.4 Site Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

2.4.1 Single Layer Without Damping . . . . . . . . . . . . . . . . . . . . 1312.4.2 Single Layer with Damping . . . . . . . . . . . . . . . . . . . . . . 1332.4.3 Multiple Layers with Damping . . . . . . . . . . . . . . . . . . . . 136

2.5 Design Ground Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1402.6 Examples of Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

3 Stochasticity of Wind Processes and Spectral Analysisof Structural Gust Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153Rüdiger Höffer and Ana Cvetkovic3.1 Short Review of the Stochasticity of Wind Processes . . . . . . . . . . 154

3.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1543.1.2 Stochastical Description of the Turbulent Wind . . . . . . . . 1543.1.3 Spectral Analysis of Wind Processes . . . . . . . . . . . . . . . . 162

3.2 Spectral Analysis of the Structural Gust Response . . . . . . . . . . . . 1693.2.1 Quasi-stationary Load Models . . . . . . . . . . . . . . . . . . . . . 1703.2.2 Aerodynamic Admittance Function . . . . . . . . . . . . . . . . . 1713.2.3 Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1733.2.4 Response Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1753.2.5 Implementation of Spectral Analysis in the Eurocode

Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1773.2.6 Modal Analysis for Structural Response Due

to the Wind Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . 180Appendix 3: MATLAB-Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

4 Earthquake Resistant Design of Structures Accordingto Eurocode 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Linda Giresini and Christoph Butenweg4.1 General Introduction and Code Concept . . . . . . . . . . . . . . . . . . . . 198

4.1.1 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1984.1.2 Performance and Compliance Criteria . . . . . . . . . . . . . . . 1994.1.3 General Rules for Earthquake Resistant Structures . . . . . . 2004.1.4 Seismic Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2034.1.5 Analysis Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2064.1.6 Torsional Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

4.2 Design and Specific Rules for Different Materials . . . . . . . . . . . . . 2314.2.1 Design of Reinforced Concrete Structures . . . . . . . . . . . . 2314.2.2 Design of Steel Structures . . . . . . . . . . . . . . . . . . . . . . . 247

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

viii Contents

5 Seismic Design of Structures and Components in IndustrialUnits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359Christoph Butenweg and Britta Holtschoppen5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3605.2 Safety Concept Based on Importance Factors . . . . . . . . . . . . . . . . 3605.3 Design of Primary Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3615.4 Secondary Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

5.4.1 Design Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3675.4.2 Example: Container in a 5-Storey Unit . . . . . . . . . . . . . . 373

5.5 Silos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3795.5.1 Equivalent Static Force Approach After Eurocode 8-4

(2006) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3815.5.2 Nonlinear Simulation Model . . . . . . . . . . . . . . . . . . . . . . 3865.5.3 Determination of the Natural Frequencies of Silos . . . . . . 3875.5.4 Damping Values for Silos . . . . . . . . . . . . . . . . . . . . . . . . 3935.5.5 Soil-Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . 3945.5.6 Calculation Examples: Squat and Slender Silo . . . . . . . . . 394

5.6 Tank Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4045.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4045.6.2 Basics: Cylindrical Tank Structures Under

Earthquake Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4045.6.3 One-Dimensional Horizontal Seismic Action . . . . . . . . . . 4095.6.4 Vertical Seismic Actions . . . . . . . . . . . . . . . . . . . . . . . . 4255.6.5 Superposition for Three-Dimensional Seismic

Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4305.6.6 Development of the Spectra for the Response

Spectrum Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4335.6.7 Base Shear and Overturning Moment . . . . . . . . . . . . . . . 4345.6.8 Seismic Design Situation and Actions for the Tank

Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.6.9 Sample Calculation 1: Slender Tank . . . . . . . . . . . . . . . . 4455.6.10 Sample Calculation 2: Tank with Medium

Slenderness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4585.6.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

Annex: Tables of the Pressure Components . . . . . . . . . . . . . . . . . . . . . 467References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

6 Structural Oscillations of High Chimneys Due to Wind Gustsand Vortex Shedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483Francesca Lupi, Hans-Jürgen Niemann and Rüdiger Höffer6.1 Gust Wind Response Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 484

6.1.1 Models for Gust Wind Loading . . . . . . . . . . . . . . . . . . . 4846.1.2 Aerodynamic Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 491

Contents ix

6.1.3 Comparison of the Models Based on CantileverResponses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

6.1.4 Conclusions and Future Outlook . . . . . . . . . . . . . . . . . . . 5116.2 Vortex Excitation and Vortex Resonance Using the Example

of High Chimneys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5176.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5176.2.2 Models, Methods and Parameters—The Eurocode

Models 1, 2 and the CICIND Model Codes . . . . . . . . . . . 5186.2.3 Worked Examples for Vortex Resonance . . . . . . . . . . . . . 5346.2.4 Structural Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551

x Contents

Authors and Contributors

About the Authors

Prof. Dr.-Ing. Konstantin Meskouris was born in Athens, Greece. He studiedCivil Engineering in Vienna and Munich, graduating in 1970 at the TechnicalUniversity (TU) Munich, from which he also received his doctoral degree. From1996 until his retirement in 2012, he served as Head of the Institute for StructuralStatics and Dynamics in the Civil Engineering Department of RWTH AachenUniversity.

Prof. Dr.-Ing. Christoph Butenweg was born in Borken, Germany. He graduatedin Civil Engineering at the Ruhr-Universität Bochum. From 1994 to 1999, heserved as Research Assistant at Essen University and subsequently as SeniorEngineer at the Institute of Structural Statics and Dynamics of RWTH AachenUniversity. Since 2006, he has been Manager of the SDA-engineering GmbH inHerzogenrath and since 2016 Full Professor for Technical Mechanics and StructuralEngineering at FH Aachen—University of Applied Sciences.

Prof. Dr. Klaus-G. Hinzen was born in Hagen, Germany. He studied Geophysicsat the Ruhr-Universität Bochum, graduating in 1979 and receiving his doctoratethere in 1984. From 1984 to 1995, he served as Researcher at the Federal Institutefor Geosciences and Natural Resources, Hanover. Since 1995, he has been Headof the Seismological Station Bensberg of Cologne University.

Prof. Dr.-Ing. Rüdiger Höffer was born in Attendorn, Germany. He graduated inCivil Engineering at Ruhr-Universität Bochum where he afterwards served as aresearch assistant. In 1996, he earned his doctoral degree in Bochum and subse-quently spent two years as a postdocoral researcher abroad. He gained practicalexperience as a project engineer and managing director of consulting offices atDüsseldorf and Bochum, Germany. In 2003, he was appointed Full Professor forWind Engineering at the Ruhr-Universität Bochum.

xi

Contributors

Ana Cvetkovic M.Sc., born at Pozarevac, Serbia, earned her bachelor diploma inStructural Engineering at Belgrade University and graduated master’s degree inComputational Engineering at the Ruhr-Universität Bochum in 2017. Since 2017,she works as Project Engineer in structural engineering.

Linda Giresini was born in Tempio, Italy. She earned her doctoral degree in CivilEngineering at the University of Pisa. She worked as Researcher at the Institute ofStructural Statics and Dynamics, RWTH Aachen University and at the Institute forSustainability and Innovation in Structural Engineering, University of Minho(Portugal) in 2013–2014. From 2014 to 2015, she was Research Assistant at theUniversity of Sassari, Italy. Since 2016, she has been working as AssistantProfessor of Structural Design at the University of Pisa.

Dr.-Ing. Britta Holtschoppen studied Civil Engineering at RWTH AachenUniversity and graduated in 2004. After a research stay at the University of Bristol,Great Britain, she returned to the Chair of Structural Statics and Dynamics ofRWTH Aachen University and focused her research on the seismic design ofindustrial facilities with special emphasis on secondary structures, tank structuresand probabilistic seismic performance assessment. Since 2016, she has worked atSDA-engineering GmbH, Herzogenrath.

Dr.-Ing. Francesca Lupi was born in Prato, Italy. She graduated master’s degreein Civil Engineering at the Universita Degli Studi di Firenze, Italy, and earned herdoctoral degree in 2013 in a joint programme of her home university and theTechnische Universität Braunschweig. In 2015, she received a research scholarshipfor postdocs from the Alexander-von-Humboldt Foundation, Germany. Since 2018,she is employed as Senior Research Assistant at the Ruhr-Universität Bochum.

Prof. Dr.-Ing. habil. Hans-Jürgen Niemann was born 1935 in Lüneburg,Germany. He studied Civil Engineering at the Universität Hannover and graduatedas Diplomingenieur. He worked as Research Assistant at the Universität Hannoverand the then newly established Ruhr-Universität Bochum, graduated as Doctor ofCivil Engineering and obtained the habilitation in Structural Engineering from theRuhr-Universität Bochum. He was appointed Full Professor of Wind Engineeringand Fluid Mechanics and served at the Department of Civil Engineering atRuhr-Universität Bochum until 2001. In the same year, he founded the EngineeringCompany Niemann und Partners, Bochum.

xii Authors and Contributors

Chapter 1Basic Theory and Numerical Tools

Konstantin Meskouris

Abstract This chapter offers an overview of the theoretical foundations and thestandard numerical methods for solving structural dynamics problems, with empha-sis placed firmly on the latter. Starting with the analysis of single degree of freedom(SDOF) systems both in the time and in the frequency domain, it includes sec-tions on the computation of elastic and inelastic response spectra, filtering in thefrequency domain, the analysis of nonlinear SDOF systems and the generation ofspectrum compatible ground motion time histories. Discrete multi-degree of free-dom (MDOF) systems, condensation techniques and dampingmodels are considerednext. Both modal analysis (“response modal analysis”) and direct integration meth-ods are employed, focussing especially on the behaviour of MDOF systems subjectto seismic excitations described by response spectra or sets of specific groundmotiontime histories. Detailed descriptions of the software used for solving the numerousexamples presented complete with full input-output parameter lists conclude thechapter.

Keywords SDOF system · Seismic excitation · Response spectrumSpectrum compatible accelerogram · Damping · MDOF system · Modal analysisDirect integration

In this section, themost important basics of structural dynamics are introduced,whichare needed in the further chapters of this book. The explanation of the theoreticalderivation is kept to a minimum, while the emphasis is set on practical applications.Formost algorithms, easy to use computing programs are provided,which applicationis illustrated by several examples.

Electronic supplementary material The online version of this chapter(https://doi.org/10.1007/978-3-662-57550-5_1) contains supplementary material, which isavailable to authorized users.

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2 1 Basic Theory and Numerical Tools

1.1 Single-Degree-of-Freedom Systems

Single-degree-of-freedom (SDOF) systems are the simplest oscillators. In spite oftheir simplicity, they are being successfully used as numerical models in many real-life cases. They are discussed in some detail in this chapter because of their wideapplication range and also because due to their “straightforwardness” they are emi-nently suitable for introducing basic structural dynamics methods and concepts.

1.1.1 Linear SDOF Systems in the Time Domain

Figure 1.1 depicts the standard case of a viscously damped SDOF system sub-ject to a time-varying external load F(t). From the free-body diagram we obtainby D’Alembert’s principle

FI + FD + FR � F(t) (1.1.1)

with FI, FD and FR as inertia, damping and restoring force, respectively. Setting theinertia force equal tomass times acceleration, the damping force equal to a coefficientc times velocity (linear viscous damping model) and the restoring force equal to dis-placement times the spring stiffness k yields the following 2nd order inhomogeneouslinear ordinary differential equation (ODE) with constant coefficients:

m · u + c · u + k · u � F(t) (1.1.2)

In order to avoid numerical errors in practice, it is advisable to use a consistentsystem of units, in which mass/force conversions are taken care of automatically.This is e.g. the case when masses are expressed in tons (1 ton�1000 kg), forces inkN, lengths in m and time in s. Accordingly, c in (1.1.2) might be given in units ofkN s/m, k in kN/m, F in kN and u in m.

The general solution of (1.1.2) is equal to the sum of the solution of the homoge-neous equation (F(t)�0) and a “particular integral”. The homogeneous ODE

u(t)

F(t)

m

k

cF(t) FI

FR

FD m

Fig. 1.1 SDOF system with free-body diagram

1.1 Single-Degree-of-Freedom Systems 3

u +c

m· u + k

m· u � 0 (1.1.3)

is satisfied by the function

u � eλt; u � λ eλt; u � λ2eλt (1.1.4)

leading to the characteristic equation

λ2 +c

mλ + ω2

1 � 0; ω21 � k

m(1.1.5)

with ω1 as circular natural frequency of the system. Its solutions are given by

λ1,2 � − c

2m±√( c

2m

)2 − ω21 (1.1.6)

The behaviour of the solution of the ODE depends on whether the radicand inEq. (1.1.6) is less than, equal to or greater than zero, corresponding to the under-damped, critically damped and overdamped case, respectively. In the latter case, λ1

and λ2 are real and no vibration occurs. The critical damping is defined as the valueof c for which the radicand is equal to zero:

c

2m� ω1 → ckrit � 2mω1 � 2

√km (1.1.7)

The dimensionless ratio D or ξ of the actual damping coefficient, c, to the criticaldamping, ckrit, called “damping ratio”, is regularly used for quantifying damping.The following expressions hold:

D � ξ � c

ckrit� c

2mω1;c

m� 2ξω1 (1.1.8)

Table 1.1 summarizes some typical values for the damping ratio for low-amplitudebuilding vibrations.

Introducing the damping ratio ξ and the natural circular frequency ω1, the differ-ential equation (1.1.3) can also be written as:

u + 2ξω1u + ω21u � 0 (1.1.9)

Table 1.1 Damping ratiosfor different structural types

Type of structure Damping ratio D or ξ (%)

Steel structure, welded 0.2–0.3

Steel structure, bolted 0.5–0.6

Reinforced concrete 1.0–1.5

Masonry 1.5–2


Its solution is

u(t) � e−ξω1t(C1 cos

√1 − ξ2ω1t + C2 sin

√1 − ξ2ω1t

)(1.1.10)

where C1, C2 are integration constants. For general initial conditions u(0) �u0, u(0) � u0 this leads to

u(t) � e−ξω1t(u0 cos√1 − ξ2ω1t +

(u0 + ξω1u0)

ω1

√1 − ξ2

sin√1 − ξ2ω1t) (1.1.11)

This expression can be further simplified by introducing the damped natural cir-cular frequency ωD (corresponding damped natural period TD):

ωD � ω1

√1 − ξ2; TD � 2π/ωD (1.1.12)

In the case of forced vibrations, Eq. (1.1.9) reads

u + 2ξω1u + ω21u � F(t)

m� f(t) (1.1.13)

The general solution of (1.1.13) is given by the sum of the homogeneous solutionEq. (1.1.11) and the particular integral (DUHAMEL integral)

up(t) � 1

ωD

t∫0

f(τ)e−ξω1(t−τ) sinωD(t − τ)dτ (1.1.14)

The DUHAMEL integral can be evaluated numerically for arbitrary forcing func-tions F(t). Alternatively, Eq. (1.1.13) can be solved by various Direct Integrationalgorithms as explained later.

Another widely used damping parameter, in addition to the critical damping ratioD or ξ according to Eq. (1.1.8), is the so-called logarithmic decrement�. It is definedas the natural logarithm of the ratio of the amplitudes of two successive positive (ornegative) peaks:

� � lnuiui+1

� lne−ξω1ti cosωDti

e−ξω1ti+1 cosωDti+1� ln e−ξω1(ti−ti+1) � ξω1(ti+1 − ti)

� ξω12π

ωD� ξω1TD � ξω1

2π

ω1

√1 − ξ2

� ξ2π√1 − ξ2

(1.1.15)

For the lightly damped systems normally encountered in structural dynamics, itis sufficiently accurate to write

ξ � D ≈ �

2π(1.1.16)


Fig. 1.2 Free vibration withviscous damping

0 0.2 0.4 0.6 0.8 1

Time, s

-0.02

-0.01

0

0.01

0.02

Dis

plac

emen

t, m

The logarithmic decrement can be experimentally determined from time-historymeasurements of free vibrations. Normally, two peaks u1 and un+1 occurring at timest1 and tn+1 and spanning n vibration cycles are considered, in which case we obtain

� � 1

nln

u1un+1

(1.1.17)

As an example, Fig. 1.2 shows the (calculated) displacement time history for aSDOF system with a natural period of T1 �0.20 s, an initial velocity at t�0 of0.6 m/s and a damping value of D�5%. The positive peak amplitudes of the firstfour cycles are given as 0.01785, 0.01304, 0.009518, 0.006949 and 0.005071 m,leading to a logarithmic decrement of

� � ln0.01785

0.01304≈ 1

4ln

0.01785

0.005071� 0.315; D ≈ �

2π� 0.05 (1.1.18)

As mentioned above, the differential equation of motion for the linear SDOFoscillator given by Eq. (1.1.2) with the general initial conditions u(0) � u0, u(0) �u0 can also be solved by Direct Integration in the time domain. There exist manysuitable algorithms for solving this classical initial-value problem.

Two issues are of central importance for the choice of an integration scheme,namely its stability and its accuracy. An unconditionally stable algorithm is presentif the solution u(t) remains finite for arbitrary initial conditions and arbitrarily large�t/T ratios, �t being the time step employed in the integration and T the natu-ral period of the SDOF system, T � 2π

√m/k. A conditionally stable algorithm

(which is generally more accurate than an unconditionally stable one) implies that


the solution remains finite only if the ratio �t/T does not exceed a certain value.For SDOF systems with known T it is easy to choose a suitable integration time step�t; however, unconditionally stable algorithms are generally preferable, especiallyif nonlinearities are to be considered.

The accuracy of a Direct Integration algorithm depends on the loading functionf(t), the system’s properties and especially on the ratio of the time step�t to the periodT. The deviation of the computed solution from the true one makes itself felt as anelongation of the period and a decay of the amplitude of the former, correspondingto a fictitious additional damping.

Integration algorithms may also be divided into single-step and multi-step meth-ods, which can also be implicit or explicit. Single-step methods, which are quitepopular in structural dynamics, furnish the values of u, u and u at time t + �t asfunctions of the same variables at time t alone, while multi-step methods requireadditional values at times t− �t, t− 2�t etc. Multi-step methods therefore involveadditional initial computations (e.g. by a single-step algorithm), while single-stepmethods are “self-starting”. Explicit algorithms furnish the solution at time t + �tdirectly, while in implicit methods the unknowns appear on both sides of algebraicequations and must be determined by solving the corresponding equation system (orjust one equation for a SDOF system). This shortcoming of implicit algorithms isoffset by their better stability properties.

The well-known NEWMARK β-γ-algorithm, to be used here, is an implicit,single-step scheme with two parameters β and γ which determine its stability andaccuracy properties. Considering the time points t1 and t2, with t2 � t1 + �t, thedynamic equilibrium of the SDOF system at time t2 is given by

mu2 + cu2 + ku2 � F(t2) � F2 (1.1.19)

Introducing increments of the displacement, velocity, acceleration and externalforce according to �u � u2 −u1,�u � u2 − u1 etc. leads to the incremental versionof Eq. (1.1.2)

m�u + c�u + k�u � �F (1.1.20)

The increments �u, �u can be given as functions of the displacement increment�u and the known values of velocity and acceleration at time t1:

�u � γ

β�t�u − γ

βu1 − �t(

γ

2β− 1)u1

�u � 1

β(�t)2�u − 1

β�tu1 − 1

2βu1 (1.1.21)

Values of β�1/4 and γ�1/2 correspond to an unconditionally stable schemewhich assumes a constant acceleration u between t1 and t2. For β�1/6 and γ�1/2the integrator is only conditionally stable and the acceleration varies linearly betweent1 and t2. The displacement increment �u is given by


�u � f∗

k∗ (1.1.22)

with

k∗ � m1

β�t2+ c

γ

β�t+ k (1.1.23)

and

f∗ � �F + m(u1

β�t+u12β

) + c

(γu1β

+ u1�t(γ

2β− 1)

)(1.1.24)

The NEWMARK algorithm is used in the programs SDOF1 and SDOF2, detailson which can be found in Appendix. The SDOF1 program deals with the case whenF(t) is an arbitrary piecewise linear function, while SDOF2 considers steady-stateexcitations of the type F(t) � A · sin(�1t) + B cos(�2t). For SDOF1 it is usuallynecessary to first use the program LININT, also described in Appendix, for deter-mining additional values of the forcing function F(t) for the chosen time step �t bylinear interpolation.

Example 1.1The task is to determine the maximum displacement u of the girder and also themaximum bending moment at the base of the central column for the frame shownin Fig. 1.3. Further data: Mass m�12 t (assumed to be concentrated in the girder),damping value D�1%, bending stiffness values EIG �1.25× 105 kNm2, EI1 �0.75× 105 kNm2, all beams and columns are considered to be inextensional (EA → ∞).

The single-story two-bay frame can be modelled using a SDOF idealization asdepicted in Fig. 1.1, with mass m�12 t and spring stiffness k in kN/m. The latter canbe determined from statics as the reciprocal of the horizontal girder displacement udue to a unit force F�1.0 kN (program FRAME) or by carrying out a static conden-sation for the horizontal girder displacement as a master DOF (program CONDEN,see Sect. 1.2.1). Here, the first approach will be used based on the discretization

4.75

m

5.00 m 5.00 m

F(t) m, EIG

u(t)

EI1

2EI 1

EI1

F(t)

t, s

1500 kN

0.0025 0.0050

Fig. 1.3 Plane frame with triangular loading function F(t)


1 2 3

4 5

1

2 3 4 5

6

Fig. 1.4 Discretization and input file for a unit load at DOF no. 2

of Fig. 1.4 (see input description for the program FRAME in Appendix), using the6-DOF beam element shown in Fig. 1.46. For more details on the use of the programFRAME and the matrix deformation method of analysis employed here see Example1.5.

The program FRAME yields a horizontal displacement of the girder due to theunit load (1 kN) equal to 6.292 × 10−5 m. The reciprocal is the spring stiffness,which in this case equals k�15,893 kN/m. With k and m known, the undampedcircular natural frequency of the frame is

ω1 �√

k

m�√15,893

12� 36.39

rad

s(1.1.25)

the corresponding natural period being equal to

T1 � 2π

ω1� 0.173 s (1.1.26)

In view of the characteristics of the load function, a time step�t of 0.5× 10−3 s ischosen and the program LININT used to create the file RHS.txt as input file for theprogramSDOF1. For 600 time points (fromzero to 0.3 s) Fig. 1.5 shows the computedtime history for the horizontal displacement of the girder (solid line). The maximumdisplacement is reached at time t�0.046 s and is equal to 0.008445 m or 8.4 mm.In view of the short duration of the excitation, a simple check on the validity of thisresult can be carried out by using the principle of impulse and momentum, whichstates that the final momentum of a mass m may be obtained by adding its initialmomentum (which, in this case, is zero) to the time integral of the force during theinterval considered. This allows the velocity at time t�0.005 s to be determined asfollows:


m · u1 +0.005∫0

F(t)dt � m · u2

→ u(0.005) �121500 · 0.005 kN s

12 t� 0.3125

m

s(1.1.27)

Equation (1.1.11) yields the following expression for the displacement u(t):

u(t) � e−0.01·36.39·t 0.3125

36.39√1 − 0.012

sin√1 − 0.012 · 36.39 · t (1.1.28)

This function, depicted as a dashed line in Fig. 1.5, is in almost perfect agreementwith the solution obtained using Direct Integration (solid line). From the maximumdisplacement umax �0.00845 m at t�0.046 s the maximum restoring force is deter-mined as FR,max �umax · k�0.00845 · 15,893�134 kN. Using the program FRAMEwith 134 kN as the load corresponding to DOF no. 2 yields a bending moment at thebase of the central column equal to 276 kNm. The bending moment diagram for theentire structure is shown in Fig. 1.6.

For a quick assessment of the maximum response of linear undamped SDOFsystems subject to impulsive loading, shock or response spectra are quite useful.They present dynamic magnification factors, defined as ratios of maximum dynamicdisplacements udyn,max to their static counterparts ustat as functions of the impulselength ratio t1/T, that is the duration t1 of the impulse divided by the natural periodof the SDOF system. Figure 1.7 shows shock spectra for three impulsive loadingshapes, namely rectangular, trapezoidal and triangular.

Fig. 1.5 Time histories forthe girder horizontaldisplacement

0 0.1 0.2 0.3Time, s

-0.012

-0.008

-0.004

0

0.004

0.008

0.012

Dis

plac

emen

t u, m


Fig. 1.6 Bending momentdiagram at t�0.00455 s

1 2 3

4 5 108

108

216

276

7272

M, kNm

Fig. 1.7 Shock spectra fordifferent impulsive loadingshapes

0 0.4 0.8 1.2 1.6 2

t1/T

0

0.5

1

1.5

2

2.5

Dyn

amic

mag

nific

atio

n fa

ctor

Rectangular

Trapezoidal

Triangular

Load

Time

1

Load

1

Load

Time

1

t1 t1 t1

A cursory look at Fig. 1.7 would seem to suggest that dynamic magnificationfactors do not exceed 2.0, which, however, is not the case. As an example, Figs. 1.8and 1.9 show some more shock spectra for piecewise linear and sinusoidal impulseshapes. In Fig. 1.8 the solid line corresponds to the positive/negative impulse shownto the left and the dashed line to the positive/positive one shown to the right, whilein Fig. 1.9 the solid line corresponds to the single half-sine and the dashed line tothe double half-sine impulse.


Fig. 1.8 Additional shockspectra for further polygonalimpulses

Load

1t1

0,5 t1

Load

1

t10,5 t1

0 0.4 0.8 1.2 1.6 2

t1/T

0

1

2

3

Dyn

amic

mag

nific

atio

n fa

ctor

Pos./neg.

Pos./pos.

A special case with significant practical importance is the linear SDOF systemsubject to stationary harmonic excitation as shown schematically in Fig. 1.10.

Its equation of motion is given by

u + 2ξω1u + ω21u � Fo

msin�t (1.1.29)

It has the general solution

u(t) � exp(−ξω1t)(A sinωDt + B sinωDt)

+Fok

1

(1 − β2)2 + (2ξβ)2[(1 − β2) sin�t − 2ξβ cos�t] (1.1.30)

where β is the ratio of the excitation frequency to the natural frequency of the system,that is

β � �

ω1(1.1.31)


Fig. 1.9 Shock spectra forsinusoidal impulses

00 .0 04 .0 08 .12

0

0.2

0.4

0.6

0.8

11

Load

t10,5 t100 .0 04 .0 08 .12

0

0.2

0.4

0.6

0.8

11

Load

t1

0 0.4 0.8 1.2 1.6 2t1/T

0

1

2

3D

ynam

ic m

agni

ficat

ion

fact

or

Single half sine

Double half sine

Fig. 1.10 SDOF systemunder harmonic excitation

u(t)

F0 sin( t)

m

k

c

The first part of the expression in Eq. (1.1.30) is the solution of the homogeneousdifferential equation; its constants A and B must be determined from the initialconditions. The second part is the particular integral which depends on the loading;this is the most important part of the system response, since the first part is eventuallydamped out, as is evident from the factor e−ξω1t. The second part of the solution canbe written down in the form

u(t) � uR sin(�t − ϕ) (1.1.32)


Fig. 1.11 Magnificationfactor V for differentdamping ratios D

0 1 2 3 4

Frequency ratio β

0

2

4

6

8

10

Dyn

amic

mag

nific

atio

n fa

ctor

V

5%

10%

20%

with

uR � Fok[(1 − β2)2 + (2ξβ)2]−0.5 (1.1.33)

and

ϕ � arctan2ξβ

1 − β2(1.1.34)

From Eq. (1.1.33) the dynamic magnification factor V defined as

V � [(1 − β2)2 + (2ξβ)2]−0.5 (1.1.35)

can be extracted. It is seen to be equal to the ratio of the dynamic to the static responseof the harmonically excited SDOF system, where the maximum dynamic responseis given by

max up(t) � uR � F0kV (1.1.36)

Figure 1.11 shows V as a function of the frequency ratio for three damping ratios,namelyD�5, 10 and 20%; clearly, for undamped systems (ξ�0), V tends to infinity.

Peak values of the magnification factor occur at the frequency ratio

β �√1 − 2ξ2 (1.1.37)


Fig. 1.12 Example 1.2, timehistory of displacement

0 1 2 3 4 5Time, s

-0.008

-0.004

0

0.004

0.008

Dis

plac

emen

t, m

They amount to

max V � 1

2

1

ξ√1 − ξ2

(1.1.38)

For a damping ratio of 1% this gives a peak value of 50, for 5% of about 10 andeven for a highly damped system with D�20% we obtain V�2.55.

Example 1.2Consider the system of Fig. 1.10 with the following data: Spring stiffness k�9000kN/m, m�10 t, F0 �25 kN, ��20 rad/s and D�5%. Determine the displacementand velocity time histories u(t) and u(t) as well as the maxima of the restoring forceand the damping force.

The circular natural frequencyω1 of the SDOF system is equal to√

km �

√900010 �

30 rads , corresponding to T1 �0.21 s. The programSDOF2produces the results shown

for the time histories of the displacement (Fig. 1.12) and the velocity (Fig. 1.13)using a time step of 0.005 s. The maximum restoring force FR occurs at time 0.25 s,corresponding to a maximum displacement of −0.00688 m, and is equal to k · umax

�−61.9 kN, while the maximum velocity of 0.161 m/s, occurring at t�0.32 s,produces a damping force equal to

2 · ξ · ω1 · m · u � 2 · 0, 05 · 30 · 10 · 0.161 � 4.83 kN (1.1.39)

Both these values were reached during the initial vibration stage, before the vis-cous damping mechanism eliminated the contribution of the “homogeneous” part


Fig. 1.13 Example 1.2, timehistory of velocity

0 1 2 3 4 5Time, s

-0.2

-0.1

0

0.1

0.2

Velo

city

, m/s

of the solution. In the subsequent steady-state harmonic vibration stage, with a fre-quency ratio of

β � �

ω1� 20

30� 0.667 (1.1.40)

the maximum displacement amounts to

max up(t) � F0kV � 25

9000V(0.667) � 25

90001.79 � 0.005m (1.1.41)

Since here u(t) is a sine wave with circular frequency �, the maximum velocityis readily determined as

max u � � · umax � 20 · 0.005 � 0.1m/s (1.1.42)

1.1.2 Linear SDOF Systems in the Frequency Domain

It can be shown that any real periodic function of time with period T

f(t + T) � f(t) (1.1.43)

can be expressed in the form


f(t) � a0 +∞∑k�1

ak cosωkt +∞∑k�1

bk sinωkt (1.1.44)

with the coefficients

a0 � ω

2π

2πω∫

0

f(t)dt � 1

T

T∫0

f(t) dt (1.1.45)

and

ak � 2

T

T/2∫−T/2

f(t) cosωkt dt (1.1.46)

bk � 2

T

T/2∫−T/2

f(t) sinωkt dt (1.1.47)

Here, ωk � k2πT , k�1, 2, …∞. The coefficients ak and bk can be regarded as the

real and imaginary part, respectively, of the harmonic component associated withthe circular frequency ωk. They can be displayed along a frequency axis at discretepoints ωk with an increment in rad/s equal to

�ω � 2π

T;

2

T� �ω

π(1.1.48)

Figure 1.14 shows such “comb spectra” consisting of discrete values of the coef-ficients ak and bk every (2π/T) rad/s.

Fig. 1.14 “Comb spectra” for periodic functions


For a0 �0 we obtain

f(t) �∞∑k�1

⎛⎜⎝ 2

T

T/2∫−T/2

f(t) · cosωktdt

⎞⎟⎠ cosωkt

+∞∑k�1

⎛⎜⎝ 2

T

T/2∫−T/2

f(t) · sinωktdt

⎞⎟⎠ sinωkt (1.1.49)

or, using Eq. (1.1.44)

f(t) �∞∑k�1

⎛⎜⎝�ω

π

T/2∫−T/2

f(t) · cosωktdt

⎞⎟⎠ cosωkt

+∞∑k�1

⎛⎜⎝�ω

π

T/2∫−T/2

f(t) · sinωktdt

⎞⎟⎠ sinωkt (1.1.50)

For an aperiodic function we can assume T → ∞, �ω → dω and

f(t) �∞∫

ω�0

1

π

⎛⎝

+∞∫−∞

f(t) · cosωt dt

⎞⎠ · cosωt dω

+

∞∫ω�0

1

π

⎛⎝

+∞∫−∞

f(t) · sinωt dt

⎞⎠ · sinωt dω (1.1.51)

Introducing

A(ω) � 1

2π

∞∫−∞

f(t) cosωt dt, B(ω) � 1

2π

∞∫−∞

f(t) sinωt dt (1.1.52)

leads to

f(t) � 2

∞∫ω�0

A(ω) · cosωt dω + 2

∞∫ω�0

B(ω) · sinωt dω (1.1.53)

and, considering that

A(ω) · cosωt � A(−ω) · cos(−ωt)

B (ω) · sinωt � B(−ω) · sin(−ωt) (1.1.54)


Fig. 1.15 Time domainfunction

to

f(t) �∞∫

−∞A(ω) · cosωt dω +

∞∫−∞

B(ω) · sinωt dω (1.1.55)

With complex coefficients

F(ω) � A(ω) − i · B(ω) (1.1.56)

we finally obtain

F(ω) � 1

2π

⎛⎝

∞∫−∞

f(t) cosωt dt

⎞⎠− i

1

2π

⎛⎝

∞∫−∞

f(t) sinωt dt

⎞⎠

F(ω) � 1

2π

⎛⎝

∞∫−∞

f(t)[cosωt − i sinωt]dt

⎞⎠ (1.1.57)

This is the formal definition of the FOURIER transform F(ω) of the time domainfunction f(t):

F(ω) � 1

2π

∞∫−∞

f(t)e−iωtdt (1.1.58)

The inverse transform is given by

f(t) �∞∫

−∞F(ω)eiωtdω (1.1.59)

F(ω) and f(t) form a “FOURIER transform pair”. As a simple practical examplefor transforming a time domain function into the frequency domain, consider the“boxcar” function shown in Fig. 1.15.

The function is even, f(t)� f(−t), so that in Eq. (1.1.52) B(ω)�0. We obtain

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Rüdiger Höffer Structural Dynamics with Applications in