D. Belleri - Displacement Based Design for Precast Concrete Structures

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University of Trento University of Brescia University of Padova University of Trieste University of Udine

University IUAV of Venezia

ANDREA BELLERI (Ph.D. Candidate)

DISPLACEMENT BASED DESIGN FOR PRECAST CONCRETE STRUCTURES

Prof. Paolo Riva (Tutor)

SUMMARY

The objective of this research is to check the suitability of the Direct Displacement Based Design (DDBD) procedure when applied to the seismic design of precast concrete structures and how the procedure is affected by taking into account the influence of foundation flexibility, beam to column and foundation to column connections. Relationships are derived to relate the hysteretic parameters used to calibrate the Equivalent Viscous Damping (EVD) equation (used in the DDBD procedure to estimate the system damping) to the momentcurvature parameters model adopted to describe the column flexural behavior. A set of experimental tests has been carried out to analyze the behavior of some foundation to column connections typical of the precast industry and to calibrate the parameters of the EVD equation associated to each connection type. A new calibration procedure is proposed and applied to determine the parameters of a modified formulation of the EVD equation, suitable for different hysteretic rules, and the dependence of the latter on the type of ground motions (near field and far field) and on the records spectral response shape is investigated. The ground motion selection and scaling have been seen to play a significant role in the design validation procedure, therefore a ground motions scaling procedure is proposed to control and limit the results variability of non linear time history analyses and the results dependence on the ground motion set chosen. This procedure could be suitable for the EVD equation calibration other than for non linear time history analyses. The study on precast concrete structures is extended to the use of rocking walls as an alternative Lateral Force Resisting System (LFRS). The main issues related to this type of system are outlined such as the base sliding, the definition of a moment rotation relationship to use in the non linear analyses and the peculiarity of the equations of motion. Rocking walls have been the lateral force resisting system of an extensive experimental campaign on precast diaphragms involving shake table tests recently concluded at the University of California at San Diego. The design and the experimental results of these walls are presented and efforts have been made to explain their dynamic behavior especially regarding the vertical and horizontal acceleration spikes associated to the wall impacts once rocking is triggered.

SOMMARIO

Lobbiettivo principale della ricerca verificare lapplicabilit del metodo di progettazione sismica chiamato Direct Displacement Based Design (DDBD) a strutture prefabbricate tipiche del panorama costruttivo italiano. In particolare dinteresse definire come la procedura viene modificata per tenere in considerazione gli effetti del terreno, delle connessioni travi pilastro e delle connessioni tra pilastro e fondazione. Sono state determinate le relazioni intercorrenti tra i parametri del modello isteretico usato per calibrare lequazione dello smorzamento viscoso equivalente (EVD) nella procedura DDBD e i parametri del legame momentocurvatura utilizzato per descrivere il comportamento flessionale dei pilastri. Test sperimentali sono stati condotti per analizzare il comportamento dal punto di vista sismico di alcune connessioni pilastro fondazione prefabbricate e per calibrare i parametri dellequazione EVD ad esse associati. Viene quindi proposta una procedura alternativa per la calibrazione dei parametri di una nuova formulazione dellequazione EVD, adatta a vari modelli isteretici, valutando la dipendenza di questultima dalle caratteristiche degli accelerogrammi adottati, se di tipo near field o far field, e della relativa forma dello spettro di risposta. La scelta e lo scaling degli accelerogrammi giocano un ruolo significativo in fase di validazione non lineare della progettazione. In questa sede proposta una procedura di scaling atta a controllare e limitare la variabilit dei risultati delle analisi non lineari dovuta alla scelta degli accelerogrammi. In particolare tale procedura pu essere applicata nella calibrazione dei parametri dellequazione EVD. Dopo avere esaminato sistemi sismo resistenti classici utilizzati nella prefabbricazione, vale a dire costituiti da pilastri isostatci considerando o meno leffetto delle connessioni trave-pilastro, la ricerca continua con lo studio di sistemi sismo resistenti innovativi, quali lo sono i rocking walls, delineandone le caratteristiche e gli aspetti principali. Sono presentati la progettazione e i risultati sperimentali dei rocking walls utilizzati come sistema sismo resistente in una campagna di prove su tavola vibrante recentemente conclusa allUniversit della California San Diego. Tali risultati permettono considerazioni significative riguardanti il comportamento di questi sistemi quando sottoposti ad eccitazione dinamica, in particolare i picchi nellaccelerazione verticale e orizzontale a cui sono soggetti.

ACKNOWLEDGEMENTS

I gratefully thank all the people who sustained me in these last three years and the people who helped directly or indirectly to extend my knowledge.

My thanks and appreciation to my advisor prof. Paolo Riva who gave me the opportunity to take this challenge and helped me handling it.

I thank all the friends, colleagues, technicians and professors who made pleasant and fruitful my year and a half permanence at the University of California at San Diego

(UCSD). At this regard special thanks to Matthew Schoettler for having been a friend and a guide and to prof. Jos Restrepo who supervised the shake table experimental tests

and the last part of this research.

alla mia famiglia

INDEX

1. INTRODUCTION ............................................................................................................. 1

1.1 Displacement Based Design methodologies ........................................................... 4

1.2 Research plan description ....................................................................................... 9

2. GROUND MOTIONS AND CASE STUDIES ..................................................................... 13

2.1 Ground motions definition .................................................................................... 13

2.2 Case studies definition .......................................................................................... 14

3. FBD AND DDBD PROCEDURES ..................................................................................... 19

3.1 FBD Procedure ....................................................................................................... 19

3.2 DDBD Procedure .................................................................................................... 20

3.3 Considerations on equivalent viscous damping .................................................... 23

3.3.1 Relationship between displacement () and curvature () ductility .............. 27 3.3.2 Relationship between r and r ............................................................................ 28

3.3.3 Relationship between and ........................................................................ 29

3.3.4 Relationship between and .......................................................................... 30 3.4 Displacement response spectrum dependence from damping ............................ 31

4. PRECAST STRUCTURES CONSIDERATIONS ................................................................... 33

4.1 Definition of the q-factor for precast concrete buildings ..................................... 33

4.2 Precast concrete structures compared to other structures .................................. 35

5. PHASE 1: FBD AND DDBD COMPARISON ..................................................................... 39

5.1 FBD-DDBD comparison: Procedure 1 .................................................................... 40

5.2 FBD-DDBD comparison: Procedure 2 .................................................................... 51

5.3 Considerations about the inelastic displacements ............................................... 56

5.4 DDBD for 2.5% drift ............................................................................................... 59

5.5 Concluding remarks ............................................................................................... 61

6. PHASE 2: DDBD AND SOIL STRUCTURE INTERACTION ................................................ 63

6.1 First approach: elastic foundation ........................................................................ 64

6.2 Second approach: inelastic foundation ................................................................. 67

6.3 Considering foundation inertia ............................................................................. 73


7. PHASE 3: DDBD AND BEAM TO COLUMN CONNECTION ............................................ 79

7.1 Analytical study ..................................................................................................... 79

7.2 Procedure application to the case studies ............................................................ 90


8. PHASE 4: DDBD AND FOUNDATION TO COLUMN CONNECTION ................................ 99

8.1 Experimental tests ................................................................................................. 99

8.2 DDBD application ................................................................................................ 116

8.3 Yield curvature equation ..................................................................................... 118

8.4 Equivalent viscous damping equation re-calibration and results ....................... 125

8.5 Concluding remarks ............................................................................................. 130

9. HYSTERTIC DAMPING EQUATION CALIBRATION ....................................................... 131

9.1 Hysteretic damping calibration results ............................................................... 132

9.2 Hysteretic model parameters influence on the damping value .......................... 142

9.3 Concluding remarks ............................................................................................. 152

10. GROUND MOTION SCALING .................................................................................... 153

10.1 Record selection and scaling ............................................................................. 153

10.2 Constant Variance Spectrum Matching procedure ........................................... 155

10.3 CVSM application .............................................................................................. 158

10.4 Concluding remarks ........................................................................................... 170

11. ROCKING WALLS IN PRECAST CONCRETE STRUCTURES .......................................... 171

11.1 Rocking walls: an introduction .......................................................................... 171

11.2 Rocking walls experimental tests in the literature ............................................ 172

11.3 Rocking wall base sliding ................................................................................... 175

11.4 Equations of motion .......................................................................................... 179

11.5 Design recommendations ................................................................................. 184

11.6 Non linear time history analyses ....................................................................... 185

12. SHAKE TABLE TESTS INVOLVING ROCKING WALLS ................................................. 191

12.1 Hybrid wall general considerations and details ................................................ 193

12.2 Hybrid wall design ............................................................................................. 200

12.3 Test sequence .................................................................................................... 204

12.4 Instrumentation layout ..................................................................................... 207

12.5 Tests results ....................................................................................................... 210

12.6 Concluding remarks ........................................................................................... 232

13. CONCLUSION AND FUTURE DEVELOPMENTS ......................................................... 233

BIBLIOGRAPHY ................................................................................................................ 237

APPENDIX A: SECTION DATA FOR NONLINEAR ANALYSES .............................................. 245

LIST OF SYMBOLS

Ad dissipation bar area

APT post tensioning strand area

B column cross section size

be confined concrete region thickness

Bf foundation dimension

ccover concrete cover

cd viscous damping coefficient

cNA neutral axis depth

de elastic displacement

di inelastic displacement

Ec concrete Young modulus

Edissipated energy dissipated in one cycle

Eelastic elastic energy at maximum response

Es steel Young modulus

fcc confined concrete strength

fck concrete cylindrical strength

Fd damping force

Fe elastic force

Fi i-floor design force

Fl maximum confining lateral stress

fl minimum confining lateral stress

Fp0 initial prestress

FPT post tensioning force

Fu ultimate lateral force

Fy yield lateral force

fyk steel yield stress

g acceleration of gravity

G soil shear modulus

Gred reduced soil shear modulus

H structure height

Heff structural effective height

Ieff effective modulus of inertia

Igross gross modulus of inertia

k structural stiffness

keff effective stiffness

ki initial stiffness force-displacement relationship

ki initial stiffness moment-curvature relationship

Ks superstructure stiffness

ku unloading stiffness force-displacement relationship

ku unloading stiffness moment-curvature relationship

Kx foundation horizontal stiffness

ky yield stiffness force-displacement relationship

Kz foundation vertical stiffness

K foundation rotational stiffness

Lp plastic hinge length

lunb_ dissipation bar unbonded length

lunb_PT tendon unbonded length

Lw wall depth

m seismic mass

meff effective seismic mass

Mu design moment

My yield moment

N axial load

P gravity load

q force reduction factor

r post-yield stiffness ratio force-displacement relationship

r post-yield stiffness ratio moment-curvature relationship

Sa spectral acceleration

SD spectral displacement

T0 structural period at secant stiffness at yield

Teff effective period

Vb base shear

vs shear wave velocity

GREEK SYMBOLS

Takeda model parameter force-displacement relationship

Takeda model parameter moment-curvature relationship

Takeda model parameter force-displacement relationship Takeda model parameter moment-curvature relationship d target displacement

f displacement due to foundation rotation

res residual displacement

s structural displacement

u ultimate displacement

y inelastic displacement

cu maximum concrete compressive strain

y steel yield strain

p plastic curvature res residual curvature u ultimate curvature y yield curvature spectrum damping dependence

displacement ductility

curvature ductility

axial load ratio

soil soil Poisson modulus

second to first order moment ratio

f foundation rotation

0 yield to gross stiffness ratio

l longitudinal steel ratio

soil soil density

angular frequency

f foundation angular frequency

s structure angular frequency

initel initial stiffness elastic damping

tangel tangent stiffness elastic damping

eq equivalent viscous damping f foundation equivalent viscous damping hyst hysteretic damping s structural equivalent viscous damping

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1. INTRODUCTION

Several efforts have been made in the last decades to address the importance of changing the focus of current seismic design codes from merely preventing collapse in major earthquakes and controlling the damage in minor earthquakes to a more general design philosophy which takes into account multiple performance objectives based on quantifiable performance criteria; this design philosophy is referred to as Performance Based Design (PBD). The Olive View Hospital in Sylmar (CA - USA) represents one of the most significant examples of the need of PBD rather than Force Based Design (FBD) approach adopted by current codes. The hospital was destroyed by the 1971 San Fernando earthquake and it was completely rebuilt in 1976 to withstand increased levels of seismic forces according to a life safety criterion. The lateral force resisting system adopted is a mix design of concrete and steel shear walls which resulted in a very strong and stiff structure. During the 1994 Northridge earthquake the sensors in the building indicated a peak ground level acceleration of 0.91 g sensibly beyond the design acceleration of 0.52 g at which the building would not be damaged badly. The roof peak acceleration was recorded to 2.31 g. From a structural point of view the hospital did not sustain damage, but it had to be evacuated because of broken water pipes and other secondary damage with sensible economic losses. This example underlines the


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inadequacy of design practice targeted only to life safety and collapse prevention criteria and the important role of nonstructural components in the functionality of a building after an earthquake: this requires the development and application of PBD methodologies. The performance objectives are statements that relate an acceptable performance level in a structure to an earthquake design level. Performance targets which can be specified limits on response parameters (like stresses, strains, displacements and accelerations among others) correspond to each performance level. As mentioned before, particular care has to be placed to non structural performance levels as well because the way the non structural components (like partitions, ceilings, elevators and electrical, plumbing, mechanical, and fire protection systems) behave during an earthquake will affect the building operability and occupancy following an earthquake. The SEAOC (1995) made an effort to relate the performance levels to the expected damage in the overall building (i.e. both structural and non structural elements) as summarized in Table 1.1.

Table 1.1. Performance Levels and Damage States

Performance Levels

Damage States

Fully Operational

No damage. Continuous service: facilities operate right after earthquake.

Operational The structure is safe for occupancy immediately after earthquake. Repair is required to restore some essential services. The structure retains a significant portion of its original stiffness and most of its strength

Life Safe Life safety is attained and the structure remains stable although damaged. Substantial damage has occurred to the structure, and it may have lost a significant amount of its original stiffness. Significant margin remains before collapse would occur.

Near Collapse Severe damage. Non structural elements may fall. If laterally deformed beyond this point, the structure can experience instability and collapse

Collapse Portions of primary structural system collapse. Or as extreme the whole structure collapses.

INTRODUCTION

3

A performance objective is a coupling of expected performance levels with levels of seismic hazard, which is represented, at a given site, as a set of earthquake ground motions with specified probabilities of occurrence. SEAOC (1995) relates four levels of seismic hazard to three sets of performance objectives, which are associated to three types of facilities: Basic Facilities, Essential Facilities and Safety Critical Facilities (Table 1.2).

Table 1.2. Performance Levels and Damage States

Objectives Earthquake Performance Level Fully

Operational Operational Life Safe Near Collapse

Earth

quake

D

esi

gn Le

vel

Frequent (50% in 30 years)

Basic Facilities

Unacceptable performance

Occasional (50% in 50 years)

Essential Facilities

Basic Facilities

(for new constructions)

Rare (5% in 50 years)

Safety Critical Facilities


Basic Facilities

Very Rare (2% in 50 years)

Safety Critical Facilities


Basic Facilities

Therefore the design method chosen has to start from the definition of the performance objective associated to the appropriate earthquake design level and performance level. The performance levels should be defined by parameters which allow a quantitative identification of the structural performance and the design methodology should deal directly with these parameters. A seismic code which addresses the seismic design in terms of equivalent seismic forces as done in the past is not suitable for the procedure just described, because the main design parameter induced by the code scheme is the structural strength which is not directly correlated to damage. The performance targets could be a level of stress not to be exceeded, a load, a displacement or a limit state. The target displacement was first applied as the response parameter (Displacement Based Design) because the structural response in terms of displacement can be related to strain based limit states, which give better indicators of damage than stresses. Based on these considerations Performance Based Design and Displacement Based Design have been used interchangeably. This assumption is an oversimplification since the level of damage is influenced by several other parameters like


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for example the number of cycles and the duration of the earthquake and the acceleration levels which affect the behavior of secondary systems. Displacement Based Design should be thought as a subset of Performance Based Design.

1.1 Displacement Based Design methodologies

A quick review of the main Displacement Based Design (DDBD) methodologies available in the literature is presented in this paragraph. According to FIB bulletin 25 (2003) the design procedures involving DBD can be organized by different criteria that will be used here as reference in the procedures presented. The first distinction is the type of analysis used in the design process:

1. Initial Stiffness Response Spectra (ISRS): the procedure utilizes elastic stiffness coupled with approximations between elastic and inelastic response.

2. Secant Stiffness Response Spectra (SSRS): the procedure adopts the secant stiffness to the maximum response and the concept of equivalent viscous damping, which will be addressed in Chapter 3.

3. Time History Analysis (THA): the procedure uses linear or non-linear time history methods to solve the equations of motion by direct integration for a given earthquake in order to evaluate the system maximum response.

The role of the displacement in the design process is the second distinction taken into account:

1. Displacement Calculations Based (DCB): the procedure involves the calculation of the maximum displacement for an already designed structure. Detailing is made to lead to a displacement capacity greater than the demand, but no attempt is made to alter the structural system in order to change the displacement demand.

2. Iterative Displacement Specification Based (IDSB): the procedure involves the maximum displacement calculation for a designed structure as before, but iterative changes are made on the structural system in order to limit the maximum displacement to a specified value.

3. Direct Displacement Specification Based (DDSB): the procedure involves a specified target displacement as a starting point. The structure design follows directly leading, as end results, to the structural strength and stiffness necessary to reach the target displacement under the specified earthquake level.

INTRODUCTION

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Panagiotakos & Fardis (1999) proposed a displacement-based seismic design of multistory reinforced concrete buildings integrated into the overall structural design process including the effects of gravity loads. The procedure is summarized in Table 1.3.

Table 1.3. Panagiotakos & Fardis procedure

Type of analysis used in the design process ISRS Role of displacement in the design process DCB Description: (1) The first step of the procedure is the elastic analysis for non-seismic actions and serviceability earthquake with an elastic spectrum and adopting un-cracked sections. (2) Then a force-based proportioning of the longitudinal reinforcement in hinge locations is carried out and the capacity design rule is applied throughout the structure. (3) An elastic analysis for life-safety earthquake is carried out with a 5% damped spectrum and using the secant to yield members stiffness. (4) The upper-characteristics of chord-rotation demands are evaluated with provided amplification factors obtained from extensive time history analyses. (5) The chord rotation demand is verified and the longitudinal and transverse re-bars are modified if necessary. (6) Finally capacity design is applied.

Freeman (1998) proposed to compare response spectra for different levels of damping with the capacity spectrum obtained from dynamic considerations on pushover analysis results of a multi degree of freedom (MDOF) system (Table 1.4).

Table 1.4. Freeman procedure

Type of analysis used in the design process SSRS Role of displacement in the design process DCB Description: (1) For a given MDOF system determine through a pushover analysis the system capacity curve in terms of roof displacement versus base shear. (2) Use the dynamic characteristics of the structure (such as period of vibrations, mode shapes and modal participation factors) to convert (3) the MDOF capacity curve to a capacity spectrum in terms of Spectral acceleration versus Spectral displacement. (4) Calculate the response spectra for various levels of damping. (5) Use ductility-damping relation to identify different damping levels along the capacity spectrum curve. (6) The intersection of the capacity spectrum with the response spectrum with the appropriate level of damping determine the seismic structural demand.


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Fajfar (2000) modified the capacity spectrum method proposed by Freeman adopting inelastic instead of elastic response spectra in the procedure (Table 1.5).

Table 1.5. Fajfar procedure Type of analysis used in the design process ISRS Role of displacement in the design process DCB Description: (1) For a given MDOF system determine through a pushover analysis the system capacity curve in terms of roof displacement versus base shear. (2) Divide the force and displacement obtained by the modal participation factor of the first mode of vibration. This will determine the force and displacement of the equivalent single degree of freedom (SDOF) system. (3) Calculate the inelastic response spectra associated to different ductility values. (4) Use the capacity spectrum and the response spectra to determine the displacement demand of the SDOF system. (5) Convert the SDOF displacement demand into MDOF maximum top displacement.

Aschheim & Black (2000) procedure differs from the previous ones because it involves the use of yield point spectra representing the yield points of oscillators with constant displacement ductility (Table 1.6)

Table 1.6. Aschheim & Black procedure

Type of analysis used in the design process ISRS Role of displacement in the design process DDSB Description: (1) Develop yield point spectra for various ductility levels. (2) Determine target displacement that satisfies limits for desired risk event. (3) Identify in the yield point spectra the admissible design region in terms of system displacement and ductility. (4) Choose ductility limit for the desired performance level. (5) Determine the structural yield displacement and the corresponding yield strength. (6) Distribute the lateral force according to conventional methods. (7) Design and detail the structure.

INTRODUCTION

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Browning (2001) procedure allows to design regular reinforced concrete frame structures to reach a predefined average drift limit (Table 1.7).

Table 1.7. Browning procedure

Type of analysis used in the design process ISRS Role of displacement in the design process IDSB Description: (1) The first step is the evaluation of the maximum target period, whose exceedance will result in the drift exceeding a specified value, using displacement response spectra. (2) Proportion the members based on gravity load requirements. (3) Adjust the member size until the structural period is less than the target period. (4) Evaluate the base shear from structural period and compare it to an acceptable minimum. (5) Ensure an appropriate hierarchy of strength. (6) Provide structural details compatible with the maximum tolerable drift.

The displacement based procedure proposed by Priestley (1997) adopts a substitute structure approach to characterize the structure by a single degree of freedom system with stiffness the secant structural stiffness at maximum displacement and with a level of equivalent viscous damping appropriate to take into account the hysteretic energy absorbed during the inelastic response.

Table 1.8. Priestley procedure

Type of analysis used in the design process SSRS Role of displacement in the design process DDBS Description: (1) Estimate the yield deformation of the system (first inelastic mode of vibration). (2) Determine the SDOF substitute structure effective height and effective mass from the system deformed shape at yield. (3) Determine the system equivalent viscous damping to represent the elastic and hysteretic damping of the system. (4) Get the substitute structure effective period from the displacement spectrum reduced according to the equivalent viscous damping. (5) Determine the substitute structure effective stiffness and the system base shear. (6) Distribute the base shear as design forces along the structure proportionally to the inelastic displacements and masses. (7) Design the structure member according to the design forces and apply capacity design.


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Chopra & Goel (2001) proposed a modified version of the method of Priestley (1997) which adopts inelastic spectra (Table 1.9).

Table 1.9. Chopra & Goel procedure

Type of analysis used in the design process ISRS Role of displacement in the design process DDBS Description: (1) Estimate the yield deformation of the system. (2) Determine the design displacement and ductility factor from acceptable plastic rotation considerations. (3) Enter inelastic constant ductility displacement spectra with the system displacement and ductility to get system initial period. (4) Determine the system initial stiffness and the required yield strength. (5) Estimate member sizes and detailing to provide the system strength required.

The procedure proposed by Restrepo (2007) introduces in the design additional factors to take into account ground motion variability and the inelastic versus elastic displacement demand variability (Table 1.10).

Table 1.10. Restrepo procedure

Type of analysis used in the design process ISRS Role of displacement in the design process DDBS Description: (1) Select an appropriate mechanism of inelastic deformation. (2) Select the level of detailing in the plastic hinge regions. (3) Calculate the reference yield displacement. (4) Calculate the theoretical ultimate lateral displacement. (5) Determine the displacement ductility and the CQR coefficient (which accounts for ground motion variability and inelastic versus elastic displacement demand variability). (6) Scale the elastic displacement spectrum by CQR and determine the period correspondent to the system ultimate displacement. (7) Determine the base shear and distribute it along the height of the building. (8) Complete the design and apply capacity design.

INTRODUCTION

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The procedure proposed by Kappos & Manafpour (2001) is the only one presented here involving time history analyses as part of the procedure (Table 1.11)

Table 1.11. Kappos & Manafpour procedure.

Type of analysis used in the design process THA Role of displacement in the design process DCB Description: (1) Apply force based design to obtain a basic strength level for serviceability earthquake combined with gravity loads. (2) Detail of the beams flexural reinforcement. (3) Use non linear time history analyses to check maximum drifts and plastic rotations in beam critical regions associated to an earthquake with probability of exceedance 50% in 50 years (beams modeled as yielding elements while columns as elastic ones). (4) Scale the ground motions to an event with correspondent probability of exceedance 10% in 50 years. (5) The time history analyses provide the critical moment and axial load combination at each column critical section. (6) Detail the column longitudinal reinforcement. (7) Design and detailing of all members for shear. (8) Detail all members for confinement, anchorages and lap splices.

In this research the procedure of Priestley (1997), usually referred as Direct Displacement Based Design (DDBD), has been adopted mainly for two reasons: the first one is that it involves a specified target displacement as a starting point, which is seen as a suitable design procedure; the second reason is that several efforts have been recently made to implement the aforementioned procedure in the Italian Seismic Design Code.

1.2 Research plan description

The main objective of this research is to check the suitability of the DDBD procedure when applied to the seismic design of precast concrete structures and how the procedure is affected by taking into account the influence of foundation flexibility, the different types of foundation to column connections and beam to column connections typical of precast buildings. The typical structural layout of Italian warehouses and commercial malls consists of concrete cantilever columns, connected by simply supported precast and prestressed beams, supporting prestressed concrete roof elements; the columns are inserted and grouted in place in isolated precast cup-footings; reducing the construction time, this solution is extremely cost effective.


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The case studies examined and the earthquake records adopted in the non linear time history analyses to validate the DDBD procedure are shown in Chapter 2. The DDBD and the Force Based Design (FBD) procedures are presented in Chapter 3; considerations are made on the equivalent viscous damping equations available in the literature and relationships are derived to relate those equations, which have been derived from force-displacement analyses of single degree of freedom systems, to the moment curvature relationship actually adopted in the column flexural description for the non linear time history analyses necessary to validate the procedure. In Chapter 4, specific considerations on the application of DDBD procedure to the precast concrete structures considered are made. Compared to traditional reinforced concrete structures the typology under exam presents lower displacement ductility demand due to the higher interstory height; this suggests to check the implications of the equivalent viscous damping equations, which have been derived for larger displacement ductility values, when applied to these structures. The suitability of the DDBD procedure when applied to the seismic design of precast concrete structures is evaluated through four phases whose schematic representation is shown in Figure 1.1. In Phase 1, Chapter 5, the equations developed in Chapter 3 are used for the comparison between FBD and DDBD procedures to outline advantages and drawbacks and the conditions under which the two procedures give compatible results. Although the comparison in not straightforward, due to the different inelastic displacement computations, two possible ways of doing it are proposed. The moment curvature relationship adopted to describe the columns flexural behavior is the same used to calibrate the equivalent viscous damping equations available in the literature. At the end of the chapter the DDBD is applied to precast structures with a different moment-curvature relationship showing the need of a more rigorous calibration of the equivalent viscous damping equations. In Phase 2, Chapter 6, the influence of Soil Structure Interaction is taken into account in the DDBD procedure adding the foundation flexibility and damping limited to the foundation rocking motion. Two possible ways of doing it are taken into account, extending results available in the literature, which consider a single degree of freedom substitute structure obtained by static condensation without considering the foundation inertia. Both analytical and non linear analyses are carried out to check the procedure suitability when foundation inertia is considered.

INTRODUCTION

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Phase 1 Phase 2

keff, eqH=

Hef

f m=meff

d

d

Kx

Kz

K

H=

Hef

f

keff, eq

m=meff

Phase 3 Phase 4

HH

eff

keff, eq

d

m

meff

keff, eqH=

Hef

f m=meff

d

Figure 1.1 Schematic approach of the DDBD procedure validation.

In Phase 3, Chapter 7, the influence of the top connection in the DDBD procedure is evaluated. The implementation of this aspect in the DDBD led to consider a substitute structure with an effective height corresponding to the point of counter flexure and with as effective mass the whole system mass. The analytical procedure developed is then applied to the case studies selected. In Phase 4, Chapter 8, the DDBD procedure is applied to systems whose hysteretic behavior is different from the ones used in the equivalent viscous damping equation calibration especially considering some of the foundation to column connections adopted in the precast industry. Experimental tests are carried out to compare different types of connections from a seismic and retrofitting point of view and to determine their hysteretic


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parameters to use in the equivalent viscous damping equation calibration. A new and faster calibration algorithm is proposed as alternative to the one available in the literature. The calibration procedure is carried out for the hysteretic relationships associated to the experimental tests and the results applied to the DDBD procedure. A new equation to relate the yield curvature to the column cross section effective depth and the axial load ratio is proposed; this formulation overcomes the drawback of the equation available in the literature especially when applied to some foundation to column connections typical of the precast industry. The equivalent viscous damping equation procedure is extended in Chapter 9 to other hysteretic rules. A new equation is proposed and the influence of the hysteretic parameters, the ground motion types and the displacement response spectrum shape is evaluated. In Chapter 10 a ground motions scaling procedure is proposed to control and limit the coefficient of variation (defined as the standard deviation divided by the mean value) of the acceleration response spectrum of the records set chosen, in order to limit the results variability of non linear time history analyses and the results dependence on the ground motion set chosen. This procedure seems suitable for the equivalent viscous damping equation calibration procedure other than for non linear time history analyses. After the application of DDBD procedure to precast structures with classical lateral force resisting systems (i.e. fixed end columns with or without the contribution of the top column to beam connection), Chapter 11 exploits the use of rocking walls as an alternative resisting system to use in precast structures. This system has self centering properties (given by post tensioning unbonded tendons) and accommodates the seismic lateral displacement demand with a base rotation which leads to only one concentrated opening of the foundation to wall joint compared to the crack spreading and damage typical of the plastic region of classical reinforced concrete walls. This chapter deals with the problem of the base sliding typical of these walls, with the definition of a moment rotation relationship to use in the non linear analyses and with the revisiting and extension of the equations of motion especially to determine the rocking period of the system whose relation to the design procedure can be exploited as an extension of this research. Rocking walls have been the lateral force resisting system of an extensive experimental campaign on precast diaphragms recently concluded at the University of California, at San Diego, and involving shake table tests. The design and the experimental results of these walls are shown in Chapter 12 where efforts have been made to explain their dynamic behavior in particular regarding the vertical and horizontal acceleration spikes associated to the wall impacts once rocking is triggered.

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2. GROUND MOTIONS AND CASE STUDIES

This chapter presents the ground motions and the case studies adopted to check the suitability of the Direct Displacement Based Design (DDBD) procedure when applied to the seismic design of precast concrete structures.

2.1 Ground motions definition

The elastic spectrum used in the design procedure, according to Eurocode 8-1:2004, is the type 1 spectrum for a soil type C with a peak ground acceleration of 0.5 g. To validate the DDBD procedure by means of non linear time history analyses, both natural and artificial ground motions (Table 2.1) have been adopted and scaled, multiplying the acceleration record by a scale factor, in order to match the Eurocode 8 design spectrum.

Table 2.1 Time history definition

Name Origin/Earthquake Duration (s) t (s) Scale factor TH1 Duzce 25.89 0.01 1.2 TH2 Kalamata 29.995 0.005 3.1 TH3 Kocaeli 1 70.38 0.02 2.1 TH4 Northridge Baldwin 60.00 0.02 4.5 TH5 Hella 60.000 0.005 2.0 TH6 SIMQKE Aritif4 19.99 0.01 TH7 SIMQKE Aritif6 19.99 0.01

As Eurocode 8 states (3.2.3.1.2.4b): in the range of periods between 0.2T1 and 2T1, where T1 is the fundamental period of the structure in the direction where the ground motion will be applied, no value of the mean 5% damping elastic spectrum, calculated from all time histories, should be less than 90% of the corresponding value of the 5% damping elastic response spectrum. In the case under consideration it is possible to note how the previous requirement is satisfied (Figure 2.1), so the ground motion records adopted seem suitable. It is important to note that the constant displacement predicted by the Eurocode 8 equation after the corner period of 2 s is not respected by the mean displacement spectrum of the records used; therefore particular care should be taken in


14

the validation of the procedure in the case the substitute structure effective period is greater than the corner period of Eurocode 8 spectrum.

Acceleration spectra comparison

0.0

0.5

1.0

1.5

2.0

2.5

0.0 1.0 2.0 3.0 4.0Period (s)

Acce

lera

tion (g)

EC8

GM mean

Displacement spectra comparison

0.00.10.20.30.40.50.60.70.80.91.0

0.0 1.0 2.0 3.0 4.0Period (s)

Dis

pla

cem

en

t (m)

EC8GM mean

Figure 2.1 Acceleration and Displacement response spectra comparison

2.2 Case studies definition

The case studies chosen are three existing buildings whose structural layout is typical of Italian precast structures:

1. One story precast concrete building with double tee roof elements (Figure 2.2). 2. One story precast concrete building with omega roof elements (Figure 2.3). 3. One story building with precast concrete columns connected at the top by wood

beams (Figure 2.4 and Figure 2.5).

GROUND MOTIONS AND CASE STUDIES

15

Plan View

A A

B

B17

5017

508785

1750

10907630

1090 1090 1090 1090 1090 109017

5017

50

Section AA Section BB

CL

750

8,40 m

1090 1090 1090

0,00 m

CL

750

95

750

1750 1750

0,00 m

Double Tee Beam Details

1860

250

51

560

6051

13011218

Figure 2.2 - Case Study 1: one story building with double tee roof beams.


16

Plan View 91

691

7

2000 2000

917

Section AA

510

7014

512

5

Section BB Omega Beam Detail

735

635

100

Figure 2.3 - Case Study 2: one story building with omega roof beams.

GROUND MOTIONS AND CASE STUDIES

17

Plan View

CL

Figure 2.4 - Case Study 3: one story building with timber beam precast column connections.

A A


18

Section AA

Timber Beam concrete column connection detail

Two dowels connection Four dowels connection

Figure 2.5 - Case Study 3: timber beam precast column connection details.

19

3. FBD AND DDBD PROCEDURES

In this chapter the Direct Displacement Based Design (DDBD) and the Force Based Design (FBD) procedures are presented and considerations are made in order to allow the procedures comparison in Chapter 5. The issue of the equivalent viscous damping (used in DDBD) is taken into account: relationships are derived to relate equations available in the literature to the moment curvature relationship actually adopted in the column flexural description, which will be used in the non linear time history analyses necessary to validate the design. The dependence of the elastic response spectra of the ground motion selected from the damping value is also checked.

3.1 FBD Procedure

For sake of simplicity the FBD procedure shown here corresponds to the case of a single degree of freedom system, which will be adopted in the FBD and DDBD procedures comparison. The FBD procedure is therefore:

1. Define a force reduction factor (q) for the structure. 2. Define an effective modulus of inertia Ieff as a percentage of the gross modulus

Igross. 3. Define the stiffness of the system and determine the system period:

2 2 mTk

pipi

= = (3.1)

4. Determine the spectral acceleration corresponding to the structural period from

the design spectrum, ( )a ,S T q . 5. Determine the base shear and the base moment as:

abV S m g= ; u bM V H= (3.2), (3.3)

6. Find the corresponding top displacement. The inelastic displacement, according

to Eurocode 8 4.3.4, is evaluated as: i d ed q d= , where qd is taken equal to q:

( )/bq V k = (3.4)


20

7. Evaluate the second order effects computing the value, defined as the ratio between the second order moment and the moment from analysis.

i

b

P dV h

=

(3.5)

- less than 0.1: second order effects negligible.

- between 0.1 and 0.2: the base moment becomes ( ) ( )/ 1u bM V H = . - between 0.2 and 0.3: second order effects to be taken into account.

- greater than 0.3: change structural dimensions.

3.2 DDBD Procedure

1. Definition of a single degree of freedom substructure system (Figure 3.1).

With:

d target displacement Heff effective height meff effective mass eq equivalent viscous damping keff effective stiffness

Figure 3.1 Substitute structure for DDBD procedure.

2. Determine the target displacement (d). The target displacement depends from both the structural deformed shape and the limit state under consideration, whose critical value can be associated either to structural components (related to material strains) either to non-structural components (related to interstory drift). A linear distribution of the yield curvature from the column base to the top (considered as the roof mass centroid) has been considered in Phase 1 of this research (Chapter 5); this represents only a first approximation, without considering the moment at top of the column due to beam connection (as it will

Heff

meff

d

keff,eq

FBD AND DDBD PROCEDURES

21

be analyzed in Phase 4 Chapter 7). A design based on the damage limitation requirement has been adopted and the critical value of the target displacement is the one associated to an interstory drift of 2.5 %. To determine the yield displacement (y) the following equations have been used (Priestley 2003):

2.1 yy B = ,

222 3 3

y yy

H HH

= = (3.6), (3.7)

Where:

y is the yield curvature B is the column cross section size

y is the steel yield strain 3. Determine the effective height (Heff).

The effective height is the point where the system ductility is evaluated; it is defined as:

1

1

n

i i ii

eff ni i

i

m HH

m

=

=

=

(3.8)

4. Determine the effective mass (meff). The effective mass represents the mass participating in the first inelastic mode of vibration and it is obtained considering the design displacement profile i of the masses mi at each floor:

1

d

n

i ii

eff

m

m =

=

(3.9)

5. Determine the equivalent viscous damping (eq). The DDBD adopts an equivalent viscous damping approach to represent the elastic and the hysteretic damping of the system eq=5+hyst%; the first term, the elastic viscous damping, takes into account material viscous damping, radiation damping due to the foundation system and damping due to the non linear behavior of the connections. The second term, the hysteretic damping, depends on the hysteretic relationship


22

of the structural elements and takes into account in somehow the capacity of the system to dissipate energy.

6. From the equivalent viscous damping determine the design displacement spectrum, reducing the displacement response spectrum by the factor :

105 eq

= + (3.10)

7. Determine the SDOF substitute structure effective period Teff, as the period corresponding to d.

8. Determine the effective stiffness keff associated to the SDOF system maximum response:

2 24 /eff eff effk m Tpi= ; 2 effeffeff

mT

kpi= (3.11), (3.12)

9. Determine the system base shear as b eff dV k= (Figure 3.2)

Figure 3.2 Base Shear estimate

10. The base shear is distributed as design forces at the different in proportion to the inelastic displacement.

1

b i ii n

i ii

V mFm

=

=

(3.13)

keff = Vb/d

Displacement (m)

Base Shear (kN)


23

3.3 Considerations on equivalent viscous damping

The equivalent viscous damping approach was first proposed by Jacobsen (1930, 1960), who considered the steady state response of SDOF non linear systems under an harmonic load and related the equivalent viscous damping to the ratio between the hysteretic and elastic energy. If a SDOF non linear system subjected to an harmonic load is considered, it is possible to follow the Jacobsen approach. By assuming a system response characterized by an elastic force

elF k u= and a damping force Fd which can

be written as d dF c u= (as it is in the viscous damping) and considering a displacement

0 sinu u t= , it follows:

0 sin cosel d d dF F F ku c u ku t c u t = + = + = + (3.14)

The energy dissipated in one cycle is:

( )2

20

T

dissipatedT

duE F t dt c udt

pi

pi

+

= = (3.15)

The elastic energy at maximum response is:

20

12elastic

E ku= (3.16)

The ratio between dissipated and elastic energy is:

2dissipated delastic

E cE k

pi = (3.17)

Considering that 2d eq critic eqc c m = = the equivalent viscous damping is: 1

4dissipated

eqelastic

EE

pi

= (3.18)

Considering this approach to describe the system behavior under an earthquake type excitation leads to underestimate the system response. In fact this approach overestimates the system damping during an earthquake because it does not consider the transient response and it is based on an harmonic excitation. Different equations exist (as reported in Blandon 2005) which relate the equivalent viscous damping to the system ductility and the post yield stiffness. The equation adopted in this research is the one proposed by Grant et al. (2004) which relates the equivalent


24

viscous damping to the target system displacement ductility (=d/y) and the substitute structure effective period (Teff):

( )1 10.05 a 1 1eq db

effT c

= + + +

(3.19)

The procedure adopted by the authors to calibrate the parameters (a, b, c, d) is based on the force displacement response of SDOF systems subjected to a ductility range from 2 to 6 and with an effective period ranging from 0.5 s to 4 s. The FBD and DDBD procedures will be compared by means of non linear analyses of SDOF systems whose non linear behavior is governed by the Takeda hysteretic rule (as reported in Carr 2006), which well describes reinforced concrete behavior, and the equivalent viscous damping parameters adopted (Eqn. 3.19) have been calibrated (Grant et al. 2004) for two sets of Takeda model parameters (Takeda fat and Takeda narrow model) as it is in Figure 3.3.

y u

Fy

F

ki

rki

pp

ku=ki

res

a b c d

Takeda fat (=0.3; =0.6; r=0.05)

0.249 0.527 0.761 3.250

Takeda narrow (=0.5; =0; r=0.05)

0.183 0.588 0.848 3.607

Figure 3.3 Equivalent viscous damping parameters for Takeda model

The elastic component of the equivalent viscous damping is related to the secant stiffness at maximum displacement, therefore this value has to be adjusted (Grant et al. 2004) to ensure compatibility between substitute and real structure in the non linear analyses. Adopting an initial stiffness or a tangent stiffness damping for the time history analyses, the elastic component of the equivalent viscous damping in the substitute structure has to be corrected (Grant et al. 2004):


25

( )tang 1 1a 1 1init e feq el el db

eT c

= + + + +

(3.20)

Where initel and tangel refer to how the damping matrix is computed to solve the equations of motion; being the former value associated to a damping matrix proportional to the initial stiffness matrix and the latter to a damping matrix proportional to the tangent stiffness matrix. The parameters of to the two Takeda models are shown in Table 3.1.

Table 3.1. Takeda parameters for analyses

a b c d e f Takeda fat (=0.3; =0.6; r=0.05) 0.305 0.492 0.790 4.463 0.312 -0.313

Takeda narrow (=0.5; =0; r=0.05) 0.215 0.642 0.824 6.444 0.340 -0.378

Referring to Figure 3.3 and Figure 3.4, it is possible to determine the equivalent viscous damping with the Jacobsen approach for the steady state response of the Takeda model, assuming the same ductility is reached for both directions of excitation.

F

Energydissipated

Energyelastic

A B

Cy u

pp

C'

B'A'

O

Figure 3.4 Takeda steady state response for Jacobsen approach

The comparison between elastic energy and the dissipated energy of Eqn. 3.18 leads to the expression:


26

( ) ( )2 21 2 2 2 22hy pi = + + + + (3.21) Where ( )1 1r = +

Figure 3.5 shows the comparison between the hysteretic portion of the equivalent viscous damping equation with the Jacobsen approach according to the expression just evaluated. It is clear how the Jacobsen approach overestimates the hysteretic damping.

Eq. viscous damping - Takeda hysteretic model

0.00

0.05

0.10

0.15

0.20

0.25

0.30

1.0 2.0 3.0 4.0 5.0 6.0Ductility ()

Eq. vi

scou

s da

mpi

ng

Jacobsen: alfa=0.3; beta=0.6; r=0.05Grant: alfa=0.3; beta=0.6; r=0.05Jacobsen: alfa=0.5; beta=0; r=0.05Grant: alfa=0.5; beta=0; r=0.05

Figure 3.5 Hysteretic damping: Jacobsen vs Grant (Teff=1s)

Considering the methodology used by Grant et al. (2004) to calibrate the equivalent viscous damping for DDBD, it is possible to note how the calibration followed the definition of an hysteretic model related to Force-Displacement relationship while in this research the Takeda model is used to describe the Moment-Curvature relationship. To compare the results obtained by the FBD and the DDBD procedures, adopting for the latter the hysteretic damping equations proposed, the relationship between , and r parameters for the Force-Displacement and Moment-Curvature Takeda model have to be defined (Figure 3.6). This will be done in the following sub-paragraphs.


27

y u

Fy

F

ki

rki

pp

ku=ki

res

y u

My

M

k'i

r'k'i

p'p

k'u=k'i'

res

Figure 3.6 Force-displacement and moment-curvature Takeda model parameters.

3.3.1 Relationship between displacement () and curvature () ductility The relationships defined in this sub-paragraph and in the following ones have been obtained considering a plastic hinge region of length Lp with constant plastic curvature p located at the element ends (Figure 3.7) as it is in the finite element program Ruaumoko (Carr 2006) adopted in the non linear analyses.

p y

H

Lp

Figure 3.7 Inelastic curvature distribution

With these considerations and adopting an elasto-plastic behavior, the relationship between the displacement and curvature ductility is

113 p

LH

= + (3.22)


28

In the general case, considering the post yield stiffness coefficient r (defined as the post yield to elastic stiffness ratio), the displacement and curvature ductility relationships are:

( )11

' 3 1 'pL

r rH

= +

+

; ( ) ( )1 ' 3 1 ' 1pLr rH

= + +

(3.23), (3.24)

3.3.2 Relationship between r and r

Referring to Figure 3.8, at the yielding point:

'y eff y iy

EI kF

H H

= = ; 2

3y

y

H = (3.25), (3.26)

y u

Fy

F

ki

rki

y u

My

M

k'i

r'k'i

Figure 3.8 Evaluation of r-r relationship

At the target displacement:

( )' 'y i u yuu

M r kMFH H

+ = = ; ( )u y u y pL H = + (3.27), (3.28)

From the definition of r:

( )( ) 2

' '

' '

iy u y y

u y ii

u y py u y p y

r kF FF F r kHr kL HL H

+

= = =

+ (3.29)

Therefore: 3

2 2

' ' 1 ' ''

3 3i i

p i p eff p

r k r k H Hr r

L H k L H EI L= = =

,

3'

pLr r

H= (3.30), (3.31)


29

3.3.3 Relationship between and

Referring to Figure 3.9, from F- considerations:

( ) ( ) ( )2 2'3 3y u ya u

res u y u y pu

rF H L H Hk

+ = = + (3.32)

y u

Fy

F

ki

rki

ku=ki

res

y u

My

M

k'i

r'k'i

k'u=k'i'

res

Figure 3.9 Evaluation of - relationship

From M- considerations: ( ) ( )( )' 'b res p res p u y u yL H L H r = = + (3.33)

Equating ( ) ( )a bres res = leads to:

( )( ) ( ) ( )' '' 3 3y u yu y u y y u yp prH H

rL L

+ + = + (3.34)

Which gives:

( )( )

'

1 1 ' 13 3

1 ' 1p p

H Hr

L Lr

+ + + =

+ (3.35)


30

Therefore:

( ) ( )( )

ln 1 1 ' 1 ln 1 ' 13 3

'

lnp p

H Hr r

L L

+ + + +

= (3.36)

( ) ( ) ( )( )

'ln 3 ln 1 ' 1 ln 1 ' 1 13

ln

pp

HL H r rL

+ + + +

= (3.37)

It is observed how - relationship is governed by the target ductility.

3.3.4 Relationship between and Referring to Figure 3.10, from F- considerations:

( ) ( ) ( )( ) ( )( )

( )( )2

1

= 1

= 13

a

c u u y y u

y y u y p

yp u y

L H

HL H

= = + =

+ + =

+

(3.38)

( ) ( )( )1c y i c y y i u yF F rk F rk = + = + (3.39)

y u

Fy

F

ki

rki

pp

c

c

y u

My

M

k'i

r'k'i

p'p

c

c

Figure 3.10 evaluation of - relationship

From M- considerations:

( ) ( )' ' 1 'c u u y y u = = + (3.40)


31

( ) ( )( )' ' ' ' 1 'c y i c y y i u yM M r k M r k = + = + (3.41) ( ) ( ) ( )( )2 1 '3b yc y c y p u y p

HL H L H

= + = + (3.42)

Equating ( ) ( )a bc c = leads to

' = (3.43)

3.4 Displacement response spectrum dependence from damping

According to the current version of Eurocode 8 (2004), the displacement spectrum amplification for damping values different from 5% can be taken into account with the factor defined before (Eqn. 3.10). Phase 1 (Chapter 5) of this research deals with the comparison between the FBD and the DDBD procedures; to reduce uncertainties in the displacement spectrum reduction due to the damping level (in the DDBD procedure), the factor has been calibrated for the records adopted. This has been done with a least square procedure applied in the period range 0 4 s to the mean displacement spectrum. The new spectrum reduction factor adopted is (Figure 3.11)

( )7.8/ 2.8 eq = + (3.44) =SD(x)/SD(5%)

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

2 3 4 5 6 7 8 9 10 11 12 13 14 15

Equivalent viscous damping (%)

SD(x)

/SD(

5%)

Computed Datag = 7.8g = 7g = 10 (EC8)

0.5( )(5%) ( 5)

SD gSD g

=

+

Figure 3.11 Spectrum reduction factor as function of the damping value

33

4. PRECAST STRUCTURES CONSIDERATIONS

In this chapter the force reduction factor (q-factor) for precast concrete structures is defined. Specific considerations on the application of DDBD procedure to the precast concrete structures considered are made: compared to traditional reinforced concrete structures the typology under exam presents lower displacement ductility demand due to the higher interstory height (Chapter 2.2); this suggests to check the implications of the equivalent viscous damping equations, which have been derived for larger displacement ductility values.

4.1 Definition of the q-factor for precast concrete buildings

According to Eurocode 8 (2004) 5.1.2 it is possible to define two possible structural types for the precast concrete structures analyzed in this research, depending on the type of connections used:

1. Frame system: structural system in which both the vertical and lateral loads are mainly resisted by spatial frames whose shear resistance at the building base exceeds 65% of the total shear resistance of the whole structural system

2. Inverted pendulum system: system in which 50% or more of the mass is in the upper third of the height of the structure, or in which the dissipation of energy takes place mainly at the base of a single building element. NOTE One-storey frames with column tops connected along both main directions of the building and with the value of the column normalized axial load d nowhere exceeding 0.3, do not belong to this category.

Thus, from the previous note, considering that the columns are connected in one direction directly by means of L-beams or inverted T-beams, and in the other direction indirectly by means of double-T or Omega beams, it is possible to consider the precast system as a frame system.

The force reduction factors for horizontal seismic actions are defined in Eurocode 8 5.2.2.2. In the case under exam the q-factor is:

0 1.5wq q k= (4.1)


34

Where 0q is taken from Table 4.1:

Table 4.1. Eurocode 8 q0 values

Ductility Class Medium (DCM)

Ductility Class High (DCH)

Frame System 13.0 /u 14.5 /u

Inverted Pendulum System 1.5 2

with 1/ 1.1u = for one story frame system buildings and wk = 1 for frame systems.

In the particular case of precast concrete structures qp=q x kp, where kp is a reduction factor depending on the energy dissipation capacity of the precast structure (kp=1 if the connections are designed on the basis of the capacity design rules otherwise kp=0.5) In this research a one story frame system with a ductility class DCM has been considered, which leads to q = 3.3. The choice to assign a ductility class DCM instead of DCH can be justified by the presence of cantilever pinned top columns as primary elements to dissipate energy and accommodate inelastic displacements, with the development of a plastic hinge at their base; therefore assigning a level of ductility high will lead to a failure in the inelastic displacement control due to the higher inelastic rotation of the column base. Regarding the connection system, Eurocode 8-1:2004 identifies three different situations:

1. Connections located outside the critical regions which do not affect the energy dissipation capacity of the structure.

2. Connections located in the critical regions designed to remain elastic and to relocate the inelastic response in other regions inside the elements.

3. Connections located in the critical regions designed to carry the inelastic response.

In the present study connections located outside the critical regions will be considered.

PRECAST STRUCTURES CONSIDERATIONS

35

4.2 Precast concrete structures compared to other structures

Considering the typical warehouse precast concrete structures layout (Figure 2.2, Figure 2.3, Figure 2.4), the story height is 2-3 times bigger than the other reinforced concrete structures. This leads to a lower amount of ductility demand as it is clear from the comparison of a precast structure to typical concrete structure with the same column cross section.

Indicating with the maximum allowed drift, equations 3.6 and 3.7 for precast concrete structures are:

2.1 yprecastyprecastB = ;

2

3precast yprecast

y

H = ; (4.2), (4.3)

While the maximum displacement and displacement ductility are:

precastu precastH = ;

32.1

u precast precastprecast

y precast y precast

BH

= =

(4.4), (4.5)

In the case of classic reinforced concrete structures the previous equations become:

2.1 yusualyusualB = ;

2

3usual yusual

y

H = ; (4.6), (4.7)

usualu usualH = ;

32.1

u usualusual usual

y usual y usual

BH

= =

(4.8), (4.9)

The ductility ratio is: precast

usualusual

precast

HH

= (4.10)

Thus, if 2 3precast usualH H= (as it usually happens), the ductility is 2 3usual

precast =

.

The low amount of ductility required leads to a limit state related to the interstory drift code control rather than a material strain limit requirement. The target ductility could be sensibly small compared to usual reinforced concrete structures, therefore the hysteretic damping equations (Grant et al. 2004) need to be checked for ductility values less than 2 (the parameters in those equations have been calibrated for ductility values between 2 and 6). This has been done by means of non


36

linear time history analyses on single degree of freedom systems with the fat Takeda hysteretic model (Grant et al. 2004). Systems compatible with case study 1 (Figure 2.2) have been chosen so that y (evaluated with Eqn. 3.7) is the same of concrete columns with H = 7.9 m and square cross section size between 60 and 110 cm. The DDBD procedure has been applied to the systems and the results from the design have been compared with the results obtained from the non linear time history analyses. Figure 4.1 compares the equivalent viscous damping computed in the design process with the one effectively obtained: the damping obtained from the design equations underestimates the system damping, which leads to a conservative estimation of the ductility. Figure 4.2 shows the maximum displacement as a function of the longitudinal reinforcement ratio: a cross section size greater than 80 cm is needed to limit the interstory drift to 2.5%.

1 1.5 2 2.5 3 3.50

2

4

6

8

10

12

14

16

18Hysteretic Damping Ductility

Hys

tere

tic D

ampi

ng (%

)

Numerical DataGrant eqn

Teff = 0.5 s

Teff = 4.0 s

1 1.2 1.4 1.6 1.8 20.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5(Target ductility Real ductility) Ratio

target

( ta

rget

) / (

o

btai

ned)

Figure 4.1 Hysteretic damping and target ductility comparisons

PRECAST STRUCTURES CONSIDERATIONS

37

0 200 400 600 800 1000 1200 1400

0.15

0.3

0.45

0.6Maximum Displacement as a function of yield force

Fy

u

1% 4%

60 cm2.5% drift

0 200 400 600 800 1000 1200 1400

0.15

0.3

0.45


Fy

u

1% 4%

70 cm2.5% drift

0 200 400 600 800 1000 1200 1400

0.15

0.3

0.45


Fy

u

1% 4%

80 cm2.5% drift

0 200 400 600 800 1000 1200 1400

0.15

0.3

0.45


Fy

u

1% 4%

90 cm2.5% drift

0 200 400 600 800 1000 1200 1400

0.15

0.3

0.45


Fy

u

1% 4%

100 cm2.5% drift

0 200 400 600 800 1000 1200 1400

0.15

0.3

0.45


Fy

u

1% 4%

110 cm2.5% drift

Figure 4.2 Maximum displacement as a function of reinforce ratio.

39

5. PHASE 1: FBD AND DDBD COMPARISON

In this chapter the equations developed in Chapter 3 are used to compare the FBD and DDBD procedures. Two possible ways of comparison are presented and applied to the design of Case Study 1. The moment curvature relationship adopted to describe the column flexural behavior is the same used to calibrate the equivalent viscous damping equations in the literature, i.e. the fat Takeda model, whose choice is justified by the usually low axial load on columns of warehouse structures. At the end of the chapter the DDBD is applied to precast structures with a different moment-curvature relationship showing the need of a more rigorous calibration of the equivalent viscous damping equations. Table 5.1 contains Case Study 1 data.

Table 5.1. Case Study 1 data

Geometric data Column height 7.9 m Number of columns 56 Total mass 4645000 kg Tributary column mass (weight) 82946 kg (814 kN)

Material data Concrete C 40/50

f ck 40 (MPa) f cd 26.5 (MPa) f ctm 3.5 (MPa) f ctk 2.45 (MPa) Rd 0.4 (MPa) Ec 34525 (MPa)

Steel FeB 44k f yk 430 (MPa) f yd 374 (MPa) y 0.00209 (MPa) Es 206000 (MPa)


40

Regarding the DDBD procedure, in this first phase of the research the model adopted to describe the structural behavior is a fixed end cantilever pin connected to the top beams. In these conditions the effective height and the effective mass are equal to the height and mass of the structure (Heff = 7.90 m, meff = 814 kN). The target displacement is the one corresponding to 2.5% drift: d = 0.1975 m.

5.1 FBD-DDBD comparison: Procedure 1

The procedure adopted is: design the column cross section with the FBD procedure,

calculate the inelastic displacement as ( )( )3 / 3u b effq V H EI = and use this value as the target displacement for the DDBD procedure. The following step is to check the procedure predictions by means of Non Linear Time History (NLTH) analyses. To obtain the NLTH model input data according to the FBD:

1. Assume, as it is usually done in practice, that at yield 0.50eff grossI I= , so

0 / 0.5eff grossk k = = (5.1)

2. Find Fy and y: 1

u eff uy

F r kF

r

=

; yyeff

Fk

= (5.2), (5.3)

3. Get ( )1u u effy u eff u

k rF r k

= =

(5.4)

4. Get (Eqn. 3.36), (Eqn. 3.43) and r (Eqn. 3.31).

To obtain he NLTH model input data according to the DDBD: 1. Get y associated to the cross section size (Eqn. 3.7). 2. Get /u y = (5.5)

3. Find ( )1 1u

yF

Fr

=

+ with r=0.05 (5.6)

4. Get yeff

y

Fk =

; 30 3

eff eff

gross gross

k kH

k EI = = (5.7), (5.8)

5. Get (Eqn. 3.36), (Eqn. 3.43) and r (Eqn. 3.31).

PHASE 1: FBD AND DDBD COMPARISON

41

In the case of a cross section size equal to 60 cm the FBD procedure leads to the results shown in Figure 5.1

0 1 2 3 40

0.5

1

1.5

2EC8 2004 type=1 soil=c PGA=0.5 q=3.30

Period (s)

Acce

lera

tion

(g)

T=1.89; S=0.14

Design spectrumElastic spectrumFBD result

FBD results: Teff 1.39 s Vb 113 kN Mu 894kNm Drift 5.16% 0.37

Figure 5.1 FBD - section size 60 cm

The inelastic displacement obtained from the FBD procedure is 0.41 m, which used as a target in the DDBD procedure leads to a displacement demand higher than the maximum values of the design displacement spectrum (Figure 5.2).

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

Period (s)

Dis

plac

emen

t (m)

EC8 2004 type=1 soil=c PGA=0.5

Design

ElasticTarget displacement

Figure 5.2 DDBD - section size 60 cm Displacement demand higher than capacity.

The DDBD procedure shows that the target displacement corresponds to a ductility level greater than the one available, therefore iterations are necessary to reduce the target


42

displacement until reaching convergence. Figure 5.3 shows the results of these iterations, which lead to a lower target displacement corresponding to a structure with as effective period the displacement spectrum corner period, in this case 2 s.

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Period (s)

Dis

plac

emen

t (m)

EC8 2004 type=1 soil=c PGA=0.5

T=2.00; Sd=0.30

Elastic SpectrumDesign SpectrumDDBD result

Figure 5.3 DDBD - section size 60 cm converged results

The comparison between FBD and DDBD procedures leads to the following results obtained varying the column cross section size from 70 cm to 110 cm (Table 5.2 to Table 5.6).

Table 5.2. Cross section size 70 cm

FBD Takeda model parameters Teff (s) 1.39 My (kNm) 1070 Vb (kN) 154 0 0.5

Mu (kNm) 1217 0.47 drift (%) 3.79 0.6

0.20 r 0.015

DDBD Takeda model parameters Teff (s) 1.92 My (kNm) 1977 Vb (kN) 266 0 0.55

Mu (kNm) 2098 0.45 drift (%) 3.79 0.6

0.12 r 0.015


43

Increasing the cross section size leads to a decrease of the inelastic displacement according to FBD, while the decrease of the target displacement in the DDBD procedure leads to a greater demand.



Mu (kNm) 1813 0.46 drift (%) 2.92 0.6

0.12 r 0.015


Mu (kNm) 2882 0.44 drift (%) 2.92 0.6

0.08 r 0.015



Mu (kNm) 2024 0.47 drift (%) 2.31 0.6

0.07 r 0.015


Mu (kNm) 3381 0.43 drift (%) 2.31 0.6

0.04 r 0.015


44



Mu (kNm) 2508 0.47 drift (%) 1.87 0.6

0.05 r 0.015


Mu (kNm) 4727 0.42 drift (%) 1.87 0.6

0.03 r 0.015



Mu (kNm) 2800 0.47 drift (%) 1.43 0.6

0.03 r 0.015


Mu (kNm) 7137 0.40 drift (%) 1.43 0.6

0.01 r 0.015

Figure 5.4 summarizes the results obtained from the FBD and DDBD comparison, where the base moment demand according to the two procedures has been plotted; the DDBD target displacement has been taken as the inelastic displacement predicted by FBD. In the dashed lines the moment capacity for longitudinal steel ratio of 1 to 4% is plotted.


45

0 1 2 3 4 5 6 70

1000

2000

3000

4000

5000

6000

7000

8000

9000Base Moment

Mom

ent (k

Nm)

Column size 1=60, 2=70, 3=80, 4=90, 5=100, 6=110 cm

1%

2%

3%

4%

5.16

% d

rift

3.79

% d

rift

2.92

% d

rift

2.31

% d

rift

1.87

% d

rift

1.43

% d

rift

FBDDDBD

Figure 5.4 FBD and DDBD procedures comparison results.

According to FBD, discarding the results of the 60 cm cross section due to the high value of theta (second order effects), it seems possible to design any cross section size greater than 70 cm. The increase of the cross section size leads to a lower value of the displacement computed by the FBD procedure and a related increase of the moment demand. Non Linear Time History (NLTH) analyses, with the Takeda hysteretic model parameters shown in Table 5.2 to Table 5.6, have been carried out. The results are shown in Table 5.7 and Figure 5.5.

Table 5.7 FBD and DDBD NLTH analyses comparison (procedure 1).

Section size Procedure Target Drift Time history results Roof Drift (%) Residual drift (%)

(cm) (%) Mean Max Mean max 70 FBD 3.79 3.06 4.29 0.18 0.38 DDBD 3.79 2.67 4.10 0.09 0.23

80 FBD 2.92 2.28 3.40 0.17 0.48 DDBD 2.92 1.98 2.77 0.12 0.27

90 FBD 2.31 1.90 3.27 0.23 0.40 DDBD 2.31 1.71 2.48 0.11 0.23

100 FBD 1.87 1.47 2.13 0.17 0.55 DDBD 1.87 1.51 2.18 0.03 0.13

110 FBD 1.43 1.33 2.04 0.10 0.24 DDBD 1.43 1.16 1.58 0.05 0.20


46

0 1 2 3 4 5 6 70

1

2

3

4

5Roof drift Mean values

Drif

t (%)

Column size 1=60, 2=70, 3=80, 4=90, 5=100, 6=110 cm

Target displacementFBD DriftFBD Residual driftDDBD DriftDDBD Residual drift

0 1 2 3 4 5 6 70

1

2

3

4

5Roof drift Max values

Drif

t (%)

Column size 1=60, 2=70, 3=80, 4=90, 5=100, 6=110 cm

Target displacementFBD DriftFBD Residual driftDDBD DriftDDBD Residual drift

Figure 5.5 FBD and DDBD THA mean and maximum values comparison (procedure 1).

The two procedures present comparable results with a slightly better control of the target displacements for the DDBD procedure. The good agreement between the displacement predicted by the FBD and the displacement obtained in the NLTH analyses is due to the column hysteretic model adopted in the latter which reflects exactly the approximation EIeff = 50% EIgross made in the design procedure. Considering in the design a model which reflects the actual displacement at yielding (Priestley 2003) leads to the values in Table 5.8.

Table 5.8 NLTH analyses results: effective stiffness dependence.

Section size Target Drift EIeff = 0.5 EIgross y (Priestley 2003) Roof Drift (%) Roof Drift (%)

(cm) (%) Mean Max Mean Max 70 3.79 3.06 4.29 3.22 4.39 80 2.92 2.28 3.39 2.79 4.15 90 2.31 1.90 3.27 2.21 3.23 100 1.87 1.47 2.13 1.79 2.43 110 1.43 1.33 2.03 1.66 2.29

It is clear how the choice of the effective stiffness affects the results. As noted before the effective stiffness in the FBD procedure is reduced usually as a percentage of the gross section stiffness and for columns with low axial load, as it is in this case study, the recommended value (Paulay and Priestley 1992) is 0.4 EIgross; this leads to a period


47

which does not take into account the actual longitudinal reinforcement ratio l. The FBD procedure considers the stiffness as a property of the section, while the system property that does not change is the yield displacement (Priestley 2003). The effective period associated to the actual yielding moment (depending on l) can be detected by:

1. yeffy

MEI = where y is taken from Eqn. 3.6

2. 2effmTk

pi= where 33 effEIk

H= (valid for fixed end cantilever)

This leads to the period values shown in Table 5.9:

Table 5.9 Steel ratio-period dependence.

Section size

Period dependence

Steel ratio - l with with

1% 2% 3% 4% EIgross 0.4 EIgross (cm) (s) (s) (s) (s) (s) (s) 60 2.85 2.16 1.90 1.73 1.34 2.11 70 2.00 1.71 1.41 1.30 0.98 1.55 80 1.56 1.27 1.08 0.97 0.75 1.19 90 1.25 0.99 0.87 0.78 0.59 0.94 100 1.05 0.83 0.71 0.62 0.48 0.76 110 0.88 0.69 0.58 0.51 0.40 0.63

It is clear how the steel ratio affects the period and therefore the demand predicted by the FBD procedure. In this specific case the period evaluated with a reduced stiffness (0.4 EIgross) is close to the period corresponding to 2% steel ratio, which is a ratio reasonably adopted in design. The period dependence on the longitudinal reinforcement ratio just shown suggests the idea of including it in the DDBD procedure. In fact knowing if the moment capacity of a given cross section size is greater than the moment demand without calculating the former will improve the DDBD procedure. This could be done with the following procedure which gives an estimation of the longitudinal steel ratio (l) based on the material characteristics and on the ratio between the effective and the gross section stiffness obtained from the DDBD procedure.


48

According to DDBD procedure, once the cross section size has been defined, the y is known (Priestley 2003); after the definition of the target displacement the displacement ductility is found and as an intermediate result of the procedure the effective period of the substitute structure Teff. From these values the period T0 (associated to the secant stiffness at yield) is:

01 ( 1)

effr

T T

+ = (5.9)

Where r is the post yield stiffness ratio. The stiffness at first yield is obtained from T0 as 2

0

4y

mkTpi

= (5.10)

Which equated to the force-displacement stiffness of a cantilever column gives: 3

3y

c y

H kE I = (5.11)

The ratio between the moment-curvature stiffness at yield and in the un-cracked conditions (considering only the concrete contribution) is:

( ) ( )0 /c y c grossE I E I = (5.12) This ratio is obtained from the DDBD procedure and, if related to the longitudinal steel reinforcement ratio, it will allow to check directly if the cross section chosen is suitable for the design in terms of Code longitudinal steel ratio limits. The other way to evaluate EcIy is with the yield curvature formula (Priestley 2003) already adopted in the DDBD procedure:

/y y y

y sy y y

M M MEI E B

B f = = = (5.13)

Where My comes from the moment-curvature bilinear approximation and is the nominal moment associated to a yield strain in the extreme tension reinforcement or a strain of 0.002 in the extreme compression fiber, whichever occurs first. indicates the factor used in the yield curvature formula which varies from the cross section size (2.1 for rectangular columns). My is now determined as a function of cross section size B, longitudinal steel rati

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