EVALUATION OF THE EFFECTS OF NONLINEAR SOIL-STRUCTURE INTERACTION ON THE INELASTIC SEISMIC RESPONSE OF
PILE-SUPPORTED BRIDGE PIERS
by
HOUMAN GHALIBAFIAN B.Sc., University of Tehran, 1993
M.A.Sc., The University of British Columbia, 2001
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Trang 3ABSTRACT
This dissertation presents an evaluation of the effects of nonlinear soil-structure interaction (SSI) on the inelastic seismic response of pile-supported bridge piers on soft soil The research was carried out by studying the dynamic responses of prototype soil- foundation-bridge pier systems subjected to earthquake ground motions The responses were obtained by performing nonlinear dynamic analyses using a commercial finite difference program The nonlinearities of the soil, the structure, and the soil-structure interface were all accounted for The numerical analysis method was carefully validated by verifying the modeling of each component of the system and by verifying the modeling of the system as a whole through analyzing an instrumented bridge pier subjected to an actual earthquake The dynamic responses of the prototype bridge piers were computed with and without consideration of SSI (i.e flexible-base versus fixed- base piers), and with and without consideration of the inelastic behaviour of the piers
This work explores the efficient implementation and practical application of the direct methods of SSI analysis with a system approach It presents the seismic demands of the prototype piers and foundations, and provides a quantified picture of the effects of SSI on the ductility and the total displacement demands of the piers as functions of their natural period This study investigates the effects of the modeling assumption of the structural elements (i.e elastic versus inelastic behaviour) on the estimated demands, and demonstrates that SSI analyses with elastic structures cannot always provide plausible predictions of the inelastic responses The effects of SSI on the seismic demands of the bridge piers are also studied probabilistically in order to consider the uncertainties in the system parameters and to account for the dispersions introduced by the variability of input ground motions and soil conditions Subsequently, SSI modification factors are proposed to estimate the demands of the flexible-base piers from their corresponding fixed-base demands The proposed method is probabilistic and quantifies the uncertainties involved in computing the modification factors Finally, this work demonstrates the shortcomings of the nonlinear static pushover analysis for seismic demand estimation of pile-supported bridge piers when SSI is significant
Trang 4TABLE OF CONTENTS
ABSTIRACCT Ác HO CĨ THÍ TH g0 .0000600448084460400058444000 04684 ii TABLE OF CONTENTS .cccccssssscsccsscssecsecescecsssssceseacsecssecsacseccsssenesnsssesessansenseasenseeses iii LIST OFE 'TABILEES o- -< cọ HH HT TH 04000 0n T9 04004 08508564085.0 vii LIST OF FIGUIRRES 2 GĨT THỌ HH HH HH HT 0 004 0640004000060640 9460980 094 viii ACKNOWLEDGMENTS .cccssssssssesosscrscssssessessssssssesseessscesssacssessessccsassccsecserssesecssaeees xiv DEDICA TION HH HH HH TT 0000066101 00400101004005609004019006 XV
1 TINTRODUCTION 0Q HH HH HH HH H6 0669060 060409004 1
L1 594340100 1
1.2 Motivations for this Researcl - HH HH TH ng HH Hy 8
E90) 9a, o0 on e 11
S92 (5°: 11 15 Organization ofthe DISS€TfafIOH - Ăn TH HH TH khu 12
2 _ DEVELOPMENT OF THE NUMERICAL MODELS -c<<<<<< 14 2.1 Prototype Soil-Foundation-Structure SYSf€T4 nhi 14
2.2 Modelling of Soil 16
2.2.1 R01 i83 nh 16
Trang 52.5 Modeling ofSoil-Foundation-Structure System Ặ 2c sec 27 2.5.1 Modeling of Soil-Foundation Interface co cccc+ Seo 29
2.5.1.1 Pile Cap-Soil Interface 29
2.5.1.2 Pile-Soil Interface ố e 30
2.5.2 Modeling of Structural Elements in Plane Strain Analysis 36
Phi co ha ố 36
2.7 _ System Physical PrOperIes Sàn HH TH HH TH H1 Hy rêu 37 2.7.1 Stiffness Of FoundafiOn - á- «c9 KH HH9 HH g1 0v, 37 2.7.2 Stiffness of Pile-Supported P1ers - LH HH 39 2.7.3 Natural Period of the Pile-Supported PIers sex 43 2.8 NH0 44
3 ANALVSIS VERIFICATION STUDY: DYNAMIC ANALYSIS OF AN INSTRUMENTED PILE-SUPPORTED BRIDGE PIER - 5<5<<<<< 45 3.1 Structure and Instrumentation of the Bridge - -.- HH vớ, 46 3.2 Modeling and Verification Procedure cccseesessceseeseceseeeeseetsessesseseseeessats 50 3.3 N8: 390/15020i1) 3200077 50
3.3.1 Soil Profile and Properties .cccccccscsesseeeeceeeseteeneceeeesecseceeeeeceeeeeeneseeensens 50 3.3.2 Nonlinear Behaviour of Soil Elements in FUAC ‹+-<5+s+<<52 33 3.3.3 08t 0/10ã19:000 0 Pe 35
3.3.4 Nonlinear Dynamic AnaÏySIS - ĩc cLn.HH* cv HS 1S HH 1g g1 8x rệt 36 3.3.5 Comparison of the Recorded and the Computed Surface Motions 57
Trang 64.2 Site Response ÀnaÌySIS -c HH HH HH HH TH HH tho 73
43 Analysis of Fixed-Base nh ae 76
44 Analysis of Soil-Foundation-Structure ŠySferms - cớ, 77 4.5 Seismic Behaviour of the System .ccecccccsscesscesscessesecceecesceesersessecseeensenes 82
4.6 SUMIMNALY 00 90
SEISMIC DEMANDS OF BRIDGE PIERS .cssssscssssesscscssssseecesesnseseeneeeers 91 5.1 Definttion of Demand Paramef€rs - HH kg he 91 5.2 Seismic Demands of the Fixed-Base PIers Ghi, 95 5.2.1 Strength Reduction FaCfOTS - HH TH HH ngu 96 5.2.2 Ductility Demands without P-A EfÍfects - ni 97 5.2.3 Ductility Demands with P-A Effects HH Hy 98 5.2.4 Comparison of Ductility and Strength Reduction Factors 100 5.3 Seismic Demands ofPile-Supported PIers - Ăn hieu 102 5.3.1 Strength Reduction FaCfOTS - HH HH ng HH KH kg 103 5.3.2 Ductility Demands ác HH ng HH HH in 104 5.3.3 Comparison of Ductility and Strength Reduction Factors 105 5.3.4 Effects of Piers Modeling Assumption on the Ductility Demands 106 5.3.5 Total Displacement Demands - - - - cà HH kg key 110 3.3.6 Effects of Piers Modeling Assumption on the Total Displacements 113
5.4 SH) 000778 114
SEISMIC DEMANDS OF FOUNDA TIONS Hee Hee 116 6.1 Translation of Foundafion ch TH HH HH ng 116
6.2 ro cugïð ion 118
Trang 77 EFFECTS OF SOIL-STRUCTURE INTERACTION ON THE SEISMIC DEMANDS OE PIERRS on HH HH HH TH HH 000060840 30 126
7.1 Effects of Soil-Structure Interaction on Ductility Demands 127 7.2 Effects of Soil-Structure Interaction on Total Displacement Demands 129 7.3 Comparison of the Effects of Soil-Structure Interaction on Ductility and Total Displacement [DDemands - - - ác ST HH HH nà ng kh 132
Sa 134
8 PROBABILISTIC ASSESSMENT OF THE EFFECTS OF SOIL-
STRUCTURE INTERACTION con HH HH HC n0 00609 1906 136
8.1 Methodology 0070777 138
TƯ AT Tin 140
8.2.1 Regression ModelÌs c4 HH HH ng HH KH KH hy 141 8.2.2 Artificial Neural NetWOrKS HH HH TH ng KH khe 144 8.3 Probabilistic Data Processing .cccsssccssecesesneceeenteeseneeesesnececeaeescearesetaneetiees 145 8.4 Performance-Based Assessment of the Effects of Soil-Structure Interaction on the Seismic Demands of the Piers cc ececcsscsseeeseccseeeesneeeseeceaeeceeeceeeeeeneseneecaneeaes 150
IS = an _— 160
9 EVALUATION OF NONLINEAR STATIC PUSHOVER ANALYSIS FOR
Trang 8LIST OF TABLES
Table 2.1: Summary of the yield properties of the piers cccecsssescesteestecsesersesseeees 26 Table 2.2: The Fixed-base and the flexible-base stiffnesses of the prototype bridge
ĐIỆF cọ, 12101121 H410 HH HH0 HH 11 0H g0 in tr 39 Table 2.3: Natural periods of the pile-supported piers and the period elongations 43 Table 3.1: Site’s nearby rock OufCTOD IOfIOTS - v.v TH H10 1 1 2tr 55 Table 3.2: Components of the recorded ground Motions ccesssesseetsstecsteesseeeeseeeeeees 55
I 0E S00ia6i 9e 1 72
Table 6.1: Mean ratio of foundation demands estimated from analyses with elastic piers to those esttmated with IinelasflC DI€TS Á Án HH yên 124 Table 8.1: Summary of the statistics of r€SuÏẲS HH HH4 1112 1 ca 141
Trang 9LIST OF FIGURES
Figure 2.1: Schematic description of the prototype soil-foundation-structure system 15
Figure 2.2: Distributions of Gmax and S, to the depth of 100 m of the prototype soil I1 18
FIigure 2.3: Shear modulus reduction CurV€ O Cây c9 TH HH ng ru 19 Figure 2.4: Equivalent viscous damping exhibited by a single soil element of FLAC 20
Figure 2.5: An example of the stress-strain relationship of a soil element at the depth of 40 m when the soil layer was subjected to an earthquake ground motion 20
Figure 2.6: Comparison of the elastic response of the pier obtained from FLAC with non- viscous damping and SAP2000 with 5% viscous dampIng -«««cc«cc<ee 23 Figure 2.7: Moment-curvature curve obtained from the cross section analysis of the 51777 25
Figure 2.8: A sample load-displacement curve of a pier when P-A effects included 26
011 27
Figure 2.10: Geometry of the FLAC model (top 30 m ưÝ soI]) cà seeeieeẽ 28 Figure 2.11: Interface elements in ELAC (Itasca 2005) HH He 29 Figure 2.12: Pile-soil coupling sprimgs 1n FUAC «7 S S ngư 30 Figure 2.13: Shear coupling springs yield ©rit€TiOn - sách Hiệu 31 Figure 2.14: Normal coupling springs yield CrIf€TIOT - Ăn HH, 32 Figure 2.15: The displacement of the pile with respect to the soil and the vectors of the soil incremental displacements in the horizontal plane 0 cece ecceeeeteeeteeseeeneees 34 Figure 2.16: An example load-displacement curve of the pile at a depth of 9.2 m 35
Figure 2.17: Validation of the normal coupling springs at a depth o£9.2 m 35
Figure 2.18: Translational stifness of Íoundafion «HH HH Hư 38 Eigure 2.19: Rotational stiffness of Íoundationn ác HH HH it 38 Figure 2.20: Stiffness of the pile-supported pier with fixed-base period of T=0.3 s 40
Figure 2.21: Stiffness of the pile-supported pier with fixed-base period of T=0.6 s 40
Figure 2.22: Stiffness of the pile-supported pier with fixed-base period of T=0.8 s 41
Figure 2.23: Stiffness of the pile-supported pier with fixed-base period of T=1.0s 4]
Trang 10Eigure 2.24: Stifness of the pile-supported pier with fixed-base period of T=1.5 s 42 Figure 2.25: Stiffness of the pile-supported pier with fxed-base period of T=2.0 s 42 Figure 3.1: Photo of the Hayward BART Elevated Section .ccccecssssssecessessessseeeens 46 Figure 3.2: Geometry and instrumentation of the bridge (After Shakal et al 1989) 48 Eigure 3.3: Foundation plan and pile patfern (per as-buIlt drawings) 49 Figure 3.4: Cross section of the foundation (per as-built drawings) ccecce<e2 49
I4 n0) ý00iii 02c 52
Figure 3.6: Soil shear modulus reduction CUTVۤ HH HH HH HH Hi ng rau 33 Figure 3.7: Shear modulus reduction exhibited by a single element in FLAC 54 Figure 3.8: Equivalent viscous damping exhibited by a single element in FLAC 54 Figure 3.9: Relative location and coordinates of the epicentre, the outcrops recording
stations, and the bridge site free field recording station .ccceccsesesssseeeteseressees 56 Figure 3.10: Comparison of the recorded and the computed response spectra 59 Figure 3.11: Comparison of transfer functions between input and recorded surface
motions and between input and computed surface motIO'S - +-cccs 59 Figure 3.12: Comparison of the computed and the recorded surface response time
DIStOLICS 61 Figure 3.13: Comparison of spectral surface responses obtained from SHAKE and
0.101 63 Figure 3.14: Surface motion computed by FLAC in response to the input motion
generated by SHAKE HH HH HH nh HT TH HH HH giết 65 Figure 3.15: FLAC model (top 30 m of soil) of the instrumented bridge pier system 66 Figure 3.16: Recorded and computed acceleration response spectra of the pier 68 Eigure 4.1: Acceleration response spectra of the input rock motIons -.- 71 Figure 4.2: Site acceleration amplification obtained from anaÌyS1s -‹ <- 73 Figure 4.3: Site acceleration amplification from Idriss (1990) cccceccceseseeeecteetseenees 74 Figure 4.4: Acceleration response spectra of the computed surface motions 75 Figure 4.5: Comparison of the acceleration response spectra of the input and the surface
MOTIONS eeccecesceeeeetseeeeaaeseceeseceeseeesseeteeesecsssceseaessecessseeeesedasseescssressseesseesenseseeenages 75 Figure 4.6: Sample displacement response at the centre of mass of a fixed-based pier with
and without P-A effects escsceseccscscccccecscestesescscesesaucsecersecseseceuccseseuausetseeuscuaevens 77
Trang 11Figure 4.7: An example of the response of a bridge pier and its foundation in time domain (T=0.8 s, Tsite=1.0 s, input motion: Nahanni Site 2) ececssessesscesseeeesceseneeees 78 Figure 4.8: Comparison of the surface spectral response of the 40 m soil layer obtained
from the analysis of a 40 m soil column and a 30 m soil column 80 Figure 4.9: Comparison of the surface spectral response of the 100 m soil layer obtained
from the analysis of a 100 m soil column and a 30 m soil column 81 Figure 4.10: Example of the response spectra of the soil-foundation-pier system
(T=0.3 s (f=3.33 Hz), Tsite=1.0 s, input motion: Nahanni Site 2) 84 Figure 4.11: Example of the response spectra of the soil-foundation-pier system
(T=0.6 s (1.67 Hz), T;u¿=1.0 s, Input motion: Nahamni Site 2) 85
Figure 4.12: Example of the response spectra of the soil-foundation-pier system
(T=0.8 s (1.25 Hz), Tsu¿=1.0 s, mput motion: Nahanni Site 2) 86 Figure 4.13: Example of the response spectra of the soil-foundation-pier system
(T=1.0 s (f=1.0 Hz), Tsite=1.0 s, input motion: Nahanni Site 2) 87 Figure 4.14: Example of the response spectra of the soil-foundation-pier system
(T=1.5 s (0.67 Hz), Tgs¡¿=1.0 s, nput motion: Nahanmi Site 2) 88
Figure 4.15: Example of the response spectra of the soil-foundation-pier system
(T=2.0 s (0.50 Hz), T;¡¿=1.0 s, input motion: Nahanni Site 2) 89 Figure 4.16: Translation response of the foundation supporting a massless pier (kinematic 79 0512))0010107177 90 Figure 5.1: Force-deformation relationship of an elastoplastic single-degree-of-freedom
Trang 12Eigure 5.11: Distribution of the strength reduction aCfOFS - - cà c2 tseevei 103 Figure 5.12: Distribution of the ductility demands of the pile-supported bridge piers 104 Figure 5.13; Distribution of the inelastic deformation ratios of the pile-supported
0112 105
Figure 5.14: Comparison of inelastic deformation ratios for the fixed-base
and flexible-base pIers; T = (.Ố § - LH HH HH HH HH tàn ng ng ưu 108 Figure 5.15: Comparison of inelastic deformation ratios for the fixed-base
and 8š [9 v115s)s001-v 1008 01 nh - 108
Figure 5.16: Comparison of inelastic deformation ratios for the fixed-base
and flexible-base piers; T = 1.0 So icc cesscseccseeeecesscssecseeeessseseaeeseaeeesseeeseeesaees 109
Figure 5.17: Comparison of inelastic deformation ratios for the fixed-base
ch 89 590151 in 109
Figure 5.18: Comparison of inelastic deformation ratios for the fixed-base
and flexible-base piers; T = 2.Ũ § LH HH TH ng gu 110 Figure 5.19: Distribution of the total displacement of the piers normalized with respect to
b"ơig o1 PP Lil Figure 5.20: Distribution o£ the local drift of the pI€FS SG ngư 112 Figure 5.21: Distribution of the piers portion of the total displacements 112 Figure 5.22: Distribution of the ratio of total displacements (SSI with elastic piers to SSI
„4108161215021 113 Figure 6.1: Distribution of peak translation of foundation - s« sec sec 117 Figure 6.2: Contribution of the foundation translation to the total displacements of the
Figure 6.4: Contribution of the foundation rotation to the total displacements of the 0= 119 Figure 6.5: Comparison of the contribution of the foundation translation and rotation to
the total displacements of the DI€FS Ác HH HH ng HH ng 121 Figure 6.6: Ratio of foundation translation demands estimated from analyses with elastic
piers to those estimated with inelastic DI€TS ĩ- - HH ng nhieu 123
Trang 13Figure 6.7: Ratio of foundation rotation demands estimated from analyses with elastic piers to those estimated with inelastic Piers cccsccscsessessesscsseseesesecssesrsesesseesess 124 Figure 7.1: Distributions of pier ductility demand ratios (DDR) ccccccseessesseesseees 128 Figure 7.2: Mean values of pier ductility demand ratios (DDR) 55552 129 Figure 7.3: Distributions of pier total displacement demand ratios (TDR) 06 131 FIigure 7.4: Mean values of pier total displacement demand ratios (TDR) 131 Figure 7.5: Mean response ratios (DDR and TDR) - - Q LH HH HH He nkệ, 132 Figure 7.6: Mean demand ratios (DDR and TDR) as functions of the period ratio
Figure 8.1: Nonlinear regression fit to mean response ratios (DDR and TDR) as
functions of the piers fixed-base perlod TÏ` c4 2112111011411 1 112g 142 Figure 8.2: Nonlinear regression fit to standard deviations of response ratios (DDR and
TDR) as functions of the piers fixed-base perIod TT - sách 142 Figure 8.3: Linear regression fit to mean response ratios (DDR and TDR) as
functions of the period ratio Tạy/T HH KH TH HT HH gu kg 143 Figure 8.4: Linear regression fit to standard deviations of response ratios (DDR and
TDR) as functions of the period ratIo Ïsy/T cong nêu 143
Figure 8.5: Probability of DDR>1 as a function of the piers fixed-base period T 147 Figure 8.6: Probability of TDR>1 as a function of the piers fixed-base period T 148 Figure 8.7: Probability of DDR>1 as a function of the period ratio Tyys/T eee 149 Figure 8.8: Probability of TDR>1 as a function of the period ratlo Tạys/T 149 Eigure 8.9: Probability of DDR>r as a function T;y/T for r values from 0.8 to 1.5 151 Figure 8.10: Probability of TDR>r as a function of T,,;/T for r values from 1.0 to 1.5.151 Figure 8.11: Probability of DDR>r as a function ofr for T;y//T of 1.02 to 1.3 152 Figure 8.12: Probability of TDR>r as a function of r for Tsy/T o£ 1.02 to 1.3 153 Figure 8.13: A bridge example with simply supported spans and pile-supported piers on
lị 82107 154 Figure 8.14: Correlation between DDR and site surface spectral acceleration at the
natural period of the fixed-base pier eccccsesecseesnetseeecceeeesneareaesteceaeeaeeaseaeeees 157 Figure 8.15: Correlation between DDR and site surface spectral velocity at the natural
period of the fixed-baS€ pI€F ch HH kiện 158
Trang 14Figure 8.16: Correlation between DDR and site surface spectral displacement at the natural period of the fixed-base pI€T - Án HH HH HH ng HH, 159 Figure 9.1: Numerical model for pushover anaÏySIS kg He, 163 Figure 9.2: Comparison of the force-displacement relationships (including SSI) obtained
from pushover and nonlinear dynamic analyses of the piers with T= 0.3 s 2.0.0 164 Figure 9.3: Comparison of the force-displacement relationships (including SSI) obtained
from pushover and nonlinear dynamic analyses of the piers with T = 0.6 s 164 Figure 9.4: Comparison of the force-displacement relationships (including SSI) obtained
from pushover and nonlinear dynamic analyses of the piers with T = 0.8 s 165 Figure 9.5: Comparison of the force-displacement relationships (including SSI) obtained
from pushover and nonlinear dynamic analyses of the piers with T= 1.05 00.00 165 Figure 9.6: Comparison of the force-displacement relationships (including SSI) obtained
from pushover and nonlinear dynamic analyses of the piers with T = 1.5 sw 166 Figure 9.7: Comparison of force-displacement relationships with SSI obtained from
pushover and nonlinear dynamic analyses of the pier with T = 2.0 s 166 Figure 9.8: Mean ratios of the global ductility demands to the local ductility demands 168 Figure 9.9: Comparison of the pier deformations with total displacements obtained from
pushover and nonlinear dynamic analyses of the piers with T = 0.3 s 169 Figure 9.10: Comparison of the pier deformations with total displacements obtained from
pushover and nonlinear dynamic analyses of the piers with T = 0.6 s 170
Trang 15ACKNOWLEDGMENTS
I would like to gratefully thank Professor Carlos Ventura for his guidance throughout the course of my research and for trusting me with my vision and research interests He provided me with the environment and the research tools I needed for the success of this work My great appreciation extends to my advisory committee members The expertise of Professor Peter Byrne was instrumental in my understanding of the geotechnical aspects of the work Professor Ricardo Foschi shared his inspiring ideas and expertise on the application of the probabilistic methods which greatly contributed to the success of this work, and Professor Robert Sexsmith provided valuable inputs I am grateful for the involvement of Professor Liam Finn upon his return to UBC and I sincerely appreciate his invaluable advice and support I would also like to thank Dr Steve Zhu of Buckland & Taylor Ltd and Dr John Cassidy of the Geological Survey of Canada for their inputs
Mr Tom Horton of San Francisco Bay Area Rapid Transit District (BART) is gratefully acknowledged for providing the soil and the structure data of the Hayward BART elevated section used as a case study in this research
My financial support came from different sources including the Postgraduate Scholarship of the Natural Sciences and Engineering Research Council of Canada (NSERC), Grant Supplement Award of the Faculty of Applied Science, and the University Graduate Fellowship (UGF) of the University of British Columbia I am indebt to Professors Sexsmith and Foschi for trusting me with the success of this work and for providing me with crucial financial support from their NSERC grants during the course of this research The additional fund from Professor Ventura supported me in the final stages of this research which is greatly appreciated
My deep gratitude goes to my parents and my sisters for the support and the peace of mind that they provided me The pressure over the course of this work would be unbearable without the joyful times that I spent with my dear friends and I would like to extend my appreciation to them for all their support
Trang 16To My Beloved Parents
Trang 171 INTRODUCTION
1.1 Background
With the evolution of performance based earthquake engineering methodologies, there is an ever increasing need for more accurate and quantifiable assessment of the complex behaviour of structures in response to earthquake loadings While the objective of seismic design codes has been to provide life safety, the recent methodologies place emphasize on various performance objectives and require the estimation of seismic demands for various levels of expected earthquake motions Seismic demand estimations must account for the response of systems in the inelastic range, so that measures of damage to the systems could be obtained Another aspect of performance-based design methodologies is the explicit consideration of the uncertainties in both capacity and demand estimations in order to optimize the design to meet various performance criteria with various levels of reliability (e.g Bertero and Bertero 2002, Krawinkler and Miranda 2004)
Seismic demand estimations involve many challenges, one of which is the evaluation of the effects of seismic soil-structure interaction (SSI) on the inelastic response of structures Seismic SSI is potentially a highly nonlinear phenomenon which causes the structural response to differ from that of the ideal structure with rigid base, often assumed by engineers SSI modifies the system response in two different ways, namely kinematic
interaction and inertial interaction Kinematic interaction occurs when the deformation of
Trang 18accounted for The inertial interaction is due to the reaction forces caused by the dynamic response of the structure These forces are transmitted to the foundation and to the soil subsequently, further displacing the foundation and its surrounding soil and thus further modifying the motion of the foundation with respect to the free field which in return further modifies the response of the structure (Gazetas and Mylonakis 1998) The flexibility at the base of the structure increases the number of the degrees of freedom of the system and lowers its stiffness causing the natural period of the system to elongate SSI dissipates the energy of vibration through hysteretic behaviour of the foundation’s surrounding soil (material damping), and by radiating the seismic waves into the soil continuum away from the structure (radiation damping)
SSI can particularly play a significant role in the response of bridge structures due to their relatively simple structural form and the low degree of redundancy of these structures that make them sensitive to the effects of SSI and SSI induced displacements This is particularly true for bridges with single-column piers and simply supported spans SSI may significantly affect the ductility demands of the bridge piers or cause large differential displacements between the piers This can be detrimental to displacement- sensitive components of bridges and may compromise the bridges structural integrity Displacement-sensitive components could be both structural and non-structural Examples of structural components include bearing seat width, restrainers or expansion joints, and examples of non-structural elements include pipe, electrical and telephone
lines
Trang 19account for all different aspects of SSI, such as the nonlinearity of soil, the nonlinearity of structure, and the nonlinearity of soil-foundation interaction, at the same time Particularly, the SSI of pile foundations is a very complicated phenomenon to model due to the complex behaviour of individual piles and the pile-soil interaction, and the complex behaviour of piles in a group of piles and the pile-soil-pile interaction
As a result of the complexities, the significance of the role of SSI on the seismic performance of structures is not yet clearly understood This is despite wealth of research that has gone into several aspects of seismic SSI The past research on SSI has largely been component-oriented and focused on the evaluation of the complex soil-foundation interaction in response to earthquake loading with less attention on the effects of this interaction on the overall system response On the other hand, researches that have dealt with the effect of SSI on the overall system response, required many simplifications in
regard to the behaviour of soil, or structure, or soil-foundation interface, in order to
reduce the size and complexity of the problem so that it could be managed with the available computing power at the time of each research
Performance-based design of pile-supported bridges requires rigorous estimation of the effects of SSI on the response of the piers, so that more accurate estimations can be made and better understanding of the role of SSI can be achieved The focus of performance-
based design, on the other hand, is on estimating the nonlinear displacements, as a better
indicator of damage in the inelastic range, rather than forces obtained from linear elastic analyses that have traditionally been used in structural and foundation design (displacement-based design versus force-based design) Therefore, estimation of SSI for performance-based design requires accurate estimation of the effects of SSI on the nonlinear displacements of systems, which in return requires simultaneous consideration
of nonlinear soil behaviour, nonlinear soil-foundation interaction, and nonlinear structural
Trang 20improved methods of SSI evaluation have been emphasized on and demonstrated by researchers in recent years
Gazetas and Mylonakis (1998) provide an overview of the methods of SSI analysis and re-explore the role of SSI on the seismic response of pile-supported bridge piers by studying the failure of the Hanshin Expressway Route 3 in Higashi-nada, with single circular columns supported on pile foundations, during the 1995 Great Hanshin Earthquake in Japan (Kobe Earthquake) They found that SSI could have contributed into the collapse of this bridge by elongating the natural period of the system which resulted in higher spectral accelerations at the shifted period Mylonakis and Gazetas (2000) further explore the effects of SSI by discussing the misconceptions in regard to ignoring SSI as a conservative assumption and explain how the spectral characteristics of ground motions, which depend on both seismic input and soil conditions, can result in higher seismic demands at the elongated natural periods of structures when SSI is accounted for, despite possible increase in damping due to SSI They demonstrate that SSI may result in increased ductility demands of bridge piers on soft soil and conclude that this may not be revealed if conventional code design spectra, as opposed to actual response spectra, are used in seismic demand estimations including SSI The possibility of increased response due to SSI was also stated previously by Jennings and Bielak (1973), and Veletsos (1993) and has been recently demonstrated by other researchers such as Sextos et al (2002) and Jeremié et al (2004)
Trang 21gapping elements and they conclude that the focus of future research should be better understanding and modeling of nonlinear foundation behaviour and to integrate such models with nonlinear structural analyses to allow overall performance assessment
Finn (2004a, 2004b) reviews the state of practice for characterization of the actions of foundations on the response of structures and evaluates the effectiveness of various approximate approaches He states that a weakness in modeling of pile-supported structures is inadequate representation of foundation by ignoring the coupling between the translational and rotational stiffnesses of the foundation represented by single valued springs He adds that most of the approximate methods in use for evaluating foundation stiffnesses are based on single pile analysis and further assumptions are made to account for the group response He states that many factors such as soil nonlinearity, kinematic interaction between piles and soil, inertial interaction of the structure with soil and piles, dynamic interaction between piles themselves, and seismically induced pore water pressures, must be taken simultaneously into account so that a complete picture of the effects of foundation on the seismic response of structures can be obtained He presents an investigation into the reliability of approximate methods for representing the rotational and translational stiffnesses of pile foundations in numerical models of pile-supported bridge piers by using a pseudo-3-dimensional nonlinear continuum soil model (Wu and Finn 1997a and 1997b, Thavaraj and Finn 2001), and highlights the importance of the relative stiffness of structure and foundation for interaction, and of including both kinematic and inertial interaction in analyses
Trang 22Kim and Roesset (2004) studied the importance of accounting for the nonlinear soil behaviour in the evaluation of the effects of SSI on the inelastic response of structures They demonstrated significant difference between the response with elastic soil and the response with inelastic soil and showed the importance of accounting for the nonlinearity of soil response especially for pile foundations
The review of literature also reveals that there is a lack of sufficient statistical description of the effects of SSI on structural response to demonstrate the effects of SSI quantitatively, rather than merely qualitatively For bridge structures, there are a few published studies on the effects of SSI on the response of the bridge piers with a statistical approach, but they are different in scope and certainly not enough to provide a complete picture For instance, Ciampoli and Pinto (1995) studied the effects of SSI on the inelastic response of bridge piers supported on spread footings without consideration of the nonlinearity of soil Hutchinson et al (2004a, 2004b) performed nonlinear dynamic analyses to estimate the inelastic response of the extended pile-shaft-supported bridge piers These foundations are different than pile groups and the results of these studies cannot be extended to bridges supported on pile groups
In summary, the above background remarks the need for the accurate evaluation of the effects of SSI on seismic response of pile-supported bridge piers on soft soil and identifies the following:
e There is a need for a quantified picture of the effects of SSI on the response of the pile-supported bridge piers, so that the circumstances under which these effects are significant can be identified, and it can be quantified what the effects of SSI are and how they can affect the overall bridge performance
Trang 23e Assessment of the effects of SSI must account for various performance objectives and various levels of earthquake ground motions
e Evaluation of SSI must be in line with displacement based design methodologies, i.e with the goal of estimating the inelastic displacements of piers and foundations as the primary demand parameters
e Estimation of SSI must account for uncertainties involved in input ground motions, soil conditions, and system properties, so that the design of the system can be optimized to meet different performance criteria with different levels of reliability tailored to a specific project
Moreover, it appears that a great number of structural engineers have not been able to take the most advantage of the available results of research on SSI due to the scatter of the current published literature and due to many instances of the lack of a link between the geotechnical aspects of the available knowledge and the practice of structural engineering Hence, the effects of SSI need to be presented in a format familiar to structural engineers and in accord with the fundamentals of structural analysis and design
Trang 24The often used “equal displacement rule” which assumes that the elastic and inelastic displacements are equivalent (i.e Cy=l or H=R), is a result of the R-p-T relations proposed by Veletsos and Newmark (1960) The equal displacement rule is valid for most practical ranges of natural period, but not for all periods Veletsos and Newmark (1960) also proposed the equal energy rule for the period range other than that of the equal displacement rule (see Chopra and Chintanapakdee 2004 and Chopra 2005 for an overview)
More refined estimation of the relationships between C, and T and their application in structural demand estimation have also been the subject of recent research due to their relevance to the displacement based design of structures with the need for more accurate estimation of the relationship between the elastic and inelastic displacements of structures (e.g Miranda 2001, Farrow and Kurama 2003, Chopra and Chintanapakdee 2004, Ruiz- Garcia and Miranda 2004) The C,-T or R-p-T relations are central to structural demand estimation and therefore consideration of these relations must be included in a study of the effects of SSI A rational approach to the presentation the effects of SSI on the response of structures would be to obtain these effects as functions of the structures natural periods so that a link between structural analysis and SSI analysis can be
established
1.2 Motivations for this Research
There are a number of motivations behind this research One is to explore the system approach to SSI analysis of pile-supported bridge piers with a) fully coupled representation of soils, foundations, and piers, b) proper consideration of the nonlinearity of soil, structure and soil-structure interface, and c) appropriate consideration of the radiation damping of the system The goal is to demonstrate the feasibility and practical application of such SSI simulation for seismic demand estimation of structures by performing nonlinear dynamic analyses
Trang 25performing finite element or finite difference analyses In the beam on a nonlinear Winkler foundation method, the nonlinear stiffness and damping characteristics of the soil-foundation system are accounted for through a series of nonlinear springs and dashpots distributed along the piles and the free field motion is applied to the ends of the springs (Gazetas and Mylonakis 1998) In the direct methods, finite element or finite
difference models of the soil continuum, the foundation and the structure are made all in
one model The material damping is accounted for by inelastic behaviour of the system and the radiation damping is considered by using non-reflecting boundaries for the soil continuum The input motion in this case is applied at the base of the soil layer
Compared to the p-y springs methods, direct methods have the advantages of being the most complete and the most transparent representation of the system Direct methods may be perceived as the more complex method However, they deal with basic physical properties of the system as apposed to the Winker foundation approach that requires complex springs representing the combination of a number of physical properties of the system with properties that may be very difficult to estimate Thus direct methods are not necessarily the most complicated ones In addition, direct methods can model the pile cap and automatically account for the pile group effects with no further approximations They can also include large deformations due to liquefaction of soil which is a great advantage for the analysis of liquefiable soils
Trang 26to demonstrate the practicality of the method and its advantages for practical earthquake engineering, and to highlight its limitations and the needs for future improvements
Another motivation for this work is to present a quantitative picture of the effects of nonlinear SSI on the seismic response of pile-supported bridge piers in a format useful to structural engineers, i.e as functions of structural period The questions to address are: 1) when SSI should be accounted for (i.e when is SSI significant); 2) how does the foundation contribute in the overall system response; and 3) how the response of the system is modified by accounting for SSI The approach to follow is to use the fully coupled, fully nonlinear numerical models developed in the first stage of this study to simulate the seismic response of prototype pile-supported bridge piers (including SSI) and by comparing the results with the results of the dynamic analyses of the same piers without accounting for SSI (i.e fixed-base piers) The goal is to study the role of SSI by providing statistical description of the response of various prototype systems with various natural periods subjected to various input ground motions with different characteristics The statistics obtained can further be processed probabilistically to include the uncertainties of the system parameters and to account for the dispersions introduced by input ground motions and soil conditions Representation of results suitable for performance-based evaluation of the effects of SSI can also be explored so that it can be used for future estimation of SSI for similar systems
By using the statistics obtained from the dynamic analyses, it is of interest to examine some aspects of the methods of structural analysis currently used in practice This includes: 1) examining the accuracy of the inelastic displacements estimated from elastic displacements of the SSI system by using inelastic displacement ratios that are obtained from the analysis of SDOF systems; 2) examining the change in the foundation demands when SSI analysis is performed with an elastic structure (i.e not accounting for the inelastic behaviour of the piers); and 3) to examine the accuracy of nonlinear static pushover analysis in demand estimation and in capturing the salient features of dynamic response of SSI systems when SSI is significant
Trang 271.3 Objectives of the Dissertation
The objectives of this dissertation are as follows:
To explore efficient implementation and practical application of the direct methods of nonlinear SSI analysis with a system approach
To evaluate the effects of nonlinear SSI on the inelastic seismic demands of pile- supported bridge piers, and to provide a quantified picture of these effects as functions of the piers natural periods
To investigate the effects of the modeling assumption of the piers (i.e elastic or inelastic behaviour) on the estimated demands of the piers and on the estimated demands of the foundations, and to examine whether analysis with an elastic structure can provide plausible prediction of the inelastic demands when SSI is significant
To provide a probabilistic representation of the effects of SSI on the seismic demands of bridge piers by including the uncertainties of the system parameters and by accounting for the dispersions introduced by the variability of input ground motions and soil conditions
By employing statistics of seismic responses, explore simplified methods for evaluating the effects of SSI on the response of piers from their estimated response without SSI
To examine the accuracy of the nonlinear static pushover analysis for seismic demand estimation of pile-supported bridge piers when SSI is significant
1.4 Scope of Work
This research was carried out by performing nonlinear dynamic analyses of prototype pile supported bridge piers on soft soils subjected to a set of input ground motions selected from historic ground motions The analysis methodology was carefully verified: the component responses were verified individually and the system response was verified by a case history analysis of an instrumented pile-supported bridge pier The analyses of the prototype piers were performed with and without accounting for SSI In each case, the analyses were carried out with assuming both inelastic and elastic behaviour of the piers The statistics of responses were summarized and further validated against available
Trang 28knowledge where possible Comparisons were made between the demands of the elastic piers and the demands of the inelastic piers to make conclusions in regard to the modeling assumption of the piers Comparisons were made between the demands with SSI and the demands without SSI to draw conclusions in regard to the effects of SSI Statistics of demands were further processed probabilistically by performing reliability analyses, and the performance-based assessment of the effects of SSI using response statistics was explored Finally, pushover analyses were carried out and the resulting demands were compared to the predicted demands by the nonlinear dynamic analyses
1.5 Organization of the Dissertation
This dissertation represents four phases of work The first phase deals with the numerical analysis and includes Chapters 2 and 3 The second phase involves application of the numerical analysis techniques discussed in the first phase to obtain the statistics of the response of SSI systems and to study the effects of SSI on the response of bridge piers This phase is presented in Chapters 4 to 7 The third phase, which is presented in Chapter 8, involves the probabilistic analyses of the results of the numerical analyses and their application for performance-based assessment of the effects of SSI on the response of bridge piers The fourth phase, presented in Chapter 9, is in regard to pushover analysis
In Chapter 2, after a description of the prototype systems, the construction of the numerical models is presented by explaining and verifying the modeling of various components of the system, and by discussing issues related to modeling of the system as a whole This chapter ends by using the constructed numerical models to obtain the system properties
Chapter 3 discusses a verification study of the modeling technique presented in Chapter 2 by performing a case history analysis of an instrumented bridge pier with recorded motions from an actual earthquake This chapter particularly verifies the capability of the numerical analysis in representing the overall system response
Trang 29The selection of input ground motions is presented in Chapter 4 This chapter also describes various stages of dynamic analysis including site response analysis, analysis of the piers without SSI, and analysis of the system as a whole including SSI The primary goal of this chapter is to examine the behaviour of the system rather qualitatively by observing the salient characteristics of the dynamic response of the system, without emphasizing on the statistics of response
Chapter 5 introduces the demand parameters and presents the statistics of the seismic demands of the piers The behaviour of the piers is studied in this chapter by means of the statistical distributions of demands
Chapter 6 presents the seismic demands of the foundations and their relation to the demands of the piers in order to better understand the role of the foundation response in the overall system response
In Chapter 7, the effects of SSI on the response of the piers are discussed by directly comparing the demands of the piers with SSI to the demands of the piers without SSI This chapter discusses the circumstances under which SSI must be accounted for, and demonstrates how SSI modifies the response of the piers
Chapter 8 presents the probabilistic assessment of the effects of SSI on the demands of the piers to account for the dispersions of demands that are related to the input ground motions and soil profiles, and to account for the uncertainties of the system properties This chapter explains how results can be used to estimate the effects of SSI on the performance of the piers with given target reliabilities and demonstrates how to use them to modify the response of the piers obtained from the analyses without SSI to account for the effects of SSI
In Chapter 9, the accuracy of nonlinear pushover analyses in estimating seismic demands of pile-supported piers is examined Finally, Chapter 10 presents a summary of the conclusions and the recommendations for future research
Trang 302 DEVELOPMENT OF THE NUMERICAL MODELS
This chapter presents the details of the numerical modeling of the prototype pile- supported bridge systems that were considered in this research The commercial program FLAC (Itasca 2005) was employed for the numerical analysis FLAC which stands for Fast Lagrangian Analysis of Continua, is a two dimensional explicit finite difference program for engineering mechanics computation It simulates the behaviour of soil, rock, steel, concrete and other materials that may undergo plastic flow when their yield limits are reached Materials are represented by elements or zones which form a grid in the shape of the object to be modeled The material can yield and flow according to several built-in constitutive models of the program or constitutive models defined by the user with the help of FISH, the built-in programming language of FLAC A useful feature of FLAC for the study of soil-structure interaction is the availability of interface elements which simulate distinct planes along which slip and/or separation can occur
2.1 Prototype Soil-Foundation-Structure Systems
The prototype soil-foundation-structure systems used in this research are illustrated in Figure 2.1 Each prototype system consists of a bridge pier, a pile foundation, and a soil layer The soil layer consists of saturated soft clay with average shear wave velocity in the upper 30 m of V,,, = 145 m/s and average strength to the depth of 30 m of S_,,, = 40 kPa The resulting prototype site is classified as Site Class E or F (NEHRP 2004, ATC-49 2003, NBCC 2005) The height of the site soil layer is 21.0 m, 40.0 m, and 100.0 m so that site periods (at low amplitude of motion) of 0.6, 1.0 and 2.0 seconds were obtained The foundation consist of a 6x6 pile group with 0.3 m square piles spaced at 1.25 m, driven to the depth of 15 m below the 7.5x7.5x1.5 m pile cap The pier has a typical
Trang 31circular cross section with a diameter of 1.5 m and with 1% longitudinal reinforcement The height of the pier varies from 3.5 m to 12.1 m to provide first mode fixed-base elastic natural periods of 0.3, 0.6, 0.8, 1.0, 1.5, and 2.0 seconds (see Table 2.1) A 3500 kN load was applied on the top of the piers to represent the weight of the bridges superstructures Details of the system properties and modelling of the individual components of the system and the system as a whole are explained in the following sections 3500 kN | "E——Ì ¥ cM | Circular Concrete Section with L 1% Longitudinal Reinforcement L=3.5 to 12.1m 15m Ts = 0.3 to 2.05 7.5X7.5x1.5 mPltle Cap ` a a H = 21.0 to 100.0 m Typical 0.3x0.3x15.0 m
Trite = 0.65 to 2.0 5 «ett Precast Reinforced
Trang 322.2 Modelling of Soil
2.2.1 Soil Profile and Properties
The stiffness, strength and mass properties of the prototype soft soil layers were chosen so that the prototype sites are classified as Site Class E or Site Class F as defined by current seismic design codes such as NEHRP (2004), ATC-49 (2003), and NBCC (2005) The material of soil was assumed to be soft clay; therefore liquefaction did not play a role in the analyses The water table was assumed to be at the surface of the soil layer The Mohr-Coulomb constitutive model (Itasca 2005) was used for the soil The parameters required by this constitutive model are stiffness parameters defined by the shear and bulk modulus of soil and by strength parameters determined by the friction angle and the cohesion of soil as described in the following subsections
2.2.1.1 Stiffness properties of Soil
The initial shear modulus of soil (G,,,,,.) was determined such that the resulting average max
shear wave velocity in the upper 30 m of soil (V5) ) was 145 m/s V,,, is used by many design codes and guidelines such as NEHRP (2004), ATC-49 (2003), and NBCC (2005) for the purpose of site classification for seismic design of structures The values of G,,,, were proportional to the square root of the mean normal effective stress of soil within the soil layer In other words, the distribution of G,,,, was proportional to /o%,/PA where
G7 m
The Gi,
is the mean normal effective stress and PA is the atmospheric pressure (100 kPa) x at the mean normal effective stress of 100 kPa (the atmospheric pressure) was (max 100 = 43 MPa The mean normal effective stresses were obtained by performing static analysis of the soil layer under its self weight and by updating the stresses in the soil before performing the dynamic analysis The G,,,, profile thus obtained is depicted in Figure 2.2 The value of V,3 can be calculated from the Equation 2.1 (e.g NEHRP 2004) In Equation 2.1, d; is the thickness of any layer i between 0 and 30 m and Ƒ(, 1s the shear wave velocity of the layer i obtained from the Equation 2.2; where p is the mass density and V, is the shear wave velocity of the soil layer
Trang 33V 24) (2.1)
V, =.|—+ (2.2)
The V,, thus calculated for the soil layer with the G,,,,, profile of Figure 2.2, and for the assumed mass density of p =1700 kg/m? was 145 m/s The bulk modulus was assumed to be ten times the shear modulus (corresponding to a Poisson ratio of 0.45) for the static analysis For the dynamic analysis, the bulk modulus was assumed to be 50 times the shear modulus for the saturated soil below the water table The nonlinear hysteretic behaviour of soil prior to yield was based on the well known modulus reduction curve proposed by Seed and Sun (1989) for clays (upper range) This modulus reduction curve is shown in Figure 2.3
2.2.1.2 Strength Properties of Soil
The shear strength of clay is defined by its cohesive strength The average cohesive strength in upper 30 m of the prototype clay of this study (S,,3)) was taken equal to 40 kPa (within the limiting values of S, for site classes E and F) The cohesive strength S, was assumed to be distributed linearly with the effective vertical stress (o,)) within the soil layer For the prototype clay layer, the ratio S/o, was assumed to be 0.381, which corresponds to a plasticity index of 73% according to the equation proposed by Skempton (Craig 1987) described by Equation 2.3, where PJ is the plasticity index The profile of S,, along the height of the soil layer is shown in Figure 2.2
Su =0.11+0.0037P7 + (2.3)
Ov
Trang 3410 + 10 + 20 + 20 ¬ 30 + 30 + ~ 40+ —~ 404 s ° s 50+ 503 o bà A 60+ a 60 + 70 + 70 + 80 + go - 90 + 90 ¬ 100 100 0 50 100 150 0 50 100 150 200 250 300
Gmax (M Pa) Su (kPa)
Figure 2.2: Distributions of G,,,, and S, to the depth of 100 m of the prototype soil layer
2.2.1.3 Nonlinear Behaviour of Soil Elements in FLAC
To verify the nonlinear behaviour of soil prior to yielding when modeled in FLAC, cyclic loading of a soil element was simulated Cyclic shear loads were applied incrementally on the element and the hysteretic response of the element was tracked Resulting hysteresis loops were then used to estimate the equivalent stiffness of the soil element corresponding to each level of the applied shear strain Then the modulus reduction curve (G/Gyax curve) of the element was constructed as a function of the shear strain The equivalent viscous damping representing the energy dissipated through the hysteresis loops was also estimated (see Chopra 1995 for methodology) Since this single element does not represent all soil elements in the model, the result of this simulation is solely for
an overall verification of the numerical model
The modulus reduction curve and equivalent viscous damping exhibited by the soil element undergoing the cyclic loading are presented in Figures 2.3 and 2.4 Figure 2.3 shows that the stiffness degradation of the soil, as exhibited by the soil element, matches
Trang 3640
—O— FLAC
35 7} wwe» Clay-Lower Bound (Sun et al 1988) 30 +1 sn Clay-Upper Bound (Sun et al 1988)
=——= C]ay-A verage (Sun et al 1988) 25 -TỊ mmm=sss Clay (Idriss 1990) 20 15 10 5 0 0.0001 0.001 0.01 0.1 1 10 Shear Strain % Equivalent Viscous Damping Ratio % Figure 2.4: Equivalent viscous damping exhibited by a single soil element of FLAC 100 ca Q j Shear Stress (kPa) oO 0.3 Shear Strain (%)
Figure 2.5: An example of the stress-strain relationship of a soil element at the depth of 40 m when the soil layer was subjected to an earthquake ground motion
Trang 372.2.2 Boundary Conditions
The lateral boundaries of the soil model must represent the infinite extent of the soil layer in the horizontal direction and therefore they should not reflect the incident waves reaching to them A non-reflecting boundary option of FLAC known as the “free field boundary” (Itasca 2005) was used in the numerical models Since these boundaries may not fully absorb the incident waves, they are placed at enough distance from the foundation so that the material damping prevents the reflected wave from returning to the vicinity of the foundation
2.3 Modeling of Bridge Piers
The bridge piers were modeled using beam elements The inelastic behaviour of the pier was accounted for by yielding of the plastic hinge at the bottom of the pier It is pointed out that a typical reinforced concrete column can have various modes of failure, including shear failure mode, flexural failure mode or a combination of shear and flexural failure modes (Priestly et al 1996) In this work, only the flexural failure mode is considered by accounting for the hysteretic behaviour of the plastic hinges Modeling of the inelastic behaviour of the piers is discussed in Section 2.3.3.1 A 3500 kN gravity load was applied at the top of the piers to represent the weight of the bridges superstructure
2.3.1 Beam Element Properties
The beam element properties included the modulus of elasticity (E), the cross sectional area of the bridge column (A), mass density (p), the cross sectional moment of inertia (1) of the column, and the yield moment (My) Typical values of E=25000 MPa and p=2400 kg/m? were used for the modulus of elasticity and mass density of the concrete, respectively The cross sectional area was that of a circular cross section with a diameter of 1.5 m The moment of inertia of the cracked concrete and the yield moment of the pier were calculated for 1% longitudinal reinforcing steel as shown in Section 2.3.3.1 The inertial mass associated with the 3500 kN load from the bridge superstructure was included by assigning high mass density for the beam elements at the top of the piers The center of mass at the top of the piers represented the centre of mass of the bridges
superstructures
Trang 382.3.2 Damping
For the dynamic analyses of the bridge piers, the damping was assumed to be equivalent to 5% of the critical viscous damping of the first mode of vibration This damping can be modeled in FLAC using Rayleigh damping However, in FLAC, when stiff structural elements are modeled along with softer soil elements, applying Rayleigh damping results in a significant decrease of the time step required by the explicit time integration scheme of the program Explicit time domain solutions are always stable but at the cost of long computational times due to small critical time steps that depend on the stiffness and the density of the material modeled Thus, the time step is chosen by FLAC for numerical stability and cannot be controlled by the user Using Rayleigh damping can result in impractically small time steps, causing the analysis runtime to become impractically long For this reason, a non-Rayleigh damping option of FLAC, called “local damping”, originally developed to increase the convergence rate of static solutions (Itasca 2005) was used This damping, however, had to be calibrated for the dynamic analysis and had to be verified against results obtained from the analysis with viscous damping
To verify the non-Rayleigh damping of FLAC, the fixed-base bridge piers were modeled using both FLAC and the well-known structural analysis program SAP2000 (Computer and Structures 2003) Results obtained from the elastic dynamic analyses of the piers using both programs were compared for various input ground motions Results demonstrated that the “local damping” represented an equivalent viscous damping of 5+0.5% of the critical damping Figure 2.6 shows a comparison between the acceleration response spectra and the acceleration time histories of one of the bridge piers obtained from FLAC and SAP2000, when it was subjected to one the selected ground motions (see Chapter 4) As can be seen in this figure, the two responses match very well with minor differences Finally, it must be mentioned that viscous damping is just a numerical convenience and it is not the real form of the physical damping of structural elements Therefore the minor differences observed in Figure 2.6 are of no significant importance
Trang 39FLAC SAP2000 Spectral Acceleration (g) 0 0.1 1 10 100 Frequency (Hz) FLAC SAP2000 Displacement (m) “0 5 10 15 20 25 30 35 Time (s)
Figure 2.6: Comparison of the elastic response of a pier obtained from FLAC with non- viscous damping and SAP2000 with 5% viscous damping
T=0.6 s, Tsite=1.0 s, Input Motion: Northridge City Terrace (Chapter 4)
2.3.3 Nonlinear Behaviour of the Bridge Piers
Both material and geometric nonlinearity of the piers response were considered in this study The material nonlinearity included the inelastic behaviour of the columns under bending moments (flexural yielding) and the geometric nonlinearity included the P-A effects Material nonlinearity under shear (shear failure) was not considered The assumption of flexural failure with no failure in shear is a reasonable assumption for especially the taller piers Shorter piers are more likely to experience flexural and shear failures, however, this consideration was beyond the scope of this work
Trang 402.3.3.1 Material Nonlinearity
The inelastic response of the piers was accounted for through hysteretic behaviour of plastic hinges under bending moment FLAC has options for modeling the plastic hinges of beam elements with elastoplastic response It was observed, however, that under
dynamic loading, the hysteretic behaviour of a plastic hinge modeled in FLAC was closer to a nonlinear-elastic response rather than an elastoplastic response Therefore, modification of the program was necessary to ensure appropriate numerical modeling of the hysteretic behaviour of plastic hinges was achieved The modifications were implemented through FLAC’s programming language FISH
An elastoplastic hysteretic model, with and without stiffness degradation, was implemented For the stiffness degrading hysteretic model, however, lack of robustness under dynamic loading was observed Therefore, only the elastoplastic model was used in this study, although the stiffness degrading model is the preferred model for reinforced concrete material Ruiz-Garcia and Miranda (2004) studied the effect of the stiffness degradation on the response of SDOF systems when subjected to ground motions recorded on soft soil (with no SSI) by comparing the results obtained from elastoplastic and stiffness-degrading systems and by plotting the mean ratios of the inelastic displacements obtained from the two models Their results showed that the variation of response was mostly less than 30%
While the aforementioned variation of response is recognized here, it is emphasized that given the comparative nature of this research between the results obtained from the fixed- base and flexible-base piers (as it will be seen in the forthcoming chapters), and given the complexity of the performed nonlinear analyses, the dispersion of results caused by using an elastoplastic hysteretic model (instead of the preferred stiffness degrading model), will not play a significant role in the findings of this research It should also be pointed out that there are still many uncertainties associated with modeling the inelastic behaviour of reinforced concrete columns, even when stiffness degrading models are used, and this is still a subject of research in structural engineering