Analysis f Curved Bridges Crssing Fault Rupture Znes R.K.Gel, B.Qu & O.Rdriguez Dept. f Civil and Envirnmental Engineering, Califrnia Plytechnic State University, San Luis Obisp, CA 93407, USA SUMMARY: This paper evaluates the adequacy f tw apprximate analysis prcedures, the fault rupture-respnse spectrum analysis (FR-RSA) prcedure and the fault rupture-linear static analysis (FR-LSA) prcedure, in estimating the seismic demands fr curved bridges crssing fault rupture znes. The FR-RSA and FR-LSA prcedures were riginally develped fr rdinary straight bridges crssing fault-rupture znes and have been prved valid in a prir investigatin. In this investigatin the tw apprximate prcedures are revisited fr evaluatin f tw Califrnian curved bridges (ne with three spans and the ther with fur spans) crssing fault rupture znes. Seismic demands btained frm the apprximate prcedures are cmpared with thse frm the nnlinear respnse histry analysis () which is mre rigrus but may be t nerus fr practical applicatin. Results shw that the FR-RSA and FR-LSA prcedures which require less mdelling and analysis effrts prvide reasnable seismic demand estimates fr curved bridges crssing fault znes. Keywrds: seismic demand, curved bridge, crssing fault rupture znes 1. INTRODUCTION While aviding cnstructin f the bridges crssing fault ruptures might be the best apprach, such situatins are nt always permissible in regins f high seismicity such as Califrnia. Currently, 5% f the Califrnian bridges crss fault ruptures r lie very clse t fault-rupture znes. In a recent earthquake in Wenchuan, China, severe damages were bserved frm bridges crssing fault ruptures, demnstrating the vulnerability f such bridges (Kawashima et al. 2009). In respnse t the bserved damages, extensive research effrts have been made t develp mre effective and well-rganized prcedures t estimate the seismic demands fr bridges crssing fault rupture znes. Gel and Chpra (2008) develped tw simplified prcedures, namely, the Fault Rupture-Respnse Spectrum Analysis (FR-RSA) and Fault Rupture-Linear Static Analysis (FR-LSA) prcedures t estimate the seismic demands fr straight rdinary bridges crssing fault ruptures. These tw prcedures estimate the seismic demands by superpsing the peak values f quasi-static and dynamic bridge respnses. The peak quasi-static respnse in bth methds is cmputed by nnlinear static analysis f the bridge under the grund displacement assciated with fault rupture. In FR-RSA and FR-LSA, the peak dynamic respnses are respectively estimated frm the cmbinatin f the peak mdal respnses using the cmplete quadratic cmbinatin (CQC) and the linear static analysis f the bridge under apprpriate equivalent seismic frces, respectively. The FR-RSA and FR-LSA prcedures have been cnfirmed t be adequate fr straight rdinary bridges thrugh result cmparisns against the nnlinear respnse histry analysis () prcedure, which is a mre rigrus apprach requiring excessive mdelling and cmputatinal effrts. This paper revisits these apprximate prcedures and further evaluates their adequacy in the analysis f tw curved bridges, which reflect the current bridge design and cnstructin practice in Califrnia. The fllwing briefly reviews the FR-RSA and FR-LSA prcedures fllwed by descriptin f the cnsidered bridges and
the crrespnding finite element mdels tgether with result cmparisns and discussins. 2. EXISTING APPROXIMATE ANLAYSIS PROCEDURES The FR-RSA and FR-LSA prcedures were develped by superpsing the peak values frm quasistatic ( r ) and dynamic ( r ) parts, t frm a ttal system respnse r fr straight rdinary s bridges. The quasi-static and dynamic cmpnents are respectively fund thrugh: (1) static analysis f the nnlinear bridge mdel with all supprt displacements applied simultaneusly; and (2) cnducting either the respnse spectrum r static prcedures n the linear bridge mdel under the fault-nrmal and fault-parallel grund mtins. Bth methds utilize special features f the spatially varying grund mtins in fault-rupture znes where displacements at supprt l can be apprximated as: u ( t) = α u ( t) (2.1) gl l g where ug ( t ) is the displacement histry at a reference lcatin, and α l is the prprtinality cnstant fr the l th supprt mtin (Gel and Chpra 2008). 2.1. FR-RSA Prcedure As described abve, the FR-RSA prcedure cmbines the bridge respnse based n: r t = r s + r FN + r FP (2.2) FN FP where r and r represent the peak bridge respnses caused by the fault-nrmal and fault-parallel grund mtins, respectively. In the FR-RSA prcedure, the perids and mde shapes f the bridge are btained based n eigenvalue analysis f the linear bridge mdel; the effective influence vectr fr fault-parallel mtin is determined based n the special grund mtins described in Eqn. 2.1 while the effective influence vectr fr the fault-nrmal mtins is the same as cnventinal respnse spectrum analysis. In this investigatin, the respnse quantities assciated with each mde are cmbined using the CQC rule. 2.2. FR-LSA Prcedure Similar t the FR-RSA prcedure, the FR-LSA prcedure als cmbines the estimates f bridge quasistatic respnse and dynamic respnse. Hwever, different frm the FR-RSA prcedure, the dynamic respnse f the bridge is determined frm a static analysis f the bridge under apprpriate equivalent seismic frces. As a result, the cmbined bridge respnse can be calculated frm Eqn. 2.3. ( ) r = r + r + r i = j = (2.3) t s FN. i FP. j max 1 r 2; 1 r 2. where FN i. r and r FP j represent the peak bridge respnses caused by the fault nrmal and fault parallel grund mtins, respectively, which can be determined frm the linear static analysis. As recgnized in Gel and Chpra (2008), dynamic respnse f the bridge may be cnservatively estimated by assuming the spectral acceleratin f the bridge equal t 2.5 times the peak grund acceleratin and distributing the seismic frce using a prper effective influence vectr shwn in Fig. 2.1, which is specifically determined fr the grund mtin described in Eqn. 2.1. t
Figure 2.1. Sketch f the effective influence vectr fr a bridge crssing fault rupture znes Given that peak dynamic respnse can be either psitive r negative, fur cases f equivalent static frce distributins are cnsidered in respnse calculatins: r caused by 2.5ml PGA r caused by - 2.5ml PGA FN.1 FN.2 eff _ FN FN eff _ FN FN FP.1 FP.2 r caused by 2.5mleff _ FPPGA FP r caused by - 2.5mleff _ FPPGA FP (2.4) where m represents the mass matrix in the equatin f mtin; l eff _ FN and eff _ FP l respectively represent the effective influence vectrs fr fault-nrmal and fault-parallel mtins; PGA FN and PGA FP respectively represent the peak grund acceleratins f the fault nrmal and fault parallel mtins. 3. MODELING OF THE SELECTED BRIDGES T further verify the adequacy f the abvementined apprximate prcedures, tw curved bridges respectively designated as Bridges 55-0837S and 55-0939G in Califrnia were selected. Bridge 55-0837S is a three-span curved bridge built in 2000; 55-0939G is a fur-span curved bridge built in 2001. Fig. 3.1 shws the selected bridges. State highway 55 555-0837S 55-0939G INTERSTATE 5 (a) plan view f 55-0837S (b) plan view f 55-0939G Abutment 1 Bent 2 Bent 3 Abutment 4 Abutment 1 Bent 2 Bent 3 Abutment 5 Bent 4 INTERSTATE 5 State highway 55 (c) elevatin f 55-0837S (d) elevatin f 55-0939G 3.1. Finite Element Mdels Figure 3.1. Selected Bridges The riginal finite element mdels f the selected bridges were set up using the Open System fr Earthquake Engineering Simulatin (OpenSees) (Mazzni et al. 2006), by the researchers frm the University f Califrnia, Irvine (UCI) fr ther research purpses. The riginal finite element mdels were mdified t be linear and nnlinear mdels in this investigatin. In bth linear and nnlinear mdels, the bridge decks were mdelled using elastic beam-clumn elements (elasticbeamclumn in OpenSees). The bents in the linear mdels were mdelled using elastic beam-clumn elements; hwever, the bents in the nnlinear mdels were cnsidered using the beam-clumn elements with
distributed plasticity and linear curvature distributin (dispbeamclumn in OpenSees). T cnsider the sil-structure interactin, spring elements were assigned at the bent bases. Prperties f the springs were determined accrding t the Califrnia Department f Transprtatin (Caltrans) Seismic Design Criteria (SDC) (Caltrans 2010). Sil springs were als assigned at the bridge abutment lcatins t take int accunt the abutment restraining effects. Alng the transverse directin, linear elastic springs with stiffness equal t 50% f the elastic transverse stiffness f the adjacent bent were defined as recmmended in the Caltrans SDC (Caltrans 2010). In the lngitudinal directin, the elastic-perfectly plastic gap springs specified in the riginal nnlinear mdels were cnverted t an elastic cmpressin nly springs using the Caltrans SDC recmmendatins (see Fig. 3.2) where stiffness f the lngitudinal spring, K, can be determined frm: eff K eff P P + P bw = = bw K eff gap bw abut (3.1) where P is the passive pressure frce resisting mvement at the abutment, and bw frm the elastic-perfectly plastic gap springs. gap is determined Figure 3.2. Simplificatin f lngitudinal abutment springs (Adapted frm Caltrans 2010) Cnceptually, the displacement restraining effect alng the lngitudinal directin due t the presence f abutments depends n the magnitude f the lngitudinal displacement f the bridge. A larger lngitudinal bridge displacement indicates mre severe damages ccurring at the abutments. As a result, a smaller stiffness shuld be assigned t the abutment lngitudinal springs t cnsider the less significant restraining actin. Accrdingly, the Caltrans SDC suggest a stiffness varying between 0.1 t K fr the lngitudinal abutment springs, which can be further determined frm an iterative eff prcess based n the lngitudinal displacement f the bridge. T validate the apprximate prcedures under a brader range f the parameters, three stiffness values -- 0.10K eff, 0.55K eff, and 0K eff -- were cnsidered in the investigatin. 3.2. Selected Grund Mtins Grund mtin pairs were selected t match the design spectrum prvided by Caltrans SDC (Caltrans, 2010). Due t the limited number f actual grund mtins recrded very clse t actual ruptured faults (less than 100m), grund mtin simulatins are the nly methd t btain time histries fr this analysis. These simulated time histries were required t incrprate the near-fault surce radiatin pattern and t accunt fr far- and near-field seismic radiatin during rupture prcess as well as the
sudden elastic rebund. A set f 10 grund mtin pairs were prvided by Caltrans fr the investigatin (Shantz and Chiu, 2011). Fig. 3.3 cmpares the gemetric means f the respnse spectra f the grund mtins and the idealized Caltrans SDC design spectrum. Figure 3.3. Caltrans design spectrum and elastic respnse spectra f the cnsidered grund mtins 3.3. Fault Lcatins and Bridge Orientatins Fr Bridge 55-0837S, the fault rupture line is assumed t be between Bents 2 and 3; fr Bridge 55-0939G, tw cases f fault lcatins were cnsidered, i.e., assuming the fault rupture lines respectively between Bents 2 and 3 and between Bents 3 and 4 as shwn in Fig. 3.4. Bent 2 Bent 3 Bent 2 Bent 3 (a) Bridge 55-0837S (fault between Bents 2 and 3) Bent 4 (b) Bridge 55-0939G (fault between Bents 2 and 3) Bent 2 Bent 3 Bent 4 (c) Bridge 55-0939G (fault between Bents 3 and 4) Figure 3.4. Illustratin f fault lcatins x X θ z Z Figure 3.5. Definitin f bridge rientatin angle Anther parameter cnsidered in the investigatin is the fault-line-t-bridge rientatin angles. In an actual case, the fault rupture may nt always rient perpendicular t the line cnnecting the tw abutments f the bridge. It is necessary t evaluate the adequacy f the apprximate prcedures under a brader ranges f the angles frm the fault rupture t the line cnnecting the tw abutments f the bridge. T this end, the rientatin angle illustrated in Fig. 3.5 is intrduced and Table 3.1 lists the selected angles cnsidered in the investigatin. It is nted that maximum and minimum angles represent the extreme cases f bridge rientatins fr which the fault will remain between the tw
adjacent bents. Table 3.1. Selected Bridge Orientatins Bridges Selected Orientatin Angles 55-0837S -51, -36, -21, -6, 0, and 9 55-0939G (fault line between Bents 2 and 3) -36, -15, 0, 18, 41, and 55 55-0939G (fault line between Bents 3 and 4) -34, -15, 0, 18, and 41 4. RESULT DISCUSSIONS AND CONCLUSIONS As described abve, three parameters were varied in the investigatin: the stiffness f the lngitudinal abutment springs (selected t be 0.10K eff, 0.55K eff, and 0K eff ), the bridge rientatin angles (selected t be the values listed in Table 3.1), and the different fault lcatins (see Fig. 3.4), t evaluate the rbustness and adequacy f the FR-RSA and FR-LSA prcedures under the practical ranges f the parameters. Due t the space cnstraint, nly results frm a few selected cases are presented in Figs. 4.1 t 4.3. Results fr ther cnsidered cases are available elsewhere (Rdriguez 2012). The bridge demands cmpared in Figs. 4.1 t 4.3 include the peak values f the lngitudinal and transverse abutment displacements and the peak values f the resultant bent displacements. In additin, the results frm the FR-RSA and FR-LSA prcedures are differentiated int tw categries: GM and SD, as shwn in the figures, which respectively represent the results btained using the spectral acceleratins frm the respnse spectra f grund mtin pairs and the idealized Caltrans SDC design spectrum. Mrever, fr the results frm the GM categry, the average values f the bridge respnses frm the 10 grund mtins pairs are presented. As cmpared in the figures, the results cnsistently shw that the FR-RSA prcedure prvides reasnable estimates fr the seismic demands f the bridge, and the FR-LSA prcedure is cnservative in sme cases due t the use f a cnservative estimate f respnse spectral acceleratin when cmpared with results frm the nnlinear prcedure. Based n the results frm this investigatin, it is fund the FR-RSA and FR-LSA prcedures can be extended in seismic analysis and design f curved bridges crssing fault znes. FR-RSA-GM FR-RSA-DS FR-RSA-GM FR-RSA-DS FR-RSA-GM FR-RSA-DS 1 4 1 4 2 3 Bent N. FR-LSA-GM FR-LSA-DS FR-LSA-GM FR-LSA-DS FR-LSA-GM FR-LSA-DS 1 4 1 4 2 3 Bent N. Figure 4.1. Result Cmparisns f Bridge 55-0837S (θ = 0 ; lngitudinal abutment stiffness= 0.1 K eff )
2.0 1.6 2 3 4 Bent N. 1.5 2.0 1.6 0.9 0.3 2 3 4 Bent N. Figure 4.2. Result Cmparisns f Bridge 55-0939G (θ = 0 ; lngitudinal abutment stiffness= 0.1 K eff, fault line between Bents 2 and 3) 2.0 1.6 2 3 4 Bent N. 1.5 2.0 1.6 0.9 0.3 2 3 4 Bent N. Figure 4.3. Result Cmparisns f Bridge 55-0939G (θ = 0 ; lngitudinal abutment stiffness= 0.1 K eff, fault line between Bents 3 and 4)
AKCNOWLEDGEMENT The research reprted in this paper is supprted by Caltrans under Cntract N. 65A0379 with Dr. Allaua Kartum as the prject manager. The supprt is gratefully acknwledged. We acknwledge Drs. Tm Shantz and Brian Chiu frm the Divisin f Research and Innvatin f Caltrans wh simulated the grund mtins used in this investigatin. We are als grateful t Mark Yashinsky, Fued Zayati, Mark Mahan, and Trak Zkaie f Caltrans fr their useful feedback n this research investigatin. Mrever, the riginal finite element bridge mdel was develped by Dr. Farzin Zareian frm UCI. REFERENCES Caltrans. (2010). Seismic Design Criteria (Versin 1.6 ed.) Califrnia Department f Transprtatin, Sacrament, CA (http://ww.dt.ca.gv) Gel, R. K., and Chpra, A. K. (2008). Analysis f Ordinary Bridges Crssing Fault-Rupture Znes. Rep. N. UCB-EERC-2008/01, Earthquake Engineering Research Center, University f Califrnia Berkeley, CA. Kawashima, K., Takahashi, Y., Ge, H., Wu, Z., Zhang, J. (2009). Damage f Bridges in 2008 Wenchuan, China Earthquake. Jurnal f Earthquake Engineering, Vl.13, 956-998. Mazzni, S., McKenna, F., Sctt, M. H., and Fenves, G. L. (2006). OpenSees Cmmand Language Manual. Rdriguez, O., (2012) Seismic Demand Evaluatin fr a Three-span Curved Bridge Crssing Fault-Rupture Znes, M.S. Thesis, Califrnia Plytechnic State University. Shantz, T., and Chiu, B. (2011). Creatin f Recrd Set fr Use in Fault Crssing Study. Divisin f Research and Innvatin. Califrnia Department f Transprtatin.