Development o Filure Probbility Anlysis Method or Conrete Piers o Multi-spn Continuous Bridges using the Probbilisti Cpity Spetrum Method Je Shin CHOI, Je Kwn KIM ABSTRACT When erthqukes our, strutures sustin dmges. But it is impossible to estimte dmges extly whih our in the strutures beuse o the unertinty o mterils nd ground motions. Thereore, probbilisti methods or estimting dmges o the strutures hve been suggested ltely. In this study, probbilisti pity spetrum method expnding estblished pity spetrum method is suggested nd this method is pplied to lultion o the ilure probbility o the multi-spn ontinuous bridges. While estblished pity spetrum method doesn't relet the unertinty o mterils nd ground motions, probbilisti pity spetrum method onsiders diretly the unertinty o the strength o the onrete in the pity spetrum nd the unertinty o the ground motion in the demnd spetrum. While the perormne point ppers one point using estblished pity spetrum method, using probbilisti pity spetrum method we obtin the probbility o the perormne point. Thereore, the ilure probbility o the bridges is lulted when erthqukes hving speii return period our. keywords : Multi-spn Continuous Bridges, Reinored Conrete Piers, Probbilisti Cpity Spetrum Method, Filure Probbility, Perormne Point Je Shin CHOI, Grdute Student, Dept. o Civil, Urbn & Geosystem Engineering, Seoul Ntionl University, Seoul, 151-742 Je Kwn KIM, Proessor, Dept. o Civil, Urbn & Geosystem Engineering, Seoul Ntionl University, Seoul, 151-742
1. INTRODUCTION Strutures ontin the unertinty bout severl tors. And yet, the struturl nlysis whih hs been rried out till now doesn t relet tht inlusively. Atully, it is resonble to relet the unertinty in the struturl designs nd i the knowledge is tken dvntge o, eonomil eiieny nd serviebility re hieved. Thereore, onsidering the unertinty o the onrete nd ground motion, the method whih the ilure probbility o bridges is lulted with under erthqukes is suggested in this pper. 2. PROBABILISTIC CAPACITY SPECTRUM METHOD Figure 1. Proedure o Probbilisti Cpity Spetrum Method
2.1 THE DAMAGE STATES In the deinition o the dmge sttes o the onrete s piers, the existing dmge levels bsed on the urvture dutility re used nd tht is deined s ollows. µ φ φ m = (1) φ y Where, φ : The Curvture t the estimted ross-setion m φ : The Curvture. o the Yield y Dmge Stte Desription Curvture Dutility No Dmge Negligible µ < 1.0 φ Light Moderte Severe Collpse Light rking & Prtil splling Dmge minly t one side Dmge minly t two opposite sides Dmge through entire ross-setion 1.0 < µ φ < 3.0 3.0 < µ φ < 5.0 5.0 < µ φ < 13.0 13.0 < µ φ Tble 1. The Deinition o the Dmge Stte (Priestley,1994) 2.2 PROBABILISTIC CAPACITY SPECTRUM When the erthqukes our, the motions o the bridges n be divided into longitudinl nd trnsverse motions. In this study, two pity spetrums whih desribe longitudinl nd trnsverse motions o the bridges re onstruted.
The seleted model o the bridge hs the one ixed pier tht stnds in the middle o the bridge nd the ross-setion o the piers is n ext squre. Reerring to the longitudinl motion, the piers behve like the ntilever bem. Also,. the MDOF (Multiple-Degree-O- Freedom) is simpliied into SDOF (Single Degree-O-Freedom) bout the trnsverse motion by ssuming the deormtion shpe o the superstruture be equl to the sine urve. The pity urve o the pier is onstruted by the struturl nlysis. The pity urve o the longitudinl diretion ppers biliner (s shown in Figure 2.) nd tht o the trnsverse diretion pper tri-liner (s shown in Figure 3). P u P y αk e K e d y d u Figure 2. Biliner Cpity Curve o the Pier (Longitudinl diretion) P u K 3 P y K 2 K 1 d y d u Figure 3. Cpity Curve o the Pier (Trnsverse diretion) And this urve is trnsormed into pity spetrum using reltive equtions o the ADRS (Aelertion-Displement Response Spetrum) [M.Shinozuk, 2001]
The untion o the pity spetrum is desribed s ollows. S = F, I, H, W, S ) (2) ( e t d Where, F is the strength o the onrete, I e is eetive moment inerti o the pier, H is height o the pier nd W t is the weight o the superstrutures. The pity spetrum through the struturl nlysis desribes the men vlue beuse o using the verge vlue o the strength o the onrete. To desribe PDF (Probbility Density Funtion) o the pity spetrum, the eqution o the pity spetrum is ombined with the PDF o the strength o the onrete. I the PDF o the strength o the onrete is ssumed to hve norml distribution, the PDF o the pity spetrum is desribed using the PDF o the strength o the onrete nd is derived s ollowing proedures. F = 1 ( S, I, H, W, S ) (3) e t d 1 1 = (4) r 2 ( ) exp[ ( ) ] F σ 2π 2 σ 1 1 1 d = (5) σ 2π 2 σ ds r 2 ( s ) exp[ ( ) ] S Where, F : Rndom Vrible o the Strength o the Conrete : The Required Strength o the Conrete (Men Vlue o the Conrete ) r 2.3 PROBABILISTIC DEMAND SPECTRUM The shpe o the demnd spetrum ollows the stndrdized demnd spetrum o the UBC ode (s shown in Figure 4).
Figure 4. The Stndrdized Demnd Spetrum o the UBC ode Inelsti demnd spetrum is obtined by deresing elsti demnd spetrum using the redution oeiients (s shown in Figure 5) Figure 5. Inelsti Demnd Spetrum Beuse the stndrd demnd spetrum o the UBC ode doesn t relet the unertinty o the ground motion, the men nd stndrd devition o the ground elertion re derived through the hzrd nlysis bsed on the erthquke dt nd re pplied to the demnd spetrum by using the reltion o the ground elertion nd spetrl elertion. I the PDF o the ground elertion is ssumed to log-norml distribution, the PDF o the demnd spetrum is derived s ollowing proedures. g C = 1 ( S, SR ( orsr ), S ) (6) v d = 1 1 1 σ 2π 2 σ (7) ln ln m 2 ( ) exp[ ( ln( _ )) ] C
g 1 1 1 1 d = (8) ds 2 ( s ) exp[ ( ln( _ )) ] s σ 2π 2 σ ln ln m 2.4 CALCULATION OF THE FAILURE PROBABILITY The probbility o the perormne point existing t the speii displement is obtined s ollowing eqution. ps ( ) = d 0 sdmx 0 0 g ds s s g ds ds s s d (9) To lulte the ilure probbility, the dmge sttes deined by the urvture dutility t setion 2.1 re trnsormed into the spetrl displement nd the probbility o the perormne point within the rnge o the ollpse stte is dded. For exmple, i the rnge o the ollpse stte is between displement nd s d mx, the ilure probbility is derived s ollows. p sdmx = 0 sdmx 0 0 g ds ds s s d g ds ds s s d ()
3. APPLICATION & RESULTS Item Vlue(m) Cross Setion 2.5*2.5 Height 14 Tble 2. The Properties o the Bridge Return Period : 00 yers Item vlue Men ( g ) 0.12 Stndrd devition ( g ) 0.4 Tble 3. The Properties o the Erthquke 3.1 RESULTS Dmge Level Cumultive Probbility Light 9.6 Moderte 9.46 Severe 8.95 Collpse 2.25 Relibility Index β = 3. 48 Tble 4. Longitudinl Diretion
Dmge Level Cumultive Probbility Light 6.57 Moderte 5.63 Severe 3.52 Collpse 7 2.65 Relibility Index β = 5. 0 Tble 5. Trnsverse Diretion 3.2 COMPARISON Cross-Setion : 2.5m*2.5m, H=15m Dmge Level Cumultive Probbility Light 6.0 Severe 1.57 Relibility Index β = 3. 6 Tble 6. Longitudinl Diretion (H. N. Cho, 2002) Cltrns' Bridges Dmge Dt Dmge Level Cumultive Probbility Minor 1.468 Moderte 5 7.86 Mjor 5 1.917 Collpse 7 2.05 Relibility Index β = 5. 1 Tble 7. Trnsverse Diretion (M. Shinozuk, 2001)
4. CONCLUSIONS In this study, the results tht re similr with existing results re quired using the Probbilisti Cpity Spetrum Method. It is the new pproh tht the onepts whih re unrelibility or ll o the strength o the onrete nd ground motion re dded to the Cpity Spetrum Method to obtin responses o strutures under erthqukes. While Cpity Spetrum Method shows one response beuse o not onsidering the unertinty, Probbility Cpity Spetrum Method shows not only the probbility o eh dmge stte but lso ilure probbility o the bridges. ACKNOWLEDGEMENTS This study ws supported prtly by the und o the Kore Erthquke Reserh Center (KEERC), prtly by the und o the Kore Bridge Design & Engineering Reserh Center (KBRC) nd prtly by the und o the BK21 Progrm o Kore Ministry o Edution. The uthors wish to express their grtitude or the support reeived. REFERENCES Alredo H-S. Ang, Wilson H. Tng, Probbility Conepts in Engineering Plnning nd Design Vol.,, John Willey & Son, 1975 Arthur H. Nilson, Design o Conrete Strutures, MGrw-Hill, 1997 Asdour H. Hdjin, A generl rmework or risk-onsistent seismi design, Erthquke Engineering nd Struturl Dynmis, Vol.31, pp.601-626, 2002 C. Allin. Cornell, "Engineering Seismi Risk Anlysis", Bulletin o the Seismologil Soiety o Ameri, Vol.58, No 5, pp.1583-1606, 1968 Jk R. Benjmin, C. Allin Cornell, Probbility, Sttistis, nd Deision or Civil Engineers, MGrw-Hill, 1970
M. Shinozuk, M. Q. Feng, H. Kim, T. Uzw, T. Ued, "Sttil Anlysis o Frgility Curves", MCEER Tehnil Report, 2001 "Seismi evlution nd retroit o onrete buildings", ATC-40 Report Vol.1, Applied Tehnology Counil, 1996