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1 6ƒ 3A Á 006 5œ pp. 497 ~ 5 ª sƒ w k w w Improved Modal Pushover Analysis of Multi-span Continuous Bridge Structures z Áy Á½ Kwak, Hyo-GyoungÁHong, Seong JinÁKim, Young Sang Abstract In this paper, a simple but effective analysis procedure to estimate seismic capacities of multi-span continuous bridge structures is proposed on the basis of modal pushover analysis considering all the dynamic modes of structure. Unlike previous studies, the proposed method eliminates the coupling effects induced from the direct application of modal decomposition by introducing an identical stiffness ratio and an approximate elastic deformed shape. Moreover, in addition to these two introductions, the use of an appropriate distributed load {P} makes it possible to predict the dynamic responses for all kinds of bridge structures through a simpler analysis procedure. Finally, in order to establish the validity and applicability of the proposed method, correlation studies between rigorous nonlinear time history analysis and the proposed method are conducted for multi-span continuous bridges. Keywords : bridges, improved modal pushover, identical stiffness ratio, equivalent modal load, elastic deformed shape š w k w k w ü sƒw wš z w w. w w z»» k x x w k w w» z g. ù ƒ w ƒ w sw w w rw w mw w dw ƒ w. w z y w» w 4ƒ w x w ry x w d w. w :, k w, w w z»», ƒ w, k x x. w w w w k x ùkü, p, (primary members) j x., w y w» w w yw x w ü sƒƒ w. y ³ (ATC, 996; ATC, 997)» y š, ATC-40 š FEMA-73 š» w rp» ry ü sƒ wš. w rp w (performance point) ¾ ƒ w w ww w rw wš w ƒ w w w w š (Chopra et al., 999). š k x ùkü š (capacity curve) rp (demand spectrum) wù v ùkü w w (performance point) w ƒ (A)- (D) x tx š rp ùküš.» rp k rp l * z w w» y œw ( khg@kaist.ac.kr) **w w» y œw ( ccalong97@naver.com) *** z w m œw ( kimys@andong.ac.kr) 6ƒ 3A 006 5œ 497

2 . š - -»(R y -µ-t) (Miranda et al., 994; assar et al., 99) w A-D x k rp yw ùkü š š ƒ w w kx w (pushover analysis) mw w - š l A- D x y. š l tw wù š ùkü w» k l l tw wù š w v w. š w ƒ š k w ƒ š. ƒ (Equivalent Single Degree Of Freedom; ESDOF) (a) ³ w s w (uniformly distributed load) g w k x x wù t x ƒ wš m ƒ yw w w (, 004). ESDOF w e ù w e w mw w dw, w e w ƒ j w w. ÿw, w e ƒ ƒ w w» ƒ ƒ x» w, ESDOF wù ƒ dw j ƒ w» (Zheng et al., 003). wr (Predominant Single Degree of Freedom; PSDOF) (b) w» ƒ ƒ j wù kw, x w x j w s w w w wwš k w w ù w w (Usanmi et al., 004). w w mw e yw dw, wƒ š w w» ƒ w ƒ e x k dw. w ƒ w d š š w w l» w (Krawinkler et al.,. š w w 498

3 998). w» w k w (Modal Pushover Analysis; MPA) (Chopra et al., 00). (c) ƒ w w s ƒ j k w ww š w, w w w» š ƒ wì w w šd dw., MPA w k, ƒ ƒ w š ƒ w ù w p ƒ w š w w w ƒ. ƒƒ ƒ w w» ƒ ƒ w xk w w ƒ y w e p MPAƒ w š w w»., š w dw» MPA w w, ù w -» rw w mw ü sƒ w š w k w wš w. ù ƒ w 4 w w ww mw w w ww. w,» MPA w w e w š w w w r y wš, x w dw mw.. k w (Modal Pushover Analysis; MPA)» k w (Modal Pushover Analysis; MPA) ƒ w w s ƒ j k w ww š w, ƒ w p. w ƒ w w š ¾ š w l dw. w k () ùký. [ m] { u } [ c] { u } { f uu, s ( )} + + = [ m] { }u g t», {u} w l, [m] w, [c] w, {f s } w l, {} w l š ü g ƒ. () k k x d z w ƒƒ {φ n } {p eff (t)} š wš M n ={φ n } T [m]{φ n }, L n ={φ n } T [m]{}, Γ n =L n /M n š {s n }= Γ n [m]{φ n } () () w z w () w {s n }. { p eff () t } { p eff n ()} s n = n=, t = n= { }u g () t = Γ m [ m] { φ n}u g t n=, n z w {p eff,n (t)} ƒw [m]{ü}+[c]{ü}+{f s }= -{s n }ü g (t), k w š» x š ƒ w. k k x {φ n } w k D n (t) j. = = { u() t } { u n () t } n= n= Γ n { φ n }D n () t k ƒ k k š» x ƒ z (coupling effect) w ƒ wì (3) k k š ω n ζ n m D n (t)ƒ w n k w. F sn = L u g () t, F sn( D D n, n) = φ n n D n ζ n ω n D n { } T { f s ( D n, D n) } k k w ww» w (4) F sn /L n x D n w. k k ¼ w {φ n } w» w s {s n* }=[m]{φ n } n w w k w ww V bn p e r u rn w, (5) w n tw A-Dx š yw. V bn V bn F sn u rn A n = = = , D * Γ M n L n L n = n Γ n φ rn n», M n* =Γ n L n z. A-Dx š F sn /L n -D n w (4) w y š mw D n0 w. MPA k x {φ n } w z w š ƒ wš z (coupling effect) w k w. ù, ƒ w z yw (a n )» ( 3(a) ) x {φ}ƒ yw š, w k x {φ n } m w z w z w., n (4) w ƒ w w w. z w š ƒ w, ƒ sw {s n *} w k w () (3) (4) (5) () 6ƒ 3A 006 5œ 499

4 3. MPA A-D x š š (pushover curve) l ƒ š 3(a) w -z»» α n ƒ, w ƒ w ƒ w xk» ƒ ƒ ƒ» š ƒ w ƒw ƒ. w, ƒ w w {sn * } x ƒ w w w p ƒ j x w w w k ƒ ƒ wš, ƒw š 3(b) ùkú., ƒ w š w F sn /L n -D n w ƒ» w ù š w» MPA j w ƒ. 3. k w (Improved Modal Pushover Analysis; IMPA) x» MPA w w w z yw x {φ n } ƒ w z ƒ w w k w w.» w w wš w r j» w w w z»» α k x x u a w. x {φ n } š z ƒ» w, x w [k 0 ], š w o x w [k(t)], š ω(t) [k(t)]=α(t)[k 0 ], ω (t)=α(t)ω o w α(t) ƒ w w yw w ùkü w., j w ƒ w w z yw α(t) w w [k(t)] j»ƒ yw wš, (6) w z x {φ} w x {φ o } w w w.», α 4 A-D x š w -z 4. k w A-Dx š»» ùkü. ([ k() t] ω [ m] ) φ { } = α() t ([ k 0 ] ω 0 [ m ]){ φ} = 0, { φ} ={ φ 0 }» MPA 3(a) ƒ α n w (6) w, š x {φ o } w ƒ š z w., (6) w w ƒ k, [φ o ] T w w w (diagonal element) w z ƒ»» MPAƒ ƒ ( (4) ) w. w w ww, k k ƒ x w w ƒ w w z ƒ x w. (6) w w z š x w, k w (pushover analysis) w w w ƒ ƒ x w x ƒ w. 5 4 ƒ x x ùkü u e,pier w ƒ w w» k x x, u pier ƒ w w z xx w. ƒ š x (6) 500

5 š w ww w. j» w w z x x u a,pier k x x g w k x x. u = apier_r, βu epier_r,», m ƒ š w β w m Min u pier_r βu epier_r, r = 5. 4 xx ( ) (7) w, u pier_r k w r ƒ š u e,pier_r u a,pier_r ƒƒ r ƒ w k k ùk ü., u pier_r u e,pier_r w u pier_r k u a,pier_r w w., w w z»» α k x x mw MPA w š» ƒ w z w, ƒ ƒ ƒ š w -z»» α ew w rw. wr» MPA w ƒ sw {s n* } w w ww» j w š, ƒw ƒ k., w w w ƒw ƒ š w sw {P} wš w. k ( () ) d z w {P} w α n, w {s n }= Γ n [m]{φ n } š ƒw w γ ƒ w { φ n } T { P} ω Dn = = n Γ n M n k k ƒƒ (Pseudo Acceleration) A n0 š w w n D n =γüa n =A n0 ƒ (8) w w w k w š ƒ w, sw {P} k k ƒƒ (A n0 ) w ({s n }) (9) tx. { P} = γ ( A i0 Γ i [ m] { φ i }) i = sw {P} w w γ ƒ j w ww k w š 6 k x x ( (7) ) w z A-D x š y j š y A n =A no γ ƒ. wr, F sn /L n -D n k k»» ω n w w γ y š w w D ny =A n0 γ y /ω n, w γ 0, A w z»» α š w n0 γ y D n0 = ω A n n0 ( γ 0 γ y ) ƒ. αω n u ar0, (8) (9) (0) r ƒ n w w u rn0 w w w (0) r ƒ u a,ro ew w. r ƒ w u ry = { φ n } T γ ( a i Γ i [ m] { φ i }) i = = γ a Γ n M n n ( ) A u n0 γ y A n0 γ 0 γ y rn0 φ rn Γ n ω n = n αω n = n = = n = ( φ rn Γ n D ny ) = n = φ rn Γ n A n0 γ y š w w () š w w z»» α. α γ 0 γ y u ry = u ry γ y u ar0, (), 6(a) k w š l ƒ w D ny š ω n 6. k w š y 6ƒ 3A 006 5œ 50

6 w w z»» α w ª wš w F sn /L n -D n w w A-Dx š w, š w y š w dw. 4. w 7. ( : m) 4. e w 4ƒ 4 gj p. ƒ ¼ ƒ e B, ƒ ¼ ƒ e B, B3 w, B3 /0 y jš ù p w w B4 w. B3 B4 x w» 4ƒ B, B B3 x 8 ùkü. ƒ v ƒ w š,.3 tonf/m 3 ƒ w. wr x ƒ ƒ w w w ƒ SD40 D9 98 e w w x 7 r. w w wš w w, ƒ w k w., ƒ w z w w p wš w., w w ƒ w w w z k k ƒ ƒ w w x wš, w x w. x p-š y(bilinear) g w, x e t r., t š w ƒ w» w z w š w. 8.»ww x ( : m) t. x e z (EI) w š (φ y ) w z (α) M-m Rayleigh» ƒ 5%ƒ w p-š š gj p w x (bilinear stiffness degrading model) w (Takeda et al., 970). k w k w OpenSees.6.(The PEER Center, 005) w w w š k w w w w» w w w w w z wš w. t 7»» w œ w.,» k rp k w š ƒ ƒ.0gƒ œ w ƒ 50

7 t. œ» o. EQ EQ EQ 3 EQ 4 EQ 5 EQ 6 EQ 7 Earthquake California El Centro Mexico City orthridge San Fernando San Francisco Taft Year Comp. S07E S00E SCTS00E CHA3 76W S09E EW q EQ l 7¾»y w w. 4. ƒ w w w š» w w w w w w w w y w» w, e x B3 B3 w w wš p w B4 w ƒ w w. š 994 w orthridge w vw w sƒ y w w m w» (Bozorgnia et al., 004), t œ ƒ t EQ4(orthridge) w w ww. w, ƒ w sw w - š z, y mw F sn /L n -D n w w t A-Dx š w B3 B4 - š l š w B3 w š w. ESDOF w s w B3 ƒw ƒ ƒ w - š w y A-Dx š w 9 y w w. wr PSDOF B3 z z 58.3% w» ƒ ƒ w š ü {s * }w w w wwš - (pushover curve) m 0(b) š yw. w MPA 9. ESDOF w B3 k w š y 0. PSDOF w B3 k w š y 6ƒ 3A 006 5œ 503

8 . MPA w B3 k w š y. IMPA w B3 k w š y ƒ ƒ ƒ xk A-Dx š ƒ š w z»» α ƒ» ƒ w w ƒ ƒ dw w ù. (b) (f) y w ƒw š w, ƒw š F sn /L n -D n w ƒ» w, š ƒ w. w w k w (IMPA) w w mw š ¾ š w w š. sw {P} w w w ww k w š (pushover curve) w ƒw š y k. ƒ - š š, ƒ - š y j (b) w š» wƒ xk š w ƒ dw ƒ w. ƒ w w w š k ƒ w w w 504

9 3. ESDOF w B3 w (orthridge EQ) w dw OpenSees x w (onlinear Time History Analysis; HA) mw w ƒ w k w., MPA ƒw š x ƒ x k¾ w wwš z w w. 3 l, ESDOF w e B3 w dw y w. ƒ w w w e š w w d ESDOF l» (Zheng et al., 003) ew. w PSDOF w 4 e B3 ƒ 3 dw w y w. exk ƒ ƒ w» w wù š w PSDOF w w w dw ƒ w. w B3 e xk ƒ» ƒ ƒ ƒ w PSDOF e ùký. 4. PSDOF w B3 w (orthridge EQ) MPA w (b) (f) ƒ 3 w z ƒ ƒ w ƒ w» 5 z w ƒ w» ƒƒ w v t w ( 5(a) 5(c) )., ƒ ƒ w w» ƒ dw y w. MPA w w» w w. k w (IMPA) w B3 w 6 w. ƒ ƒ w w w j š q, w w B3 d š w. IMPA w ƒ w e w w w y w. ƒ w e B4 w ƒ w w 7~0 ùkü. ƒ w» B3 6ƒ 3A 006 5œ 505

10 5. MPA w B3 w (orthridge EQ) ƒ ƒ w wš, ƒ ƒ»» ƒ ùkü. ƒ w w B4 dw, PSDOF MPA B3 w w w ƒ, k w (IMPA) w w d w y w. w, B4 w ESDOF w d w 7 w. w mw y w w., ESDOF e xk w w w w w ƒ., PSDOF e xk w., MPA ƒw š w» w w ww., IMPA w w e xk k w w. 4.3 k w k w (IMPA) k w (MPA) w 7 œ w e B e B 6. IMPA w B3 w (orthridge EQ) d w. MPA š ƒw (α = (b) (f) ) w w ƒ w k w w w š w w z k p ƒ ƒ (α=0)w MPA xk w w ƒ w w. MPA IMPA w w ƒ š p e t 3 4 ùkü. MPA j w ww w, ƒ ƒ ƒ w z»» a ƒ š., IMPA w w mw ƒ w š ƒ. w w z»» αƒ w» rw š w dw. ƒ w w t 3 4 š w ( (4) ) w ww, ( (3) ) w ƒ ƒ dw. š x w (HA) mw ƒ ƒ š MPA IMPA dw ƒ ƒ y w 506

11 7. ESDOF w B4 w (orthridge EQ) d wš w w. ƒ ƒ w ùkü., e B ƒ 3 w» ƒ 3 w w. w ƒ w () k w, ƒ ƒ w ƒ w s³ (mean square root of square sum) w 3 ùkü. ( ) = Error Rate MSRSS EQ EQ ( Relative Error) n ( HA ) n n= () ƒ ¼ ƒ ¼ d ƒ w, e B B e j ƒw w 3 y w. š MPA w IMPA d w ù w, p MPA ƒ w w (αƒ α=0 ey) m w š w wš IMPA z sƒ w. 8. PSDOF w B4 w (orthridge EQ) 5. š w w m w š dw š w,» k w (MPA) k w (IMPA) w w w w» w e Bƒ EQ4(orthridge EQ) w š mw d w. š w» w A-D x rp v w. w, rp w» w» R y w z»» α=0~ 0% p k š š w š(miranda et al., 994; assar et al., 99), w rp x A-Dx A y -D x ùküš (Chopra et al., 999). w, w z»»» R y ƒ š w α α=0~0. ù t 3 4 y w.,» w A y -D x rp zw š q w w 6ƒ 3A 006 5œ 507

12 9. MPA w B4 w (orthridge EQ) t 3. MPA A-Dx š p e ƒ»»» w w z»» (ω n ) (D ny ) (α) B B 0.33 st mode rd mode ƒ st mode nd mode ƒ rd mode ƒ α µ w A-D x rp w. MPA w š B k 0. IMPA w B4 w (orthridge EQ) t 4. IMPA A-Dx š p e»»» w w z»» (ω n ) (D ny ) (α) B B st mode rd mode st mode nd mode rd mode ~6 ùküš. ƒ ƒ ƒ ƒ ƒ w z»» α w A-D x rp µ w wš ƒ rp w w š e ƒ ew ƒ D n0. α j» rp», ƒ w z»» ƒ MPA(t 3 ) w š w ù 4~6 y w., IMPA š g 7 w ùkü. IMPA ƒ w š š, B ƒ ƒ w w z»» α=0.407 ƒ» 508

13 . B d. B d 3. ƒ d s³ (t 4 ) α=0.407 w wƒ rp š ƒ w. 7 wù rp ƒ w š w ƒ ew ƒ D n0. 4~7, ƒ D n0 w ƒ U rn0 =Γ n φ rn D n0 w, SRSS w dwš w B ƒ 6ƒ 3A 006 5œ 509

14 4. B ƒ w MPA š 5. B ƒ w MPA š 6. B ƒ 3 w MPA š (U pier_r ) max t 5. B x w (HA) w, MPA IMPA w mw š w x dw, ƒ w IMPA d w ƒ w ƒ y w. ww w, š w 50

15 7. B w IMPA š MPA IMPA ƒ w w x dw ƒ w. w IMPA š wì w MPA w rw w mw w x dw, rw w š IMPAƒ w š w. p MPA w ƒ w (α ƒ )ƒ w w IMPA sƒw. 6. ü rw sƒw š w v w š w w w. š w» w» w w w y w š, w w wì w mw w. w, w š mw» k w (MPA) k w (IMPA) d x w (HA) w š w..» w w ƒ. ƒ (ESDOF) w w e xk dw w ƒ w, (PSDOF) e xk d j ƒ. k w (MPA) ƒw š j» w w w ƒ w..» w, w p w š w ww ƒ w. w, w w z»» α k x x w x w z (coupling effect) w d w w ƒ w y w. 3. w w k w mw š w. w ry, w wù š ¾ š w w ƒ. 4. w w š w w z»» a ƒ ƒ w», w š w sƒ v w w w. 5. w w y š wì w w d., e j ƒ( w ƒ ¼ ƒ ) ¼ ƒ ƒ w d 5% ƒ w», e p e w w ƒ v w q. SMART z» l (SISTEC) w w ¾. ƒ t 5. š B w d ( : m) HA MPA IMPA (U pier_r ) max D n0 (U pier_r ) max D n0 (U pier_r ) max st mode nd mode rd mode st mode nd mode rd mode st mode nd mode rd mode ƒ 3A 006 5œ 5

16 š x (004) rp w k sƒ, wm wz, wm wz, 4«3Ay, pp Applied Technology Council (996) Seismic evaluation and retrofit of concrete buildings, ATC 40, Redwood City, CA. Applied Technology Council (997) EHRP guidelines for the seismic rehabilitation of buildings, (FEMA 73); and HERP commentary on the guidelines for the seismic rehabilitation of buildings, (FEMA 74), ATC-33, Redwood City, CA. Bozorgnia, Y. and Bertero, V.V. (004) Earthquake engineering from engineering seismology to performance-based engineering, CRC Press, Boca Raton at Florida Chopra, A.K. and Goel, R.K. (999) Capacity-demand-diagram methods for estimating seismic deformation of inelastic structures: SDF systems, Report o. PEER-999/0, Pacific Earthquake Research Center, University of California at Berkeley, Chopra, AUK. and Goel, RUK. (00) A modal pushover analysis procedure for estimating seismic demands for buildings, Earthquake Engineering and Structural Dynamics, Vol. 3, pp Krawinkler, H. and Seneviratna, G.D.P.K. (998) Pros and cons of a pushover analysis of seismic performance evaluation, Engineering Structures, Vol. 0, o. 4-6, pp Miranda, E. and Bertero, V.VU (994) Evaluation of strength reduction factors for earthquake-resistant design, Earthquake Spectra, Vol. 0, pp assar, A.A. and Krawinkler, H. (99) Seismic demands for SDOF and MDOF systems, Report o.95, John A. Blume Earthquake Engineering Center, Stanford University, CA. Takeda, T., Sozen, M.A., and ielsen,.. (970) Reinforced concrete response to simulated earthquake, Journal of the Structural Division, Vol. 96, o. ST-, pp Usanmi, T., Lu, Z., Ge, H., and Kono, T. (004) Seismic performance evaluation of steel arch bridges against major earthquakes. Part : Dynamic analysis approach, Earthquake Engineering and Structural Dynamics, Vol. 33, pp Usanmi, T., Lu Z., Ge, H.,G and Kono, T. (004) Seismic performance evaluation of steel arch bridges against major earthquakes. Part : Simplified verification procedure, Earthquake Engineering and Structural Dynamics, Vol. 33, pp Zheng, Y.S Usanmi, T., and Ge, H. (003) Seismic response prediction of multi-span bridges through pushover analysis, Earthquake Engineering and Structural Dynamics, Vol. 3, pp ( :005..9/ : / : ) 5