THREE-DIMENSIONAL NON-LINEAR EARTHQUAKE RESPONSE ANALYSIS OF REINFORCED CONCRETE STRUCTURES

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1 HREE-DIMESIOAL O-LIEAR EARHQUAKE RESPOSE AALYSIS OF REIFORCED COCREE SRUCURES K. ishiura 1), K. akiguhi 2), and H. H. guen 3) 1) Assistant Professor, Dept. of Arhiteture and Building Engineering, oko Institute of ehnolog, Japan 2) Professor, Dept. of Mehanial and Environental Inforatis, oko Institute of ehnolog, Japan 3) Graduate Student, Dept. of Mehanial and Environental Inforatis, oko Institute of ehnolog, Japan Abstrat: In this paper, three-diensional non-linear earthquake responses of R/C strutures were onsidered with one-ass-sste. he restoring fore harateristis were odeled based on the theor of plastiit, whih was one of the aro odels. wo tpes of the restoring fore odel were adopted. One was flexural tpe that ould be seen in rigid frae strutures, and the other was shear tpe that ould be seen in R/C box wall strutures. Paraeters of the response analsis were natural period of the sste and tpes of analsis those were three and two diensional analses. As a result, oparison between three and two diensional responses, onsideration of vertial response aeleration, and estiation of total energ input were arried out. 1. IRODUCIO An earthquake response analsis of building is one of the effetive ethods in order to estiate the seisi safet. One-diretional analsis is popular in the seisi design, and lateral two-diretional effets are often onsidered. However, three-diretional analsis is hardl ade in the seisi design instead of its neessit. In this paper, three-diensional non-linear earthquake responses of R/C strutures were onsidered with one-ass-sste. he restoring fore harateristis an be represented with skeleton urve and hsterisis loop. It an be said that the harateristis of R/C strutures an be lassified into two tpes. One is flexural tpe that has large area in hsterisis loop, and the other is shear tpe that has sall area in the loop, where the area eans energ dissipation. he forer one an be seen in rigid frae strutures, and the later one an be seen in R/C box wall strutures. he restoring fore odel presented b ishiura and akiguhi (23), whih was based on the theor of plastiit so-alled analog odel that was one of the aro odels, was used for both tpes with odifiation of stiffness of hsterisis loop. his odel an desribe axial deforation behavior on unloading that was one of the probles on analog odels of R/C strutures (akiguhi and Gao 2). 2. RESORIG FORCE MODEL 2.1 Skeleton Curves he two tpes of restoring fore odel, whih are alled F-odel and S-odel in this paper, are represented in tri-linear skeleton urves as shown in Figure 1. he F-odel has large area and the

2 S-odel has sall area in hsterisis loops. Figure 2 shows raking surfae and ield surfae those are orresponded to first and seond orners of the skeleton urve, respetivel. he F-odel has parabola surfaes and the S-odel has ellipse, and the raking surfae and the ield surfae are expressed as F = and F =, respetivel. he harateristis of Y-axis are assued to be the sae with -axis. A fore point and a deforation point are expressed as {P}=(Q, Q Y, ) and {δ}=(δ, δ Y, δ ), respetivel, where and Y axes are lateral axes and -axis is vertial axis. he restoring fore odels of one-diretion shown in Figure 1 were expressed when = C. he ield surfaes and the raking surfaes obe the ixed hardening rule that onsist of isotropi hardening and Prager's kineati hardening rule (Shield and iegler 1958). Q Q Q K p. K K e K Q Q Q K p. K β. K K el K δ δ p. K p. K Q -δ relationship = C Q -δ relationship = C K p. K K e K K p. K β. K K p. K δ p. K -δ relationship Q = and Q Y = -δ relationship Q = and Q Y = (a) F-odel (b) S-odel Figure 1 Restoring Fore Model on One-axis el K ield surfae ield surfae δ Q Q raking surfae Q Y ield surfae raking surfae Q Y ield surfae Q Q Q Q Q (a) F-odel (b) S-odel Figure 2 Craking Surfae and Yield Surfae Q

3 2.2 Loading Surfae Loading surfae that is orresponded to orner of hsterisis loop of F-odel is assued as F l = (ishiura and akiguhi 23). As shown in Figure 3, the loading surfaes are assued on a virtual plane and -axis those are expressed as b F l = and F l =, respetivel. b F l = is represented as irle where { b P}=( b Q, b Q Y ), and F l = deide the range of oponent of a fore point. Q bq Y loading surfae b F l = δ δ bq On Virtual Plane range of loading surfae F l = δ Restoring Fore Charateristis of RC Colun under a Constant Axial Fore δ δ Modeling of Deforation Q {l} δ δ Y On -Axis Figure 3 Loading Surfae δ {dδ p } Modeling in ri-axial Case Figure 4 Model of Diretion of Plasti Deforation he fore point {P} in tri-axial fore spae is transfored to the point { b P} on the virtual plane with the following equation, where {P'}=( b Q, b Q Y, ) (ishiura and akiguhi 23). { l} { n} ' n (1) { n} { l} { dp } = k I { dp} I is unit atrix, {l} is noral vetor of the orn shown in Figure 4, and {n} is a unit vetor lies in positive diretion of -axis. In this paper, k n is assued as follows. k n = 1 {( ) } 2 for F-odel, and k 1 { } 2 C n = for S-odel. (2) C A enter of the loading surfae is also transfored in the sae wa as the fore point. 2.3 Rigidit Degradation Elasti rigidit degradation of F-odel is assued as shown in Figure 5. he elasti rigidit before ielding is deided as a point diret to the axiu point experiened. he rigidit after ielding is deided aording to ratio between axiu ield deforation and initial ield deforation those are expressed as δ and Y δ in Figure 5, respetivel. he following equation is given to express these rules.

4 γ e δ α K = α K (3) Y δ α is the lower value between α and i Q /( Y δ K ), where i Q is initial value of Q. δ, whih initial value is equal to Y δ, is assued to inrease as ield surfae expands. α el shown in Figure 5 is assued as follows, whih is onsidered axial fore effets b using k n that is equal to Eqn.(2). α el el kn α el l l e =, where α = ( Q Q ) ( Q α Q α ). (4) α e of S-odel is assued as follows, whih is also onsidered axial fore effets. α e kn α =, where k n is equal to Eqn.(2). (5) Rigidities of Y and axes are alulated the sae as -axis. 2.4 op of Corn A top of the orn shown in Figure 4 is ade sooth b parabola to have no orner as shown in Figure 6 beause the orner ake alulation diffiult. In this paper the range of parabola is δ /5 in -diretion, and the range in Y- diretions are alulated the sae as -diretion. Q Q Q e K δ δy Yδ δ δ K el K δ range of parabola Figure 5 Rigidit Degradation Figure 6 op of the Corn 3. EARHQUAKE RESPOSE AALYSIS 3.1 uerial Progra Earthquake responses of R/C strutures were exained with the one-ass-sste and the two tpes of restoring fore odel those were F-odel and S-odel. ewark ethod β=1/4 was used for the response analsis, and three earthquake ground otions those were 1999, 1995, and 194 were inputted. Constants of the F-odel and the S-odel were deided based on experiental results of R/C oluns (akiguhi et al. 21) and R/C box wall strutures (orita et al. 1998), as shown in able 1. Coeffiient of daping was alulated with daping fator and instant stiffness of the sste on eah step. Paraeters of the analsis were natural period ranged fro.1 to.6 and tpes of analsis those were three and two diensional analses. he natural period orresponds to initial elasti stiffness of the odel. he restoring odel of two-diensional analsis is the sae to the odel of three-diensional analsis exept for having no -diretional oponents. A total plasti deforation an be given as shown in Figure 7. A total plasti deforation ratio η

5 on one-diension an be estiated b h if p, β, and ξ. he equation shown in Figure 7 was adopted in three and two diensional analses. In this paper, earthquake response analses were arried out as η get to 2. in F-odel and.5 in S-odel those are deided based on the past experiental stud (akiguhi et al. 21, and orita et al. 1998). h orresponded to η was alulated, and then analses were ade as h get to the alulated values. able 1 Constants of the sste F-odel S-odel β (=β Y =β ) p (=p Y =p ).1.1 Axial fore ratio.25.1 Ratio of ield strength to raking strength Axial ield strength ratio of tension to opression Axial raking strength ratio of tension to opression.1.1 γ shown in Eqn Daping fator.2.5 Ratio of vertial natural period to lateral natural period.3.3 δ 1 δ 3 h 2 Q Q δ h 1 Q h 3 Q η = otal plasti (=δ deforation 1 +δ 2 +δ 3 ) Yield deforation (=δ ) where h = h 1 + h 2 + h 3 and = h/p ξ + (1-ξ)/β ξ = C Q YQ, δ 2 Figure 7 otal Plasti Deforation 3.2 uerial Results Figure 8 shows responses in ase of inputting ground otion. Initial natural periods of the sste are.2 for F-odel and.1 for S-odel. UD-axis takes opressive side as positive diretion. As shown in Figure 8, the restoring fore odel shown in this paper ould represent behaviors on unloading stage, whih UD-diretional deforation diret to opressive side. Figure 9 to 12 show nuerial results when,, and ground otions were inputted. Cirle and diaond arks represent the results of F-odel and S-odel, respetivel. Blak and White arks represent three and two diensional analses, respetivel. D, A, A V,, and express axiu lateral response deforation, axiu lateral absolute response aeleration, axiu vertial absolute response aeleration, equivalent veloit of total energ input (Akiaa 1985), and initial natural period of the sste, respetivel. D is equal to a square root of su of δ S 2 and δ EW 2. A is alulated in the sae wa as D. an be given as follows, where E and M are energ input and ass of the sste, respetivel. = 2E M (6) D-, A-, and - relationships of three-diensional analsis alost agree with the relationships of two-diensional analsis as shown in Figure 9 to 11. hese results indiate possibilit that it is enough to onsider two diretional input of earthquake otion when we estiate lateral external fore and axiu deforation in seisi design. It is iportant to know the axiu deforation to judge whether non-strutural laddings of buildings an follow the deforation.

6 (a) F-odel with.2 se natural period (b) S-odel with.1 se natural period Figure 8 Responses inputted wave D D D Figure 9 Maxiu Lateral response deforation A se A se A se Figure 1 Maxiu Lateral Absolute Response Aeleration.2.4.6

7 se se se Positive and negative sides of the axiu vertial absolute aelerations are both plotted in Figure 12. As shown in this figure, the values of positive and negative side were alost equal. he biggest values were about 94 /se 2 on the F-odel of.3 se natural period inputted wave, and about 19 /se 2 on the S-odel of.5 se natural period inputted wave. It an be said that the nuerial results of A V were obtained in the range of 3 to 11 /se 2. Although this result a not be serious fro the viewpoint of ollapse of strutures, we have to onsider this result fro the viewpoint of influene on inside of buildings. A V se Figure 11 Equivalent Veloit of otal Energ Input A V se A V se Figure 12 Maxiu Vertial Absolute aeleration Evaluation of otal Energ Input An evaluation of the total energ input was onsidered with a result of elasti response analsis, whih daping fator is.1 (Akiaa 1985). Figure 13 show - relationships where a solid urve express a result of three-diensional elasti response analsis, whih daping fator is.1. Cirle and diaond arks are represented the results of F-odel and S-odel, respetivel. he results were plotted b averaged period of initial period given b initial elasti stiffness and last period given b using last value of α shown in Figure 1. of F-odel and S-odel show good agreeent with the elasti analsis results. here are soe differenes between the results of the elasti analsis and non-linear analsis in the range of longer period, however those are safet side and those differenes aren t large. herefore, it an be said that of elasti analsis at averaged natural period have possibilit to be able to evaluate of F-odel and S-odel. se se se Figure 13 Estiation of otal Energ Input

8 4. COCLUSIOS hree-diensional non-linear earthquake responses of R/C strutures were exained with one-ass sste. he restoring fore harateristis were odeled based on the theor of plastiit, whih was one of the aro odels. wo tpes of the restoring fore odel were adopted. One was flexural tpe that had large area in hsterisis loop, and the other was shear tpe that had sall area in the loop. he forer one an be seen in rigid frae strutures, and the later one an be seen in R/C box wall strutures. Paraeters of the analsis were natural period ranged fro.1 to.6 and tpes of analsis those were three and two diensional analses. he natural period orresponds to initial elasti stiffness of the odel.,, and earthquakes were eploed for input data. As a result, the following onlusions were found. 1) Maxiu lateral response deforation, axiu lateral absolute response aeleration, and total energ input of three-diensional analsis were alost equal to the responses of lateral two-diensional analsis. hese results indiate possibilit that it is enough to onsider two diretional input of earthquake otion to estiate lateral external fore and axiu deforation in seisi design. 2) Maxiu vertial absolute response aelerations of three-diensional analses were obtained in the range of 3 to 1,1 /se 2. Although this result a not be serious fro the viewpoint of ollapse of strutures, it is neessar to onsider this result fro the viewpoint of influene on inside of buildings. 3) otal energ input of elasti response analsis with.1 daping fator showed good agreeent with result of three-diensional non-linear response at an averaged period that was an average of two periods assoiated with initial elasti stiffness and last stiffness that onneted two axiu points of hsterisis loop diagonall. It an be said the result of elasti response analsis with.1 daping fator has possibilit to be able to evaluate the total energ input of three-diensional non-linear earthquake response b using the averaged periods. Referenes: ishiura, K. and akighhi, K. (23), ri-axial on-linear Restoring Fore Model of R/C Strutures b Using an Analog to the Plasti heor, Journal of Strutural and Constrution Engineering, Arhitetural Institute of Japan, 566, , (in Japanese). akiguhi, K., and Gao,. (2), ri-axial non-linear restoring fore odel of R/C struture using the theor of plastiit, Pro. of 12th World Conferene on Earthquake Engineering, Aukland, ew ealand, Paper ID 693. Shield, R.., and iegler, H. (1958), On Prager's Hardening Rule. eitshrift fur angewandte Matheatik und Phsik, Vol.9a, akiguhi, K., ishiura, K., and Hirai, K. (21), Experiental stud on two-diensional restoring fore harateristis of R/C oluns under varing axial fore, Journal of Strutural and Constrution Engineering, Arhitetural Institute of Japan, 539, , (in Japanese). orita, H., ishikawa,, Saito, H., Ishikawa, Y., and Kitada, Y. (1998), Stud on Restoring Fore Charateristis of a RC Box Wall Subjeted to Diagonal Horizontal Fore. Suaries of ehnial Papers of Annual Meeting AIJ, C-2, , (in Japanese). Akiaa, H. (1985). Earthquake-Resistant Liit-State Design for Buildings, Universit of oko Press, oko, Japan.