RESIDUAL DISPLACEMENTS IN CAPACITY DESIGNED REINFORCED CONCRETE STRUCTURES

Transcription

1 3 th World Coferece o Earthquake Egieerig Vacouver, B.C., Caada August -6, 4 Paper No. 399 RESIDUAL DISPLACEMENTS IN CAPACITY DESIGNED REINFORCED CONCRETE STRUCTURES Alessadro DAZIO SUMMARY Large-scale tests o reiforced cocrete walls ad bridge piers performed at the Swiss Federal Istitute of Techology (ETH), Zurich ad at the Uiversity of Califoria, Sa Diego (UCSD) cofirmed that usig capacity desig priciples the ielastic deformatio capacity of reiforced cocrete structure ca be greatly improved. However, the tests clearly idicated residual permaet deformatios as possibly the major drawback of ductile structures. Structural elemets with a high value of the ew desig parameter α, defied as the ratio betwee the bedig stregth of the elemet due to axial load oly ad its total bedig stregth, showed much lower permaet deformatios upo uloadig. The hysteretic behavior of such elemets substatially differed from the commoly used elasto-plastic or Takeda-type hysteretic models, i.e. showig a reduced eergy dissipatio capacity ad differet stiffess degradatio. These differeces raised questios o the applicability of commoly used desig tools like the equivalet force method, the capacity spectrum method or the direct-displacemet desig to the desig of such structures. To aswer these questios a fiber-elemet, able to carefully predict the behavior of structural elemets with differet values of α was developed ad checked agaist experimetal evidece. Subsequetly, a extesive parametric study usig oliear time-history aalyses was performed, showig that elemets with a high value of α had a larger ductility demad oly at very high ductilities, makig the use of such elemet extremely appealig i performace based desig. The paper presets the coducted large-scale tests, the developed fiber-elemet, ad the results of the performed time-history aalyses. I coclusio recommedatios o the optimum value of α are give ad strategies to reduce permaet deformatio are outlied both for buildigs ad bridge piers. I the latter case presetig as a example the West Achor Pier of the New Sa Fracisco-Oaklad Bay Bridge. INTRODUCTION Reiforced cocrete (RC) structural wall buildigs are popular i Switzerlad ad Cetral Europe. Such buildigs, as show i Figure a, are ofte coceived as structural wall systems cosistig of flat slabs, colums desiged for gravity loads oly ad RC structural walls. Flat slabs are beamless cocrete slabs typically with spas of 6 to 9m ad thickess of to 3cm, which are ofte stregtheed aroud colums to prevet puchig shear failure. Colums are i most cases moolithic with the slabs ad have a small cross sectio with typical dimesios ragig from to 4cm desiged to carry axial forces. Structural walls are relatively sleder reiforced cocrete walls fixed i a very stiff RC foudatio box structure with oe or more basemet stories. The structural walls have to resist horizotal wid ad earthquake forces Istitute of Structural Egieerig (IBK). Swiss Federal Istitute of Techology (ETH), Zurich.

2 ad, by meas of capacity desig priciples, they ca be desiged to behave i a ductile maer. A lot of research o the behavior of ductile reiforced walls has bee coducted all over the world. However, i Switzerlad ad i other regios of Cetral Europe the coceptual desig ad the costructio methodologies, the ratio betwee seismic iertia forces ad gravity loads ad especially the mechaical properties of the reiforcig steel are quite differet as i coutries like New Zealad, USA ad Japa. For this reaso research results gaied i these coutries with high seismicity caot be simply applied to Cetral Europe; they have to be adapted. To reach this goal ad to give the practisig structural desiger recommedatios to desig better structural wall systems, several reiforced cocrete walls were tested by Dazio, Lestuzzi ad Thiele at the Swiss Federal Istitute of Techology [-3]. I the followig sectios some of the test results are briefly preseted ad selected basic aspects of the behavior of reiforced cocrete structural walls are discussed. Figure : Reiforced cocrete structural walls buildig (a) ad relevat floor pla (b). STATIC CYCLIC TESTS ON REINFORCED CONCRETE STRUCTURAL WALLS The six uits tested by Dazio ad preseted i [] represet the lower part of the reiforced cocrete structural walls of the six story referece buildig show i Figure at 5% scale. The test setup pictured i Figure a reproduced the same sectioal forces i the plastic regio of the test uit as i the structural walls of the referece buildig uder seismic actio. Figure : Setup for static cyclic tests of RC structural walls (a). Reiforcemet of test uits WSH3 (b) ad WSH5 (c).

3 Average drift [%] Average drift [%].4 a) Wall WSH3 b) Wall WSH5 Base shear [MN]....4 =.5 ρt =.8 α = Top displacemet [mm] =. ρt =.39 α = Top displacemet [mm] Figure 3: Hysteretic behavior of test uits WSH3 (a) ad WSH5 (b) []. The test matrix icluded variatio of the ductility of the reiforcig steel, the logitudial reiforcemet ratio, the axial load ratio ad the desig method amog the test uits. A detailed discussio of all the test results is beyod the scope of this paper ad oly a compariso betwee the hysteretic behavior of the test uits WSH3 ad WSH5 is give here. Both walls had the same legth l w, hece almost the same yield displacemet y. Figure 3 shows that also the bedig stregth of both walls was almost the same. I the case of wall WSH3 the bedig stregth was esured by a total logitudial reiforcemet cotet ρt =.8% ad a axial load ratio ' = N /( A f g c ) =. 5. The same bedig stregth of wall WSH5 was esured maily by a high axial load ( =. ), because oly the miimum reiforcemet cotet was provided ( ρ =.39% ). The reiforcemet pla of both walls is pictured i Figures b ad c. t Bedig Momet [MNm] Despite the similar mootoic behavior (same yield displacemet ad same bedig stregth) the two walls showed a fairly differet hysteretic behavior: ) The maximum residual displacemet of wall WSH3 upo uloadig was sigificatly larger. Eve cosiderig shake dow effects, it is expected wall WSH3 to experiece larger residual displacemets after a earthquake, leadig to a poorer performace. ) The iitial stiffess of both walls was similar. However, after plastic deformatios occurred, the reloadig stiffess of wall WSH5 was sigificatly larger because the high axial load was able to almost fully close flexural cracks durig load reversal. Such a characteristic meas that after a earthquake cracks are closed ad do ot eed ay repair. Furthermore, almost the etire iitial stiffess of the wall is available, esurig for example full serviceability for wid actio. 3) Eergy dissipatio occurs maily due to yieldig of the reiforcemet. Wall WSH3 had a larger reiforcemet cotet ad was able to dissipate 67% of the iput eergy while wall WSH5 could oly dissipate 45% of the iput eergy. Therefore, the hysteresis curve of Wall WSH5 is characterized by a lower equivalet viscous dampig, what accordig to moder desig methodologies leads to larger displacemet demads. 4) Because of the higher reiforcemet cotet, wall WSH3 showed a higher post-yield stiffess that yielded to a larger plastic hige legth, hece a larger displacemet capacity. However, i this particular case, wall WSH5 experieced a sigificatly lower displacemet capacity maily because of the poor material properties of the D6 web logitudial reiforcemet (see Figure ). Wall with bedig stregth due mostly to high axial load have sigificat advatages i terms of residual displacemets ad residual stiffess after a earthquake; two key parameters i the performace assessmet of structures. However, the reduced eergy dissipatio capacity could lead to a icreased displacemet ductility demad. This issue is ivestigated i the followig sectios by meas of fiite elemet aalyses.

4 SOFTWARE To perform all umerical simulatios preseted i the ext sectios two differet programs were implemeted. I the followigs they will be briefly preseted. The Takeda sigle degree of freedom (SDOF) system The Takeda rules implemeted to describe the hysteretic behavior of reiforced cocrete structures are pictured i Figure 4 ad correspod to the oes proposed i [4]. While the rules for cycles with large amplitude, Figure 4a, were derived from observatios made durig tests, the rules for small amplitude cycles are based maily o egieerig judgmet ad are set up to avoid clearly urealistic behaviors durig small cycles, i.e. to avoid very large or eve egative reloadig stiffesses. Recogizig that the hysteretic rules assumed for the small amplitude cycles play a importat role i the computatio of the residual displacemet it is importat to always specify which rules are assumed for the aalyses. Figure 4: Hysteretic rules of the Takeda SDOF system: large amplitude cycles (a) ad small amplitude cycles (b). The fiber elemet program Rechebrett D Rechebrett-D is a simple program developed by Dazio [5] that allows the modelig of reiforced cocrete structures with two-odes Beroulli beam fibre elemets. This kid of elemets are equivalet to a sectioal aalysis program performig momet curvature aalysis, itegratig curvatures alog the elemet legth ad automatically accoutig for the iteractio betwee momet ad axial load. 6 5 a) Cofied cocrete Ucofied cocrete 8 6 b) Cocrete stress [MPa] 4 3 Steel stress [MPa] Cocrete strai [mm/m] Steel strai [mm/m] Figure 5: Simplified costitutive laws for cocrete (a) ad reiforcig steel (b).

5 The simplified uiaxial cyclic costitutive laws for reiforcig steel ad cocrete used by the elemets are pictured i Figure 5 to allow a better iterpretatio of the results of the umerical simulatios. The cocrete costitutive law is a simplified versio of the well kow Mader s model, i which o tesile stregth of the cocrete is cosidered ad where small uloadig ad reloadig cycles occur with o eergy dissipatio as show i Figure 5a. The also well kow Meegotto-Pito s steel costitutive law is pictured i Figure 5b. It allows a fairly accurate descriptio of the Baushiger s effect, while o bucklig of the logitudial reiforcemet is cosidered. To esure a good degree of robustess ad to allow the computatio of time history aalyses, the costitutive laws for the materials were kept as simply as possible. Despite these simplificatios, Rechebrett D was able to predict fairly accurately the dyamic behavior of a 3-story reiforced cocrete wall tested by Lestuzzi [] o the ETH shake-table. The time history of the relative displacemet of the wall is pictured i Figure 6a. The locatio of the amplitude maxima ad their magitude is well predicted. Discrepacies betwee test ad simulatio occurs i the free vibratio phase at the ed of the earthquake ( t > s ) maily because i the test a frictio dampig occurred while i the simulatio a viscous dampig of the same magitude was cosidered. The residual displacemet predicted by the model is smaller tha the measured oe. Figures 6a ad 6b show the measured ad the computed cyclic momet-curvature relatioship at the base of the wall. While the magitude of momets ad the curvatures is i good agreemet, the shape of the reloadig braches is differet. Due to the presece of rough cracks the reloadig stiffess of the test uit is smoother compared to the umerical simulatio where a sharp bet ca be observed whe the smooth cracks of the model suddely close shortly past the zero-curvature lie ad the stiffess rapidly icreases. Realtive displacemet [mm] a) Wall WDH4: FE simulatio Wall WDH4: shake table test Time [s] b) Experimet c) FE simulatio Bedig momet [knm] 5 5 Wall WDH4 Wall WDH4 3 3 Curvature [ 3 m ] 3 3 Curvature [ 3 m ] Figure 6: Simulatio of the dyamic behavior of the wall WDH4 []. Time history of the top displacemet (a) ad cyclic momet-curvature relatioships at the base of the wall (b, c).

6 ANALYSES Chose walls To characterize reiforced cocrete wall with differet axial loads ad differet logitudial reiforcemet cotets a ew parameter α is itroduced. α quatifies what part of the total bedig stregth of a wall is esured by the axial load: N α () w =. 45 l M The umerator of Equatio (), where N is the axial load of the wall ad l w its legth, is a good approximatio for the cotributio of the axial load to the omial bedig stregth of the wall M. α is zero whe there is o axial load ad about oe whe there is o logitudial reiforcemet. To ivestigate the ifluece of α o the behavior of reiforced cocrete walls, a parametric study usig eleve walls listed i Table was carried out. The walls were all 4.m log ad.3m wide. The cylider ' stregth of the cocrete was f c = 4MPa ad the yield stregth of steel f y = 5MPa with hardeig b =.8 typical of Europea Tempcore steels. All the walls had the same bedig stregth M = 5MNm. Walls N to N8 showed axial load ratios ad reiforcemet cotets that ca be foud i real buildig, while walls N, N, N9 ad N represeted too extreme combiatios of these parameters. Therefore, they were ot further cosidered i parametric study. Wall M [knm] α [-] N [kn] [-] α x [-] ρ t [-] ρ w [-] ρ e [-] A se [mm ] A sw [mm /m] N N N N N N N N N N N Table : Eleve walls with the same bedig stregth give by differet combiatios of axial load ad logitudial reiforcemet. (α x = relative depth of the eutral axis, ρ t, ρ w, ρ e = logitudial reiforcemet cotet of the wall, of the web regio ad of the ed regio, A se, A sw = steel area of the ed ad web regios of the wall). Momet curvature aalyses Momet curvature diagrams for walls N to N8 are pictured i Figure 7 showig the depedece of the post-yield hardeig o logitudial reiforcemet cotet. The omial yield curvature was almost the 3 same for all walls ad correspoded to φ y = m. Figure 7b shows that up to omial yield the secat stiffess of the walls is iflueced by the axial load ad that the taget stiffess is highly oliear already i elastic regio of the momet curvature diagram. The assumed steel properties leaded to a post-yield stiffess of the walls ragig betwee.3 ad.% of the elastic stiffess, i.e. very low. Assumig a hardeig b =. 3 of the reiforcemet - correspodig to typical US Grade 6 steel - the post-yield stiffess rages betwee.5 ad.5% of the elastic stiffess.

7 a) 5 b) Bedig momet [MNm] 5 5 N N3 N4 N5 N6 N7 N Curvature [ 3 m ] Bedig momet [MNm] N N3 N4 5 N5 N6 N7 N Curvature [ 3 m ] Figure 7: Momet-curvature relatioship of the seve walls N to N8 (a). Magified view of the elastic regio (b). Static cyclic aalyses The eergy dissipatio capacity of the differet walls ad its ifluece o the dyamic behavior of the latter was first aalyzed by meas of static cyclic umerical aalyses. Sigle degree of freedom (SDOF) systems equivalet to the 4-DOF system show i Figure b were implemeted i Rechebrett D ad static cyclic aalyses were performed usig a covetioal loadig history with symmetric cycles of icreasig amplitude. The hysteretic behavior of the walls N, N5 ad N7 is pictured i Figures 8a to 8c. The ifluece of the parameter α o the hysteretic behavior is evidet ad cofirms the experimetal observatios preseted i the previous sectios. Rechebrett D does ot take ito accout shear deformatios. Therefore, the pichig i the hysteresis loops is due to the axial load aloe. Pichig meas reduced eergy dissipatio capacity ad while i correspodece of displacemet ductility µ = 6 wall N was able to dissipate 8% of the iput eergy, wall N7 dissipated just 45% of it. The equivalet viscous dampig ζ eq is also a idicator of eergy dissipatio capacity ad is plotted for all walls i Figure 8d. The dotted lie is a proposal by Priestley [6] to estimate the equivalet viscous dampig of reiforced cocrete walls. The proposal correspods basically to the average value of all computed cases. However, it does ot cosider the depedecy betwee ζ eq ad α. Accordig to moder desig methodologies, a lower value of ζ eq leads to larger displacemet demads. This icrease i displacemet demad ca be estimated by meas of Equatio () where Sd is the spectral displacemet of a SDOF system of period T ad dampig ζ. ( T, ζ ) + ζ = ( T, ζ ) + ζ Sd () Sd Let be wall N desiged usig direct displacemet desig priciples for a target displacemet ductility µ =, the wall N7 - desiged to have the same mootoic behavior as N (eglectig the fact that by idetical target displacemet the secat stiff of wall N7 would be slightly smaller that wall N) - would reach a displacemet ductility µ =. 4. I the same way it ca be show that whe wall N is desiged for a target displacemet ductility µ = 5, wall N7 would reach µ = The equivalet viscous dampig ζ eq plotted i Figure 8d cosiders hysteretic dampig oly. However, to compute the target displacemets of walls N ad N7 a additioal 5% dampig to accout for system dampig was used. Cosiderig that the desig ductility of a structure seldom exceeds 4 because of serviceability ad performace cosideratios, the assumptio of a average equivalet viscous dampig value idepedet from α for all wall types seems reasoable. Such a assumptio leads theoretically to differeces i the estimated displacemet demad of the order of %, what is small compared to the ucertaities related to the seismic actio. However, i the followig sectios, this issue will be ivestigated by meas of trasiet aalyses.

8 a) Wall N b) Wall N5 Base shear [MN] µ = 5 µ = 5 Base shear [MN] µ = 5 µ = Top Displacemet [m]..... Top Displacemet [m] Base shear [MN] c) Wall N7 µ = 5 µ = Top Displacemet [m] Equivalet viscous dampig [%] d) N N3 N4 N5 N6 N7 N8 Accordig to Priestley [...] Displacemet ductility [ ] Figure 8: Force-displacemet relatioship of walls N, N5, N7 (a-c) ad relevat equivalet viscous dampig (d). Groud Motios I the followig sectio time history aalyses are performed usig 5 differet groud motios. The respose spectra of these groud motios are pictured i Figure 9 for a viscous dampig ζ = 5%. The differet groud motios were scaled usig differet factors ragig from. to 4.63 depedig o the target displacemet ductility assumed for each computatio. Sa [g]..5. a) Erzica El Cetro Llollelo Northridge Loma Prieta Sd [m] b) ζ = 5%.5 ζ = 5% Period [s] Period [s] Figure 9: Acceleratio (a) ad displacemet (b) respose spectra of the groud motios cosidered i the time history aalyses.

9 Dyamic aalyses o multi degrees of freedom systems The dyamic behavior of multistory walls is ivestigated by meas of time-history aalysis. Totally 48 cases give by permutatio of dyamic systems (4-DOF ad 7-DOF), 3 effective atural periods (.6,.9 ad.8 secods), reiforcig steel hardeig ratios b (.8 ad.3), target displacemet ductilities µ (.5 ad 4.5) ad 5 groud motios were aalyzed. The elevatio of the cosidered dyamic systems is pictured i Figures a ad b; each story has the same mass ad its magitude is calculated i fuctio of the target period of the system. The axial load geerated by each story is the same ad the total axial load i the plastic hige regio of each wall is give i Table. The discretizatio of the sectio ad the relevat material properties are give i Figure c. The groud motios pictured i Figure 9 were scaled by meas of the equal displacemet priciple so that the wall would reach the targeted displacemet ductility. Figure : Dyamic systems (a, b) ad typical cross-sectio with relevat material properties (c). Amog all the results produced by the aalyses, here oly the oes relevat to maximum ductility demad ad residual displacemet are displayed i Figure. The computatios performed o the 7-DOF system produced basically the same results as for the 4-DOF system. Therefore, oly the latter will be further discussed. I the case of target ductility µ =. 5, wall N reached a average ductility of. while the average value for wall N8 was.43, i.e. a icrease of %. For a target ductility of µ = 4. 5 the same values were 4.33, 4.96 ad 5%, respectively. The scatter is similar to the oe predicted usig the equivalet viscous dampig approach preseted before. These results cofirm previous ivestigatio performed by Dazio [5] ad is i excellet agreemet with the results obtaied by Christopoulos ad coworkers while aalyzig the displacemet demad of flag-shaped hysteretic models [7, 8, 9]. Furthermore, i [5] it is show that sigificatly higher ductility demads - i the order of about 3 to 4% - for walls with higher axial load occur oly whe a target ductility of µ = 6. 5 or higher is assumed. However, it is very ulikely that a real structure would be desiged for such a large target ductility, therefore the meaig of this last fidig is of relative importace. Figure c displays the ratio betwee the residual displacemet of the wall at the ed of the time history to the maximum displacemet reached by the wall durig the same time history. The target displacemet ductility, i.e. the maximum displacemet reached by the wall, had little ifluece o the ratio. The same fidig was reported i [7]. O the other had ad as expected, the ratio is iversely proportioal to the parameter α. It has to be oted that the magitude of the ratio resultig from these computatio is, especially for wall with low axial load, somewhat smaller compared to the results obtaied by other researchers. This effect is probably due to the shape of the reloadig braches produced by the fiber

10 model. The sudde closure of the smooth cracks ad the relatively high reloadig stiffess of the cocrete material model (see Figure 5a) geerates reloadig braches startig with low stiffess that suddely icreases just past zero horizotal displacemet. This creates a kid of artificial self-ceterig mechaism that i real reiforced cocrete elemets is also preset, however, it is ot so proouced. This example shows the importace of the defiitio of the hysteretic rules for small amplitude cycles. Max. displacemet ductility [ ] a) 4 DOF R y =.5 R y = Alpha [ ] Max. displacemet ductility [ ] b) 7 DOF R y =.5 R y = Alpha [ ] Residual / maximum displ. [ ] c) 4 DOF R y =.5 R y = Alpha [ ] Residual / maximum displ. [ ] d) 7 DOF R y =.5 R y = Alpha [ ] Figure : Average maximum displacemet ductility (a, b) ad average relative residual displacemet (c, d) i fuctio of the parameter α. Dyamic aalyses o sigle degree of freedom systems The dyamic behavior of ductile reiforced cocrete walls was further ivestigated by meas of oliear SDOF systems. Due to the sigificatly smaller computatioal burde it was possible to ru a higher umber of cases as i the previous sectio. Totally 44 cases give by permutatio of 4 atural periods (.6,.9,.8 ad 3.64 secods), 3 hardeig ratios r (, 5 ad %), 5 Takeda α -parameters (.,.,.4,.6 ad.8), 6 Takeda β -parameters (.,.,.4,.6 ad.8), target ductilities µ (.5 ad 4.5) ad groud motios were aalyzed. Before the results o the aalyses are preseted, a brief commet o the shape of small amplitude cycles of SDOF systems is due. Figure shows o the top the time history of 3 Takeda-type SDOF systems with the same mootoic backboe curve, the same uloadig stiffess factor α but differet reloadig stiffess factors β. Each oe of these time histories is compared with the time history of a SDOF system with the same mootoic backboe curve but modeled with Rechebrett D (dotted lie). O the bottom of Figure, the hysteresis loops of the same SDOF systems are pictured. For the computatio plotted i Figure a, a Takeda β =. ad the hysteresis rules o Figure 4 were used. While the compariso of the large amplitude cycles betwee the two SDOF systems is excellet (up to the fourth large peak), the reloadig stiffess of the Takeda model is sigificatly lower leadig to quite

11 differet small amplitude cycles ad residual displacemets. For the computatio plotted i Figure b, a Takeda β =. 9 ad the hysteresis rules o Figure 4 were used. While the magitude of the large amplitude cycles betwee the two SDOF systems is very similar up to the third large peak, the compariso of the reloadig braches of the large amplitude cycles are ot very good. O the other had the small cycle behavior of the two SDOF systems is very similar ad the residual displacemets correspod. For the computatio plotted i Figure c, a Takeda model with β =. 9 for large amplitude cycles ad β =. for small amplitude cycles was used. This shows the importace of the defiitio of small amplitudes cycles o residual displacemets. While the large cycle respose of the Takeda SDOF systems i Figures b ad c are coicidet, the small cycles are completely differet leadig also to completely differet residual displacemets. Figure 3 is similar to Figure. I both cases the residual displacemet ratio ad the maximum displacemet ductility are plotted agaist a parameter that strogly affects the uloadig stiffess of the hysteresis loops ad the results show the same treds. A typical reiforced cocrete structure ca be modeled usig the Takeda parameter α =. ad β =. 4. Accordig to Figure 3a, the average residual displacemet ratio for such SDOF systems is about.85. Chagig β from.4 to -. would icrease the average residual displacemet ratio by a factor.8 ad chagig α from. to.6 would decrease the average residual displacemet ratio by a factor.8. The ifluece of both Takeda parameters o residual displacemets is therefore sigificat. O the other had, Figure 3b shows that both parameters have a limited ifluece o the maximum displacemet ductility like it was the case i Figure. It has to be oted that i Figure 3 for sake of completeess all the combiatios of the Takeda parameters α ad β are plotted. However, whe both parameters are large ad specially whe the hardeig parameter r = % is used, odd hysteresis loops ca results ad the iterpretatio of the results has to be carried out with cautio. I a similar study preseted i [7] the same coclusios regardig the ifluece of the parameter Takeda β o residual deformatios were draw. However, i that study the parameter Takeda α had a smaller ifluece o residual deformatios has it has here.. a). b). c) Displacemet [m].... Takeda, beta =. Fiber model 3 Time [s]. Takeda, beta =.9 Fiber model 3 Time [s]. Takeda, beta =.9/. Fiber model 3 Time [s] Force [MN] Displacemet [m].... Displacemet [m].... Displacemet [m] Figure : Ifluece of the Takeda small-cycles hysteretic rules o the residual displacemet of sigle degree of freedom systems.

12 Residual / maximum displ. [ ] a) Takeda beta =.8 Takeda beta =.6 Takeda beta =.4 Takeda beta =. Takeda beta =. Takeda beta = Takeda alpha [ ] Max. displacemet ductility [ ] b) R y =.5 R y = Takeda alpha [ ] Figure 3: Ifluece of the Takeda parameters α ad β o the relative residual displacemets (a) ad o the ductility demad (b) of sigle degree of freedom systems. RECCOMENDATIONS It was show that walls with bedig stregth due mostly to high axial load have sigificat advatages i terms of residual displacemets ad residual stiffess after a earthquake; while the reduced eergy dissipatio capacity has i most practical cases egligible effects. Therefore, a structure should be desiged targetig walls with α as large as possible. To reach this goal, the tributary area of the walls ca be icreased by icreasig the spa of the slabs or by choosig a appropriate locatio of the wall withi the floor pla (see Figure ). However, eve if ot specifically ivestigated i this paper, it has to be oted that walls with a high value of α develop a reduced plastic hige legth leadig to a lower maximum displacemet capacity. Therefore the value of α should be limited to: α.7 (3) or to: α.35 + (4) '.75 / f c whichever is smaller. I Equatio (4) = N ' /( A f g c ) is the axial load ratio of the wall ad ' f c is the cylider stregth of the cocrete. The limit give by Equatio (4) esures that the omial bedig stregth of the wall is at least twice the crackig momet of the wall as recommeded by Priestley i [9]. PERMANENT DISPLACEMENT OF A BRIDGE PIER: AN EXAMPLE I the framework of the proof testig i support to the desig of the ew East Spas of the Sa Fracisco- Oaklad Bay Bridge performed i the laboratories of the Uiversity of Califoria, Sa Diego; a quarter scale model of the West Achor Pier (Pier W) of the Mai Spa Self-achored Suspesio Bridge was tested by Seible ad Dazio [, ]. Two of the eight colum formig Pier W were modeled i the laboratory ad tested usig the setup pictured i Figure 4a. The test uit showed a excellet hysteretic behavior (Figure 4b) ad could easily outperform all the performace criteria required by the desig specificatio. A computatio by meas of Rechebrett D was able to predict the global behavior of the test uit fairly accurately, while beig able to exactly predict the permaet deformatios. Durig the so-called Safety Evaluatio Earthquake (SEE), Pier W is expected to udergo plastic deformatios up to a displacemet ductility of about two. Upo uloadig after such a displacemet, the maximum possible residual displacemet of the stad-aloe Pier W measured durig the test was

13 mm, correspodig to 4mm i the prototype structure. This value does ot coform with the performace requiremet limitig permaet deformatios to 3mm. Time history aalyses of the bridge uder the SEE Evet showed that takig ito accout shake-dow effects, the expected residual displacemet is well below the permitted limit. However, cosiderig that the maximum displacemet capacity of the Pier was ot a issue beig more tha three times larger tha the seismic demad, this example shows that residual displacemets are importat desig parameters for ductile bridge piers. Base shear [kn] Drift (%) b) SFOBB Pier W Test F y F y, µ =6 µ =4 µ =3 µ = µ = µ = µ = µ =3 µ =4 µ =6 F y Top displacemet [mm] F y, Experimet FE Model Figure 4: Setup for Pier W Test (a). Measured ad predicted hysteretic behavior of the test uit (b). I the previous sectios it was show that icreasig the axial load will reduce the maximum possible residual displacemet of reiforced cocrete elemets. I the followigs, the ifluece of the shape of bridge piers o permaet deformatios is ivestigated. Figure 5a shows the cross-sectio of the uit tested i the laboratories. The two circular colums had a diameter of.88m ad a clear height of.5m. The colums were fixed at the base, separated by a 7mm gap alog the height ad coected at the top by a rigid cap beam. To esure visual cosistecy with the rest of the prototype bridge, the colums were provided with a petagoal shaped architectural cocrete. Figures 5b ad 5c show desig alteratives of the test uit. The circular sectio ad the hollow sectio were desiged i such a way that the correspodig Pier would have a almost idetical mootoic behavior as the Test Uit, assumig that the actig axial load was the same o the three piers (See Figure 6a). Figure 5: Desig alteratives of the Pier W Test Uit.

14 The three piers were modeled with Rechebrett D ad subjected to the same loadig as the Pier W Test Uit durig the laboratory testig. Figure 6b shows the computed hysteretic behavior ad while the reloadig stiffess of the three piers is similar, the uloadig stiffess of the piers with circular ad hollow sectios is larger leadig to larger maximum possible residual deformatios. Upo uloadig from a displacemet ductility of two, the residual deformatio of the latter piers was 4mm, i.e. 4% larger tha the oe of the test uit. Base shear [kn] 5 5 a) Drift (%) Test Uit (FE) Pier with circular sectio Pier with hollow sectio Top displacemet [mm] Base shear [kn] Drift (%) b) SFOBB Pier W Test F y F y, µ =6 µ =4 µ =3 µ = µ = µ = µ = µ =3 µ =4 µ = Top displacemet [mm] F y, F y Test Uit (FE) Pier w. circular sectio Pier w. hollow sectio Figure 6: Predicted mootoic (a) ad cyclic (b) behavior of the three desig alteratives of Pier W Test Uit. CONCLUSIONS Walls with bedig stregth due mostly to high axial load are characterized by a large value of the ewly defied momet ratio α. Hysteresis loops geerated usig large values of α are almost flag-shaped esurig self-ceterig properties of such walls. The dyamic behavior of flag-shaped hysteretic models has already bee ivestigated ad researchers report that despite the limited eergy dissipatio capacity of this type of ielastic oscillators, their maximum displacemet ductility demad is oly slightly larger compared to systems with larger equivalet viscous dampig. These fidigs were cofirmed by the aalyses preseted i this paper. Post-tesioed precast walls have a value of α close to uity ad perform extremely well uder seismic actio. I capacity desiged walls the value of α has to be limited to prevet a premature failure of the logitudial reiforcemet. Suggested limits for α are give i the relevat sectio of the paper. Time history aalyses showed that maximum displacemets of RC structures subjected to earthquake are rather isesitive to the shape of the hysteresis loops, provided the mootoic backboe curve is the same. O the other had, residual displacemets are very sesitive to the shape of the hysteresis loops. Therefore, hysteresis rules for small amplitude cycles have to be carefully chose. Fiber models are very suitable to predict the behavior of structures, however, to correctly deal with residual deformatios the costitutive laws of the cocrete should properly take ito accout crackig. The costitutive laws of the steel should properly take ito accout the Bauschiger s effect. I cases where it is ot appropriate to icrease the axial load, a reductio of the permaet displacemet potetial ca be achieved by optimizig the shape of the elemet s cross sectio. Twi colum or twi wall sectios seem to provide a reductio of permaet displacemet, this because of the additioal axial load due to frame actio betwee the colums or the walls.

15 AKNOWLEDGEMENTS The author is highly idebt to Professor Dr. Dr h.c. Hugo Bachma, Professor Emeritus, Istitute of Structural Egieerig (IBK), Swiss Federal Istitute of Techology (ETH), Zurich, ad to Professor Dr. Frieder Seible, Dea Jacobs School of Egieerig, Uiversity of Califoria Sa Diego (UCSD) for the support ad guidace provided to the author durig his stay at the respective Istitutios. REFERENCES. Dazio A, Wek Th, Bachma H. Versuche a Stahlbetotragwäde uter zyklisch-statischer Eiwirkug. (Tests o RC Structural Walls uder Cyclic-Static Actio). IBK-ETH Zurich. IBK- Report No. 39, ISBN Birkhäuser Verlag, Basel Lestuzzi P, Wek Th, Bachma H. Dyamische Versuche a Stahlbetotragwäde auf dem ETH-Erdbebesimulator. (Dyamic Tests o RC Structural Walls o the ETH Shakig Table). IBK-ETH Zurich. IBK-Report No. 4, ISBN X. Birkhäuser Verlag, Basel Thiele K, Wek Th, Bachma H. Versuche a Stahlbetotragwäde uter pseudodyamischer Eiwirkug. (Pseudo-Dyamic Tests o RC Structural). IBK-ETH Zurich. IBK-Report No. 57, ISBN Birkhäuser Verlag, Basel. 4. Allahabadi R, Powell G. Drai-DX, User Guide. Report UBC/EERC-88/6. Earthquake Egieerig Research Ceter, Uiversity of Califoria, Berkeley Dazio A. Etwurf ud Bemessug vo Tragwadgebäude uter Erdbebeeiwirkug (Desig ad Detailig of RC-Wall Buildigs uder Seismic Actio). IBK-ETH Zurich. IBK Report Nr. 54. ISBN Birkhäuser Verlag, Basel. 6. Priestley MJN. Performace Based Seismic Desig. Proceedigs of the XII World Coferece o Earthquake Egieerig, Aucklad, New Zealad 3 Jauary - 4 February. 7. Christopoulos C, Pampai S, Priestley MJN. Performace-Based Seismic Respose of Frame Structures Icludig Residual Deformatios. Part I: Sigle-Degree of Freedom Systems. Joural of Earthquake Egieerig 3; 7(): Christopoulos C, Filiatrault A, Folz B. Seismic respose of self-ceterig hysteretic SDOF systems. Earthquake Egieerig ad Structural Dyamics, 3: Pampai S, Christopoulos C, Priestley MJN. Performace-Based Seismic Respose of Frame Structures Icludig Residual Deformatios. Part II: Multi-Degree of Freedom Systems. Joural of Earthquake Egieerig 3; 7(): Priestley MJN, Seible F, Calvi GM. Seismic Desig ad Retrofit of Bridges. Joh Wiley & Sos, 996. Dazio A, Seible F. Structural Testig of the Sa Fracisco-Oaklad Bay Bridge Mai Spa Pier W. SSRP Report /. Departmet of Structural Egieerig. Uiversity of Califoria, Sa Diego. La Jolla, CA,.. Seible F, Dazio A. Proof Testig i Support of Seismic Bridge Desig: Example of the Sa Fracisco-Oaklad East Bay Safety Project. Proceedig of the FIB 3 Symposium Cocrete Structures i Seismic Regios. Athes May 6-9, 3