An Analytical Model to Predict the Inelastic Seismic Behavior of Shear-wall, Reinforced Concrete...

Engineering Structures 24 (2002) 8598www.elsevier.com/locate/engstruct

An analytical model to predict the inelastic seismic behavior ofshear-wall, reinforced concrete structures

P.A. Hidalgo *, R.M. Jordan, M.P. MartinezDepartment of Structural and Geotechnical Engineering, Catholic University of Chile, Av. Vicuna Mackenna 4860, Casilla 306, Santiago, Chile

Received 10 July 2000; received in revised form 5 June 2001; accepted 11 June 2001

Abstract

The development of an analytical model to predict the inelastic seismic response of reinforced concrete shear-wall buildings,including both the flexural and shear failure modes is presented. The use of shear-wall buildings is quite common in a number ofseismic countries as a result of their successful seismic behavior during past severe earthquakes. The objective of this study hasbeen to develop a computer model capable of predicting the seismic behavior of shear-wall buildings. Such model would allowbetter estimations to be obtained of both the ultimate lateral strength of these buildings as well as their inelastic deformation demandunder severe ground motions. Such information may be used in the implementation of performance-based design procedures, andto improve present code design procedures. To fulfill this objective, a shear failure mode model based on experimental results hasbeen added to the computer program larz. This paper discusses the most relevant problems and solutions devised during thedevelopment of this model. Validation of the model proposed to predict the inelastic seismic response of shear-wall structures wascarried out by comparing its results with the actual response of two real buildings during the March 3, 1985 Chilean earthquake. Inspite of the fact that the model is two-dimensional and, hence, it ignores the torsional response, the results obtained are satisfactory. 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Shear-wall buildings; Reinforced concrete buildings; Inelastic behavior; Shear failure model

1. Introduction

Properly designed multistory R/C shear-wall buildingsshould behave in a ductile flexural manner when sub-jected to severe earthquake ground motions. Conse-quently, design forces are usually much smaller thanthose required to design a structural system without thecharacteristics of ductility and toughness typical ofbuildings with predominant flexural failure mode.Nevertheless, there are cases where this ductile failuremode may not be achieved due to the large flexuralstrength as compared with the shear strength of thewalls. In such cases, an undesired shear failure mode islikely to develop. This may be the case of structural sys-tems that have a large wall area relative to the floor planarea. This situation may also happen in shear walls

* Corresponding author. Tel.: +56-2-686-4207; fax: +56-2-686-4243.

E-mail address: [email protected] (P.A. Hidalgo).

0141-0296/02/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S0141- 02 96 (01)00 06 1- X

coupled by stiff lintels that may induce bending momentto shear force ratios in the wall too small relative to thelength of the wall. On the other hand, even in the casethat ultimate strength is controlled by a ductile flexuralbehavior of the shear walls, structural damage in theform of mild or extensive shear cracking may affect theobjectives of performance-based design. Severalexamples of these situations have been found in theresponse of R/C shear-wall buildings after severe earth-quakes like the March 3, 1985 Chilean earthquake.

The use of shear-wall buildings is quite common insome earthquake-prone countries such as Chile; theirseismic behavior has been successful during past severeearthquakes, both, from a serviceability as well as asafety standpoint [1]. Therefore, their use has been rec-ommended in earthquake-resistant design [2] as long asits true behavior is included in building modeling.Consequently, the objective of this study has been todevelop a computer model capable of predicting the seis-mic behavior of such buildings. The model proposedallows better estimations to be obtained of both the ulti-

86 P.A. Hidalgo et al. / Engineering Structures 24 (2002) 8598

mate lateral strength as well as the inelastic deformationdemand of shear-wall buildings under severe groundmotions. The use of this information helps in theimplementation of performance-based design proceduresand the improvement of code design procedures.

A model capable of predicting the seismic behaviorof this type of structures has to include the possibilityof developing flexural as well as shear failure modes inthe shear walls. Most research studies have addressedthe problem of including flexural failure modes in themodeling of structural shear-wall elements. For instance,Saiidi and Sozen [3] studied several hysteretic modelsassociated to the flexural failure mode; in one of them,pinching effects, typical of shear behavior, wereincluded and labeled as the SINA model. Later, Lindeand Bachmann [4] developed an element to represent theinelastic seismic behavior of shear walls controlled byflexure, with a modest influence of shear cracking in thehysteretic response.

The model developed in this study has beenimplemented in the larz computer program [5]. Theshear-wall model has been modified to allow the wall todevelop a shear failure mode when its lateral shearstrength is smaller than the strength required to developa flexural failure mode. The shear model characteristicsare determined from experimental results obtained fromcyclic test of shear walls and beams, as described below.larz is a standard inelastic analysis software forreinforced concrete buildings, like drain or other avail-able software, and was developed by Saiidi [5] twentyyears ago. Nevertheless, the available software do notrecognize the possibility of developing a shear mode offailure in the structural elements, which precludes shearcracking or shear failure in these elements.

The definition of the uncoupled hysteresis models forshear and flexure described previously allows for theupdate of the flexural and shear tangent stiffnesses ofany wall element in a step by step integration procedure.A wall element as shown in Fig. 1, defined at each story,is divided into several sub-elements in height, each ofthem having the lateral and vertical degrees of freedom

Fig. 1. Typical wall element.

Fig. 2. Hysteretic model for flexural mode of failure (beams, col-umns, shear walls).

at each node and the end rotations. By assuming a con-stant inter-story shear and a linear variation of bendingmoments along the height of the wall, the shear andmoment at the center of each sub-element are computed.Hence, the tangent flexural stiffness EI and shear stiff-ness AG are determined from the moment-curvature andthe shear-deformation relationship for the wall as definedin Figs. 2 and 3. The tangent stiffness matrix for eachsub-element is then computed and assembled into theglobal stiffness matrix of the wall. By static conden-sation of the interior degrees of freedom associated withthe wall sub-elements, the tangent stiffness matrixrelated to the external degrees of freedom is computed.Thus, only the horizontal and vertical displacements andthe rotation at the ends of the wall are used to assemblethis element into the structural model.

The paper discusses the development and implemen-tation of this analytical model. The model proposed isvalidated by evaluating the results predicted by themodel for two buildings that underwent shear crackingduring the 1985 Chilean earthquake. The true horizontalacceleration components recorded at nearby sites and

Fig. 3. Hysteretic model for shear mode of failure (shear walls).

87P.A. Hidalgo et al. / Engineering Structures 24 (2002) 8598

similar soil conditions as for the buildings were used asinput for the computer model.

2. Model for flexural failure mode

The SINA hysteresis model implemented in the larzcomputer program (Fig. 2) was adopted in this study tomodel the non-linear flexural behavior and the moment-curvature hysteretic relations for wall elements. Asshown in Fig. 2, pinching effects and stiffness andstrength reductions due to repeated cycles at the samedeformation level were not implemented in the modelfor flexural behavior.

The model operates on a primary Mf envelope curveconsisting of four linear segments for positive and nega-tive bending as shown in Fig. 2. The primary curves neednot be symmetric about the origin, but a single straightline must be specified for moments below the crackingmoment in both directions (line CC in Fig. 2). PointsY and U (and Y and U) correspond to moments asso-ciated with first yielding and a concrete compressivestrain of cu=0.003, respectively. An horizontal line fol-lowing point U is assumed, and collapse is defined bya maximum curvature fmax, associated to a concretecompressive strain of cmax=0.01.

For moments below the cracking level, loading andunloading follow the primary curve. For moments abovethe cracking moment, unloading follows a line con-necting the unloading point with the cracking point inthe other direction (line PC in Fig. 2). If the yieldmoment is exceeded and unloading takes place at pointP1, the slope of the unloading branch P1P2 is taken as

KunKycfyfma

(1)

where fm is the maximum curvature attained in the load-ing direction and Kyc the slope of the line connectingthe yield point in the loading direction with the crackingpoint in the opposite direction. The exponent a controlsthe slope of the unloading branch after yielding, and wastaken equal to 0.5 as suggested by Saiidi and Sozen [3].A detailed description of all the hysteresis rules can befound elsewhere [3].

The bending moment and curvature values weredefined using the standard theory for reinforced concreteelements; both the boundary reinforcement and the dis-tributed vertical reinforcement are taken into account indefining the primary Mf curve for wall elements.Further, the axial load force values due to gravity,assumed to remain constant throughout the seismicresponse, are considered in the calculation of momentand curvature associated with points C, Y and U of theprimary curve. This constitutes an approximation forshear walls coupled by spandrel beams, since these

beams may develop significant seismic shear forces thatinduce variable axial loads on the walls, but the resultingerror in the axial loads becomes smaller as the flexuralstrength of the coupling elements decreases.

The evaluation of the bending moment for the collapsepoint defined by fmax , assumes the same compressivestress block in the concrete that was used for point U.This is obviously an approximation since it always yieldsto fmax=3.33fu in Fig. 2. This assumption is verified bycomputing the moment-curvature relationship for someof the walls using a realistic stressstrain curve for theconcrete. In all cases a slightly larger value of fmax isobtained by using the more exact method. Neverthe-less, when the model was used in this study to predictthe inelastic seismic behavior of real buildings, themaximum curvature never exceeded the value of fu.

3. Model for shear failure mode

Shear dominated behavior was also modeled using theSINA hysteresis model as shown in Fig. 3. Pinchingeffects and strength reduction due to repeated cycles atthe same deformation level were now implemented inthe hysteresis model. The model for the shear failuremode assumes independence of the shear strength ofwalls on both the bending moment and the axial forcepresent in the wall. This is also an approximation, butto neglect interaction between shear and axial force isconsistent with the current ACI design provisions forwalls [6].

The model was initially developed for squat shearwalls with an aspect ratio M/(VLw) of 1.0 or smaller,where M is the bending moment at the base of the wall,V the shear force, and Lw the length of the wall. It wasthen extended to the case of slender shear walls withaspect ratio larger than 1.0, as explained later. In Fig. 3,point C represents the point where a change in the slopeof the envelope of the loaddisplacement relationship isexperimentally observed; the new value of the stiffnessof the specimens is about 60% of the initial stiffness.Point C was generally very close to the point at whichthe first diagonal crack from corner to corner of the wallswas developed during the tests. Point Y corresponds tothe largest value of shear load attained during the test,while point U may be associated with the ultimate con-dition under which the element may still be consideredas an effective part of the resisting mechanism of thestructure. The definitions of points C, Y and U in theenvelope curve (Fig. 3) are based on the experimentalresults obtained from the cyclic test of 26 full scale,shear wall specimens. All these specimens weredesigned to exhibit a shear mode of failure and hadaspect ratios M/(VLw) between 0.35 and 1.0. The loadingsequence of each test consisted of sets of two cycles ata given displacement amplitude, which increased gradu-


ally and followed a sequence determined by the aspectratio of the specimen. All tests were finished when lat-eral strength of the specimen had dropped to 75% of themaximum value, approximately. More details of this testprogram may be found elsewhere [7]. On the other hand,the model characteristics for slender shear walls wereobtained from test results of reinforced concrete beams[8,9]. The main difference between the shear behaviorof beams and that of squat walls is that strength of beamsis lost after point Y (Fig. 3) has been attained.

The actual slope of branch YU obtained from tests ofsquat walls showed decreasing shear strength withincreasing displacement. This fact leads to a problem inthe software that cannot handle structures with a degrad-ing stiffness and, hence, with negative semi-definite tan-gent stiffness matrices occurring at some points in theresponse. For this reason, the branch YU of the modelwas taken as nearly constant, but keeping the actual ulti-mate displacement dult obtained from experimentalresults. Once this displacement is eventually attained ina wall during the response history analysis, the elementis removed from the structure and the stiffness matrix isre-evaluated. Fig. 3 also shows the eight hysteresis rulesfor this model, that also follow the SINA model [3]. Inthe shear failure model, a crack closing point (dcg, Vcg)is defined to account for the pinching effect alwayspresent in the hysteretic behavior after a shear crack hasdeveloped. In order to consider the shear strengthreduction due to repeated cycles at the same deformationlevel observed in the test specimens, a strength reductionfactor of 0.15 was adopted, as shown by point E in Fig.3, i.e. after unloading takes place from point D in Fig.3, the target point for subsequent load cycles is set verti-cally under D at a shear V=0.85VD. The model for slen-der walls is similar to that for squat walls, with the onlydifference that points Y and U (Fig. 3) become the samepoint in the tests of beams when a shear failure is attain-ed.

Figs. 411 show comparisons between experimental

Fig. 4. Shear model to estimate drift at first cracking.

Fig. 5. Shear model to estimate drift at maximum strength.

Fig. 6. Shear model to estimate drift at ultimate strength.

Fig. 7. Comparison of VC (ACI) with experimental cracking strengthof squat walls.


Fig. 8. Comparison of shear model to estimate cracking strength withexperimental values.

Fig. 9. Comparison of Vu (Arakawa) with experimental maximumstrength of squat walls.

Fig. 10. Comparison of Vu (New Zealand) with experimentalmaximum strength for slender walls.

Fig. 11. Comparison of shear model to estimate maximum strengthwith experimental values.

results and model envelope curve definitions for bothsquat and slender shear-walls under a shear failure mode.Experimental results for walls with aspect ratios lessthan or equal to one were obtained from the tests perfor-med by Hidalgo et al. [7], while those for elements withaspect ratios larger than one were taken from the test ofbeams [8,9]. Fig. 4 shows the definition of the drift ratioat cracking point C (Fig. 3), DRcr as a function of theaspect ratio of the wall. Similarly, Figs. 5 and 6 showthe estimations of the drift ratios at point Y, DRu, andat point U, DRult, respectively. Experimental results forbeams may be obtained only for DRcr and DRu sincethere is no residual shear strength after point Y (Fig. 3);therefore, as shown in Fig. 6, DRult was assumed to be0.014 for walls with aspect ratio larger than one.

Same information as before but for strength is shownin Figs. 7 and 8 for shear strength at cracking point C,Vcr. Different models were adopted for squat and slendershear walls. Fig. 7 shows the comparison between theexperimental value Vcr for squat walls and the value ofVc proposed in the ACI Code [6] to estimate the contri-bution of concrete to the shear strength of such walls.As shown in Fig. 7, the Vc value prescribed in ACI forshear walls underestimates the Vcr value obtained in thetests conducted earlier [7]. In order to improve the corre-lation between Vc (ACI) and Vcr for walls with aspectratio less than or equal to one, the value of Vc has to beamplified by a factor of 1.12. For more slender walls nocorrection of the ACI Vc value is necessary, as shownin Fig. 8.

Likewise as for point C, Figs. 911 show the esti-mation of maximum shear strength Vu at point Y (Fig.3). The best correlation with experimental results forsquat walls (Fig. 9) was obtained for the contribution ofconcrete Vc as proposed by Arakawa for beams, with thecontribution of shear reinforcement as proposed in theACI Code [7]. As before with the cracking value, thecorrelation of the Vc value proposed by Arakawa and the


Fig. 12. Structural floor plans of buildings studied. (Dimensions in centimeters). (a) Villa Real building. (b) Sermena building.

experimental values obtained from the test of squat walls[7] may be improved by multiplying Vc by a factor of1.41. The estimation of Vu for slender walls was takenas the ACI formula to predict the shear strength ofbeams, but using as contribution of concrete theexpression proposed by Paulay and Priestley [10] (Fig.10). This model has been labeled as N.Z. model in Fig.11 that shows the complete correlation between modelvalues and experimental results. Finally, the value of Vultshown in Fig. 3 was arbitrarily taken as 1.01 times Vuto avoid instability in the numerical model analysis dueto negative stiffness.

4. Prediction of inelastic seismic behavior of realbuildings

The models of flexural and shear failure modes wereimplemented into the larz computer program, such that

either a flexural or a shear failure mode could bedeveloped in each of the walls of the structure under asevere ground motion. The failure mode predicted by theprogram is based on the fact that the flexural responseof a particular wall element follows the hysteresis curvefor flexure (Mf curve, see Fig. 2) while the shearresponse follows the hysteresis curve for shear (Vdcurve, see Fig. 3). Therefore, at each integration step,the curvature f and shear displacement d demands asso-ciated with the calculated values of M and V are obtaineddirectly from the hysteresis curves, making possible thedevelopment of either a flexural or a shear failure mode.Since the shear model parameters depend on the aspectratio M/(VLw), the value of this ratio for each wall wastaken as the inter-story height h divided by the lengthof the wall Lw. This assumption is based on time-historyresults of M/(VLw) which was found to be closer to h/Lw,than to the total wall height divided by Lw. The analytical


Fig. 13. Acceleration record and base shear response, Villa Realbuilding. (a) Vina del Mar acceleration record, S20W component. (b)Base shear, X-direction analysis. (c) Base shear, Y-direction analysis.

model developed so far is two-dimensional, but a three-dimensional version is currently being investigated.

The model was validated by predicting the inelasticseismic behavior of two real buildings that developedinelastic behavior and shear cracking during the March3, 1985 Chilean earthquake. The 10-story plus a base-ment Villa Real building, designed in 1981, and the 7-story plus a basement Sermena building, designed in1971, are presented in Fig. 12(a) and (b), respectively.Both are shear wall reinforced concrete structural sys-tems located in the coastal cities of Vina del Mar andValparaiso, respectively, about 30 km from the epicenterof the 1985 earthquake. The distribution in plan of shearwalls for both buildings is quite symmetric with respectto horizontal axes X and Y. The typical ratios of shearwall area to floor plan area are 0.027 in the X-directionand 0.033 in the Y-direction, respectively, for the Villa

Fig. 14. Acceleration record and base shear response, Sermena build-ing. (a) El Almendral acceleration record, N50E component. (b) Baseshear, X-direction analysis. (c) Base shear, Y-direction analysis.

Real building, and 0.008 and 0.015, in the X- and Y-directions, respectively, for the Sermena building. Thelatter, despite being three stories smaller than the VillaReal building, has lower values of this ratio, which isconsistent with the more extensive shear crackingdeveloped by this building during the 1985 earthquake.

The earthquake response histories along both principalhorizontal directions of the Villa Real building wereobtained using the S20W component of the accelerationrecord obtained in downtown Vina del Mar. The site ofthe record was at about 1000 m from the site of thebuilding. In the case of the Sermena building, the N50Ecomponent of the El Almendral acceleration record wasused for the analysis of the building. This was recordedat about 2600 m from the building. In both cases, the


Fig. 15. Hysteretic response examples, Villa Real building. (a) Hys-teretic shear behavior wall M5, axis 2, story 2. (b) Hysteretic flexuralbehavior at base of wall M5, axis 2, story 2.

soil at the recording site had the same characteristics asthose of the corresponding building site.

Structural and architectural drawings of both build-ings, strength characteristics of both concrete and rein-forcing steel obtained after the earthquake, and a detaileddocumentation of damage and cracking patterns areavailable to conduct this study [11,12]. In order to ascer-tain the validity of structural model assumptions associa-ted with the elastic behavior of the buildings, the com-puted first mode translational periods of vibration arecorrelated with the values measured after the earthquake.For the Villa Real building the nominal periods are0.74 s (0.71) and 0.53 s (0.51), in the X- and Y-direc-tions, respectively; and for the Sermena building 0.48 s(0.45) and 0.42 s (0.41), in these directions, respectively.The numbers in parenthesis correspond to the periodsmeasured experimentally.

Let us consider first the prediction of base shearobtained with the computer program along each of theprincipal directions of resistance as shown in Figs. 13and 14. It is interesting to note that the original designbase shear for the Villa Real building was 2390 kN(11.5% of total weight), using the current Chilean Codewhen the design was carried out. Maximum values pre-dicted for the X- and Y-directions of the Villa Real build-

Fig. 16. Hysteretic response examples, Sermena building. (a) Hyster-etic shear behavior wall M3, axis A, story 1. (b) Hysteretic flexuralbehavior at base of wall M3, axis A, story 1.

ing, were 5520 kN (26.6% of weight) and 7020 kN(33.9% of weight), respectively, that is, 2.3 and 2.9 timeslarger than the design base shear. Likewise for the Ser-mena building, the design base shear was 2830 kN (12%of weight), while predicted maximum values were8370 kN (35.5% of weight) and 7310 kN (31% ofweight), in the X- and Y-directions, respectively, i.e. 3.0and 2.6 times the design base shear. Maximum valuesof inter-story drift ratios predicted for these buildingswere 0.0073 and 0.0051, in the X- and Y-directions,respectively, for the Villa Real building. Correspondingvalues for the Sermena building were 0.0048 and0.0044, respectively.

Examples of hysteretic behavior, both in flexure andshear, developed at some critical shear-wall segments areshown in Fig. 15 for the Villa Real building and in Fig.16 for the Sermena building. Fig. 15(a) and (b) showsthe hysteresis loops developed in wall M5 of the VillaReal building, shown in Fig. 12(a), both for the shearbehavior and the flexural behavior, respectively. It canbe observed from Fig. 15(a) that shear behavior almostreached the cracking point, with very limited energy dis-sipated through this type of behavior. On the other hand,the formation of a plastic hinge is apparent from Fig.15(b), with larger energy dissipation through the flexural


Fig. 17. Experimental results for Specimen 23 (M/VLw=0.69, rh=0.25%, rv=0). (a) Actual cracking patterns. (b) Hysteretic shear behavior.

behavior than through the shear behavior. The sameinformation is shown in Fig. 16 for wall M3 of the Ser-mena building (Fig. 12(b)); in this case, more energy isdissipated, through both the shear and the flexuralbehavior. It is worthwhile to note that minimum valuesof the aspect ratio M/(VLw) of walls that experiencedshear cracking in the Villa Real building, at any instantof the time-history response, reached 0.72 and 0.60, forthe X- and Y-directions, respectively.

The capability of the model to predict the inelastic

seismic response was evaluated by comparing the crack-ing that would be obtained from the predicted internalforces in each of the structural elements with the actualcracking patterns due to flexure and shear. The cracksdue to flexure are sometimes difficult to observe afteran earthquake, but those due to shear may be easily reco-gnizable. In order to correlate the predicted internal shearforces with cracking patterns, the experimental infor-mation obtained from a previous research program wasused, [7]. Figs. 17 and 18 show two examples of the


Fig. 18. Experimental results for Specimen 27 (M/VLw=0.50, rh=0.25%, rv=0). (a) Actual cracking patterns. (b) Hysteretic shear behavior.

relation between the hysteretic shear behavior of shearwalls and the resulting cracking patterns. A moredetailed analysis of the results of this comparison foreach building follows.

4.1. Villa Real building

The inelastic response of the structure was determinedby using the acceleration record indicated above, first asbase acceleration history along the X-direction, and thenalong the Y-direction. To evaluate the inelastic shear

behavior of walls, the attention was mainly focused onthe walls that developed shear cracking during the 1985earthquake. In order to validate the results from theanalysis with the actual performance of the structure, theexperimental results of cracking patterns, like thosepresented in Fig. 17, are used as follows. Consider forinstance wall M14 (Fig. 19) which shows a maximumshear of 58% of the cracking shear (Fig. 19(b)). There-fore, this wall has attained a point that is below point 1in Fig. 17 and, hence, should present less cracking thanpattern 1 in that figure. This coincides with the actual


Fig. 19. Axis D, Villa Real Building. (a) Observed cracking pattern.(b) Quality of shear behavior prediction.

behavior exhibited by wall M14 (Fig. 19(a)). On theother hand, wall M34 in Fig. 20 attained a maximumshear of 1.04 times the cracking value, i.e. it is slightlyabove cracking, say point 3, in the force-deformationloop of Fig. 17. The cracking associated with this levelof shear (cracking pattern 3 in Fig. 17) is similar to thatobserved in the structure after the earthquake (Fig. 20(a))and, thus, there is a good correlation between predictionof the model and the actual behavior. Similar analysisis performed for each wall with actual cracking and theresults presented in Figs. 1922. The ratios of maximumshear obtained from the inelastic, dynamic analysis andVcr and Vu defined for each wall are included in Figs.19(b)22(b). Such a procedure would enable us to vali-date the analytical model developed in this study.

Figs. 19 and 20 illustrate the results of this damageidentification procedure for walls in two of the X-direc-tion resisting planes. Given the uncertainties involved inthe analytical prediction of the inelastic shear behavior,as will be discussed below, the quality of the predictionis only rated in general terms as adequate, too high,or too low. A quality of prediction rated as adequate,means that shear cracking observed in the walls afterthe 1985 earthquake is similar to that observed in theexperimental program for the same level of shear force.On the other hand, too high or too low means that pre-dicted shear force is above or below the values consistent

Fig. 20. Axis E, Villa Real Building. (a) Observed cracking pattern.(b) Quality of shear behavior prediction.

with the actual cracking pattern observed in the wall,respectively. Figs. 21 and 22 do the same for the wallsin two of the Y-direction resisting planes. The rating ofthe quality of the prediction has to consider severalaspects that do not allow the prediction of the crackingpattern to be more exact. First, the actual base acceler-ation history experienced by this particular building isnot exactly known; the actual seismic excitation was bi-directional and not unidirectional as it has been assumed,and the vertical seismic action has not been included inthe analysis. Secondly, the experimental information thatcorrelates the hysteretic shear force with the resultingcracking patterns for shear walls generally shows a sig-nificant degree of scattering. Thirdly, the strong assump-tions related to the definition of the structural model.And finally, the fact that this analytical model is stilltwo-dimensional, and torsional effects have not beenconsidered to predict the earthquake response of thestructure.

The analytical model also showed the development ofbending moments larger than the yield moment in coup-ling beams and in some of the walls, as shown in theexample of Fig. 15(b). The prediction of bendingmoments and the development of plastic hinges in thecoupling beams have a reasonably good correlation withthe actual amount of flexural cracking experienced by


Fig. 21. Axis 2, Villa Real Building. (a) Observed cracking pattern.(b) Quality of shear behavior prediction.

Fig. 22. Axis 5, Villa Real Building. (a) Observed cracking pattern.(b) Quality of shear behavior prediction.

this building. Nevertheless, the actual flexural crackingis sometimes difficult to observe after an earthquakeevent, particularly if yielding has been moderate.Besides, the emphasis of the prediction capability of thismodel has been placed in the inelastic shear behaviorrather than in the flexural behavior.

Generally speaking and based on the results presentedin Figs. 19(b)22(b), it may be stated that this analyticalmodel has been able to reproduce reasonably well theinelastic shear behavior experienced by the Villa Realbuilding during the 1985 earthquake.

4.2. Sermena building

The procedure followed for this building was the sameas for the Villa Real building. Figs. 23 and 24 illustratethe results of this procedure for walls in two of the X-direction resisting planes, while Figs. 25 and 26 illustratethe results of this procedure for walls in four of the Y-direction resisting planes. In general terms, limitationsassociated with the analytical prediction of the earth-quake inelastic behavior of the structure and quality of

Fig. 23. Axis A, Sermena Building. (a) Observed cracking pattern.(b) Quality of shear behavior prediction.


Fig. 24. Axis B, Sermena Building. (a) Observed cracking pattern.(b) Quality of shear behavior prediction.

Fig. 25. Axes 2, 3 and 5, Sermena building. (a) Observed crackingpattern Axis 2. (b) Observed cracking pattern Axis 3. (c) Observedcracking pattern Axis 5. (d) Quality of shear behavior prediction, Axes2, 3 and 5.

the results are about the same as those found for theVilla Real building. The distinctive feature of the resultsobtained for the Sermena building relates to the resultsobtained for the seismic analysis in the X-direction. Theanalytical model assumes that the structure is longitudi-nally symmetric, and therefore, predicts the same inelas-tic behavior for resisting planes along both axes A andD (Fig. 12(b)). Nevertheless, the resisting plane alongaxis D showed much less damage than that along axisA and was practically free of shear cracks in the firststory walls after the earthquake. The reasons to explain

Fig. 26. Axis 5b, Sermena Building. (a) Observed cracking pattern.(b) Quality of shear behavior prediction.

this difference between the actual behavior of the build-ing and the prediction of the model may be one of thefollowing: (1) the analytical model did not include thestructure for the elevator shaft adjacent to axis A; (2)there is evidence of existing cracking in the first storyshear walls along axis A, prior to the 1985 earthquake[12]; (3) some unknown facts related to the constructionof the building; and (4) inelastic torsion of the buildingplan as reported elsewhere [13].

5. Conclusions

This study has proposed and implemented a modelto include the shear failure mode for walls in availablecomputer programs. The model developed in this studyis a macro-model validated with the experimental resultsof cyclic tests of shear walls. Though this model maystill be refined, it may be used to predict the inelasticseismic behavior of reinforced concrete, building struc-tures as long as they have nominally symmetric struc-tural plans, thus providing a useful tool to estimate seis-mic demands on this type of buildings.

Acknowledgements

This research was funded by the Chilean SuperiorCouncil for Technological Development under GrantNo. 1980953. The sponsorship of this institution is grate-fully acknowledged. The authors also thank ProfessorJuan C. De la Llera for his valuable comments and con-


tributions to the writing of the manuscript, and ProfessorP. Bonelli from the Universidad Tecnica F. Santa Maria,who provided the structural details and damage docu-mentation of the buildings used in this study.

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An Analytical Model to Predict the Inelastic Seismic Behavior of Shear-wall, Reinforced Concrete...

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