Probabilistic Models for Seismic Design and Assessment of RC Structural Walls

Transcription

1 robabilisti Models for Seismi Design and Assessment of RC Strtral Walls Abstrat Mehrdad Sasani Northeastern Universit Boston, Massahsetts robabilisti models for estimating lateral flexral displaement apait, shear strength apait, and shear deformation of reinfored onrete strtral walls are presented. In developing all the models, available experimental data are tilized and the Baesian parameter estimating tehniqe is sed. The model for estimating the shear deformation of strtral walls is onstrted based on the opling between the flexral and shear inelasti deformations. Comparisons with some rrent models in seismi odes are made and signifiant improvements are shown. Kewords: Displaement Capait, Shear Strength Capait, Shear Deformation, Strtral Walls Introdtion In probabilisti design and assessment of a strtre, aonting for sores of nertainties, the probabilit of demands being more than orresponding apaities (i.e. probabilit of failre) is estimated. In order to have meaningfl estimation of the reliabilit of the strtre nder external ations (loads), one needs to not onl aont for nertainties assoiated with the external ations and materials, bt also onsider other sores of nertainties affeting estimations of demands and apaities. Frthermore, there is a need for the seletion of proper measres of demands and apaities. In seismi engineering, these measres in addition to strength, old inlde displaement, energ, or more generall damage. In this paper strength and displaement apaities and demands are estimated and ompared. To estimate strtral apaities, different models at setion, element, and strtre levels are reqired whose nertainties have to be aonted for. Frthermore, these models wold estimate apaities more realistiall, if the have been alibrated sing available experimental data. To estimate strtral demands proper models that inorporate both flexral and shear strengths and deformations are needed. In this paper different probabilisti models for estimations of seismi apaities of reinfored onrete (RC) strtral walls are presented.

2 Flexral Displaement Capait Flexral apait of a strtral wall is limited b its strength as well as its displaement apaities. Mehanial models for estimating the flexral strength apait of the strtral wall are well developed and the sall predit the strength within a small error. Some models for estimating the displaement apait that are available and even implemented in odes (UBC, 997), however, ma not reasonabl estimate the apait of the wall (Sasani, 998). The flexral displaement apait of the wall is limited b the maximm aeptable onrete and steel strains. Displaement apait of strtral walls has been stdied b Sasani and Der Kireghian (00) and below a smmar of the std is presented. Figre shows the elasti and inelasti deformations of a strtral wall. The maximm displaement apait at the top of a strtral wall that has developed a plasti hinge near its base is approximatel given b Fig.. Deformation of strtral wall the expression ˆ = a Φ f H + ( Φ Φ ) L ( H L / ) where H is the height of the wall, L is the length of the plasti hinge, Φ and Φ respetivel are the ield rvatre (at the first ielding of the flexral reinforement ) and the ltimate li rvatre of the setion near the base of the wall, and a is a oeffiient that depends on the distribtion of bending moments and the flexral stiffness along the height of the wall. The sperposed hat on ˆ f is sed to signif the fat that the above model is not exat. The first term in () represents the ontribtion of the elasti deformation at the first ielding of the flexral reinforement, whereas the seond term represents the ontribtion from the loalized plasti deformation near the base of the wall. The varios terms in the model are frther disssed or developed below. (). Coeffiient a The oeffiient a depends on the distribtions of the flexral stiffness and lateral load along the height of the wall. If one assmes a niform flexral stiffness eqal to that of the raked setion at the base of the wall, and a linear relation between the setion rvatre and bending moment, nder an inverted trianglarl distribted lateral load, Fig., one obtains a = / 40, whereas for a onentrated lateral load at the top one obtains a = / 3. In realit, the top portion of the wall ma not be raked. While it is possible to se the nraked setion for the top portion, the effet on the oeffiient a and, hene, the estimated top displaement is sall insignifiant nless the amont of the flexral reinforement and the axial ompressive load of the wall are

3 small (Sasani 998). Therefore, in the following analsis, raked onrete setions are assmed and the inverted trianglar load distribtion is emploed. revios investigations have shown that this approah provides fairl arate estimates of the elasti ontribtion to the top displaement (Sasani and Anderson 996).. Yield Crvatre For a given ross setion of the wall and for known stress-strain relations of onrete and reinforing steel, the ield rvatre Φ is easil determined b emploing a kinemati assmption for the deformation of the wall, sh as the assmption that plane setions remain plane. Usall the raked setion of the onrete is sed for this analsis. This tpe of analsis is rotine and needs no frther investigation here. Note, however, that the predition of Φ is not free from error. The ontribtion of this error will be aonted for along with the errors in the other terms when the overall error in model () is assessed..3 lasti Hinge Length A term in () that needs to be developed is the plasti hinge length, L (see Fig. ). Aording to Corel (966), L is a distane sh that when mltiplied b the average plasti rvatre (over a distane eqal half the setion depth) at the base of a antilever member gives the plasti rotation of the hinge. Based on experimental reslts, Corle (966) sggested the empirial expression L = 0.5d +.6 H d, where d is the effetive depth of the setion and H (the wall height) is the distane between the points of zero and maximm moment, in meters. In his disssion of Corle's paper, Mattok (967) sggested the expression L = 0.5d H. In order to develop a probabilisti model for L that is appropriate for RC walls, 9 of the test reslts reported b Corle (966) and Mattok (967) are seleted that orresponded to beams with effetive depths greater than 0.5m. This data was sed in onjntion with the Baesian method to estimate the parameters of the following: (a detailed desription of the appliation of the method is presented later in this paper, where a shear strength apait model is developed) 3 / L H ls = α + α + ε () L d d whih provides a good fit to the data. In (), l s is a standard length eqal to meter (=39.4 inhes) that is inserted to make the model parameters dimensionless, and ε L in eah eqation is a model error term that is assmed to have the normal distribtion with zero mean (so that one obtains nbiased models) and nknown standard deviation σ L. (Stritl speaking, L being non-negative, the normal distribtion for ε L is not appropriate. However, the variabilit in the model is small in relation to its mean and the probabilit of having negative L is virtall zero.) The above models are linear in terms of the nknown parameters α and α. For sh a ase, with non-informative priors on α, α and σ, losed form soltions of the posterior statistis are available (Box and L 3

4 Tiao, 99). The posterior mean vales of the parameters based on the 9 test data are α = 0.47, α = and σ L = The standard deviations are 0.088, 0.09, and 0.0, respetivel. The onl onsiderable orrelation oeffiient is between α and α, whih is eqal to Cli Crvatre Capait The li rvatre apait, Φ, of the ross-setion of the RC wall is determined b modifing its monotoni rvatre apait, Φ. The monotoni rvatre apait is determined b allating the moment-rvatre relationship for the ross setion. The ltimate monotoni rvatre of the setion is assmed to have been reahed when an one of the following three riteria is satisfied: () onrete reahes its maximm sable strain, () an steel reinforing bar reahes its fratre strain, (3) the moment strength of the setion drops to 80% of the moment apait. In addition to the geometr of the ross setion and the plaement and area of reinforing bars, the monotoni rvatre apait depends on the stress-strain relationships of onrete and steel and on the magnitde of the axial load. For the present std, the modified Kent and ark model (ark et al. 98) is emploed to desribe the stressstrain relations for nonfined and onfined onrete. In the following setion, a model for the maximm sable onrete strain is examined..4. Maximm Useable Conrete Strain The maximm rvatre apait of a RC wall setion ma be limited b the maximm sable onrete strain, ε max, inlding the effet of onfinement b the transverse reinforement. A good measre for the onfining ation of the transverse reinforement is f hρ sh, where f h is the ield stress and ρ sh is the volmetri ratio of the onfining steel hoops. Most available measrements of ε max are for olmns nder axial loads with niform strain distribtion, whih is not representative of the strain distribtion in the ompressive zone of strtral walls. Kaar et al. (976) have tested speimens that have been speifiall designed to model the ompression zones of strtral walls. To develop a probabilisti model based on this data, the idealized model max f h ρ ε sh ε ε = e β + β (3) f s is onsidered, where β and β are the model parameters and f s = 43 Ma (=60 ksi) is the ield stress of grade 60 steel, whih is sed to make the model parameters dimensionless. Note that parameter β is idential to the maximm sable strain of nonfined onrete. The error term, ε ε, is assmed to have a normal distribtion with zero mean and an nknown standard deviation, σ ε. The nknown parameters of the model are β, β and σ ε. 4

5 In the Baesian approah, one an easil inorporate an prior information on the model parameters. In the present ase, while there is no information available abot β and σ ε, prior information on β, i.e., the maximm seable strain of nonfined onrete is available. Most investigators wold se a vale of to for β. To inorporate sh information, β is assmed to have a lognormal prior distribtion with mean and a standard deviation For β and σ ε, non-informative priors are sed (Box and Tiao 99), whih essentiall impl loall niform distribtions for β and ln σε. Using the experimental reslts of Kaar et al. (976) and the ompter program BUM for Baesian pdating developed b Geskens et al. (993), the posterior statistis of the parameters are ompted. The posterior mean vales of the parameter are β = , β = 0.8, and σ ε = The standard deviations are , 0.080, and 0.057, respetivel. The onl onsiderable orrelation oeffiient is fond to be eqal to 0.3 between β and β. Effet of Compressive Strain Conentration The traditional assmption that plane setions remain plane in flexre is not appliable to strtral walls with deep setions, partilarl within the hinging region. Unfortnatel, sffiient data are not available to onstrt a probabilisti model for this mehanism. Instead, to aont for the effet of strain onentration in the ompression zone of onrete, the maximm seable onrete strain is modified to obtain max ( ) = θ ε ε mod max (4) where θ is a orretion parameter having a vale less than nit. With this reded onrete strain apait, setion analsis with a linear strain distribtion is arried ot to determine the moment-rvatre relationship and, thereb, the ield rvatre Φ and the monotoni rvatre apait Φ. Sine no data is available to diretl assess the model in (4), θ will be estimated in the orse of assessing the global model for the displaement apait of the wall, as desribed below. Model for the Cli Crvatre Capait Using the ark and Ang (985) damage model, it an be shown that the redtion in the rvatre apait of a RC setion de to the li natre of the load depends on the rvatre dtilit of the setion, µ Φ = Φ / Φ. Using that model with parameter vales sggested b Fajfar (99), Sasani (998) sggested an empirial model for the rvatre apait nder li displaement having the form ( γ γµ Φ ) Φ = Φ (5) where γ and γ are nknown parameters. Unfortnatel, no reliable data is available to diretl estimate the parameters of this model. Hene, the will be estimated along with parameter θ of the model in (4) in the orse of assessing the global model for the displaement apait of the strtral wall, as desribed below. The reader will 5

6 note that the error terms are not inlded in the above two sb-models. This is bease the errors in these sb-models are inorporated in the overall error term of the global model..5 robabilisti Model for Flexral Displaement Capait Motivated b () and the sb-models desribed above, and noting that the displaement apait mst be non-negative, the following global probabilisti model is onsidered for the displaement apait of RC strtral walls: f L ln = ln a Φ H + ( Φ Φ ) L + f H H ε (6) In this model a, Φ and H are as desribed earlier and ontain no model parameters; L is the mean estimate of the plasti hinge length obtained b sing the mean vales of α and α and setting ε L = 0 in (); Φ is ompted from (5) and involves the nknown parameters γ and γ as well as the monotoni rvatre apait Φ, whih in trn involves the modified maximm seable onrete strain in (4) involving the nknown parameter θ. As sal, ε f is the random orretion fator of the model, whih is assmed to have the normal distribtion with zero mean and nknown standard deviation σ f. This orretion term inldes not onl the error in the form of the global model (6), bt also the errors inherent in the sb-models for max L, Φ, Φ, ε. Φ and ( ) mod The model in (6) involves for nknown parameters: θ, γ, γ andσ f. These parameters are estimated sing data for 8 strtral wall models, whih were tested liall nder displaement ontrol and failed in flexre. At eah displaement level, the walls were sbjeted to either or 3 les. The relevant referenes and essential parameters of the tested wall models an be fond in Sasani and Der Kireghian (00). Note that the volme fration of the onfining reinforement varies from ρsh = 0. to.08, the perentage of total longitdinal reinforement in the setion varies from ρ t = 0.48 to.95, the aspet ratio, i.e., the height from the base to the point of zero bending moment divided b the length of the wall in the plan, varies from Z / L W =.8 to 3., and the axial ompressive load divided b the gross setion area times the ompressive strength of onrete varies from 0.3 to 0.. In other words, the data overs a wide range of these important variables. The measred flexral displaement apait of eah wall is fond b linearl interpolating between the displaement experiened b the wall in the le where the fore-displaement relationship shows a drop of more than 0% in the lateral load apait, and the displaement experiened in the previos stable le. The interpolation is arried ot based on the nmber of stable les before the mentioned drop in lateral load apait. For example, if onl ot of 3 les at displaement level = 0. 0m is stable and the previos le is at = m, then 6

7 the displaement apait of the wall is f = ( ) (/ 3) = m. For the analtial preditions, the stress-strain relation for the steel is obtained from tension reslts reported for eah tested wall. As indiated earlier, the onrete stressstrain relation was based on the modified Kent and ark model (ark et al. 98) with the ompressive strength of onrete as measred in eah test. Baesian analsis b se of the program BUM revealed strong orrelation between γ and γ, sggesting that these parameters are approximatel linearl dependent. Using the posterior statistis, the linear estimate γ = 0.030γ was obtained. Sbstitting this relation in (5), one obtains the simplified model [ γ 0.030γ 0.04) µ Φ ] Φ = ( Φ (7) where γ is replaed b γ. The nmber of nknown parameters inherent in the model in (6) is now reded to 3, i.e., θ, γ and σ. Repeating the Baesian analsis with the reded model, the posterior mean vales of the parameter are θ = 0.796, γ = and σ f = The standard deviations are 0.03, 0.06, and 0.05, respetivel. All the orrelation oeffiients are negligible. Figre shows a omparison of the measred verss predited top displaement apaities for the 8 walls tested. On the horizontal axis the measred displaement apait is shown. The vertial axis shows the predited displaement apaities. Solid irlar dots indiate median (50% fratile) estimates, whereas the I-bars indiate the 5-85% fratile ranges. It is noted that almost all the I- bars over the / line. Also shown in Fig., as solid sqare marks, are estimates of the displaement apait of the 8 tested walls obtained b sing the rrent provisions of the Uniform Bilding Code (997). These estimates are fond to be grossl on the nonservative side. The athor believes the reason for this overestimation of the displaement apait b the UBC ode provisions redited Top Displaement Capait (m) is the fat that these provisions neglet the effet of ompressive strain onentration in the ompression zone of onrete and the effet of the li load in reding the rvatre apait of the wall setion UBC % 5%~85% Range SW6 SW4 RW Measred Top Displaement Capait (m) Fig.. Measred verss predited displaement apaities R B R4 RW B3

8 3 Shear Strength Capait The shear failre of strtral walls ma arise from an ombination of sliding shear, web rshing and shear-ompression failre of the ompression zone (Fig. 3). In order to develop a shear strength apait model for strtral walls, sixteen strtral walls tested nder li loads are stdied (see Sasani et. al., 00). Among the sixteen walls, nine failed in shear and the remaining seven had flexral failres. (a) (b) () Fig. 3. Shear modes of failre: (a) sliding shear; (b) web rshing and () shear-ompression failre (Oesterle et. al., 976) reliminar stdies with a shear strength model revealed weak orrelation between the displaement dtilit at the failre and the shear strength of the wall. Based on this observation, the following probabilisti model is onsidered for shear strength apait, V ap, of RC strtral walls: [( ν a + A f ) ( f f ) bl V ] ε V Vap = e ν asp g s + Vs = ρ h f h b l ν 3 ( f f ) b l s s In the above, ν, ν and ν 3 are the model parameters and ε V is a normall distribted model error with zero mean and nknown standard deviation σ V. a asp aonts for the aspet ratio of the wall and linearl varies from.5 to.0 as the aspet ratio inreases from.5 to.5. For aspet ratios larger than.5, a asp is set eqal to.0. f s is a saling stress eqal to Ma (or its eqivalent in other nits), whih is emploed to make the parameters of the model dimensionless. is the axial ompression on the wall and A g is the gross setion area, b is the width of the web, l is the total length of the setion, f is the ompressive strength of onrete and f h is the ield stress of the horizontal reinforement in the web. Finall, V s is the shear strength orresponding to the horizontal reinforement and has an pper bond in order to inhibit web rshing of the wall de to large amont of shear reinforement. The Baesian parameter estimation tehniqe provides an effetive tool for the development of probabilisti models (Der Kireghian 999). In this paper, the Baesian tehniqe is emploed to develop different probabilisti models reqired for seismi design and assessment of reinfored onrete (RC) strtral walls at the life safet level. (8) 8

9 3. Baesian model assessment Details of the Baesian tehniqe an be fond in the existing literatre (Box and Tiao 99, Der Kireghian 999). Here, onl a brief otline is presented. Let = gˆ ( x, θ ) + ε (9) be a mathematial model for prediting variable in terms of a set of observable variables x = ( x, x, K), in whih g ˆ( x, θ ) is an idealized model (signified b the sperposed hat), θ = ( θ, θ, K) is a set of nknown model parameters, and ε is a random variable representing the nknown error in the model. We will assme that ε has a normal distribtion (normalit assmption) and that it has a onstant standard deviation σ. (homoskedastiit assmption). If, for a given model g ˆ( x, θ ), these assmptions are not satisfied, then it is possible to make a transformation of the model sh that these assmptions are at least approximatel satisfied. Box and Cox (964) sggest a parametri famil of transformations for this prpose. In the experimental reslts tilized in this paper, it is expeted that the error in the apait model will inrease linearl with the apait. Frthermore, the apait being nonnegative is well represented b a lognormal distribtion. Therefore, a logarithmi transformation is seleted to approximatel satisf the normalit and homoskedastiit assmptions. Finall, with the aim of developing an nbiased model, we assign a zero vale to the mean of ε. The set of nknown parameters of the model, ths, are Θ = (θ,σ). The model is assessed b estimating Θ based on the available information, whih tpiall onsists of a set of measred vales of x and the orresponding, and possibl sbjetive information on the likel vales of the parameters. In the Baesian approah, this is done b the se of the well-known pdating rle ( Θ ) L( Θ ) p( Θ ) f = (0) where p(θ) denotes the prior distribtion on Θ refleting the sbjetive information, L(Θ) is the likelihood fntion, whih is a fntion proportional to the onditional probabilit of making the observations on x and for a given vale of the parameters and reflets the objetive information gained from the data, is a normalizing fator, and f(θ) is the posterior distribtion refleting or pdated information abot Θ. This rle is sed to onstrt apait and demand models and estimates of the fragilit for RC strtral walls based on observed laborator test data. Formlations of the prior distribtion and the likelihood fntion for speifi models are presented throghot the paper. 3. robabilisti Model for Shear Strength Capait The experimental information available for prediting the shear strength apait of the walls are of two kinds: Measred shear strength, when shear failre has been 9

10 observed, and measred lower bond to the shear strength when the wall has failed in flexre. These two tpes of information are refleted in the likelihood fntion. Let Vˆ ap [( a + A f ) ( f f ) bl + V ] = ν ν () asp g s s denote the predited shear strength apait exlding the error term. In the k-th experiment, given the set of observable variables (a asp,, A g,b,l, f, f h,v s ) k, ( V ˆ ap ) is k allated from (). Having the measred vale of the maximm applied shear fore on the setion, the k-th realization of the error term is ( V ) = ln( Vap ) ln( Vˆ ap ) k k k ε () Considering the normal distribtion of the error term with a zero mean, and assming statistial independene between the observations, the likelihood fntion takes the form ( ε ) V ( ε ) k V k L( ν ) =, ν, ν 3, σ V ϕ x Φ Shear Failre σ V σ (3) V FlexralFailre σv where the first prodt is for all the walls that failed in shear and the seond prodt is for all the walls that failed in flexre. In the above expression ϕ (.) is the standard normal probabilit densit fntion and Φ (.) is the standard normal mlative distribtion fntion. Not having prior information on the parameters of the model, a non-informative prior distribtion is sed (Box and Tiao, 99). This essentiall implies loall niform distribtions for ν, ν, ν 3, and ln(σ V ). This prior distribtion together with the likelihood fntion in (3) is sed in the Baesian pdating formla to estimate the posterior statistis of the parameters. The ompter program BUM (Geskens et al., 993) is sed for this prpose. The posterior mean vales of the parameters based are ν = 0.067, ν =.40, ν 3 = 0.500, and σ V = The standard deviations are 0.03, 0.44, 0.00 and 0.00, respetivel. The onl onsiderable orrelation oeffiient is between ν and ν, whih is eqal to The standard deviation of the model error is small (eqivalent to a oeffiient of variation of abot 0.05 in the apait), whih is an indiation of the ara of the model. Based on the omparison between the mean vales of ν and ν, for a /(A g f ) vale of onl 0.06, the effet of the axial load on the shear strength apait of the wall is twie that of the first term on the right hand side of (). The importane of the axial load on the shear strength of the wall is also refleted in the signifiant orrelation between the shear deformation and the term /(A g f ). The large negative orrelation between ν and ν implies that the two terms an be ombined with little loss of ara. This simplifiation is not sed in this std. 0

11 Figre 4 ompares the measred and predited median shear strength apaities for the tested walls. As an be seen, the data points for walls that failed in shear are losel lined p along the : line that represents eqal vales for the measred and predited shear strengths. The data points for walls that did not fail in shear fall below the diagonal line, indiating that the predited median shear strength apaities are larger than the maximm applied shear fore. V measred (kn) Shear failre Flexral failre V predited (kn) Fig. 4. Measred verss mean predited shear strength 4 Shear Displaement The form of the shear deformation of a RC strtral wall is different from the form of the flexral deformation over its height. Fig. 5 shows an idealized shear distortion pattern of a strtral wall. Experimental data shows that a signifiant part of the inelasti shear deformation takes plae at the base of the wall over a height almost eqal to the total depth of the setion, length of the wall in the plan, L W, (Oesterle et. al., 976 and Vallenas et al., 979). Therefore, in this setion a model is proposed for estimating the shear distortion of LW LW the wall over this length, denoted as Drift = L. S The test reslts show that the shear ielding (i.e. signifiant drop in shear stiffness) oinides with flexral ielding, whih is not neessaril aompanied b ielding of horizontal reinforement (Oesterle et. al., 976 and 979). Therefore, inelasti flexral and shear deformations are opled. In a trss analog, nder the lateral loads, the longitdinal reinforement (mainl in the bondar element region) forms the tensile element of the assmed trss sstem. Therefore, the ielding of the flexral reinforement implies the ielding of the tensile element of the assmed trss sstem. This is demonstrated in Fig. 6. Figre 6(a) shows the deformation of a trss model for a strtral wall de onl to the ielding of the bottom left vertial element. This deformation is deomposed to flexral (Fig. 6(b)) and shear (Fig. 6()) deformations. S W Fig. 5. Shear deformation of RC strtral wall

12 (a) (b) () = + B B' C C' 0.5 B' B C C' 0.5 B B' C C' A B A B A Total re Flexre re Shear Fig. 6. Effet of flexral ielding in shear deformation B Frthermore, after ielding of the flexral reinforement, the raks (flexral and shear raks) widen and the stiffness of the shear-transferred mehanism throgh aggregate interlok drops. As explained b Oesterle et. al. (976), this is aompanied b a redtion in the dowel stiffness of the tensile bondar element. Test reslts show a signifiant orrelation between the amont of axial load and the shear distortion. Oesterle et. al. (984) sggest the following relationship between W the shear drift, L LW Drift S and total drift, Drift t, over the height L W L W LW Drift s = Driftt 0. 5 Drift Ag f Another parameter that ma affet the shear deformation of strtral walls is the level of shear fore demand, V d, on the wall. Therefore, the following model is sed to estimate the shear distortion of strtral walls in the regions lose to the base of the walls: LW t (4)

13 L VCap W ε Ds LW Drift s = e λ Drift λ λ3 t Ag f V d Set = λ if A f (5) Set V g Cap V d 4 λ 4 Ag f V = λ Cap 5 if λ 5 Vd W where Drift L f is the flexral drift over the height L W, V Cap is the shear strength apait of the setion as given in (8), exlding V Cap. λ to λ 5 are the parameters of the model and ε Ds is a random variable representing the nknown error in the model having the normal distribtion with zero mean and nknown standard deviation σ Ds. Using the Baesian parameter estimation tehniqe, the posterior statistis of the model parameters are estimated and given in Table. Table. osterior statistis of shear distortion model parameters arameter Mean Correlation Coeffiient Standard Deviation λ λ λ 3 λ 4 λ 5 σ Ds λ λ λ λ λ σ Ds Figre 7 ompares the measred and predited mean shear distortion vales for the tested walls. The data for two walls are not inlded bease of lak of information on the measred shear distortion. As an be seen, exept for one wall (the onl barbellsetion wall with low flexral and high onfining reinforement bondar element and nder low axial load that failed in flexre), in whih the measred shear distortion is onsiderabl larger than the estimated vale, the reslts of other walls fairl losel follow the : line. Drift meas s Flexral failre Shear failre Drift pred s Fig. 7. Measred verss mean predited shear drift 3

14 5 Appliation JCSS Workshop on Reliabilit Based Code Calibration The apait models presented in this paper an be tilized in the probabilisti seismi design and assessment of RC strtral walls. The models an also be sed in the estimation of seismi fragilit of RC strtral wall. The seismi fragilit of a strtral sstem is defined as the onditional probabilit of failre of for a given intensit of the grond motion. roper measres of the grond motion intensit need to be seleted, in order to find better orrelation between the seismi grond motion intensit and the response of strtres. For long period RC strtral walls it is fond that the elasti response spetrm is a reliable measre of the grond motion intensit (Sasani and Der Kireghian, 00). For short period RC strtral walls, a new measre of grond motion intensit, alled signifiant peak grond aeleration, is fond to be well orrelated with the response of strtres nder severe plse-tpe grond motions (Sasani et. al., 00). Having the probabilisti models for demands and apaities and the proper measres of grond motion intensit, the fragilit of RC strtral walls an be estimated. 6 Smmar Inorporating mehanis of the shear and flexral behavior of RC walls, sing the Baesian parameter estimating tehniqe, and tilizing available experimental data, apait models for flexral deformation, shear strength, and shear deformation of RC strtral walls are developed. Signifiant errors observed in some available models in rrent seismi odes sggest a need for aonting for the model errors in probabilisti design of strtres. 7 Referenes Box, G. E.., and Cox, D. R., (964). An analsis of transformation, Jornal of the Roal Statistial Soiet. Series B (Methodologial), 6(), -5. Box, G. E.., and Tiao, G. C., (99). Baesian inferene in statistial analsis. Addison-Wesle, Reading, MA. Corle, W. G. (966). Rotational apait of reinfored beams, Jornal of the Strtral Division, ASCE, 9(0), -46. Der Kireghian, A. (999). A Baesian framework for fragilit assessment, ro. 8th Int. Conf. On Appliations of Statistis and robabilit (ICAS) in Civil Engineering Reliabilit and Risk Analsis, Sdne, Astralia, Deember 999, R. E. Melhers and M.G. Stewart, Eds.,, Fajfar,. (99). Eqivalent dtilit fators, taking into aont low-le fatige, Earthqake Engineering and Strtral Dnamis,, Geskens,., Der Kireghian, A., and Monteiro,., (993). BUM: Baesian pdating of model parameters. Report UCB/SEMM-93/06, Strtral 4

15 Engineering, Mehanis and Materials, Department of Civil Engineering, Universit of California, Berkele, CA. Kaar,. H., Fiorato, A. E., Carpenter, J. E., and Corle, W. G., (976). Confined onrete in ompression zones of strtral walls designed to resist lateral loads de to earthqakes, ro. International Smposim on Earthqake Strtral Engineering, St. Lois, MI, Mattok, A. H. (967). Disssion of "Rotational apait of reinfored onrete beams," b W.G. Corle, Jornal of the Strtral Division, ASCE, 93(): Oesterle, R. G., Fiorato, A. E., Johal, L. S., Carpenter, J. E., Rssell, H. E. and Corle, W. G., (976). Earthqake resistane strtral walls - tests of isolated walls, Constrtion Tehnolog Laboratories, CA, Skokie, IL, 35pp. Oesterle, R. G., Aristizabal, J. D., Shi, K. N., and Corle, W.G., (984). "Web Crshing of Reinfored Conrete Strtral Walls," ACI Jornal, 8(), 3-4. ark, R., riestle, M. J. N. and Gill, W. D. (98). Dtilit of sqare-onfined onrete olmns, Jornal of the Strtral Division, ASCE, 08(4), ark, Y. J. and A. H-S. Ang (985). Mehanisti seismi damage model for reinfored onrete, Jornal of Strtral Engineering, ASCE, (4), Sasani, M. (998). A two-level performane-based design of reinfored onrete strtral walls, ro. 6 th US National Conferene on Earthqake Engineering, Seattle, Washington, aper No. 78. Sasani, M. and Anderson, D. L. (996). Displaement-based design verss forebased design for strtral walls, ro. th World Conferene on Earthqake Engineering, Mexio, aper No. 3. Sasani, M. and Der Kireghian, A., (00). Fragilit of reinfored onrete strtral walls: displaement approah, Jornal of Strtral Engineering, ASCE, 7(), 9-8. Sasani, M., Der Kireghian, A., and Bertero, V. V. (00). Seismi fragilit of short period reinfored onrete strtral walls nder near-sore grond motions Strtral Safet, 4(-4), UBC (997). Uniform Bilding Code, Volme. International Conferene of Bilding Offiials, Whittier, CA. Vallenas, J. M., Bertero, V. V., and opov, E.. (979). Hstereti behavior of reinfored onrete strtral walls. Report UCB/EERC-79/0, Earthqake Engineering Researh Center, Universit of California, Berkele, CA. 5