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Seismic fragility of short period reinforced concrete structural walls under near-source ground motions
Structural Safety 24 (2002) 123–138 www.elsevier.com/locate/strusafe
Seismic fragility of short period reinforced concrete structural walls under near-source ground motions Mehrdad Sasania,*, Armen Der Kiureghianb, Vitelmo V. Berteroc a
Northeastern University, 441 Snell Engineering Center, Boston, MA 02115, USA b University of California, 721 Davis Hall, Berkeley, CA 94720, USA c University of California, PEER, 1301 S 46th St. Richmond, CA 94804, USA
Abstract The Bayesian parameter estimation technique is used to develop probabilistic displacement and strength capacity and demand models for reinforced concrete structural walls. Experimental data are used to develop the capacity models, and nonlinear dynamic analysis is employed to develop the demand models. Both flexural and shear failures are accounted for. These models are used to assess the seismic fragility of an example RC structural wall. As a new measure of the ground motion intensity, the significant peak ground acceleration is defined and incorporated in the probabilistic demand models and fragility assessment. It is shown that, for short period structures, this measure better correlates with the inelastic response than the elastic response spectrum. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Seismic fragility; Structural walls; Near-source ground motions; Probabilistic models; Shear strength; Displacement capacity; Bayesian technique; Short period structures; Measure of ground motion intensity
1. Introduction The seismic fragility of a structural system is defined as the conditional probability of failure of the system for a given intensity of the ground motion. In a performance-based seismic design or assessment approach, the ‘‘failure’’ event is said to have occurred when the structure fails to satisfy the requirements of a prescribed performance level, e.g. serviceability or life safety. If the intensity of the ground motion is expressed as a single variable, e.g. the peak ground acceleration or the spectral displacement corresponding to the period of the structure, the conditional probability of failure expressed as a function of the ground motion intensity is called a seismic fragility curve. The probability of failure over a specified period of time can then be obtained by an integration of the fragility and the seismic hazard curve at the location of the structure.
* Corresponding author. Tel.: +1-617-373-5222; fax: +1-617-373-4419. E-mail address: [email protected] (M. Sasani). 0167-4730/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0167-4730(02)00021-8
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The assessment of fragility ideally should employ as much objective information as possible. Such information is gained from fundamental laws of nature, e.g. laws of mechanics, and from laboratory and field observations. However, such information is often shrouded in uncertainties that arise from imperfections in our mathematical models, from measurement errors, and from the finite size of observed samples. When the available objective information is incomplete or insufficient, one must also rely on subjective information that is derived from engineering expert opinion. The Bayesian parameter estimation technique provides an effective tool for the development of probabilistic models and assessment of fragility under these conditions [1]. The method is capable of incorporating both types of information and it properly accounts for all prevailing uncertainties. In this paper, the Bayesian technique is employed to develop a seismic fragility model for short-period reinforced concrete (RC) structural walls under near-source ground motions at the life safety level. In assessing the seismic fragility of a structural system, three major pieces of information are required: One or more measures of the ground motion intensity, the seismic demand on the structural system, and the corresponding capacity of the system. Appropriate measures of the ground motion intensity depend on the characteristics of the ground motion and the structure, and on the performance level of interest. Under seismic ground motions, the damage in a structure depends on the force-deformation histories that are developed in its members. In general, the damage depends on the strength and deformation capacities and demands of each member, as well as the input and dissipated energies. However, for near-source events, where the ground motion is dominated by a few pulses, the effect of accumulative energy dissipation is less significant. Thus, with focus on near-source ground motions, in this paper probabilistic models are developed for the strength and deformation capacities and demands of structural walls, but the effect of dissipated energy is not considered. Data from experiments conducted on scaled wall models are used. The capacity models for both flexural behavior and shear strength are developed. The probabilistic model of the demand is developed by use of simulated data generated by nonlinear dynamic analyses of multi-degreeof-freedom (MDOF) structural wall systems and single-degree-of-freedom (SDOF) systems under recorded near-source ground motions. The deformation of a RC structural wall is contributed by both flexure and shear effects. Although the shear deformation in the plastic region of the wall can be considerable, for welldesigned typical walls with aspect ratios (ratio of wall height to plan length) greater than about 2, the contribution of the flexural deformation to the top displacement is much greater than that of the shear deformation. Other factors contributing to the displacement at the top of the wall include tension stiffening and bond slip of the reinforcement. However, these contributions are usually small and in opposite directions, so they tend to cancel out. On these bases, in the present study only the contribution of the flexural deformation to the lateral displacement of the wall is considered. In this sense, the capacity and demand models developed in this paper, as well as the resulting fragility estimates, are appropriate for RC structural walls with a medium to large aspect ratio.
2. Bayesian model assessment In this paper we make use of the Bayesian parameter estimation technique to develop probabilistic capacity and demand models for structural walls, which can be used to assess the seismic
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fragility of any specific wall with short period and medium to large aspect ratio. Details of the Bayesian technique can be found in the existing literature [1,2]. Here, only a brief outline is presented. Let y ¼ g^ ðx; Þ þ " be a mathematical model for predicting variable y in terms of a set of observable variables x ¼ ðx1 ; x2 ; . . .Þ, in which g^ðx; Þ is an idealized model (signified by the superposed hat), ¼ ð1 ; 2 ; . . .Þ is a set of unknown model parameters, and " is a random variable representing the unknown error in the model. We will assume that " has the normal distribution (normality assumption) and that it has a constant standard deviation (homoskedasticity assumption). If, for a given model g^ðx; Þ, these assumptions are not satisfied, then it is possible to make a transformation of the model such that these assumptions are at least approximately satisfied. Box and Cox [3] suggest a parametric family of transformations for this purpose. In the experimental results utilized in this paper, it is expected that the error in the capacity model will approximately increase linearly with the capacity. Furthermore, the capacity being non-negative, it is well represented by a lognormal distribution. Therefore, a logarithmic transformation is selected to approximately satisfy the normality and homoskedasticity assumptions. Finally, with the aim of developing an unbiased model, we assign a zero value to the mean of ". The set of unknown parameters of the model are H ¼ ð; Þ. The model is ‘‘assessed’’ by estimating H on the basis of available information, which typically consists of a set of measured values of x and the corresponding y, and possibly subjective information on the likely values of the parameters. In the Bayesian approach, this is done by the use of the well-known updating rule f ðH Þ ¼ cLðH ÞpðH Þ
ð2Þ
where p(H) denotes the prior distribution on H reflecting the subjective information, L(H) is the likelihood function, which is a function proportional to the conditional probability of making the observations on x and y for a given value of the parameters and reflects the objective information gained from the data, c is a normalizing factor, and f (H) is the posterior distribution reflecting our updated information about H: This rule is used to construct capacity and demand models and estimates of the fragility for RC structural walls on the basis of observed laboratory test data. Formulations of the prior distribution and the likelihood function for specific models are presented throughout the paper.
3. Seismic demand As defined earlier, seismic fragility is the conditional probability of failure given one or more measures of ground motion intensity. At the life safety performance level, structures undergo considerable inelastic displacement. One commonly used measure of the ground motion for estimating the seismic response of inelastic structures is the elastic response spectrum. In this section the applicability of this measure for short period structures subjected to near-source ground
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motions is critically reviewed and a more robust measure of the ground motion intensity is introduced. Bertero et al. [4] have discussed the characteristic difference between critical ground motions for elastic and inelastic responses. They point out that, whereas for elastic structures the critical ground motions are the ones that have large energy contents around the fundamental frequency of the structure (similar to harmonic motions), for inelastic structures the critical ground motions are those that have large acceleration pulses. Such pulses often occur in near-source ground motions. In order to investigate the importance of the acceleration pulses, the sixteen near-source ground motions listed in Table 1 are considered in this study. For short period structures, say with periods less than about 0.8 s, the inelastic displacement can be considerably larger than the displacement obtained based on an elastic model with the same initial period. This can be explained by the effect of acceleration pulses in the ground motion. To account for this effect, in this paper we introduce the significant peak ground acceleration (SPGA) as a new measure of the ground motion intensity. The SPGA is defined as the significant variation of ground velocity (SVGV) divided by its duration. The SVGV, in turn, is defined as the maximum variation of the ground velocity over a time period no less than 0.35 s. Fig. 1 shows the recorded ground acceleration and the calculated ground velocity during Erzincan, Turkey 1992 earthquake and the corresponding SPGA and SVGV. The last column of Table 1 lists the SPGA values for each of the recorded ground motions. Fig. 2(a) shows the relationship between the elastic acceleration response spectra (Sa) and the yield coefficient, Cy (inelastic acceleration spectra) for a SDOF system with period T=0.7 s and a displacement ductility =4 for the 16 ground motions. Fig. 2(b) shows the same relationship between the SPGA and Cy. In spite of the fact that the SPGA is independent of the period of the structure, there is a much stronger correlation between the SPGA and Cy than between Sa and Cy. While the correlation coefficient between the SPGA and Cy is about 0.93, the correlation Table 1 Sixteen near-source ground motions No.
Earthquake name
Year
Station
Comp.
PGA (g)
PGV (m/s)
PGD (m)
SPGA (g)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Tabas Imperial Valley Morgan Hill North Palm Spring Superstition Loma Prieta Cape Mendosino Cape Mendosino Landers Turkey Northridge Northridge Northridge Northridge Kobe Kobe
1978 1979 1984 1986 1987 1989 1992 1992 1992 1992 1994 1994 1994 1994 1995 1995
Tabas Array 10 Coyote North Palm Parachute Los Gatos Cape. Mend Petrolia Lucerne Erzincan Jensen (Gen) NewHall Rinaldi Sylmar (OV) JMA Takatori
TR 230 285 210 225 000 000 090 260 000 292 000 228 000 000 000
0.85 0.44 1.30 0.59 0.46 0.56 1.52 0.66 0.68 0.52 0.99 0.59 0.89 0.73 0.82 0.79
1.21 1.10 0.81 0.73 1.12 0.95 1.28 0.90 1.37 0.84 0.59 0.97 1.74 1.22 0.81 1.74
0.95 0.66 0.10 0.12 0.53 0.41 0.41 0.30 2.30 0.28 0.19 0.38 0.39 0.31 0.18 0.56
0.30 0.22 0.38 0.25 0.33 0.33 0.35 0.33 0.25 0.35 0.27 0.43 0.71 0.34 0.45 0.64
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Fig. 1. Erzincan ground motion.
coefficient between Sa and Cy is about 0.55. For a family of SDOF systems with 0.4 s
ð3Þ
where 1 , 2 , and 3 are model parameters. The superposed hat on Cy indicates that the model is idealized and subject to error. In order to have a normally distributed error term (with zero mean, if the model is to be unbiased) linearly added to the model and to have a constant coefficient of variation for Cy, (3) needs to be properly transformed. Considering the non-negative nature of Cy, the error term is added after taking logarithm from both sides of (3), yielding LnðCy Þ ¼ Ln½ð1 þ 2 T þ 3 ÞSPGA þ "cy
ð4Þ
To estimate the parameters of the model, first the inelastic response Cy of elastic-perfectly plastic SDOF systems with periods T=0.4, 0.5, 0.6, 0.7, and 0.8 s and displacement ductility ratios =4,
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Fig. 2. Correlation between Sa and Cy and SPGA and Cy for SDOF systems with T=0.7 s, =4, and =0.05.
6, and 8 under the 16 near-source ground motions listed in Table 1 are calculated. These data are then used in conjunction with the Bayesian updating rule to estimate the statistics of the model parameters 1 , 2 , 3 and the standard deviation cy of "cy . The posterior estimated means, standard deviations, and correlation coefficients are listed in Table 2. The application of the Bayesian technique is presented in more detail later when the shear strength capacity is modeled. As can be seen in Table 2, the posterior mean of cy is 0.191. If in (4) SPGA is replaced by Sa, the mean of the standard deviation increases to 0.362. Fig. 3 compares the errors in estimating Cy from (4) using the SPGA and Sa as the measures of the ground motion intensity. It is again evident that SPGA is a much better predictor of seismic demand than Sa. As shown above, SPGA can be used as an effective measure of the ground motion intensity for short-period structures at the life safety performance level. Although the same model can reasonably predict the response of structures with periods down to 0.2 s, for very stiff structures, PGA is a better measure of the ground motion intensity. It has been found that for such structures better predictions can be obtained by use of SPGA and PGA together. Specifically, for very shorter period structures, say with T40.2 s, the value of Cy can be predicted by an equation similar to (4) that depends on the PGA rather than the SPGA. For structures with 0.2 s< T<0.3 s one may interpolate between the value predicted by such an equation and that predicted by (4). Although for a short-period structure the displacement response can be represented by an equivalent SDOF system, this is not true for the estimation of the base shear [5]. Nonlinear
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M. Sasani et al. / Structural Safety 24 (2002) 123–138 Table 2 Posterior statistics of yield coefficient model parameters Parameter
1 2 3 cy
Mean
2.011 1.002 0.088 0.191
Standard deviation
0.0636 0.0781 0.0068 0.0088
Correlation coefficient 1
2
3
cy
1 0.73 0.64 0.00
1 0.06 0.00
1 0.00
1
Fig. 3. Error in predicting Cy, using: (a) Sa and (b) SPGA as measures of ground motion intensity.
analyses of short period structural walls under the sixteen near-source ground motions given in Table 1 is performed and it is found that the maximum base shear demand, Vd , can be estimated from the base moment capacity, M, by use of the model Vd ¼ e"Vd M=H
ð5Þ
where H is the total height of the wall. The posterior estimated means, standard deviations, and correlation coefficients of and Vd (standard deviation of Vd) are listed in Table 3.
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Table 3 Posterior statistics of maximum base shear demand model parameters Parameter
Mean
1.954 0.160
Vd
Standard deviation
0.271 0.036
Correlation coefficient
Vd
1 0.00
1
4. Seismic capacity The two main modes of failure of a structural wall are flexural and shear failures. Below, capacity models for these two modes are presented. 4.1. Flexural displacement and strength capacities As discussed in the introduction, the seismic response of a short-period structural wall is controlled by its strength and displacement capacities. A probabilistic displacement capacity model is developed by Sasani and Der Kiureghian [6]. Below a brief review of the model is presented. The flexural displacement capacity, f, can be estimated from the probabilistic model f ¼ e f ½ay H 2 þ ðu y ÞL p ðH L p =2Þ
ð6Þ
In this model, a is a displacement shape coefficient, y is the yield curvature, H is the height of the wall, L P is the mean estimate of the plastic hinge length from a model discussed in the above reference, u is the ultimate curvature capacity of the wall section that involves one parameter, , and the monotonic curvature capacity 0 u, which in turn involves the modified maximum useable concrete strain, ð"max c Þmod . The latter involves another unknown parameter, . Finally, "f is the random correction factor of the model, which is assumed to have the normal distribution with zero mean and unknown standard deviation f. This correction term includes not only the error in the form of the global model (6), but also the errors inherent in the sub-models for LP , y, u, 0 u and ð"max c Þmod . Posterior statistics of the parameters , and f are given in Sasani and Der Kiureghian [6]. Assuming that the critical section of the wall is at the base, the strength capacity associated with the displacement capacity is calculated from the flexural strength of the wall section at the base and by the first mode approximation. 4.2. Shear strength capacity The shear failure of structural walls may arise from any combination of sliding shear, web crushing and shear-compression failure of the compression zone. In order to develop a shear strength capacity model for structural walls, sixteen structural walls tested under cyclic loads are studied (see Table 4). The first nine walls failed in shear and the remaining seven had flexural failures.
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Preliminary studies with a shear strength model revealed weak correlation between the displacement ductility at the failure and the shear strength of the wall. Based on this observation, the following probabilistic model is considered for shear strength capacity, Vcap, of RC structural walls: qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi "Vc 0 0
1 aasp þ 2 P=Ag f c Vcap ¼ e f c fsc bl þ VS qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi VS ¼ h fyh bl 4 3 f 0c fsc bl ð7Þ In the above, 1, 2 and 3 are the model parameters and Vc is a normally distributed model error with zero mean and unknown standard deviation Vc. aasp accounts for the aspect ratio of the wall and linearly varies from 1.5 to 1.0 as the aspect ratio increases from 1.5 to 2.5. For aspect ratios larger than 2.5, aasp is set equal to 1.0. fsc is a scaling stress equal to 1 MPa (or its equivalent in other units), which is employed to make the parameters of the model dimensionless. P is the axial compression on the wall and Ag is the gross section area, b is the width of the web, l is the total length of the section, f 0c is the compressive strength of concrete and fyh is the yield stress of the horizontal reinforcement in the web. Finally, Vs is the shear strength corresponding to the horizontal reinforcement and has an upper bound in order to inhibit web crushing of the wall with a large amount of shear reinforcement. As mentioned in the introduction, the Bayesian parameter estimation technique can incorporate all the available information in the analysis. In this case, the experimental information available for predicting the shear strength capacity of the walls are of two kinds: Measured shear strength, when shear failure has been observed, and measured lower bound to the shear strength
Table 4 Sixteen structural walls and their failure modes No
Specimen
Reference
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
PCA/B2 PCA/B5 PCA/B6 PCA/B7 PCA/B8 PCA/B9 PCA/B11 PCA/B12 UCB/SW4 PCA/R1 PCA/R4 PCA/B1 PCA/B3 UCB/SW6 CU/RW1 CU/RW2
[16] [16] [16] [16] [16] [16] [17] [17] [18] [18] [17] [16] [16] [18] [19] [19]
Cross section
Failure mode Shear Shear Shear Shear Shear Shear Shear Shear Shear Flexural Flexural Flexural Flexural Flexural Flexural Flexural
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when the wall has failed in flexure. These two types of information are reflected in the likelihood function. Let qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 ^ ð8Þ Vcap ¼ 1 aasp þ 2 P=Ag f c f c fsc bl þ Vs denote the predicted shear strength capacity excluding the error term. In the kth experiment, given the set of observable variables (aasp, P, Ag, b, l, f 0c , fyh,Vs)k, ðV^ cap Þk is calculated from (8). Having the measured value of the maximum applied shear force on the section, the k-th realization of the error term is ð"Vc Þk ¼ ln Vcap k ln V^ cap ð9Þ k
Considering the normal distribution of the error term with a zero mean, and assuming statistical independence between the observations, the likelihood function takes the form Lð 1 ; 2 ; 3 ; Vc Þ ¼
Y Shear failure
Y 1 ð"Vc Þk ð"Vc Þk ’ 1 Vc Vc Vc Flexural Failure
ð10Þ
where the first product is for all the walls that failed in shear and the second product is for all the walls that failed in flexure. In the above expression ’ð Þ is the standard normal probability density function and ð Þ is the standard normal cumulative distribution function. Not having prior information on the parameters of the model, a non-informative prior distribution is used [2]. This essentially implies locally uniform distributions for 1, 2, 3 and ln( Vc). This prior distribution together with the likelihood function in (10) are used in the Bayesian updating formula to estimate the posterior statistics of the parameters. The computer program BUMP [7] is used for this purpose. The posterior statistics of the model parameters are listed in Table 5. As can be seen in this table, the standard deviation of the model error is small (equivalent to a coefficient of variation of about 0.051 in the capacity), which is an indication of the accuracy of the model. Based on the comparison between the mean values of 1 and 2, for a P/(Ag f 0c ) value of only 0.06, the effect of the axial load on the shear strength capacity of the wall is twice that of the first term on the right hand side of (8). The importance of the axial load on the shear strength of the wall is also reflected in the significant correlation between the shear deformation and the term P/(Ag f 0c ) [8]. The large negative correlation between a1 and a2 implies that the two terms can be combined with little loss of accuracy. This simplification is not used in this study. Fig. 4 shows the deviation between the measured and median predicted values (obtained by setting "vc ¼ 0) for the tested walls. The left part of the figure shows the errors for walls that failed in shear. The right part shows the deviation between the mean predicted value and the measured lower bound for walls that did not fail in shear. As expected, for walls that did not fail in shear, the corresponding points fall below the zero error line. The data points for walls numbers 10 and 15 fall below the limit of the figure and are
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M. Sasani et al. / Structural Safety 24 (2002) 123–138 Table 5 Posterior statistics of shear strength model parameters Parameter
1
2
3 Vc
Mean
0.067 2.240 0.500 0.051
Standard deviation
0.013 0.244 0.010 0.002
Correlation coefficient
1
2
3
Vc
1 0.64 0.02 0.00
1 0.00 0.00
1 0.00
1
Fig. 4. Normalized deviation between measured and mean predicted values.
not shown. Fig. 5 compares the measured and predicted median shear strength capacities for the tested walls. As can be seen, the data points for walls that failed in shear are closely lined up along the 1:1 line that represents equal values for the measured and predicted shear strengths. The data points for walls that did not fail in shear fall below the diagonal line, indicating that the predicted median shear strength capacities are larger than the maximum applied shear force.
5. Fragility assessment Having determined probabilistic models for displacement and shear capacity and demand, the seismic fragility of a given structural wall system is obtained as
[
Vcap Vd 4 0 SPGAÞ FðSPGAÞ ¼ P ðSPGAÞcap SPGA 4 0
ð11Þ
where (SPGA)cap is estimated from (4) using Cy, T, S and of the structure, PðEjSPGAÞ denotes the probability of event E for a given SPGA and denotes the union of events. Let g1 x; "cy ; "f ; H ; SPGA ¼ ðSPGAÞcap SPGA
ð12Þ
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Fig. 5. Measured versus mean predicted shear strength.
g2 x; "Vc ; "Vd ; "cy ; "f ; H; SPGA ¼ Vcap Vd
ð13Þ
denote the limit state functions for the failure event, where x represents the set of random variables describing the material properties of the wall, "cy, "f, "vc and "vd are the random error terms for the capacity and demand models, respectively, and H includes the standard deviation of the above four error terms along with nine parameters of the capacity and demand models in (4)–(7), as described earlier. Two distinct types of uncertainty are present in the above expressions: intrinsic variabilities, i.e. those present in x and "=("cy, "f, "vc, "vd), and epistemic (knowledgebased) uncertainties, i.e., those present in the model parameters H . Depending on how these uncertainties are treated, different estimates of the fragility can be obtained [1]. The simplest fragility estimate is obtained by using point-estimates of the model parameters, e.g. the posterior mean values H . The corresponding fragility point-estimate, denoted F ðSPGAÞ, is obtained by computing the integral ð f ðxÞ’ð"cy Þ’ð"f Þ’ð"Vc Þ’ð"Vd Þdxd" ð14Þ F ðSPGAÞ ¼ S fg1 4 0g fg2 4 0g 4 0 where fðxÞ is the probability density function of x. Note that the integration domain is the union of the two failure modes. This estimate does not include the effect of the epistemic uncertainties. Probability integrals such as (14) are common in structural reliability theory. Effective methods for their evaluation are available, including the first- and second-order reliability methods (FORM and SORM) and various simulation techniques [9]. Results shown in the subsequent application are obtained from FORM analysis, which in all cases were close to values obtained by Monte Carlo simulation. The general-purpose reliability analysis program CalREL [10] was used for the analysis. In some applications, it is desirable to treat the intrinsic and epistemic uncertainties separately. Specifically, it is desirable to determine the uncertainty in the fragility estimate arising from the
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epistemic uncertainty. This is done by treating the fragility (14) as a function F(SPGA,H ) of the uncertain model parameters H , and then computing its distribution using the posterior density f ðHÞ of the parameters. This requires nested reliability analysis. A simpler approach is to use first-order approximations to compute the variability in F(SPGA,H ). As pointed out by Der Kiureghian [11], it is more appropriate to carry out this kind of approximation for the reliability index functional b(SPGA,H )1[1F(SPGA,H )], in which 1[.] denotes the inverse of the standard normal probability function. This is because b(SPGA,H ) tends to be less nonlinear in H than F(SPGA,H ). Based on the mean-centered, first-order approximation method [12], the variance of b(SPGA,H ) for uncertain H is given by 2 ðSPGAÞ ffi rH T '(( rH
ð15Þ
where rH denotes the gradient vector of b(SPGA,H ) with respect to H at its posterior mean , and '(( denotes the posterior covariance matrix of H . The gradient vector is easily comH puted in the first-order reliability method [9]. Alternatively, it can be computed by finite-difference estimate are calculations. Computing , approximate 16 and 84% fractiles of the fragility Þ þ ðSPGAÞ and FðSPGAÞ ¼ ðSPGA; H obtained from the expressions: FðSPGAÞ 16% 84% Þ ðSPGAÞ respectively, whereas the 50% fractile is approximately given ¼ ðSPGA; H Þ. by FðSPGAÞ50% ¼ FðSPGA; H 6. Application As an example, the seismic fragility of a five-story RC structural wall system having the symmetric cross section shown in Fig. 6 is evaluated. The height of the wall is H ¼ 15 m and the section length is 4.0 m. A fixed axial compressive gravity load at the lower portion of the wall equal to 0:2ABE f 0c is assumed, where ABE is the area of one boundary element and f 0c is the nominal compressive strength of concrete. The mass associated with the lateral tributary area is assumed to be 2.5 times that of the vertical tributary area. The wall is designed based on seismic forces calculated from UBC [13] for a site 3.9 km (the average distance from the fault for the sixteen records used in this study) away from an active fault type A on a soil condition type D. Considering the nominal properties of the steel and concrete, the period of the structural system
Fig. 6. Cross section of example wall.
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is 0.6 s. The total lateral force coefficient was found to be 0.26. The wall is designed based on the conventional method described in ACI 318–99 [14]. The required flexural reinforcement ratio in the boundary elements is found to be BE=0.015. In developing the capacity models (6) and (7), we assumed that the laboratory measured values of the tensile yield strength, fy , and ultimate strength, fu , of the reinforcing steel, and of the nominal compressive strength of concrete, f 0c , accurately represented the material properties of the tested walls. In assessing the fragility for an existing or new wall, the exact values of these material property constants may not be known. For the example wall, it is assumed that the nominal value of the tensile yield strength of the steel reinforcement is 413 MPa. In order to account for the uncertainty in the yield strength, a lognormal distribution for fy with a mean value of 460 MPa and a coefficient of variation of 0.11 is used. The average ultimate strength of steel is assumed to be about 1.55 times its yield strength with a coefficient of variation equal to that of the yield strength. Also concrete with a nominal compressive strength f 0c ¼ 28 MPa is assumed. The modified Kent and Park model [15] is used to describe the stress–strain relation for both unconfined and confined concrete. The compressive strength of concrete is assumed to have a normal distribution with mean 26.5 MPa and a coefficient of variation of 0.15. (for further details on material properties see Sasani and Der Kiureghian [6]). 6.1. Fragility estimates Fig. 7 shows fragility estimates for the example wall including the effect of material variabilities. The thick solid curve shows the point-estimate of the fragility based on the mean values of the
Fig. 7. Fragility estimate for example wall.
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parameters, i.e. based on (14), which neglects the effect of the epistemic uncertainties. The thin solid curves are computed using the procedure employing (15). They approximately represent the 16 and 84% fragility bounds, reflecting the effect of the epistemic uncertainties on the fragility estimate. The dotted curve shows the fragility with respect to shear failure alone, i.e. limit state function given by (13). Since the shear capacity is dependent only on the strength, its fragility curve shows a steep slope at relatively low levels of the SPGA. Up to a SPGA of about 0.4 g, the failure is practically a shear one. For larger values of SPGA the effect of the flexural failure (dashed curve), i.e. limit state function given by (12), becomes more important.
7. Summary and conclusions Probabilistic models for displacement and strength capacities and for corresponding demands for RC structural walls are developed by use of the Bayesian model assessment techniques and employing laboratory test data. The model for the demand is developed by use of simulated data obtained by inelastic dynamic analysis with 16 near-source recorded ground motions. A new measure of the ground motion intensity called SPGA is defined and shown to be more strongly correlated with the inelastic response of short-period structures than elastic response spectra. Formulations for estimating fragility, with proper account of intrinsic and epistemic uncertainties, are presented. The methodology developed in this paper for assessing the fragility of RC structural walls is unique in the sense that it is based on experimental data, it properly and rigorously accounts for all the prevailing uncertainties of both intrinsic and epistemic nature, and it is transportable in the sense that the probabilistic models developed can be used to assess the fragility of any RC structural wall within the limitations of the assumed models, i.e., structural walls with short periods and with medium to large aspect ratios. An example demonstrates the application. References [1] Der Kiureghian A. A Bayesian framework for fragility assessment. In: Melchers RE, Stewart MG, editors. Proc. 8th Int. Conf. On Applications of Statistics and Probability (ICASP) in Civil Engineering Reliability and Risk Analysis, Sydney, Australia, December 1999, vol. 2. p. 1003–10. [2] Box GEP, Tiao GC. Bayesian inference in statistical analysis. Reading, MA: Addison-Wesley; 1992. [3] Box GEP, Cox DR. An analysis of transformation. Journal of the Royal Statistical Society Series B (Methodological) 1964;26(2):211–52. [4] Bertero VV, Herrera RA, Mahin SA. Establishment of design earthquakes-evaluation of present methods. In: International Symposium on Earthquake Structural Engineering, St. Louis, MI; 1976. [5] Eberhard MO, Sozen MA. Behavior-based method to determine design shear in earthquake-resistant walls. Journal of Structural Engineering, ASCE 1993;119(2):619–40. [6] Sasani M, Der Kiureghian A. Seismic fragility of reinforced concrete structural walls: a displacement approach. Journal of Structural Engineering, ASCE 2001;127(2):219–28. [7] Geyskens P, Der Kiureghian A, Monteiro P. BUMP: Bayesian updating of model parameters. Report UCB/ SEMM-93/06, Structural Engineering, Mechanics and Materials. Berkeley (CA): Department of Civil Engineering, University of California; 1993. [8] Sasani M. Reliability and performance-based seismic design, assessment, and rehabilitation of RC structures located near active faults. PhD dissertation, University of California, Berkeley; 2001.
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[9] Ditlevsen O, Madsen HO. Structural reliability methods. New York (NY): J. Wiley & Sons; 1996. [10] Liu P-L, Lin HZ, Der Kiureghian A. CALREL user manual. Report No. UCB/SEMM-89/18, Structural Engineering, Mechanics and Materials. Berkeley (CA): Department of Civil Engineering, University of California; 1989. [11] Der Kiureghian A. Measures of structural safety under imperfect states of knowledge. Journal of Structural Engineering, ASCE 1989;115(5):1119–40. [12] Ang AH-S, Tang W-H. Probability concepts in engineering planning and design, vol. I—basic principles. New York (NY): John Wiley & Sons; 1975. [13] UBC. Uniform Building Code, Volume 2. International Conference of Building Officials, Whittier (CA); 1997. [14] ACI 318–99. Building code requirements for structural concrete and commentary. Farmington Hills (MI): American Concrete Institute; 1999. [15] Park R, Priestley MJN, Gill WD. Ductility of square-confined concrete columns. Journal of the Structural Division, ASCE 1982;108(4):929–50. [16] Oesterle RG, Aristizabal JD, Fiorato AE, Russell HE, Corley WG. Earthquake resistance structural walls—tests of isolated walls-Phase II. Skokie, IL: Construction Technology Laboratories, PCA; 1979 325pp. [17] Oesterle RG. Inelastic analysis for in-plane strength of reinforced concrete shear walls. PhD dissertation, Northwestern University, Evanston, IL, 1986. 322 pp. [18] Vallenas JM, Bertero VV, Popov EP. Hysteretic behavior of reinforced concrete structural walls. Report UCB/ EERC-79/20. Berkeley (CA): Earthquake Engineering Research Center, University of California; 1979. [19] Thomsen IV JC, Wallace JW. Displacement-based design of RC structural walls: an experimental investigation of walls with rectangular and T-shape cross-sections, CU/CEE-95/06. Postdam (NY): Department of Civil Engineering, Clarkson Universit; 1995.
Seismic fragility of short period reinforced concrete structural walls under near-source ground motions Mehrdad Sasania,*, Armen Der Kiureghianb, Vitelmo V. Berteroc a
Northeastern University, 441 Snell Engineering Center, Boston, MA 02115, USA b University of California, 721 Davis Hall, Berkeley, CA 94720, USA c University of California, PEER, 1301 S 46th St. Richmond, CA 94804, USA
Abstract The Bayesian parameter estimation technique is used to develop probabilistic displacement and strength capacity and demand models for reinforced concrete structural walls. Experimental data are used to develop the capacity models, and nonlinear dynamic analysis is employed to develop the demand models. Both flexural and shear failures are accounted for. These models are used to assess the seismic fragility of an example RC structural wall. As a new measure of the ground motion intensity, the significant peak ground acceleration is defined and incorporated in the probabilistic demand models and fragility assessment. It is shown that, for short period structures, this measure better correlates with the inelastic response than the elastic response spectrum. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Seismic fragility; Structural walls; Near-source ground motions; Probabilistic models; Shear strength; Displacement capacity; Bayesian technique; Short period structures; Measure of ground motion intensity
1. Introduction The seismic fragility of a structural system is defined as the conditional probability of failure of the system for a given intensity of the ground motion. In a performance-based seismic design or assessment approach, the ‘‘failure’’ event is said to have occurred when the structure fails to satisfy the requirements of a prescribed performance level, e.g. serviceability or life safety. If the intensity of the ground motion is expressed as a single variable, e.g. the peak ground acceleration or the spectral displacement corresponding to the period of the structure, the conditional probability of failure expressed as a function of the ground motion intensity is called a seismic fragility curve. The probability of failure over a specified period of time can then be obtained by an integration of the fragility and the seismic hazard curve at the location of the structure.
* Corresponding author. Tel.: +1-617-373-5222; fax: +1-617-373-4419. E-mail address: [email protected] (M. Sasani). 0167-4730/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0167-4730(02)00021-8
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The assessment of fragility ideally should employ as much objective information as possible. Such information is gained from fundamental laws of nature, e.g. laws of mechanics, and from laboratory and field observations. However, such information is often shrouded in uncertainties that arise from imperfections in our mathematical models, from measurement errors, and from the finite size of observed samples. When the available objective information is incomplete or insufficient, one must also rely on subjective information that is derived from engineering expert opinion. The Bayesian parameter estimation technique provides an effective tool for the development of probabilistic models and assessment of fragility under these conditions [1]. The method is capable of incorporating both types of information and it properly accounts for all prevailing uncertainties. In this paper, the Bayesian technique is employed to develop a seismic fragility model for short-period reinforced concrete (RC) structural walls under near-source ground motions at the life safety level. In assessing the seismic fragility of a structural system, three major pieces of information are required: One or more measures of the ground motion intensity, the seismic demand on the structural system, and the corresponding capacity of the system. Appropriate measures of the ground motion intensity depend on the characteristics of the ground motion and the structure, and on the performance level of interest. Under seismic ground motions, the damage in a structure depends on the force-deformation histories that are developed in its members. In general, the damage depends on the strength and deformation capacities and demands of each member, as well as the input and dissipated energies. However, for near-source events, where the ground motion is dominated by a few pulses, the effect of accumulative energy dissipation is less significant. Thus, with focus on near-source ground motions, in this paper probabilistic models are developed for the strength and deformation capacities and demands of structural walls, but the effect of dissipated energy is not considered. Data from experiments conducted on scaled wall models are used. The capacity models for both flexural behavior and shear strength are developed. The probabilistic model of the demand is developed by use of simulated data generated by nonlinear dynamic analyses of multi-degreeof-freedom (MDOF) structural wall systems and single-degree-of-freedom (SDOF) systems under recorded near-source ground motions. The deformation of a RC structural wall is contributed by both flexure and shear effects. Although the shear deformation in the plastic region of the wall can be considerable, for welldesigned typical walls with aspect ratios (ratio of wall height to plan length) greater than about 2, the contribution of the flexural deformation to the top displacement is much greater than that of the shear deformation. Other factors contributing to the displacement at the top of the wall include tension stiffening and bond slip of the reinforcement. However, these contributions are usually small and in opposite directions, so they tend to cancel out. On these bases, in the present study only the contribution of the flexural deformation to the lateral displacement of the wall is considered. In this sense, the capacity and demand models developed in this paper, as well as the resulting fragility estimates, are appropriate for RC structural walls with a medium to large aspect ratio.
2. Bayesian model assessment In this paper we make use of the Bayesian parameter estimation technique to develop probabilistic capacity and demand models for structural walls, which can be used to assess the seismic
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fragility of any specific wall with short period and medium to large aspect ratio. Details of the Bayesian technique can be found in the existing literature [1,2]. Here, only a brief outline is presented. Let y ¼ g^ ðx; Þ þ " be a mathematical model for predicting variable y in terms of a set of observable variables x ¼ ðx1 ; x2 ; . . .Þ, in which g^ðx; Þ is an idealized model (signified by the superposed hat), ¼ ð1 ; 2 ; . . .Þ is a set of unknown model parameters, and " is a random variable representing the unknown error in the model. We will assume that " has the normal distribution (normality assumption) and that it has a constant standard deviation (homoskedasticity assumption). If, for a given model g^ðx; Þ, these assumptions are not satisfied, then it is possible to make a transformation of the model such that these assumptions are at least approximately satisfied. Box and Cox [3] suggest a parametric family of transformations for this purpose. In the experimental results utilized in this paper, it is expected that the error in the capacity model will approximately increase linearly with the capacity. Furthermore, the capacity being non-negative, it is well represented by a lognormal distribution. Therefore, a logarithmic transformation is selected to approximately satisfy the normality and homoskedasticity assumptions. Finally, with the aim of developing an unbiased model, we assign a zero value to the mean of ". The set of unknown parameters of the model are H ¼ ð; Þ. The model is ‘‘assessed’’ by estimating H on the basis of available information, which typically consists of a set of measured values of x and the corresponding y, and possibly subjective information on the likely values of the parameters. In the Bayesian approach, this is done by the use of the well-known updating rule f ðH Þ ¼ cLðH ÞpðH Þ
ð2Þ
where p(H) denotes the prior distribution on H reflecting the subjective information, L(H) is the likelihood function, which is a function proportional to the conditional probability of making the observations on x and y for a given value of the parameters and reflects the objective information gained from the data, c is a normalizing factor, and f (H) is the posterior distribution reflecting our updated information about H: This rule is used to construct capacity and demand models and estimates of the fragility for RC structural walls on the basis of observed laboratory test data. Formulations of the prior distribution and the likelihood function for specific models are presented throughout the paper.
3. Seismic demand As defined earlier, seismic fragility is the conditional probability of failure given one or more measures of ground motion intensity. At the life safety performance level, structures undergo considerable inelastic displacement. One commonly used measure of the ground motion for estimating the seismic response of inelastic structures is the elastic response spectrum. In this section the applicability of this measure for short period structures subjected to near-source ground
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motions is critically reviewed and a more robust measure of the ground motion intensity is introduced. Bertero et al. [4] have discussed the characteristic difference between critical ground motions for elastic and inelastic responses. They point out that, whereas for elastic structures the critical ground motions are the ones that have large energy contents around the fundamental frequency of the structure (similar to harmonic motions), for inelastic structures the critical ground motions are those that have large acceleration pulses. Such pulses often occur in near-source ground motions. In order to investigate the importance of the acceleration pulses, the sixteen near-source ground motions listed in Table 1 are considered in this study. For short period structures, say with periods less than about 0.8 s, the inelastic displacement can be considerably larger than the displacement obtained based on an elastic model with the same initial period. This can be explained by the effect of acceleration pulses in the ground motion. To account for this effect, in this paper we introduce the significant peak ground acceleration (SPGA) as a new measure of the ground motion intensity. The SPGA is defined as the significant variation of ground velocity (SVGV) divided by its duration. The SVGV, in turn, is defined as the maximum variation of the ground velocity over a time period no less than 0.35 s. Fig. 1 shows the recorded ground acceleration and the calculated ground velocity during Erzincan, Turkey 1992 earthquake and the corresponding SPGA and SVGV. The last column of Table 1 lists the SPGA values for each of the recorded ground motions. Fig. 2(a) shows the relationship between the elastic acceleration response spectra (Sa) and the yield coefficient, Cy (inelastic acceleration spectra) for a SDOF system with period T=0.7 s and a displacement ductility =4 for the 16 ground motions. Fig. 2(b) shows the same relationship between the SPGA and Cy. In spite of the fact that the SPGA is independent of the period of the structure, there is a much stronger correlation between the SPGA and Cy than between Sa and Cy. While the correlation coefficient between the SPGA and Cy is about 0.93, the correlation Table 1 Sixteen near-source ground motions No.
Earthquake name
Year
Station
Comp.
PGA (g)
PGV (m/s)
PGD (m)
SPGA (g)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Tabas Imperial Valley Morgan Hill North Palm Spring Superstition Loma Prieta Cape Mendosino Cape Mendosino Landers Turkey Northridge Northridge Northridge Northridge Kobe Kobe
1978 1979 1984 1986 1987 1989 1992 1992 1992 1992 1994 1994 1994 1994 1995 1995
Tabas Array 10 Coyote North Palm Parachute Los Gatos Cape. Mend Petrolia Lucerne Erzincan Jensen (Gen) NewHall Rinaldi Sylmar (OV) JMA Takatori
TR 230 285 210 225 000 000 090 260 000 292 000 228 000 000 000
0.85 0.44 1.30 0.59 0.46 0.56 1.52 0.66 0.68 0.52 0.99 0.59 0.89 0.73 0.82 0.79
1.21 1.10 0.81 0.73 1.12 0.95 1.28 0.90 1.37 0.84 0.59 0.97 1.74 1.22 0.81 1.74
0.95 0.66 0.10 0.12 0.53 0.41 0.41 0.30 2.30 0.28 0.19 0.38 0.39 0.31 0.18 0.56
0.30 0.22 0.38 0.25 0.33 0.33 0.35 0.33 0.25 0.35 0.27 0.43 0.71 0.34 0.45 0.64
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Fig. 1. Erzincan ground motion.
coefficient between Sa and Cy is about 0.55. For a family of SDOF systems with 0.4 s
ð3Þ
where 1 , 2 , and 3 are model parameters. The superposed hat on Cy indicates that the model is idealized and subject to error. In order to have a normally distributed error term (with zero mean, if the model is to be unbiased) linearly added to the model and to have a constant coefficient of variation for Cy, (3) needs to be properly transformed. Considering the non-negative nature of Cy, the error term is added after taking logarithm from both sides of (3), yielding LnðCy Þ ¼ Ln½ð1 þ 2 T þ 3 ÞSPGA þ "cy
ð4Þ
To estimate the parameters of the model, first the inelastic response Cy of elastic-perfectly plastic SDOF systems with periods T=0.4, 0.5, 0.6, 0.7, and 0.8 s and displacement ductility ratios =4,
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Fig. 2. Correlation between Sa and Cy and SPGA and Cy for SDOF systems with T=0.7 s, =4, and =0.05.
6, and 8 under the 16 near-source ground motions listed in Table 1 are calculated. These data are then used in conjunction with the Bayesian updating rule to estimate the statistics of the model parameters 1 , 2 , 3 and the standard deviation cy of "cy . The posterior estimated means, standard deviations, and correlation coefficients are listed in Table 2. The application of the Bayesian technique is presented in more detail later when the shear strength capacity is modeled. As can be seen in Table 2, the posterior mean of cy is 0.191. If in (4) SPGA is replaced by Sa, the mean of the standard deviation increases to 0.362. Fig. 3 compares the errors in estimating Cy from (4) using the SPGA and Sa as the measures of the ground motion intensity. It is again evident that SPGA is a much better predictor of seismic demand than Sa. As shown above, SPGA can be used as an effective measure of the ground motion intensity for short-period structures at the life safety performance level. Although the same model can reasonably predict the response of structures with periods down to 0.2 s, for very stiff structures, PGA is a better measure of the ground motion intensity. It has been found that for such structures better predictions can be obtained by use of SPGA and PGA together. Specifically, for very shorter period structures, say with T40.2 s, the value of Cy can be predicted by an equation similar to (4) that depends on the PGA rather than the SPGA. For structures with 0.2 s< T<0.3 s one may interpolate between the value predicted by such an equation and that predicted by (4). Although for a short-period structure the displacement response can be represented by an equivalent SDOF system, this is not true for the estimation of the base shear [5]. Nonlinear
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M. Sasani et al. / Structural Safety 24 (2002) 123–138 Table 2 Posterior statistics of yield coefficient model parameters Parameter
1 2 3 cy
Mean
2.011 1.002 0.088 0.191
Standard deviation
0.0636 0.0781 0.0068 0.0088
Correlation coefficient 1
2
3
cy
1 0.73 0.64 0.00
1 0.06 0.00
1 0.00
1
Fig. 3. Error in predicting Cy, using: (a) Sa and (b) SPGA as measures of ground motion intensity.
analyses of short period structural walls under the sixteen near-source ground motions given in Table 1 is performed and it is found that the maximum base shear demand, Vd , can be estimated from the base moment capacity, M, by use of the model Vd ¼ e"Vd M=H
ð5Þ
where H is the total height of the wall. The posterior estimated means, standard deviations, and correlation coefficients of and Vd (standard deviation of Vd) are listed in Table 3.
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Table 3 Posterior statistics of maximum base shear demand model parameters Parameter
Mean
1.954 0.160
Vd
Standard deviation
0.271 0.036
Correlation coefficient
Vd
1 0.00
1
4. Seismic capacity The two main modes of failure of a structural wall are flexural and shear failures. Below, capacity models for these two modes are presented. 4.1. Flexural displacement and strength capacities As discussed in the introduction, the seismic response of a short-period structural wall is controlled by its strength and displacement capacities. A probabilistic displacement capacity model is developed by Sasani and Der Kiureghian [6]. Below a brief review of the model is presented. The flexural displacement capacity, f, can be estimated from the probabilistic model f ¼ e f ½ay H 2 þ ðu y ÞL p ðH L p =2Þ
ð6Þ
In this model, a is a displacement shape coefficient, y is the yield curvature, H is the height of the wall, L P is the mean estimate of the plastic hinge length from a model discussed in the above reference, u is the ultimate curvature capacity of the wall section that involves one parameter, , and the monotonic curvature capacity 0 u, which in turn involves the modified maximum useable concrete strain, ð"max c Þmod . The latter involves another unknown parameter, . Finally, "f is the random correction factor of the model, which is assumed to have the normal distribution with zero mean and unknown standard deviation f. This correction term includes not only the error in the form of the global model (6), but also the errors inherent in the sub-models for LP , y, u, 0 u and ð"max c Þmod . Posterior statistics of the parameters , and f are given in Sasani and Der Kiureghian [6]. Assuming that the critical section of the wall is at the base, the strength capacity associated with the displacement capacity is calculated from the flexural strength of the wall section at the base and by the first mode approximation. 4.2. Shear strength capacity The shear failure of structural walls may arise from any combination of sliding shear, web crushing and shear-compression failure of the compression zone. In order to develop a shear strength capacity model for structural walls, sixteen structural walls tested under cyclic loads are studied (see Table 4). The first nine walls failed in shear and the remaining seven had flexural failures.
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Preliminary studies with a shear strength model revealed weak correlation between the displacement ductility at the failure and the shear strength of the wall. Based on this observation, the following probabilistic model is considered for shear strength capacity, Vcap, of RC structural walls: qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi "Vc 0 0
1 aasp þ 2 P=Ag f c Vcap ¼ e f c fsc bl þ VS qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi VS ¼ h fyh bl 4 3 f 0c fsc bl ð7Þ In the above, 1, 2 and 3 are the model parameters and Vc is a normally distributed model error with zero mean and unknown standard deviation Vc. aasp accounts for the aspect ratio of the wall and linearly varies from 1.5 to 1.0 as the aspect ratio increases from 1.5 to 2.5. For aspect ratios larger than 2.5, aasp is set equal to 1.0. fsc is a scaling stress equal to 1 MPa (or its equivalent in other units), which is employed to make the parameters of the model dimensionless. P is the axial compression on the wall and Ag is the gross section area, b is the width of the web, l is the total length of the section, f 0c is the compressive strength of concrete and fyh is the yield stress of the horizontal reinforcement in the web. Finally, Vs is the shear strength corresponding to the horizontal reinforcement and has an upper bound in order to inhibit web crushing of the wall with a large amount of shear reinforcement. As mentioned in the introduction, the Bayesian parameter estimation technique can incorporate all the available information in the analysis. In this case, the experimental information available for predicting the shear strength capacity of the walls are of two kinds: Measured shear strength, when shear failure has been observed, and measured lower bound to the shear strength
Table 4 Sixteen structural walls and their failure modes No
Specimen
Reference
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
PCA/B2 PCA/B5 PCA/B6 PCA/B7 PCA/B8 PCA/B9 PCA/B11 PCA/B12 UCB/SW4 PCA/R1 PCA/R4 PCA/B1 PCA/B3 UCB/SW6 CU/RW1 CU/RW2
[16] [16] [16] [16] [16] [16] [17] [17] [18] [18] [17] [16] [16] [18] [19] [19]
Cross section
Failure mode Shear Shear Shear Shear Shear Shear Shear Shear Shear Flexural Flexural Flexural Flexural Flexural Flexural Flexural
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when the wall has failed in flexure. These two types of information are reflected in the likelihood function. Let qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 ^ ð8Þ Vcap ¼ 1 aasp þ 2 P=Ag f c f c fsc bl þ Vs denote the predicted shear strength capacity excluding the error term. In the kth experiment, given the set of observable variables (aasp, P, Ag, b, l, f 0c , fyh,Vs)k, ðV^ cap Þk is calculated from (8). Having the measured value of the maximum applied shear force on the section, the k-th realization of the error term is ð"Vc Þk ¼ ln Vcap k ln V^ cap ð9Þ k
Considering the normal distribution of the error term with a zero mean, and assuming statistical independence between the observations, the likelihood function takes the form Lð 1 ; 2 ; 3 ; Vc Þ ¼
Y Shear failure
Y 1 ð"Vc Þk ð"Vc Þk ’ 1 Vc Vc Vc Flexural Failure
ð10Þ
where the first product is for all the walls that failed in shear and the second product is for all the walls that failed in flexure. In the above expression ’ð Þ is the standard normal probability density function and ð Þ is the standard normal cumulative distribution function. Not having prior information on the parameters of the model, a non-informative prior distribution is used [2]. This essentially implies locally uniform distributions for 1, 2, 3 and ln( Vc). This prior distribution together with the likelihood function in (10) are used in the Bayesian updating formula to estimate the posterior statistics of the parameters. The computer program BUMP [7] is used for this purpose. The posterior statistics of the model parameters are listed in Table 5. As can be seen in this table, the standard deviation of the model error is small (equivalent to a coefficient of variation of about 0.051 in the capacity), which is an indication of the accuracy of the model. Based on the comparison between the mean values of 1 and 2, for a P/(Ag f 0c ) value of only 0.06, the effect of the axial load on the shear strength capacity of the wall is twice that of the first term on the right hand side of (8). The importance of the axial load on the shear strength of the wall is also reflected in the significant correlation between the shear deformation and the term P/(Ag f 0c ) [8]. The large negative correlation between a1 and a2 implies that the two terms can be combined with little loss of accuracy. This simplification is not used in this study. Fig. 4 shows the deviation between the measured and median predicted values (obtained by setting "vc ¼ 0) for the tested walls. The left part of the figure shows the errors for walls that failed in shear. The right part shows the deviation between the mean predicted value and the measured lower bound for walls that did not fail in shear. As expected, for walls that did not fail in shear, the corresponding points fall below the zero error line. The data points for walls numbers 10 and 15 fall below the limit of the figure and are
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M. Sasani et al. / Structural Safety 24 (2002) 123–138 Table 5 Posterior statistics of shear strength model parameters Parameter
1
2
3 Vc
Mean
0.067 2.240 0.500 0.051
Standard deviation
0.013 0.244 0.010 0.002
Correlation coefficient
1
2
3
Vc
1 0.64 0.02 0.00
1 0.00 0.00
1 0.00
1
Fig. 4. Normalized deviation between measured and mean predicted values.
not shown. Fig. 5 compares the measured and predicted median shear strength capacities for the tested walls. As can be seen, the data points for walls that failed in shear are closely lined up along the 1:1 line that represents equal values for the measured and predicted shear strengths. The data points for walls that did not fail in shear fall below the diagonal line, indicating that the predicted median shear strength capacities are larger than the maximum applied shear force.
5. Fragility assessment Having determined probabilistic models for displacement and shear capacity and demand, the seismic fragility of a given structural wall system is obtained as
[
Vcap Vd 4 0 SPGAÞ FðSPGAÞ ¼ P ðSPGAÞcap SPGA 4 0
ð11Þ
where (SPGA)cap is estimated from (4) using Cy, T, S and of the structure, PðEjSPGAÞ denotes the probability of event E for a given SPGA and denotes the union of events. Let g1 x; "cy ; "f ; H ; SPGA ¼ ðSPGAÞcap SPGA
ð12Þ
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Fig. 5. Measured versus mean predicted shear strength.
g2 x; "Vc ; "Vd ; "cy ; "f ; H; SPGA ¼ Vcap Vd
ð13Þ
denote the limit state functions for the failure event, where x represents the set of random variables describing the material properties of the wall, "cy, "f, "vc and "vd are the random error terms for the capacity and demand models, respectively, and H includes the standard deviation of the above four error terms along with nine parameters of the capacity and demand models in (4)–(7), as described earlier. Two distinct types of uncertainty are present in the above expressions: intrinsic variabilities, i.e. those present in x and "=("cy, "f, "vc, "vd), and epistemic (knowledgebased) uncertainties, i.e., those present in the model parameters H . Depending on how these uncertainties are treated, different estimates of the fragility can be obtained [1]. The simplest fragility estimate is obtained by using point-estimates of the model parameters, e.g. the posterior mean values H . The corresponding fragility point-estimate, denoted F ðSPGAÞ, is obtained by computing the integral ð f ðxÞ’ð"cy Þ’ð"f Þ’ð"Vc Þ’ð"Vd Þdxd" ð14Þ F ðSPGAÞ ¼ S fg1 4 0g fg2 4 0g 4 0 where fðxÞ is the probability density function of x. Note that the integration domain is the union of the two failure modes. This estimate does not include the effect of the epistemic uncertainties. Probability integrals such as (14) are common in structural reliability theory. Effective methods for their evaluation are available, including the first- and second-order reliability methods (FORM and SORM) and various simulation techniques [9]. Results shown in the subsequent application are obtained from FORM analysis, which in all cases were close to values obtained by Monte Carlo simulation. The general-purpose reliability analysis program CalREL [10] was used for the analysis. In some applications, it is desirable to treat the intrinsic and epistemic uncertainties separately. Specifically, it is desirable to determine the uncertainty in the fragility estimate arising from the
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epistemic uncertainty. This is done by treating the fragility (14) as a function F(SPGA,H ) of the uncertain model parameters H , and then computing its distribution using the posterior density f ðHÞ of the parameters. This requires nested reliability analysis. A simpler approach is to use first-order approximations to compute the variability in F(SPGA,H ). As pointed out by Der Kiureghian [11], it is more appropriate to carry out this kind of approximation for the reliability index functional b(SPGA,H )1[1F(SPGA,H )], in which 1[.] denotes the inverse of the standard normal probability function. This is because b(SPGA,H ) tends to be less nonlinear in H than F(SPGA,H ). Based on the mean-centered, first-order approximation method [12], the variance of b(SPGA,H ) for uncertain H is given by 2 ðSPGAÞ ffi rH T '(( rH
ð15Þ
where rH denotes the gradient vector of b(SPGA,H ) with respect to H at its posterior mean , and '(( denotes the posterior covariance matrix of H . The gradient vector is easily comH puted in the first-order reliability method [9]. Alternatively, it can be computed by finite-difference estimate are calculations. Computing , approximate 16 and 84% fractiles of the fragility Þ þ ðSPGAÞ and FðSPGAÞ ¼ ðSPGA; H obtained from the expressions: FðSPGAÞ 16% 84% Þ ðSPGAÞ respectively, whereas the 50% fractile is approximately given ¼ ðSPGA; H Þ. by FðSPGAÞ50% ¼ FðSPGA; H 6. Application As an example, the seismic fragility of a five-story RC structural wall system having the symmetric cross section shown in Fig. 6 is evaluated. The height of the wall is H ¼ 15 m and the section length is 4.0 m. A fixed axial compressive gravity load at the lower portion of the wall equal to 0:2ABE f 0c is assumed, where ABE is the area of one boundary element and f 0c is the nominal compressive strength of concrete. The mass associated with the lateral tributary area is assumed to be 2.5 times that of the vertical tributary area. The wall is designed based on seismic forces calculated from UBC [13] for a site 3.9 km (the average distance from the fault for the sixteen records used in this study) away from an active fault type A on a soil condition type D. Considering the nominal properties of the steel and concrete, the period of the structural system
Fig. 6. Cross section of example wall.
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is 0.6 s. The total lateral force coefficient was found to be 0.26. The wall is designed based on the conventional method described in ACI 318–99 [14]. The required flexural reinforcement ratio in the boundary elements is found to be BE=0.015. In developing the capacity models (6) and (7), we assumed that the laboratory measured values of the tensile yield strength, fy , and ultimate strength, fu , of the reinforcing steel, and of the nominal compressive strength of concrete, f 0c , accurately represented the material properties of the tested walls. In assessing the fragility for an existing or new wall, the exact values of these material property constants may not be known. For the example wall, it is assumed that the nominal value of the tensile yield strength of the steel reinforcement is 413 MPa. In order to account for the uncertainty in the yield strength, a lognormal distribution for fy with a mean value of 460 MPa and a coefficient of variation of 0.11 is used. The average ultimate strength of steel is assumed to be about 1.55 times its yield strength with a coefficient of variation equal to that of the yield strength. Also concrete with a nominal compressive strength f 0c ¼ 28 MPa is assumed. The modified Kent and Park model [15] is used to describe the stress–strain relation for both unconfined and confined concrete. The compressive strength of concrete is assumed to have a normal distribution with mean 26.5 MPa and a coefficient of variation of 0.15. (for further details on material properties see Sasani and Der Kiureghian [6]). 6.1. Fragility estimates Fig. 7 shows fragility estimates for the example wall including the effect of material variabilities. The thick solid curve shows the point-estimate of the fragility based on the mean values of the
Fig. 7. Fragility estimate for example wall.
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parameters, i.e. based on (14), which neglects the effect of the epistemic uncertainties. The thin solid curves are computed using the procedure employing (15). They approximately represent the 16 and 84% fragility bounds, reflecting the effect of the epistemic uncertainties on the fragility estimate. The dotted curve shows the fragility with respect to shear failure alone, i.e. limit state function given by (13). Since the shear capacity is dependent only on the strength, its fragility curve shows a steep slope at relatively low levels of the SPGA. Up to a SPGA of about 0.4 g, the failure is practically a shear one. For larger values of SPGA the effect of the flexural failure (dashed curve), i.e. limit state function given by (12), becomes more important.
7. Summary and conclusions Probabilistic models for displacement and strength capacities and for corresponding demands for RC structural walls are developed by use of the Bayesian model assessment techniques and employing laboratory test data. The model for the demand is developed by use of simulated data obtained by inelastic dynamic analysis with 16 near-source recorded ground motions. A new measure of the ground motion intensity called SPGA is defined and shown to be more strongly correlated with the inelastic response of short-period structures than elastic response spectra. Formulations for estimating fragility, with proper account of intrinsic and epistemic uncertainties, are presented. The methodology developed in this paper for assessing the fragility of RC structural walls is unique in the sense that it is based on experimental data, it properly and rigorously accounts for all the prevailing uncertainties of both intrinsic and epistemic nature, and it is transportable in the sense that the probabilistic models developed can be used to assess the fragility of any RC structural wall within the limitations of the assumed models, i.e., structural walls with short periods and with medium to large aspect ratios. An example demonstrates the application. References [1] Der Kiureghian A. A Bayesian framework for fragility assessment. In: Melchers RE, Stewart MG, editors. Proc. 8th Int. Conf. On Applications of Statistics and Probability (ICASP) in Civil Engineering Reliability and Risk Analysis, Sydney, Australia, December 1999, vol. 2. p. 1003–10. [2] Box GEP, Tiao GC. Bayesian inference in statistical analysis. Reading, MA: Addison-Wesley; 1992. [3] Box GEP, Cox DR. An analysis of transformation. Journal of the Royal Statistical Society Series B (Methodological) 1964;26(2):211–52. [4] Bertero VV, Herrera RA, Mahin SA. Establishment of design earthquakes-evaluation of present methods. In: International Symposium on Earthquake Structural Engineering, St. Louis, MI; 1976. [5] Eberhard MO, Sozen MA. Behavior-based method to determine design shear in earthquake-resistant walls. Journal of Structural Engineering, ASCE 1993;119(2):619–40. [6] Sasani M, Der Kiureghian A. Seismic fragility of reinforced concrete structural walls: a displacement approach. Journal of Structural Engineering, ASCE 2001;127(2):219–28. [7] Geyskens P, Der Kiureghian A, Monteiro P. BUMP: Bayesian updating of model parameters. Report UCB/ SEMM-93/06, Structural Engineering, Mechanics and Materials. Berkeley (CA): Department of Civil Engineering, University of California; 1993. [8] Sasani M. Reliability and performance-based seismic design, assessment, and rehabilitation of RC structures located near active faults. PhD dissertation, University of California, Berkeley; 2001.
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[9] Ditlevsen O, Madsen HO. Structural reliability methods. New York (NY): J. Wiley & Sons; 1996. [10] Liu P-L, Lin HZ, Der Kiureghian A. CALREL user manual. Report No. UCB/SEMM-89/18, Structural Engineering, Mechanics and Materials. Berkeley (CA): Department of Civil Engineering, University of California; 1989. [11] Der Kiureghian A. Measures of structural safety under imperfect states of knowledge. Journal of Structural Engineering, ASCE 1989;115(5):1119–40. [12] Ang AH-S, Tang W-H. Probability concepts in engineering planning and design, vol. I—basic principles. New York (NY): John Wiley & Sons; 1975. [13] UBC. Uniform Building Code, Volume 2. International Conference of Building Officials, Whittier (CA); 1997. [14] ACI 318–99. Building code requirements for structural concrete and commentary. Farmington Hills (MI): American Concrete Institute; 1999. [15] Park R, Priestley MJN, Gill WD. Ductility of square-confined concrete columns. Journal of the Structural Division, ASCE 1982;108(4):929–50. [16] Oesterle RG, Aristizabal JD, Fiorato AE, Russell HE, Corley WG. Earthquake resistance structural walls—tests of isolated walls-Phase II. Skokie, IL: Construction Technology Laboratories, PCA; 1979 325pp. [17] Oesterle RG. Inelastic analysis for in-plane strength of reinforced concrete shear walls. PhD dissertation, Northwestern University, Evanston, IL, 1986. 322 pp. [18] Vallenas JM, Bertero VV, Popov EP. Hysteretic behavior of reinforced concrete structural walls. Report UCB/ EERC-79/20. Berkeley (CA): Earthquake Engineering Research Center, University of California; 1979. [19] Thomsen IV JC, Wallace JW. Displacement-based design of RC structural walls: an experimental investigation of walls with rectangular and T-shape cross-sections, CU/CEE-95/06. Postdam (NY): Department of Civil Engineering, Clarkson Universit; 1995.