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Quantification of the Approximations Introduced by Assumptions on the Marginal Distribution of the Demand for Highway Bridge Fragility Analysis Downloaded from ascelibrary.org by LEHIGH UNIVERSITY on 12/14/15. Copyright ASCE. For personal use only; all rights reserved.
Aman Karamlou1 and Paolo Bocchini2 1
Department of Civil and Environmental Engineering, Advanced Technology for Large Structural Systems (ATLSS) Engineering Research Center, Lehigh University, 117 ATLSS Drive, Bethlehem, PA 18015, USA; PH (610) 758 6108; FAX (610) 738 5553; email:
[email protected] 2 Department of Civil and Environmental Engineering, Advanced Technology for Large Structural Systems (ATLSS) Engineering Research Center, Lehigh University, 117 ATLSS Drive, Bethlehem, PA 18015, USA; PH (610) 758 3066; FAX (610) 738 5553; email:
[email protected] ABSTRACT Fragility analysis is one of the most popular tools for the computation of the probability of reaching each investigated limit state for a given value of the external load intensity. Because of its versatility and computational efficiency, fragility analysis is particularly appropriate for regional loss estimations and risk analyses. Several different techniques have been proposed in the literature to compute the fragility of buildings and bridges. However, most of these techniques rely on simplifications and assumptions which, in some cases, are made more for analytical convenience than for adherence to reality. This paper investigates the effect of three of these common assumptions made on the engineering demand parameters for the case of seismic fragility analyses of bridges: (1) the marginal distribution is lognormal, (2) the median of such distribution follows a power law of the seismic intensity measure of choice, (3) the dispersion of such distribution is constant for any value of the intensity measure. Numerical analyses have been performed running extensive Monte Carlo simulations on computational clusters, focusing on structural models of different complexity. The results suggest that assumption (1) is not realistic in general, but it does not induce a significant error on the final results, whereas assumptions (2) and (3) corrupt significantly the quality of the results, introducing substantial errors. A comprehensive methodology for the assessment of fragility curves for structural components and systems is then proposed, which does not rely on the above mentioned assumptions. INTRODUCTION Seismic fragility analysis has a key role in seismic risk assessment procedures, as it provides the probability of exceeding a damage threshold given a certain level of an intensity measure (IM). Different methodologies have been
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proposed to compute the fragility of a structure. However, typically three major tasks need to be integrated in order to generate seismic fragility curves. These include the selection of a representative suite of ground motions, the probabilistic analysis of the seismic structural demand, and the evaluation of the capacity limit states. Among the mentioned tasks, the probabilistic seismic demand analysis (PSDA) is meant to relate the investigated demand parameter (e.g. story drift) with the selected IM (e.g. peak ground acceleration). Cornell and Jalayer (2002) proposed a seminal framework for the PSDA of structures and a closed-form probabilistic seismic demand model (PSDM) which is obtained on the basis of three major simplifying assumptions. First, it was assumed that the distribution of the demand at any level of IM follows a lognormal distribution. Second, a power model was adopted to relate the medians of such distributions with the investigated IM. Third, the dispersion of the demand about the medians is assumed to remain constant for all values of IM. This methodology is very effective. For its simplicity and clearness became quickly popular among researchers and practitioners, and gave a paramount contribution in creating momentum for modern seismic engineering and performance-based design. Several researchers have implemented this methodology to perform the PSDA on different types of structures. To mention just a few, Mackie and Stojadinović (2001) studied the PSDM of the California bridges. Padgett et al. (2008) and Shafieezadeh et al. (2012) evaluated the performance of different IMs on the PSDA of highway bridges. Tondini and Stojadinovic (2012) assessed the influence of deck radius and column height on the probabilistic seismic response of curved bridges. Fragility curves have been generated in different studies by taking advantage of the mentioned PSDM and other necessary assumption made on the capacity of the structure (or its components). Padgett and DesRoches (2007) modeled the capacity of the vulnerable components of the Central and Southern United States (CSUS) with lognormally distributed random variables by updating the physics-based limit states through a Bayesian process. By integrating the PSDM and the lognormal limit states, Nielson (2005) computed the fragility curves for the typical bridges of the CSUS. Padgett (2007) calculated fragility curves of the typical CSUS bridges considering also different retrofit measures. The underlying assumptions of the discussed PSDM (and of all studies based on it) are mostly due to analytical or computational convenience, but the veracity of the model is yet to be examined (Shome 1999; Lagaros et al. 2009; Shafieezadeh et al. 2012). To this purpose, the present paper evaluates the effect of these assumptions on the results of the PSDA and fragility curves. A PSDA is performed on a cantilever reinforced concrete column, and on a Multi-Span Simply Supported Steel Girder bridge. The impact of each assumption is evaluated by comparing the results with a reference simulation-based technique that does not need any of these assumptions. PROBABILISTIC SEISMIC DEMAND AND FRAGILITY ANALYSES Assuming that the demand follows a lognormal distribution, Cornell and Jalayer (2002) formulated a PSDM in the form of the following conditional probability function:
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ln(𝑑)−ln(𝑆 )
]
𝛽 |
(1)
where D is the seismic demand parameter, 𝑆𝐷 is the median of the demand, 𝛽𝐷|𝐼𝑀 is the conditional dispersion of the demand about its median, and Φ[∙] is the standard normal cumulative distribution function. Also, the median of the demand (𝑆𝐷 ) is assumed to be a power function of the selected IM as shown in Equation (2). 𝑆𝐷 = 𝑎(𝐼𝑀)𝑏
(2)
in which a and b are computed by linear regression performed on the demand and IM. The third assumption typically made in the implementation of this model, is that 𝛽𝐷|𝐼𝑀 is constant over the investigated range of IM and can be estimated as follows: 𝛽𝐷|𝐼𝑀
√∑
[ln(𝑑 )−ln(𝑎𝐼𝑀 )]
(3)
−
where N is the number of samples. If also the limit state capacities of the structure follow a lognormal distribution, a closed-form fragility function can be derived from Equation (1). In this way, the fragility function is presented in the form a lognormal cumulative function shown in Equation (4). 𝑃[𝐷 ≥ 𝐶|𝐼𝑀] = Φ [
ln(𝑆 )−ln(𝑆 ) √𝛽 |
+𝛽
]
(4)
where 𝑆𝐶 and 𝛽𝐶 are median and dispersion of the capacity, respectively. In order to compute fragility curves using Equation (4), the first step is calculating the parameters of the power model of Equation (2) and the dispersion (𝛽𝐷|𝐼𝑀 ) through a PSDA. Thus, a detailed analytical model of the structure needs to be developed.
ln(Demand)
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𝑃[𝐷 ≥ 𝑑|𝐼𝑀] = 1 − Φ [
1323
Linear Regression
ln(Intensity Measure)
Figure 1.Probabilistic seismic demand model
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Section A-A
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Unconfined concrete Confined concrete Rebar
A
A
(a)
(b)
Figure 2.Cantilever column (a) geometry; (b) section discretization Samples of the material and geometrical properties of the model are generated. The analytical models should be analyzed under proper ground motion records. The maximum response of each investigated demand parameter from the time history analyses is plotted in the logarithmic scale (Figure 1). After calculating the regression parameters of Equation (2) and dispersion (𝛽𝐷|𝐼𝑀 ), fragility curves can be estimated by Equation (4). As an alternative, fragility can be computed more accurately by removing the common assumptions made on the distribution of the demand. To achieve this, a simulation-based technique is presented in the following section. NUMERICAL EXAMPLES The methodology presented in the previous section is one of the most popular techniques for computing fragility curves and has been implemented by several researchers (Nielson and Desroches 2007; Padgett and DesRoches 2008; Ghosh and Padgett 2011; Ramanathan et al. 2012; Chang et al. 2012). Nowadays, in the light of the widely accessible computational resources, complex numerical analyses have become feasible for most institutions and companies. Therefore, large-scale simulation techniques can be employed to evaluate the accuracy of the results of probabilistic seismic demand models and fragility curves generated considering such assumptions. To this purpose, two structural models with different degrees of complexity have been developed and probabilistic seismic demand and fragility analyses have been carried out. These case studies are (i) a single cantilever reinforced concrete column (Figure 2), and (ii) a Multi-Span Simply Supported (MSSS) Steel Girder bridge (Figure 3), both modeled in OpenSees (McKenna et al. 2000).
Abutment
Bearing
Figure 3. MSSS Steel Girder bridge
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In order to account for the uncertainty in the seismic behavior, 1000 samples of 15 selected material and modeling parameters of the cantilever column and bridge models have been generated using Latin hypercube (Ayyub and Lai 1989). The random variables include damping ratio, mass, steel reinforcement yield stress, and concrete compressive stress, among others. These samples have been used to build 1000 analytical cantilever column and bridge models to undergo a complete PSDA. The models have been paired with 7 ground motion records with peak ground accelerations (PGA) ranging from 0.1(g) to 0.7(g). The 7000 resulting time history analyses have been carried out and the maximum responses of the selected vulnerable components have been recorded from each analysis. The representative demand parameter for the column model is the curvature ductility ratio of the column (𝜇𝑐 ) which is the ratio between the maximum and yield curvature of the column. For the bridge model, the demand parameters consist in the maximum deformation of the fixed and rocker bearings (𝑓𝑏𝑑 and 𝑟𝑏𝑑 ), maximum displacement of abutments in active and passive directions (𝐴𝑏𝐴𝑑 and 𝐴𝑏𝑃𝑑 ), as well as the maximum curvature ductility of the columns. The probabilistic seismic demand and fragility analyses performed on the results of the simulations are discussed in the following sections. EFFECT OF THE COMMON ASSUMTIONS ON THE PSDA The first assumption made in typical probabilistic seismic demand analyses is the lognormalty of the demand. Based on this assumption, the distribution of the demand at any level of the IM follows a lognormal distribution. To assess the veracity of this hypothesis, Figure 4 shows the normalized histogram of the curvature ductility demand of the cantilever column model at two levels of the IM, namely 0.2(g) and 0.6(g). These plots also contain the empirical distribution as well as the lognormal distribution fitted to the data. By comparing the empirical and the fitted lognormal distributions in Figure 4, it can be observed that although the lognormal distribution seems to be a good approximation for the recorded demand at the intensity of 0.2(g), there is a considerable amount of discrepancy between the empirical and lognormal
Kernel Smoothing 0.35 Column Curvature Column Curvature 0.3 Ductility Ratio Ductility Ratio @ PGA=0.2(g) @ PGA=0.6(g) 0.25
Histogram
1.6
Probability density
1.4 Probability density
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1.2 1 0.8 0.6 0.4
(a)
0.2 0.15 0.1
0.05
0.2 0
Lognormal fit
0
1
2
3
4
5
6
7
0
8
2
4
6
8
10
12
(b)
Figure 4.Cantilever column curvature ductility ratio demand
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3 2
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Linear regression
Medians
Column PSDM
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ln( )
1 0 -1
-2 -3 -2.5
-2
-1.5 -1 ln(PGA(g))
-0.5
0
Figure 5.Pobabilistic seismic demand model for cantilever column distributions at 0.6(g). Similar trends have been observed for the other demand parameters at the various levels of IM for both column and bridge models. This means that the distribution of the demand parameter does not necessarily follow a lognormal distribution, in general. In the next stage, the PSDA has been performed on the columns and other vulnerable components of the models using the methodology discussed earlier. To this purpose, the recorded maximum demand of the vulnerable components at each level of IM has been plotted in logarithmic scale and a linear regression has been performed. Figure 5 shows the PSDM of the single column ductility ratio demand. In this figure, the dots are the maximum curvature ductility ratios computed by the Monte Carlo simulation. The solid line is the representative of the power model obtained by means of regression analysis. This line is supposed to estimate the medians of the demand at each level of IM. The dashed line connects the actual medians of the data. The figure shows that in general the power model does not give an acceptable approximation of the logarithmic median of the demand. In this case, the error is larger at the mid-range IMs with a maximum of 36% at 0.3(g). In order to evaluate the third assumption (constant dispersion) made on the PSDM, the shaded area in Figure 5 shows the region enclosed between the mean plus and minus one standard deviation of the associated normal distribution of the data. From these results, it is qualitatively clear that the dispersion changes substantially across the IM range. EFFECT OF THE COMMON ASSUMTIONS ON THE FRAGILITY Fragility curves have been generated for the cantilever column as well as the components of the bridge model using 4 methods with different levels of assumptions in order to evaluate the effect of each assumption on the computed fragility. Table 1 shows the assumptions considered for each of the fragilities presented in this section. Figures 6 and 7 show the fragilities of the cantilever column and fixed bearings of the bridge for slight and moderate damage states, respectively. In these figures, the solid lines (HP12) are the fragility curves calculated by using Equation
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Table 1.Fragility Solutions and Underlying Assumptions Common Assumptions Solutions Demand lognormal Power model Constant dispersion distribution HP12 Included Included Included HP12a Included Included ̶ HP1 Included ̶ ̶ NHP ̶ ̶ ̶ (4) and the result of the PSDA presented in the previous section, including all three assumptions made on the distribution of the demand. The cross markers (HP12a) are the fragilities computed by removing the constant dispersion assumption. To this end, the dispersion of the demand about its median (𝛽𝐷|𝐼𝑀 ) has been calculated separately at each level of the IM using Equation (3) and used in Equation (4) to compute the fragility. The dashed lines are the lognormal cumulative distributions fitted to these set of fragility points. HP1 fragility values shown by square markers in the figures are calculated considering only the lognormalty assumption. To achieve this, lognormal distributions were fitted to the demand data at each level of IM. Next, 105 samples were generated from the demand (using the fitted lognormal parameters) and capacity. The capacity of components at different limit states was assumed to follow lognormal distributions. The parameters of these distributions can be found in the literature (Nielson 2005). The probabilities of failure have been estimated by calculating the ratio of the negative safety margins at each level of IM. Circle markers indicate the fragilities calculated with no major assumption on the marginal distribution of the demand (NHP). In this case, 105 samples have been generated from the empirical distribution of the demand at each level of IM by using classic translation theory (Grigoriu 1998). Finally, an estimate of the probability of failure has been calculated by computing the ratio of the negative safety margins. By comparing the reference fragility points (NHP) with fragilities computed 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 (a)
HP12
HP12a
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 (b) PGA (g)
HP1
NHP
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 PGA (g)
Figure 6.Cantilever column fragility: (a) slight damage, (b) moderate damage
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1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 00 (a)
HP12
HP12a
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0 (b) PGA (g)
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HP1
NHP
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 PGA (g)
Figure 7.Bridge fixed bearing fragility: (a) slight damage, (b) moderate damage with the other methodologies, it can be seen that the results of HP1 fragilities are close to the reference values. Hence, the lognormalty assumption does not introduce a significant amount of error. On the other hand, the power model and constant dispersion assumptions yield a large error in the fragility curves. CONCLUSIONS Large-scale simulation has been performed in order to evaluate the error introduced to the results of probabilistic seismic demand and fragility analyses due to the common assumptions made on the distribution of seismic demand, in particular for the case of highway bridges. To this purpose, detailed numerical models of a concrete cantilever column and a Multi-Span Simply Supported Steel Girder bridge have been developed and analyzed considering different levels of assumptions on the marginal distribution of the demand. The results of PSDA showed that lognormal distribution is not a good representative of the distribution of the demand for several values of IM. For middle ranges of the IM, the power model provided a poor approximation of the medians of the demand. Furthermore, in contrast with the constant dispersion assumption, the dispersion of the demand changes considerably at different levels of IM. In the case of fragility analysis, the fragilities calculated considering the power model and constant dispersion assumption differs considerably from the reference values. Better approximation of fragility can be obtained if lognormalty is the only assumption made on the distribution of the demand. The large-scale simulation used to compute the reference solution NHP is today a feasible alternative for the assessment of fragility curves for these types of structures, without the need of simplifications and assumptions. In fact, all the assumptions were originally introduced to limit the number of required simulations, by combining the results at all IM levels for the estimation of the distribution parameters. Nowadays, simulations can be performed faster and these limitations can be overcome.
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REFERENCES Ayyub, B. M., and Lai, K.-L. (1989). “Structural reliability assessment using latin hypercube sampling.” Structural Safety and Reliability, 1177–1184. Chang, L., Peng, F., Ouyang, Y., Elnashai, A., and Spencer, B. (2012). “Bridge Seismic Retrofit Program Planning to Maximize Postearthquake Transportation Network Capacity.” Journal of Infrastructure Systems, 18(2), 75–88. Cornell, C., Jalayer, F., Hamburger, R., and Foutch, D. (2002). “Probabilistic Basis for 2000 SAC Federal Emergency Management Agency Steel Moment Frame Guidelines.” Journal of Structural Engineering, 128(4), 526–533. Ghosh, J., and Padgett, J. E. (2011). “Probabilistic seismic loss assessment of aging bridges using a component-level cost estimation approach.” Earthquake Engineering & Structural Dynamics, 40(15), 1743–1761. Grigoriu, M. (1998). “Simulation of Stationary Non-Gaussian Translation Processes.” Journal of Engineering Mechanics, 124(2), 121–126. Lagaros, N. D., Tsompanakis, Y., Psarropoulos, P. N., Georgopoulos, E. C. (2009). “Computationally efficient seismic fragility analysis of geostructures.” Computers & Structures, 87(19), 1195–1203. Mackie, K., and Stojadinović, B. (2001). “Probabilistic Seismic Demand Model for California Highway Bridges.” Journal of Bridge Engineering, 6(6), 468–481. McKenna, F., Fenves, G. L., Scott, M. H., and Jeremic, B. (2000). Open System for Earthquake Engineering Simulation (OpenSees). Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA. Nielson, B. G. (2005). “Analytical fragility curves for highway bridges in moderate seismic zones.” Georgia Institute of Technology. Nielson, B. G., and DesRoches, R. (2007). “Seismic fragility methodology for highway bridges using a component level approach.” Earthquake Engineering & Structural Dynamics, 36(6), 823–839. Padgett, J. E., and DesRoches, R. (2008). “Methodology for the development of analytical fragility curves for retrofitted bridges.” Earthquake Engineering & Structural Dynamics, 37(8), 1157–1174. Padgett, J. E. (2007). “Seismic vulnerability assessment of retrofitted bridges using probabilistic methods.” Georgia Institute of Technology. Padgett, J. E., and DesRoches, R. (2007). “Bridge Functionality Relationships for Improved Seismic Risk Assessment of Transportation Networks.” Earthquake Spectra, 23(1), 115–130. Padgett, J. E., Nielson, B. G., and DesRoches, R. (2008). “Selection of optimal intensity measures in probabilistic seismic demand models of highway bridge portfolios.” Earthquake Engineering & Structural Dynamics, 37(5), 711–725. Ramanathan, K., DesRoches, R., and Padgett, J. E. (2012). “A comparison of preand post-seismic design considerations in moderate seismic zones through the fragility assessment of multispan bridge classes.” Engineering Structures, 45, 559–573. Shafieezadeh, A., Ramanathan, K., Padgett, J. E., and DesRoches, R. (2012). “Fractional order intensity measures for probabilistic seismic demand
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modeling applied to highway bridges.” Earthquake Engineering & Structural Dynamics, 41(3), 391–409. Shome, N. (1999). “Probabilistic seismic demand analysis of non-linear structures.” Stanford University. Tondini, N., and Stojadinovic, B. (2012). “Probabilistic seismic demand model for curved reinforced concrete bridges.” Bulletin of Earthquake Engineering, 10(5), 1455–1479.
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