Impact on loss/risk assessments of inter-model variability in vulnerability analysis

Abstract

Fragility curves (FCs) constitute an emerging tool for the seismic risk assessment of all elements at risk. They express the probability of a structure being damaged beyond a specific damage state for a given seismic input motion parameter, incorporating the most important sources of uncertainties, that is, seismic demand, capacity and definition of damage states. Nevertheless, the implementation of FCs in loss/risk assessments introduces other important sources of uncertainty, related to the usually limited knowledge about the elements at risk (e.g., inventory, typology). In this paper, within a Bayesian framework, it is developed a general methodology to combine into a single model (Bayesian combined model, BCM) the information provided by multiple FC models, weighting them according to their credibility/applicability, and independent past data. This combination enables to efficiently capture inter-model variability (IMV) and to propagate it into risk/loss assessments, allowing the treatment of a large spectrum of vulnerability-related uncertainties, usually neglected. As case study, FCs for shallow tunnels in alluvial deposits, when subjected to transversal seismic loading, are developed with two conventional procedures, based on a quasi-static numerical approach. Noteworthy, loss/risk assessments resulting from such conventional methods show significant unexpected differences. Conventional fragilities are then combined in a Bayesian framework, in which also probability values are treated as random variables, characterized by their probability density functions. The results show that BCM efficiently projects the whole variability of input models into risk/loss estimations. This demonstrates that BCM is a suitable framework to treat IMV in vulnerability assessments, in a straightforward and explicit manner.

Appendix: Vulnerability assessment through fragility models

Recently, new analytical fragility curves for shallow metro tunnels have been proposed based on numerical simulation, considering both structural parameters, local soil conditions and variation of input ground motion (Argyroudis and Pitilakis 2012). The quantification of the damage states is based on a damage index (DI) that is defined as the exceedance of strength capacity of the most critical sections of the tunnel (i.e., ratio of the developing moment (M) to the moment resistance (MRd) of the tunnel lining). The definition of damage states is then based on the range of damage index values (Table 1, col. 1–3). From the evaluated damage index, as a function of the PGA at the ground surface, the set of fragility curves relative to a discrete number of Damage States can be derived. Three different damage states are considered due to ground shaking: minor, moderate and extensive-to-complete damage (d ₁, d ₂ and d ₃ respectively). Fragility curves are usually represented as a two-parameter (median and log-standard deviation) lognormal cumulative distribution functions. The development of FCs requires the definition of 4 parameters, 3 medians m _i and 1 value of β, which are estimated in the literature following different procedures.

Two procedures are adopted here: (1) a linear regression method (e.g., Nielson and DesRoches 2007; Pinto 2007), herein referred to as M1 and (2) a maximum likelihood method (ML, e.g., Saxena et al. 2000; Shinozuka et al. 2000, 2003; Kim and Feng 2003; Straub and Der Kiureghian 2008), herein referred to as M2.

M1 has been recently published in Argyroudis and Pitilakis (2012). Such fragility functions are reported in Table 1, col. 4–5 and 8–9 and plotted in Fig. 2 (light blue) for the case of circular (bored) tunnel in soil type C and D. As regards M2, while ML is normally used starting from real data (Kalbfleisch 1977), with the same philosophy, it is here used with synthetic data produced by a model. In particular, as for M1, the starting database for M2 consists of the result of the coupled numerical analysis, that is, the earthquake parameter and the consequent damage index for the modeled tunnel (PGA _i , DI _i ). By defining one threshold in DI for each damage state (t ₁, t ₂ and t ₃), the data can be transformed as the result of a Bernoulli trial experiment, associating each PGA to the consequent expected damage state, that is, (PGA _i , y _i ), where y _i is equal to 1 or 0 depending on whether or not the tunnel section sustains the damage state, that is, equal to 1 if it is observed the ith damage state, 0 otherwise. To account for the uncertainty on damage state definition, for each starting datum (PGA _i , DI _i ), a Monte Carlo simulation is performed, by producing N = 500 couples of (PGA _i , y _i ) data, each one obtained by comparing out the observed value for the damage index (DI _i ) with randomly sampled thresholds. The thresholds are sampled from uniform distributions in their confidence intervals (Table 1, col. 2). The fragility curves are assumed to be log-normally distributed, with different medians m _j and equal β-value. The best guess values for the parameters (m _i ′ and β′) are obtained by numerically maximizing, as a function of m _j , and β, the likelihood function L. The obtained values (m′ and β′) account for the demand uncertainty, since different seismic records as input for the coupled numerical analysis are used, and the damage state definition uncertainty by randomly selecting the DI thresholds. Among the principal sources of uncertainties, only the capacity uncertainty is not yet considered. Thus, it is added to the results of the analysis as the square root of the sum of squares of β′ and 0.3 (e.g., NIBS 2004).

The obtained β″-value, which includes also capacity uncertainty, is then put into the likelihood function that, this time, is a function of the medians m _j only. The best guess medians (m _i ″) are obtained by numerically maximizing ln(L′) and, together with the total β″-value, represent the best guess parameters for the log-normal distribution. From this analysis, we obtain the final parameters for M2, as reported in Table 1, col. 6–7 and 10–11 and plotted in Fig. 2 (dark blue).