Pierre Gehl

Vector-valued fragility functions for seismic risk evaluation

This article presents a method for the development of vector-valued fragility functions, which are a function of more than one intensity measure (IM, also known as ground-motion parameters) for use within seismic risk evaluation of buildings. As an example, a simple unreinforced masonry structure is modelled using state-of-the-art software and hundreds of nonlinear time-history analyses are conducted to compute the response of this structure to earthquake loading. Dozens of different IMs (e.g. peak ground acceleration and velocity, response spectral accelerations at various periods, Arias intensity and various duration and number of cycle measures) are considered to characterize the earthquake shaking. It is demonstrated through various statistical techniques (including Receiver Operating Characteristic analysis) that the use of more than one IM leads to a better prediction of the damage state of the building than just a single IM, which is the current practice. In addition, it is shown that the assumption of the lognormal distribution for the derivation of fragility functions leads to more robust functions than logistic, log-logistic or kernel regression. Finally, actual fragility surfaces using two pairs of IMs (one pair are uncorrelated while the other are correlated) are derived and compared to scalar-based fragility curves using only a single IM and a significant reduction in the uncertainty of the predicted damage level is observed. This type of fragility surface would be a key component of future risk evaluations that take account of recent developments in seismic hazard assessment, such as vector-valued probabilistic seismic hazard assessments.

Influence of the number of dynamic analyses on the accuracy of structural response estimates

Nonlinear dynamic analysis is often used to develop fragility curves within the framework of seismic risk assessment and performance-based earthquake engineering. In the present article, fragility curves are derived from randomly generated clouds of structural response results by using least squares and sum-of-squares regression, and maximum likelihood estimation. Different statistical measures are used to estimate the quality of fragility functions derived by considering varying numbers of ground motions. Graphs are proposed that can be used as guidance regarding the number of calculations required for these three approaches. The effectiveness of the results is demonstrated by their application to a structural model. The results show that the least-squares method for deriving fragility functions converges much faster than the maximum likelihood and sum-of-squares approaches. With the least-squares approach, a few dozen records might be sufficient to obtain satisfactory estimates, whereas using the maximum likelihood approach may require several times more calculations to attain the same accuracy.

Accounting for site characterization uncertainties when developing ground-motion prediction equations

Current ground-motion prediction equations invariably assume that site conditions at strong-motion stations, often characterized by the average shear-wave velocity to a depth of 30 m (VS30), are known to a uniform accuracy. This is, however, rarely the case. In this article, we present a regression procedure based on generalized least-squares and maximum-likelihood approaches that take into account the varying uncertainties on VS30. Assuming that VS30s for various groups of stations are known to different accuracies, application of this procedure to a large set of records from the Japanese KiK-net shows that the regression coefficients are largely insensitive to the assumption of nonuniform uncertainties. However, this procedure allows the computation of a site-specific standard deviation (σ) that should be used for sites where VS30 is known to different accuracies (e.g., a site only specified by class or a site with a measured VS profile). For sites with a measured VS profile, this leads to lower sitespecific σ than for a site that is poorly characterized because this technique explicitly models the separation between the epistemic uncertainty in VS30 and the aleatory variability in predicted ground motion.

Bayesian Networks and Infrastructure Systems: Computational and Methodological Challenges

This chapter investigates the applicability of Bayesian Network methods to the seismic assessment of large and complex infrastructure systems. While very promising in theory, Bayesian Networks tend to quickly show limitations as soon as the studied systems exceed several dozens of components. Therefore a benchmark study is conducted on small-size virtual systems in order to compare the computational performance of the exact inference of various Bayesian Network formulations, such as the ones based on Minimum Link Sets. It appears that all formulations present some computational bottlenecks, which are either due to the size of Conditional Probability Tables, to the size of clique potentials in the junction-tree algorithm or to the recursive algorithm for the identification of Minimum Link Sets. Moreover, these formulations are limited to connectivity problems, whereas the accurate assessment of infrastructure systems usually requires the use of flow-based performance indicators. To this end, the second part of the chapter introduces a hybrid approach that presents the merit of accessing any type of system performance indicator: it uses simulation-based results and generates the corresponding Bayesian Network by counting the outcomes given the various combinations of events that have been sampled in the simulation. The issue of the system size is also addressed by a thrifty-naive formulation, which limits the number of the components that are involved in the system performance prediction, by applying a cut-off threshold to the correlation coefficients between the components and system states. A higher resolution of this thrifty-naive formulation is also obtained by considering local performance indicators, such as the flow at each sink. This approach is successfully applied to a realistic water supply network of 49 nodes and 71 pipes. Finally the potential of this coupled simulation-Bayesian approach as a decision support system is demonstrated, through probability updating given the observation of local evidences after an event has occurred.

Specification of the vulnerability of physical systems

The general methodology presented in Chap. 2 of this book, has been conceived in order to be general enough to be adequate for each system. The purpose of this chapter is to decline this methodology to the specificity of each physical system considered: i.e., Buildings, Water Supply System, Waste Water Network, Electrical Power Network, Oil and Gas Network, Transportation Network, Health Care System and Harbours. Each system is described based on its structure and taxonomy, on the dependencies it shares with the other systems, on the available methods to describe its systemic vulnerability and, finally, on the existing indicators to evaluate its performance, but also its functionality according to the societal needs.

Fragility Functions of Gas and Oil Networks

The present chapter aims to present and review fragility curves for components of gas and oil system networks. These fragility functions need to be applicable to the specific European context and they should be available for a variety of network components such as buried pipelines, storage tanks and processing facilities (i.e. compression and reduction stations). Based on a literature review, it is found that the available fragility functions are mostly empirical and should be applied to the European context, given the current lack of data needed to validate potential analytical methods of vulnerability assessment. For buried pipelines, fragility relations are reviewed with respect to both wave propagation and ground failure. Existing fragility curves for storage tanks and processing facilities are also critically appraised, according to the modelling assumptions and the derivation techniques (e.g. fault-tree analysis, numerical simulation or empirical relation).

Stress tests for a road network using fragility functions and functional capacity loss functions

A quantitative approach to conduct a specific type of stress test on road networks is presented in this article. The objective is to help network managers determine whether their networks would perform adequately during and after the occurrence of hazard events. Conducting a stress test requires (i) modifying an existing risk model (i.e., a model to estimate the probable consequences of hazard events) by representing at least one uncertainty in the model with values that are considerably worse than median or mean values, and (ii) developing criteria to conclude if the network has an adequate post-hazard performance. Specifically, the stress test conducted in this work is focused on the uncertain behavior of individual objects that are part of a network when these are subjected to hazard loads. Here, the relationships between object behavior and hazard load are modeled using fragility functions and functional capacity loss functions. To illustrate the quantitative approach, a stress test is conducted for an example road network in Switzerland, which is affected by floods and rainfall-triggered mudflows. Beyond the focus of the stress test, this work highlights the importance of using a probabilistic approach when conducting stress tests for temporal and spatially distributed networks.

Inferring earthquake ground motion fields with Bayesian Networks

Bayesian Networks (BNs) have the ability to perform inference on uncertain variables given evidence on observed quantities, which makes them relevant mathematical tools for the updating of ground-motion fields based on strong-motion records or macroseismic observations. Therefore the present article investigates the use of BN models of spatially correlated Gaussian random fields as an accurate and scalable method for the generation of ground-motion maps. The proposed BN model is based on continuous Gaussian variables, as opposed to discrete variables as in previous formulations, and it is built to account for cross-correlated ground-motion parameters as well as macroseismic observations. This approach is validated with respect to the analytical solution (i.e., conditional multivariate normal distributions) and it is also compared to the USGS ShakeMap method, thus demonstrating a better ability to model jointly the inter- and intra-event error terms of ground-motion models. The scalability of the approach, i.e. its capacity to be applied to large grids, is ensured by a grid sub-division strategy, which appears to be computationally efficient and accurate within an error rate of a fraction of percent. Finally, the BN implementation is demonstrated on a real-world example (the Mw 6.2 Kumamoto, Japan, 2016 foreshock), where vector-valued shake-maps of cross-correlated intensity measures are generated, along with the integration of macroseismic observations.

A multi hazard RISK assessment methodology accounting for cascading hazard events

The INFRARISK project is developing reliable stress tests on European Critical Infrastructure using integrated tools for decision-support. This aims to achieve higher infrastructure network resilience to rare and low probability extreme events. As part of the project, a hazard assessment methodology is developed to account for extreme natural hazards with cascading effects. Often hazard scenarios arising from cascading effects lead to disastrous consequences because such hazards are not prepared for. In particular, this paper focuses on the cascading hazard scenario involving earthquake triggered landslides. Traditional risk analysis considers each risk source as independent from the others. As a consequence, values for risk are usually defined regardless of interactions among the multiple risks present in a region. The current approach accounts for interaction between the two hazards in such a way that the probabilities of occurrence can be aggregated as part of an overall risk assessment methodology. The methodology is then demonstrated on a virtual road network case study as a proof of concept.

System loss assessment of bridge networks accounting for multi-hazard interactions

Abstract This paper details an integrated method for the multi-hazard risk assessment of road infrastructure systems exposed to potential earthquake and flood events. A harmonisation effort is required to reconcile bridge fragility models and damage scales from different hazard types: this is achieved by the derivation of probabilistic functionality curves, which express the probability of reaching or exceeding a loss level given the seismic intensity measure. Such probabilistic tools are essential for the loss assessment of infrastructure systems, since they directly provide the functionality losses instead of the physical damage states. Multi-hazard interactions at the vulnerability level are ensured by the functionality loss curves, which result from the assembly of hazard-specific fragility curves for local damage mechanisms. At the hazard level, the potential overlap between earthquake and flood events is represented by a time window during which the effects of one hazard type on the infrastructure may still be present: the value of this temporal parameter is based on the repair duration estimates provided by the functionality loss curves. The proposed framework is implemented through Bayesian Networks, thus enabling the propagation of uncertainties and the computation of joint probabilities. The procedure is demonstrated on a bridge example and a hypothetical road network.