An efficient approach for computing analytical non-parametric fragility curves
Introduction
Seismic fragility curves describe how the failure probability of structural systems or components evolves for different given levels of a seismic intensity measure parameter (IM) adopted to characterize the ground motion record (e.g. peak ground acceleration, spectral acceleration at the fundamental frequency). They are gaining popularity especially in the last years due to their importance in the framework of a performance based earthquake engineering (PBEE) approach [1], [2]. For example, by combining the fragility curve and the seismic hazard curve it is possible to compute the mean annual rate of the failure state [3], [4].
The dataset of the structural response under different seismic actions, needed to compute fragility curves, can be obtained from both empirical data (observed failures) (e.g. [5], [6]) and analytical model evaluations (e.g. [7], [8]). In the first case data are collected by recording the damage state of a set of structures along with the intensity seismic levels, that can be estimated from a ground motion instrument or in absence of that from a ground motion map [9], [10] or a proper attenuation law.
In case of analytical fragilities curves (i.e. computed on the basis of nonlinear dynamic analyses), different procedures have been proposed to efficiently perform nonlinear dynamic structural analyses for increasing seismic intensity measures. Classic analytical parametric methodologies are based on the assumption that fragility curves can be defined by means of a lognormal cumulative distribution function so that two parameters, median and standard deviation , are sufficient to derive conditional seismic reliability measures for any given failure criterion [11], [12], [13]. The statistical parameters and refer to the dataset of IMs corresponding to the failure state, since the randomness of the seismic input can lead to different dynamic response even having the same IM. The estimation of and is commonly obtained by means of three different approaches: method of moments, maximum likelihood method or by adopting a regression demand model (mostly linear) in the log-scale between the IMs and the structural response [14], [15].
One of the most common parametric approach performs a sequence of dynamic analysis by repeatedly scaling the ground motion until the failure state is reached (Incremental Dynamic Analysis – IDA) [11]. However, IDA may require significant computational efforts and the final fragility curve may be affected by the scaling stage since the ground motion frequency content may be not representative of the corresponding intensity level [16], [17]. In this regard, [18] proposes an hybrid procedure to reduce the number of analysis required by a classic IDA by previously estimating the seismic intensity level corresponding to the failure state by means of the so-called Cloud Analysis [19], [13] that provides a simple linear relation between the un-scaled seismic inputs and the corresponding Engineering Demand Parameter (EDP) values.
Alternatively, the so-called Multiple Stripes Analysis (MSA) allows estimating the median and the dispersion of a sequence of ‘stripes’ of structural responses for increasing constant IMs, until the collapse is reached along the whole stripe [20], [21].
In general, parametric fragility models provide a smooth fragility curve easy to estimate. However they can lead to a lack of fit with respect the real data distribution due to the restriction on the final shape [22]. While parametric methods have been widely addressed due their robustness and simplicity, this is not the case for non-parametric procedures that result mainly based on the use of kernel smoothing methods (KSMs) [22], [23], [24] or the copula function [25]. These numerical approaches turn out to be extremely flexible and efficient to compute the joint probability or the conditional probability for the fragility curve estimation. However, KSMs can suffer of over-fitting or show bias near the boundaries of the analyzed domain, while a copula function cannot be always properly fitted and the selection of the optimal copula remains somehow murky [26]. The approach introduced in this article does not work directly on the joint probability and does not require an ambiguous tuning stage.
More specifically, this study proposes a methodology able to derive seismic fragility curves without any assumption on the final conditional cumulative density function, through a failure region mapping process that employs a Multinomial Logistic Regression coupled with a polynomial kernel. The absence of a predefined final shape reduces the risk of lack of fitting, while the use of probabilistic scores instead of a classical indicator function makes the approach more flexible and applicable even in case of noise along the failure region.
The paper is organized as follows. Section 2 provides a brief description of three parametric approaches employed to validate and test the proposed procedure. In Section 3 the proposed methodology is described into details along with the failure region mapping process. In Section 4 the problem raised by the seismic dataset selection is presented and two alternatives discussed. In Section 5 two different case studies are presented and finally, all the results are reported and the conclusions drawn.
Section snippets
Fragility curves and parametric models
Analytical fragility curves allow selecting the IM levels for the seismic dataset definition along with the number of dynamic analysis required for each IM value [12]. This aspect contributes to define methods increasingly able to perform an efficient dataset selection by reducing the number of analyses performed.
Generally, by assuming that the IMs causing the failure are lognormal distributed then the final fragility curve can be derived from just 2 parameters, the median capacity (i.e. the
Proposed non-parametric approximation
The proposed approach aims at reducing the computational effort by providing a non-parametric approximated solution to compute conditional reliability measures. For the sake of clarity, small letters are used to indicate parameters while capital letters refer to random variables or subsets of the sample space.
Given a failure event F and a sequence of subsets in the 2-dimensional sample space [moment magnitude - M, epicentral distance - R], the failure probability for a
Seismic demand
Seismic reliability analyses are based on ground motion records consistent with the local seismic hazard. Despite the seismic record selection strategy is not the focus of this study, two possible approaches for the seismic demand definition are described below, both compatible with the proposed methodology allowing a probabilistic description of the magnitude M, the epicentral distance R and the associated seismic intensity measure IM.
Case studies
Two different structural systems are analyzed to test and validate the proposed approach. Both models do not require an excessive computational effort in order to derive the target seismic fragility curve through a Latin Hypercube simulation. The opensource Matlab toolbox OpenCossan [37], [38] is used for the probabilistic analysis.
Conclusions
An efficient non-parametric methodology to compute seismic fragility curves has been proposed. A stochastic earthquake model is employed to generate the ground motion dataset. The failure region mapping process based on the use of surrogate PDFs and a Multinomial Logistic Regression allows decreasing the number of samples required. The method is able to manage noise along the failure region boundaries thanks to the adoption of probabilistic scores as indicator function, while more complex
Acknowledgements
The authors would like to acknowledge the gracious support of this work through the EPSRC and ESRC Centre for Doctoral Training on Quantification and Management of Risk and Uncertainty in Complex Systems and Environments Grant No. (EP/L015927/1) and the EPSRC project A Resilience Modelling Framework for Improved Nuclear Safety (NuRes) - EPSRC EP/R020558/2.
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2023, Engineering StructuresCitation Excerpt :However, the final fragility curves may not represent the real data distribution due to the predefined shape of the curves[16]. The classical parametric methods are widely used because of their simplicity and robustness[17]. With the development of artificial intelligence (AI), machine learning (ML) methods [60] have been extensively adopted to derive seismic fragility curves[18,19].