[Paper Review] Assessment of the lognormality assumption of seismic fragility curves using non-parametric representations

Bruno Sudret, Chu Mai+1 more|arXiv (Cornell University)|Mar 21, 2014

Wind and Air Flow Studies17 citations

TL;DR

This paper proposes non-parametric methods—binned Monte Carlo simulation and kernel density estimation—to assess the validity of the widely used lognormal assumption in seismic fragility curves. It demonstrates that lognormal curves significantly deviate from non-parametric references, especially at higher drift thresholds, revealing substantial inaccuracies in traditional parametric approaches for nonlinear steel structures under synthetic ground motions.

ABSTRACT

Fragility curves are commonly used in civil engineering to estimate the vulnerability of structures to earthquakes. The probability of failure associated with a failure criterion (e.g. the maximal inter-storey drift ratio being greater than a prescribed threshold) is represented as a function of the intensity of the earthquake ground motion (e.g. peak ground acceleration or spectral acceleration). The classical approach consists in assuming a lognormal shape of the fragility curves. In this paper, we introduce two non-parametric approaches to establish the fragility curves without making any assumption, namely the conditional Monte Carlo simulation and the kernel density estimation. As an illustration, we compute the fragility curves of a 3-storey steel structure, accounting for the nonlinear behavior of the system. The curves obtained by the proposed approaches are compared with each other and with those obtained using the classical lognormal assumption.

Motivation & Objective

• To evaluate the validity of the classical lognormal assumption in seismic fragility curves, which is widely used but rarely questioned.
• To develop and apply non-parametric alternatives—binned Monte Carlo simulation and kernel density estimation—without assuming a specific distributional shape.
• To compare non-parametric fragility curves with traditional lognormal fits to quantify discrepancies in failure probability estimates.
• To assess the impact of epistemic uncertainty on non-parametric fragility curves using bootstrap resampling.
• To explore the feasibility of extending non-parametric methods to fragility surfaces for multi-intensity measure risk assessment.

Proposed method

• Binned Monte Carlo simulation (bMCS) computes crude Monte Carlo estimators for small subsets of ground motions with similar intensity measure values (e.g., PGA or Sa), enabling local probability estimation.
• Kernel density estimation (KDE) non-parametrically estimates the conditional probability density function of structural response given an intensity measure, avoiding distributional assumptions.
• The two non-parametric methods are applied to a 3-storey steel frame subjected to synthetic ground motions, with fragility curves computed for three inter-storey drift limit states.
• Lognormal fragility curves are generated via maximum likelihood estimation and linear probabilistic seismic demand modeling in log-scale for comparison.
• Bootstrap resampling is used to quantify epistemic uncertainty in non-parametric fragility estimates, assessing their stability and reliability.
• The framework is extended to suggest non-parametric computation of fragility surfaces using KDE, enabling assumption-free vulnerability modeling with two intensity measures.

Experimental results

Research questions

• RQ1 How do non-parametric fragility curves based on binned Monte Carlo simulation and kernel density estimation compare with classical lognormal curves in terms of shape and failure probability?
• RQ2 To what extent do the lognormal assumptions lead to significant errors in fragility estimates for nonlinear steel structures under synthetic ground motions?
• RQ3 How does epistemic uncertainty in non-parametric fragility curves vary between intensity measures such as PGA and spectral acceleration (Sa)?
• RQ4 Can non-parametric methods provide a stable and reliable reference for calibrating or validating parametric fragility models?
• RQ5 What is the potential of non-parametric approaches for constructing assumption-free fragility surfaces in multi-intensity measure seismic risk assessment?

Key findings

• The binned Monte Carlo simulation and kernel density estimation methods produce highly consistent non-parametric fragility curves across all cases, validating their reliability as reference methods.
• For the two higher inter-storey drift limit states, the lognormal fragility curves differ significantly from the non-parametric references, indicating substantial inaccuracies in the classical parametric approach.
• Discrepancies between the two lognormal fitting methods (maximum likelihood vs. linear probabilistic seismic demand model) are large, showing that parametric results are sensitive to estimation technique.
• Lognormal-based fragility estimates may be either conservative or unconservative depending on the intensity measure and drift threshold, highlighting the risk of systematic bias in decision-making.
• Epistemic uncertainty in non-parametric fragility curves is lower for spectral acceleration (Sa) than for peak ground acceleration (PGA), suggesting Sa is more stable as a structure-specific intensity measure.
• The proposed non-parametric methods provide a robust, assumption-free basis for fragility assessment and open a path toward non-parametric fragility surfaces using KDE with multiple intensity measures.