top of page

Risk & reliability assessment

Fragility functions, indicating the probability of a system exceeding a certain damage state for a given intensity measure (e.g. peak ground acceleration for seismic events, wind/water velocities for hurricanes, etc.), are typically modeled through log-normal cumulative distribution functions.


In some cases, including fragility functions generated through multiple stripes analysis, cloud method, or incremental dynamic analysis, consideration of multi-state functions is not straightforward, whereas in the presence of multi-variate Intensity Measures (IMs) this task becomes even more challenging since many methods rely on discrete representations of the IM space. Regardless of the fitting approach, in the case of fragility functions based on cumulative log-normal distributions, avoidance of function crossings which erroneously imply a negative probability of the system being at a certain damage state, is typically accomplished by assuming identical variance among fragility functions of different damage states.

This restrictive single-variance assumption can be however lifted, by tackling the fragility fitting task as a damage state classification problem through softmax regression. More importantly, it can be formally shown that softmax-based fragility functions hold the mathematically accurate form for fragility functions under broad probabilistic assumptions (i.e. the likelihood function of intensity measures belonging to the broad family of exponential distributions). Below are shown the fragility curves of a steel two-story moment-resisting frame, with fragility crossings avoided without the need for the common variance assumption.

Fragility crossings.png

Below are examples of seismic fragility functions based on the above-described approach for a bi-variate IM case. The examples include both linear and nonlinear (quadratic) predictors, as well as different assumptions regarding the nature of the damage states (nominal, ordinal, hierarchical). The ordinal case essentially corresponds to the single-variance assumption, for the predictor following the logit instead of the probit link in the log-space of intensity measures.

Fragilities with linear and quadratic pr



Andriotis, C.P., and Papakonstantinou, K.G., “Extended and generalized fragility functions”, Journal of Engineering Mechanics, 144 (9), 04018087, 2018. [Link]

Andriotis C.P., and Papakonstantinou, K.G., “Probabilistic structural performance assessment in hidden damage spaces”, Proceedings of Computational Stochastic Mechanics (CSM) Conference, Greece, Paros, June 2018. [Link]


Data, Documentation, Presentations


Faculty of Architecture & the Built Environment

Delft University of Technology

Julianalaan 134, 2628 BL, Delft 

email: c.andriotis [at] tudelft [dot] nl

Copyright © 2020-21 by C.P. Andriotis

bottom of page