Risk & reliability assessment

Structural capacity degrades with time. Causes of this degradation are often chronic stressors, such as corrosion and fatigue, which naturally change structural fragility towards more vulnerable configurations. Such aging effects have been widely studied in the literature, indicating that long-term planning for resilient infrastructure systems should not ignore time-dependent fragility properties.


Another important factor significantly contributing to fragility deterioration, however lacking similar investigation, is the deterioration induced due to structural systems being subject to multiple sequential high-intensity events. In such cases, structures undergo multiple cycles of loading and unloading that is characterized by substantial plastic deformations and energy dissipation. This cyclic excitation is associated with low-cycle fatigue phenomena, which reduce the strength and stiffness of structural members, thus forcing them to more easily reach their limit states. Even ignoring low-cycle fatigue, the presence of residual plastic deformations per se, after a high-intensity event, makes a violation of limit states in subsequent events more probable. This description is conceptualized in the figure below.

Structure subject to multiple events.png
Markov fragility concept.png

Such deterioration effects due to multiple sequential damaging events can be efficiently captured through Dependent Hidden Markov Models (DHMMs). Generalizing the frameworks of softmax-based fragility to account for multiple timesteps (or, more precisely, event steps) with Markovian transitions between hidden states, we can obtain the following transition matrices for the two-story steel frame structure shown above.

Gaussian mixture.png
sample event sequence.png
HMM and RNN.png

After training the RNN for obtaining the parameters of the hidden nonlinear functions, the hidden space can be mined so that stateful representations that have been learned are extracted. This is shown in the figure below where the high-dimensional hidden space of the LSTM RNN is embedded in a 2D space. This extraction is achieved through unsupervised learning algorithms (k-means in this case).


The transitions among the clusters are then collected and their parametric approximation is learned through a Markov model. The above-described RNN approach is motivated by mining approaches in the field of computer science, mainly developed for rule extraction in grammars and construction of finite-state automata. Below is the resulting transition matrix.

Markov Transitions.png

In both RNN and DHMM cases, we can observe that the learned transitions reveal irreversible dynamics between states. This is an important property verifying that the two approaches can adequately describe damage- and degradation-consistent phenomena, which are fundamentally irreversible. 

The learned hidden states represent probability distributions which are here assumed to be Gaussian in the log-IM space. Hence, the hidden state of the system at every step is represented by a Gaussian mixture, as more illustratively shown for the case of DHMM.



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]

Andriotis C.P., and Papakonstantinou, K.G., “Dependent Markov models for long-term structural fragility”, 11th National Conference on Earthquake Engineering (NCEE), Los Angeles, California, USA, June, 2018. [Link]


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Faculty of Architecture & the Built Environment

Delft University of Technology

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

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