Abstract
Recent literature about life-cycle models for earthquake resistant structures considers that damage accumulation and failure are possibly due to subsequent shocks occurring during the time period of interest. In fact, most of these models only consider the effect of mainshocks. On the other hand, it is well known that earthquakes occur in clusters in which the mainshock represents only the principal (e.g., prominent magnitude) event. Because there is a chance that aftershocks can also cause deterioration of structural conditions, it may be appropriate to include this effect in the life-cycle assessment. Recently, stochastic processes describing the occurrences of aftershocks and their effect on cumulative structural damage have been formalized. These can be employed to develop stochastic damage accumulation models for earthquake resistant structures, accounting for the cluster effect. In the paper, such a model is formulated with reference to simple elastic-perfectly-plastic single degree of freedom systems. Temporal distribution of mainshocks is modeled via a homogeneous Poisson process. Occurrence of aftershocks is modeled by means of non-homogeneous Poisson processes conditional to the characteristics of the triggering mainshock. Approximate closed-form solutions are derived for the reliability assessment under the two hypotheses that total damages produced by events pertaining to different clusters can be assumed to be independent and identically distributed gamma or inverse-Gaussian random variables. An application illustrates the implications of the model on the life-cycle assessment when compared to the case where the effect of damaging aftershocks is ignored.
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Notes
Models used in this study consider that the aftershock source zone depends on the magnitude and location of the mainshock. Considering magnitude and distance, instead, is equivalent herein. It is also to note that both \(f_{IM_A \left| {M_E ,R_E } \right. }\) and \(f_{IM_E \left| {M_E ,R_E } \right. }\) should be indicated as \(f_{IM_{A,ij} \left| {M_E ,R_E } \right. }\) and \(f_{IM_{E,i} \left| {M_E ,R_E } \right. }\), yet the notation is intentionally simplified due to the i.i.d. features of these RVs. Actually, while also damages are i.i.d., subscript are kept there to avoid confusion, as it will be clarified in the following.
In Eq. (15), and in the others above, the distribution of damage is always indicated as a PDF, for simplicity of notation. However, it is not perfectly appropriate because the damage in a single event is not a continuous RV.
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Acknowledgments
The study was partially developed in the framework of AMRA—Analisi e Monitoraggio dei Rischi Ambientali scarl (http://www.amracenter.com), within the Strategies and tools for Real-Time Earthquake Risk Reduction project (REAKT; http://www.reaktproject.eu) funded by the European Commission via the FP7 programme; Grant No. 282862. Partial support was also from the ISLAR project granted by the AXA Research Fund in 2011. Finally, authors want to thank Racquel K. Hagen of Stanford University who proofread the manuscript, and the anonymous reviewers, as well as the guest editor Christoph Adam, whose comments improved quality and readability of the manuscript.
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Iervolino, I., Giorgio, M. & Polidoro, B. Reliability of structures to earthquake clusters. Bull Earthquake Eng 13, 983–1002 (2015). https://doi.org/10.1007/s10518-014-9679-9
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DOI: https://doi.org/10.1007/s10518-014-9679-9