This paper proposes a reliability-based framework for quantifying structural robustness considering the occurrence of a major earthquake (mainshock) and subsequent cascading hazard events, such as aftershocks that are triggered by the mainshock. These events can significantly increase the probability of failure of buildings, especially for structures that are damaged during the mainshock. The application of the proposed framework is exemplified through three numerical case studies. The case studies correspond to three SAC steel moment frame buildings of three, nine, and 20 stories, which were designed to pre-Northridge codes and standards. Two-dimensional nonlinear finite-element models of the buildings are developed with the Open System for Earthquake Engineering Simulation framework (OpenSees), using a finite length plastic hinge beam model and a bilinear constitutive law with deterioration, and are subjected to multiple mainshock-aftershock seismic sequences. For the three buildings analyzed herein, it is shown that the structural reliability under a single seismic event can be significantly different from that under a sequence of seismic events. The reliability based robustness indicator shows that the structural robustness is influenced by the extent to which a structure can distribute damage

The use of empirically calibrated moment-rotation models that account for strength and stiffness deterioration of steel frame members is paramount in evaluating the performance of steel structures prone to collapse under seismic loading. These deterioration models are typically used as zero-length springs in a concentrated plasticity formulation; however, a calibration procedure is required when they are used to represent the moment-curvature (M−χ) behavior in distributed plasticity formulations because the resulting moment-rotation (M−θ) response depends on the element integration method. A plastic hinge integration method for using deterioration models in force-based elements is developed and validated using flexural stiffness modifications parameters to recover the exact solution for linear problems while ensuring objective softening response. To guarantee accurate results in both the linear and nonlinear range of response, the flexural stiffness modification parameters are computed at the beginning of the analysis as a function of the user-specified plastic hinge length. With this approach, moment-rotation models that account for strength and stiffness deterioration can be applied in conjunction with force-based plastic hinge beam-column elements to support collapse prediction without increased modeling complexity

Finite-length plastic hinge (FLPH) models have shown advantages over the concentrated plasticity hinge (CPH) models. However, empirical phenomenological relationships, such as Modified Ibarra–Medina–Krawinkler (ModIMK) deterioration model, were mainly calibrated for use in CPH models. ModIMK relationships are versatile and have been applied to steel, reinforced concrete, and timber structures. Herein, a calibration procedure of FLPH models and a unified algorithm for use of ModIMK relationships in CPH and FLPH models are presented. Results from included examples validate the proposed algorithms, which were implemented in OpenSees. Additionally, results highlight that FPLH models avoid errors and convergence pitfalls of CPH models.

The safety and resilience of school buildings against natural disasters is of paramount importance since schools represent a reference point for communities. Such significance is not only related to the direct consequences of collapse on a vulnerable part of the population, but also due to the importance of schools in the post-disaster recovery. This work is focused on the risk and resilience assessment of school buildings in Lisbon (Portugal) under seismic events. The results of this study, in which a subset of 32 schools are analyzed, are used to define a prioritization strategy to mitigate the seismic risk of the Lisbon City Council school building portfolio and to assess the overall resilience of the school network. Numerical modeling of the school buildings is performed in order to estimate losses in terms of the built-up area of the schools and recovery times associated with different seismic scenarios, which are probabilistically defined specifically for the sites of the buildings, accounting for the local soil conditions and associated amplification effects. Based on the obtained risk estimates, which are compared to reference values established on international guidelines and specialized literature, the Lisbon City Council and LNEC jointly defined a short- and medium-term risk mitigation plan, starting with a detailed inspection and assessment of the most vulnerable school buildings and continuing to the implementation of retrofitting measures.