NDP with Element Removal

Traditional structural analysis software takes the earthquake analysis of a building to the point of incipient lateral instability.  At that point, the analysis stops, so the evolution of collapse remains unknown:  does  the building continue in sidesway, ultimately impacting an area as long as its height?

The progressive collapse of tall buildings can be analyzed using a nonlinear dynamic procedure (NDP) with the element removal-based method, which consists of the direct removal of structural elements from the structural model upon their failure (Talaat and Mosalam, 2007, 2009, Mosalam and Günay, 2014). This approach, that is based on dynamic equilibrium and the resulting transient change in the system kinematics and the corresponding progressive collapse algorithm (McKenna et al., 2010) for automated removal of collapsed elements during an ongoing time history simulation, Figure 1. The implementation is carried out as an OpenSees module, designed to be called by the main analysis module after each converged integration time step to check each potential failing element for possible violation of its respective removal criteria, which are defined according to relevant models for each element type (e.g. Elwood and Moehle, 2005, for shear critical columns or Kadysiewski and Mosalam, 2008, for infill walls considering in-plane/out-of-plane interaction). A violation of a pre-defined removal criterion triggers the activation of the algorithm on the violating element before returning to the main analysis module. Activation of the element removal algorithm includes updating nodal masses, checking if the removal of the collapsed element results in leaving behind dangling nodes or floating elements, which must be removed as well as removing all associated element and nodal forces, imposed displacements, and constraints. It is noted that the gravity loads at the node of a column, which is common with the other elements, is not removed. Accordingly, the gravity loads on the structure are not reduced upon removal of a column, allowing for the analysis model to capture the redistribution of the gravity loads to the other intact columns.

Figure 1  Element removal algorithm (Talaat and Mosalam, 2007).

Element Removal allows the complete development of the failure mechanism, with the result that the zone of impact can be estimated more reliably.  Other models (e.g., Varieschi & Kamiya) can be used to estimate the energy of the collapsing tower when it strikes a neighboring structure.

Specific examples from the analysis of detailed structural models can allow us to generalize to rules for collapse and the consequence of collision, so that efficient models of a large urban area can be assembled for analysis using Robust Simulation.

Robust Simulation

Probabilistic analysis for the portfolios of buildings (the city) makes use of a large catalog of earthquake simulations, considering the variability and spatial correlation of the ground motions.   Using geo-spatial processing and a framework called Robust Simulation (Taylor, 2013; Lee, 2014), urban risk analysis can make use of rules generalized from expert opinion and the NDP methods described above.  Diachronic simulations can be used to randomize collapse directions and modes, and project the consequences of building-to-building impacts, to estimate statistical distributions for the life-safety and economic consequences from tower collapses for the earthquake.  Consequences can be compared between models that include the collateral consequences of collapse and identical models that exclude these consequences, and buildings associated with large collateral consequences identified.

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