**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.