A mathematical approach to computing structural-failure boundaries
Abstract
Earthquakes can cause substantial damage to buildings in ways that are still not well understood. The magnitude and principal frequency of an earthquake are two primary components that affect the extent of the damage, and they are the basis for many design specification guidelines. We investigate how an external force with varying magnitude and principal frequency affects structurural stability. As an example we consider a model of a planar, post-tensioned frame that exhibits dynamics quite similar to the experimental measurements of a scaled model on a shake table. Our goal is to predict behaviour of models subject to an aperiodic external force (an earthquake). Here, we consider a periodic external force, which is a simplifying but common choice. Many results in the literature are obtained from performing a large number of simulations over a range of magnitudes and frequencies. Our approach is much more efficient and uses a novel computational method that approximates the failure boundary directly. We find that failure can occur in profoundly different ways, due to inherent nonlinearities in the system. Stability is particularly affected if the natural frequency of the structure is close to that of the external forcing.
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