Shaken but not stirred: using mathematics in earthquakes
Earthquakes can cause substantial damage to buildings in ways that are still not well understood. The amplitude 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 amplitude and principal frequency affects structurural integrity. 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. The frame remains structurally sound as long as the tilt angle of the frame does not exceed a certain maximum. Many results in the literature are obtained from performing a large number of simulations over a range of amplitudes and frequencies. Such a brute-force approach establishes a region in the frequency-amplitude plane for which the structural stability of the frame eventually fails. Our approach is much more efficient and uses a novel computational method that approximates the failure boundary directly. This method is based on continuation of a suitable two-point boundary value problem. Our computations demonstrate that the failure boundary is only piecewise smooth and the results highlight further interesting details of how the dynamics is organised in the frequency-amplitude plane. We find that failure can occur in profoundly different ways, due to inherent nonlinearities in the system. Stability is particularly affected in a nonlinear way if the natural frequency of the structure is close to that of the external forcing.
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