Some mathematical modelling problems of seismic response of structures

Abstract

In this study, several mathematical modelling problems of traditional methods in seismic analysis of structures have been identified. It has been shown that assuming the motions of seismic response of structures to be harmonic or exponentially decaying with time, and thus transforming the governing partial differential equation of hyperbolic type to elliptic type, would lead to questionable results in earthquake engineering. Similarly, assuming the velocity of wave to be infinite or assuming the incompressibility of water and then transforming the wave equation or the horizontal dynamic hydraulic pressure on a reservoir's dam excited by an earthquake with the same form as the wave equation into a Laplace equation to obtain the global solution are also questionable. For the example of the waterdam system a traditional solution should contain singularity at the origin by nature as a result of imposing analytic value at infinity, and thus induced relative error may reach infinity at this important location, since a local solution for a hyperbolic partial differential equation will be analytic at its defined domain, including the origin.