Estimate of spectral and pseudo-spectral acceleration proximity

This paper deals with the deterministic relationship between spectral and pseudo-spectral acceleration, S A ( ξ , ω 0 ) and PS A ( ξ , ω 0 ) respectively, and gives the reason for their proximity on theoretical basis. The key step is to estimate the velocity at the acceleration extrema using the exact integral representation of the solution. When the product ξ T 0 < 1 , each maximum displacement occurs ξ T 0 / π s after each acceleration maximum. In this case the maximum displacement is evaluated in closed form as a simple function of the maximum acceleration: the well-known proximity between spectral and pseudo-spectral acceleration is thus quantified for the first time. As an alternative, a very simple procedure is presented using a “first-order” version of the motion equation; despite its coarseness the result agrees with the more complex analysis discussed previously. On the other hand, in Seismic Base Isolation systems (with large modal periods and damping ratios) a discussion of the relationship between PS A ( ξ , ω 0 ) –giving the forces on the structure–and S A ( ξ , ω 0 ) –governing the forces on the foundation–becomes crucial. Comparison of the obtained results with spectral quantities derived from a number of real accelerograms, along with a description of a typical counterexample in which displacement and acceleration extrema are not correlated, end the paper.