Critical cross power spectral density models for earthquake loads on multi-supported structures

Abstract

The response of a doubly supported MDOF system under differential earthquake ground motions has been studied in this chapter using the principles of linear randomvibration theory. The earthquake ground motions are modeled as a vector of nonstationary Gaussian random processes. The spatial variability of the ground motions is characterized in terms of the crossPSD (power spectral density) functions. It has been demonstrated that the response variance is significantly influenced by the choice of crossPSD functions. The nature of the crossPSD functions that lead to the highest and lowest response variances has been established. These optimal crossPSD functions correspond neither to fully correlated motions nor to statistically independent motions. Instead, these spectra depend on:

system parameters, the nature of information available on the input, and the response variables chosen for optimization.

The findings offer a valuable counterpoint to traditional responsespectrumbased approaches to seismic response analysis of multiply supported structures. The study reported in this chapter focuses on linear system behavior. However, structural nonlinearities are an important aspect of aseismic design. The question of generalizing the analysis developed here to nonlinear structural behavior is addressed in the next chapter.

The method of equivalent linearization has been developed to analyze the response of a nonlinearly supported beam subjected to stationary random differential support motions. The response analysis uses beam dynamicstiffness matrices, and an iterative method for evaluating equivalent linear parameters has been outlined. The performance of these approximations was assessed by conducting digital simulation studies based on the finiteelement method. Satisfactory agreement between theoretical and simulated results has been demonstrated over a wide range of system parameters. Furthermore, the nature of crossPSD functions that lead to the highest and lowest response variances has also been established for nonlinear systems. These optimal CPSD functions share qualitative features with those for linear systems: the extreme responses are produced neither by fully correlated motions nor by independent motions. The response variance is significantly influenced by the choice of input CPSD functions, and the upper bound on the response variance can be as much as 100% higher than the lower bound.

With the study reported in this chapter, the research work undertaken in this thesis comes to an end. The next chapter summarizes the contributions made and offers suggestions for further research.