Reliability-based critical earthquake load models. Part 2: nonlinear structures

Abstract

The problem of determining critical stochastic earthquake excitation models for simple nonlinear systems under single-point or multi-point nonstationary seismic inputs is considered. The earthquake acceleration components are obtained by multiplying known deterministic enveloping functions with zero mean Gaussian stationary random processes with unknown auto-power spectral density functions (for single-point excitations) and power spectral density matrix (for multi-point excitations). The definition of critical earthquake input is based on the notion of a performance function. The system is considered to have failed if the maximum response over a given time interval exceeds specified limits. The critical excitations are defined as those that minimize the Hasofer–Lind reliability index associated with this performance function. The computation of this index, in turn, is based on the use of response surface to model the limit surface near the check point. Here the quantity to be optimally determined is taken to be the unknown input power spectral density function (for single-point excitations) or the input power spectral density matrix (for multi-point excitations). The excitations are taken to satisfy constraints on total average energy, zero crossing rate, lower bounds on entropy rate and other positivity and bounding requirements that are of mathematical nature. The resulting constrained nonlinear optimization problems are solved using the sequential quadratic programming method. Illustrative examples for computing random critical excitations for singly supported and multiply supported oscillators that have cubic force–displacement relations are provided.