| A mathematical model for the response of multi-degree of freedom systems to natural hazards which accounts for the randomness of the excitation, and the nonlinear, hysteretic, and potentially degrading behavior of structures, is presented. The key element of the model is a first order nonlinear differential equation for a typical hysteretic element. System degradation is governed by the total energy dissipation of the hysteresis. The model is quite versatile, being able to represent multi-degree of freedom shear beam, or discrete hinge structures. Response statistics are obtained via stochastic equivalent linearization, without recourse to the Krylov-Bogoliubov method. The resulting zero time lag covariance matrix agrees more closely with Monte Carlo simulations than do other available approximate solutions. Using modal decomposition, the response power spectral density function of the linearized system is obtained. Several available first passage probability estimates are compared, with the individual response variables treated as scalars. Extensive numerical studies are presented for shear beam models. Excitations considered include stationary white noise, Kanai filtered white noise, and non-stationary excitations obtained by a temporal multiplier. Several prototypical studies for framed structures with discrete hinge regions show that implementation is feasible for such structures, and provides, in addition to the usual zero time lag covariance response, a joint by joint breakdown of the energy dissipation requirements. |
