Linear Dynamic Analysis Of SDOF SystemsWith Random Properties Subject ToEarthquake Excitations

Abstract

The present paper deals with the dynamic analysis of linear single degree of freedom (SDOF) systems with random mass, random stiffness and random damping subject to random seismic loading. The importance of considering random system model is implemented with a detailed examination of the impulse response function of SDOF systems and with several exact solutions. Response characteristics are initially determined on condition of a specific realization of the structural system. Then, total probability theorem is applied to evaluate unconditional expectations of the response. Conventional zeroth, first and second order perturbation solutions are compared to the exact solutions for the response statistics. Random damping and random stiffness effects are investigated separately considering eigenvibrations, stationary response under harmonic loadings and horizontal nonstationary earthquake excitations. For eigenvibrations, in both cases, perturbation solutions carry divergent secular terms. When only random stiffness is considered, due to the secular divergent terms, perturbation solutions blow up with time, even for very small variabilities of the random variables. In the case of only random damping, as the divergent secular terms are under the governing control of the exponential decay, the existing deviations in the perturbed solutions become neither observable nor important. Hence, perturbation solutions don't diverge from the exact solution and are observed to be very good approximations. Under harmonic loadings, after the dissipation of eigen vibrations, perturbation solutions don't possess any secular terms and are very good approximations except for the cases in which the system is excited at or close to its resonance frequency. Finally, the nonstationary random response of stochastic SDOF systems subject to nonstationary random seismic exciTransactions on the Built Environment vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509 776 Soil Dynamics and Earthquake Engineering tations are studied. It is concluded that perturbation solutions are good approximations in the case of random damping. However, in the case of random stiffness, significant deviations are observed, especially for short duration earthquakes. On the other hand, perturbations solutions seem to be admissable and useful in the estimation of the maximum variance of the response which is expected within the first few natural periods.