The response of simple structural systems to stationary random excitation is considered under two criteria of failure. When failure is specified as the crossing of a maximum tolerable threshold by the response, the reliability of a structure is commonly measured by means of response spectra. These give the expected maximum value of the response parameter for a given excitation level. The statistical variations in these spectra are obtained here for viscously damped linear and elastoplastic single-degree of freedom systems by electronic analog simulation. The results obtained are compared with approximate statistical analyses; for example, the threshold crossing statistics of narrow-band oscillators. It is concluded that such methods give satisfactory, but conservative, estimates of the mean spectral values. It is significant that all the spectra obtained showed a very wide distribution about the mean. This was also true of the Fourier amplitude spectrum of the excitation. For responses that are so large that structures actually collapse, the linear model was replaced by an elastoplastic system, and the effect of gravity on the collapse time was considered. Experimental simulation showed that the structural response in this case is essentially that of a linear oscillator with yielding occurring at intermittent intervals. Gravity acts to increasingly bias this yielding in one direction, eventually causing instability in the system. Collapse of the system was sensitive to the distribution of peaks in the excitation and it was found that the wide dispersion in the collapse time can be reasonably represented by a Gamma distribution function. An analytic method for estimating the mean collapse time was derived by considering the energy distribution of the excitation and its effect on the yielding of the structure. The response process was thus modelled by that of an equivalent linear oscillator whose baseline is biased by the yielding in the structure. It was concluded that this procedure gives a good estimate of the failure time for excitations strong enough to cause failure in less than 20 seconds. A two- degree of freedom elastoplastic hysteretic system with gravity was also simulated. In a certain sense, the qualitative behavior is similar to that of the single-degree of freedom system. It was thus possible to estimate the failure time of the structure from that of a single-degree of freedom system once the transmission of vibration is accounted for by considering a linear two-degree of freedom system.