The motion of a shallow arch subjected to random loading is investigated. It is expected that the arch will vibrate about its initial stable configuration under low intensity loading, but will snap back and forth about its horizontal axis under high intensity loading. The parameter of interest is the time of first snap-through of the shallow arch under random load. This is defined as the time taken for the arch to snap to the other side of the horizontal axis. The statistics of the time to first snap-through, such as the mean time to failure as well as its probability distribution. are determined. Most of the work treats one response mode. In the first part of the study, the critical random loading for dynamic snap-through of the shallow arch was investigated using the method of computer simulation. The random excitation was assumed to be a stationary white-noise process. The primary object was to determine the critical power spectral density parameter of the random excitations. The vanishing or diminishing of the average frequency of snap-through was used to estimate this parameter. An exact value of the critical random loading parameter could not be obtained using this criterion since it was based on numerical integration of the non-linear equation of motion and computer simulation which is expensive and time-consuming. However, the critical value or range of critical values of intensity of random excitations could be estimated with a reasonable degree of accuracy. The second part of the study dealt with the first-passage problem. The exact solution of the first-passage problem is available for only a limited class of problems. In this study, the solution was obtained using numerical approximation techniques and computer simulation. For an oscillator subjected to white noise, the displacement and velocity process are governed by the Fokker-Planck-Kolmogorov (FPK) equation. The finite difference method was used to solve the derived FPK equation of the energy envelope of the equation of motion of the arch subjected to white-noise excitation. Solutions were obtained in terms of the mean time to failure, and the probability distribution function of the time to failure. A computer program for the Monte Carlo simulation of the response of the arch subjected to random loads was also developed. A large number of records of the random excitations were simulated and these were used as input in the numerical integration of the equation of motion. The Runge-Kutta method was used to obtain the time history of the displacement response, and the time at which the response exceeded the critical threshold was recorded. Statistics of the time to first snap-through were obtained and these were then used to select an empirical distribution model for the first-passage time. The results of the approximate analysis were compared with those from the simulation. Results of both methods were in close agreement. The effect of including more than one mode in the equation of motion was also studied. Multi-mode approximations of up to four modes were considered in the analysis. It was found that the results of the multi-mode approximations are significantly different from the one-mode approximation. The effect of nonstationary random excitation on the time to first snap-through was also investigated using computer simulation. / Ph. D.