The problem of determining the probability distribution function of extremes of Von Mises stress, over a specified duration, in linear vibrating structures subjected to stationary, Gaussian random excitations, is considered. In the steady state, the Von Mises stress is a stationary, non-Gaussian random process. The number of times the process crosses a specified threshold in a given duration, is modeled as a Poisson random variable. The determination of the parameter of this model, in turn, requires the knowledge of the joint probability density function of the Von Mises stress and its time derivative. Alternative models for this joint probability density function, based on the translation process model, combined Laguerre-Hermite polynomial expansion and the maximum entropy model are considered. In implementing the maximum entropy method, the unknown parameters of the model are derived by solving a set of linear algebraic equations, in terms of the marginal and joint moments of the process and its time derivative. This method is shown to be capable of taking into account non-Gaussian features of the Von Mises stress depicted via higher order expectations. For the purpose of illustration, the extremes of the Von Mises stress in a pipe support structure under random earthquake loads, are examined. The results based on maximum entropy model are shown to compare well with Monte Carlo simulation results.