The stability of elastic/plastic columns and circular rings with random geometric imperfections is investigated. Columns are analyzed for axial loading and rings for uniform external pressure. A procedure is developed to evaluate the reliability of imperfect elastic/plastic columns and rings against instability. The geometric imperfections are modeled as Gaussian random field in one dimension with given mean, variance, and covariance functions. The random field is discretized by the method of orthogonal series expansion using a Fourier series. The weak form of the boundary value problems for column and ring is formulated using Galerkin's method. A mixed finite element is used in which the primary degrees of freedom are transverse deflection and bending moment. For illustration purposes the material behavior is taken as elastic/perfectly plastic. The computationally efficient first- and second-order reliability methods are used to evaluate the failure probabilities and Monte Carlo simulation is used as a check. The variation of the probability of failure of columns and rings over a range of applied loads is presented for different amounts of random geometric imperfections.