Many structures in civil engineering are excited by dynamic forces, which may be generated by earthquakes and winds. While deterministic models have traditionally been used to describe these forces, stochastic models offer a more realistic representation, including white noise processes, real noise processes, or bounded noise processes. The moment Lyapunov exponent, a key tool for analyzing the stochastic stability of structures, has primarily been applied to linear systems. However, its applicability to non-linear structures remains underexplored. In this paper, the stochastic stability of strongly non-linear structural systems subject to parametric excitations of white noise processes is investigated through moment Lyapunov exponents. The method of stochastic averaging is formulated to derive a system of stochastic differential equations. Then Khasminskii’s transformation is applied to determine the moment Lyapunov exponent. The proposed procedure is applied to study the stochastic stability of the plane motion of a beam under axial stochastic compressive load. The stability conditions are determined by examining the behaviors of the averaged square-root of total energy at its boundaries. Understanding structures’ stability and their response to dynamic loads will lead to efficient structural design.