The nonlinear stochastic vibrations of a beam with a varying cross-section are investigated. The nonlinearity is caused by midplane stretching and cubic in nature, and the forcing function is wide band white noise. The analysis is carried out by expanding the deflection curve in terms of the undamped linear modes. Substituting this expansion into the partial differential equation yields a set of ordinary differential equations in terms of the modal response functions, which are coupled through the nonlinear terms. The normal modes are found by the finite element method. The differential equations are then converted to a set of Ito's equations, from which a set of first-order differential equations for the response joint moments is found using the Fokker-Planck equation. These equations form an infinite hierarchy which is closed by the quasi-moment method. The solution is investigated near an internal resonance condition and the effects of higher order cumulants in the closure scheme and of additional modes to the expansion arc considered. It is shown that the second order solution is inadequate in the presence of internal resonances, but the fourth order solution proves to be adequate. The one mode approximation underestimates the nonlinear stiffening, and a multiple mode approach is necessary. It is also shown that the effect of an internal resonance of the stochastic vibration is to transfer of energy from the higher modes involved to the lower modes involved. / M.S.