The current work presents a formulation of stochastic B-spline wavelet on the interval (BSWI)-based wavelet finite element method (WFEM) for analysis of beams incorporating von Kármán nonlinear strains. The spatial variation of the modulus of elasticity is modelled as a homogeneous random field. The proposed formulation is given for both Euler–Bernoulli beam theory and Timoshenko beam theory. For the discretization of both random field and response, BSWI scaling functions are used. A set of three nonlinear equations is derived based on the perturbation approach for evaluating the derivatives of the field variable with respect to random variables. Numerical examples under different boundary conditions based on the proposed formulations are solved and compared with solutions obtained from Monte Carlo simulation (MCS). The effect of the coefficient of variation and correlation length parameter on the response statistics is examined. In addition, a comparison of normalized computational times obtained from the perturbation approach and MCS is carried out. © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature.