The current paper presents the formulation of stochastic B-spline wavelet on the interval (BSWI) based wavelet finite element method (WFEM) for beams wherein, the spatial variation of modulus of elasticity is modelled as a homogeneous random field. Stochastic beam element formulations based on both Euler-Bernoulli beam theory and Timoshenko beam theory are proposed. BSWI scaling functions are used for the discretization of the random field and the response statistics are obtained using the perturbation approach. Numerical examples are solved and the results from perturbation approach are compared with that obtained from Monte Carlo simulation (MCS). A parametric study is also done to understand the effect of different coefficient of variation (CV) values and correlation length parameters on the response statistics. The study concludes that the proposed BSWI WFEM based perturbation approach for beams produce accurate response statistics for values of CV less than 15%. A comparative study is carried out between the results obtained from the proposed stochastic WFEM with stochastic finite element method (SFEM) wherein the random field discretization is done using Lagrange shape functions. Furthermore, normalized computational times for the execution of perturbation approach and MCS based on WFEM are evaluated and compared with those obtained for SFEM. © 2020 Elsevier Ltd