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The Monte Carlo method is widely used for the estimation of uncertainties in mechanical engineering design. However, while flexible, this method remains impractical in terms of computational time and scalability. To bypass these limitations, other more efficient approaches such as the Galerkin stochastic finite element method (GSFEM) or the collocation method have been proposed. GSFEM provides accurate output statistics, has the advantage of being sampling independent and can be modular in terms of operations, albeit code intrusive. While linear elasticity has been extensively covered in the literature, the application of GSFEM to nonlinear mechanical behaviour remains relatively unexplored, in part due to the difficulty to capture nonlinear effects with the traditional GSFEM interpolants. To this end, we propose a seamless and efficient modular framework avoiding the need to know a priori the material law. In particular, the method makes use of a wavelet based formulation able to capture simultaneously continuous and discontinuous behaviours, and stochastic operators are proposed to straightforwardly adapt any material model. Finally, the flexibility of this approach is illustrated with two problems: (i) a 3D hyperelastic example with nonlinear behaviour arising from buckling uncertainty, and (ii) the evaluation of the displacement of a point within a structure based on how much of the external accessible structural deformation is known. • A stochastic finite element method framework is presented. • A stochastic algebra library is provided for easy integration in FE code. • Polynomial chaos, Haar and wavelet expansions are considered. • Wavelet expansion is shown to capture mixed smooth/non-smooth problems. [ABSTRACT FROM AUTHOR]