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We develop a stochastic Galerkin finite element method for nonlinear elasticity and apply it to reinforced concrete members with random material properties. The strategy is based on the modified Newton-Raphson method, which consists of an incremental loading process and a linearization scheme applied at each load increment. We consider that the material properties are given by a stochastic expansion in the so-called generalized polynomial chaos (gPC) framework. We search the gPC expansion of the displacement, which is then used to update the gPC expansions of the stress, strain and internal forces. The proposed method is applied to a reinforced concrete beam with uncertain initial concrete modulus of elasticity and a shear wall with uncertain maximum compressive stress of concrete, and the results are compared to those of stochastic collocation and Monte Carlo methods. Since the systems of equations obtained in the linearization scheme using the stochastic Galerkin method are very large, and there are typically many load increments, we also studied iterative solution using preconditioned conjugate gradients. The efficiency of the proposed method is illustrated by a set of numerical experiments.