Due to the inherent uncertainties in manufacturing properties and intrinsic variability of materials, the assumption of homogeneous input variables is generally not justified. As a result, stochastic forward problems have emerged as a tool to incorporate these uncertainties into numerical simulations, improving model prediction capability in structural analyses. Although most of the existing methods focus on the consideration of stochastic loading, the recently developed statFEM employs a Bayesian paradigm to incorporate data and propagate uncertainties from random physical properties in finite element models. This tool, however, is not developed for cases when Gaussian assumptions are inadequate. The present work provides a copula-based approach embedded into the statFEM methodology to propagate uncertainty from arbitrarily distributed physical properties. The random variables are defined in terms of a Gaussian Copula Process, where samples are drawn from a latent variable governed by a Gaussian Process and then brought respectively to copula and marginal spaces, producing random variables with the desired distribution while retaining the usually desired smooth Gaussian dependence in the spatial domain. The quality of the approximated results is then assessed in a simplified 1D Poisson problem by comparing with Monte Carlo sampling results for different random diffusion coefficients, demonstrating that the method is capable of providing good responses for non-Gaussian physical parameters.