This paper presents a finite element methodology for the static analysis of infinite periodic structures under arbitrary loads. The technique hinges on the method of representative cell which through the discrete Fourier transform reduces the original problem to a boundary value problem defined over one module, the representative cell. Starting from the weak form of the transformed problem, or from the FE equations of the infinite structure, the equilibrium equations are written in terms of the complex-valued displacement transforms which are considered as the displacements in the representative cell. Having found the displacements in the transformed domain, the real displacements anywhere in the real structure are obtained by numerical integration of the inverse transform. The theory, which is valid for spatial structures with 1D up to 3D translational symmetry, is illustrated with examples of periodic structures having 1D translational symmetry under general static loading.