A formulation for the stochastic finite element method is presented which is a natural extension of the deterministic finite element method. Discretization of the random dimension is achieved via two spectral expansions. One of them is used to represent the coefficients of the differential, equation which model the random material properties, the other is used to represent the random solution process. The method relies on viewing the random aspect of the problem as an added dimension, and on treating random variables and processes as functions defined over that dimension. The versatility of the method is demonstrated by discussing, as well, some non-traditional problems of stochastic mechanics.