Membrane structures play an important role in a wide variety of applications, ranging from solar sails, satellites and aircraft, to pneumatic structures, lightweight temporary constructions and biological tissues.  In this research project a numerical model has been developed for stress analysis in isotropic elastic membranes undergoing finite deformations, while partly or totally subjected to pressure loading of the hydrostatic type. The possibility of wrinkling is accounted for by employing a Relaxed Strain Energy Density. The numerical procedure is based on the Dynamic Relaxation Method. This is an explicit, iterative technique in which the static solution is obtained as the steady state part of the damped dynamic response of the structure. The numerical scheme is constructed by applying Green's theorem differencing method for the spatial discretization of the partial differential equations describing the damped motion of the membrane. The resulting system of ordinary differential equations is further integrated in time by a central difference time integrator. Solutions of typical boundary value problems are obtained and analyzed. The effects of various loading and boundary conditions on the response of the membrane are examined. The predictions of the numerical method are in good agreement with the results obtained with experimental models.