A unified approach in solving equilibrium problems of standard cells of membrane structures made of various absolutely flexible film and fabric materials is presented in this paper. The objects of study are rectangular membranes. The problems were considered in a geometrically nonlinear formulation, with the deformations and the squares of the rotation angles thereunder being considered to be comparable with each other, but small compared to unity. A resolving system of differential equations in partial derivatives expressed in a mixed form is obtained therewith. These equations combining with the presented boundary conditions are numerical models of a number of fragments of real membrane structures. The closed nonlinear system of equations was integrated using the continuation method. Therewith, the known solution for a square isotropic membrane was used to select the initial values of stresses and displacements. The problem of equilibrium of a square isotropic membranes rigidly fixed under a uniformly distributed load is presented as an example. The resulting graphs and tables show the distribution of forces and displacements. They may be used for calculating the membrane structures. The developed technique may be applied to those values of the initial parameters under which the calculations have not yet been made.