This paper presents the equilibrium analysis of a membrane with rigid boundary. The idea of using membranes in applications such as footbridges, a new technology being developed in Spain, implies a more accurate analysis procedure. Due to the tension stresses, membrane is identified to a negative gaussian curvature surface. Equilibrium is directly expressed by means of partial differential equations, in terms of the membrane shape and stress tensor. Starting from these equations, two dual approaches can be defined, namely direct problem and dual problem. Both problems are analyzed, studying their possible solutions in order to obtain practical results. In particular, the main analytical aspects of the direct problem are discussed, a numerical resolution procedure is proposed and, finally, analytical and numerical solution examples are presented.