Finite deformation of elastic curves and surfaces

Cable and membrane structures have always been under consideration by engineers as structural elements because of advantages like high strength to weight ratio and low cost over other structures. But due to the highly nonlinear geometrical and material behavior of such structures, their design and analysis is a difficult task. Also, treatments that consider both elements in a structure and their interaction is rare to find. In this work, the equilibrium equations for the coupled finite deformations of perfectly flexible elastic membranes and cables is established. In order to incorporate the necessary conditions for stability of the equilibrium configuration, relaxed strain energy functions for cable and membrane are used which eliminate the possibility of the existence of compressive stresses (which cannot be carried by cables and membranes) in the structures and introduce wrinkles. The difference form of the equations of equilibrium is derived using Green's theorem and a numerical method called dynamic relaxation (which considers the problem as a damped dynamic one) is used to obtain the equilibrium configuration as the steady state response to the dynamical problem. Several examples are solved using this method. Two interesting cases among them are neutral holes (elliptic and circular) and inclusion. It is seen that the cable reinforcement reduces the sharp strain gradients specially at singular points. Qualitative behavior of cable reinforcement is shown in an experiment.