Method for calculation of the stress-strain state for cable-membrane space reflector structures

Cable-membrane space reflectors are widely used in the modern space industry. They are essential for communication, monitoring, and observation of the Earth and space objects. Experiments with actual reflector structures are quite expensive. Thus, effective calculation techniques should be applied to describe the reflector behavior under operative loads. A specific feature of such structures is its geometrical non-linear behavior, i.e., significant displacements of the elements under loads. Therefore, geometrical nonlinear governing equations of elasticity theory should be applied in describing the mathematical model of the reflector. The exact solution of these equations could be found only in the simplest cases. Thus, numerical methods for such equations should be used. This paper presents a two-stage calculation method of the stress-strain state for reflector structures based on force density and finite-element methods. The first stage embraces the calculation of the cable element shapes for reflector frontal (rear) nets by the nonlinear force density method. It has been proved that, in some cases, calculating the force density vector iteration step could be challenging due to the ill-conditioned matrix being a component part of this vector. To exclude this problem, the Moore-Penrose pseudoinverse matrix was applied. In the second stage, the calculated reflector frontal (rear) net shapes and corresponding values of cable tension were used as an initial estimate in determining the reflector node displacement field via the nonlinear finite-element method. The reflector stress-strain state is determined using a solution sequence in which every next solution involves the previous one as an initial estimation.