High-precision nonlinear comprehensive analysis of curved beams with shear effects under extreme loads via advanced mixed finite elements

This work presents novel mixed-type finite elements for geometrically nonlinear analysis of curved structures in high precision. In nonlinear problems, it is possible to approach exact results using a numerical method when an energy-based functional is used. However, the generation of such a functional structure has mathematical difficulties. In this study, these difficulties are overcome and a functional that is perfectly close to exact results is presented. The functional structure of the finite elements is derived based on the fact that the operator which corresponds to the equilibrium equations, constitutive equations and boundary conditions is potential. A necessary and sufficient condition that the operator is potential is used in two different aims. Instead of using the virtual work expression, both the equilibrium equations consistent with generalized strains subject to the first-order approximation of strain tensor and the proper relations for the generalized strains consistent with the equilibrium equations written directly on the deformed configuration are obtained. It is thought that the structure presented in this study provides highly accurate numerical results that can be obtained with the finite element method in the geometric nonlinear analysis of the beams.