Development of finite elements for two-dimensional structural analysis using the integrated force method

Abstract

The integrated force method has been developed in recent years for the analysis of structural mechanics problems. In the intgrated force method all independent forces are treated as unknown variables, which are calculated by simultaneously imposing equations of equilibrium and compatibility conditions. The development of a finite element library for the analysis of two-dimensional problems using the integrated force method is presented in this paper. Elements of triangular and quadrilateral shapes, capable of modeling arbitrary domain configurations are developed. The element equilibrium and flexibility matrices are derived by discretizing expressions for corresponding potential and complementary energies, respectively. Independent approximations of displacement and stress fields within finite elements are performed. Interpolation of the displacement field is done similarly as in the standard displacement method. The stress field is approximated using full polynomials of correct orders. A procedure for deriving the stress interpolation polynomials that utilizes the definitions of stress components in terms of Airy's stress function is developed. Such derived stress fields identically satisfy equations of equilibrium, and the resulting element matrices are insensitive to the orientation of local coordinate systems. A method to calculate the number of rigid body modes is devised, and it is shown that the present elements do not possess spurious zero energy modes. A number of example problems are solved using the present library and the results are compared with corresponding analytical solutions and those obtained from the standard displacement finite element method. A good agreement of the results, and better performance of the integrated force method, compared to the displacement method, in stress calculations, is observed.