Study On The Finite Element Method Based On The Complementary Energy Principle For Large Elastic Deformation

Based on the complementary energy principle (CEP) for large elastic deformation proposed by Gao yuchen, the generalized complementary energy principles (GCEP) are achieved, and the finite element method (FEM) formulas including the rigid rotation are established, which are applied to the calculation of the plane problems for large deformation. The main studies in text are mentioned as follows:1. Based on the CEP proposed by Gao yuchen, the GCEP (1) is deduced by relaxing the limitation of the constraint conditions to the CEP functional, in which the base forces and the displacements are independent functions. Dividing the complementary energy density into the pure deformation part and the rotation part, and using the rigid rotation displacements as independent function, the GCEP functional (2) can be obtained.2. The complementary energy denoted the rigid rotation of an element is added to the finite element model proposed by Gao yuchen. Considering the force equilibrium conditions between elements, the FEM formulations based on the modified GCEP (2) are obtained.3. The constraint conditions for the CEP functional of plane rods structure are equilibrium equations on nodes. By these equations, some unknown inner forces can be represented with other ones. So there are unknown inner forces only in the complementary functional, and the pure CEP is obtained which can be used to structural analysis for large deformation.4. It is considered that the inner forces of rods are all along the axis directions, and the equilibrium conditions on nodes are that ones on the given force boundary, the GCEP with plane structure is achieved in terms of the GCEP (1). According to its stationary condition, a nonlinear equation set can be gained, which is simple and can be assembled easily by programming, and all inner forces and node displacements can be obtained only if the necessary information be input into the program edited by the author. From examples, it can be found that both the pure CEP and the GCEP can solve the geometrical nonlinear problems.5. The FEM formulations of quadrilateral plane element are given base on the modified GCEP (2), with which the simple shear deformation, the large deflection problems of shallow beams and circle arches are calculated. By the GCEP (2), it has been discussed the influences on deformation of the various shallow beams under various loads. When the circle arch is under uniformly distributed radial loads, it has been discussed the influence on deformation of the subtending angle, and the influence on the relation between loads and deflections of the slenderness ratio and the cross section height. The correctness and the feasibility of the CEP and the FEM proposed by Gao yuch have been validated.