An 8-Node Plane Hybrid Element for Structural Mechanics Problems Based on the Hellinger-Reissner Variational Principle

The finite element method (FEM) plays a valuable role in computer modeling and is beneficial to the mechanical design of various structural parts. However, the elements produced by conventional FEM are easily inaccurate and unstable when applied. Therefore, developing new elements within the framework of the generalized variational principle is of great significance. In this paper, an 8-node plane hybrid finite element with 15 parameters (PHQ8-15,8) is developed for structural mechanics problems based on the Hellinger-Reissner variational principle. According to the design principle of Pian, 15 unknown parameters are adopted in the selection of stress modes to avoid the zero energy modes. Meanwhile, the stress functions within each element satisfy both the equilibrium and the compatibility relations of plane stress problems. Subsequently, numerical examples are presented to illustrate the effectiveness and robustness of the proposed finite element. Numerical results show that various common locking behaviors of plane elements can be overcome. The PH-Q8-15,8 element has excellent performance in all benchmark problems, especially for structures with varying cross sections. Furthermore, in bending problems, the reasonable mesh shape of the new element for curved edge structures is analyzed in detail, which can be a useful means to improve numerical accuracy.