A study on hybrid finite elements for plate bending

Abstract

Alternative to the conventional finite element method, there is another version named hybrid finite element in which the displacement fields and the stress fields are assumed independently and simultaneously for the element stiffness formulation. The method regarded as systematic and effcient in formulating the stiffness is the variation of the energy functional. The Hellinger-Reissner energy functional and its modified version are employed in the present study. Numerous hybrid plate elements based on these energy functionals have been proposed in the past. Some four-node quadrilateral elements with interior displacement assumptions were formulated. The stress classification method was used as a guideline in order to obtain the stress matrix. The stress matrix, which is free from kinematic deformation modes in the element stiffness, can easily be obtained. Numerical studies confirmed that using too many stress parameters in the assumed stresses wouldresult in an overstiff model and more computational effort. Proof of the optimization method through the penalty-equilibrium matrix showed that a penalty constant tends to eliminate the stress modes which do not satisfy the equilibrium equations from the element stiffness formulation-as if those stress modes are not present in the stress matrix. Therefore, kinematic deformation modes may be introduced into an element this way. Various types of plates with different boundary conditions and loading such as square plates, circular plates and thin rhombic plates were tested. Tests were concentrated on convergence of displacement and moment, shear-locking as well as invariance property. Tests on accuracy of displacement showed that most of the elements gave more than 95 percent accuracy regardless of variation in thickness and aspect ratio for moderate mesh refinement. Some good elements gave more than 99 percent accuracy despite some shear-locking. Accuracy of moments averaged more than 90 percent. All elements,however, lack inveriance to some degree. Comparison of the overall performance ofthe hybrid elements studied indicates that the element HBP1 and HBP2 are the most efficient among those elements compared. These versatile elements perform very well in all tests, especially, the shear-locking test.