An efficient shear and bending-locking-free quadrilateral plate element using a modified Hellinger-Reissner functional and the Bergan free formulation

This paper presents a general variational principle as theoretical support for creating a simple and efficient quadrilateral plate finite element called BKWA, having only a single displacement and two rotations at corner nodes according to the Reissner-Mindlin plate theory. The functional is a modified Hellinger-Reissner in terms of the kinematic variables and independent transverse shear strains. Hence, we consider quadratic rotations, as in DKQ, and linear independent shear strains with constant tangential components on sides, as in MITC4. The quadratic rotation contributions and the mixed shear strain components are wiped out by static condensation. To adequately satisfy the constant curvature patch tests, we also consider the orthogonality conditions between the linear and quadratic bending modes, as in the Bergan free formulation, to avoid ‘spurious bending energy’ for thick plates (also called bending locking). The paper contains comprehensive patch tests and convergence tests with regular and irregular meshes (displacements, internal forces, and strain energy) of circular and square plates. The element is compared with other existing elements (MITC4, DKMQ). It is concluded that BKWA is an efficient element regarding several criteria.

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