In order to develop robust finite element models for analysis of thin and moderately thick plates, a simple hybrid displacement function element method is presented. First, the variational functional of complementary energy for Mindlin-Reissner plates is modified to be expressed by a displacement function F , which can be used to derive displacement components satisfying all governing equations. Second, the assumed element resultant force fields, which can satisfy all related governing equations, are derived from the fundamental analytical solutions of F . Third, the displacements and shear strains along each element boundary are determined by the locking-free formulae based on the Timoshenko's beam theory. Finally, by applying the principle of minimum complementary energy, the element stiffness matrix related to the conventional nodal displacement DOFs is obtained. Because the trial functions of the domain stress approximations a priori satisfy governing equations, this method is consistent with the hybrid-Trefftz stress element method. As an example, a 4-node, 12-DOF quadrilateral plate bending element, HDF-P4-11ˇ, is formulated. Numerical benchmark examples have proved that the new model possesses excellent precision. It is also a shape-free element that performs very well even when a severely distorted mesh containing concave quadrilateral and degenerated triangular elements is employed. 204S. CEN ET AL. the improved shear strain interpolation schemes derived from the formulae of the Timoshenko's beam [21][22][23][24][25][26], the degenerated shell element method [27], the geometrically exact shell element method [28,29], and so on. Recently, new Mindlin-Reissner elements are still emerging in many literatures: Nguyen-Xuan et al. [30] proposed a smoothed curvature method to develop quadrilateral elements; Castellazzi et al. [31] used nodal integration method to construct displacement-based models; Hu et al. [32] presented a combined hybrid method; Hansbo et al. [33] formulated new element with discontinuous rotations; Nguyen-Thanh et al. [34] proposed an alternative alpha finite element; Falsone et al. [35] developed a new model using the Kirchhoff-like solution; Nguyen-Thoi et al. [36] proposed a cell-based smoothed discrete shear gap method; Ribaric et al.[37] studied the quadrilateral elements using higher-order linked interpolation; Vu-Quac et al. [38] used efficient hybrid-EAS solid element for accurate stress prediction in thick plates; and so on. It is very interesting that this topic attracts attentions from so many researchers and has become a platform for testing new numerical methods [39].Besides the aforementioned shear locking problem, a good plate bending element should also satisfy the general requirement for finite element technologies: (1) tolerance to various mesh distortions and (2) high-precision results for stress/resultant-force solutions, as well as displacement. In 2006, Cen et al. [40] developed a quadrilateral Mindlin-Reissner plate bending element with the generalized conforming co...

SummaryIn bending problems of Mindlin–Reissner plate, the resultant forces often vary dramatically within a narrow range near free and soft simply‐supported (SS1) boundaries. This is so‐called the edge effect or the boundary layer effect, a challenging problem for conventional finite element method. In this paper, an effective finite element method for analysis of such edge effect is developed. The construction procedure is based on the hybrid displacement function (HDF) element method [1], a simple hybrid‐Trefftz stress element method proposed recently. What is different is that an additional displacement function f related to the edge effect is considered, and its analytical solutions are employed as the additional trial functions for the first time. Furthermore, the free and the SS1 boundary conditions are also applied to modify the element assumed resultant fields. Then, two new special elements, HDF‐P4‐Free and HDF‐P4‐SS1, are successfully constructed. These new elements are allocated along the corresponding boundaries of the plate, while the other region is modeled by the usual HDF plate element HDF‐P4‐11 β [1]. Numerical tests demonstrate that the present method can effectively capture the edge effects and exactly satisfy the corresponding boundary conditions by only using relatively coarse meshes. Copyright © 2015 John Wiley & Sons, Ltd.

Summary A recent unsymmetric 4‐node, 8‐DOF plane element US‐ATFQ4, which exhibits excellent precision and distortion‐resistance for linear elastic problems, is extended to geometric nonlinear analysis. Since the original linear element US‐ATFQ4 contains the analytical solutions for plane pure bending, how to modify such formulae into incremental forms for nonlinear applications and design an appropriate updated algorithm become the key of the whole job. First, the analytical trial functions should be updated at each iterative step in the framework of updated Lagrangian formulation that takes the configuration at the beginning of an incremental step as the reference configuration during that step. Second, an appropriate stress update algorithm in which the Cauchy stresses are updated by the Hughes‐Winget method is adopted to estimate current stress fields. Numerical examples show that the new nonlinear element US‐ATFQ4 also possesses amazing performance for geometric nonlinear analysis, no matter whether regular or distorted meshes are used. It again demonstrates the advantages of the unsymmetric finite element method with analytical trial functions.

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Summary The unsymmetric finite element method is a promising technique to produce distortion‐immune finite elements. In this work, a simple but robust 4‐node 12‐DOF unsymmetric quadrilateral membrane element is formulated. The test function of this new element is determined by a concise isoparametric‐based displacement field that is enriched by the Allman‐type drilling degrees of freedom. Meanwhile, a rational stress field, instead of the displacement one in the original unsymmetric formulation, is directly adopted to be the element's trial function. This stress field is obtained based on the analytical solutions of the plane stress/strain problem and the quasi‐conforming technique. Thus, it can a priori satisfy related governing equations. Numerical tests show that the presented new unsymmetric element, named as US‐Q4θ, exhibits excellent capabilities in predicting results of both displacement and stress, in most cases, superior to other existing 4‐node element models. In particular, it can still work very well in severely distorted meshes even when the element shape deteriorates into concave quadrangle or degenerated triangle.