Hybrid displacement function element method: a simple hybrid‐Trefftz stress element method for analysis of Mindlin–Reissner plate

Abstract

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...

Two generalized conforming quadrilateral Mindlin–Reissner plate elements based on the displacement function

“…The solutions of the deflection, rotations and strains can be readily derived from the fundamental analytical solutions of displacement function F [29]. For the general (homogeneous) part of F, the displacement components and strains are listed in Table 1.…”

“…The deflections w j (j¼5-8) are determined by using the locking-free formulae of Timoshenko's beam [29]:…”

“…These two quadrilateral elements are denoted as GCP4-12α and GCP4-14α, respectively, indicating that they are "Generalized Conforming Plate element with 4 nodes and 12 or 14 displacement coefficients". Since the displacement components derived from the displacement function F can satisfy all related governing equations, these new elements also possess the advantages from analytical methods, as the HDF elements [29] do.…”

“…By combining the displacement function F [29,30] with the genera lized conforming element method [1], this work develops two highperformance displacement-based Mindlin-Reissner plate elements. The concept of the generalized conforming element was first proposed by Long et al [31], and since then has been successfully applied to plane problems [32,33], plate problems [34][35][36][37] and shell problems [38] as well.…”

“…Ribatić et al [28] constructed two 9-node distortion-immune displacement-based quadrilateral thick plate elements by using bubble parameters. Cen et al [29] proposed a hybrid displacement function (HDF) method for formulating Mindlin-Reissner plate elements, where they took the displacement function F [30] to derive the stress trial functions that satisfy all governing equations and employed the Timoshenko's beam formulae to determine boundary displacement modes. The HDF element enables superior precision for both displacements and resultants, even in cases where a severely distorted mesh containing concave quadrilateral or degenerated triangular elements is used.…”