A New Triangular Hybrid Displacement Function Element for Static and Free Vibration Analyses of Mindlin-Reissner Plate

Abstract

A new 3-node triangular hybrid displacement function Mindlin-Reissner plate element is developed. Firstly, the modified variational functional of complementary energy for Mindlin-Reissner plate, which is eventually expressed by a so-called displacement function F, is proposed. Secondly, the locking-free formulae of Timoshenko's beam theory are chosen as the deflection, rotation, and shear strain along each element boundary. Thirdly, seven fundamental analytical solutions of the displacement function F are selected as the trial functions for the assumed resultant fields, so that the assumed resultant fields satisfy all governing equations in advance. Finally, the element stiffness matrix of the new element, denoted by HDF-P3-7β, is derived from the modified principle of complementary energy. Together with the diagonal inertia matrix of the 3-node triangular isoparametric element, the proposed element is also successfully generalized to the free vibration problems. Numerical results show that the proposed element exhibits overall remarkable performance in all benchmark problems, especially in the free vibration analyses.

Shape‐free arbitrary polygonal hybrid stress/displacement‐function flat shell element for linear and geometrically nonlinear analyses

“…For linear formulations, two recent advanced FEMs with analytical trail functions are employed to formulated the new element, in which the bending part of the element is constructed by the HDF element method for Mindlin-Reissner plate, [32][33][34][35][36][37][38] and the membrane part with drilling DOF is formulated by the HSF element method. 32,[39][40][41][42][43][44] Through a best-fit analytical trial functions selection scheme for different number of element edges, the final linear element formulations can be obtained. Then, the linear flat shell element is generalized to F I G U R E 26 Load-displacement curves of hinged cylindrical shell geometrically nonlinear applications by employing the CR method, in which a best-fit corotated frame for polygonal elements is defined to achieve the best accuracy.…”

“…On the other hand, the hybrid-Trefftz element methods is also an effective way for developing various FEMs in many fields, such as the T-Trefftz Voronoi cell finite elements, 24,25 and the hybrid-Trefftz FEMs for poroelastic media, 26 plate bending, 27,28 elastoplassticity, 29 Helmholtz problem, 30,31 heat conduction, 32 and so forth. In the past 10 years, Cen et al proposed two hybrid-Trefftz FEMs, a hybrid stress-function (HSF) [33][34][35][36][37][38][39] and a hybrid displacement-function (HDF) 33,[40][41][42][43][44][45] element methods for developing 2D plane and Mindlin-Reissner plate elements, respectively. The constructions of these element are simply extended from the original hybrid stress element proposed by Pian.…”

“…Then, by adopting proper element boundary displacement modes and applying the principle of minimum complementary energy, the F I G U R E 1 Examples for discretely distributed loading and materials: (A) Industrial silos on floor plate; (B) crystal materials 3 element stiffness matrix related to the conventional nodal displacement degrees of freedoms (DOFs) can be obtained. Following above procedure, Cen et al proposed several 8-and 12-node plane quadrilateral HSF elements, 34 a 4-node plane quadrilateral HSF element with drilling degree of freedom, 35 and a plane HSF element for anisotropic material analyses by employ the anisotropic analytical solutions 36 ; Cen et al, 40 Shang et al, 41 Bao et al, 42 and Huang et al 43 developed several triangular and quadrilateral HDF elements for Mindlin-Reissner plate. Shang et al 44 developed a 4-node quadrilateral flat shell element by combining 4-node HSF plane and HDF plate elements.…”

An efficient three-node triangular Mindlin–Reissner flat shell element

“…The obtained results show all cases have acceptable performance; moreover, the bending element with the first eleven polynomials (forth-order completeness) provides better performance when combined with the considered membrane element than the other cases. The main difference between the proposed bending element and the one presented by Huang et al [25] is the order of considered polynomial approximations in the trial function. These polynomials are illustrated in Table 1.…”

“…Furthermore, using generalized conforming theory they improved this method [24] for orthotropic Mindlin-Reissner plate. For free vibration analysis of Mindlin-Reissner plate, Huang et al [25] proposed a triangular bending element by employing hybrid displacement function element method. The results of these studies proved the efficiency of the hybrid displacement function element method.…”