High-order Hybrid Stress Mindlin Plate Element And Enhanced Patch Test

The patch test which can be used to examine the convergence of the element and construct a convergence element, has been seen as a criterion for assessing the convergence of the finite element for a long time. However, the applications of the existing patch test are limited in problems with homogeneous differential equations such as 2D,3D and thin plate problem et al. It is hard to accurately assess the convergence of the problem with non-homogeneous differential equations, for example, none of the current existing Mindlin plate elements can pass the non-zero constant shear stress patch test. There has been no test method on Mindlin plate convergence, unlike the elastic plane and space element has constant stress patch test methods. This is a legacy issue of Mindlin plate element. The existing Mindlin plate elements can only pass the zero shear stress patch test; they can not pass the non-zero constant shear stress patch test. The convergence of element is an unavoidable issue which has been a longstanding problem. Investigators who know this problem on Mindlin plate have devoted themselves to this issue, but could not find a feasible solution, while quite a few researchers may not be consciously aware of the difficulty of the problem. Mindlin plate elements are widely used by large commercial software, many authors think there is no problem, but it is not rigorous in theory. The programs of this commercial software have no proof of convergence. There has been a way to prove the convergence and the basic of the approach is the enhanced patch test. This test is stronger than the original test; the original constant stress patch test is just a special case of it. This dissertation establishes the triangular and quadrilateral Mindlin plate elements which can pass the non-zero constant shear stress patch test, so as to establish the convergence of Mindlin plate element, the main works include as follows:1) Based on the variational principle, the enhanced patch test for Mindlin plate theory is analyzed. It should be noted that the individual element condition is the sufficient condition for passing the constant stress patch test of homogeneous differential equations and without any external forces. The current existing Mindlin plate element cannot pass the non-zero constant shear patch test with the 3rd order test function.2) A 6-node triangular hybrid stress element is presented for Mindlin plate in this paper. The proposed element, denoted by TH6-27?, can pass both the zero shear stress patch test and the non-zero constant shear stress enhanced patch test and, it can be employed to analyze very thin plate. To accomplish this purpose, special attention is devoted to selecting boundary displacement interpolation and stress approximation in domain. The arbitrary order Timoshenko beam function is used successfully to derive the displacement interpolation along each side of the element. According to the equilibrium equations, an appropriate stress approximation is rationally obtained. The assumed stress field is modified by using 27? instead of 15? to improve the accuracy. Numerical results show that the element is free of shear locking, and reliable for thick and thin plates. Moreover, it has no spurious zero energy modes and with geometric invariance (coordinate invariance, node sequencing independence).3) An 8-node quadrilateral assumed stress hybrid Mindlin plate element with 39? is presented. The proposed element is free of shear locking and is capable of passing all the patch tests, especially the non-zero constant shear enhanced patch test. To accomplish this purpose, special attention is devoted to selecting boundary displacement interpolation and stress approximation in domain. The arbitrary order Timoshenko beam function is successfully used to derive the boundary displacement interpolation. According to the equilibrium equations, an appropriate stress approximation is rationally derived. Particularly, in order to improve element's accuracy, the assumed stress field is derived by employing 390 rather than conventional 21?. The resulting element can be adopted to analyze both moderately thick and thin plates, and the convergence for the very thin case can be ensured theoretically. Excellent element performance is demonstrated by a wide of experimental evaluations..4) This paper investigates the free vibration of plate with different boundary conditions and thicknesses, and the lumped mass matrix is adopted. Numerical results show that the element can be used to analyze both moderately thick and thin plates and it is efficient and accurate.