On a mixed finite element method for clamped anisotropic plate bending problems

Abstract

The paper deals with a new mixed finite element method of solution of the bending problem of clamped anisotropic/orthotropic/isotropic plates with variable/constant thickness. This new mixed method gives simultaneous approximations to displacement u and bending and twisting moment :ensor ($*J, i, jp 2. Computer implcmentttion procedures for this mixed method are given along with results of numerical experiments on a good number or interesting problems. bent anisotropic plate such that (PI: Au =fin a ulr = (i?u/r3n)(, = 0 where thc anisotropic plate operator A is defined by (in (2) and ulso in the sequel, the Einstein's summation convention with respect 10 twice repeated indices i, j , k, 1 = I , 2, is to be understood), aijkt = aijkl(x 1, x2) V (x , x2 j c Tz = n u r are anisotropic platc coeficients3 satisfying the following conditions: Vi,J k, I = 1,2, a i j k l E C"(a); rzijk,(x) =a#.,ij(x) = aklji(x) alkji(x)b x = ( X I , X 2 ) E a (33 and explicitly given in the notations of References 15 and 23 as foilows: uiiii=Dii(i= 1,2); a1212=a1221 = n 2 1 1 2 = u 2 1 2 1 = D S 6 ; u1122 = a 2 2 1 1 = D I 2 (4 Dij=Dij(xl,x2)tl(x,,x2)~~denoterigiditics15defined byDij= Ri,it3/12(i= 1,2j= I, 2,6), the Bi,:s being expressions in terms of elastic constants of the generalized Hooke's law for the anisotropic material of the thin plate given in Reference 15, r .t ( x l , x 2 ) being the thickness of the plate at the point ( x , , x2)€fi, such that =a121l = a I l l 2 = ~l l t l = a 2 ] l l =DIG; a1222=02322 = ~2 2 1 2 = u2221 = DM D11, Dtt, D66 > 0 , D 1 2 =~1 1~2~= v 2 D 1 1 (OG vi < 1/2), O < Di6 < (1 -~j ) D , i (i # j ) 1 < i,j < 2, D,,+Djzc, < D66 (5) CL4MPED ANISOTROPIC PLATE BFNDING PROBLEMS

Alternativno rješenje za križno uslojene kompozitne debele ploče

“…They can be written as follows: (4) with E i being the Young's modulus in the i th material direction, μ ij is the Poisson ratio, and G ij , is the shear modulus of the i-j plane. Constitutive equations for the k th orthotropic layer are transformed to the laminate coordinates (x, y, z) as follows: (5) where are the transformed elastic constants or stiffness matrix with respect to the laminate coordinates (x, y, z) [2]. The elements of the matrix can be written as follows: (6) where θ is the angle between the global axis and the local x-axis of each layer.…”

“…Static behaviour of orthotropic rectangular thick plates is examined in [4] using initial functional methods to analyse uniformly loaded simply supported square plates for various thickness and material properties. The mixed finite element method is used in [5] for the bending problem of clamped anisotropic / orthotropic / isotropic plates with variable/ constant thickness. A mesh-free method is used in [6] for the static and free vibration analysis of shear deformable laminated composite plates with various side to thickness ratios, material coefficients, boundary conditions or ply angles.…”

Error estimates for isoparametric mixed finite element solution of 4th order elliptic problems with variable coefficients

“…But such estimates for classical, standard finite element methods for 2nd order elliptic problems have been obtained, for example, in [16,18,[33][34][35][38][39][40][41] and 4th order problems, for example, in [8,27]. For mixed/hybrid finite element schemes for this fourth order elliptic problem in convex, polygonal domains under the assumptions that no boundary approximation is necessary and an exact evaluation of the integrals of the corresponding bilinear forms is possible, we refer to [5,7,9,10,21,31].…”