A C⁰ triangular plate element with one‐point quadrature

Abstract

SUMMARYA triangular plate element of the Mindlin type, which uses linear fields for rotations and transverse deflections, is developed. Success is achieved, in contrast to previous failures with this element, by decomposing the displacements into well-defined bending and shear modes which are associated with bending and shear energies, respectively. The element only requires a single quadrature point, which is very attractive for nonlinear analysis. Results are presented for a variety of plate problems; the accurac is similar to the 4-node bilinear Mindlin plate and its rate of convergence is approximately of order h . YIn the analysis of nonlinear problems, particularly transient nonlinear problems, computation time and memory are often crucial factors. Since element stiffness computations are repeated many times it is advantageous to have efficient and simple elements. Consequently, much research is aimed at formulating accurate elements with these characteristics.'-* In the analysis of thin flexible structures, perhaps the most promising approach for developing simple and efficient elements is that based on independent approximations of the rotations and displacements combined with a reduced order of shear i n t e g r a t i~n .~-~'~-'~ As opposed to the C' continuity required in the Kirchhoff-type theory, only C' continuity need be satisfied in this approach. Consequently, lower order shape functions can be used which enhance simplicity. However, the use of the low-order shape functions necessitates reduced integration of the shear contribution to the stiffness m a t r i~;~ otherwise the elements are considerably too stiff. Fortunately this necessity further contributes to the efficiency of the C" element; reducing the number of the integration points reduces the number of computations and, along with the simple shape functions, results in extremely efficient elements. While this approach also has its drawbacks, such as possible zero energy modes, these appear only for certain boundary conditions and then can be effectively eliminated. '7"8 In some cases reduced integration may fail. For instance Batoz, Bathe and Ho' examine three different approaches to triangular elements and in their study the SRI (selective/reduced integration) triangular element was found to be ineffective. On the other hand, Hughes and Taylor" have developed a more successful one-point quadrature triangular element by overlapping two nodes of the quadrilateral:' but the element's performance is mediocre for certain element arrangements.The results of References 7 and 19 show that to develop a successful C" element it is not sufficient to use reduced shear integration; it is necessary in fact to identify those mechanisms which are associated with excessive shear-strain energy and hence shear locking. This will be t o n leave from the Institute

A new shell formulation using complete 3D constitutive laws

“…n iy (28) where n ix and n iy are the in-plane director cosines of the outward pointing normals n i . So, the previous equations can be written differently:…”

“…Among several constant strain elements, it is the one proposed by Belytschko et al [28] holds here mainly because it is available in Pam-Stamp, a code which covers the entire industrial stamping process. The rotations X and Y at a current point (x, y) of the mid-plane are interpolated via triangular area co-ordinates:…”

A basic thin shell triangle with only translational DOFs for large strain plasticity

“…Note that for plate bending analysis only one DOF per node is required [10; 14]. For example in Reference [19] for the simple supported case and a mesh with 208 DOFs, the two elements presented in that paper show an error in the central displacement between 1.5 per cent (mesh A) and 0.6 per cent (mesh B). For the present element with only 64 DOFs the errors are of 0.05 per cent (mesh A) and 1.1 per cent (mesh B).…”

A Simple High-Order Shear Deformation Triangular Plate Element with Incompatible Polynomial Approximation

“…Unfortunately, the FSDT elements suffer from the shear-locking problem when the thickness to length ration of the plate becomes very small, due to inadequate dependence among transverse deflection and rotations using an ordinary low-order finite element [4]. Quite a large number of techniques have been developed to overcome this problem, such as the assumed shear strain approach, the discrete Kirchhoff/Mindlin representation, the mixed/hybrid formulation, and the reduced/selected integration [5][6][7][8][9][10][11][12][13][14][15]. These formulations are free from shear locking and are applicable to a wide range of practical engineering problems, but in general, it is rather complex and time consuming to include the transverse shear effects for thick plates, which would also lead to complexity and difficulty in the programming.…”