DEVELOPMENT AND VERIFICATION OF HIGH PERFORMANCE TRIANGULAR PLATE BENDING ELEMENT WITH NINE-DOF

Abstract

 1. Introduction The plate bending problem of flat plate is the basis of not only static analysis but also analyses of three dimensional wall structures. Development of elements having high accuracy with few elements is one essential of the finite element method. A triangular flat plate bending element is indispensable to analyze plates with curved shape boundaries. But shape function is not determined in a simple way, because of using area coordinates. The generalized coordinates of each node of the bending plane plate element are three degrees of freedom, displacement, and rotations in two directions. A triangular three-node element has a total of nine degrees of freedom. The BCIZ element defines the shape function using area coordinate in three order terms. The element is nonconforming and gives static solution with weak constraint. Conforming elements between neighboring elements are developed and show they have strong constraints for models of small element mesh.

 In order to develop a high-performance triangular plate bending element, it is necessary to find a basis function that can control the rotation of the boundary side between adjacent elements. First, the basis functions corresponding to the six degrees of freedom essential for the plate bending element and a particular basis function that can be selected as appropriate are clearly analyzed. Although the latter have three degrees of freedom, there are three boundaries, and if we select a basis function that affects the rotation of one boundary, it is necessary to arrange the basis functions with the same effect on the remaining boundaries. After all, there is only one degree of freedom available to improve the performance of the triangular plate bending element. The purpose of this paper is to add high-order polynomials to this basis function to develop high-performance elements.

 2. Area Coordinate and Basis Functions

 3. Particular Basis Function and its Control

 4. Numerical analyses and verification

 In order to examine the performance of the triangular element developed here, four benchmark tests are carried out using six element meshes. Considering biaxial symmetry condition, 1/4 of the analysis model is set as numerical analyses. Element mesh in two directions 1 × 1, 2 × 2, 3 × 3, 5 × 5, 7 × 7, and 9 × 9 are used. (The numerical integration formula used here is inappropriate, and correction based on analysis obtained by applying high precision formula would be reported in the next journal.)

 5. Conclusion For a triangular plate bending element with nine degrees of freedom, six basis functions corresponding to rigid body motions and constant strains are necessarily determined. Considering the invariance condition of a triangular element, the remaining three degrees of freedom substantially determine one kind of basis function. First, we find this particular basis function and add three kinds of higher-order terms to this function to construct a basis function. Using the element basing on those new basis functions, the typical benchmark test examples of the plane plate were analyzed and compared with the solution of other triangular plane plate elements. It has been demonstrated that new elements give stable and accurate solutions from a region where the element mesh is sparse to a region where the element mesh is dense.