Development of Shape Functions and Stiffness for Higher-Order Cubic Elements to Enhance the the Accuracy in FEM Based Industrial Designs

Abstract

We found extensive textin FEM (Finite element method) to develop shape functions and corresponding stiffness matrix forone-dimensionallinear and quadratic elements.The development of shape functions and corresponding stiffness matrix forhigher-order 4 noded (cubic)elementsis hardly seen in the literature. In this article, an attempt has been made to develop shape functions and corresponding stiffness matrix for cubic elements usingmathematical modeling. Graphical representation of shape functions along with important properties of the stiffness matrix is discussed in detail. The stiffness matrix is found to be symmetric, singular and positive semi-definite.Results are checked using the MATLAB Symbolic toolbox.Developed shape functions are applied to model a vertically hanging rod subjected to self-weight. Results are compared with 7 cases in which the same rod is modeled in increasing order of linear elements. It was found that one cubic element is far better than 64 linear elements. At last, it is concluded that the developed shape functions and corresponding stiffness matrix can be utilized to increase the accuracy of results during finite element analysis of mechanicalcomponents by design engineers, industrial engineering experts, and research professionals. Further higher order 5 noded and 6 noded elements can also be added as a module in FEM software to enhance their capability and efficiency.