Question

Explain the concept of shape functions and compare and contrast the shape functions for the different element types introduced in this module. Use diagrams as necessary. Relate this to the capabilities and limitations of each element type. (Truss, beam, plane stress, shell and 3d solid elements)

Answer 1

There are two ways to group shape functions in FEM:

• by polynomial order:
  • linear
  • quadratic
  • cubic
  • …
• by interpolation type:
  • Lagrange - used when we know only the values of the function
  • Serendipity - used when we know both values and their derivatives
  • Hermite - used when we don’t want to have nodes inside higher order (other than linear) elements

Shape functions or interpolation functions are functions used to represent behaviour of a field variable within an element. In finite element analysis we deal with different elements. The elements can be linear, quadratic, 8- noded, 9-noded etc. The shape function for these elements is required to be determined to draw a relationship between the nodal displacements and displacement function chosen for the entire element. Starting from a two noded linear element we can calculate the shape function for 4 noded square element. Quadratic or 3 noded element can be used for determining the shape functions for 9- noded square element. 8- noded element is quiet tricky and we can use the solutions for 2 noded square element to get its shape function. I have presented in the attached literature, a simple way of arrive at shape function for 4, 8 and 9 noded square finite elements.