Lect19
- 1. Dr. A. S. Sayyad Professor & Head Department of Structural Engineering Sanjivani College of Engineering, Kopargaon 423603. (An Autonomous Institute, Affiliated to Savitribai Phule Pune University, Pune) Finite Element Method In Civil Engineering Shape functions/Interpolation Function 2 1 1 2 and x x x x N N L L
- 2. Shape functions/Interpolation Function Shape functions In FEM analysis, the displacement model we assume the variation of displacements within the element since the true variation of displacements are not known. e.g. But in higher engineering mathematics, analytical solution of some problems is either not known or difficult to find out. In such cases we replaced that function by another function which is easy to solve mathematically. That function is called as Shape function or Interpolation Function. 1 2 u x 1 2 3 u x y 1 1 2 2 u N u N u 1 1 2 2 3 3 u N u N u N u
- 3. 1 2 u x 1 2 3 u x y 1 1 2 2 u N u N u 1 1 2 2 3 3 u N u N u N u 1 2 3 4 u x y xy 1 1 2 2 3 3 4 4 u N u N u N u N u
- 4. Important properties of shape functions The magnitude of shape function at each node is Unity. Number of shape functions are equal to number of nodes (Except bending element) The sum of shape functions is always unity (Except bending element) Element No. Shape Functions Properties of shape function N1 and N2 N1=1 at node 1 and N2=1 at node 2 N1+N2 = 1 N1, N2 and N3 N1=1 at node 1, N2=1 at node 2 and N3=1 at node 3 N1+N2+N3 = 1 N1, N2, N3 and N4 N1=1 at node 1, N2=1 at node 2, N3=1 at node 3 and N4 = 1 at node 4 N1+N2+N3+N4 = 1
- 5. Methods for deriving shape functions Shape functions using polynomials in Cartesian coordinate system Shape functions using polynomials in natural coordinates Shape functions using Lagrange interpolation function in natural coordinates