Shape functions or interpolation functions

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