This document discusses displacement functions or polynomial functions for various finite elements. It provides examples of the displacement functions for: 1) A two-noded bar element with two degrees of freedom. 2) A three-noded bar element with three degrees of freedom. 3) A two-noded beam element with four degrees of freedom. 4) A constant strain triangular element with six degrees of freedom. It explains that the displacement functions are selected from Pascal's triangle and include terms for x- and y-coordinates for two-dimensional elements.

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
How to write Displacement
Functions or Polynomial
Functions for finite elements

2. Displacement functions or Polynomial functions
for finite elements
1) Two nodded bar element
DOF per node: 01 (u), Total DOF: 02
Select two elements from Pascal triangle to write displacement function. Select
only x or y-coordinate
Displacement Function or Polynomial Function:

3. 2) Three nodded bar element
DOF per node: 01 (u), Total DOF: 03
Select three elements from Pascal triangle to write displacement function.
Select only x or y-coordinate
Displacement Function or Polynomial Function:
2
1 2 3
u x x
  
  

4. 3) Two nodded beam element
DOF per node: 01 (w, θ), Total DOF: 04
Select four elements from Pascal triangle to write displacement function. Select
only x –coordinate because beam has only length. (width = 1)
Displacement Function or Polynomial Function:

5. 4) Constant Strain Triangular
(CST) element
Displacement Function or Polynomial Function:
1 2 3
4 5 5
u x y
v x y
  
  
  
  
DOF per node: 02 (u, v), Total DOF: 06
03 DOF in x-direction (u1 u2 u3)
03 DOF in y-direction (v1 v2 v3)
Select three elements from Pascal triangle to
write displacement function. Since it is 2D
element, select x and y coordinates both

6. 5) Linear Strain Triangular
(LST) element
Displacement Function or Polynomial Function:
DOF per node: 02 (u, v), Total DOF: 12
06 DOF in x-direction (u1 u2.... u6)
06 DOF in y-direction (v1 v2….. v6)
Select six elements from Pascal triangle to
write displacement function. Since it is 2D
element, select x and y coordinates both

7. 6) Four nodded rectangular
element
Displacement Function or Polynomial Function:
DOF per node: 02 (u, v), Total DOF: 08
04 DOF in x-direction (u1 u2 u3 u4 )
04 DOF in y-direction (v1 v2 v3 v4)
Select four elements from Pascal triangle to
write displacement function. Since it is 2D
element, select x and y coordinates both

8. 7) Four nodded Rectangular
plate bending element
Displacement Function or Polynomial Function:
DOF per node: 03 (w, θ1, θ2), Total DOF: 12
Select 12 elements from Pascal triangle to write displacement function. Since it is 2D
element, select x and y coordinates both

9. 8) Four nodded tetrahedron
element (3D)
Displacement Function or Polynomial Function:
DOF per node: 03 (u,v,w),
Total DOF: 12
Select 12 elements from
Pascal triangle to write
displacement function. Since it
is 2D element, select x and y
coordinates both