Page 112 - Finite Element Analysis with ANSYS Workbench

6.2 Finite Element Method 103

Since there are 3 displacement unknowns at each node,
a problem containing only few elements is almost impossible to
solve by hands. A computer program is needed to carry out the
analysis for solutions.
For ease of understanding, the four-node tetrahedral
element is explained herein. The element contains 12 displacement
unknowns as shown in the figure.
4 w
4 4 v
4 u

3 w

1 w 3 3 v
z 1 v 3 u
1 u 1 2 w
y
x 2 v
2 u 2

Distribution of the u displacement component over the
element is assumed in the form,
(, , ) 
ux yz N N N N     u
1 
2
3
4
where the interpolation functions are,
1
a 


N   b x c y d z i  1,2,3,4
i
6V i i i i
In the equation above, V is the element volume, the parameters a
,
i
y
,
b i , , d depend on the nodal coordinates x , z . The
c
i
i
i
i
i
element vector   u contains the nodal displacements in the x-
coordinate direction,
 u T  u u u u  
1 
3
2
4
Distribution of the v and w displacement components
over the element are in the same form as,
vx y z NN N N     v
(, , )  
1
3
2
4