Page 94 - Finite Element Analysis with ANSYS Workbench

5.2 Finite Element Method 85

We will consider the four-node rectangular element as
shown in the figure because it is one of the simplest element type.
The element has dimensions of ab with the thickness of h. Each
node contains three unknowns which are the deflection w in the z-
direction and the rotations  and  about the x- and y-
y
x
coordinates, respectively. Thus, the element has a total of 12
unknowns.
y
3
4

z
b
w 1  1 y h

1 x
 1 x 2
a

Distribution of the deflection is assumed in the form,

wx y NN N N N N N N N N 1 0 N 1 1 N  1 2   
1 
(, )  
4
9
3
5
6
8
7
2
where   T  w   1  1  1 x w  2 y x 2  2 w  3 y 3  3 x w  4 y x 4  y  4 
The element interpolation functions, N i  1 to 12 ,
,
i
are rather complicated. As an example,
2
xy x y xy 2 x y xy 3
3
N   3  3  2  2
7
3
ab a b ab 2 a b ab 3
2
This leads to a complicated element stiffness matrix with lengthy
coefficients, such as,
D  a 2 b 2 
K   60  60  30  42(1  )


15ab  77 b 2 a 2 
Derivation of element matrices must be performed carefully.
Symbolic manipulation software can help alleviating such task.