Page 74 - Finite Element Analysis with ANSYS Workbench

4.2 Finite Element Method 65

1
A   x )y  x (y  )y  x ( y  y )y   (
2 2 3 1 1 2 3 3 1 2
In the above equations, x i , y i  1, 2, 3 are coordinates of the
;
i
three nodes. The parameters a b c i  1, 2, 3 depend on the
,
,
;
i
i
i
nodal coordinates as follows,
a  1 x y  23 x y b  y  y c  x  x
3
32
1
2
2
3
1
a  2 x y  31 x y b  y  y c  x  x
1 3
2
3
1
2
3
1
a  3 x y  12 x y b  y  y c  x  x
2 1
3
1
2
1
2
3
The finite element equations corresponding to the
three-node triangular element above are,
     
K 
F



(6 6) (6 1) (6 1)
where the element stiffness matrix   can be determined from,
K
T
       t A
K
B
B
C




(6 6) (6 3) (3 3) (3 6)
b 1 0  b 2 0 b 3 0 
where   B  1  0 c 1 0 c 2 0 c 3 

(3 6) 2A  
c  1  b 1 c 2 b 2 c 3 b  
3
  uv u v u v  
T

1 
3
3
2
1
2

(6 1)
and    FF 1x F 2y F 2x F 3y F 
T
F
1 
y 
3x

(6 1)
Similarly, the finite element equations for the four-node
quadrilateral element can be determined in the same way, except
the process is more complicated. These element matrices suggest
that it is nearly impossible to solve plane stress problems by hands
even though they contain only few elements. Developing a finite
element computer program is thus required. A model with few
hundred elements can be solved easily by using a computer
program. We will employ ANSYS through its Workbench to
analyze plane stress problems in the following sections. We will
find that the software can provide solutions conveniently and
effectively for model containing a large number of elements.