Page 52 - Finite Element Modeling and Simulations with ANSYS Workbench

Bars and Trusses 37

Assemble the structure FE equation,

u 1 v 1 u 2 v 2 u 3 v 3
 1 1 − 1 − 1 0 0 u 1   F F X 
1
 − −    
1
 1 1 1 1 0 0  1 v   F Y 
EA −  1 − 1 2 0 − 1 1 u 2  =   F X  
2

2 L − 1 − 1 0 2 1 − 1   2 v   F Y 
 

2
 0 0 − 1 1 1 − 1 u 3   F X 
3
 0 0 0 1 − 1 − 1 1   v   
3
  3   F Y 
Load and boundary conditions (BCs):
u 1 = v 1 = u 3 = v 3 = 0, F X2 = P 1 , F Y2 =
P 2
Condensed FE equation,
EA 2  0 u 2   P 1 

P 2  
2 L 0 2   v 2    =  




Solving this, we obtain the displacement of node 2,
 u 2  L  
P 1
  =  

 v 2   EA P 2  
Using Equation 2.34, we calculate the stresses in the two bars,
 
0
 
0
E 2 L   2
2 P
σ=  −  1 −1 1  1    = ( 1 P + )
1
1 P
L 2 EA   2A
 
2 P
 
 
1 P
 
2 P
  =
σ= E 2 1 −1 −1  1  L   2 ( 1 P − )
2 P

2
L 2 EA   2A
0
 
0
 
Check the results: Again, we need to check the equilibrium conditions, symmetry, anti-
symmetry, and so on, of the FEA results.
EXAMPLE 2.4: (MULTIPOINT CONSTRAINT)
y
x
P 3
2 2
1
L Y
3
45°
1
X