Page 45 - Finite Element Modeling and Simulations with ANSYS Workbench

30 Finite Element Modeling and Simulation with ANSYS Workbench

The nodal forces are transformed in the same way,

f′ = Tf (2.30)

In the local coordinate system, we have

′
′

EA 1 − 1    
u i 


f i 
′
′
L   − 1 1    =  
f j 
u j
      
Augmenting this equation, we write
′
 1 0 − 1 0    
′
u i
f i
  

′
EA 0 0 0 0  v i   
0
 


′
L  − 1 0 1 0   =  
′
u j 
 

f j

 0 0 0 0    

′
0
v j
     
or,
k′ u′ = f′
Using transformations given in Equations 2.29 and 2.30, we obtain
k′ Tu = Tf
Multiplying both sides by T and noticing that T T = I, we obtain
T
T
T k′ Tu = f (2.31)
T
Thus, the element stiffness matrix k in the global coordinate system is
k = T k′ T (2.32)
T
which is a 4 × 4 symmetric matrix.
Explicit form is
j v
u i v i u j
 l 2 lm l − 2 − lm
 2 
EA lm m 2 − lm − m  (2.33)

k =
L  l − 2 − lm l 2 lm 
  2 2 
 −  lm − m lm m  
Calculation of the directional cosines l and m:
X j − X i Y j − Y i
l = cosθ = , m = sin θ =
L L