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76 Finite Element Modeling and Simulation with ANSYS Workbench

3.5 Examples with Beam Elements
EXAMPLE 3.1
y
P
1 M 2
1 E, I 2 3 x
L L

Given:
The beam shown above is clamped at the two ends and acted upon by the force P and
moment M in the midspan.

Find:
The deflection and rotation at the center node and the reaction forces and moments at
the two ends.

Solution
Element stiffness matrices are

1 θ 2 θ
v 1 v 2
 12 6 L − 12 6 L 

EI 6 L 4 L 2 − 6 L 2 L 2 
k 1 = L   − 12 − 6 L 12 − 6 L 
3
 2 2 
   6 L 2 L − 6 L 4 L  
v 2 θ 2 v 3 θ 3
 12 6 L −12 6 L 

EI 6 L 4 L 2 − L6 2 L 2  

2 k = L  −12 − L6 12 −6L
3
6
 2 − 2 
  6L 2L 6L 4L  
Global FE equation is

1 θ 2 θ 3 θ
v 1 v 2 v 3
v

 12 6 L − 12 6 L 0 0  v 1   F Y 
1
 2 − 2    
 6 L 4 L 6 L 2 L 0 0  1 θ   M 1 

EI −   12 − 6 L 24 0 − 12 6 L  v 2   F Y 



2 
L 3  6 L 2 L 2 2 0 8L 2 − 6L 2L 2   2 θ  =  M 2  

 
 
 0 0 − 12 − 6L 12 − 6L    

 2 2   v 3   F Y 
3
 
3 
  0 0 6L 2L − 6L 4L  θ    M 3  
Loads and constraints (BCs) are
F Y2 =− P, M 2 = M, v 1 = v 3 =θ =θ = 0
3
1