Page 97 - Finite Element Modeling and Simulations with ANSYS Workbench

82 Finite Element Modeling and Simulation with ANSYS Workbench

3000 lb 3000 lb

72,000 lb-in. 72,000 lb-in.
3000 lb 1 1 2

2 3

3 4

In local coordinate system, the stiffness matrix for a general 2-D beam element is

i θ j θ
u i v i u j v j
 EA EA 
 L 0 0 − L 0 0 
 
 0 12 EI 6 EI 0 − 12 EI 6 EI 
 L 3 L 2 L 3 L 2 
 6 EI 4 EI I 6 EI 2 EI 
 0 2 0 − 2 
k =   L L L L  
−  EA 0 0 EA 0 0 
 L L 
 12 EI 6 EI 12 EI 6 EI 
 0 − 3 − 2 0 3 − 2 
 L L L L 
 0 6 EI 2 2EI 0 − 6EI 4EI 
  L 2 L L 2 L  

Element Connectivity Table

Element Node i (1) Node j (2)
1 1 2
2 3 1
3 4 2

For element 1, we have
1 θ 2 θ
u 1 v 1 u 2 v 2
 141 7 . 0 0 − 141 7 . 0 0 
 
.
0 784
4
 0 0 784 56 4 . 0 − . 56.4 
 0 564 . 5417 0 − 564 . 2708 
4
k 1 = k′ 1 = 10 ×  
 − 141 7 . 0 0 141 7 . 0 0 
 0 − 0 784 − 56 4 . 0 0 784 − 56.4 4 
.
.
 
  0 564 . 2708 0 − 564 . 5417  