Page 24 - Finite Element Modeling and Simulations with ANSYS Workbench

Introduction 9

which is a function of the three nodal displacements (u ,u ,u ). According to the principle
3
1
2
of minimum potential energy, for a system to be in equilibrium, the total potential energy
must be minimum, that is, dΠ= 0, or equivalently,
∂Π ∂Π ∂Π
= 0, = 0, = 0, (1.11)
∂u 1 ∂u 2 ∂u 3

which yield the same three equations as in Equation 1.6.

1.2.3 Boundary and Load Conditions

Assuming that node 1 is fixed, and same force P is applied at node 2 and node 3, that is

u = 0 and F = F = P
2
1
3
we have from Equation 1.6
 k 1 − k 1 0  0   F 1 

 − k 1 + −     
P

 k 1 k 2 k 2  u 2 =  



 0 − k 2  u 3   
P
 k 2  
which reduces to
 k 1 + k 2 − k 2  u 2   P
 −    =  
P
 k 2 k 2  u 3   
and

F = −k u
1 2
1
Unknowns are

u 2 
u =  
u
 3 
and the reaction force F (if desired).
1
Solving the equations, we obtain the displacements

 u 2   2 / 
Pk 1
  =  
Pk 1 + /
 u 3   2 / Pk 2 
and the reaction force

F = −2P
1