Page 39 - Finite Element Modeling and Simulations with ANSYS Workbench

24 Finite Element Modeling and Simulation with ANSYS Workbench

u i u j

f i i x A, E j f j

L

FIGURE 2.4
Notation for a bar element.

Assuming that the displacement u is varying linearly along the axis of the bar, that is, in
terms of the two nodal values u and u, we can write
j
i
 x x
ux() = 1 −  u i + u j (2.4)

 L L

We have
∆
u j
− u i
ε= = ∆ ( = elongation ) (2.5)
L L
E∆
σ= E ε= (2.6)
L

We also have

F
σ= F ( = force in bar ) (2.7)
A

Thus, Equations 2.6 and 2.7 lead to

EA
F = ∆ = k∆
L

where k = EA/L is the stiffness of the bar. That is, the bar behaves like a spring in this case
and we conclude that the element stiffness matrix is
 EA − EA 
 k −  k  L L 
k =   =  
 −k k   − EA EA 
  L L  

or

EA  1 −  1
k =   (2.8)
L  −1 1 