Page 41 - Finite Element Modeling and Simulations with ANSYS Workbench

26 Finite Element Modeling and Simulation with ANSYS Workbench

where B is the element strain–displacement matrix, which is

d d ξ d
ξ  ⋅
B =  N i ξ () N j ()   N i ξ () N j () 
ξ  =
dx d ξ dx
that is,

B =−  1/L 1/L  (2.14)
 

Stress can be written as

σ= E ε= EBu (2.15)

Consider the strain energy stored in the bar

1 1
T
U = ∫ σ εdV = ∫ ( uB E Bu dV
T
T
)
2 2
V V
 
∫
1
T
= u T  ( B E )  u (2.16)
B dV
2  
V 
where Equations 2.13 and 2.15 have been used.
The potential of the external forces is written as (this is by definition, and remember the
negative sign)
T
Ω= − fu =−uf (2.17)
j
ii − fu j
The total potential of the system is
Π = U + Ω
which yields by using Equations 2.16 and 2.17

 
∫
1
Π= u T  ( B E B)dV  u − uf (2.18)
T
T
2  
V 
Setting dΠ = 0 by the principle of minimum potential energy, we obtain (verify this)
 
 ∫ ( B E ) u = f
B dV 
T
 
V  
or

ku = f (2.19)