Variational formulation of the FEM
Principle of Stationary Potential Energy:
Among all admissible displacement functions u, the actual ones are those
which render the total potential energy P stationary
P= W + P ,
1
W =  σ T .ε.dV
2
where
is strain energy function,
P =   uT .o.dV   uT .p.dS

external forces potential.
p
Meaning of the variables:
- displacement
uT = [u, v, w]
-strain
εT = [ x ,  y ,  z ,  xy ,  yz ,  zx ]
-stress
σ T = [ x , y ,  z , xy , yz , zx ]
-external body forces
oT = [ox , o y , oz ]
-external surface forces
pT = [ px , p y , pz ]
Illustrative Example:
Using the principle of stationary potential energy, evaluate the displacement u 0 of the end
point of spring on the Fig.1
Given:
spring stiffness k, loading force F
2
Accumulated strain energy in the spring: W = k. u 2
Potential of external force:
P = F.u
2
Total potential energy: P = 12 ku  F . u
dP
= 0 = ku  F
Its stationary value
du
renders the trivial result u0 = F k
Fig.1 Loaded spring
Fig.2 shows clearly, that the equilibrium
displacement u0 corresponds to minimum
potential energy:
Fig.2 Displacement vs. energy
Discretization of continuum – basic idea
State and characteristics of continuum are described by continuous functions – e.g.
displacements u(x,y,z), v(x,y,z), w(x,y,z). To solve any problem on digital computer, it
must be first discretized – continuous functions must be expressed by finite number of
scalar parameters.
In the most popular displacement version of FEM, unknown functions of displacement
~ v~ w
~
are approximated with the help of apriori selected, known simple functions u
j
i
k
- so called shape functions, which are defined on each element:
l
u=

i =1
m
ai . u~i ; v =

n
b j . v~ j ; w =
j =1
 c . w~
k
k
k =1
Shape functions are multiplied by unknown coefficients ai, bj, ck, or deformation
parameters, which must be evaluated.
Inserting the approximation into the functional P(u,v,w), we obtain P as a function of a
finite number of parameters P(a1,a2,a3, ...). Principle of stationary value of P then leads
to a system of equations for unknown values of deformation parameters:
P

= 0
 a1


  a1 , a 2 ,  , c n
P
= 0

 cn
