FEM: Stationary Functional Approach

FEM: Stationary Functional Approach
Mohammad Tawfik
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Introduction to the Finite
Element Method
Stationary Functional Approach

Mohammad Tawfik
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Few definitions
Over-simplified versions!

Mohammad Tawfik
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A functional …
• A functional is a “function of functions” that
produces a real/complex number
• In mechanics problems, usually, the
functional used is the total energy
functional which contains the potential
energy, the kinetic energy, and the
externally work done on the system

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A functional …
• A functional may be presented in the form:
    
    

Domain
nnmn
nmn
dxdxxxfxxfG
xxfxxfI
...,...,,...,,...,
,...,,...,,...,
1111
111

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Variation …
• Variation of a functional is the “differentiation” of
the functional with respect to one or more of its
entries (functions)
• Note that the Variation of the functional with
respect to the independent variables is always
equal to zero
 
 







Domain
nm
m
m
dxdxf
df
dG
f
df
dG
f
df
dG
fffI
......
,...,,
12
2
1
1
21



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Strain Energy …
• Strain energy is the amount of mechanical
energy stored in a structure, due to the
deflection of the structure.
• An expression for the strain energy may
be given by

Volume
dVU 
2
1

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Applications

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The bar tensile problem
• The total energy of the elastic structure is
given as the difference between the strain
energy and the work done by the
externally applied forces

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• An expression for the total energy for a
bar, may be given by the following integral
  
















BarLength
dxxFu
x
u
EA .
2
1
2

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• For equilibrium, the total energy needs to
be at a minimum value, that is to say, its
variation is zero
  0. 





















 BarLength
dxxFu
x
u
x
u
EA 



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• Now, let us perform integration by parts,
we get
• Which indicates that
  0.2
2
0
























 BarLength
l
dxxFu
x
u
uEA
x
u
uEA 
lxx
l
x
u
uEA
x
u
uEA
x
u
uEA

























 &00
00

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• The other term becomes
• Then, the integrand should be equal to
zero:
  0.2
2
















BarLength
dxxFu
x
u
uEA 
  02
2
















xF
x
u
EAu

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• And, since the variation of the
displacement is an arbitrary function, it can
not be equal to zero everywhere which
yields
• This is the original differential equation for
the displacement function of a bar subject
to distributed loading along its axis
  02
2








xF
x
u
EA

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A Conclusion …
• So, using the stationary functional
approach, we could start from the total
energy and go all the way through
obtaining the governing differential
equation!
• However, we are more interested in the
week form … so let’s get back to FEM.

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Deriving the finite element
model!

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Proposing an approximate
solution …
• We get:
     e
uxNxu 
     e
uxNxu  

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Energy variation relation:
• But the nodal values of the function are
independent of the integration
             0  gthElementLen
Tee
xx
Te
dxxFNuuNNuEA 
           0
0

l
e
xx
Te
dxxFNuNNEAu

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Element Equation
• Variation is arbitrary, therefore, it can not
be zero; hence:
         0
0

l
e
xx dxxFNuNNEA
         
l
e
l
xx dxxFNudxNNEA
00
    ee
fuk 

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Beam Bending Problem

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• Obtaining the strain energy expression for
the beam under transverse loading, we
get:
• Giving:
  














l
dxxFw
dx
wd
EI
0
2
2
2
.
2
1
  0.
0
2
2
2
2






 
l
dxxFw
dx
wd
dx
wd
EI 



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• Using the approximate solution into the
above expression
• We get:
     e
wxNxw 
             0
0
 
l
Tee
xxxx
Te
dxxFNwwNNwEI 

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• Using the same procedure as for the bar
example above, we get
    ee
fwk 
           
l
e
l
xxxx dxxFNfdxNNEIk
00
&

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In conclusion …
• We were able to derive the differential
equation governing the mechanics of an
elastic body, including the boundary
conditions, starting from the total energy
expression!

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In conclusion …
• Using the stationary functional approach,
we could get the same FE model without
having to have the governing differential
equstion!

FEM: Stationary Functional Approach

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FEM: Stationary Functional Approach