Overview of Finite Element Methods

1
Overview of Finite Element Methods
Group Members Daniel Braley Heriberto
Cortes Adam Hollrith
2
Outline

• Fundamental Concept of FEM
• Reasons for Using FEM
• Various Steps in FEM Analysis
• Examples
• Brief Outline of Algor
• Explanation of Homework Problem

3
The Fundamental Concept of FEM

• A continuous field ? of a domain ? and an
  infinite number of degrees of freedom is broken
  up and approximated by a set of piecewise
  continuous functions of a finite number of
  degrees of freedom.
• The piecewise functions are then defined over a
  set of subdomains called elements. The unknown
  ?s are defined at nodes and are evaluated using
  the equation, K? F

4
Concept of FEM
Discrete Element
Node
Continuous function
Finite Degrees of Freedom
Infinite Degrees of Freedom
5
Reasons For Using FEM

• Serves as a tool for
• - Stress and vibration analysis
• - Fluid flow analysis
• - Electrostatic analysis
• - Displacement analysis
• Allows for an approximation of otherwise
  impossible calculations

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Various Steps in FEM Analysis

• 1) Discretize the Structure
• a) identify and label nodes
• i) must be at points where loads
  act
• ii) must be at points where
  geometry
• changes
• b) identify and label elements
• c) identify symmetry conditions

7
Steps in FEM Analysis Cont.

• 2) Select a displacement function that is
  defined within the element, using the nodal
  values of the element.

1
1
2
6
2
3
6
3
5
4
4
5
8
Steps in FEM Analysis Cont.

• 3) Define the strain and stress displacement
  relationships

In the case of one-dimensional deformation,
strain in the x-direction
Now apply Hookes Law for the stress analysis
?x
is the stress in the x direction
E is the modulus of elasticity
9
Steps in FEM Analysis Cont.

• 4) Derive the element stiffness matrix by the
  work or energy methods, or by methods of weighted
  residuals
• fkd
• Utilizes the method of weighted residuals
• f is the vector of the element nodal forces
• k is the element stiffness matrix
• d is the vector of unknown element nodal
  degrees of freedom or generalized displacements

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Steps in FEM Analysis Cont.

• 5) Assemble the element equations to obtain the
  global or total equations, and also introduce
  boundary conditions
• FKd
• F is the vector global nodal forces
• K is the structure global or total stiffness
  matrix
• d is now the vector of known and unknown
  structure nodal degrees of freedom or generalized
  displacements

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Steps in FEM Analysis Cont.

• 6) Solve for unknown degrees of freedom or
  generalized displacements by such methods as the
  Gauss-Elimination Method
• 7) Solve for the element strains and stresses
• 8) Interpret the results and analyze them for use
  in the design/analysis process.
• Lets Look at an Example!!

12
Example

• Find the nodal displacements at points 1,2, and
  3, and find the stress in each element

3
1
1
2
2
Where P is a load applied at node 2 to the center
of the bar, the bar has a constant area A, and an
elastic modulus of E
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Example Continued
1) Discretize the function
1
3
2
1
2
1
2
2
3
1
2
u1
u2
u2
u3
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Example Continued
2) Select a displacement function for bar
1
2
2
3
1
2
u1
u2
u2
u3
Note that this a one-dimensional problem and the
displacement is only in the x-direction
15
Example Continued

• 3) Define the stress and strain relationships

Now apply Hookes Law for the stress analysis
?x is the stress in the x direction
E is the modulus of elasticity
16
Example Continued
4) Derive the element stiffness matrix for each
element

• Model Using a 1-D bar of the following
  dimensions
• Given
• sx Eex.
  
• 
  sx P/A
• 
  ex du/dx
• du/dx (d2x
  d1x)
• By substitution Eex P/A , P
  EA ex , P f
• -f1 EA (d2x d1x)
• f1 EA (d1x d2x)
• L
• f2 EA (d2x d1x)
• L

L
A
f2
f1
d1x
d2x
L
L
The stiffness matrix k can now be found by

• -1
• -1 1

k EA
L
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Example Continued

• f kd
• For element 1
• For element 2

1
3
2
1
2
u1
u2
1 -1 -1 1
EA
f1 f21
u1 u2
u1 u2

L
1
2
2
3
1
2
u1
u2
u2
u3
u2
u3
1 -1 -1 1
EA
f22 f3
u2 u3

L
18
Example Continued

• 5) Construct the global matrix and introduce the
  boundary conditions and known values

• -1 0
• -1 2 -1
• 0 -1 1

f1 f21f22 f3
EA
u1 u2 u3

L
u10 , u30 , f2 f21f22 P

• -1 0
• -1 2 -1
• 0 -1 1

f1 P f3
EA
0 u2 0

L
19
Example Continued

• 6) Solve for the Unknowns

• -1 0
• -1 2 -1
• 0 -1 1

f1 P f3
EA
0 u2 0

L
Reaction 1 f1 (-EAu2)/L
P (2EAu2)/L, so u2 (PL)/(2EA)
Reaction 3 f3 (-EAu2)/L
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Example Continued

• 7) Solve for the element strains and stresses

u1 u2
1/L -1 1
0 u2
0 u2
?x1 1/L -1 1
?x1 E/L -1 1
u2 0
u2 0
?x2 E/L -1 1
?x2 1/L -1 1
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End of Example

• 8) Analyze the results found
• - Do they make sense?
• - In being a static problem, do all of
  the
• forces add up to 0?

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Algor

• A method for computer aided analysis (i.e. can
  import Pro/E files into Algor)
• Saves time and money for companies by not having
  to calculate everything by hand

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References

• Chandrakath, Shet. FEM Class Notes.
  http//www.eng.fsu.edu/chandra/courses/eml4536
  2003.
• Logan, Daryl L. A First Course in the Finite
  Element Method Third Edition. Brooks/Cole.
  2002.
• Fancher, Darren, et.al., IAS2 Spring Report
  Integrated Advanced Surveillance System Final
  Report. 2003.