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

6. 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 6 3 5 3 4 5 4

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 Hooke’s 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
{f}=[k]{d}
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

10. Steps in FEM Analysis Cont.
5) Assemble the element equations to obtain the global or total equations, and also introduce boundary conditions
{F}=[K]{d}
{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

11. 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.
Let’s 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

13. Example Continued 1) Discretize the function 1 3 1 2 2 1 2 2 3 1 2 u1

14. Example Continued 2) Select a displacement function for bar 1 2 2 3 1
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 Hooke’s 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:
σx = Eεx.
σx = P/A
εx = du/dx
du/dx = (d2x – d1x)
By substitution: Eεx = P/A , P = EA εx , P = f
-f1 = EA (d2x – d1x)
f1 = EA (d1x – d2x) L
f2 = EA (d2x – d1x) L A f2 f1
d1x
d2x L L
The stiffness matrix [k] can now be found by
[k] = EA -1 L

17. Example Continued {f} = [k]{d} For element 1: For element 2: 1 3 1 2 2EA u1 u2 u1 u2 = L 1 2 2 3 1 2 u1 u2 u2 u3 u2 u3
f22 f3 EA u2 u3 = L

18. Example Continued
5) Construct the global matrix and introduce the boundary conditions and known values f1
f21+f22 f3 EA u1 u2 u3 = L
u1=0 , u3=0 , f2 = f21+f22 = P f1 P f3 EA u2 = L

19. Example Continued 6) Solve for the Unknowns -1 0 -1 2 -1 0 -1 1 f1 P f1 P f3 EA u2 = L
Reaction 1= f1 = (-EAu2)/L
P = (2EAu2)/L, so u2 = (PL)/(2EA)
Reaction 3 = f3 = (-EAu2)/L

20. Example Continued 7) Solve for the element strains and stresses u1 u2u2 u2
x1 = 1/L [-1 1]
x1 = E/L [-1 1] u2 u2
x2 = E/L [-1 1]
x2 = 1/L [-1 1]

21. 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?

22. 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

23. References
Chandrakath, Shet. “FEM Class Notes”
Logan, Daryl L. A First Course in the Finite Element Method: Third Edition. Brooks/Cole
Fancher, Darren, et.al., IAS2 Spring Report: Integrated Advanced Surveillance System Final Report