1. Finite Element Analysis
MEEN 5330
Dustin Grant
Kamlesh Borgaonkar
Varsha Maddela
Rupakkumar Patel
Sandeep Yarlagadda

2. Introduction What is finite element analysis, FEA?
What is FEA used for?
1D Rod Elements, 2D Trusses

3. Basic Concepts
Loads
Equilibrium
Boundary conditions

4. Development of Theory Rayleigh-Ritz Method Galerkin’s Method
Total potential energy equation
Galerkin’s Method

5. 1D Rod Elements
To understand and solve 2D and 3D problems we must understand basic of 1D problems.
Analysis of 1D rod elements can be done using Rayleigh-Ritz and Galerkin’s method
To solve FEA problems same are modified in the Potential-Energy approach and Galerkin’s approach

6. 1D Rod Elements
Loading consists of three types : body force f , traction force T, point load Pi
Body force: distributed force , acting on every elemental volume of body i.e. self weight of body.
Traction force: distributed force , acting on surface of body i.e. frictional resistance, viscous drag and surface shear
Point load: a force acting on any single point of element

7. 1D Rod Elements Element strain energy Element stiffness matrix
Element Element-2
Element strain energy
Element stiffness matrix
Load vectors
Element body load vector
Element traction-force vector

8. Example 1D Rod Elements Example 1
Problem statement: (Problem 3.1 from Chandrupatla and Belegunda’s book)
Consider the bar in Fig.1, determine the following by hand calculation: 1) Displacement at point P 2) Strain and stress
3) Element stiffness matrix 4) strain energy in element
Given:

12. 2D Truss 2 DOF Transformations Modified Stiffness Matrix
Methods of Solving

13. 2D Truss
Transformation Matrix
Direction Cosines

14. 2D Truss
Element Stiffness Matrix

15. Methods of Solving Elimination Approach Penalty Approach
Eliminate Constraints
Penalty Approach
Will not discuss Today

16. Elimination Method
Set defection at the constraint to equal zero

17. Elimination Method
Modified Equation
DOF’s 1,2,4,7,8 equal to zero

18. 2D Truss
Element Stresses
Element Reaction Forces

19. 2D Truss Development of Tables Coordinate Table Connectivity Table
Direction Cosines Table

20. 2D Truss
Coordinate Table

21. 2D Truss
Connectivity Table

22. 2D Truss
Direction Cosines Table

23. Example 2D Truss

29. MATLAB Program TRUSS2D.M

30. 3D Truss Stiffness Matrix
3D Transformation Matrix
Direction Cosines

31. 3D Truss Stiffness Matrix
3D Stiffness Matrix

32. Conclusion
Good at Hand Calculations, Powerful when applied to computers
Only limitations are the computer limitations

33. References

34. Homework