FEA_Min_P.E._Method_Fall2024 analysis .pptx

Finite Element analysis description .

1. Finite Element Analysis (FEA) (Other FEA Methods) 1

2. Minimum Total Potential Energy Formulation/Method After having used the Direct Formulation/Method, the method of minimum potential energy (P.E.) can now be used. 2

3. Basic FEA Steps Preprocessing Solution Postprocessing 3

4. FEA One Dimensional Problem Field of Study = Solid Mechanics 4

5. Theory of Minimum Total P.E. 5

6. Theory of Minimum Total P.E. 6

7. Theory of Minimum Total P.E. 7 Here is the general equation for the minimum total P.E.;

8. Theory of Minimum Total P.E. 8 Example 1.5

9. Theory of Minimum Total P.E. 9 Now let’s apply the minimum Total P.E. method in Example 1.1

10. Examples 1.1 10

11. Bar Under Axial Loading 11

12. Stiffness Equivalence 12

13. Preprocessing Phase 13

14. Minimum Total P.E. Formulation 14

15. Minimum Total P.E. Formulation 15

16. Discretization 16

17. Discretization 17

18. Free Body Diagram For Each Node 18

19. Equations and Matrices Derived 19

20. Equations and Matrices Derived 20

21. Equations and Matrices Derived 21

22. Equations and Matrices Derived 22

23. Equilibrium 23

24. Equations and Matrices Derived 24

25. Equations and Matrices Derived 25

26. Solution Phase 26

27. Equations and Matrices Derived 27

28. Equations and Matrices Derived 28

29. Equations and Matrices Derived 29

30. Equations and Matrices Derived 30

31. Equations and Matrices Derived 31

32. Postprocessing Phase 32

33. Equations and Matrices Derived After having obtained the necessary solution in term of nodal displacements, student/researchers may be interesting in obtaining other crucial information e.g., normal stresses, shear stresses, factor of safety, etc. These information are acquired via postprocessing phase. 33

34. Equations and Matrices Derived OR Let us apply these equations for each element 34

35. Equations and Matrices Derived 35

36. Equations and Matrices Derived Let consider again the equilibrium at node 1 to calculate finally the reaction i.e. R1. 36

37. Equations and Matrices Derived So, the reaction matrix will be as follows: 37