Mechanics of Thin Structure

1. Mechanics of Thin Structure Lecture 15 Wrapping Up the Course Shunji Kanie

2. Mechanics of Thin Structure What you learned are; Introduction for Linear Elasticity Stress and Strain with 3D General Expressions Plane Stress and Plane Strain Principle of Energy Principle of Virtual Work Calculus of Variations Theory of Beams Theory of Plates

3. ExamplePure bending problem Solve the problem Equilibrium Compatibility Governing Equation Energy

4. ExamplePure bending problem Solve the problem Equilibrium Equilibrium Compatibility Vertical Moment

5. ExamplePure bending problem Solve the problem Equilibrium Equilibrium Compatibility X direction

6. ExamplePure bending problem Solve the problem Equilibrium Compatibility Compatibility

7. ExamplePure bending problem Solve the problem Equilibrium Compatibility Governing Equation

8. ExamplePure bending problem Compatibility Energy Energy should be Minimum, so that Energy is calcualted such as;

9. ExamplePure bending problem Energy for section

10. ExamplePure bending problem Potential Energy You have the Functional !!

11. ExamplePure bending problem You have the Functional !! Apply the Euler’s Equation

12. ExamplePure bending problem Governing Equation for pure bending beam from the Euler’s Equation

13. ExamplePure bending problem Boundary Condition See eq. (6-31) Shearing force Bending Moment

14. ExamplePure bending problem

15. ExampleSolution 1 : Direct Integration

16. ExampleSolution 2 : Virtual Work Equilibrium No Energy produced if the Equilibrium is satisfied Virtual Displacement Virtual Work What is the requirement for the virtual displacement?

17. Example Solution 2 : Virtual Work Assumption Virtual Displacement

18. Example Solution 2 : Virtual Work Assumption Virtual Displacement

19. Example Solution 2 : Virtual Work Assumption Virtual Displacement

20. ExampleSolution 2 modified Assumption Virtual Displacement

21. ExampleSolution 2 modified Fourier Transform It’s same as the solution introduced for Plate Theory

22. ExampleSolution 3 :Point Collocation Assume you would like to have the value at the center Dirac Delta Function

23. ExampleSolution 3 :Point Collocation Assumption

24. ExampleSolution 4 :Least Square Method Equilibrium Error should be minimum Assumption Estimated

25. ExampleSolution 4 :Least Square Method Error Squared Error In order to expect the Error minimized, an appropriate value for a must be

26. ExampleSolution 4 :Least Square Method

27. Example Solution 2 : Virtual Work Assumption Virtual Displacement

28. ExampleSolution 4 :Least Square Method

29. ExampleSolution 5 :Galerkin Method 1 Starting Equation is Same Assumption Weighting Function

30. ExampleSolution 5 :Galerkin Method 2 Starting Equation is Same

31. ExampleSolution 5 :Galerkin Method 2

32. ExampleSolution 5 :Galerkin Method 2

33. ExampleSolution 5 :Galerkin Method 2

34. ExampleSolution 5 :Galerkin Method 2

35. ExampleSolution 5 :Galerkin Method 3

36. ExampleSolution 5 :Galerkin Method 3

37. ExampleSolution 5 :Galerkin Method 3

38. ExampleSolution 5 :Galerkin Method 3

39. Mechanics of Thin Structure What you learned are; Introduction for Linear Elasticity Stress and Strain with 3D General Expressions Plane Stress and Plane Strain Principle of Energy Principle of Virtual Work Calculus of Variations Theory of Beams Theory of Plates