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. Example Pure bending problem Solve the problem Equilibrium Compatibility Energy Governing Equation

4. Example Pure bending problem Solve the problem Equilibrium Compatibility Vertical Moment Equilibrium

5. Example Pure bending problem Solve the problem Equilibrium Compatibility X direction Equilibrium

6. Example Pure bending problem Solve the problem Equilibrium Compatibility Compatibility

7. Example Pure bending problem Solve the problem Equilibrium Compatibility Governing Equation

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

9. Example Pure bending problem Energy for section

10. Potential Energy You have the Functional !!

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

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

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

14. Example Pure bending problem Mechanical Boundary Condition Geometrical Boundary Condition or

15. Example Solution 1 : Direct Integration

16. Example Solution 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 Virtual Displacement Assumption

18. Example Solution 2 : Virtual Work Virtual Displacement Assumption

19. Example Solution 2 : Virtual Work Virtual Displacement Assumption

20. Example Solution 2 modified Virtual Displacement Assumption

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

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

23. Example Solution 3 :Point Collocation Assumption

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

25. Squared Error Error In order to expect the Error minimized, an appropriate value for a must be

26. Example Solution 4 :Least Square Method

27. Example Solution 2 : Virtual Work Virtual Displacement Assumption

28. Example Solution 4 :Least Square Method

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

30. Example Solution 5 :Galerkin Method 2 Starting Equation is Same

35. Example Solution 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