AERSP 301 Plate Theory

1. AERSP 301Plate Theory Jose Palacios August 2008

2. Today • Plate Theory • Analyzing energy method (Stationary Principle of Total Potential Energy) • Similar to process for FEM • Consider plate as a whole (not discretized) • Displacement Functions change • Final Exam: August 14 2008 @ 10:00 RCOE

3. Plate Theory • Beam • Plate

4. Plate Theory

5. Plate Theory

6. Plate Theory

7. Plate Theory • Non-zero strains (in vector form): • These are the strains/deformations that we are concerned with when considering analysis of a plate

8. Plate Theory

9. Plate Theory • Stress-Strain Relations for plates: But for a plate (2 in-plane directions) we must consider Poisson’s effects

10. Plate Theory • Strains in each direction are functions of stress in both directions:

11. Plate Theory • For an isotropic material, there are only 2 independent constants relating stress to strain • For isotropic materials we need to know Young’s Modulus, Shear Modulus, and poisson’s ratio. If we know 2 of these constants, we can calculate the third.

12. Plate Theory • Put strain/stress relations into matrix form:

13. Plate Theory

14. Plate Theory • Now that we have stress-strain relations, we use Energy methods (Stationary Principle) to finish the analysis • Start with Strain Energy • We only had to consider 1 axial stress and strain • For plates we are concerned with two axial stresses and strains as well as shear stress and strain

15. Plate Theory • For plates:

16. Plate Theory • Integrate over thickness, and area of plate

17. Plate Theory (Example of non-isotropic plate)

18. Plate Theory • Similar to plates and bars

19. Plate Theory • At this point, for bars and beams we discretized the structure using the finite element method

20. Plate Theory • For the plate, like the beam or bar, we need to assume a displacement function Assumed Modes Method – assume the displacement of the plate can be written as the sum of a number of mode shapes, mn, and corresponding coefficients, Amn.

21. Plate Theory • The assumed displacements must match the geometric boundary conditions of the plate and be 2 times differentiable. • Mode shapes must match BCs • 2 times differentiable since strain energy is a function of curvatures • Simply-supported plate (BCs)

22. Plate Theory Clamped plate

23. Plate Theory • Once the displacement of the plate is assumed, then the Total strain energy of the plate can be determined. • As before the work potential is a function of the external loading and assumed displacement. • We can then write out the total potential energy and apply the stationary principle to get equilibrium equations and determine the resulting displacements, strains, and stresses of the plate

24. Plate Theory example • Example

25. Plate Theory example We have D already, need to determine the curvatures!

26. Plate Theory example

27. Plate Theory example • Look at the integrand kTDk in the expression for U • Since the our assumed displacement has only one term (one mode) the product, kTDk, is a scalar

28. Plate Theory example

29. Plate Theory example • Perform the integrals for each term: • Using these expressions, the strain energy can be evaluated as

30. Plate Theory example

31. Plate Theory example • What about W, work potential?

32. Plate Theory example

33. Plate Theory example Know our final results depends on the loading (work potential)

34. Plate Theory example

35. Plate Theory example

36. Plate Theory • To get accurate results • Use multiple mode shapes • Displacement due to loading should match the mode shapes (examples)