F inite Element Method

1. Finite Element Method for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 8: FEM FOR PLATES & SHELLS

2. CONTENTS • INTRODUCTION • PLATE ELEMENTS • Shape functions • Element matrices • SHELL ELEMENTS • Elements in local coordinate system • Elements in global coordinate system • Remarks

3. INTRODUCTION • FE equations based on Mindlin plate theory will be developed. • FE equations of shells will be formulated by superimposing matrices of plates and those of 2D solids. • Computationally tedious due to more DOFs.

4. PLATE ELEMENTS • Geometrically similar to 2D plane stress solids except that it carries only transverse loads. Leads to bending. • 2D equilvalent of the beam element. • Rectangular plate elements based on Mindlin plate theory will be developed – conforming element. • Much software like ABAQUS does not offer plate elements, only the general shell element.

5. PLATE ELEMENTS • Consider a plate structure: (Mindlin plate theory)

6. PLATE ELEMENTS • Mindlin plate theory: In-plane strain: where (Curvature)

7. PLATE ELEMENTS Off-plane shear strain: Potential (strain) energy: In-plane stress & strain Off-plane shear stress & strain

8. PLATE ELEMENTS Substituting , Kinetic energy: Substituting

9. PLATE ELEMENTS , where

10. Shape functions • Note that rotation is independent of deflection w (Same as rectangular 2D solid) where

11. Shape functions where

12. Element matrices Substitute into  Recall that: where (Can be evaluated analytically but in practice, use Gauss integration)

13. Element matrices Substitute into potential energy function from which we obtain Note:

14. Element matrices (me can be solved analytically but practically solved using Gauss integration) For uniformly distributed load,

15. SHELL ELEMENTS • Loads in all directions • Bending, twisting and in-plane deformation • Combination of 2D solid elements (membrane effects) and plate elements (bending effect). • Common to use shell elements to model plate structures in commercial software packages.

16. Elements in local coordinate system Consider a flat shell element

17. Elements in local coordinate system Membrane stiffness (2D solid element): (2x2) Bending stiffness (plate element): (3x3)

18. Elements in local coordinate system Components related to the DOF qz, are zeros in local coordinate system. (24x24)

19. Elements in local coordinate system Membrane mass matrix (2D solid element): Bending mass matrix (plate element):

20. Elements in local coordinate system Components related to the DOF qz, are zeros in local coordinate system. (24x24)

21. Elements in global coordinate system where

22. Remarks • The membrane effects are assumed to be uncoupled with the bending effects in the element level. • This implies that the membrane forces will not result in any bending deformation, and vice versa. • For shell structure in space, membrane and bending effects are actually coupled (especially for large curvature), therefore finer element mesh may have to be used.

23. CASE STUDY • Natural frequencies of micro-motor

24. Mode Natural Frequencies (MHz) 768 triangular elements with 480 nodes 384 quadrilateral elements with 480 nodes 1280 quadrilateral elements with 1472 nodes 1 7.67 5.08 4.86 2 7.67 5.08 4.86 3 7.87 7.44 7.41 4 10.58 8.52 8.30 5 10.58 8.52 8.30 6 13.84 11.69 11.44 7 13.84 11.69 11.44 8 14.86 12.45 12.17 CASE STUDY

25. CASE STUDY Mode 1: Mode 2:

26. CASE STUDY Mode 3: Mode 4:

27. CASE STUDY Mode 5: Mode 6:

28. CASE STUDY Mode 7: Mode 8:

29. CASE STUDY • Transient analysis of micro-motor F Node 210 x x F Node 300 F

30. CASE STUDY

31. CASE STUDY

32. CASE STUDY