The Finite Element Method A Practical Course CHAPTER 7: FEM FOR PLATES & SHELLS

CONTENTS INTRODUCTION PLATE ELEMENTS SHELL ELEMENTS CASE STUDY Shape functions Element matrices SHELL ELEMENTS Elements in local coordinate system Elements in global coordinate system Remarks CASE STUDY

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

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

(Reissner-Mindlin plate theory) PLATE ELEMENTS Consider a plate structure: (Reissner-Mindlin plate theory)

PLATE ELEMENTS Reissner-Mindlin plate theory: In-plane strain: where (Curvature) in which

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

PLATE ELEMENTS Substituting , Kinetic energy: Substituting

PLATE ELEMENTS where ,

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

Shape functions where

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

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

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

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.

Elements in local coordinate system Consider a flat shell element

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

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

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

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

Elements in global coordinate system where

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.

CASE STUDY Natural frequencies of micro-motor

CASE STUDY 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 3 7.87 7.44 7.41 4 10.58 8.52 8.30 5 6 13.84 11.69 11.44 7 8 14.86 12.45 12.17 CASE STUDY

CASE STUDY Mode 1: Mode 2:

CASE STUDY Mode 3: Mode 4:

CASE STUDY Mode 5: Mode 6:

CASE STUDY Mode 7: Mode 8:

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

CASE STUDY

CASE STUDY

CASE STUDY