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**Finite Element Method CHAPTER 8: FEM FOR PLATES & SHELLS**

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

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**CONTENTS INTRODUCTION PLATE ELEMENTS SHELL ELEMENTS Shape functions**

Element matrices SHELL ELEMENTS Elements in local coordinate system Elements in global coordinate system Remarks

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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.

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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.

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**(Mindlin plate theory)**

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

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**PLATE ELEMENTS Mindlin plate theory: In-plane strain: where**

(Curvature)

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**PLATE ELEMENTS Off-plane shear strain: Potential (strain) energy:**

In-plane stress & strain Off-plane shear stress & strain

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PLATE ELEMENTS Substituting , Kinetic energy: Substituting

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PLATE ELEMENTS where ,

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**Shape functions Note that rotation is independent of deflection w**

(Same as rectangular 2D solid) where

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Shape functions where

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**Element matrices Substitute into Recall that: where**

(Can be evaluated analytically but in practice, use Gauss integration)

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**Element matrices Substitute into potential energy function**

from which we obtain Note:

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

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**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.

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**Elements in local coordinate system**

Consider a flat shell element

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**Elements in local coordinate system**

Membrane stiffness (2D solid element): (2x2) Bending stiffness (plate element): (3x3)

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**Elements in local coordinate system**

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

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**Elements in local coordinate system**

Membrane mass matrix (2D solid element): Bending mass matrix (plate element):

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**Elements in local coordinate system**

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

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**Elements in global coordinate system**

where

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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.

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CASE STUDY Natural frequencies of micro-motor

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**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

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CASE STUDY Mode 1: Mode 2:

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CASE STUDY Mode 3: Mode 4:

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CASE STUDY Mode 5: Mode 6:

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CASE STUDY Mode 7: Mode 8:

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

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CASE STUDY

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CASE STUDY

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CASE STUDY

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