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

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Finite Element Method by G. R. Liu and S. S. Quek 2 CONTENTS INTRODUCTION PLATE ELEMENTS – Shape functions – Element matrices SHELL ELEMENTS – Elements in local coordinate system – Elements in global coordinate system – Remarks

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Finite Element Method by G. R. Liu and S. S. Quek 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.

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Finite Element Method by G. R. Liu and S. S. Quek 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.

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Finite Element Method by G. R. Liu and S. S. Quek 5 PLATE ELEMENTS Consider a plate structure: (Mindlin plate theory)

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Finite Element Method by G. R. Liu and S. S. Quek 6 PLATE ELEMENTS Mindlin plate theory: In-plane strain: where(Curvature)

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Finite Element Method by G. R. Liu and S. S. Quek 7 PLATE ELEMENTS Off-plane shear strain: Potential (strain) energy: In-plane stress & strain Off-plane shear stress & strain

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Finite Element Method by G. R. Liu and S. S. Quek 8 PLATE ELEMENTS Substituting, Kinetic energy: Substituting

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Finite Element Method by G. R. Liu and S. S. Quek 9 PLATE ELEMENTS where,

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Finite Element Method by G. R. Liu and S. S. Quek 10 Shape functions Note that rotation is independent of deflection w where (Same as rectangular 2D solid)

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Finite Element Method by G. R. Liu and S. S. Quek 11 Shape functions where

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Finite Element Method by G. R. Liu and S. S. Quek 12 Element matrices Substituteinto where Recall that: (Can be evaluated analytically but in practice, use Gauss integration)

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Finite Element Method by G. R. Liu and S. S. Quek 13 Element matrices Substituteinto potential energy function from which we obtain Note:

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Finite Element Method by G. R. Liu and S. S. Quek 14 Element matrices (m e can be solved analytically but practically solved using Gauss integration) For uniformly distributed load,

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Finite Element Method by G. R. Liu and S. S. Quek 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.

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Finite Element Method by G. R. Liu and S. S. Quek 16 Elements in local coordinate system Consider a flat shell element

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Finite Element Method by G. R. Liu and S. S. Quek 17 Elements in local coordinate system Membrane stiffness (2D solid element): Bending stiffness (plate element): (2 x2 ) (3 x 3)

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Finite Element Method by G. R. Liu and S. S. Quek 18 Elements in local coordinate system (24 x 24) Components related to the DOF z, are zeros in local coordinate system.

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Finite Element Method by G. R. Liu and S. S. Quek 19 Elements in local coordinate system Membrane mass matrix (2D solid element): Bending mass matrix (plate element):

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Finite Element Method by G. R. Liu and S. S. Quek 20 Elements in local coordinate system Components related to the DOF z, are zeros in local coordinate system. (24 x 24)

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Finite Element Method by G. R. Liu and S. S. Quek 21 Elements in global coordinate system where

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Finite Element Method by G. R. Liu and S. S. Quek 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.

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Finite Element Method by G. R. Liu and S. S. Quek 23 CASE STUDY Natural frequencies of micro-motor

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Finite Element Method by G. R. Liu and S. S. Quek 24 CASE STUDY Mode Natural Frequencies (MHz) 768 triangular elements with 480 nodes 384 quadrilateral elements with 480 nodes 1280 quadrilateral elements with 1472 nodes

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Finite Element Method by G. R. Liu and S. S. Quek 25 CASE STUDY Mode 1: Mode 2:

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Finite Element Method by G. R. Liu and S. S. Quek 26 CASE STUDY Mode 3: Mode 4:

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Finite Element Method by G. R. Liu and S. S. Quek 27 CASE STUDY Mode 5: Mode 6:

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Finite Element Method by G. R. Liu and S. S. Quek 28 CASE STUDY Mode 7: Mode 8:

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Finite Element Method by G. R. Liu and S. S. Quek 29 CASE STUDY Transient analysis of micro-motor F F F x x Node 210 Node 300

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Finite Element Method by G. R. Liu and S. S. Quek 30 CASE STUDY

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Finite Element Method by G. R. Liu and S. S. Quek 31 CASE STUDY

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Finite Element Method by G. R. Liu and S. S. Quek 32 CASE STUDY

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