Finite 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.

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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

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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
F
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CASE STUDY
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CASE STUDY
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CASE STUDY