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F inite Element Method. for readers of all backgrounds. G. R. Liu and S. S. Quek. CHAPTER 8:. FEM FOR PLATES & SHELLS. CONTENTS. INTRODUCTION PLATE ELEMENTS Shape functions Element matrices SHELL ELEMENTS Elements in local coordinate system Elements in global coordinate system

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f inite element method

Finite Element Method

for readers of all backgrounds

G. R. Liu and S. S. Quek

CHAPTER 8:

FEM FOR PLATES & SHELLS

contents
CONTENTS
  • INTRODUCTION
  • PLATE ELEMENTS
    • Shape functions
    • Element matrices
  • SHELL ELEMENTS
    • Elements in local coordinate system
    • Elements in global coordinate system
    • Remarks
introduction
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.
plate elements
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.
plate elements1
PLATE ELEMENTS
  • Consider a plate structure:

(Mindlin plate theory)

plate elements2
PLATE ELEMENTS
  • Mindlin plate theory:

In-plane strain:

where

(Curvature)

plate elements3
PLATE ELEMENTS

Off-plane shear strain:

Potential (strain) energy:

In-plane stress & strain

Off-plane shear stress & strain

plate elements4
PLATE ELEMENTS

Substituting

,

Kinetic energy:

Substituting

shape functions
Shape functions
  • Note that rotation is independent of deflection w

(Same as rectangular 2D solid)

where

element matrices
Element matrices

Substitute

into

Recall that:

where

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

element matrices1
Element matrices

Substitute

into potential energy function

from which we obtain

Note:

element matrices2
Element matrices

(me can be solved analytically but practically solved using Gauss integration)

For uniformly distributed load,

shell elements
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
Elements in local coordinate system

Consider a flat shell element

elements in local coordinate system1
Elements in local coordinate system

Membrane stiffness (2D solid element):

(2x2)

Bending stiffness (plate element):

(3x3)

elements in local coordinate system2
Elements in local coordinate system

Components related to the DOF qz, are zeros in local coordinate system.

(24x24)

elements in local coordinate system3
Elements in local coordinate system

Membrane mass matrix (2D solid element):

Bending mass matrix (plate element):

elements in local coordinate system4
Elements in local coordinate system

Components related to the DOF qz, are zeros in local coordinate system.

(24x24)

remarks
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
CASE STUDY
  • Natural frequencies of micro-motor
case study1

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
case study2
CASE STUDY

Mode 1:

Mode 2:

case study3
CASE STUDY

Mode 3:

Mode 4:

case study4
CASE STUDY

Mode 5:

Mode 6:

case study5
CASE STUDY

Mode 7:

Mode 8:

case study6
CASE STUDY
  • Transient analysis of micro-motor

F

Node 210

x

x

F

Node 300

F