F inite Element Method

1 / 32

# F inite Element Method - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'F inite Element Method' - bell

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Finite Element Method

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
• Remarks
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
• 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 ELEMENTS
• Consider a plate structure:

(Mindlin plate theory)

PLATE ELEMENTS
• Mindlin plate theory:

In-plane strain:

where

(Curvature)

PLATE ELEMENTS

Off-plane shear strain:

Potential (strain) energy:

In-plane stress & strain

Off-plane shear stress & strain

PLATE ELEMENTS

Substituting

,

Kinetic energy:

Substituting

Shape functions
• Note that rotation is independent of deflection w

(Same as rectangular 2D solid)

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)

SHELL ELEMENTS
• 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)

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

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 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

F