Finite Element Method

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30 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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

1

F
inite

Element Method


INTRODUCTION TO MECHANICS

FOR SOLIDS AND STRUCTURES

for readers of all backgrounds


G. R. Liu and S. S. Quek

CHAPTER 2:


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

2

INTRODUCTION


Solids and structures are stressed when they
are subjected to
loads

or
forces
.


The
stresses

are, in general, not uniform as
the forces usually vary with coordinates.


The stresses lead to
strains
, which can be
observed as a
deformation

or
displacement
.


Solid mechanics

and
structural

mechanics

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

3

Statics and dynamics


Forces can be static and/or dynamic.


Statics

deals with the mechanics of solids and
structures subject to static loads.


Dynamics

deals with the mechanics of solids and
structures subject to dynamic loads.


As statics is a special case of dynamics, the
equations for statics can be derived by simply
dropping out the dynamic terms in the dynamic
equations.

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

4

Elasticity and
p
lasticity


Elastic
:

the

deformation

in

the

solids

disappears

fully

if

it

is

unloaded
.


Plastic
:

the

deformation

in

the

solids

cannot

be

fully

recovered

when

it

is

unloaded
.


Elasticity

deals

with

solids

and

structures

of

elastic

materials
.


Plasticity

deals

with

solids

and

structures

of

plastic

materials
.

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

5

Isotropy and
a
nisotropy


Anisotropic
: the material property varies
with direction.


Composite materials: anisotropic, many
material constants.


Isotropic

material: property is not direction
dependent, two independent material
constants.

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

6

Boundary conditions


Displacement (
essential
) boundary
conditions


Force (
natural
) boundary conditions


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

7

Different structural components


Truss and beam structures

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

8

Different structural components


Plate and shell
structures

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

9

EQUATIONS FOR 3D SOLIDS


Stress and strain


Constitutive equations


Dynamic and static equilibrium equations


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

10

Stress and strain


Stresses at a point in a 3D solid:

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

11

Stress and strain


Strains

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

12

Stress and strain


Strains in matrix form

where

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

13

Constitutive equations


=
c



or

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

14

Constitutive equations


For isotropic materials

,

,

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

15

Dynamic equilibrium equations


Consider stresses on an infinitely small
block

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

16

Dynamic equilibrium equations


Equilibrium of forces in
x

direction
including the inertia forces

Note:

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

17

Dynamic equilibrium equations


Hence, equilibrium equation in
x
direction


Equilibrium equations in
y
and
z
directions

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

18

Dynamic and static equilibrium equations


In matrix form

or


For static case

Note:

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

19

EQUATIONS FOR 2D SOLIDS

Plane stress

Plane strain

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

20

Stress and strain

(3D)

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

21

Stress and strain


Strains in matrix form

where

,

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

22

Constitutive equations


=
c



(For plane stress)

(For plane strain)

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

23

Dynamic equilibrium equations

(3D)

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

24

Dynamic and static equilibrium equations


In matrix form

or


For static case

Note: