Finite Element Method

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Oct 29, 2013 (3 years and 7 months ago)

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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

CONTENTS


INTRODUCTION


Statics and dynamics


Elasticity and plasticity


Isotropy and anisotropy


Boundary conditions


Different structural components


EQUATIONS FOR THREE
-
DIMENSIONAL (3D) SOLIDS


EQUATIONS FOR TWO
-
DIMENSIONAL (2D) SOLIDS


EQUATIONS FOR TRUSS MEMBERS


EQUATIONS FOR BEAMS


EQUATIONS FOR PLATES

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

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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

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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

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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

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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

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Boundary conditions


Displacement (
essential
) boundary
conditions


Force (
natural
) boundary conditions


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

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Different structural components


Truss and beam structures

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

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Different structural components


Plate and shell
structures

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

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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

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Stress and strain


Stresses at a point in a 3D solid:

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

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Stress and strain


Strains

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

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Stress and strain


Strains in matrix form

where

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

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Constitutive equations


=
c



or

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

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Constitutive equations


For isotropic materials

,

,

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

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Dynamic equilibrium equations


Consider stresses on an infinitely small
block

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

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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

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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

19

Dynamic and static equilibrium equations


In matrix form

or


For static case

Note:

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

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EQUATIONS FOR 2D SOLIDS

Plane stress

Plane strain

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

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Stress and strain

(3D)

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

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Stress and strain


Strains in matrix form

where

,

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

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Constitutive equations


=
c



(For plane stress)

(For plane strain)

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

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Dynamic equilibrium equations

(3D)

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

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Dynamic and static equilibrium equations


In matrix form

or


For static case

Note:

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

26

EQUATIONS FOR TRUSS
MEMBERS

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

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Constitutive equations


Hooke’s law in 1D




=
E




Dynamic and static equilibrium equations

(Static)

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

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EQUATIONS FOR BEAMS


Stress and strain


Constitutive equations


Moments and shear forces


Dynamic and static equilibrium equations

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

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Stress and strain


Euler

Bernoulli theory

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

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Stress and strain

Assumption of thin beam

Sections remain normal

Slope of the deflection curve

where


xx

=
E


xx



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

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Constitutive equations


xx

=
E


xx


Moments and shear forces


Consider isolated beam cell of length d
x

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

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Moments and shear forces


The stress and moment

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

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Moments and shear forces

Since

Therefore,

Where

(Second moment of area about
z

axis


dependent on shape and
dimensions of cross
-
section)

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

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Dynamic and static equilibrium equations

Forces in the
x

direction

Moments about point A





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

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Dynamic and static equilibrium equations



Therefore,

(Static)

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

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EQUATIONS FOR PLATES


Stress and strain


Constitutive equations


Moments and shear forces


Dynamic and static equilibrium equations


Mindlin plate


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

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Stress and strain


Thin plate theory or Classical Plate Theory (CPT)

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

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Stress and strain

Assumes that

xz

= 0,


yz

= 0

,

Therefore,

,

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

39

Stress and strain


Strains in matrix form


=

z
L
w

where

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

40

Constitutive equations



=
c




where
c

has the same form for the plane
stress case of 2D solids

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

41

Moments and shear forces


Stresses on isolated plate cell

z

x

y

f
z

h


xy


xx


xz


yx


yy


yz

O

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

42

Moments and shear forces


Moments and shear forces on a plate cell d
x

x
d
y

z

x

y

O

dx

dy

Q
y

M
y

M
yx

Q
y
+d
Q
y

M
yx
+d
M
yx

M
y
+d
M
y

Q
x

M
x

M
xy

Q
x
+d
Q
x

M
xy
+d
M
xy

M
x
+d
M
x

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

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Moments and shear forces


=
c






=


c
z
L
w

Like beams,

Note that

,

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

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Moments and shear forces

Therefore, equilibrium of forces in
z

direction

or

Moments about A
-
A

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

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Dynamic and static equilibrium equations

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

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Dynamic and static equilibrium equations



where

(Static)

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

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Mindlin plate

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

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Mindlin plate

,

Therefore, in
-
plane strains



=

z
L


where

,

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

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Mindlin plate

Transverse shear strains

Transverse shear stress