# Finite Element Method

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29 Οκτ 2013 (πριν από 4 χρόνια και 8 μήνες)

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

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F
inite Element Method

INTRODUCTION TO MECHANICS

FOR SOLIDS AND STRUCTURES

G. R. Liu and S. S. Quek

CHAPTER 2:

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

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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

or
forces
.

The
stresses

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

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

Dynamics

deals with the mechanics of solids and

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

.

Plastic
:

the

deformation

in

the

solids

cannot

be

fully

recovered

when

it

is

.

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

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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

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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

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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

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

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

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Stress and strain

Strains in matrix form

=

z
L
w

where

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

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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

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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

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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

-
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