Finite Element Method

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

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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
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

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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

(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

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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

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