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
4
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
8
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
16
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
18
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
20
EQUATIONS FOR 2D SOLIDS
Plane stress
Plane strain
Finite Element Method by G. R. Liu and S. S. Quek
21
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
23
Constitutive equations
=
c
(For plane stress)
(For plane strain)
Finite Element Method by G. R. Liu and S. S. Quek
24
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
27
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
28
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
29
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
31
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
34
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
36
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
38
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
43
Moments and shear forces
=
c
=
c
z
L
w
Like beams,
Note that
,
Finite Element Method by G. R. Liu and S. S. Quek
44
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
45
Dynamic and static equilibrium equations
Finite Element Method by G. R. Liu and S. S. Quek
46
Dynamic and static equilibrium equations
where
(Static)
Finite Element Method by G. R. Liu and S. S. Quek
47
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
49
Mindlin plate
Transverse shear strains
Transverse shear stress
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