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Chapter 2 Stress and Strain
-- Axial LoadingStatics – deals with undeformable bodies (Rigid bodies)
Mechanics of Materials – deals with deformable bodies
-- Need to know the deformation of a boy under various stress/strain state
-- Allowing us to computer forces for statically indeterminate problems.
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The following subjects will be discussed:
Stress-Strain Diagrams
Modulus of Elasticity
Brittle vs Ductile Fracture
Elastic vs Plastic Deformation
Bulk Modulus and Modulus of Rigidity
Isotropic vs Orthotropic Properties
Stress Concentrations
Residual Stresses
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2.2 Normal Strain under Axial Loading
normal strainL
0lim
x
dx dx
For variable cross-sectional area A, strain at Point Q is:
The normal Strain is dimensionless.
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2.3 Stress-Strain Diagram
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Ductile Fracture Brittle Fracture
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Some Important Concepts and Terminology:
1. Elastic Modulus
2. Yield Strength – lower and upper Y.S. -- y
0.2% Yield Strength
3. Ultimate Strength, ut
4. Breaking Strength or Fracture Strength
5. Necking
6. Reduction in Area
7. Toughness – the area under the - curve
8. Percent Elongation
9. Proportional Limit
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2.3 Stress-Strain Diagram
100%B o
o
L LL
0100% B
o
A AA
Percent elongation =
Percent reduction in area =
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( / ) t L L
2.4 True Stress and True Strain
Eng. Stress = P/Ao True Stress = P/A
Ao = original area A = instantaneous area
Eng. Strain = True Strain = oL
o
L
t Lo
dL Ln
L L(2.3)
Lo = original length L = instantaneous length
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Where E = modulus of elasticity or Young’s
modulus
2.5 Hooke's Law: Modulus of Elasticity
E (2.4)
Isotropic = material properties do not vary with
direction or orientation.
E.g.: metals
Anisotropic = material properties vary with direction or
orientation. E.g.: wood, composites
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2
2.6 Elastic Versus Plastic Behavior of a Material
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Some Important Concepts:
1. Recoverable Strain
2. Permanent Strain – Plastic Strain
3. Creep
4. Bauschinger Effect: the early yielding behavior in the
compressive loading
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Fatigue failure generally occurs at a stress level that is much
lower than y
The Endurance Limit = the stress for which fatigue failure does not occur.
2.7 Repeated Loadings: Fatigue
The -N curve = stress vs life curve
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2.8 Deformations of Members under Axial Loading
E P
E AE
L PLAE
i i
i i i
PLAE
Pdxd dx
AE
(2.4)
(2.5)
(2.6)
(For Homogeneous rods)
(For various-section rods)
(For variable cross-section rods)
P
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L
o
PdxAE
/ B A B A
PLAE
(2.9)
(2.10)
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2.9 Statically Indeterminate Problems
A. Statically Determinate Problems:
-- Problems that can be solved by Statics, i.e. F = 0
and M = 0 & the FBD
B. Statically Indeterminate Problems:
-- Problems that cannot be solved by Statics
-- The number of unknowns > the number of equations
-- Must involve “deformation”
Example 2.02:
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Example 2.02
1 2
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Superposition Method for Statically Indeterminate Problems
1. Designate one support as redundant support
2. Remove the support from the structure & treat it as an unknown load.
3. Superpose the displacement
Example 2.04
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Example 2.04
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0 L R
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2.10 Problems Involving Temperature Changes
( ) T T L
T T ( ) T T L
P
PLAE
2(.21)
= coefficient of thermal expansion
T + P = 0
0( ) T P
PLT L
AE
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Therefore:
( ) P
E TA
( ) P AE T
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2.11 Poisson 's Ratio
/ x x E
' lateral strain
Poisson s Ratioaxial strain
y z
x x
X X
x y zE E
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Cubic rectangular parallelepiped
Principle of Superposition:
-- The combined effect = (individual effect)
2.12 Multiaxial Loading: Generalized Hooke's Law
Binding assumptions:
1. Each effect is linear 2. The deformation is small and does not change the overall condition of the body.
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Generalized Hooke’s Law
2.12 Multiaxial Loading: Generalized Hooke's Law
y zxx
y zxy
y zxz
E E E
E E E
E E E
Homogeneous Material -- has identical properties at all points.
Isotropic Material -- material properties do not vary with direction or orientation.
(2.28)
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Original volume = 1 x 1 x 1 = 1
Under the multiaxial stress: x, y, z
The new volume =
2.13 Dilation: Bulk Modulus
1 1 1( )( )( ) x y z
1 x y z
1 1 1
2 30( . )
x y z
x y z
e the hange of olume
e
Neglecting the high order terms yields:
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Eq. (2.28) Eq. (2-30)
e = dilation = volume strain = change in volume/unit volume
( )X y z X y zeE E
2
1 2( )X y ze
E
3 1 2( ) e p
E 3 1 2( )
E
pe
= bulk modulus = modulus of compression +
(2.31)
(2.33)
(2.33)
Special case: hydrostatic pressure -- x, y, z = p
Define:
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3E
3e p
E
3 1 2( )
E
Since = positive,
Therefore, 0 < < ½
(1 - 2) > 0 1 > 2 < ½
= 0
= ½3 1 2 0( )
e pE
0e
-- Perfectly incompressible materials
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2.14 Shearing Strain
xy xyG
yz yz zx zxG G
(2.36)
(2.37)
If shear stresses are present
Shear Strain = xy (In radians)
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y zXx
y zXy
y zXz
xy yz zxxy yz zx
E E E
E E E
E E E
G G G
The Generalized Hooke’s Law:
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12EG
2 1( )E
G
2.18 Further Discussion of Deformation under Axial Loading: Relation Among E, , and G
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Saint-Venant’s Principle:
-- the localized effects caused by any load acting on the body will dissipate or smooth out within region that are sufficiently removed form the location of he load.
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2.16 Stress-Strain Relationships for Fiber-Reinforced Composite Materials
y zxy xz
x x
and
-- orthotropic materials
xy y zx zXx
x y z
xy X y zx zy
x y z
xy X yz y zz
x y z
E E E
E E E
E E E
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xy yx yz zy zx xz
x y y z z xE E E E E E
xy yz zxxy yz zxG G G
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2.17 Stress and Strain Distribution Under Axial Loading: Saint-Venant's Principle
( ) y y ave
PA
If the stress distribution is uniform:
In reality:
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2.18 Stress Concentrations
max
ave
K
-- Stress raiser at locations where geometric discontinuity occurs
= Stress Concentration Factor
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2.19 Plastic Deformation
Elastic Deformation Plastic Deformation
Elastoplastic behavior
yY C
A D
Rupture
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max ave
AP A
K
Y
Y
AP
K
U YP A
UY
PP
K
max
ave
K max ave K
For ave = Y
For max = Y
For max < Y
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2.20 Residual Stresses
After the applied load is removed, some stresses may still remain inside the material
Residual Stresses
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