Chapter 2 Stress and Strain

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Chapter 2 Stress and Strain -- Axial
Loading
Statics 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
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
Percent elongation
Percent reduction in area
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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

Lo original length
L instantaneous length

(2.3)
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2.5 Hooke's Law Modulus of Elasticity
(2.4)
Where E modulus of elasticity or Youngs
modulus
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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2.6 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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2.7 Repeated Loadings Fatigue
Fatigue failure generally occurs at a stress
level that is much lower than ?y
The ? -N curve stress vs life curve
The Endurance Limit the stress for which
fatigue failure does not occur.
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2.8 Deformations of Members under Axial Loading
(2.4)

(2.5)
(2.6)
(For Homogeneous rods)
(For various-section rods)
(For variable cross-section rods)
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(2.9)
(2.10)

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2.9 Statically Indeterminate Problems

• 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 gt the number of
equations -- Must involve deformation
Example 2.02
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Example 2.02
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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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2.10 Problems Involving Temperature Changes
2(.21)

? coefficient of thermal expansion
?T ?P 0

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Therefore



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2.11 Poisson 's Ratio
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2.12 Multiaxial Loading Generalized Hooke's Law
? Cubic ? rectangular parallelepiped
? Principle of Superposition
-- The
combined effect ? (individual effect)
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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2.12 Multiaxial Loading Generalized Hooke's Law
Generalized Hookes Law
(2.28)
Homogeneous Material -- has identical properties
at all points.
Isotropic Material -- material properties do not
vary with direction or orientation.
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2.13 Dilation Bulk Modulus
Original volume 1 x 1 x 1 1 Under the
multiaxial stress ?x, ?y, ?z The new volume


Neglecting the high order terms yields
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e dilation volume strain change in
volume/unit volume
Eq. (2.28) ? Eq. (2-30)
(2.31)
Special case hydrostatic pressure -- ?x, ?y,
?z p
Define
(2.33)
(2.33)
? bulk modulus modulus of compression
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Since ? positive,
(1 - 2?) gt 0 1 gt 2 ? ? lt ½
Therefore, 0 lt ? lt ½
? 0
? ½
-- Perfectly incompressible materials
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2.14 Shearing Strain

If shear stresses are present
Shear Strain
(In radians)
(2.36)
(2.37)
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The Generalized Hookes Law
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2.18 Further Discussion of Deformation under
Axial Loading Relation Among E, ?, and G
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Saint-Venants 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
-- orthotropic materials
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2.17 Stress and Strain Distribution Under Axial
Loading Saint-Venant's Principle
If the stress distribution is uniform
In reality
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2.18 Stress Concentrations
-- 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
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For ?max lt ?Y
For ?max ?Y
For ?ave ?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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