44558176 chapter-2-stress-and-strain-axial-loading
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Transcript of 44558176 chapter-2-stress-and-strain-axial-loading
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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
δε = =normal strainL
0lim
δ δε∆ →
∆= =∆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. -- σ y
0.2% Yield Strength
3. Ultimate Strength, σut
4. Breaking Strength or Fracture Strength
5. Necking
6. Reduction in Area
7. Toughness – the area under the σ-ε 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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( / )ε ε= Σ∆ = Σ ∆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 = δoL
lε = =∫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
Eσ ε= (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 σy
The Endurance Limit = the stress for which fatigue failure does not occur.
2.7 Repeated Loadings: Fatigue
The σ -N curve = stress vs life curve
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2.8 Deformations of Members under Axial Loading
Eσ ε=P
E AEσε = =
Lδ ε=PLAE
δ =
i i
i i i
P LAE
δ = ∑
Pdxd dxAE
δ ε= =
(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
δ = ∫
/δ δ δ= − =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. ΣF = 0
and ΣM = 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 2δ δ=
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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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0δ δ δ= + =L R
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2.10 Problems Involving Temperature Changes
( )δ α= ∆T T L
ε α= ∆T T ( )δ α= ∆T T L
PPLAE
δ =
2(.21)
α = coefficient of thermal expansion
δT + δP = 0
0( )δ δ δ α= + = ∆ + =T P
PLT L
AE
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Therefore:
( )σ α= = − ∆PE T
A
( )α= − ∆P AE T
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2.11 Poisson 's Ratio
/ε σ=x x E
'υ = = − lateral strainPoisson s Ratioaxial strain
ε ευε ε
= − = −y z
x x
σ υσε ε ε= = = −X X
x y zE E
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• Cubic → rectangular parallelepiped
• Principle of Superposition:
-- The combined effect = Σ (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
υσ υσσε
σ υσυσε
υσ συσε
= + − −
= − + −
= − − +
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: σx, σy, σ z
The new volume =
2.13 Dilation: Bulk Modulus
1 1 1( )( )( )υ ε ε ε= + + +x y z
1 x y zυ ε ε ε= + + +
1 1 1
2 30( . )
x y z
x y z
e the hange of olume
e
υ ε ε ε
ε ε ε
= = − = + + + −
∴ = + +
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
σ σ σ υ σ σ σ+ + + += −
2
1 2( )X y ze
Eυ σ σ σ−= + +
3 1 2( )υ−= −e pE 3 1 2( )
κυ
=−E
κ= − pe
κ = bulk modulus = modulus of compression +
(2.31)
(2.33)
(2.33)
Special case: hydrostatic pressure -- σx, σy, σz = p
Define:
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3Eκ =
3e pE= −
3 1 2( )κ
υ=
−E
Since κ = positive,
Therefore, 0 < υ < ½
(1 - 2υ) > 0 1 > 2 υ υ < ½
υ = 0
υ = ½3 1 2 0( )
e pEυ−
= − = 0eκ = ∞ =
-- Perfectly incompressible materials
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2.14 Shearing Strain
xy xyGτ γ=
yz yz zx zxG Gτ γ τ γ= =
(2.36)
(2.37)
If shear stresses are present
Shear Strain = xyγ (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
υσ υσσε
σ υσυσε
υσ συσε
τ τ τγ γ γ
= + − −
= − + −
= − − +
= = =
The Generalized Hooke’s Law:
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12EG
υ= +
2 1( )E
Gυ
=+
2.18 Further Discussion of Deformation under Axial Loading: Relation Among E, υ , 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
andε ευ υε ε
= − = −
-- 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
υ σ υ σσε
υ σ σ υ σε
υ σ υ σ σε
= + − −
= − + −
= − − +
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xy yx yz zy zx xz
x y y z z xE E E E E E
υ υ υ υ υ υ= = =
xy yz zxxy yz zxG G G
τ τ τγ γ γ= = =
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2.17 Stress and Strain Distribution Under Axial Loading: Saint-Venant's Principle
( )σ σ= =y y ave
PA
If the stress distribution is uniform:
In reality:
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2.18 Stress Concentrations
maxσσ
=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
σ
σyY C
A Dε
Rupture
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maxσσ= =ave
AP A
K
σ= YY
AP
K
σ=U YP A
= UY
PP
K
maxσσ
=ave
K maxσσ =ave K
For σ ave = σY
For σmax = σY
For σmax < σY
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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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