chapter 2 stress and strain --- axial loading
TRANSCRIPT
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.
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
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.
2.3 Stress-Strain Diagram
Ductile Fracture Brittle Fracture
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
2.3 Stress-Strain Diagram
100%B o
o
L LL
0100% B
o
A AA
Percent elongation =
Percent reduction in area =
( / ) 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
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
2
2.6 Elastic Versus Plastic Behavior of a Material
Some Important Concepts:
1. Recoverable Strain
2. Permanent Strain – Plastic Strain
3. Creep
4. Bauschinger Effect: the early yielding behavior in the
compressive loading
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
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
L
o
PdxAE
/ B A B A
PLAE
(2.9)
(2.10)
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:
Example 2.02
1 2
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
Example 2.04
0 L R
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
Therefore:
( ) P
E TA
( ) P AE T
2.11 Poisson 's Ratio
/ x x E
' lateral strain
Poisson s Ratioaxial strain
y z
x x
X X
x y zE E
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.
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)
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:
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:
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
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)
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:
12EG
2 1( )E
G
2.18 Further Discussion of Deformation under Axial Loading: Relation Among E, , and G
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.
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
xy yx yz zy zx xz
x y y z z xE E E E E E
xy yz zxxy yz zxG G G
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:
2.18 Stress Concentrations
max
ave
K
-- Stress raiser at locations where geometric discontinuity occurs
= Stress Concentration Factor
2.19 Plastic Deformation
Elastic Deformation Plastic Deformation
Elastoplastic behavior
yY C
A D
Rupture
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
2.20 Residual Stresses
After the applied load is removed, some stresses may still remain inside the material
Residual Stresses