1. 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.

2. 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

3. 2.2 Normal Strain under Axial Loading
For variable cross-sectional area A, strain at Point Q is:
The normal Strain is dimensionless.

4. 2.3 Stress-Strain Diagram

5. Ductile Fracture
Brittle Fracture

7. 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

8. Percent reduction in area =
2.3 Stress-Strain Diagram
Percent elongation =
Percent reduction in area =

9. 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)

10. 2.5 Hooke's Law: Modulus of Elasticity
(2.4)
Where E = modulus of elasticity or Young’s 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

12. 2.6 Elastic Versus Plastic Behavior of a Material

13. Some Important Concepts:
1. Recoverable Strain
2. Permanent Strain – Plastic Strain
3. Creep
4. Bauschinger Effect: the early yielding behavior in the compressive loading

14. 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.

16. 2.8 Deformations of Members under Axial Loading
(2.4)
(2.5)
(2.6)
(For Homogeneous rods)
(For various-section rods) P
(For variable cross-section rods)

17. (2.9)
(2.10)

18. 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 > the number of equations
-- Must involve “deformation”
Example 2.02:

19. Example 2.02

20. 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

21. Example 2.04

23. 2.10 Problems Involving Temperature Changes
2(.21)
 = coefficient of thermal expansion
T + P = 0

24. Therefore:

25. 2.11 Poisson 's Ratio

26. 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 The deformation is small and does not change the overall condition of the body.

27. 2.12 Multiaxial Loading: Generalized Hooke's Law
(2.28)
Homogeneous Material -- has identical properties at all points.
Isotropic Material -- material properties do not vary with direction or orientation.

28. 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:

29. Special case: hydrostatic pressure -- x, y, z = p
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 +

30. (1 - 2) > 0 1 > 2   < ½  = 0  = ½ Since  = positive,
(1 - 2) > 0 1 > 2   < ½
Therefore, 0 <  < ½
 = 0
 = ½
-- Perfectly incompressible materials

31. 2.14 Shearing Strain Shear Strain = (In radians)
If shear stresses are present
Shear Strain =
(In radians)
(2.36)
(2.37)

32. The Generalized Hooke’s Law:

33. 2. 18 Further Discussion of Deformation under Axial Loading:
2.18 Further Discussion of Deformation under Axial Loading: Relation Among E, , and G

34. 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.

35. 2. 16 Stress-Strain Relationships for Fiber-Reinforced
2.16 Stress-Strain Relationships for Fiber-Reinforced Composite Materials
-- orthotropic materials

37. 2. 17 Stress and Strain Distribution Under Axial Loading:
2.17 Stress and Strain Distribution Under Axial Loading: Saint-Venant's Principle
If the stress distribution is uniform:
In reality:

38. 2.18 Stress Concentrations
-- Stress raiser at locations where geometric discontinuity occurs
= Stress Concentration Factor

40. 2.19 Plastic Deformation Elastic Deformation  Plastic Deformation
Elastoplastic behavior  y Y C A D 
Rupture

41. For max < Y
For max = Y
For ave = Y

42. 2.20 Residual Stresses
After the applied load is removed, some stresses may still remain inside the material
 Residual Stresses