1. Chapter 7
Transformations of Stress and Strain

2. 7.1 Introduction Goals: determine: 1. Principal Stresses
2. Principle Planes
3. Max. Shearing Stresses
3 normal stresses
-- x, y, and z
General State of Stress
3 shearing stresses -- xy, yz, and zx

3. z = 0, yz = xz = yz = xz = 0 z  0, xy  0
Plane Stress condition
2-D State of Stress
Plane Strain condition
A. Plane Stress State:
z = 0, yz = xz = yz = xz = 0
z  0, xy  0
B. Plane Stress State:
z = 0, yz = xz = yz = xz = 0
z  0, xy  0

4. Examples of Plane-Stress Condition:

5. Thin-walled Vessels In-plane shear stress Shear stress
Out-of-plane shear stress

6. Max. x & y
(Principal stresses)
Max. xy

7. 7.2 Transformation of Plane Stress

9. After rearrangement:
(7.1)
(7.2)
Knowing

10. Eqs. (7.1) and (7.2) can be simplified as:
(7.5)
(7.6)
Can be obtained by replacing  with ( + 90o) in Eq. (7.5)
(7.7)

11. 2. max and min are 90o apart. max and min are 90o apart.
1. max and min occur at  = 0
2. max and min are 90o apart. max and min are 90o apart.
3. max and min occur half way between max and min

12. 7.3 Principal Stresses: Maximum Shearing Stress
Since max and min occur at x’y’ = 0, one can set Eq. (7.6) = 0
(7.6)
It follows,
(a)
Hence,
(b)

13. This is a formula of a circle with the center at:
Substituting Eqs. (a) and (b) into Eq. (7.5) results in max and min :
(7.14)
This is a formula of a circle with the center at:
and the radius of the circle as:
(7.10)

14. Mohr’s Circle

15. The max can be obtained from the Mohr’s circle:
Since max is the radius of the Mohr’s circle,

16. Since max occurs at 2 = 90o CCW from max,
Hence, in the physical plane max is  = 45o CCW from max.
 In the Mohr’s circle, all angles have been doubled.

17. 7.4 Mohr’s Circle for Plane Stress

18. Sign conventions for shear stresses:
CW shear stress =  and is plotted above the -axis,
CCW shear stress = ⊝ and is plotted below the -axis

20. 7.5 General State of Stress – 3-D cases
Definition of Direction Cosines:
with

23. Dividing through by A and solving for n, we have
(7.20)
We can select the coordinate axes such that the RHS of Eq. *7.20) contains only the squares of the ’s.
(7.21)
Since shear stress ij = o, a, b, and c are the three principal stresses.

24. 7.6 Application of Mohr’s Circle to the 3-D Analysis of Stress
A > B > C
= radius of the Mohr’s circle

27. 7.9 Stresses in Thin-Walled Pressure Vessels

28. Hoop Stress 1
(7.30)

29. Longitudinal Stress 2 Solving for 2 Hence
Assuming the end cap or the fluid inside takes the pressure
Solving for 2
(7.31)
Hence

30. Using the Mohr’s circle to solve for max

33. 7.8 Fracture Criteria for Brittle Materials under Plane stress

37. 7.10 Transformation of Plane Strain

39. 7.11 Mohr’s Circle for Plane Strain

41. D Analysis of Strain

43. 7.13 Measurements of Strain : Strain Rosette

46. 1 2

47. 1 2

48. 1 2

49. 1 2

50. 1 2