Chapter 7 Transformations of Stress and Strain

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. 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. 1. maxand min occur at  = 0 2. max and min are 90o apart. max and min are 90o apart. 3. max and minoccur 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. 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 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. 7.12 3-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