Chapter 7 Transformations of Stress and Strain

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

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

Examples of Plane-Stress Condition:

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

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

7.2 Transformation of Plane Stress

After rearrangement: (7.1) (7.2) Knowing

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)

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

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)

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)

Mohr’s Circle

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

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.

7.4 Mohr’s Circle for Plane Stress

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

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

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.

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

7.9 Stresses in Thin-Walled Pressure Vessels

Hoop Stress 1 (7.30)

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

Using the Mohr’s circle to solve for max

7.8 Fracture Criteria for Brittle Materials under Plane stress

7.10 Transformation of Plane Strain

7.11 Mohr’s Circle for Plane Strain

7.12 3-D Analysis of Strain

7.13 Measurements of Strain : Strain Rosette

1 2

1 2

1 2

1 2

1 2