Question

Consider a rectangular plate with in-plane loads. The stress at an arbitrary angle \( \theta \) is given by \( \sigma_x \), \( \sigma_y \), and \( \tau_{xy} \) as shown in the figure. If the principal plane is at \( \theta = 45^\circ \), and the principal stresses are \( \sigma_x = 8 \, \text{N/mm}^2 \) and \( \sigma_y = 3 \, \text{N/mm}^2 \), then the corresponding \( \tau_{xy} \) is …………… \( \text{N/mm}^2. \) 

Hint:

For principal planes: 1. The shear stress is always zero on a principal plane, regardless of the values of \( \sigma_1 \) and \( \sigma_2 \).
2. Verify the orientation of the principal plane (angle \( \theta \)) to ensure calculations are consistent.
3. Use this property to simplify stress analysis problems.

Answer 1

Step 1: Understanding the condition of the principal plane. 
The principal plane is defined as the plane where the shear stress \( \tau_{xy} \) is zero. This is a key property of the principal plane in stress analysis. Step 2: Identify the angle of the principal plane. 
The given angle \( \theta = 45^\circ \) corresponds to the orientation of the principal plane. By definition, the shear stress on the principal plane is: \[ \tau_{xy} = 0 \, \text{N/mm}^2. \] 

Conclusion: The corresponding \( \tau_{xy} \) is \( 0 \, \text{N/mm}^2 \).