Question

For the state of plane stress shown, the components of normal and shear stresses are given in terms of stress \( \sigma \) and unknown constants \( m \) and \( n \). If the normal and shear components of stress on a 45° plane are \( 2\sigma \) and zero, the values of \( m \) and \( n \) would be: \includegraphics[width=0.5\linewidth]{imagee10.png}

• A. \( m = 1, n = 2 \)
• B. \( m = 2, n = 1 \)
• C. \( m = 1, n = 1 \)
• D. \( m = 2, n = 2 \)

Hint:

For the plane stress state, the components of stress on a plane at an angle can be derived using Mohr's circle or stress transformation equations.

Answer 1

The Correct Option is C

In this problem, we are given a state of plane stress where the normal and shear components of stress are expressed in terms of the unknown constants \( m \) and \( n \). The normal stress on a 45° plane is \( 2\sigma \), and the shear stress on that plane is zero.

Step 1: Analyze the normal and shear stress components.
For a plane stress state, the stress components on a plane at an angle \( \theta \) are given by the following equations: \[ \sigma_{\theta} = \frac{1}{2} \left( \sigma + \sigma \right) + \left( \sigma - \sigma \right) \cos(2\theta), \] \[ \tau_{\theta} = \frac{1}{2} \left( \sigma + \sigma \right) \sin(2\theta). \] Substituting the values for \( \theta = 45^\circ \), we can solve for the constants \( m \) and \( n \). After solving, we find that \( m = 1 \) and \( n = 1 \).

Final Answer: \text{(C) \( m = 1, n = 1 \)}