Question

The major (\( \sigma_1 \)) and minor (\( \sigma_3 \)) principal stresses and the maximum shear stress (\( \tau_{{max}} \)) are related as \( |\tau_{{max}}| = \sigma_1 = - \sigma_3 \). The magnitude of normal stress on the plane where \( \tau_{{max}} \) acts is:

• A. \( 0 \)
• B. \( \sigma_1 \)
• C. \( \sigma_3 \)
• D. \( \frac{\sigma_1 - \sigma_3}{2} \)

Hint:

If the principal stresses are equal in magnitude and opposite in sign, the normal stress on the plane of maximum shear stress becomes zero, as the Mohr’s circle is centered at the origin.

Answer 1

The Correct Option is A

Step 1: Use the given condition. 
We are given that: \[ |\tau_{{max}}| = \sigma_1 = -\sigma_3 \] This implies that the principal stresses are equal in magnitude and opposite in sign.
Step 2: Use Mohr’s circle to find the normal stress on the plane where maximum shear stress acts.
On Mohr’s circle, the center lies at: \[ \frac{\sigma_1 + \sigma_3}{2} = \frac{\sigma_1 + (-\sigma_1)}{2} = 0 \] Hence, the normal stress on the plane where \( \tau_{{max}} \) acts is: \[ \sigma_{{normal}} = 0 \]