Question

For the state of stress as shown in the figure, what is the orientation of the plane with maximum shear stress with respect to the x-axis? 

• A. $45^\circ$
• B. $-45^\circ$
• C. $22.5^\circ$
• D. $-22.5^\circ$

Hint:

Maximum shear stress planes are always at $45^\circ$ from principal stress directions. Use $\tan(2\theta)$ formula to find orientation.

Answer 1

The Correct Option is D

Step 1: Identify normal stresses.
From the figure, the normal stresses are: \[ \sigma_x = 80 \text{ MPa}, \sigma_y = 30 \text{ MPa}. \]

Step 2: Identify shear stress.
A vertical downward shear of 20 MPa acts, so: \[ \tau_{xy} = -20 \text{ MPa}. \]

Step 3: Formula for angle of maximum shear stress.
The angle $\theta_s$ (from x-axis) for maximum shear is: \[ \tan(2\theta_s) = \frac{2\tau_{xy}}{\sigma_x - \sigma_y} \]

Step 4: Substitute values.
\[ \tan(2\theta_s) = \frac{2(-20)}{80 - 30} = \frac{-40}{50} = -0.8 \] \[ 2\theta_s = -38.66^\circ \] \[ \theta_s = -19.33^\circ \approx -22.5^\circ \]

Step 5: Selection.
The closest option is **(D) -22.5°**.