Question

Mohr’s circle is a graphical method to find

• A. Bending stresses
• B. Principal stresses
• C. Torsional shear stresses
• D. None

Hint:

Mohr’s circle is especially useful when stresses act simultaneously in two perpendicular directions.

Answer 1

The Correct Option is B

Step 1: Understand Mohr’s circle.
Mohr’s circle is a graphical representation of the state of stress at a point. It helps visualize normal and shear stresses acting on different planes.
Step 2: Identify what Mohr’s circle provides.
Using Mohr’s circle, we can determine:
- Principal stresses
- Maximum shear stress
- Orientation of principal planes
Step 3: Evaluate the options.
(A) Bending stresses: These are obtained using bending theory, not Mohr’s circle.
(B) Principal stresses: Correct, Mohr’s circle is primarily used to determine principal stresses graphically.
(C) Torsional shear stresses: These are calculated using torsion theory, not Mohr’s circle directly.
(D) None: Incorrect, since Mohr’s circle has clear applications.
Step 4: Conclusion.
Therefore, Mohr’s circle is a graphical method to find principal stresses.

Answer 2

Instead of stating the definition of Mohr's circle directly, we can look at what the underlying stress-transformation equations actually produce when plotted, and use that to test each option.

For a 2D stress state with normal stresses \(\sigma_x, \sigma_y\) and shear stress \(\tau_{xy}\), the normal and shear stress on a plane inclined at angle \(\theta\) are given by

\[ \sigma_\theta = \frac{\sigma_x+\sigma_y}{2} + \frac{\sigma_x-\sigma_y}{2}\cos 2\theta + \tau_{xy}\sin 2\theta, \qquad \tau_\theta = -\frac{\sigma_x-\sigma_y}{2}\sin 2\theta + \tau_{xy}\cos 2\theta \]

If \(\sigma_\theta\) is plotted on the horizontal axis and \(\tau_\theta\) on the vertical axis as \(\theta\) is varied, these two equations trace out a circle centered at \(\left(\dfrac{\sigma_x+\sigma_y}{2},0\right)\) with radius \(\sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2}\) — this circle is exactly Mohr's circle.

1. Bending stresses: Bending stress at a section is obtained from the flexure formula \(\sigma = My/I\); it is a direct computation from bending moment and section geometry, and does not require plotting a transformation circle, so this is not what Mohr's circle is built for.
2. Principal stresses: On the circle traced above, the two points where the circle crosses the horizontal (\(\tau=0\)) axis correspond to planes where shear stress vanishes — by definition, these are the principal planes, and their normal-stress values (center plus or minus radius) are the principal stresses. This is precisely the graphical output of Mohr's circle.
3. Torsional shear stresses: Torsional shear stress is computed from the torsion equation \(\tau = Tr/J\), a direct formula based on applied torque and shaft geometry; it does not depend on constructing a circle from a pair of transformation equations.
4. None: Since the circle constructed from the transformation equations clearly has a well-defined graphical purpose (locating principal stresses), this option cannot be correct.

The two intercepts of the circle on the shear-free axis are, by construction, the principal stresses of the stress state.

Therefore, the correct answer is Principal stresses.