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: Plot two points from the known stress state: \((\sigma_x, \tau_{xy})\) and \((\sigma_y, -\tau_{xy})\) on normal-stress vs shear-stress axes.
Step 2: Join these two points with a straight line; the point where this line crosses the normal-stress axis is the center of the circle, at \(\left(\dfrac{\sigma_x+\sigma_y}{2}, 0\right)\).
Step 3: Draw a circle centered there passing through both plotted points; its radius equals \(\sqrt{\left(\dfrac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2}\).
Step 4: The two points where this circle crosses the shear-stress-free (horizontal) axis give the values of the principal stresses directly, without further calculation.
So Mohr's circle is the graphical method used to find the principal stresses.

Answer 2

Another way to settle this is by matching each stress type in the options to the specific analysis tool engineers actually use to calculate it, and checking whether that tool is the graphical circle described in this question.

1. Bending stresses: engineers calculate bending stress directly from the bending moment and section modulus of a beam; this is a formula-based calculation from beam theory, unrelated to a graphical stress-transformation circle.
2. Principal stresses: at a general point under combined loading, direct formulas for principal stress exist, but the standard graphical alternative to those formulas is precisely the circle constructed by plotting stress pairs, whose axis intercepts hand over the principal stress magnitudes without solving the formulas explicitly.
3. Torsional shear stresses: these are obtained from shaft torsion theory using torque and polar moment of inertia; again a direct formula, not a graphical transformation.
4. None: ruled out since a specific graphical technique for principal stresses does exist and is exactly the one being described.

Matching each stress type to its proper analysis method confirms the graphical technique in question is used only for principal stresses.

Therefore, the correct answer is Principal stresses.