Mohr’s circle and maximum shear stress\n\nIn strength of materials, Mohr’s circle is a graphical tool to visualize 2D stress states. In this representation, the magnitude of the maximum shear stress at a point is equal to which geometric feature of the circle?

Introduction / Context:
Mohr’s circle is widely used in mechanics of materials to determine principal stresses and maximum shear stress from a given plane stress state. Understanding which element of the circle corresponds to the maximum shear stress helps in fast, visual problem solving for shafts, plates, and pressure vessels.

Given Data / Assumptions:

• Two-dimensional stress state with normal stresses σx, σy and in-plane shear τxy.
• Elastic, small-strain behavior; sign convention standard for Mohr’s construction.
• Circle drawn in σ–τ coordinates with center at C and radius R.

Concept / Approach:
The general principal stress formula yields σ1 and σ2. The shear stress on any plane is obtained graphically from Mohr’s circle. The maximum in-plane shear stress equals half the difference of principal stresses, which is the circle’s radius.

Step-by-Step Solution:
Circle center: C = (σx + σy) / 2.Radius: R = sqrt[ ( (σx − σy) / 2 )^2 + τxy^2 ].Principal stresses: σ1,2 = C ± R.Maximum shear stress: τmax = (σ1 − σ2) / 2 = R.

Verification / Alternative check:
If τxy = 0 and only σx ≠ σy, the circle collapses to a line segment of length |σx − σy|. The maximum shear is half that length, i.e., the radius, confirming τmax = R.

Why Other Options Are Wrong:
Diameter equals 2R, not τmax. Circumference and area are geometric measures unrelated to shear magnitude. Hence only the radius matches τmax.

Common Pitfalls:
Confusing radius with diameter; misplacing the circle center; using τmax = R only for in-plane shear, not 3D where τmax may differ.

Final Answer:
radius