Mohr’s circle – Normal stress on an oblique plane (unequal like principal stresses) Given Mohr’s circle for two unequal and like principal stresses σx and σy acting on orthogonal planes, the normal stress on a plane making angle θ with the minor principal plane equals which distance in the circle representation?

Introduction / Context:
Mohr’s circle graphically relates normal and shear stresses acting on rotated planes. For any plane at a physical angle θ from a reference principal plane, the corresponding point on Mohr’s circle lies at an angle 2θ from the principal stress point.

Given Data / Assumptions:

• Principal stresses σ1 (major) and σ2 (minor) are unequal and of the same sign (like).
• The circle center C lies at ( (σ1 + σ2)/2 , 0 ) on the normal-stress axis.
• Point P on the circle represents the stress state on the plane at angle θ; its projection on the σ-axis is Q.

Concept / Approach:
On Mohr’s circle, the horizontal axis is normal stress (σ), and the vertical axis is shear stress (τ). The normal stress on a plane equals the abscissa of the representative point, i.e., the horizontal projection from the origin to that point’s σ-coordinate.

Step-by-Step Solution:
Locate center C at σ_avg = (σ1 + σ2)/2.Construct the circle with radius R = (σ1 − σ2)/2.Rotate from the minor principal plane by angle 2θ to find point P.Drop a horizontal projection from P to the σ-axis → this distance from origin is OQ, the normal stress.

Verification / Alternative check:
Coordinates of P satisfy σ = σ_avg + R cos(2θ) and τ = R sin(2θ). The normal stress is the σ-coordinate only, which is exactly the horizontal projection OQ.

Why Other Options Are Wrong:
OC is the offset to the circle center, not the stress on a plane; OP mixes σ and τ components; PQ represents shear magnitude; CQ is partial and not referenced to the origin.

Common Pitfalls:
Confusing physical angle θ with 2θ on the circle; reading radial length instead of horizontal projection for σ; sign conventions for clockwise/counter-clockwise rotations.

Final Answer:
OQ (the abscissa/projection of the plane point on the normal-stress axis)