Mohr’s circle – Normal stress on an oblique plane (unequal like principal stresses)\n\nGiven 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)