Mohr’s circle relationship for plane stress: If p1 and p2 are mutually perpendicular principal stresses in a soil mass, the normal stress σ on a plane at angle θ to the p1-plane is given by which expression?

Introduction / Context:
Stress transformation in soils and solids is routinely performed using Mohr’s circle or equivalent equations. Knowing the normal stress on a plane inclined to principal directions is essential for earth pressure, bearing capacity, and slope stability problems.

Given Data / Assumptions:

• Principal stresses p1 (major) and p2 (minor) act on mutually perpendicular planes.
• Angle θ is measured from the plane carrying p1.
• Plane stress condition.

Concept / Approach:
For any plane at angle θ from the p1-plane, the normal stress is obtained from standard transformation equations. The fundamental identity is σ = p1 cos^2θ + p2 sin^2θ. Equivalent forms using double-angle identities also exist but must be used carefully with correct trigonometric factors.

Step-by-Step Solution:
Start with transformation: σ = p1 cos^2θ + p2 sin^2θ.Recognize that shear stress transformation involves sin 2θ terms, but the normal stress here uses squared trigs.Confirm angle is referenced properly to p1-plane as stated.Select the matching expression (option a).

Verification / Alternative check:
Using double-angle identity: σ = (p1 + p2)/2 + (p1 − p2)/2 cos 2θ, which is algebraically identical to option (a). Among choices, (d) is the double-angle equivalent; since the stem defines angle from the p1-plane, either (a) or the correct double-angle form works. The explicit squared-trig version is unambiguous, hence chosen.

Why Other Options Are Wrong:

• (b) swaps sin^2θ and cos^2θ, not matching the reference plane stated.
• (c) uses sin 2θ for normal stress, which is incorrect; sin 2θ appears in shear stress.

Common Pitfalls:
Confusing sign conventions and angle reference; mixing up the normal and shear transformation formulas.

Final Answer:
σ = p1 cos^2θ + p2 sin^2θ