Planes of maximum shear under biaxial stress:\nFor a biaxial state with principal planes defined, what is the orientation of the planes of maximum shear stress?

Introduction / Context:
Understanding the orientation of maximum shear stresses relative to principal planes is fundamental for failure analysis and for interpreting Mohr’s circle in biaxial stress states.

Given Data / Assumptions:

• Biaxial stress condition with defined principal planes (no in-plane shear on those planes).
• Linear elasticity; standard sign conventions.

Concept / Approach:
Principal planes carry only normal stress; shear is zero there. Maximum shear planes occur midway between principal planes. In physical space, this corresponds to a 45° inclination to principal planes, and two orthogonal maximum-shear planes exist, 90° apart from each other.

Step-by-Step Solution:

Verification / Alternative check:
Transforming stresses via the stress transformation equations and differentiating τ(θ) with respect to θ yields the same 45° condition for extremum shear.

Why Other Options Are Wrong:

• Any angle other than 45° contradicts the transformation result.
• Coincidence with principal planes is impossible because shear there is zero by definition.

Common Pitfalls:
Mixing Mohr’s-circle angle (2θ) with the physical angle θ; forgetting the two orthogonal maximum-shear planes; sign convention errors.

Final Answer:
True