Define a principal plane in stress analysis: on a principal plane, the shear stress is what?

Introduction / Context:Principal stresses and principal planes are foundational in mechanics of materials. They simplify complex stress states by finding orientations where shear vanishes and only normal stress acts.

Given Data / Assumptions:

• Plane stress at a point in a continuum.
• Principal planes are the physical planes where normal stress is extremal.
• We must recall the definition regarding shear stress on such planes.

Concept / Approach:By definition, on principal planes, shear stress τ is zero and the normal stress σ reaches principal values (maximum and minimum). This follows from extremising σ_n with respect to plane angle in the stress transformation equations or by Mohr’s circle where principal points lie on the σ-axis (τ = 0).

Step-by-Step Solution:

Verification / Alternative check:Differentiating σ_n(θ) with respect to θ shows extrema occur when τ(θ) = 0, confirming the zero-shear condition on principal planes.

Why Other Options Are Wrong:“Minimum but not zero” and “maximum” misstate the defining property; principal planes are shear-free.

Common Pitfalls:Confusing maximum shear planes (where τ is extreme and σ is intermediate) with principal planes (where τ = 0).

Final Answer:

Zero