Octahedral shear stress from principal stresses At a point in a solid, the three principal stresses are σ1 = 100 kgf/cm², σ2 = 100 kgf/cm², and σ3 = −200 kgf/cm². Compute the octahedral shear stress τ_oct (in kgf/cm²).

Introduction / Context:Failure criteria that depend on distortional energy (e.g., von Mises) and octahedral stresses are widely used in design. The octahedral shear stress τ_oct is a scalar measure derived from the three principal stresses and represents the shear acting on planes equally inclined to the principal axes (the octahedral planes).

Given Data / Assumptions:

• Principal stresses: σ1 = 100 kgf/cm², σ2 = 100 kgf/cm², σ3 = −200 kgf/cm².
• Small-strain continuum; standard octahedral definitions.

Concept / Approach:

The octahedral shear stress is computed from the principal stresses via the invariant-based expression: τ_oct = sqrt( ( (σ1 − σ2)² + (σ2 − σ3)² + (σ3 − σ1)² ) ) / (3sqrt(2)). This comes from resolving the stress state on the octahedral planes (planes making equal angles with all three principal directions).

Step-by-Step Solution:

Verification / Alternative check:

Using the alternative form τ_oct = (1/3) * sqrt( ( (σ1 − σ2)² + (σ2 − σ3)² + (σ3 − σ1)² ) / 2 ) yields the same numerical result since 1/(3sqrt(2)) = (1/3) * (1/sqrt(2)).

Why Other Options Are Wrong:

• 141.4 corresponds to dividing by 3 only (missing sqrt(2)).
• 173.2, 200, 244.9 are inconsistent with the invariant formula.

Common Pitfalls:

• Using maximum shear ( (σ_max − σ_min)/2 = 150 ) instead of octahedral shear; they are not the same.
• Dropping the sqrt(2) factor in the denominator.

Final Answer:

100 kgf/cm².