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).
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A100
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B141.4
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C173.2
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D200
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E244.9
Answer
Correct Answer: 100
Explanation
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:
Compute differences: σ1 − σ2 = 0; σ2 − σ3 = 100 − (−200) = 300; σ3 − σ1 = −200 − 100 = −300.Square and sum: 0² + 300² + (−300)² = 0 + 90000 + 90000 = 180000.Take square root: sqrt(180000) = 424.264.Divide by (3sqrt(2)) = 31.4142 = 4.2426.τ_oct = 424.264 / 4.2426 ≈ 100 kgf/cm².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.