Difficulty: Medium
Correct Answer: k ≈ (2 * b^2 * d^2 * t) / (b + d)
Explanation:
Introduction / Context:The torsion constant k (often denoted J for Saint-Venant torsion) governs the elastic twist of prismatic members under pure torsion: θ' = T * L / (G * k). For thin-walled closed sections, warping is restrained, and the torsional rigidity is substantially higher than for open sections. A compact approximate expression exists for rectangular thin boxes of uniform thickness t, useful in steel and aerospace structures.
Given Data / Assumptions:
Concept / Approach:For thin closed sections, the standard approximation is k ≈ 4 * A_m^2 / Σ(s/t), where A_m is the area enclosed by the midline of the wall, and Σ(s/t) is the summation of (segment length / thickness) around the perimeter. For a rectangle with uniform thickness, A_m ≈ b * d and Σ(s/t) ≈ 2 * (b + d) / t. Substituting yields a compact formula for k in terms of b, d, and t.
Step-by-Step Solution:
Start with thin-walled closed section formula: k = 4 * A_m^2 / Σ(s/t).Use A_m ≈ b * d for a rectangle (midline area).Use Σ(s/t) ≈ 2 * (b + d) / t for uniform thickness t.Compute: k ≈ 4 * (b * d)^2 / (2 * (b + d) / t) = (2 * b^2 * d^2 * t) / (b + d).Verification / Alternative check:
Dimensional check: k has units of length^4. Right-hand side gives b^2 * d^2 * t / (b + d) → L^4, which is correct.Why Other Options Are Wrong:
2 * b * d * t * (b + d) and 4 * b * d * t: scale as L^4 but ignore the squared area dependence, underestimating stiffness.(b * d * t) / (b + d): scales as L^3, dimensionally incorrect for k.(b^2 * d^2 * t) / (2 * (b + d)): off by a factor of 4 relative to the correct derivation.Common Pitfalls:
Using open-section approximate J instead of the closed-section formula.Forgetting that A_m is the midline area, not the gross area minus wall thickness corrections (difference negligible for thin walls).Final Answer:
k ≈ (2 * b^2 * d^2 * t) / (b + d)
Discussion & Comments