For a thin-walled rectangular box (closed) of depth d, width b, and uniform wall thickness t, what is the approximate torsion constant k (Saint-Venant torsional rigidity parameter)?

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:

• Section: closed rectangular thin-walled box.
• Dimensions: width b, depth d, uniform thickness t.
• Thin-wall assumption: t << b and t << d.

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.

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