Compute the torsion constant J for circular shafts, thin-walled open and closed sections, solve for shaft size from a target J, and convert J units.
J = π·d⁴/32 · Hollow shaft: J = π·(D⁴−d⁴)/32For circular sections the torsional constant equals the polar moment of inertia. In solve mode: solid d = (32J/π)1/4; hollow D = (32J/(π(1−k⁴)))1/4 with the same k.
Membrane-analogy approximation, valid when the wall is thin (t/b ≤ 0.1). Unlike circular shafts, the torsional constant of open sections is much smaller than their polar moment of inertia.
Bredt's first formula J = 4Am²/Σ(li/ti) for single-cell thin-walled closed sections (t/r ≤ 0.1). For a thin circular tube this reduces to J = 2πrm³t.
Exact powers of the length conversion: 1 in⁴ = (0.0254)⁴ m⁴, 1 ft⁴ = (0.3048)⁴ m⁴.
The torsional constant J measures a cross-section's resistance to twisting under torque. For a solid circular shaft J = π·d⁴/32. Example: a solid shaft with diameter 60 mm has J = 1.2723×10^-6 m⁴ = 1272345.02 mm⁴. Use the "Circular sections" tab to compute it, or solve for the diameter from a target J.
For a hollow circular shaft J = π·(D⁴ − d⁴)/32, where D is the outer and d the inner diameter; equivalently J = π·D⁴·(1 − k⁴)/32 with k = d/D. Example: D = 80 mm, d = 60 mm gives J = 2.7489×10^-6 m⁴. Enter a k-ratio directly, or invert the formula to size a hollow shaft from a required J.
For thin-walled open sections (flat bars, channels, I-beams) J ≈ Σ lᵢ·tᵢ³/3 over the rectangular components — far smaller than the polar moment of inertia. A single narrow rectangle uses the series J = b·t³·[1/3 − 0.21·(t/b)·(1 − (t/b)⁴/12)]. Example: a 100 mm × 5 mm flat bar has J = 4.0354×10^-9 m⁴. The "Thin-walled open" tab also handles built-up I/channel/T sections with an optional junction factor.
For single-cell thin-walled closed sections (tubes, box beams), Bredt's first formula gives J = 4·Aₘ²/Σ(lᵢ/tᵢ), where Aₘ is the area enclosed by the wall centerline. For a circular tube this reduces to J = 2·π·rₘ³·t. Example: a tube with centerline radius 50 mm and wall 3 mm has J = 2.3562×10^-6 m⁴.
J has units of length⁴. Convert by the fourth power of the length factor: 1 in⁴ = (0.0254)⁴ m⁴ and 1 ft⁴ = (0.3048)⁴ m⁴. Example: J = 1.2723×10^-6 m⁴ equals 3.0568 in⁴. The "J unit converter" tab converts between m⁴, mm⁴, cm⁴, in⁴ and ft⁴.
Only for circular sections: there J = π·d⁴/32 is both the torsional constant and the polar moment of inertia. For non-circular sections they differ — a thin open section's torsional constant is much smaller than its polar moment. The related angle-of-twist calculator uses J together with torque, length and shear modulus (φ = T·L/(G·J)).
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