Calculate the torsional constant (J) for solid circles, hollow tubes, rectangles, and I-beams. Includes shear stress, twist angle, and formula reference.
The torsional constant (J) measures a cross-section's resistance to twisting. For circular shafts, J equals the polar moment of inertia (π D⁴/32). For non-circular sections — rectangles, I-beams, channels — J is not the polar moment but a distinct geometric property calculated from different formulas depending on whether the section is closed or open.
Closed sections (tubes, boxes) are far more efficient in torsion than open sections (I-beams, channels) of the same material and area. A thin-wall tube can have 100× the torsional constant of an open channel with the same cross-sectional area. This explains why drive shafts, roll cage members, and helicopter blades all use closed tubular sections.
The maximum shear stress under torsion is τ = Tr/J (for circular sections, exact; for others, approximate). The angle of twist per unit length equals T/(GJ), where G is the shear modulus. This calculator provides J for five common cross-section types, plus the resulting stress and twist under applied torque.
Use this calculator when you need to compare shaft, tube, or beam sections before selecting a part for torque transmission. It gives the geometric torsional constant, then shows the resulting shear stress and twist so you can judge whether the section is stiff enough for the load.
Solid circle: J = πD⁴/32. Hollow circle: J = π(D⁴−d⁴)/32. Thin-wall tube: J = 2πr³t. Shear stress: τ = Tr/J. Twist: φ/L = T/(GJ).
Result: J = 613,600 mm⁴, τ = 20.4 MPa, twist = 0.0585°/m
A 50mm solid steel shaft under 500 N·m torque: J = π(50)⁴/32 = 613,600 mm⁴. τ = 500000×25/613600 = 20.4 MPa — well below steel's 250 MPa yield.
Closed sections such as tubes and boxes are far more efficient in torsion than open sections such as channels and I-beams. When possible, compare sections with the same material and similar mass before deciding which one to use.
For circular shafts, J is exact and the stress estimate is straightforward. For non-circular shapes, the calculator uses standard approximations for thin-wall or open-section behavior, so the result is best treated as a design check rather than a finite-element substitute.
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J (sometimes called the St. Venant torsion constant) is a geometric property that quantifies how well a cross-section resists twisting. Larger J = less twist for the same torque.
Only for circular sections. For rectangles, I-beams, and other non-circular shapes, J is a different quantity computed from different formulas.
Closed sections develop a continuous shear flow around the perimeter, making them extremely torsion-efficient. Open sections lack this flow and rely on much less effective warping resistance.
For steel shafts: typically ≤0.6 × yield shear strength. Yield shear ≈ 0.58 × tensile yield. For common mild steel (250 MPa yield), allowable torsional stress is about 87 MPa.
Excessive twist causes misalignment and vibration. Many codes limit twist to 0.25°/m for general machinery and 0.1°/m for precision shafts.
For constrained open sections (I-beams bolted at ends), warping resistance adds significant stiffness beyond simple St. Venant torsion. This calculator covers St. Venant only.