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.
Mechanical engineers need torsional constants for shaft design, spring calculations, and structural analysis. This calculator handles five common shapes without needing to look up individual formulas, and immediately shows whether the stress is acceptable. Keep these notes focused on your operational context. Tie the context to the calculator’s intended domain. Use this clarification to avoid ambiguous interpretation.
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.
Use consistent units, verify assumptions, and document conversion standards for repeatable outcomes.
Most mistakes come from mixed standards, rounding too early, or misread labels. Recheck final values before use. ## Practical Notes
Use concise notes to keep each section focused on outcomes. ## Practical Notes
Check assumptions and units before interpreting the number. ## Practical Notes
Capture practical pitfalls by scenario before sharing the result. ## Practical Notes
Use one example per section to avoid misapplying the same formula. ## Practical Notes
Document rounding and precision choices before you finalize outputs. ## Practical Notes
Flag unusual inputs, especially values outside expected ranges. ## Practical Notes
Apply this as a quality checkpoint for repeatable calculations.
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.