Torsion of Sections Calculator — Shear Stress, Angle of Twist & Fatigue (Saint-Venant / Bredt-Batho)

None — and that is the honest answer. There is no ISO, EN or ASME standard governing the torsion of general prismatic sections, so this tool implements established textbook theory and says so. Solid circular and hollow circular sections use the exact elasticity result (τ = T·c/J; φ = T·L/(G·J)) as given in Roark's Formulas for Stress and Strain and Timoshenko's Theory of Elasticity. Non-circular solid sections (square, rectangle) use the Saint-Venant torsion coefficients tabulated by Roark/Timoshenko. Closed thin-walled sections (hollow square, rectangular box, circular tube) use Bredt-Batho shear-flow theory, which is valid while the wall is thin (r/t ≥ 10) and under-predicts the peak shear outside that — the tool warns when your section leaves the range. Torsional fatigue follows the Soderberg, Modified Goodman and Sines criteria from Shigley's Mechanical Engineering Design, from a Marin-corrected endurance limit (surface, size and reliability applied; temperature and notch effects are not). The governing method and coefficients are shown in the generated PDF report.

For circular sections (solid or hollow) the torsion constant J is identical to the polar second moment of area (J = π·D⁴/32). For non-circular sections, however, the polar moment does not govern torsion — the Saint-Venant torsion constant (also called J or C) is a different, smaller quantity that must be obtained from Roark's tables or computed from the Timoshenko α/β coefficients. This calculator uses the correct constant for each section type, not the naive polar moment.

A closed cross-section (hollow box or tube) carries torque by a continuous shear flow q = T/(2·A_m) around the perimeter — the shear stress is low and the torsional stiffness is high. If the wall is cut (open section), the shear flow is interrupted and the section can only resist torque by Saint-Venant warping shear, which is proportional to t³. The penalty grows as the wall gets thinner: for a 60×40 box in 4 mm wall the calculator measures about 23× the shear stress and 135× the twist, and for a 200×100 box in 1 mm wall about 240× and 16,000×. The Bredt-Batho comparison panel shows both ratios for your own dimensions.

Yes — the combined-loading tab takes a bending moment M, axial force N and torque T for whichever of the seven sections you have selected (it uses that section's real area, second moment and torsional modulus, not a round-shaft substitute) and computes the Von Mises equivalent stress, the Tresca equivalent stress and the maximum principal stresses. It also draws Mohr's circle for the combined state and recommends a minimum diameter for a SOLID ROUND shaft carrying the same bending and torsion — that recommendation covers M and T only, not the axial force, and the page says so. A separate sub-panel sizes a solid round shaft from a bending moment, a torque and shock/fatigue factors at an allowable of Sy/2; that is a max-shear-stress sizing, not an ASME code check, whose allowable is lower.

When the wall stops being thin. Bredt-Batho assumes the shear flow is constant through the wall, which holds while the mid-line radius is at least ten times the thickness (r/t ≥ 10). Outside that it UNDER-predicts the peak shear, and the error is one-sided: measured against the exact solution it is about 4.5 % low at r/t = 10, 8 % at 5, and 15 % at 2. For a closed box it gets worse than a smooth degradation — past a wall of roughly one seventh of the outer width the formula reports a hollow section as LESS stressed than the solid bar of the same envelope, which is impossible. The calculator warns you when your section leaves the range, demotes a green verdict to CHECK while the warning is live, and discloses the approximation whenever Bredt-Batho runs at all. For a thick-walled round tube use the Hollow Circular section instead, which is exact.

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