Saint-Venant uniform torsion — torsional stiffness K, angle of twist θ and maximum shear stress τ_max for 9 section types
This calculator finds the torsional response of a straight bar of uniform cross-section under a pure torque T, using classical Saint-Venant uniform torsion theory. The two key outputs are the torsional stiffness factor K (in mm⁴) and the maximum torsional shear stress \tau_{max} (in MPa). From K the angle of twist follows as \theta = TL/(KG), where L is the bar length and G is the shear modulus. The factor K equals the polar moment of area J only for a circular cross-section; for all non-circular sections K < J, and the difference can be large (for a thin open section, K is far smaller than J).
Nine section types are covered with closed-form expressions. For solid compact sections (circle, ellipse, square, rectangle, equilateral triangle) the formulas come from rigorous Saint-Venant torsion analysis or from well-established polynomial approximations (the rectangular section carries a ≤ 4 % error). For hollow sections, the concentric circular and elliptical tubes use exact scaling of the solid solution by a wall factor (1-q^4) or (r_o^4 - r_i^4). For thin-wall sections, the closed tube uses the Bredt formula K = 4A^2 t/U and \tau = T/(2tA) (where A is the area enclosed by the median line and U its perimeter); the open circular tube uses the standard open-section formula K = 2\pi r t^3/3.
The angle of twist increases linearly along the bar: \theta(x) = (T/KG)\,x (Saint-Venant uniform torsion, no warping restraint). The figure shows the cross-section shape with the location of \tau_{max} marked, and a chart of \theta(x) from zero at the fixed end to \theta_{total} at x = L.
Model limits: uniform cross-section; linear-elastic isotropic material; pure torque only (no bending, axial force, or combined loading); small deformation; warping fully free at both ends. If one end is warping-restrained or the torque is applied away from the ends, the open thin-wall sections require a warping-torsion correction (Vlasov theory), which is outside the scope of this calculator. The outputs are structural response quantities only — there is no pass/fail safety check.
Covered cases of this table:
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