Torsion & twist calculator
Calculate torsional shear stress and angle of twist of straight bars under a torque per Saint-Venant torsion theory. Choose the cross-section – solid circle, circular tube, thin-walled closed Bredt section or rectangle –, enter torque, length and shear modulus, and the calculator returns W_t, I_t, tau and phi plus the shear check against the yield limit with traffic-light rating, live with every input.
Torsion calculator (Saint-Venant)
Model: straight, prismatic bar, pure torsion per Saint-Venant (free warping, linear-elastic). No restrained warping, no superimposed bending, no notch effect, no buckling or fatigue check. For rotating shafts additionally use DIN 743 (shaft calculator).
Results
Calculating …
Formulas and fundamentals
The basis is Saint-Venant torsion theory of pure torsion with free warping: the torque T produces a shear stress tau = T/W_t and an angle of twist phi = T·L/(G·I_t) over the bar length L. W_t is the torsional section modulus (governs the stress), I_t the torsion constant (governs the twist), G the shear modulus (steel ≈ 81000 N/mm²). Only for the circle and circular tube does I_t equal the polar second moment of area I_p; for all other sections I_t is smaller because the cross-section warps.
The section properties follow from closed-form expressions. Solid circle: W_t = π·d³/16 and I_t = π·d⁴/32. Circular tube: W_t = π·(D⁴−d⁴)/(16·D) and I_t = π·(D⁴−d⁴)/32. Thin-walled closed hollow sections use the two Bredt formulas: the shear stress is approximately constant across the wall, tau = T/(2·A_m·t), so W_t = 2·A_m·t, and I_t = 4·A_m²/(∮ds/t), for constant wall thickness I_t = 4·A_m²·t/U. Here A_m is the area enclosed by the wall centreline and U its perimeter.
The solid rectangular section twists per Saint-Venant with form factors that depend on the aspect ratio: I_t = a·b³·(1/3 − 0.21·(b/a)·(1 − b⁴/(12·a⁴))) and W_t = a²·b²/(3·a + 1.8·b) with long side a and short side b. The shear check compares the maximum torsional shear stress tau with the shear yield limit tau_F = R_e/√3 (von Mises); the available safety factor is S = tau_F/tau. The key difference from simple calculators is the closed thin-walled section: the Bredt hollow section carries torsion many times stiffer and stronger than an open section of equal area.
Worked example
A solid shaft of 40 mm diameter in steel (G = 81000 N/mm²) transmits a torque of 500 Nm over a length of 1000 mm. The section properties are W_t = π·40³/16 = 12566 mm³ and I_t = π·40⁴/32 = 251327 mm⁴.
This gives a torsional shear stress tau = T/W_t = 500000/12566 = 39.79 N/mm² and an angle of twist phi = T·L/(G·I_t) = 500000·1000/(81000·251327) = 0.02456 rad = 1.407°. Against the shear yield limit tau_F = 355/√3 = 205 N/mm² of an S355 shaft the available safety factor is about 5 – the check passes comfortably.
Replacing the solid shaft with a thin-walled closed square tube of 95 mm centreline side length (A_m = 9025 mm²) and 5 mm wall thickness under 1000 Nm, the shear stress per Bredt is tau = T/(2·A_m·t) = 1000000/(2·9025·5) = 11.08 N/mm². This shows the strength of closed hollow sections: at low mass they carry torsion efficiently – a case simple calculators usually do not cover.
Frequently asked questions
What is the difference between the torsion constant I_t and the polar moment I_p?
Only for the circle and circular tube are the two equal, because these sections do not warp under torsion. For all other sections (rectangle, open and closed profiles) the I_t that governs the twist is smaller than I_p. Using I_p for a rectangle or hollow section significantly overestimates the stiffness – the calculator always uses the correct I_t.
What do the two Bredt formulas mean?
They apply to thin-walled closed hollow sections. The first Bredt formula gives the shear stress that is constant across the wall, tau = T/(2·A_m·t), with A_m the area enclosed by the wall centreline. The second Bredt formula gives the twist via I_t = 4·A_m²/(∮ds/t), for constant wall thickness I_t = 4·A_m²·t/U. Note that A_m is the centreline area, not the outer contour.
Why is a closed hollow section so much better than an open one?
In a closed section the shear flow runs as a closed loop around the cavity and produces a large internal moment (Bredt). An open section (slit tube, U- or L-section) cannot form this loop and is orders of magnitude softer and more highly stressed in torsion. Even a narrow longitudinal slit turns a stiff tube into a very torsionally soft open section.
Which shear yield limit is used for the check?
The calculator uses the shear yield limit per the distortion energy hypothesis (von Mises): tau_F = R_e/√3 ≈ 0.577·R_e. Alternatively the shear stress hypothesis (Tresca) gives the more conservative tau_F = R_e/2. The available safety factor is S = tau_F/tau, compared with the required safety factor S_req.
What does the calculation cover – and where are its limits?
Straight, prismatic bars under pure torsion (Saint-Venant) with free warping, linear-elastic. Not included are restrained warping (e.g. at fixed ends of open sections), superimposed bending, notch effects at shoulders and keyways, and local buckling of thin-walled sections. For rotating, fatigue-loaded shafts an additional fatigue check per DIN 743 is required.
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