Torsion Calculator

Torsion of a circular shaft is one of the few elasticity problems with an exact and simple closed-form solution, which is why the formulas here look deceptively general. Because a circular cross section is axisymmetric, plane sections remain plane and radii remain straight as the shaft twists, so shear strain varies linearly with radius. That gives a shear stress of T times c divided by the polar moment of inertia J, zero at the axis and maximum at the outer fibre, and a twist angle of TL divided by GJ. For a solid shaft J is pi times the radius to the fourth power over two, and for a tube it is the difference of the outer and inner values. The fourth-power dependence explains why hollow shafts are so efficient: boring out 60 percent of the diameter removes about 36 percent of the weight while retaining roughly 87 percent of the torsional stiffness, since the discarded core was barely stressed. The shear modulus G equals E divided by twice the quantity one plus Poisson ratio, which puts structural steel near 77 to 80 GPa, aluminum alloys near 26 GPa, and copper near 41 to 48 GPa. Practical design usually limits twist as well as stress; a common machine design rule of thumb caps rotation at about one degree per twenty shaft diameters of length. The critical caveat is that none of this applies to non-circular sections. Square, rectangular, and especially open thin-walled shapes such as I-beams and channels warp out of plane when twisted and are dramatically weaker in torsion than their polar moment of inertia would imply, which is why steel design uses a separate torsional constant. The formulas also ignore stress concentrations at keyways, fillets, and shoulders, where factors of 1.5 to 3 are typical, and they do not address fatigue or combined bending and torsion.

Torsion entered quantitative science through Charles Augustin Coulomb, who published a memoir on the torsion of metal wires in 1784. Working with a suspended horizontal bar, he established that the restoring torque is proportional to the angle of twist and to the fourth power of the wire diameter and inversely proportional to its length, results that amount to the modern TL over GJ relationship in experimental form. The apparatus mattered as much as the theory: the torsion balance became the most sensitive instrument of its era, and Henry Cavendish used one in 1798 to weigh the Earth by measuring the gravitational attraction between lead spheres. Extending torsion beyond circular shafts took another seventy years and proved considerably harder, because the assumption that plane sections stay plane simply fails. Adhemar Jean Claude Barre de Saint-Venant resolved it in his 1855 memoir on the torsion of prisms, showing that non-circular sections warp longitudinally and formulating the problem in a way that yields a torsional constant distinct from the polar moment of inertia. His solution was mathematically demanding, and Ludwig Prandtl made it far more accessible in 1903 with the membrane analogy, which showed that the stress function for a twisted bar has the same governing equation as a soap film stretched over a hole of the same shape. Engineers could literally see where torsional stress concentrated, and the analogy remained a standard design aid well into the computational era.