Torsion Shear Stress Calculator

When a circular shaft is twisted, plane cross-sections remain plane and radii remain straight, so the shear strain at any point grows in direct proportion to its distance from the axis. That kinematic fact is what makes circular torsion so tractable: shear stress varies linearly from zero at the centre to a maximum at the surface, giving tau = T c / J with c the outer radius and J the polar second moment of area. For a solid section J = pi D^4 / 32; for a tube J = pi (D^4 - d^4) / 32. Twist over a length L follows from the same derivation as theta = T L / (G J), where G is the shear modulus - about 79 to 84 GPa for carbon steel, 25 to 28 GPa for aluminium alloys and 41 to 45 GPa for titanium. G is not independent of the tensile modulus; for an isotropic material G = E / (2(1 + nu)), which for steel with nu near 0.29 reproduces the familiar 80 GPa from E of 205 GPa. The fourth-power dependence explains why hollow sections are so efficient in torsion. Material near the axis is lightly stressed and contributes almost nothing to J, so removing it costs little strength: a tube with a bore 80 percent of the outer diameter keeps roughly 59 percent of J while carrying only 36 percent of the solid section area. Design is usually governed by two separate checks - peak shear stress against an allowable value, and angle of twist against a stiffness limit, commonly 0.25 to 1.0 degrees per metre for transmission shafting and far tighter for machine tool spindles. Important limits apply: the linear formulas hold only for elastic behaviour in circular or annular sections. Non-circular sections warp and require Saint-Venant torsion constants instead of J, thin-walled tubes can buckle locally before reaching the stress limit, and keyways, cross-holes and shoulders introduce stress concentrations this calculation does not capture.

The experimental foundation of torsion belongs to Charles-Augustin de Coulomb, who in 1784 published his research on the torsional force in wires. Working with the suspended torsion balance he had built, Coulomb measured the restoring couple of fine metal wires and established that it varied with the fourth power of the diameter and inversely with length - the geometric relationship still embedded in the polar moment of inertia. The instrument was as consequential as the law: Coulomb used the same torsion balance to measure the force between electric charges, and Henry Cavendish adapted it in 1798 to weigh the Earth. The general theory arrived in the following century as continuum mechanics matured. Augustin-Louis Cauchy formalised the concept of stress at a point in the 1820s, giving torsion a rigorous description in terms of shear stress components rather than a bulk restoring couple. Adhemar Jean Claude Barre de Saint-Venant then solved the far harder problem of non-circular shafts, publishing his semi-inverse treatment of torsion in 1855; he showed that only circular sections stay plane, that all others warp out of plane, and that the elementary formula therefore cannot be applied to rectangular or I-shaped members. Ludwig Prandtl's membrane analogy of 1903 later gave engineers a physical way to visualise those warping solutions. The result is the practical division still used today - the simple tau = T c / J for round shafts and tubes, and section-specific torsion constants for everything else.