Shaft Torsion (Solid Round)
Max shear stress τ = Tr/J for a solid circular shaft, plus optional twist angle.
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The engineering
Shear stress in torsion lives at the surface — the core of a solid shaft is on a paid vacation, carrying almost nothing, which is why hollow shafts win every strength-per-kilogram contest and your driveshaft is a tube. The d⁴ in J is the number that bites in reverse: turning a shaft down 10% for a bearing seat costs 27% of its torsional strength at that shoulder.
The twist angle matters even when stress doesn't. Long drill strings, steering columns, and machine-tool drives are usually stiffness-limited, not strength-limited — a shaft can be perfectly safe and still wind up like a torsion bar, arriving degrees late at the far end.
Where this math comes from
Charles-Augustin de Coulomb worked out the torsion of thin wires in 1784 because he needed a spring so soft it could weigh the force between charges — the torsion balance measured electricity before torsion theory served machinery. The elegant τ = Tr/J result holds exactly only for circular sections.
That fine print is Adhémar Barré de Saint-Venant's contribution: his 1855 memoir solved torsion for non-circular sections and showed square shafts shear worst at mid-face, not corners — one of elasticity theory's genuinely counterintuitive results, and the reason this card insists on 'solid round'.
- 1784Charles-Augustin de CoulombTorsion of wires quantified — the torsion balance.
- 1855Saint-VenantGeneral torsion theory; circular-section formula's limits mapped.
- 1930Stepan Timoshenko (circa)Torsion design practice standardized in the classic texts.
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