Torsionsgleichung

Torsionsgleichung is the German term for the governing equation that describes the relationship between torque, shear stress, and angular displacement in a twisted structural element, most commonly a cylindrical shaft. In its simplest elastic form it is expressed as τ = T r / J, where τ is the shear stress at radius r, T the applied torque, and J the polar moment of inertia of the cross‑section. For a solid circular shaft J equals π d⁴/32, with d the diameter. When the same torque is applied along a length L of the shaft, the angular twist θ is given by θ = TL / (JG), where G is the shear modulus of the material. These two equations together constitute the basic Torsionsgleichung used in mechanical and civil engineering to design drive shafts, gears, and other rotating components.

In the inelastic or plastic range the equation must be modified to account for the yielding behaviour of the material. An empirical strength‑strain relationship, such as the Hollomon law σ = K εⁿ, is used to find the shear flow, q = τ r, and then integrated over the cross‑section to obtain the torque capacity. For non‑circular cross‑sections a more complicated expression for J is required, often derived via conformal mapping or numerical methods, because the maximum shear stress occurs at a point rather than uniformly around the perimeter. Elliptical, rectangular, or hollow shapes lead to a factor of safety that can be found from standard tables or through finite element simulations.

Because the Torsionsgleichung is based on principles of static equilibrium and Hooke's law, it assumes uniform material properties, a constant cross‑section, and small deformations. In real-world applications additional effects such as material anisotropy, torsional fatigue, temperature dependence, and axial twisting couplings are taken into account by more sophisticated models. Nonetheless, the classic Torsionsgleichung remains the foundational tool for calculating twisting stresses in engineering design, often referenced alongside related concepts like shear flow, polar torsion, and warping in standard textbooks and industry standards.