torsion-stress

name	torsion-stress
description	Torsion analysis — circular shafts, non-circular sections (Prandtl membrane analogy), thin-walled closed sections (Bredt's formula), power transmission, angle of twist, statically indeterminate torsion.
metadata	{"priority":8,"promptSignals":{"phrases":["torsion","torsional stress","torque","angle of twist","shear flow","Bredt","torsional rigidity"],"minScore":4}}

Torsion Analysis — Complete Skill

Circular Shaft (Solid)

Shear stress: τ = Tc/J = Tr/J [max at outer fiber r=c] τ(r) = Tr/J [varies linearly with radius]

Where:

• T = torque [N·m or lbf·in]
• c = outer radius [m or in]
• J = polar moment of inertia = πc⁴/2 (solid circle)

Angle of twist: φ = TL/(GJ) [radians] G = shear modulus: steel=79.3GPa, Al=26.2GPa, brass=37GPa, Ti=44GPa

Torsional rigidity: GJ [N·m²] Torsional stiffness: k_T = GJ/L [N·m/rad]

Hollow Circular Shaft

J = π(c_o⁴ - c_i⁴)/2 τ_max = T·c_o/J φ = TL/(GJ) (same formula)

Material efficiency: Hollow vs. solid for same J: hollow uses less material (mass ∝ outer/inner ratio) For same τ_max: hollow shaft is always more efficient (higher J/mass)

Power Transmission

P = T × ω = T × 2πn/60 T = P/ω = P×60/(2πn) [N·m, where P in W, n in rpm] T = 63,025×HP/n [lbf·in, n in rpm] T = 9549×kW/n [N·m, n in rpm]

Non-Circular Sections (Prandtl Membrane Analogy)

No simple closed-form — use stress function or tabulated results.

Solid rectangle (b×t, b ≥ t): τ_max = T/(α·b·t²) [at midpoint of longer side] φ = T·L/(β·b·t³·G) α, β from tables: b/t = 1: α=0.208, β=0.1406 b/t = 1.5: α=0.231, β=0.1958 b/t = 2: α=0.246, β=0.2289 b/t = 3: α=0.267, β=0.2633 b/t = 5: α=0.291, β=0.2923 b/t → ∞: α=β=1/3

Note: τ at corners = 0 (stress concentration at re-entrant corners, not at free corners)

Thin rectangular strip (b >> t): τ_max = 3T/(b·t²) (same as b/t→∞ above, α=1/3) τ at ends (short side) = much smaller

Thin-Walled Closed Sections (Bredt's Formula)

Shear flow: q = T/(2Am) [N/mm, constant around section] Am = enclosed area (bounded by median line of wall)

Shear stress: τ = q/t(s) = T/(2·Am·t(s)) [varies with local thickness t] Maximum τ at thinnest wall

Angle of twist: φ = TL/(4Am²·G) × ∮(ds/t) [where ∮ is integral around median line] For constant thickness t, perimeter p: φ = TLp/(4Am²·G·t)

Applications: rectangular tubes, HSS sections, closed airfoil shapes Compare to open vs. closed: closed section is 100-1000× stiffer in torsion

Open Thin-Walled Sections (I-beams, channels, angles)

Very low torsional stiffness — avoid if possible J_open = Σ(b·t³/3) [sum of all flat elements] τ_max = T·t_max/J [at thickest element] φ = TL/(GJ) (but J is very small)

Warping torsion: non-uniform warping creates additional bending-like stress T = T_sv + T_w (Saint-Venant + warping torsion) Full analysis: T = GJ·φ' - ECw·φ''' (differential equation — see Timoshenko)

Statically Indeterminate Torsion

Same approach as indeterminate axial: Compatibility: φ₁ = φ₂ (twist at junction) T_total = T₁ + T₂ (equilibrium) T₁/T₂ = (G₁J₁/L₁)/(G₂J₂/L₂) (stiffness sharing)

Combined Torsion + Bending

σ = Mc/I (bending normal stress) τ_T = Tc/J (torsional shear) τ_V = VQ/Ib (transverse shear — usually neglected if L>>d)

Principal stresses: σ₁,₂ = σ/2 ± √((σ/2)² + τ_T²) τ_max = √((σ/2)² + τ_T²) σ_von = √(σ² + 3τ_T²) (von Mises for combined bending+torsion)

Torsion in Curved Bars / Springs

Helical spring: T = F×r (r = coil radius), stress = K_w×8FD/πd³ (Wahl factor) Curved beam: additional flexural stress — use Winkler-Bach theory

Output

Provide: τ_max [MPa/ksi], φ [degrees], GJ [N·m²], location of max stress, safety factor vs. Ssy = 0.577Sy.