Torsion of Circular Shafts mechanism

Model scope

NEEDS_REVIEW because the specification does not define whether r is an arbitrary evaluation radius or the outer radius required for τmax, does not specify whether τmax is signed or absolute, and does not define how solid versus hollow shaft selection is represented. The honest_label field is blank; it should state the elastic circular-shaft assumptions and the required input constraints.

• τ=Tr/J is the standard shear-stress formula at radius r, but it is not inherently τmax. For τmax, r must be the outer radius c (d/2 for a solid shaft or do/2 for a hollow shaft), and the output should use the magnitude if torque may be signed.
• The formulas are otherwise standard for circular elastic shafts: θ=TL/(GJ), Jsolid=πd⁴/32, and Jhollow=π(do⁴−di⁴)/32.

Units and conventions

• T, dimensions, and G must use a mutually consistent unit system.
• Using T in N·mm, dimensions in mm, and G in N/mm² produces τ in N/mm² (MPa), J in mm⁴, and θ in radians.
• Using T in N·m, dimensions in m, and G in Pa produces τ in Pa, J in m⁴, and θ in radians.
• For τmax, r is assumed to be the outer radius: r=d/2 for solid shafts or r=do/2 for hollow shafts.
• The formulas assume a circular shaft under Saint-Venant torsion with linear-elastic, homogeneous, isotropic material and Saint-Venant warping conditions.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "shaft_type": "solid",
    "T_Nmm": 2000000,
    "d_mm": 40,
    "L_mm": 1000,
    "G_N_per_mm2": 80000
  },
  "expected_outputs": {
    "J_mm4": 251327.4123,
    "taumax_N_per_mm2": 159.1549,
    "theta_rad": 0.09947184
  },
  "working": "d⁴ = 40⁴ = 2,560,000 mm⁴. J = πd⁴/32 = π(2,560,000)/32 = 80,000π = 251,327.4123 mm⁴. Outer radius r = d/2 = 20 mm. τmax = Tr/J = (2,000,000)(20)/251,327.4123 = 40,000,000/251,327.4123 = 159.1549 N/mm². θ = TL/(GJ) = (2,000,000)(1,000)/[(80,000)(251,327.4123)] = 2,000,000,000/20,106,192,984 = 0.09947184 rad."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

A hollow shaft with do=di produces J=0 and therefore undefined shear stress and angle of twist through division by zero; the instrument must reject the zero-wall-thickness input. A separate convention hazard is the possible radius-versus-diameter misinterpretation, which changes the calculated stress by a factor of 8 and twist by a factor of 16.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche4/tranche4-results.json
• BOUNDARY_HAZARD_CATALOG.json