{
 "title": "Torsion of a tube through a kinematically coupled reference point",
 "kind": "theory",
 "source": "Saint-Venant torsion of a circular tube, φ = TL/(GJ), τ = Tr/J with J = π(b⁴ − a⁴)/2; Timoshenko and Goodier, Theory of Elasticity, 3rd ed., 1970, §104",
 "quantities": [
  {
   "id": "twist",
   "quantity": "rotation of the reference point about z",
   "measure": {
    "from": "solve",
    "pointer": "/reference_points/0/displacement/5"
   },
   "value": 0.05607084715974633,
   "tolerance": {
    "relative": 0.005
   },
   "reason": "Saint-Venant torsion of a circular tube is exact in elasticity; the rigid end coupling only stiffens a short zone near the tip; quadratic elements with 16 divisions around are within 0.1%"
  },
  {
   "id": "shear",
   "quantity": "surface shear stress τ_θz = σ_yz at (10, 0, 20)",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "Twist",
     "at": [
      [
       10.0,
       0,
       20.0
      ]
     ]
    },
    "pointer": "/nodes/0/stress/5"
   },
   "value": 1078.285522302814,
   "tolerance": {
    "relative": 0.01
   },
   "reason": "stress extrapolated to the outer surface and averaged where two blocks meet: about 0.3% with four radial quadratic elements (two gave 1.1%)"
  }
 ]
}