{
 "title": "Rotating disc with a central hole, plane stress",
 "kind": "theory",
 "source": "Timoshenko and Goodier, Theory of Elasticity, 3rd ed., McGraw-Hill 1970, §32 (rotating disc of uniform thickness with a hole)",
 "quantities": [
  {
   "id": "hoop_bore",
   "quantity": "hoop stress σ_θ = σ_yy at the bore (20, 0)",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "Spin",
     "at": [
      [
       20.0,
       0,
       0
      ]
     ]
    },
    "pointer": "/nodes/0/stress/1"
   },
   "value": 65.312,
   "tolerance": {
    "relative": 0.01
   },
   "reason": "stress extrapolated to the bore surface, where the a²b²/r² term is steepest: 16 radial quadratic elements graded towards the bore leave about 0.2% (8 uniform: −2.4%; 32 graded: −0.05%)"
  },
  {
   "id": "radial_max",
   "quantity": "radial stress σ_r = σ_xx at r = √(ab), its maximum",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "Spin",
     "at": [
      [
       44.721359549995796,
       0,
       0
      ]
     ]
    },
    "pointer": "/nodes/0/stress/0"
   },
   "value": 20.724,
   "tolerance": {
    "relative": 0.01
   },
   "reason": "the nearest node to √(ab) is within half an element of it, where σ_r is stationary (an error of order (h/r)², under 0.5%), and nodal averaging adds a few tenths of a percent"
  },
  {
   "id": "hoop_rim",
   "quantity": "hoop stress σ_θ = σ_yy at the rim (100, 0)",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "Spin",
     "at": [
      [
       100.0,
       0,
       0
      ]
     ]
    },
    "pointer": "/nodes/0/stress/1"
   },
   "value": 16.328,
   "tolerance": {
    "relative": 0.01
   },
   "reason": "as at the bore"
  }
 ]
}