{
 "title": "Pure bending through a distributing coupling",
 "kind": "theory",
 "source": "Euler–Bernoulli pure bending, σ = M_y z/I, θ = ML/(EI) (exact in elasticity for ν = 0); Timoshenko and Goodier, Theory of Elasticity, 3rd ed., 1970, §102",
 "quantities": [
  {
   "id": "sigma_top",
   "quantity": "bending stress σ_xx at mid-span, top fibre (10, 0.5, 1)",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "Bend",
     "at": [
      [
       10.0,
       0.5,
       1.0
      ]
     ]
    },
    "pointer": "/nodes/0/stress/0"
   },
   "value": 15.0,
   "tolerance": {
    "relative": 0.005
   },
   "reason": "with ν = 0 pure bending is a quadratic displacement field that hex20 represents exactly; the coupling's nodal force pattern differs from the linear traction only within a depth or two of the end (Saint-Venant), five depths from the probe"
  },
  {
   "id": "rotation",
   "quantity": "rotation of the reference point about y",
   "measure": {
    "from": "solve",
    "pointer": "/reference_points/0/displacement/4"
   },
   "value": 0.30000000000000004,
   "tolerance": {
    "relative": 0.01
   },
   "reason": "the reference point follows the best-fit rigid motion of the end section, which the end zone's local deformation disturbs by well under 1%",
   "known_failure": "1.2% high: the distributing coupling weights every node of the end face equally, corner and mid-side nodes of hex20 alike, which is not the consistent linear traction; the end face deforms locally and the fitted rotation includes it. Area-consistent weights would remove it"
  }
 ]
}