{
 "title": "Large-deflection cantilever under a dead tip load against the elastica (solids)",
 "kind": "theory",
 "source": "The elastica of a cantilever under a vertical end load: K. E. Bisshopp, D. C. Drucker, Large deflection of cantilever beams, Quarterly of Applied Mathematics 3(3):272–275, 1945 (elliptic-integral solution); values computed by integrating θ″ = −(P/EI) cos θ by shooting (our own RK4, 20000 steps), agreeing with the table of K. Mattiasson, Numerical results from large deflection beam and frame problems analysed by means of elliptic integrals, IJNME 17(1):145–153, 1981, to all six digits",
 "quantities": [
  {
   "id": "v1",
   "quantity": "tip deflection δ_v/L at P L²/(E I) = 1",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "K1",
     "at": [
      [
       10.0,
       0,
       0.25
      ]
     ]
    },
    "pointer": "/nodes/0/displacement/2",
    "scale": -0.1
   },
   "value": 0.301721,
   "tolerance": {
    "relative": 0.01
   },
   "reason": "hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica"
  },
  {
   "id": "u1",
   "quantity": "tip shortening δ_h/L at P L²/(E I) = 1",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "K1",
     "at": [
      [
       10.0,
       0,
       0.25
      ]
     ]
    },
    "pointer": "/nodes/0/displacement/0",
    "scale": -0.1
   },
   "value": 0.056433,
   "tolerance": {
    "relative": 0.02
   },
   "reason": "as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's"
  },
  {
   "id": "v2",
   "quantity": "tip deflection δ_v/L at P L²/(E I) = 2",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "K2",
     "at": [
      [
       10.0,
       0,
       0.25
      ]
     ]
    },
    "pointer": "/nodes/0/displacement/2",
    "scale": -0.1
   },
   "value": 0.493457,
   "tolerance": {
    "relative": 0.01
   },
   "reason": "hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica"
  },
  {
   "id": "u2",
   "quantity": "tip shortening δ_h/L at P L²/(E I) = 2",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "K2",
     "at": [
      [
       10.0,
       0,
       0.25
      ]
     ]
    },
    "pointer": "/nodes/0/displacement/0",
    "scale": -0.1
   },
   "value": 0.160642,
   "tolerance": {
    "relative": 0.02
   },
   "reason": "as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's"
  },
  {
   "id": "v5",
   "quantity": "tip deflection δ_v/L at P L²/(E I) = 5",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "K5",
     "at": [
      [
       10.0,
       0,
       0.25
      ]
     ]
    },
    "pointer": "/nodes/0/displacement/2",
    "scale": -0.1
   },
   "value": 0.713792,
   "tolerance": {
    "relative": 0.01
   },
   "reason": "hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica"
  },
  {
   "id": "u5",
   "quantity": "tip shortening δ_h/L at P L²/(E I) = 5",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "K5",
     "at": [
      [
       10.0,
       0,
       0.25
      ]
     ]
    },
    "pointer": "/nodes/0/displacement/0",
    "scale": -0.1
   },
   "value": 0.387628,
   "tolerance": {
    "relative": 0.02
   },
   "reason": "as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's"
  },
  {
   "id": "v10",
   "quantity": "tip deflection δ_v/L at P L²/(E I) = 10",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "K10",
     "at": [
      [
       10.0,
       0,
       0.25
      ]
     ]
    },
    "pointer": "/nodes/0/displacement/2",
    "scale": -0.1
   },
   "value": 0.810609,
   "tolerance": {
    "relative": 0.01
   },
   "reason": "hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica"
  },
  {
   "id": "u10",
   "quantity": "tip shortening δ_h/L at P L²/(E I) = 10",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "K10",
     "at": [
      [
       10.0,
       0,
       0.25
      ]
     ]
    },
    "pointer": "/nodes/0/displacement/0",
    "scale": -0.1
   },
   "value": 0.554996,
   "tolerance": {
    "relative": 0.02
   },
   "reason": "as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's"
  }
 ]
}