{
  "schema": "bent-offset-frequency/v1",
  "id": "c09-bent-offset-frequency",
  "basis": "empty carbon-steel historical preliminary method; k inches, lengths feet, weights lbf",
  "equal_leg_equation": "fn=(Co*k/L**2)/sqrt(1+Cp*P/W)",
  "equal_leg_W": "total distributed weight of illustrated span; uniform w gives sum of leg lengths times w",
  "equal_leg_configurations": [
    {
      "panel": "A",
      "shape": "right angle",
      "lengths": [
        "L",
        "L"
      ],
      "boundary": "fixed at both outer ends in illustrated plane",
      "P_location": "midpoint horizontal leg",
      "in_plane": {
        "Co": 3430,
        "Cp": 0.28
      },
      "out_of_plane": {
        "Co": 830,
        "Cp": 0.28
      }
    },
    {
      "panel": "B",
      "shape": "portal",
      "lengths": [
        "L",
        "L",
        "L"
      ],
      "boundary": "both feet fixed in illustrated plane",
      "P_location": "midpoint top leg",
      "in_plane": {
        "Co": 690,
        "Cp": 0.61
      },
      "out_of_plane": {
        "Co": 450,
        "Cp": 0.7
      }
    },
    {
      "panel": "C",
      "shape": "stepped",
      "lengths": [
        "L",
        "L",
        "L"
      ],
      "boundary": "both outer ends fixed in illustrated plane",
      "P_location": "midpoint vertical leg",
      "in_plane": {
        "Co": 620,
        "Cp": 0.66
      },
      "out_of_plane": {
        "Co": 500,
        "Cp": 0.68
      }
    }
  ],
  "offset_method": {
    "direction": "in-plane only",
    "L1": "shortest anchored leg",
    "L2": "horizontal connecting leg",
    "L3": "opposite leg",
    "ell": "distance from lower end of L3 to concentrated weight; 0<=ell<=L3",
    "w1": "distributed weight per unit length, lbf/ft; graphical W1 notation distinguished from total-weight W",
    "q": "ell/L3",
    "mu": "P/(w1*L1)",
    "D_boundary": "short leg fixed, opposite lower end simple as drawn; in-plane rotation permitted",
    "E_boundary": "short leg and opposite lower end fixed in illustrated plane",
    "CD": "sqrt(1/(1+4.24*(L2/L1+L3/(3*L1)+q**2*mu)))",
    "CE": "sqrt((1+(L1/L3)**3)/(1+L3/L1+4.24*(L2/L1+0.25*(3*q**2-q**3)**2*mu)))",
    "frequency": "785*k*C/L1**2",
    "scope": "uniform pipe-section illustration; actual boundary compliance and other freedoms separately modeled"
  },
  "corrections": {
    "stainless": "historical 7 percent lower coefficient basis; offset sheet applies prior-sheet corrections to configuration factor",
    "water": "diameter/wall curve required; not numerically recovered",
    "weight_table_unknowns": "15 scalars remain unknown in current source; not zero; examples use declared hypothetical known weights"
  },
  "service_observation": "shortest anchored leg has experienced fatigue in metering-pump service; proposed horizontal axial restraint changes the modeled configuration and thermal response",
  "acceptance": "no generic resonance, fatigue or metering-pump acceptance"
}