Cable/Catenary Sag-Tension Calculator mechanism

Model scope

The uniform-load parabolic formulas are mathematically sound under the stated assumptions, and the exact catenary test uses a fully specified symmetric equal-support interpretation. However, the specification omits the cable-length formula, exact-catenary parameter relations and convergence details, and claims point-load capability without providing a point-load algorithm. These ambiguities require review before catalog approval.

• The parabolic formulas for horizontal tension and maximum tension are standard for a level-span cable with uniform load w per unit horizontal span, but that load convention is not stated.
• The cable-length formula is missing. Under the parabolic approximation, the standard length is S=(L/2)sqrt(1+(4d/L)^2)+(L^2/(8d))asinh(4d/L).
• The exact catenary equation y=a·cosh(x/a) requires a vertical datum and horizontal origin; for a symmetric cable with its lowest point at x=0, the sag-referenced equation is y=a[cosh(x/a)-1].
• Exact catenary mode must define the relation used to solve a: d=a[cosh(L/(2a))-1], assuming equal-level supports and uniform load per unit horizontal length. It must also define T_h=w·a, T_max=w·a·cosh(L/(2a)), and S=2a·sinh(L/(2a)).
• Point loads are claimed in the description, but no point-load statics equations, load locations, or piecewise cable geometry are specified. The listed uniform-load parabolic and catenary formulas do not handle arbitrary point loads.
• The parabolic approximation becomes inaccurate for large d/L; no numerical validity threshold is specified despite the honest label referring to small sag-to-span ratio.

Units and conventions

• L and d use the same length unit.
• w is force per unit horizontal span, such as kN/m or N/m; it is not force per unit cable length.
• T_h and T_max have the same force unit as w multiplied by the length unit.
• For the exact catenary vectors, the cable is symmetric, supports are at equal elevation, and the lowest point is at midspan.
• Inverse hyperbolic sine is dimensionless and is evaluated as asinh(x)=ln(x+sqrt(1+x^2)).

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "mode": "parabolic",
    "L_m": 100,
    "sag_d_m": 5,
    "load_w_kN_per_m": 1
  },
  "expected_outputs": {
    "T_horizontal_kN": 250,
    "T_max_kN": 254.950976,
    "cable_length_m": 100.662723
  },
  "working": "T_h=wL^2/(8d)=1·100^2/(8·5)=10000/40=250 kN. 4d/L=20/100=0.2. T_max=T_h·sqrt(1+(4d/L)^2)=250·sqrt(1+0.2^2)=250·sqrt(1.04)=250·1.019803903=254.950976 kN. Using the standard parabolic arc-length formula, S=(L/2)sqrt(1+(4d/L)^2)+(L^2/(8d))asinh(4d/L)=50·1.019803903+250·asinh(0.2)=50.9901952+250·0.198690110=100.662723 m."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The calculator breaks on zero span because T_max and cable length become undefined through division by zero; it also has a physical-convention hazard when a signed negative load is supplied and the interface does not specify whether load means signed force or nonnegative magnitude.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche6/tranche6-results.json
• BOUNDARY_HAZARD_CATALOG.json