Cable/Catenary Sag-Tension Calculator

Inspectable records

Ledger state

The current ledger marks this instrument verified with errata that must be applied at binding time.

Area

Mechanical, Civil & Structural Engineering — Statics/Dynamics, Strength of Materials, Machine Design, Civil, HVAC, Vibration

Tranche

tranche-6

Evidence ID

t6-140

Ledger rows

1

No ledger note is attached.

Review findings

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.

Formula and scope findings

• 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.

Unit assumptions

• 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)).

Boundary hardening

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.

Errata and corrections

Tranche errata

Inspectable example

4 review vectors are recorded. One source-bound sample is shown below.

{
  "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."
}

Related workflows

No source-defined multi-instrument workflow membership is recorded.

Sources and evidence

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

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