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Category Math & Algebra

Solve for:

Horizontal Span (L)

Mid-span Sag (D)

Total Cable Length (S)

Unit Weight (w)

N/m

Horizontal Tension (H)

Catenary Parameter (a) — m

Mid-span Sag (D) — m

Arc Length (S) — m

Horizontal Tension (H) — N

Max Tension at Supports (T_max) — N

Vertical Reaction (V) — N

Sag-to-Span Ratio (D/L) —

Cable Angle at Support — °

x (m)	y (m)

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About

A catenary is the curve formed by a uniform chain or cable hanging under its own weight between two supports. It follows y = a ⋅ cosh(x÷a), where the catenary parameter a = H÷w relates horizontal tension to distributed weight. Confusing a catenary with a parabola leads to structural errors in cable-stayed bridges, power line design, and architectural arches. The parabola approximation diverges by over 5% when sag-to-span ratio exceeds 0.1. This tool solves the transcendental catenary equations numerically using Newton-Raphson iteration to machine precision.

Inputs accept three modes: known sag, known total cable length, or known horizontal tension with unit weight. The calculator returns sag, arc length, minimum and maximum tensions, curve coordinates, and a scaled interactive visualization. Results assume uniform cable density, negligible bending stiffness, and static loading. For wind or ice loads, multiply w by the appropriate load factor per your regional code (ASCE 7, EN 50341).

Formulas

The fundamental catenary equation describes the shape of a flexible, inextensible chain hanging under uniform gravity:

y = a ⋅ cosh(xa)

where a = H ÷ w is the catenary parameter. The sag D at mid-span for a symmetric catenary of horizontal span L:

D = a ⋅ (cosh(L2a) − 1)

Total arc length S of the cable between supports:

S = 2a ⋅ sinh(L2a)

Tension at any point along the curve decomposes into horizontal and vertical components:

T_max = w ⋅ (a + D) = H ⋅ cosh(L2a)

The catenary parameter a is found iteratively via Newton-Raphson on the transcendental equation. When sag D is known, solve f(a) = a ⋅ cosh(L÷(2a)) − a − D = 0. When cable length S is known, solve g(a) = 2a ⋅ sinh(L÷(2a)) − S = 0.

Variable legend: y = vertical coordinate, x = horizontal coordinate, a = catenary parameter m, H = horizontal tension N, w = weight per unit length N/m, L = horizontal span m, D = mid-span sag m, S = total arc length m, T_max = maximum tension at supports N.

Reference Data

Application	Typical Span	Sag-to-Span Ratio	Unit Weight w	Material	Design Standard
High-Voltage Transmission (110 kV)	300 - 500 m	3 - 5%	10 - 25 N/m	ACSR Aluminum	IEC 60826 / EN 50341
Distribution Lines (10-35 kV)	50 - 150 m	2 - 4%	3 - 10 N/m	AAC / AAAC	NESC C2
Suspension Bridge Main Cable	500 - 2000 m	8 - 12%	100 - 500 N/m	Galvanized Steel Wire	AASHTO LRFD
Catenary Wire (Railway)	50 - 70 m	0.5 - 1.5%	6 - 12 N/m	CuMg / BzII	EN 50119
Zip Line / Aerial Ropeway	50 - 500 m	2 - 5%	15 - 80 N/m	Steel Wire Rope	ANSI B77.1
Clothesline / Light Cable	5 - 20 m	5 - 15%	0.2 - 2 N/m	Nylon / Polypropylene	-
Fiber Optic Cable (ADSS)	100 - 300 m	1 - 3%	2 - 5 N/m	Aramid / Fiberglass	ITU-T G.652
Decorative Chain / Railing	1 - 5 m	10 - 30%	5 - 50 N/m	Steel / Bronze Chain	-
Anchor Chain (Marine)	50 - 300 m	Variable (catenary mooring)	200 - 2000 N/m	Studlink Steel Chain	DNV GL / ABS
Gateway Arch (Inverted Catenary)	192 m (width)	Aspect ratio 1:1	Self-weight structure	Carbon Steel / Concrete	Architectural
Spider Silk (Natural)	0.1 - 2 m	5 - 40%	≈0.00001 N/m	Silk Protein	-
Undersea Power Cable	1000 - 50000 m	Follows seabed terrain	50 - 300 N/m (submerged)	Copper / Lead Sheath	IEC 63026

Frequently Asked Questions

The parabola y = wx²/(2H) is a second-order Taylor expansion of cosh(x/a). It matches the catenary within 1% error when the sag-to-span ratio D/L is below 0.04 (4%). Above D/L ≈ 0.1, the parabola underestimates arc length by over 5% and overestimates tension at the lowest point. For power lines with typical D/L of 3-5%, the parabola is acceptable. For suspension bridge main cables (D/L of 8-12%) or decorative chains, always use the full catenary equation.

Thermal expansion changes cable length by ΔS = S · α · ΔT, where α is the coefficient of thermal expansion (about 23 × 10⁻⁶ /°C for aluminum, 11.5 × 10⁻⁶ /°C for steel). A 50°C temperature rise on a 400 m aluminum span can increase sag by 15-25%. This calculator computes the static catenary at a reference temperature. To account for thermal effects, increase the cable length input by the thermal elongation and recalculate.

The transcendental equation a·cosh(L/(2a)) − a − D = 0 has a singularity as a → 0 (infinite sag-to-span ratio) and becomes flat as a → ∞ (taut string). The solver may fail if the sag exceeds half the span (physically impossible for a symmetric catenary without the cable touching the ground) or if the cable length S is less than the span L (the cable must be longer than the straight-line distance). The tool validates these constraints before iterating and caps iterations at 200 with a tolerance of 10⁻¹².

This version computes the symmetric catenary where both supports are at the same elevation. For unequal supports, the catenary equation still applies but the lowest point shifts horizontally. The parameter a remains the same, but the x-origin moves by an offset calculated from the height difference Δh: the shifted position satisfies a·cosh((L/2 + δ)/a) − a·cosh((L/2 − δ)/a) = Δh. A future update could add asymmetric support input.

The catenary parameter a = H/w has units of length and represents the radius of curvature at the lowest point of the curve. A large a means high horizontal tension relative to weight - the cable is taut with little sag. A small a means the cable hangs deeply. Horizontal tension H = w·a is constant along the entire cable. Vertical tension increases from zero at the lowest point to V = w·(S/2) at each support. Maximum tension T_max = w·(a + D) occurs at the supports.

The mathematical model is exact for an ideal catenary: a perfectly flexible, inextensible, uniform cable under gravity with no wind load. Real cables have finite bending stiffness (which matters near clamps), elastic stretch under tension (important for long spans), non-uniform mass (due to ice/connectors), and aerodynamic loading. For preliminary design and educational purposes, this tool provides results within 1-2% of finite element analysis. Final engineering design should use specialized software (PLS-CADD, SAG10) and comply with local codes.