Cable and catenary
A flexible cable between two supports, level or at different heights, under its own weight, uniform loads along the span and point loads: horizontal and largest tension, sag, length, elastic stretch, the effect of a temperature change, and the largest tension against the breaking strength. Use it for guy and stay cables, catenary supports of canopies and nets, messenger cables and lifelines. The cable is solved exactly (catenary, parabola or the polygon of the point loads), with the parabolic approximation shown beside it.
Inputs of the cable #
- Span L and support heights: the horizontal distance between the supports and how far the right support is above the left (negative when lower).
- Sag under dead load: the vertical distance from the chord to the cable, at midspan or at a point you give, under the dead load at the installation temperature. This is the state the cable is erected to; it fixes the cable's unstressed (fabrication) length.
- Cable weight q: along the cable, as dead load.
- Minimum breaking strength, E and metallic area A: from the manufacturer or the strand and rope standards (ASTM A586, A603). A 1 in. structural strand is prefilled as an example (122 kips, 24,000 ksi, 0.596 in²): enter your cable. Elastic stretch can be turned off for an inextensible cable.
- Thermal coefficient α and the temperature rise and drop: α in millionths per degree, in the same degree scale as the temperature changes (°F in US units, °C in metric). Steel is about 6.6 per °F (12 per °C). The rise and drop are your design range from the installation temperature.
- Loads: vertical, downward: uniform loads per horizontal length over the whole span or from From to To, and point loads at From, each with its load type (D, L, Lr, S, R).
- Load combinations: ASCE 7-22 2.3.1 (LRFD) or 2.4.1 (ASD) by the method switch, or one unfactored load set. ASCE 19 (structural applications of steel cables) may add its own requirements: to confirm against its text.
- Resistance factor φ and safety factor Ω: 0.65 and 2.2 are prefilled; ASCE 19 and the fittings set them, and you choose them (to confirm against ASCE 19-16, which is not on hand).
- Largest service sag allowed: for clearance; 0 turns the check off.
The cable has no auto-design: change the cable or the sag and run it again.
Checks performed #
| Check | Basis |
|---|---|
| Reference state: H, tensions at both supports, length, stretch, unstressed length | General cable theorem H y = M of the simple beam (Irvine 1981; Udoeyo 2020, 6.2.1), iterated with the weight along the cable |
| Parabola against the exact shape | H, tension and length of the parabolic approximation (weight along the chord taken as uniform along the span) |
| Temperature rise and drop | Same unstressed length, grown by α ΔT and stretched by ∫ T ds / EA: the sag and H after each |
| Cable tension | Largest tension of the strength combinations, at the installation temperature and after the drop, against φ Tb or Tb / Ω |
| Largest service sag | ASD combinations (or the load set) at the installation temperature and after the rise, against your limit |
| Advisory: sag ratio | Outside about 1/40 to 1/7 |
Every combination is solved as its own equilibrium: the cable is not linear, so the combinations are not superposed. The unstressed length is the one installed, so a heavier load sags the cable more and raises H less than in proportion when the cable stretches.
Limitations of the cable #
- Vertical gravity loads only: no wind on the cable, no uplift, no ice accretion shape, no dynamic effects (galloping, vortex shedding) or fatigue.
- Rigid supports at fixed positions; no prestress from a turnbuckle other than through the sag entered; the sag up to 30 % of the span.
- The cable is perfectly flexible and linear elastic; strand constructional stretch and creep are not counted (use the manufacturer's prestretched modulus).
- Fittings, sockets, anchorages and the supporting structure are not designed. ASCE 19-16 is not on hand: its factors and load combinations are to be confirmed.
Verification examples #
Verification status: 18 values compared in 6 examples (2 published, 4 hand calculations from the cited equations): 18 within tolerance, 0 recorded deviations. Thin: only 2 published examples. Code basis: written without the code document on hand, and confirmed only in part through published examples.
| Example | Source | Values |
|---|---|---|
| Three point loads, sag at a load | Udoeyo, Structural Analysis (2020), Example 6.6 | H, two sags, tensions at both supports |
| Uniform load along a 20 m span | Udoeyo Example 6.8 | H, largest tension, length (the book's series value) |
| Catenary under its own weight | Hand calculation, catenary closed form (Irvine 1981) | H, largest tension, length, the parabola's H |
| Elastic stretch of a parabolic cable | Hand calculation | H, stretch H/EA ∫(1 + y′²) dx |
| Right support 20 ft higher | Hand calculation, general cable theorem | H unchanged, the two end tensions |
| Temperature rise, inextensible | Hand calculation with the length series of Udoeyo Eq. 6.20 | Sag increase |
Udoeyo's Example 6.8 prints a horizontal reaction of 9,000 N that does not agree with its own T0 of 3,000 N (the theorem is misapplied there); the 3,000 N is the right value and is the one used. Irvine's Cable Structures is not on hand: the catenary relations used are the standard closed forms, quoted from knowledge.