Cables

Cables are among the most efficient structural elements ever devised. Because a cable can carry only tension, every bit of its material works at the same stress along its length, with no material wasted resisting bending. That is why the longest bridge spans in the world are suspension bridges, why power and telecommunication lines are strung on cables between towers, and why cables appear in guyed towers, cable-stayed roofs, ski lifts, cranes, elevators, mooring lines and zip lines. The price of that efficiency is that a cable changes its shape to suit its load: it sags into whatever shape lets it carry the load in pure tension. This lesson shows how to analyze cables carrying concentrated loads, cables carrying loads uniformly distributed along the horizontal (which hang as parabolas), and cables hanging under their own weight (catenaries), and how sag, tension and length are linked, with consequences for power lines on hot days and for the design of arches.

Assumptions

In statics, a cable is treated as perfectly flexible, so it can transmit only a tension force along its own length; it cannot carry bending or compression. It is also usually treated as inextensible: its stretch under load is small enough to ignore when finding its shape, although the power-line example later in this lesson shows that small changes in length can still matter a great deal for sag. Because the cable is a flexible, pin-ended member, the tension force at every point is tangent to the cable.

Cables with concentrated loads

When a light cable carries point loads, it forms straight segments between the loads, with a kink at each one. Each segment is a two-force member (Lesson 4) in tension. The supports provide both horizontal and vertical reactions, and if no horizontal loads act, the horizontal component of tension H is the same in every segment, since each joint is in horizontal equilibrium.

The problem has one more unknown than equilibrium can supply, because the shape is not yet known. Some piece of geometry must be given, usually the sag at one point. Then the method is: find the vertical reactions by taking moments for the whole cable (exactly as for a simply supported beam carrying the same loads); cut the cable at the point of known sag and take moments for one side to find H; then use moments at the other load points to find their sags.

That last observation is general and important. The shape a cable takes under a set of loads, called the funicular shape, is the bending moment diagram of a simply supported beam with the same loads, scaled by 1/H. A deeper sag means a smaller H; a tight cable with little sag needs an enormous horizontal tension, the same lesson as the taut line in Lesson 2.

Funicular shapes and arches

Turn a hanging cable upside down and its tension becomes compression: an arch built to the inverted funicular shape of its loads carries them in pure compression, with no bending. The English scientist Robert Hooke stated this in 1675 ("as hangs the flexible line, so but inverted will stand the rigid arch"). Masonry, which is strong in compression but weak in tension, depends on it: Roman aqueducts, Gothic cathedrals and stone bridges stand because their arches follow, closely enough, the inverted funicular of their loads. The Catalan architect Antoni Gaudí designed the Sagrada Família using hanging models of strings and weights, photographed and turned upside down. Modern concrete shells and arch bridges are shaped the same way. If the load pattern changes, for example a heavy truck on one half of an arch bridge, the funicular shape changes too, and the arch must resist the difference in bending.

Cables with a load uniform along the horizontal: the parabola

The main cables of a suspension bridge carry the deck, which is hung from them by closely spaced vertical hangers. The deck weighs much more than the cable, and its weight is uniform along the horizontal span. A cable carrying a load w per unit horizontal length takes the shape of a parabola. Taking the origin at the lowest point, with horizontal tension \(T_0\) there:

For level supports a span L apart with sag h at midspan, setting y = h at x = L/2 gives the horizontal tension, which is the same everywhere. The tension is largest at the supports, where the cable is steepest and its vertical component equals half the total load:

The first result is the beam formula wL²/8 divided by the sag, exactly as the funicular idea predicts. The length of a shallow parabolic cable is close to \(S \approx L\left[1 + \tfrac{8}{3}(h/L)^2\right]\).

Cables under their own weight: the catenary

A cable carrying only its own weight, such as a power line between towers or a chain hanging between two posts, has a load that is uniform along its curved length rather than along the horizontal. Its shape is a catenary (from the Latin for chain):

where w is the weight per unit length of cable, and the tension at any height y above the lowest point is T₀ + wy, so it is greatest at the higher support. Given the span and sag, T₀ must be found by trial or numerically, because it appears inside the hyperbolic cosine. For shallow cables, with sag less than about one-tenth of the span, the catenary and parabola are almost identical, and the simpler parabolic formulas are used. For deep sags, such as a long chain between closely spaced posts or a mooring line, the difference matters.

import math

w, L, h = 16.0, 300.0, 6.0          # conductor weight N/m, span m, sag m (supports level)

# Parabolic approximation (load uniform along the horizontal)
T0_par = w * L ** 2 / (8 * h)
Tmax_par = math.hypot(T0_par, w * L / 2)

# Exact catenary: find T0 so that (T0/w)(cosh(wL/(2 T0)) - 1) equals the sag
def sag(T0):
    return T0 / w * (math.cosh(w * L / (2 * T0)) - 1)
lo, hi = 100.0, 1e7
for _ in range(100):                # sag falls as T0 rises
    mid = (lo + hi) / 2
    if sag(mid) > h:
        lo = mid
    else:
        hi = mid
T0 = (lo + hi) / 2
Tmax = T0 + w * h                   # catenary: T = T0 + w y at height y
S = 2 * T0 / w * math.sinh(w * L / (2 * T0))
print(f"Parabola: T0 = {T0_par / 1000:.3f} kN, Tmax = {Tmax_par / 1000:.3f} kN")
print(f"Catenary: T0 = {T0 / 1000:.3f} kN, Tmax = {Tmax / 1000:.3f} kN, length {S:.3f} m")

# Hot day: conductor lengthens by alpha * S * dT (elastic stretch relief ignored)
alpha, dT = 19e-6, 50.0
S_hot = S + alpha * S * dT
h_hot = math.sqrt(3 * L * (S_hot - L) / 8)   # parabolic length-sag relation
print(f"Extra slack {S - L:.3f} m -> {S_hot - L:.3f} m after a {dT:.0f} C rise")
print(f"Sag grows from {h:.2f} m to {h_hot:.2f} m; tension falls to {w * L ** 2 / (8 * h_hot) / 1000:.1f} kN")

Parabola: T0 = 30.000 kN, Tmax = 30.096 kN
Catenary: T0 = 30.016 kN, Tmax = 30.112 kN, length 300.320 m
Extra slack 0.320 m -> 0.605 m after a 50 C rise
Sag grows from 6.00 m to 8.25 m; tension falls to 21.8 kN

For this conductor, with a sag of only 2 % of the span, the parabola and catenary differ by less than 0.1 %. The second part of the output is more striking. The conductor is only 0.32 m longer than the span. A temperature rise of 50 °C, from a cool morning to a hot afternoon with heavy current flowing (PHYS-131), adds only 0.29 m of length, less than 0.1 %, yet nearly doubles the slack and increases the sag from 6.0 m to 8.3 m. Because sag depends on the square root of the slack, small changes in length produce large changes in sag. (In a real line, the lower tension also lets the conductor contract elastically a little, so the increase is somewhat less; utilities use detailed sag-tension software that includes it.)

This matters for safety. Electrical codes require minimum clearances between conductors and the ground, roads, buildings and trees at the conductor's maximum design temperature and under ice loading, the conditions of greatest sag. Lines that sag into trees on hot days have started wildfires and triggered major blackouts, including the August 2003 blackout that left about 50 million people in Ontario and the northeastern United States without power, which began when overloaded, sagging lines in Ohio contacted trees that had not been trimmed.

Sag, tension and design choices

Every cable design trades sag against tension. A tight cable with little sag has large tensions, needing a stronger cable and heavier anchorages and towers; a slack cable has lower tension but needs taller towers or more clearance. Several further points arise in practice:

• Supports at different heights. The lowest point of the cable shifts toward the lower support, and may lie beyond it, in which case the cable rises all the way from the lower support. The same equations apply, measured from the lowest point.
• Ice and wind. Ice accretion adds weight, and wind adds horizontal load per unit length; codes specify combinations of radial ice thickness and wind pressure for each region, and the resulting tension often governs the design of towers and anchors.
• Vibration. Wind can make cables oscillate, through vortex shedding on slender conductors (fatigue at the clamps, controlled by dampers) or galloping of ice-coated lines. Cable-stayed bridge stays are fitted with dampers for the same reason.
• Guy wires. Guyed masts and towers are held by pretensioned cables whose sag reduces their effective stiffness; designers pretension guys so that they stay taut under the design wind.