Cables & Suspension Bridges Notes

Last reviewed 16 Sept 2026 · 6 min read

Cables as structural members

A cable is a flexible member that can carry only tension — it has no bending or compression resistance. Under any load it takes the shape in which every part is in pure tension (the funicular shape). Cables are used in suspension and cable-stayed bridges, ropeways, transmission lines, guyed masts and cable roofs.

Assumptions in elementary analysis: the cable is perfectly flexible and inextensible (or its elongation is small), self-weight is either neglected or treated as a UDL, and loads are vertical.

Cable under point loads

Between loads the cable is straight. At each load point, equilibrium gives a polygon. Because loads are vertical, the horizontal component of tension $H$ is the same everywhere in the cable.

For supports at the same level, the reactions follow from treating the cable like a beam, and $H$ follows from one known sag.

General cable theorem

FormulaGeneral cable theorem

At any point on a cable supporting vertical loads, the product of the horizontal tension $H$ and the vertical distance between the cable and the chord joining the supports equals the bending moment at that point in a simply supported beam of the same span under the same loads:

H y = μ_{x}

It is the arch relationship with $M = 0$ : a cable is an arch turned upside down that can only be in tension.

Cable under a UDL (parabolic cable)

A UDL $w$ per unit horizontal length (e.g. the deck of a suspension bridge, which is much heavier than the cable) makes the cable a parabola. With span $L$ and central sag $h$ (supports at the same level):

H h = \frac{w L ^{2}}{8} \Rightarrow H = \frac{w L ^{2}}{8 h}

V = \frac{w L}{2} at each support, T_{ma x} = H^{2} + V^{2} = \frac{w L}{2} 1 + \frac{L ^{2}}{16 h ^{2}}

Tension is minimum at the lowest point ( $= H$ ) and maximum at the supports.

Remember

Halving the sag doubles the horizontal tension. Engineers pick sag ≈ span/10 to span/12 for suspension bridges as a balance between cable tension and tower height.

A cable hanging under its own weight per unit length of cable forms a catenary; for small sag–span ratios the parabola is an excellent approximation.

Supports at different levels

Let the lowest point be at horizontal distances $x_{1}$ and $x_{2}$ from the supports, with sags $h_{1}$ and $h_{2}$ below them ( $x_{1} + x_{2} = L$ ). Each side behaves like half a parabola:

H = \frac{w x _{1}^{2}}{2 h _{1}} = \frac{w x _{2}^{2}}{2 h _{2}} \Rightarrow \frac{x _{1}}{x _{2}} = \frac{h _{1}}{h _{2}}

Vertical reactions: $V_{1} = w x_{1}$ , $V_{2} = w x_{2}$ .

Length of a parabolic cable

For supports at the same level:

ℓ = L + \frac{8 h ^{2}}{3 L} (approximately, for small h / L)

Each half of a cable with unequal supports: $ℓ_{i} \approx x_{i} + \frac{2 h _{i}^{2}}{3 x _{i}}$ .

Change of sag with length or temperature

From $ℓ = L + 8 h^{2} /3 L$ : $δ ℓ = \frac{16 h}{3 L} δ h$ → $δ h = \frac{3 L}{16 h} δ ℓ$ .

A temperature rise $T$ lengthens the cable by $δ ℓ = α T ℓ \approx α T L$ , which increases the sag and reduces $H$ . Elastic stretch under tension does the same.

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Cables & Suspension Bridges