Cable-Stayed Bridges

How Bridges Work

Chapter 8 · Cable-Stayed Bridges

A suspension bridge hangs the deck from a draped main cable. A cable-stayed bridge skips the drape: each stay is a straight steel cable that runs from a tower directly to a point on the deck. The towers are still in compression and the stays in tension, but now every stay is its own short support, so there is no huge main cable and no giant anchorage at the ends. The third Forth bridge, the Queensferry Crossing, opened in 2017, is the one this chapter follows. It also brings the course's main theme to a close: the same steel, forces and mathematics, arranged in a third way across the same water.

What was run, and what to trust

Every Python snippet in this chapter and in the three exercise solutions was executed, and the output shown in the comments is what Python printed. The facts about the Queensferry Crossing come from the sources named. The loads, tower height above the deck, number of stays and segment weights in the calculations are illustrative assumptions, labelled where they appear, so the numbers show how the principle works, not the designers' figures.

How a Stay Works

A stay holds up one point of the deck. It is tilted at an angle θ above the deck, so its tension T has two parts: a vertical part that lifts the deck (T sin θ = P, the load it carries) and a horizontal part that pulls the deck towards the tower (T cos θ = P / tan θ). The horizontal parts, added up along the deck, put the whole deck into compression, like a long strut. The tower receives the vertical parts and carries them down in compression.

              tower
               ▲
              ╱│╲
            ╱  │  ╲
          ╱    │    ╲
        ╱      │      ╲
  ────┬────────┼────────┬────  deck
      P                  P
  each stay: tension T, with the load P taken by its vertical part
  and a horizontal push on the deck that adds up along it
    

Fan and Harp

Two arrangements are the classic pair. In a fan the stays all converge on one point at the top of the tower. In a harp the stays are parallel to one another, like the strings of a harp, each meeting the tower at a different height. A semi-fan is a compromise: the stays are spread down the upper part of the tower. The sources I found say that the fan gives the smaller moment on the tower but puts a large concentration of force where the stays meet it, and that the semi-fan is often preferred when there are many cables. The code below compares the first two in a simple model of one side of a tower, with the deck load shared equally by ten stays. The tower height above the deck, the stay spacing and the load are illustrative assumptions:

import math

def stays(layout, n, d, height, p):
    """Forces in the n stays on one side of a tower. Stay i holds a deck point
    i*d metres from the tower, and carries a vertical load p (kN)."""
    out = []
    for i in range(1, n + 1):
        x = i * d
        if layout == "fan":
            y = height                          # every stay meets the tower top
        else:
            y = height * x / (n * d)            # harp: parallel stays
        length = math.hypot(x, y)
        t = p * length / y                      # T sin(theta) = p
        out.append((x, y, length, t, t * x / length))   # last item: horizontal part
    return out

n, d, height, p = 10, 32.5, 150, 8000           # illustrative
for layout in ("fan", "harp"):
    rows = stays(layout, n, d, height, p)
    h_sum = sum(r[4] for r in rows)             # total push along the deck, kN
    steel = sum(r[3] * r[2] for r in rows) / 600e3   # m3, at 600 MPa
    print(layout, round(max(r[3] for r in rows)), round(h_sum), round(steel, 1))
# fan 19090 95333 56.1
# harp 19090 173333 62.6
    

The columns are the layout, the largest stay force (kN), the total horizontal push the stays put on the deck (kN), and the steel volume of all ten stays (m³) at an assumed 600 MPa. In this model the two layouts have the same largest stay force (the outermost stay is the same in both), but the harp pushes the deck 82% harder (173,333 against 95,333 kN) and needs about 12% more steel. The reason is geometry: a fan has many steep stays near the tower, which are efficient, whereas every harp stay has the same shallow slope. Real bridges choose layouts for many reasons (looks, the room on the tower to connect to, how the loads change along the span), so this is a simple comparison, not a verdict.

The Best Stay Angle

Why not make every stay as steep as possible? A steep stay needs a taller tower. A shallow stay is cheaper in tower but pulls harder and pushes the deck harder. For a stay holding a deck point at a fixed horizontal distance, there is a best angle, where the stay's steel volume (tension × length ÷ stress) is smallest:

def stay_volume(theta_deg, p=8000, x=100, stress=600e3):
    """Steel volume (m3) of one stay holding load p at horizontal distance x."""
    th = math.radians(theta_deg)
    t = p / math.sin(th)                  # tension, kN
    length = x / math.cos(th)             # metres
    return t * length / stress

for deg in (20, 30, 40, 45, 50, 60, 70):
    th = math.radians(deg)
    print(deg, round(8000 / math.sin(th)), round(8000 / math.tan(th)), round(stay_volume(deg), 2))
# 20 23390 21980 4.15
# 30 16000 13856 3.08
# 40 12446 9534 2.71
# 45 11314 8000 2.67
# 50 10443 6713 2.71
# 60 9238 4619 3.08
# 70 8513 2912 4.15
    

The columns are the angle (degrees), the tension (kN), the horizontal push on the deck (kN) and the steel volume (m³). The volume is lowest at 45° (2.67 m³) and rises either side: at 20° the stay needs 4.15 m³ and pushes the deck with 21,980 kN, nearly three times the push at 45°. This is a useful guide, because it explains why stays on real bridges tend to be neither very flat nor very steep. It ignores the cost of the tower and of the deck's compression, so designers adjust it.

The Queensferry Crossing

Fact	Figure
Opened	to traffic on 30 August 2017; formally opened by Queen Elizabeth II on 4 September 2017, 53 years after she opened the Forth Road Bridge
Construction	preparatory works began in September 2011; finished 27 August 2017
Design	Ramboll leading a design joint venture with Sweco and Leonhardt, Andrä and Partners; Naeem Hussain of Arup was the lead design engineer
Contractor	Forth Crossing Bridge Constructors (Dragados, Hochtief, American Bridge, Morrison Construction)
Type	a three-tower cable-stayed bridge; at opening, the longest triple-tower cable-stayed bridge in the world
Towers	207 m (679 ft) high each
Longest span	650 m (2,133 ft)
Total length	2,700 m (8,858 ft), including approaches
Deck	149 segments, each 12 m long, built in China and Spain and shipped in October 2013 (the page also gives the width as 40 m, which looks wrong for a road deck)
Cost	£1.35 billion (infobox)
Traffic	motorcycles, cars and heavy goods vehicles; public transport, cyclists and pedestrians use the Forth Road Bridge

(Source: the Wikipedia article on the Queensferry Crossing, read in full by a subagent apart from its reference list. The same page does not state the number of spans or stays, the stay layout, the deck type, the construction method or the foundations.) For those, I used a separate search, with the reliability noted: the stays cross at midspan (Hochtief's project page describes a zone of crossed cables at mid-span that gives extra strength and stiffness, allowing a lighter deck and slimmer towers, with strands replaceable one at a time); one source gives 288 stays in total; and the deck was built by balanced cantilever. I read search summaries, not the full pages, so treat these items as less certain than the table.

Building by Balanced Cantilever

How Bridges Work 6's balanced cantilever idea appears again here, for a different reason. The deck of a cable-stayed bridge is built out from each tower, one segment at a time, with each new segment hung from a new stay as soon as it is in place. A search turned up descriptions of segments being lifted from barges by equipment on the towers and added at alternate ends, so that the cantilever stays balanced and the tower is not overloaded, and a report that the pair of freestanding cantilevers at the central tower reached about 644 m tip to tip, which was recognised as a world record. That agrees with the 650 m longest span. The key calculation is how much out-of-balance load the tower can stand:

seg_len, seg_weight = 12, 3000          # m and kN (illustrative: 250 kN/m x 12 m)

def unbalance(arm_m, segments_ahead=1):
    """Moment on the tower (kN m) when one side is ahead by whole segments."""
    return segments_ahead * seg_weight * arm_m

for arm in (96, 192, 288, 324):
    print(arm, unbalance(arm), unbalance(arm, 2))
# 96 288000 576000
# 192 576000 1152000
# 288 864000 1728000
# 324 972000 1944000
    

The columns are the arm length (m), the moment on the tower if one side is one segment ahead (kN·m) and two segments ahead. The moment is proportional to the arm: one segment ahead at 324 m (972,000 kN·m) puts 3.4 times the load on the tower of the same segment at 96 m (288,000). This is why the segments are added in turn from alternate ends, and why the whole job is sensitive to the weather. The same search source gave a wind limit of 11 m/s (about 25 mph) while the stays were being installed, and the Wikipedia article reports 25 days lost to high winds in April and May 2016 alone.

Wind Shielding

The Forth Road Bridge was often restricted in high winds, and the Wikipedia article says that wind shielding "has been built into the design" of the Queensferry Crossing so that it can be used in high winds. The sources I found do not agree on the details: one says 3 m high louvred shields run the full length of the deck, another describes 3.6 m high sides; one says cars can cross in winds up to 100 mph where the old bridge's restriction was 80 mph, and another says 115 mph for all vehicles. What they all rely on is simple physics: wind pressure grows with the square of the speed.

rho = 1.225                              # air, kg/m3
MPH = 0.44704                            # metres per second per mph

def pressure_kpa(mph):
    v = mph * MPH
    return 0.5 * rho * v**2 / 1000

for mph in (50, 80, 100, 115):
    print(mph, round(mph * MPH, 1), round(pressure_kpa(mph), 2))
# 50 22.4 0.31
# 80 35.8 0.78
# 100 44.7 1.22
# 115 51.4 1.62
print(round(pressure_kpa(100) / pressure_kpa(80), 2))
# 1.56
    

The columns are the speed in mph, the speed in m/s and the dynamic pressure (kPa) of the wind. At 100 mph the pressure is 1.22 kPa against 0.78 kPa at 80 mph: a 25% increase in speed gives a 56% increase in force. A solid barrier cuts the wind the vehicles feel, but it also takes some of that pressure itself, and the deck and towers must be designed for it. The same Wikipedia article records a different kind of weather problem: on 11 February 2020 the bridge was closed for the first time since it opened, when ice fell from the towers and damaged eight vehicles.

Three Bridges, Three Ideas

Bridge	Opened	Type	Longest span	How the load travels
Forth Bridge	1890	cantilever	about 521 m	deep steel arms in tension (top) and compression (below), balanced over each tower
Forth Road Bridge	1964	suspension	1,006 m	deck hangs from a draped main cable, which runs to towers and anchorages
Queensferry Crossing	2017	cable-stayed	650 m	straight stays from three towers to the deck, which is in compression

Common mistakes

Forgetting that the stays push the deck in compression (the deck is not just carried: it is also a strut); using the stay's vertical load as its tension (the tension is larger by 1 / sin θ); treating a model result as the designers' figure (the numbers here are illustrations); and relying on one source for a detail (the stay count and the wind speeds in this chapter each come from one or two web summaries).

Where This Leads

You now have all three Forth bridges in the toolkit. The chapters so far have asked whether a bridge is strong enough for the loads we expect. The next chapter asks what happens when it meets loads we did not expect or understand: wind that sets a deck swinging, water that undermines a pier, and cables that corrode out of sight. It looks at the Tacoma Narrows bridge, the myths about resonance, scour, and what the Forth Road Bridge's cable story taught engineers.

Exercise 1

A tower 150 m above the deck carries a half-span of 325 m, with a total deck load of 80,000 kN on that side shared equally by n evenly spaced stays. For n = 6, 10 and 14, compare the fan and the harp: the largest stay force, the total horizontal push on the deck and the steel volume of all the stays at 600 MPa. What happens to the push in the harp as n changes, and why?

📄 View solution

Exercise 2

A stay holds a vertical load of 5000 kN at a deck point 150 m from the tower base. For angles from 20° to 70° in 5° steps, find the tension, the horizontal push on the deck and the steel volume of the stay (tension × length ÷ 600 MPa). Find the angle with the smallest volume, and say what a shallower stay costs.

📄 View solution

Exercise 3

Each deck segment weighs 3000 kN, and the tower can take an out-of-balance moment of 700,000 kN·m (both illustrative). If one side gets k segments ahead, the moment is k × 3000 × arm length. For arms of 96, 192, 288 and 324 m, find the largest whole number of segments one side may be ahead and the moment it causes. What does the result say about building long cantilevers?

📄 View solution

Quick Reference

Stay: a straight cable from tower to deck; tension T = P / sin θ; horizontal part P / tan θ
Deck: pushed into compression by the stays; the tower carries their vertical parts
Fan: stays meet at the tower top; harp: parallel stays; semi-fan: a compromise
Stay angle: steel volume is lowest near 45° (a simple guide)
Queensferry Crossing (2017): three towers of 207 m, longest span 650 m, total length 2,700 m
Balanced cantilever construction: segments added from alternate ends; out-of-balance moment grows with arm length
Wind pressure: 0.5 × air density × speed²; a 25% rise in speed is a 56% rise in pressure

Coming next

How Bridges Work 9 is about wind, water and failure: the Tacoma Narrows bridge, the myths about resonance, scour, fatigue, the Forth Road Bridge's cable corrosion findings, and what engineers learned from failures.