Cantilevers and the Forth Bridge

This is the chapter the whole course has been walking towards. The Forth Bridge, opened on 4 March 1890 by the Duke of Rothesay (the future Edward VII), is designed by Sir John Fowler and Sir Benjamin Baker, and the sources describe it as the first major structure in Britain built of steel. Its three great steel towers, each with long arms reaching out to either side, are the most famous cantilever bridge in the world. This chapter explains what a cantilever is, why balancing two of them works, and what the engineers did to show it to a doubting public.

What a Cantilever Is

A cantilever is a beam supported at one end only, like a diving board or a balcony. The fixed end must resist both the load's weight and the turning effect it has, which is the whole load times the distance to it. Compare it with the simply supported beam of How Bridges Work 2 using the same span and load:

A 10 m cantilever under 20 kN/m has a root moment of 1000 kN·m and a shear of 200 kN. The simply supported beam of the same length has only 250 kN·m and 100 kN, so the cantilever works four times as hard in bending. It also sags far more: 1,562 mm at the tip against 163 mm for the beam, nearly ten times. A lone cantilever is therefore a poor use of material, and the clever idea of the Forth Bridge is to give each cantilever a partner that balances it.

The Balanced Cantilever

Put a tall tower on a pier and send an arm out to each side. If the two arms are equal, their turning effects about the tower cancel, just as two children of equal weight balance on a seesaw. The tower then carries only the two weights, pressing straight down. The arms are in tension along the top (they are trying to bend downwards at the tips) and compression along the bottom, exactly like the chords of a truss in How Bridges Work 4. The Forth Bridge has three of these "double cantilevers". Between two of them a shorter girder, the suspended span, rests on the tips of the arms and carries traffic across the gap.

The numbers for one main span of the Forth Bridge agree between the sources: two cantilever arms of 680 feet (207.3 m) carry a central girder of 350 feet (106.7 m). Add them up: 207.3 + 106.7 + 207.3 = 521.3 m, which matches the reported main span of about 521 m. (The Wikipedia article gives it variously as 1,700 and 1,709 feet; the 1,709 feet figure is the 521 m one.) The table collects the other facts I could check:

Working Out the Balance

To get a feel for the forces, take the superstructure's average weight spread along the whole bridge: 51,324 tonnes over 2,467 m is about 204 kN per metre. This is a rough average (the cantilever parts are heavier than the approach spans, so the real figure at the towers is higher), but it is grounded in the bridge's own numbers. The suspended span puts half its weight on each arm tip. The moment at the tower is the arm's own load plus the tip load times the arm length:

The moment at the tower is about 6.64 million kN·m, the tower must carry roughly 106,000 kN (about 10,800 tonnes), and a pair of chords 100 m apart would each carry about 66,000 kN. Look at the middle result: one beam across the whole 521 m gap would suffer a moment of 6.93 million kN·m, only 4% more. The cantilever does not reduce the bending much. What it does is change where the bending is. In the double cantilever the moment peaks at the tower, where the structure can be made enormously deep (110 m tall), and falls to zero at the tips, where it can be made slim:

The columns are the fraction of the way from the tip, the distance from the tip (m), the moment (kN·m) and the moment as a fraction of the value at the tower. Halfway along the arm the moment is only a third of its peak; a quarter of the way in, only 13%. A designer can follow the moment with the depth of the structure: very deep at the tower, tapering to a slender point at the tip. This is the shape of the Forth Bridge, and it is the same logic as the I-beam of How Bridges Work 2, where material goes where the stress is. It also explains the use of steel in two forms: the compression members of the great arms are large tubes, and the tension members are lattice (general knowledge, noted in How Bridges Work 4).

Why a Cantilever, and Why Then

The story of the Forth Bridge begins before Fowler and Baker. In 1871 Thomas Bouch, the engineer whose Tay Bridge later fell, proposed a stiffened steel suspension bridge on roughly the same line. He allowed for a wind pressure of only 10 pounds per square foot on the advice of the Astronomer Royal, the Forth Bridge Railway Act was passed on 5 August 1873, and work began on a brick pier at Inchgarvie in 1878. After the first Tay Bridge collapsed in December 1879, the inquiry found it "badly designed, badly constructed and badly maintained", and Bouch's design for the Forth was formally abandoned on 13 January 1881. The sources I could read do not say in so many words why the cantilever was chosen; what they establish is the order of events. Later, the Board of Trade required all bridges to be designed for a wind pressure of 56 pounds per square foot, more than five times Bouch's figure.

One reading, which is my own interpretation and not something the sources state, is that the engineers wanted a design that was visibly stiff, with no reliance on a thin cable, and sturdy against wind. A cantilever's deep, braced steel arms and wide base (the structure is 120 ft, about 37 m, wide at the piers) provide exactly that. Another practical gain, which belongs to How Bridges Work 8 as much as this one, is that each arm can be built out from the tower in balance, so that no temporary supports are needed in the water below (general knowledge, not checked in the sources for this chapter).

The Human Cantilever

Fowler and Baker knew that people would doubt a bridge whose arms seemed to hang in the air. In 1887 they gave a famous demonstration, now called the human cantilever. The sources describe it this way: the Japanese engineer Kaichi Watanabe sat in the middle, supported between Fowler and Baker, who sat on chairs. Fowler and Baker represent the cantilevers, with their arms in tension and the sticks under compression, and the bricks represent the end piers, weighted with cast iron.

It is a brilliant piece of teaching: anyone can feel their own arms pulling and the sticks pushing, and see that nothing needs to hold the middle man except the two arms and the bricks behind them. The counterweights at the ends matter as much as the arms, which Exercise 3 explores with numbers. (I describe the demonstration only as far as the source states it; the text does not give more detail than this.)

Where This Leads

The cantilever passes load through deep, rigid steel arms. The next chapter looks at the opposite approach, the one that Bouch proposed for the Forth: hang the deck from flexible cables. It follows the catenary curve of a cable, the anchorages that hold it, the towers that lift it, and the Forth Road Bridge of 1964, which stands next to its older neighbour.