Materials

How Bridges Work

Chapter 5 · Materials

A bridge's shape tells you which forces it uses (beam, arch, truss), but the material decides whether it can survive them. The history of bridges is as much a history of materials as of ideas: timber and stone, then cast iron, wrought iron and steel, then concrete. Each was better than the last at one job and worse at another, and each failed in its own characteristic way. This chapter compares them with a simple measure of strength for weight, then looks at the two slow enemies that no bridge escapes, fatigue and corrosion, and at what the Forth Bridge's famous paint is really for.

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 material strengths in the code are rounded, illustrative figures (real values vary with grade, age and condition), so use the table to see the pattern, not as design data.

The Six Materials

Material	Likes	Dislikes	In bridges
Timber	light; good in tension and compression along the grain	rot, insects, fire; weak across the grain	early beams and trusses, temporary works
Stone and brick	compression; durable	tension	arches, piers, abutments
Cast iron	compression; can be cast into shapes	tension and shock; brittle	early iron arches and columns
Wrought iron	tension; ductile, less brittle than cast iron	costly to make in large pieces; slag inside	mid-1800s girders, tubes and trusses
Steel	tension and compression; strong, ductile	rust; fatigue; fire	almost every large modern bridge
Concrete	compression; mouldable, cheap	tension (cracks)	piers, decks, towers; reinforced or prestressed

The most useful fact in this table is the split between tension and compression, which How Bridges Work 3 already used: stone, brick and plain concrete are almost useless when pulled, and cast iron is nearly as bad. A bridge made of them has to be built so that the load squeezes them. A material that is good in tension (wrought iron, steel, and timber along its grain) can be used in a cable, a hanger or a tie.

Strength for Weight

A bridge has to carry itself before it carries anything else (How Bridges Work 1). A fair measure of a material is therefore how long a uniform bar of it can be before it just carries its own weight: length = strength / (density × g). The longer the length, the less of a material's strength is wasted just holding itself up. Here it is worked out for the six materials, in tension and in compression, with rounded illustrative values:

G = 9.81
# name: (density kg/m3, tension MPa, compression MPa): rounded, illustrative
MATS = {
    "timber (along grain)": (500, 20, 25),
    "granite":              (2700, 8, 150),
    "plain concrete":       (2400, 3, 40),
    "cast iron":            (7200, 150, 600),
    "wrought iron":         (7700, 250, 250),
    "structural steel":     (7850, 355, 355),
}

def length_km(mpa, rho):
    """Length of a uniform bar that just carries its own weight, in km."""
    return mpa * 1e6 / (rho * G) / 1000

for name, (rho, t, c) in MATS.items():
    print(f"{name:22} {length_km(t, rho):6.2f} {length_km(c, rho):6.2f}")
# timber (along grain)     4.08   5.10
# granite                  0.30   5.66
# plain concrete           0.13   1.70
# cast iron                2.12   8.49
# wrought iron             3.31   3.31
# structural steel         4.61   4.61
    

The two columns are the length in kilometres in tension and in compression. Read them like this. Steel is the best in tension (4.61 km), which is why every cable-supported bridge uses it. Granite has a compression length of 5.66 km but a tension length of only 0.30 km, nineteen times less: stone is a perfect column and a useless cable. Plain concrete manages only 0.13 km in tension, the weakest of all. Timber is a surprise: at about 4 km it is almost as good as steel for its weight, which is why wooden structures can be so light, although it rots and burns. Cast iron is the champion in compression (8.49 km) and poor in tension (2.12 km), which is exactly the weakness that catches it out. Real design adds buckling, joints and a safety factor, so treat these as comparisons, not predictions.

A Short Tour: From Timber to Steel

Timber was the first bridge material and is still used. It is light and easy to work, and a good truss built from it needs no great skill (the early Pratt trusses of How Bridges Work 4 paired wooden compression members with iron tension rods). Its enemies are rot and insects.

Stone and brick gave the long-lived arches of How Bridges Work 3. They last for centuries, but only while the load keeps them squeezed.

Cast iron was the first new bridge material of the industrial age. The Iron Bridge at Coalbrookdale in Shropshire (about 1779) is the famous early example, an arch of cast-iron ribs (general knowledge, not checked for this chapter). Cast iron is strong when squeezed, which suits an arch, but it is brittle when pulled, and it breaks without warning. That weakness features in a famous disaster: the first Tay Bridge collapsed in December 1879 with a train on it, and investigators found that the cast-iron lugs that held parts of the structure together were the critical weak point; in tests for the inquiry, lugs broke at only about 20 long tons, rather than the 60 expected. Some accounts add that fatigue cracks had formed in the lugs before the disaster, and the inquiry blamed faults of design, construction and maintenance.

Wrought iron is tougher: it stretches and deforms before it fails, so it gives a warning. In the 1840s Robert Stephenson used it for the great box girders of the Britannia Bridge over the Menai Strait (How Bridges Work 2), and for many railway girders and trusses.

Steel combines the best of both: strong in tension and compression, and ductile. Once it could be made cheaply and in quantity, it took over the long-span bridges, and the Forth Bridge of 1890, the subject of How Bridges Work 6, was built of it (general knowledge).

Concrete is the modern stone. It is poured to any shape, but plain concrete is weak in tension (see the table). Two cures exist: put steel bars inside where tension will occur (reinforced concrete), or squeeze the concrete in advance with stretched steel cables so that the load never puts it in tension (prestressed concrete). Both are standard techniques.

Fatigue: Death by a Thousand Cycles

A steel bar that holds a steady load for ever may fail if the load is applied and removed over and over, even when each cycle is well below the strength. This is fatigue: tiny cracks start at a notch, a weld toe or a rivet hole and grow a little with every cycle until the piece suddenly breaks. What matters is the range of stress in each cycle and the number of cycles. A widely used simplified model says the number of cycles to failure falls with the cube of the stress range. The code uses a "detail category" of 71 MPa, meaning that a stress range of 71 MPa lasts 2 million cycles; the category and the model are simplified and illustrative:

def cycles_to_fail(delta_sigma, category=71):
    """Simplified S-N curve: N = 2e6 * (category / range)^3."""
    return 2e6 * (category / delta_sigma) ** 3

for s in (35, 71, 142):
    print(s, round(cycles_to_fail(s)))
# 35 16695557
# 71 2000000
# 142 250000

trains_per_day = 150
n = cycles_to_fail(45)
print(round(n), round(n / (trains_per_day * 365), 1))
# 7855385 143.5
    

Halving the stress range multiplies the life by eight, and doubling it divides the life by eight (16.7 million cycles at 35 MPa, 250,000 at 142 MPa). At 150 trains a day, each causing one 45 MPa cycle, the detail would last about 143.5 years. That is why engineers spend effort on the details of a bridge: a welded or riveted joint with a poor shape raises the local stress range and cuts the life sharply, and why a small increase in axle load matters more than it looks (Exercise 2).

Corrosion: Rust Never Sleeps

Steel rusts when it meets oxygen and water, and rust takes material from the section. A thinner plate carries the same force at a higher stress. The code below follows a plate 200 mm wide and 12 mm thick, carrying a steady 240 kN, that loses 0.05 mm from each face every year (an assumed rate; real rates depend on the climate and the sheltering of the member):

width, thick0, force = 200, 12.0, 240e3        # mm, mm, N
rate = 0.05                                    # mm per year lost from each face (assumed)

def stress(years):
    thick = thick0 - 2 * rate * years
    return force / (width * thick)

for years in (0, 20, 40, 60, 80):
    print(years, round(stress(years), 1))
# 0 100.0
# 20 120.0
# 40 150.0
# 60 200.0
# 80 300.0

limit = 165                                    # MPa
t_min = force / (width * limit)
print(round(t_min, 2), round((thick0 - t_min) / (2 * rate), 1))
# 7.27 47.3
    

The stress climbs slowly and then faster, because losing a millimetre from a thin plate matters more than from a thick one: it is 100 MPa new, 150 MPa after 40 years, and 300 MPa after 80, which is already beyond the limit. The plate reaches the 165 MPa limit after 47.3 years. Corrosion rarely strikes evenly, either. Pits, crevices and the places where water collects (joints, drainage points, the underside of a deck) wear much faster than the average, so inspectors look at those first.

Why the Forth Bridge Is Painted

Paint is the cheapest protection against corrosion: it keeps water and air off the steel. The Forth Bridge, with a huge area of exposed steel and a salt-laden sea climate, needs protection on a vast scale. The saying "painting the Forth Bridge" came to mean a task that is never finished, the idea being that as soon as one end had been painted, it was time to start again at the other. That ended in 2011: Network Rail announced on 9 December 2011 that a ten-year, £130 million refurbishment was complete, ahead of schedule, and the bridge was free of scaffolding for the first time in a decade.

The new coating was a glass flake epoxy, a type used in the offshore oil industry, designed to last about 25 years (one trade report quotes 30). One source gives the layers as a zinc-based primer of 35 microns, an intermediate coat of glass flake epoxy 400 microns thick, and a polyurethane gloss top coat of 35 microns, applied over an area of more than 230,000 square metres. The glass flakes make a layered barrier that water finds hard to cross. The zinc primer is a standard feature of steel protection: zinc corrodes in preference to the steel beneath it, so the primer protects even where the coat above has been scratched (general knowledge). The numbers in this paragraph come from one source each and have not been cross-checked.

Common mistakes

Using one strength figure for tension and compression (the two are very different for stone, concrete and cast iron); judging a bridge only by its peak load and forgetting the number of cycles; treating a corrosion rate as a fixed constant (it depends on climate, drainage and coating); and assuming a coating lasts for ever (it is a maintenance cycle, not a one-off).

Where This Leads

You now have the toolkit for forces (How Bridges Work 1), beams (2), arches (3), trusses (4) and materials (5). The next chapter turns to the structure that has made the Forth famous: the cantilever bridge, a design that balances huge steel arms over their piers, and the story of the engineers who built it in 1890.

Exercise 1

Using the table of materials in this chapter and adding brick (1900 kg/m³, 1 MPa in tension, 20 MPa in compression), work out the self-weight length in tension and compression for each material, print the table with formatted columns, then rank the materials for a hanging cable (tension) and for a column (compression). Which materials appear near the top of both rankings, and which only in one?

📄 View solution

Exercise 2

A bridge detail has fatigue category 71 MPa (N = 2×10⁶ cycles at a range of 71 MPa, with N proportional to 1 / range³). Every day 100 light trains cause a range of 45 MPa and 10 heavy trains cause 70 MPa, one cycle each. Using Miner's rule (damage = cycles applied ÷ cycles to failure, summed over the loads, with failure at 1), find the life in years, and then the life if the heavy trains double to 20 a day. How much does one heavy cycle cost compared with one light cycle?

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Exercise 3

A 200 mm × 12 mm plate carries a steady 240 kN and loses material from both faces at 0.01, 0.05 or 0.1 mm per face per year (all assumed). The allowable stress is 165 MPa. For each rate find how many years it takes to reach that stress and the stress after 50 years. Then suppose a coating stops corrosion completely for the first 25 years: when does the plate reach the limit for each rate?

📄 View solution

Quick Reference

Stone, brick, plain concrete, cast iron: far stronger in compression than in tension
Wrought iron and steel: good in tension; steel is best for cables and ties
Timber: light and efficient along the grain; rots and burns
Reinforced and prestressed concrete: steel bars or stretched cables supply the tension
Self-weight length: strength ÷ (density × g); a fair measure of strength for weight
Fatigue: repeated cycles crack a detail; life falls with the cube of the stress range (simplified)
Corrosion: thins the section, so the stress rises; coatings buy years but need renewing

Coming next

How Bridges Work 6 is about cantilevers and the Forth Bridge: the balanced cantilever principle, the 1890 Forth Bridge and its engineers Fowler and Baker, and the famous human-model demonstration.