Arches and Early Bridges

How Bridges Work

Chapter 3 · Arches and Early Bridges

How Bridges Work 2 ended with a problem: a beam carries load by bending, and bending wastes material. Stone makes it worse, because stone is strong when squeezed and weak when stretched, and bending always stretches one face. Builders who had only stone, brick and timber found a way round that, and it lasted two thousand years: shape the structure so that the load squeezes every part of it. That shape is the arch. This chapter looks at the simplest stone bridge (a slab), the arch that replaced it, the sideways push that every arch creates, and the Scottish bridges that show all of these at work.

What was run

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. Material strengths and some dimensions are illustrative round figures, and are marked where they appear.

Stone Likes to Be Squeezed

Materials differ in which force they prefer (How Bridges Work 1). Stone, brick and mass concrete are all strong in compression and weak in tension: a block of stone will carry an enormous squeezing load, but a pull of only a small fraction of that tears it apart. So a stone structure must be shaped to avoid tension. A beam cannot do that, since bending puts the underside in tension. An arch can, because its curve turns the vertical load into a squeeze along the ring of stones.

The Clapper Bridge: A Stone Beam

The simplest stone bridge is a flat slab laid from bank to bank, or from pier to pier, which is a beam with all the problems of How Bridges Work 2. These are called clapper bridges, and Scotland has many of them. Historic Environment Scotland describes Achnamara clapper bridge in Argyll as the last survivor of a group of mid-Argyll clapper bridges, and the national record (Canmore) classes it as 17th century. The slab fails when the tension on its underside reaches what the stone can take. The code finds the longest span for a slab 1 m wide, carrying only its own weight, using an illustrative working tension of 3 MPa (real stone varies widely, and a single flaw can break a slab):

import math

RHO, G = 2600, 9.81          # stone density kg/m3, gravity
ALLOW = 3.0                  # MPa, illustrative working tension for stone

def slab_span(thick_m, width_m=1.0):
    w = RHO * G * thick_m * width_m / 1000          # kN/m, own weight
    z = width_m * 1000 * (thick_m * 1000) ** 2 / 6  # section modulus, mm3
    m_cap = ALLOW * z / 1e6                         # kN m
    return w, m_cap, math.sqrt(8 * m_cap / w)

for t in (0.2, 0.3, 0.4):
    w, m, s = slab_span(t)
    print(t, round(w, 2), round(m, 1), round(s, 1))
# 0.2 5.1 20.0 5.6
# 0.3 7.65 45.0 6.9
# 0.4 10.2 80.0 7.9
    

The columns are the thickness (m), the slab's own weight (kN/m), the moment it can carry (kN·m) and the longest span (m). A 0.2 m slab reaches only 5.6 m with nothing on it, and doubling the thickness gains only about 40% (7.9 m), because the span grows with the square root of the thickness and the slab gets heavier as it gets thicker. A clapper bridge therefore crosses wide water with several slabs resting on stone piers in the stream, never with one long slab.

How an Arch Works

An arch is a ring of wedge-shaped stones called voussoirs (say "voo-SWAHR"). The one at the top is the keystone. Each stone is squeezed between its neighbours, and the load travels around the curve to the foundations at each end. Because each joint is squeezed shut, the arch needs little or no tension capacity, and the stones are held in place by the squeeze rather than by the mortar between them. Roman builders used the semicircular arch throughout their empire. The Pont du Gard, the Roman aqueduct bridge in southern France, stacks three tiers of arches, and its large stones are laid without mortar.

There is one practical catch. Until the keystone is in, the arch cannot carry anything: each half would simply fall inward. Builders therefore built the arch over a temporary timber frame called centring, laid the voussoirs on it, set the keystone, and only then removed the frame. The arch then carried itself.

Thrust: The Hidden Sideways Push

The curve of an arch does not only press down. It also pushes outwards at the two ends (the springings), like the sides of a bowed bow trying to spring open. This sideways push is called thrust, and it is the key fact about every arch. For a parabolic arch under a uniform load w across a span L, with a rise (height from springing to crown) of f, the horizontal thrust is H = wL²/(8f). The vertical reaction is the same as for a beam: wL/2.

def thrust(w, span, rise):
    """Horizontal thrust (kN) of a parabolic arch under a uniform load w (kN/m)."""
    return w * span**2 / (8 * rise)

w, span = 50, 12
v = w * span / 2                       # vertical reaction at each springing
print(v)                               # 300.0
for rise in (1.5, 3, 6):
    h = thrust(w, span, rise)
    print(rise, h, round((h**2 + v**2) ** 0.5, 1))
# 1.5 600.0 670.8
# 3 300.0 424.3
# 6 150.0 335.4
print(w * span**2 / 8)                 # beam moment for comparison: 900.0
    

The columns are the rise (m), the thrust (kN) and the resultant force on the springing (kN). The vertical reaction is 300 kN whatever the shape. But the thrust is inversely proportional to the rise: flatten the arch to a quarter of the height and the thrust quadruples, from 150 kN to 600 kN. The last line shows what the arch has done to How Bridges Work 2's problem: the beam with the same span and load would have a bending moment of 900 kN·m, and the arch replaces it with a thrust times a lever arm (the rise), since 600 × 1.5 = 900. The arch trades bending for a horizontal push that the ground must resist.

Abutments: Holding the Push

The ends of an arch rest on abutments (end supports) or piers (supports between arches), and these must resist the thrust without sliding sideways or tipping over. A heavy block does both: weight gives friction against sliding, and weight gives a restoring moment against tipping. The code tests a masonry abutment block 4 m high, per metre length of the bridge, with a vertical load of 300 kN and a thrust of 150 kN acting at the top. The friction coefficient of 0.6 and the density of 2200 kg/m³ are illustrative:

G, RHO, MU = 9.81, 2200, 0.6     # gravity, masonry density, friction

def abutment(b, height, v, h_thrust, h_arm):
    """Safety factors for a block b m wide, per metre run of bridge."""
    wt = RHO * G * b * height / 1000          # kN/m, block weight
    sliding = MU * (wt + v) / h_thrust
    overturn = (wt + v) * b / 2 / (h_thrust * h_arm)   # moments about the outer toe
    return round(sliding, 2), round(overturn, 2)

print(abutment(2.0, 4, 300, 150, 4))   # (1.89, 0.79)
print(abutment(3.0, 4, 300, 150, 4))   # (2.24, 1.4)
print(abutment(4.0, 4, 300, 150, 4))   # (2.58, 2.15)
    

A factor below 1 means the block moves. The 2 m block has a comfortable sliding factor (1.89) but an overturning factor of only 0.79, so it would tip. At 3 m it is still short (1.4), and at 4 m both pass (2.58 and 2.15). Notice that tipping, not sliding, sets the width. This simple model, with the loads through the middle of the block, is only a first check: real masonry design also asks where the line of thrust passes through the stones.

Where the line of thrust goes

In a real masonry arch the load path follows a curve called the line of thrust. If it stays inside the ring of stones, every joint is in compression and the arch stands. If it escapes the ring, a joint opens, and a few such openings turn the arch into a mechanism that collapses. This is why masonry arches fail from moving supports or heavy point loads rather than from "crushed stone", and why abutments that settle or spread are dangerous.

Early Bridges in Scotland

Scotland's old bridges show every idea in this chapter. A few examples, with what the search I made for this chapter supports and what comes from general knowledge:

Bridge	Type	What it shows
Achnamara (Argyll)	clapper (stone slabs)	classed as 17th century by Canmore; the last survivor of a group of mid-Argyll clapper bridges
Brig o' Balgownie (Aberdeen)	single arch	reported as the earliest surviving single-arch bridge in Scotland, about 1320
Old Stirling Bridge	multi-arch	reported as the oldest surviving multi-arch bridge in Scotland, from the 1400s or 1500s; it crosses the River Forth, upstream of the three bridges this course follows
Carrbridge packhorse bridge	single arch	over the River Dulnain, finished in 1717; the great flood of 1829 (the "Muckle Spate") swept away its parapets and it survived
Tay Bridge, Aberfeldy	five arches	one of General Wade's Highland bridges, designed by William Adam (general knowledge: it dates from the 1730s)

A packhorse bridge like Carrbridge is deliberately narrow, with low parapets so that the loaded baskets (panniers) on a horse's flanks could pass, and it is often steeply humped; the formula above shows the benefit of a high rise, since the taller the arch, the smaller the thrust the abutments must resist. The Carrbridge survival in 1829 is a nice example of the arch's strength: a flood that destroyed many bridges broke the parapets off this one and left the arch standing. The same strength has a weakness: arches resist floods and loads well but dread movement of their abutments and piers, a problem that comes back in How Bridges Work 9, when we look at scour and failure.

Where This Leads

The arch solved the problem of stone, but it needs a large mass of masonry to hold the thrust, and it is limited to the span a ring of stones can safely reach. The next idea is a clever one: instead of curving a heavy ring, build a light, deep frame out of straight members arranged in triangles, so that each piece carries only tension or only compression. That is the truss, the subject of How Bridges Work 4.

Exercise 1

A clapper slab is 1 m wide and may be 0.2, 0.3 or 0.4 m thick. Stone density is 2600 kg/m³. For working tensions of 1, 2 and 3 MPa (all illustrative), find the longest span (a) when the slab carries only its own weight and (b) when it also carries 3 kN/m of people and animals. Print a table and explain what happens when the thickness doubles.

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Exercise 2

A parabolic arch of span 20 m carries a uniform load of 40 kN/m. For rises of 2, 4, 5 and 10 m, print the vertical reaction, the horizontal thrust and the resultant force, and check that the beam moment wL²/8 equals the thrust times the rise. Then work out the smallest rise that keeps the thrust at or below 400 kN. What fraction of the span is it?

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Exercise 3

An arch pushes on a masonry abutment block 4 m high with a vertical force of 300 kN and a horizontal thrust acting at the top. Density is 2200 kg/m³ and the friction coefficient is 0.6. Write a loop that finds the narrowest block (in 0.1 m steps from 1 m) with a sliding factor of at least 1.5 and an overturning factor of at least 2.0, for a thrust of 150 kN and then 300 kN. Which check decides the width, and what does the result say about flat arches?

📄 View solution

Quick Reference

Stone, brick, mass concrete: strong in compression, weak in tension
Clapper bridge: a stone slab acting as a beam; short spans only
Arch: a ring of voussoirs with a keystone; the load travels around the curve as compression
Centring: the temporary frame that carries an arch until the keystone is in
Thrust: H = wL²/(8f) for a parabolic arch; flatter means more thrust
Beam moment = thrust × rise: the arch swaps bending for a horizontal push
Abutments: must resist sliding (friction) and tipping (weight); tipping often decides

Coming next

How Bridges Work 4 turns to trusses: why triangles make a rigid frame, the Warren and Pratt layouts, which members are in tension and which in compression, and the method of joints for finding the force in each.