Arch | Beam | Cable-Stayed | Cantilever | Suspension | Truss

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Arch Bridges

The essence of an arch is that ideally there should be no tendency for it to bend, except under live loads. It should be purely in compression, and for that reason it can be made of materials such as, masonry, cast iron and concrete, that perform poorly in tension. Of course, in a trussed arch there will be some tension members, but the main ones are always in compression. These main members are always much thicker than than the cross-members.  

On the other hand, in a deck-stiffened arch, the deck is much thicker than the arch, because the deck is resisting any tendency to bend or buckle, leaving the arch chord to resist pure compression. In such a bridge, the deck can be very much thinner than a simple beam across the gap, because its weight is supported by the arch, and the arch can be very much thinner than a simple arch, because it is stiffened by the beam.

These two types of arch are shown below.

In any structure, except a simple pier or column, it is impossible to have compression without tension. In the case of an arch, the tension is in the ground, which is therefore a member that costs nothing. If we take this argument further, it can prove that arch spans can be made longer than beam spans. Although the ground under an arch is in tension, the ground just outside the abutments is compressed by the thrust of the arch. Between the regions of tension and compression, the ground is subject to complicated mixtures of tension, compression and shear stresses.

Although an arch is generally not under stress to make it bend, it has curvature designed in, because it is in a gravitational field. The amount of curvature at any point is designed so that the whole thing is perfectly balanced, neither tending to increase the curvature or to decrease it. The ideal shape is called the funicular, the exact shape of which depends on the weight distribution, so the funicular is not necessarily a simple mathematical curve such as a circle or a parabola. The arch and the suspension bridge are generally closer to the funicular, or natural curve, than any other type. In this they imitate the path of projectiles, which also follow curved natural paths, and even light, which curves in a gravitational field. Nevertheless, although the cause, gravity, is the same for both arches and projectiles, the detailed reasons for the curvature are different. We must always beware of making false analogies, though similarities have on occasion been valuable in science and mathematics in finding solutions to problems.

Why must an arch be curved? If we consider any section of an arch, the forces comprise two distinct kinds - those pulling down (the weight of the section pulling down, and the load, if any) - and the forces from the sections on either side. In order to balance the downward forces, the forces from the side must not be exactly in line: the angle between them, repeated throughout the arch, is the reason for the curvature. A beam, because it is straight, cannot work like this - it has to balance the downward forces by means of shear stress.

In one sense, the arch is one of the simplest of all bridges, because if you build it against hard rock, you only need the arch, and no other parts. The rock acts as abutments, provided that you cut the rocks to the right shape so that they are at right angles to the arch. In practice, abutments would be used to spread the load so that the stresses in the ground would be safely small. Here is a simple example.

In less suitable ground, you may need to provide massive abutments to spread the load, otherwise the arch may simply push into the ground like a shovel or a high heeled shoe.

This is an over-simplification, because the shape of an arch bridge is very inconvenient, and in fact impossible for wheeled traffic, apart from mountain bikes. So a real arch has to have a flat deck that is propped above it or hung below it, or even a bit of both.

The thrusts on the ground add up to more than the weight of the arch, because it they are composed of vertical parts, which take the weight, and horizontal parts, which keep the arch from spreading.

Whatever the type of ground, we might ask what happens to the thrust as it enters the ground. It does something like this -

The pressure decreases as we look further from the springing of the arch. The purpose of abutments is to take the pressure until it has decreased to the point where the ground can be relied upon to take it. The next picture hints at this, the white outline representing the abutment.

Because the ground is not moving under the compressive forces (after everything has settled down), we know that some opposing forces must exist. The ground under the arch is in tension, but the stresses are very weak, because the forces are diffused through a large volume. This illustrates a general principle - in an isolated object, in this case the earth, it is not possible for tensile (or compressive) forces to exist without compressive (or tensile) forces. Usually there will be shear stresses as well, as there are in this case, in the regions near the abutments.

The simple funicular arch is one of the simplest of all structures, apart from a vertical column. Like the vertical column, it is purely in compression parallel to its axis, all complexities such as shear being exported to places where they need not be calculated. All we need to know is that these stresses are small enough for the ground to bear.

Why is there a sideways thrust? We can think of the two halves of the arch as leaning on one another, each pushing the other so that they do not fall. Since the two halves do not move, we know that there must be an equal and opposite force on each one. Because there is no other connection to the arch, these forces can only come from the abutments.

The longest arch span is the New River Gorge bridge, in the USA, with a span of 1700 feet, 518 metres.

Arches have been made of stone, brick, cast iron, wrought iron, wood, steel and aluminium. The largest are in steel, while somewhat smaller ones may be in concrete. The oldest arches were in masonry, brick or stone, of which some ancient examples, and many medieval ones, are still standing. The famous cast iron Ironbridge, shown below, was a good advertisement for the local cast iron, which was very suitable for arches, given its ability to take compression. It must have worked, because it was followed by a large number of iron bridges in Britain.  The great weakness of cast iron is apparent under tension. Cast iron was superseded by wrought iron, and then by steel, both of which are strong in tension.

Here are pictures of some arches, hinting at the wealth of different ways of building arches. For more pictures, click here.

HerefordOldCS.jpg (241687 bytes) BredwardineCS.jpg (280151 bytes) IronB2002B.jpg (332843 bytes) AlpViaZ.jpg (343209 bytes) BayonneBr.jpg (37648 bytes) M4A46.jpg (36043 bytes) M42TiedV.jpg (82301 bytes)

Advantages of arches

The entire arch is in compression. The compression is transferred into the abutments, and ultimately resisted by tension in the ground under the arch. The absence of tension in the arch means that it can sustain much greater spans than beams can achieve, and it can use materials that are not strong in tension, such as masonry and cast iron. In older times, before the advent of wrought iron, many cast iron arches were built, some of which are still in use. Relatively few arches are now built in masonry, and none are built in cast iron. Masonry is labour intensive, and for the shorter spans, over roads and railways, it is often cheaper and simpler to lift whole beams and cantilevers into position.

Disadvantages of arches

An arch cannot stand until it is complete. Therefore it must either rest on falsework (centring) until it is complete, or the two halves must be cantilevered from the springing, using cables. The cantilever method cannot be used for masonry arches or concrete arches. Clickt o see photographs of centring for the Nicholson bridge.

The thrust of a big arch has a horizontal component, which the abutments must withstand without significant movement. The pictures below show the results of movement.

ChepstowSag293Y.jpg (264715 bytes) Over1.jpg (28255 bytes) Burford325B.jpg (139555 bytes) ArchesJYZ.jpg (136544 bytes)

ArchBristolA.jpg (179968 bytes)When spanning a road or a railway, the round arch must be wider than the railway or roadway in order to maintain clearance according to the loading gauge. In this example, the plate girder bridge behind is actually higher than the arch.

CulhamEllipse215.jpg (118662 bytes)An elliptical arch provides clearance with less elevation of the road.

Click on the links to download a program about arch behaviour and a program about deck-stiffened arch behaviour. You can also download programs about catenary, circular, and parabolic arches.

Lists of the longest arch spans in concrete and steel.

For more information please see the main pages about arches. The diagram at left gives the names of the main parts of a masonry arch.

See also these pages -

 

How Arches Work

Arches One

Arches Two

Arch Variations

Arches and Keystones

Arches and Domes

Arches Miscellany

Arched Railway Viaducts

Arch Shapes

Arch Footbridges

Concrete Arches

Masonry Arches

Skew Arches

Arch Funicular

Arch Simulation

Arches in Architecture One

Arches in Architecture Two

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Why is there a horizontal component of thrust in an arch? We can look at this in several ways, of which one will be given here. The diagram below shows a simple arch. In the second part we imagine the two halves of the arch being separated. (Artists imagine things, so why not engineers?)

The weight (green arrow) tends to produce an anti-clockwise rotation about the abutment, so we need a force to counteract it (red arrow). Thus the two halves push each other with just enough force to stabilize the system. But there is a problem. With the two forces shown, the half-arch would accelerate upwards and rightwards: there must be something more, as we see in the next diagram.

Now we have no net force in any direction, and no rotation. But wait: we still have a problem: the arch is pushing on the abutment, but what pushes on the abutment to stop it sliding? The foundations, which themselves are pushed by the ground. Does the series of forces never end? No, it doesn't. The forces just spread out and the stresses become weaker, but the total force remains the same in any direction. In principle the whole solid part of the earth would take part, though of course the forces are immeasurably weak at a long distance from the abutment. All we really need to be sure of is that by the time the forces emerge from the structures and foundations that we build, the stresses are small enough that we can ignore the deflections they produce in the ground. Sometimes the abutments do not spread the forces enough for the ground in which they are placed, and the arch does spread and sag, as we saw in some earlier pictures. We must never forget that the ground is involved in all our constructions. This is perhaps most important when building dams.

Earlier, we mentioned the several ways of proving that there is a horizontal thrust. Another way is to imagine that the arch is built on ice. The weight of the arch would pull it downwards, and it could only move by spreading outwards, again showing that a force must be provided to keep it in position.

The horizontal component of thrust in any arch is the same throughout: only the vertical forces depend on position. If the spandrels, the volumes above the arch, are filled with masonry, the mass of stone or brick may push inwards as well as downwards, so some thrust may come from positions above the springing, but the principle remains the same, all arches create horizontal thrust on the abutments, unless the ends are tied together, making a tied arch. This can only be done if the tie gives enough clearance for traffic, or if it is below the ground. A steel tie below the ground might be subject to corrosion unless it were carefully protected, and in practice it is usually easier to retain the thrust by means of abutments which transfer the thrust to the ground.

Here are some diagrams showing the forces on one voussoir in arches with different numbers of voussoirs.

The point here is that for each part of a structure, however big or small, the resultant of all the external forces on it must be zero when the structure is in equilibrium. The same is of course true for the whole structure.

CailleBigHF.jpg (225080 bytes)M42TiedV.jpg (82301 bytes)The forces in arches have been described here as being smoothly along the direction of the arch, at least when the arch is perfectly funicular, but if we look at the two arches pictured at left, we see that this cannot be the case at all points. Where the weight of the deck is injected into the arch by the vertical members, there must be local deformation of the lines of force within both the arch and the vertical members. Far from the junctions the forces will be aligned with the members. This is true of any structure, and the design and construction of attachments is of crucial importance.

MultiArchA.gif (5993 bytes) MultiArchB.gif (7695 bytes) ArchSteelA.gif (3883 bytes)