What is the strongest kind of bridge?

This question sometimes appears as a school project which is to be solved by building models. If you look in a list of long bridge spans you will find that all spans of greater than 890 m are suspension spans. Above 890 m, the given question of the strongest has no meaning, because no other kind of bridge exists. Clearly, other types could be made with spans above 890 m, but they would probably be very expensive, and their load carrying capacity would probably be small. It is very unlikely that a cantilever span will ever be made longer than that of the Quebec bridge, which is 550 m long. In fact, for any type of construction, there is a maximal span that can only just hold up, even with no load. The question of longest spans is discussed in another page, and use of models is discussed in the page about model bridges.

Whatever the longest possible spans are, we can see that there is a range for which only suspension bridges can exist, and below that there is another range in which cable-stayed bridges can exist as well. In fact, cable-stayed bridges are becoming very popular for spans where a suspension bridge would once have been chosen.

If we look at short spans, for example 30 metres, all types of bridge are feasible. To carry a railway across a very wide motorway we could in principle choose a suspension bridge. But the concentration of weight produced by a locomotive is so great that very deep and rigid trusses are needed to spread the load. Otherwise the train would find itself constantly climbing out of a valley, and the flexure of the bridge would lead to early fatigue. The beam would probably be so strong that the suspension cables and towers might well be disposed of. In other words, the suspension bridge is not the best type of bridge for that purpose. Yet we have seen that for very long spans no other bridge can even be built.

Perhaps we should change our question from "What is the strongest type of bridge?" to "What is the most economic type of bridge for the span, type of ground, and type of loads that we have to deal with?" There are no simple answers and no simple formulas in engineering. This leads to the next question.

What formulas are used in bridges?

This question is asked most often in connection with suspension bridges and arches.  Semicircles, parabolas and catenaries are the curves usually mentioned. While it is true that arches have use arcs of circles for hundreds of years, this has been done to simplify the construction, particularly of expensive centring, rather than from any theoretical considerations.

Formulas do exist in engineering, often giving the results for such calculations as deflection of a beam of given cross section with a given distribution of loading, distribution of stresses within a member, and so on.

But the overall shape and size of a bridge and its parts are more likely to emerge from long series of calculations which are intended to optimize a large number of variables, one of which may be the cost. That is not to say that an individual engineer may not have preferences in terms of type of bridge, and shapes to be used. But the idea of a formula guiding the design is generally wide of the mark. Engineering is far removed from geometry, and even when something turns out to look elegant, this may be accidental.

Take the airliner Concorde, for instance, reckoned to be elegant or beautiful by many people who otherwise take no interest in aircraft. Yet its shape is actually the result of a compromise between the need to fly efficiently at Mach 2, and the need to fly slow enough to take off and land safely in a reasonable distance. In subsonic airliners this is taken care of by a vast choice of slow flying aids, such as flaps, slats and droops.  But those solutions were no good for Concorde.  If it could have been launched from a ramp at 500 miles per hour, it might have looked more like a huge cruise missile, with a large fuselage and quite small wings and tail surfaces. But how would it have landed?

To return to the original question, what do engineers use, if not formulas?  Suppose you had only two variables to control. You could imagine a sort of computational landscape in which the altitude would represent efficiency, and distances north and east would represent the values of the two variables. Somewhere there would be hill of which the summit would represent the best design: there might even be smaller hills, and if your computer program found one of these, you might never find the highest one. In practice, there are large numbers of variables, and computer programs exist to find the maxima in the resulting multi-dimensional space.

If we look at other creative processes, we don't see formulas either. The musical composer, the novelist, and the poet may conceive overall plans for their work, but it is certain that when they try to create the actual work, all sorts of compromises may be needed, and, of course, the need to find a rhyme can even provoke the creation of an idea that would not otherwise have come. Even the most geometrical of artists tend to introduce asymmetry and variability, to make the work "look right". As an aeronautical engineer said - "If it looks right, it probably is right."  

Even scientific theories are not perfectly symmetrical, mainly because nature seems to make them so. Broken symmetry could be the motto of the last fifty years in physics: in fact, even Maxwell's equations of the 19th century are not symmetrical between electricity and magnetism. The existence of magnetic monopoles could make them so, but no confirmed sign of them has ever been found, in spite of their properties having been predicted.

Where should bridges be built?

There is a page about bridge location in this website.

What are the longest bridge spans in the world?

Please see these two websites.

A site listing long bridge spans of many types - Click here

Another site listing the longest spans - Click here

What makes a bridge hold up?

This is either easy to answer or difficult to answer, depending on the kind of answer you are seeking.

The easy answer is -

A     The designers designed it to hold up under the likely conditions

       at the site, and they got it right.

B     The construction procedure was designed correctly, and the

       construction process was adequately supervised and executed.

       Construction includes the preparation of all the parts and the

       materials from which they are made.

C     The maintenance schedule was well planned and correctly

       carried out.

This all seems obvious, but there have been famous examples in which A, B or C was not done right, resulting in problems, or even collapses.

An apparently different answer for A concerns energy. The lowest possible energy state of a bridge occurs when it is lying on the bottom of the river, or of the valley, having collapsed. One job of the designer is to ensure that between this condition and the actual state of the bridge as built, there are energy states so high that they will never be reached under any foreseeable loads. This has to be the case both for the whole bridge and for every individual part that isn't redundant.

This can be symbolized by a simple graph, showing the energy of a system versus the force applied.

The graph above applies to any linear, elastic structure. But what happens if we apply more force?

The difference here is that when a certain critical condition is reached, something breaks or buckles and the structure collapses, and the energy plummets down on one of the vertical lines. This happened to the first Quebec bridge during construction.  In the case of the first Tay bridge, the structure actually got built, but the effect of strong winds had been greatly underestimated, and the construction had not been performed adequately.

The red line has been drawn as a possible limit to the load, giving a safety factor.  The structure would be designed so that no foreseeable loading could make the energy go past the red line. The curve is symmetrical in this example. In the case of the Tay bridge, the wind could have blown from either side, with the same disastrous result.

For actual live loads, such as traffic, the loading would seem to be always be in the same direction. This is not necessarily true of every individual member. It is even possible that the stress in a member could change from compressive to tensile as a load moved, or a bending or a torsion could change direction. In steady flight, the wings of an aircraft are always pushed upwards, but on the ground, the wings hang down.

In time of war, bridges may be among the early structural casualties, the energy needed to overcome the stability being provided by explosives.

Another way to look at this is that the forces in a bridge act in such a way as to produce equilibrium at all points. If at any points there is no equilibrium, the structure will deflect until equilibrium is reached. If a structural member reaches its limit before complete equilibrium is reached then the structure is not viable.

What is the best kind of bridge?

If you need build across a deep valley, a simple crossing could be made by building an embankment. If you do it properly, it will be very strong. This is, of course, seldom practical, as there is usually a river, a road or a railway in the valley, which would have to go through a tunnel in the embankment. There is a also the question of economics, and in a wide V-shaped valley, there may be embankments at the ends, and a viaduct in the middle. The use of embankments might become cheaper if the road had required a cutting nearby, making large quantities of material available.

So we probably have to build a bridge. The type of bridge will depend on the length of the longest gap that has to be cleared by a single span, on the type and volume of traffic, on the type of ground, on environmental considerations, on the available budget, and on the relative cost of different designs. An extreme example of having to build a bridge, rather than an embankment is found at Millau, in France, where some of the piers of the bridge are taller than the Tour Eiffel.

The relative costs of different designs does not necessarily remain the same under a change of scale. If you scaled the Forth railway bridge down by a factor of ten in length, you would have the same number of parts, but vast numbers of them would be ridiculous complications for such a small structure. If you look at a very short truss, all the members will almost certainly be very simple struts and ties, but if you look at a really large truss bridge, many of the members will structures in their own right, such as trusses, tubes or box girders.

Scale is a major factor in the choice of structure. In fact, for spans above 890 metres, there are at present only suspension bridges. Galileo explained the effects of changing scale very well. Compare, for example, aphids, dragonflies and eagles, or insects and elephants.

The great variety of bridges tells us that there is more to the art of bridge building than span and load. Technical and aesthetic change, for example, must be taken into account. Travel north from London on the M1 motorway and you travel in time.  Look at the Forth railway bridge and the Quebec bridge. Why has this type of bridge not been repeated on that grand scale? Was it because railways had reached their peak of expansion? Was it because there were no more wide and deep gaps for them to cross? Was it because better and cheaper types of bridges were invented?

See also strongest bridges.

Why are the longest spans all in suspension bridges?

The longest component of a suspension bridge is the set of main cables, which are all in tension, a stable state. In an arch, the main members are all in compression, which means that they are liable to buckling. Hence the thickness of the members and the large amount of bracing that you see in structures such as Sydney Harbour bridge. Suspension bridges do have to include anchorages and towers, which are massive constructions, but they are relatively simple compared with the trussed arch.

Worse still are beams, which suffer both tension and compression, unlike the suspension bridge and the arch, which have mainly the one type of stress. In a sense, a beam is like a very shallow tied arch, or a very shallow self-anchored suspension bridge.

The cantilever bridge can just about compete with the arch, but the Quebec bridge remains the longest span attempted. With road traffic dominating, the cable-stayed bridge has taken over much of what cantilever bridges used to do.

Click here for a more detailed discussion of this topic.

Why are there so many small beams and cantilevers?

Vast numbers of modern bridges over roads and railways are beams and cantilevers, often plate girder/I-beam or concrete. Why is this? These bridges have the following advantages -

The spans can often be built off-site and lifted into place.

The designs are relatively simple and the principles are well understood.

The piers can be vertical, because there is no horizontal reaction, so the ground area needed is small.

Clearance between road and bridge can be maintained high throughout, as compared with the arches which were commonly built in the 19th century.

How can I measure small deflections of my model?

Some projects require you to measure the deflections of a structure under various load conditions.  There are several difficulties.

The deflections are likely to be small.

The measured position is likely to be distant from any fixed point.

These two difficulties may in some cases be met by the same mechanism. If the deflection involves a change in angle, a simple method is to glue a very small mirror on the structure. A beam of light reflected on this will respond to the movement of the mirror by being deflected by twice the angle that the mirror moves. Catching the light on a distant screen will provide a long "lever" arm. The patch of light may then be fuzzy, making measurement difficult. The measurements may be made easier by the careful use of a small laser, of the type that is used as a pointer for slide shows. The laser beam should of course never be allowed to enter an eye.

The second point may also be met by the principle of the lever. It is usually a bad principle to measure small changes in large values. For example, if you can measure 1 metre with an accuracy of 1 mm, you are measuring to 0.1 %. But if your maximum deflection is only 10 mm, you can never do better than 10 %. Suppose that you built two copies of the structure, side by side, with the non-deflected ends rigidly fixed together. The one that will not be deflected need not be as detailed as the one to be tested. The point is to get to places close together, which can be compared, and to arrange things so that if the equipment becomes displaced or vibrated, the distance between these two points does not change. If you now rest a strip of material like a little beam between the moving place and the fixed place, it will change its angle as the deflections change. If you glue a very long and light pointer to this little beam, you will see the movements amplified. You can place a measuring scale behind the pointer. But all your problems are not yet solved: how are you going to fix your scale? Look at the little moving beam and its supports. The arrangement could be better. What could you do with these parts to increase the pointer movement?

Whatever method you use, remember that it is better to measure changes directly, rather than by subtraction from a big value, or at least to arrange that the equipment does the subtraction for you.

Does a force have a direction?

This looks like a silly question. When you lift something, or you push a car that has broken down, or you pull something along, you are very sure of which way the forces act. This is true, but what about the forces inside an object. In the three cases already mentioned, it is you and something else, and you feel the force that the object exerts on you.

Inside an object, we can still think of tension and compression acting along lines, but we cannot assign a direction along those lines. All we can say is that if we cut the object at right angles to the lines, the two parts either push each other (compression) or pull each other (tension). But as soon as we reconnect the parts, this idea becomes meaningless. In fact, all mechanical forces are inside objects. What about electrical and magnetic fields in empty space? They can push or pull objects.  Suppose you pull a nail with a magnet. The nail must pull the magnet with an equal force. But what about the empty space between the nail and the magnet? Is there a material that also experiences a force? Physics theory works perfectly well without requiring the existence of such a medium.

So lines of mechanical force are not the same as electric lines of force and magnetic lines of force. Electric fields have direction because there are positive and negative charges. No magnetic charges have been detected, but the polarity of magnetic fields is created by the direction of the electric currents that produce them.

Does stress cause strain or vice versa?

Stress is defined as force per unit area, while strain is the change in dimension divided by the whole dimension, in other words, the relative change in dimension. If we bend a piece of balsa wood or we stretch a piece of elastic material, we are the cause of the force, which creates the stress, and as a result, the object changes its shape. So is it obvious that stress causes strain? No.

We must be very clear as to what we mean by the statements made above. We are using words to describe the changes to a physical object. The words are not the physical reality: even a mathematical formula is only a metaphor. Nobody believes that when we pull something, it somehow thinks it has suddenly to obey Hooke's "Law", and stretch according to the force.

Let us analyse what happens when we stretch an object. At some moment we start to pull our hands apart, and the effect of this cause is that the object starts to elongate. This is cause and effect as we normally understand them. But if we ask about stress and strain, we find that we cannot decide which comes first. If one comes before the other, by how much? Perhaps there is a speed at which the stretching happens? Yes, there is. If you strike the end of a very long bar of steel in the longitudinal direction, a wave of compression travels along the bar at the speed of sound.  

But this does not help us answer our question. As the wave of compression strain travels along the bar, there is a wave of compression stress as well. They are inseparable, and in fact, we cannot measure either of them inside the material.

Let us try another thought experiment. We clamp a steel bar in an extremely strong frame made of invar, which does not change size with temperature change. Now we heat or cool the steel bar. It cannot change its size, but it will experience compression or expansion stress. This example is not so very unreal, because continuous railway tracks experience just these effects. We can now say that the bar has not been strained, because its length is the same. Not so. Suppose we took an identical bar in free air and heated it, and then we compressed it in a vice until it was the same length as the invar frame, it would in every way be in the same condition as the bar in the invar frame, except that it has changed its length in the process.

The conclusion must be that strain is not to be measured by the actual change in dimension: it must be related to the change in dimension that would have occurred if the bar had not been constrained.

So stress and strain are not cause and effect, or effect and cause: they occur together. This has the advantage that strain gauges on the surface of an object can tell us about the strain and stress at the measured locations. But we can only know about the inside by calculations, which can be very complicated.

How many bridges exist?

People sometimes ask how many bridges there are in a particular country. This question probably cannot be answered. Even if some authorities keep good records of railway bridges, road bridges, canal bridges, etc, there may be some bridges on private land that are not recorded. Official maps probably record almost every bridge, but counting off a map, even if digitally stored, might be difficult.

Another problem is the definition of a bridge. Many small streams run under roads, and there must be a size below which we would call the structure a pipe and not a bridge. So the number of bridges depends on our definition by size.