What is the Longest Possible Span ?
November 2003 Back to Home Page
Link to a site listing the longest spans of many types of bridge - Click here
Link to another site listing long bridge spans of many types - Click here
The chart below shows the longest actual spans of several bridge types at a recent date. Note the dates. What do they tell us about the various types of bridge? And what do we learn from the six lengths?
Assuming that there is a place to build it, enough people to use it, and enough money to build it, what would be the longest possible span? To make things reasonably realistic we will consider a suspension bridge in which the towers rise above the road by one tenth of the span.
We will take the extreme case in which the cable is only just strong enough to support itself. Then we can reduce the result to allow for a useful load. The diagram below, not to scale, shows the situation, with a deck drawn in, but assumed, like the hangers, to weigh nothing.
|The span is S, the
length of the cable is L, and the height of the tower above the road is H.
The angle between the cable and the vertical at the tower is A. That takes care of the
geometry of the "bridge".
Let's assume that the tension of the cable at the tower is T. Elsewhere it will be less. Why?
We need to know the maximum stress M that we can ask of the cable, and we need to know its density D, and its cross-sectional area C (temporarily), so that we can calculate its mass, and hence its weight, W. For that we also need the acceleration due to gravity, g, which is about 10 m/s/s.
The tension in the cable is T = W / cos(A), so W = Tcos(A).
But T = MC just before the cable breaks.
Therefore W = MCcos(A), or W / C = Mcos(A).
But W = DCLg, and so W / C = DLg.
Equating the values for W / C,
we have DLg = Mcos(A), so that L = (M/Dg)cos(A).
|Does this make sense? It says
that increasing the allowed stress or decreasing the density will increase
L. That seems reasonable . Since cos(A) increases as A decreases it also
says that increasing the sag of the cable increases L. This also seems
reasonable. So we may as well continue.
All we need to do now is to calculate the span from the length of the cable. The span is clearly somewhat less than the length of the. For the purposes of the exercise, calculating the cable will suffice, as we only need an approximate result.
The density of steel is about 8000 kg/cubic metre, and g is about 10 m/s/s.
So Dg is about 80000 SI units.
Now we need to calculate the angle A.
This depends on H and L, or on H and S.
Very roughly, A = pi / 2 - 4H / L in radians.
Let's assume that the sag to length ratio is H/L is 1/10, so that A = 1.2 radians,
which is about 67 degrees, and the cosine is 0.4.
We now have L = M (0.4 / 80000) = 0.000005 M.
Taking a guess for steel, M = 1000 million N/square metre, we get L = 5000 metres, which is about 3 miles. If we remember that the span will hang from two cables, so that each cable has to carry only half the weight, it looks as though a span of about two miles is not impossible.
The calculation was only very rough, and if the exact value mattered we would need to look up the actual strength of some steels, and also do the calculation properly. But at least we can guess that 10 miles (and most probably 3 miles) is not feasible. Nobody is going to bridge La Manche in one span.
It's amazing what you can do with simple arithmetic, but if you start with the wrong assumptions you will certainly end up with the wrong answers. Was this calculation any good?
The next diagram shows the suspension spans of over 700 m versus date. It gives a false impression because there must be many smaller suspension spans being built all over the world, right down to small footbridges.
If we plot the number of suspension spans in brackets of 100 m, we see the chart below, which has a logarithmic vertical scale. Unfortunately, the data are sparse, but we can see the expected result that shorter spans are more common than long ones. We could extrapolate this chart to get a rough estimate for shorter spans.
The chart illustrates the difficulty of answering questions like "How many bridges are there in region X?". Even if some authority had a list, there would be the difficulty of deciding what was a small bridge and what was a wide pipe, though this problem would not occur with suspension spans. Similarly, if we want to work out the probability of very high floods or other extreme events, all we can do is to extrapolate from what we have had in the past, assuming some simple distribution of events. Even then, we cannot guess when the next big event will happen. "We are due for a big one." is generally a meaningless statement.
Here is an extrapolation for suspension spans. By calculating the area under the line we would obtain an estimate of the number of suspension bridges in the whole world, but the estimate would be totally dependent on the accuracy of the extrapolation.
The calculation gives a total of about 400 suspension bridges in the whole world. Do you think this is reasonable? Do you think there are probably more than this? Looking at the chart, it is clear that the line could have been drawn steeper or less steep, given the scatter of the data. What is needed is to obtain data down to perhaps 500 metres instead of 700 m. What probably happens is that the curve becomes steeper for lower spans, and the estimate of 400 is a bad underestimate.
There are several reasons for this. One is that short crossings are far more numerous than long ones. Another is that short spans are far cheaper than long ones - the cost of a span rises as a high power of the length, and expensive constructions are not undertaken lightly. There has to be a high probability that the bridge will pay for itself in tolls, or that it will generate a tangible benefit in some other way. The Humber bridge seems to have been an exception. Another possible reason is the distribution of wide rivers and estuaries. Although towns and cities have tended to grow up on the banks of rivers, the largest ones are seldom found where the estuaries are very wide, the cost of bridging having probably contributed to this. An interesting school project would be to look at the fifty longest bridge spans, and to look at the distance to the nearest city, or the population living within a certain distance of each bridge.
If we consider the case of beam bridges, we can see how this type of extrapolation would fail, because there are hundreds of thousands of beam bridges over roads, railways and small rivers, and a chart of the longest beams would be completely unable to give a guide to this. For suspension bridge this type of analysis does not apply, because such bridges are totally uneconomic for short spans, and are found mainly in the form of footbridges over small rivers.
The next two charts show the longest spans of the following types - concrete arch, steel arch, I-beam, cable-stayed, cantilever, suspension and truss, plotted in decreasing order of size. On the logarithmic chart (2nd one), the graphs all look fairly similar, but two of them, those for concrete arches and trusses, cross over. Perhaps this is related to the cost or the convenience of trusses for the shorter spans.
We can see that when the chart was made, early in 2004, about 20 suspension bridges had spans greater than that of the longest cable-stayed bridge. This number may not change much, because there is only a finite number of places where a long span is needed enough to make it economic. These charts show nothing about dates: perhaps it would be interesting to find out how the choice of spans has differed at different times.
The diagram below shows the results of a better calculation for steel bridges than the one at the top of the page. The simple calculation used a fairly conservative figure for the strength of steel, though so the result was not too bad. The second diagram shows some current longest spans, which are only about 0.3 to 0.4 of the theoretical maxima.
What if we try to do the same sort of calculation for an arch?
The maximum span will be much shorter than that of the suspension bridge because an arch has to be thick, and therefore heavy, to be rigid. The cable is statically stable, whereas the arch is statically unstable; if bent far enough it will collapse.
Even by using a truss arch to gain stiffness, we will lose out on weight well before the maximum suspension span is reached.
For a horizontal beam or truss the situation is even worse. In the page on beams we see that the forces in a beam or a truss are fighting each other. As we make the span bigger, the beam has to be deeper in order to resist bending. The arch wins by exporting thrust to the abutments. The beam contains the thrust within itself, exporting only the weight. Long before we reach the state where depth equals span, we have reached absurdity. This is why the maximal beam is much shorter than the maximal arch, and why the maximal cable-stayed bridge is much shorter than the maximal suspension bridge.
Here are some diagrams to illustrate the way that beams include greater stresses than the equivalent arches. Firstly we show an idealised I-beam.
Next we change the shape to make the stresses more uniform.
Now we increase the depth of the beam. The black area representing the web has been omitted to allow the feebler colours to show better. These feebler colours represent weaker stresses.
Finally we remove the web and the lower flange, and use the abutments to take the thrust that had been held by the tension in the lower flange.
Thus we have an arch that includes stresses that are much weaker than those in the original beam. So we can increase the span beyond that which would be feasible for a beam, though a limit is reach because of resistance to buckling and for economic reasons.
The long span box-girders are usually continuous between spans, so that the long central spans can in a sense export some bending moment to the sides. A single beam, hinged at each end, could not achieve as long a span.
Finally, how does the cantilever beat the truss and the box girder? Although its force on the ground is purely vertical, a span like that of the Quebec bridge gets resistance to bending moment from the connection to the side-spans, using a very deep truss, deeper than would normally be used for a simple truss. The outer ends of the side-spans are anchored down. The relationships between arches, beams and cantilevers are discussed in cantilevers.
In fact, many box girders, starting with Stephenson's Britannia Bridge, exchange stresses with other spans not only by simple connection, but by pre-stressing.
The cable-stayed bridge does better than the beam because the span is supported by cables at regular intervals, and it does better than the cantilever because the tension members are light cables. Like the cantilever, it has great depth.
Here is a preliminary version of a chart showing progress with different types of bridges during the last two centuries. The categories are -
concrete arch bridges steel arch bridges
cantilever bridges box girder bridges
cable-stayed bridges suspension bridges
Can you work out which graph applies to each type of bridge?
The laws of scaling apply to any object, as Galileo explained. If we look at the legs of a fly, a gazelle, and an elephant we see this very clearly. And the elephant could never spread its legs like a fly. A giraffe can spread its legs to drink, and a crocodile has well spread legs, though its belly usually rests on the ground. But these animals may be near the weight limit for such behaviour. Looked at as a bridge, the span of an elephant from front to back feet is not very large, in relation to its weight, compared with those of a weasel and a fly.
This picture shows a whisker from a domestic cat. If we look at bigger cats, up to tigers and lions, we find that on the bigger cats, the whiskers are much shorter, relative to the length of the cat, and much thicker, relative to the length of the whisker.
A whisker is a cantilever, and if we look at cantilever bridges, we see the same progression. These pictures show a footbridge and a part of the Forth bridge. It is clear that if we tried to make bigger and bigger bridges like the Forth bridge, we would soon find that the height and reach of the cantilevers would be the same - an impractical and uneconomic situation. The reach of the cantilevers at Forth is 675 feet (204 m) , and the height is 340 feet (104 m), giving a ratio of about 2. The ratio for the Quebec bridge, at 1.85, is even lower. When the reach of a structure is only twice the height, we are clearly near the practical limit.
Of course, if you look at the heights and half-spans of big steel arches you might say the same thing, but in fact the real depth of the steel work in an arch remains quite shallow throughout. The picture shows the Bayonne bridge. Comparing this with a cantilever is a little "unfair", because the halves of an arch have the advantage of being propped by each other.
A flea can develop an acceleration of 100 g, a gazelle can pronk, a highly trained human can perform ballets or high jumps, a horse can get over a fence, but an elephant can never leave the ground.
In fact, for most human activities, there are specialist animals that can do better.
We see very clearly in the case of aircraft how the construction reflects the size. A model can be made of thin pieces of balsa wood, but the largest aircraft have to be very robust indeed, exceptions being man-powered planes, which must be very light, and specialist solar-powered planes.
Returning to the question of leaping, the mean acceleration as a multiple of g is given by the ratio of the rise in the centre of gravity to the vertical extension of the legs or other appendages.
If a flea can extend its legs by 0.5 mm and jump 5 cm in the air, the mean acceleration would be 100 g. A human doing the high jump can only manage around 2 m, raising the centre of gravity a metre or so. The acceleration is less than 3 g. Why is it that a flea can withstand 100 g, whereas a human already feels uncomfortable at 3 g?
A humpback whale can leave the water completely, because it can gather speed under water and then convert its kinetic energy into potential energy. Obviously animals can run and then jump, but their vertical acceleration is still limited by leg extension. But the whale can steer itself upwards and burst through the surface with a considerable vertical velocity, even though the acceleration is low compared with that of a flea, because the whale is very long, even if we only include the section at the back that contributes to motion.
Why can a flea rest on dry land, whereas a whale dies on land, even though a 100 ton whale is happy in water?
The longest bridge that could possibly be built would be useless, because it could carry no load. This is general principle: extremes don't work. The wandering albatross is one of the largest birds that can fly, but in some ways it is not a good flier: it has trouble getting off the ground, and alights on land very clumsily. What it is good at, very good indeed, is gliding near the surface of the sea, making use of the sloping waves and the wind gradient. Similarly, an aircraft at its maximum possible height has can make only one manoeuvre - descent.
Arch Beam Box Girder Cable Stayed Cantilever Pre-Stressed Truss Oscillation
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