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Note the relative thickness of the horizontal tube at the bottom of the tower, the lower chord of the cantilever, and the almost vertical tube of the tower. These tell us something about the relative magnitudes of the forces. Let’s try a very much over-simplified calculation. We will neglect the weights of individual members.
A is the angle of the lower chord from the vertical. C is the force in the lower chord, and we will assume that the force in the top chord is the same. This cannot be exactly the case. B is the force in the bottom tube of the tower. P is the force in the pier. T is the force in the leg of the tower. At the top of the tower, to make the vertical forces balance, we have C cos(A) = T, that is, C = T/cos(A). For various angles A, we can work out the ratio C / T. A = 60 C / T = 2.00 A = 65 C / T = 2.37 A = 70 C / T = 2.92 A = 75 C / T = 3.86 As the chords become more nearly horizontal, the forces in them increase rapidly. This is one reason why beams and cantilevers cannot be made very thin. Using the sine of angle A, you can also work out the ratio of C and B, by balancing the horizontal forces.
C = weight of tower and two cantilevers S = weight of suspended span T = weight of train, assumed to act at a point L = length of cantilever W = width of tower In the condition where the tower is about to rotate about the pivot, we can write down values for all the moments as follows. Clockwise TL + 0.5SL Anticlockwise 0.5CW + 0.5S(L+W) And we must equate these as follows. TL + 0.5SL = 0.5CW + S(L+W) 0.5CW = TL + 0.5SL – S(L+W) 0.5CW = TL + S(0.5L-0.5W-0.5L) CW = 2TL – SW If CW is less than this value, the system will rotate. The width W of this central tower is much larger than the widths of the other two towers. The deficiency in W for those towers is compensated by massive weights that are hidden in the piers at the ends of the outer cantilevers. Some people say that the Forth bridge was over-engineered, as a result of the collapse of the Tay bridge. We could equally well ask these people what they would have built, given that the size of the gap to be spanned and the state of knowledge and materials at the time, especially as steel was relatively new. A counter example is the de Havilland Comet, which was not over-engineered, though it explored new territory in terms of speed and altitude; it succumbed catastrophically to metal fatigue. After the disasters, no doubt some people were saying that it should have been engineered more sturdily. The reality is that when you commence a project, you have only these sources of knowledge – Histories of previous projects, both failed and successful, Current knowledge of structures and materials, The laws of physics. To these you can add – New knowledge gained by research and testing, New ideas gained by creative imagination. Sometimes, in spite of all our efforts, things go wrong. The designers of the Millennium bridge in London could not have foreseen a mode of oscillation that had not to their knowledge been seen before. They knew about vertical and torsional oscillations, but not the swaying induced by pedestrians, and especially not the falling into to step of pedestrians, induced by the oscillations. The Tacoma Narrows bridge was, in a sense, a part of a logical progression in boldness of design: the problem there was that a threshold was crossed in terms of aerodynamic forces. The idea that you can go on changing things in a given direction is insidious when all goes well: it is only when disaster strikes that you realise that you had continued to change things, without continuing to test them adequately. Testing in itself is not enough: you have to test for all possible contingencies, and if there is something you don’t know about, you won’t do the test. It is even possible to believe that things will work out because you want this badly enough, but king Knut pointed up this fallacy a long time ago to convince his courtiers that he could not do the impossible . . .
Here is one of the two 350 foot (107 m) suspended spans, which are substantial trusses in their own right, though, as we shall see later, they are small relative to the bridge. You can see the rather sudden taper of the lower chords of the cantilevers towards the ends. Tubes are very efficient, but they presented problems in design and construction, especially where they had to be attached to other members with large forces being transferred.
One of the problems with many bridges is that the deck does not conform to the structural part of the bridge. In a suspension bridge it hangs below the main cable: in a steel arch it may hang below or be supported above the arch, or it may even do both. In a cantilever bridge we again have to find a means of fitting a deck, unless, as in the Jacques Cartier bridge and the Carquinez Strait bridge, the lower chords are nearly horizontal. In the Forth bridge, the railway tracks are carried by a bridge within the bridge, which does not actually conform comfortably at all points to the main structure.
In a very large structure, the eye can take in only parts of the system, making differences in appearance easier to assimilate. You see this well in a cathedral such as the one in Gloucester, which was built over a long period in a variety of styles. Only where the styles meet are we made aware that there were problems, and we can see how they were more or less solved. When structures are as old as that, people are often willing to accept as "quaint" or "charming", things that they would find unacceptable if built now. Geelong sewerage bridge
Quebec Bridge
The Quebec bridge suffered two catastrophes during construction. In the first one, a lower chord of the south cantilever began to buckle, and it fell into the river, with the loss of eighty-two men, who had not been withdrawn when the ominous signs were seen. It is possible that the details of the design had been extrapolated from successful smaller examples that had been successful. The new design was much heavier and stronger, but during the raising of the suspended span, something broke, and the span, carrying eleven men, fell into the river. For comparison – Forth bridge – main span 1700 feet – suspended span 350 feet Quebec bridge – main span 1800 feet – suspended span 640 feet. The suspended span of the Quebec bridge weights 5200 tons: remember that its supports are 580 feet from the feet of the cantilevers. Compare this with the small balsa trusses that are entered into competitions: a load : weight ratio of over 2000:1 has been achieved. It’s all a matter of scale. As another page shows, there is a limit to bridge spans at which they can only just hold themselves up, with nothing in hand for a load. For every structure that you see, you can ask at least one question. Here is a question. How do designers work out what fraction of the span is to be filled by the suspended span? Two examples have already been given. Here, in outline, are five possible configurations, from no suspended span at all to no cantilevers at all. The first design is very rare: the Queensborough bridge being the most well-known.
Here is a table showing data for some examples of steel truss cantilevers, giving the following lengths – Main span cantilever reach suspended span. The final numbers represent the fraction of the main span occupied by the suspended span. 1800 580 640 0.356 Quebec 1710 680 350 0.204 Forth 1644 Commodore Barry 1500 468 564 0.376 Howrah 1180 590 000 0.000 Queensborough 1098 332 433 0.394 Carquinez Strait 1095 378 358 0.345 Jacques Cartier 0469 234 000 0.000 Middlesbrough transporter The Firth of Forth railway bridge and the Queensborough bridge are apparently exceptional. The data are shown graphically below. If you want a simple number to remember, most of the ratios are not far from 3/8 or 0.375.
We could add to this list a number of long truss spans, counting them as cantilever bridges with cantilevers of zero length. The Structurae web-site includes a long list of truss bridges. From this, you can find out whether the spans listed above are typical in length for truss spans in general. As well as the difference in the suspended spans, the Forth bridge and the Quebec bridge have another obvious difference: the Quebec bridge has straight lower chords, while the Forth bridge has segments that form part of a huge polygon. Is it possible that the collapse of the first Quebec bridge, which had curved lower chords, had some influence on the new designers? The two differences may not be entirely unrelated – had the Forth bridge employed straight lower chords, the width for navigation of the tallest ships would have been much less. You might think 350 feet to be adequate, but with strong winds and tidal flows, you cannot be too careful when piloting a great ship. For spans of less than a certain length, the complication of a cantilever is uneconomic: for spans of greater than a certain other length, a continuous truss is impossible. Can you find out what these two lengths are? Here is another diagram, comparing bridges with much shorter spans than those in the previous diagram. You might think that with shorter spans there would be fewer constraints on dimensions.
The purple square represents a steel truss bridge, while the five green triangles represent steel plate girder bridges. The six red diamonds represent concrete road bridges. For the concrete bridges you can almost imagine a trend, shown by the black points, leading to a value of 1.0 for zero span, in other words, as the spans get shorter, the cantilever arms shrink faster. In practice, of course, for very short spans the complication of cantilevers may not be worthwhile. These few data are of course insufficient to do more than suggest the possibility of a trend. One of the three blue diamonds representing concrete footbridges, does not come near the trend of the road bridges. This is not surprising, since the lightness of live loads on footbridges leaves more scope for the designer than in almost any other type of bridge. The three comparative diagrams above, along with others in this web-site. suggest that the division between the standard types of bridges are not completely rigid. Nevertheless, the types are distinct, and they remain so, because many of the intermediate designs, even if possible, are simply not practicable or economic. On the other hand, many designers, from Brunel to Calatrava, have explored bridge design with imagination, and have produced unusual solutions to problems. The art of bridge building is like any other sort of art – there is room for many kinds of people. In the 19th century, the names of Brunel, Locke and Stevenson represent rather different approaches to engineering. These names also remind us that historical fame is not always a good guide. Before looking at some other bridges, how about a few charts. The next pictures show a uniform beam, such as an I-beam, the turning moment caused by each part of the beam, and the total moment at each point caused by all the parts outside that point.
The forces in the beam will be proportional to the values in the last graph. Now this is not very efficient, because the beam has equal thickness and strength throughout, though in fact the required force varies. In the next example, the thickness of I-beam varies along its length. (The diagram is exaggerated). You can see this construction in many old plate girder bridges. We ignore the weight of the vertical web in these calculations.
Better still, we can vary the depth of the girder, and in the last graph we see the effect on the forces in the top and bottom of the I-beam.
Note that the vertical scales in these graphs are not all the same, so only the shapes are important. These calculations are grossly oversimplified, but there is little point in going into great detail for this purpose. As the depth of the cantilever has been increased about 4.5 times at the root, the forces are reduced by the same factor, as in the next graph, which compares the forces in the parallel cantilever and the deep cantilever.
The shapes of actual bridges differ considerably from those considered here, because we ignored the presence of the suspended span and of live loads. Those are two reasons why the chords of the Quebec bridge and the Forth railway bridge do not taper as much as in our example.
We return now to looking at actual bridges. |
