zzzzzzzz suspension bridge 1 suspension bridges 1
Part One - Theory - This page
Part Two - Bridges - Click here
=== suspension bridges 1
Here are some ideas about suspension bridges, copied from the basic bridges page. As we all know, the standard suspension bridge looks like the picture below (showing one half only) with the attendant cost of the towers and anchorages.
The basic functions of the various parts are easy to understand.
The towers hold the cable up.
The anchorages pull the cable outwards and downwards.
The hangers connect the deck to the main cable.
And the deck is there to carry the traffic.
This diagram shows an anchorage that relies on its weight to hold the cable. The moment of the weight about the toe must be greater than the moment of the pull of the cable about that same point. The diagram includes the inequality. The strands of the cable splay out from the saddle to a very large number of attachments into the concrete. This type of anchorage may be used when the ground is not good enough for a buried anchorage. In good rock, the saddle and the attachments are in a cavity in the ground. The anchorages of the Clifton bridge are in tapering tunnels that are filled with masonry which acts as wedges.
The behaviour of the cables on the saddles depends on the detailed design. In some older bridges the assumption was that as the live loads varied, the cables might need to move, as different parts were stretched or relaxed. They could be clamped at the saddles or they could be free to slide, overcoming friction. Either they or the saddles could rest on rollers. In practice, friction may well be enough to prevent sliding, though vertical divider plates may be fitted in the saddles to increase friction. In addition to the variations caused by changes in loading and temperature, the tension on either side of a tower will differ if the gradients of the cable differ. The difference may be quite large if the side spans are much shorter than half the main span, as in the George Washington bridge and one side of the Humber bridge. In the case of the Humber bridge, the cables at the short end include extra strands which pass from the tower to the anchorages. The number of extra strands divided by the number of main strands agrees with a simple calculation based on trigonometry.
The deck has to possess enough local rigidity in bending and torsion to prevent undue flexure as vehicles pass. Locally, around each vehicle, it acts as a beam with rather diffuse supports, namely the hangers for some distance in each direction. This same rigidity must be sufficient to help in the task of preventing undesirable amplitudes of oscillation. Some hangers are provided with small devices that help to damp oscillations in them. Oscillation is such an important subject that it has its own page. Perhaps the most important feature of oscillation, apart from the inconvenience and unpleasantness for travellers, is that it is the result of collection and storage of energy by a structure. Should this result in any part of the system being called upon to store more energy than it is capable of, that member is likely to fail. And once a member has failed, some other part is very likely to be immediately over-stressed, and so on. Even if the ultimate limit is not reached, repetitive lower energy levels may result in failure by fatigue.
Functions of the Parts - Summary
The towers hold up the cables. They have to be rigid enough to act as struts between the downward forces from the cables and the upward forces from the foundations. But modern cables are fixed to the towers at the saddles: there is no sliding. So the towers have to be flexible enough to allow for changes in length due to live loads and temperature.
The anchorages have to hold the ends of the cables against the enormous tension, either by sheer weight, or by taking the tension into the ground. At the time of construction, they have to include means of adjusting each strand to the correct tension.
The cables hold up the deck and the traffic, via the hangers or suspenders. They have to be strong enough to do this without undue stretching. They have to withstand vibration and be resistant to corrosion and wind-borne dust.
The deck has to be as light as possible, but rigid enough to prevent a dip as each vehicle passes. It is very difficult to make a suspension deck stiff enough so that a railway locomotive doesn't spend the whole crossing in trying to climb out of a valley. The deck must also be stable in winds of any possible direction and magnitude, whether steady or gusting.
== .============ suspension bridges 2
Why are the longest spans all in suspension bridges? There are two reasons.
Firstly, apart from the towers, which are in principle simple struts, all the most highly stressed parts of a suspension bridge are in tension. A cable, though flexible, is inherently stable against perturbations, and need only be thick enough to withstand the tension, with a safety factor. A strut is inherently unstable, and needs to be thick enough to prevent buckling. So an arch can never be as light as a suspension span of the same span. But an arch has one great advantage over the suspension bridge - its thrust goes straight into the ground. The forces of a suspension bridge are carried to the tops of high towers, which have to be resistant to buckling, flexure and oscillation. In a sense, the arch is the most perfect structure, because the dead loads can be made to follow the line of the arch. This is discussed in the page about the funicular.
Secondly, unlike a beam, a truss, and a cable-stayed bridge, a suspension bridge does not rely on on internally-cancelling forces to produce the required effects. The horizontal components of the tensions within it are resisted by a component which is very large and very strong, yet costs nothing This component is the ground. Somewhere between the anchorages, the ground must be in compression, though just outside the anchorages the ground is in tension. Under an arch, the ground is in tension, while the arch itself is in compression. This is a poor division of labour between the expensive and narrow arch, and the cheap and thick ground.
The diagram below suggests how the compressive forces in the ground underly the bridge. This is not an exact calculation: it is only a rough idea. The lines of force would be distorted by variations in the ground. The stress field is actually continuous: the lines are only a visual aid. The downforce from the towers has not been included. The ground stresses are very weak, except near the anchorages, because the area covered is so great elsewhere. If the bridge spans a deep gorge, like the Clifton bridge shown above, the lines of force will be very different, and the ground near the bridge will be in tension, the compressive forces being very diffuse.
Spiders manage to export all the compressive forces to the external world, so that an entire orb-web consists of threads in tension.
The lines of force are spread sideways and vertically, as if they repel each other. Why do the lines of force not simply run straight along under the bridge? The energy density at a place is proportional to the square of the stress, for elastic material. Therefore the minimum energy state is found when the stress field is diffuse. Halving the stress at a place divides the energy density by four. The distribution of stress and strain is the one that minimises the total energy. Spreading it or shrinking it would increase the strain energy. The stresses near the anchorages are more complicated, because the cables induce tensions, which are present along with the compressions already described.
This diagram makes clear that the structure includes not only the visible structure, but any part of any other object that is subject to significant stresses.
.suspension bridges 5
Spinning the Cables
Building a suspension bridge across a wide and deep gorge is the human equivalent of building a spider web across the space between two plants. It looks difficult, if not impossible.
The first job is to get some kind of rope or wire across. In older times this has even been done using a kite. Other means are boats and helicopters. Once a wire has been placed, progressively stronger cables can be pulled across. The next step is to build aerial walkways along the intended paths of the suspension cables. Fixed cables are also provided for carrying materials, including strands of the eventual suspension cables.
The usual technique then is to pull strands of cable across the gap, two by two, until they are all present. Then they can be tightly bound together, forming strong cables.
When the main cables are complete, vertical or almost vertical wires are hung at regular intervals from the suspension cables. Then the sections of the deck are fitted, one by one.
Spinning the cables in place was used by John Roebling for the Brooklyn bridge in New York, in the 19th century. The method uses something like an alpine cable-car system. A pulley is hung from a moving cable. The pulley has one or more grooves for cable strands. One end of each strand is pulled off a large drum, passed around the pulley, and returned to an anchorage near the drum. To see a really beautiful picture of Brooklyn bridge by Anney Bonney, click here.
You can read about the building of the Brooklyn bridge in this magnificent book - The Great Bridge David McCullough Simon & Schuster/Touchstone ISBN 0-671-45711-X A superb book in every way. Far more than just a book about the Brooklyn bridge.
When the trolley is pulled across the river at a speed of S miles per hour, the strand comes off the drum at 2S mph, forming two strands across the gap. The trolley climbs up to the top of the first tower, goes down the other side, back up to the top of the second tower, and finally down to the far anchorage. This is reminiscent of the cable car system that crosses from Chamonix to Italy, except that it is all done in one continuous journey.
When the strands have been pulled right across the gap, they are anchored. At the original end the cable is cut from the drum and anchored also. The strands need to be adjusted to the right tension before being finally fixed. This is necessary in order that all strands take more or less the same share of the eventual load.
The trolley can now be used to pull more strands back across the gap. And so it shuttles back and forth across the gap for weeks, or even months, until the job is done.
The spider doesn't need all this complication - it has a store of fluid in its body, which it can extrude through spinnerets to form a very strong thread that can absorb a lot of energy from struggling insects. The capturing strands carry blobs of sticky, viscous fluid which trap the insects. This is described elsewhere in this web-site.
. suspension bridges 6
Catenary and parabola
note that the name "catenary" is applied to a specific
mathematical curve, but some people use it to refer to any hanging
cable, of whatever shape. The second use is legitimate, given that
the word is derived from a Latin word for a chain.
The big picture at the top of this page shows the suspension bridge across the River Severn from Aust to Beachley. The picture at shows the Clifton suspension bridge near Bristol. The suspension bridge is easy to understand. Strong cables hang from massive towers. Smaller cables hang from the main cables and support a deck which carries a road. When the main cables have been laid, they hang in a curve called a catenary.
To make a catenary, imagine a compound interest or exponential curve, imagine reflecting it left-to-right in a vertical mirror, and imagine adding the two curves together. That makes a curve called a hyperbolic cosine, or cosh. A catenary is a special case of this curve.
. suspension bridges 8
|But the final shape is not a
mathematical catenary, though some people use the word for any hanging
shape. If the deck were vastly heavier than the cable, making the load
per horizontal metre the same throughout, the correct curve would be a
parabola. In fact this is not strictly true - if the cable were very
light, the connection points of the hangers would lie on a parabola, but the
main cable would be stretched almost straight in between them. If the curve does not get too steep, a parabola looks very
similar to a catenary. You can see this by writing the polynomial series for the
two exponentials of the catenary, and then using only small values on the
horizontal scale. Only the x2 term remains significant, which generates a
In an actual bridge, neither the cable or the deck is vastly heavier than the other. So the actual curve is a compromise between a catenary and a parabola. As stated above, the cable does not actually form a smooth curve. It is slightly kinked at each hanger attachment, and between attachments it follows catenaries which are less curved than than the curve on which the attachments lie. A flexible cable can never be perfectly straight unless it is vertical, since an infinite tension is impossible.
The picture at left shows the chains of the Clifton suspension bridge on the Clifton side. Nothing hangs on them this side of the tower, and so the curve you see is almost a catenary. Only almost, because the bars are straight, and they are in sets of alternately ten and eleven bars, and so the chain is not uniform in weight per unit length. You see also that the chains are much less curved here than over the gorge, where the deck hangs on them. The curvature is directly related to the load. You could in principle calculate the ratio of the weights per unit length of the deck and the chains by comparing these curvatures.
Here is a link to an excellent page by Xah Lee about the catenary.
This picture has been expanded vertically by a factor of five times. The black lines show the changes in direction at the hangers, slightly exaggerated, because the cables are in fact slightly curved between the attachments, as they are not weightless. This corresponds nicely with the segmented arches by Menn which were mentioned in a previous paragraph. So the curve of the cables is not only not a catenary or parabola - it is not a mathematically simple curve at all.
suspension bridges 9
The picture below shows a parabola and a catenary with the same span and sag. When a cable has been spun into place, it is a catenary, but as the sections of the deck are added, the weight distribution changes. If the cable had negligible weight compared with the deck, the cable would follow a parabola, or rather, it would have short sections of barely curved catenaries between hanger points lying on a parabola, with slight kinks at those points.
At an intermediate stage of construction, the line of the deck can look very strange, especially if the deck is added starting from the towers.
An Unusual Curved Bridge
On the cover of "Ekiben - The art of the Japanese box lunch" (Kamekura, Bosker and Watanabe - Chronicle Books - San Francisco - ISBN 0-87701-490-6) there is a picture of a pontoon bridge. The bridge makes a sweeping curve across a bay or estuary, pushed by the flow of the tide or the river. Had the builders tried to make it straight, the tension would have been too great. The curve is not a catenary because the speed of the water is greater near the middle than it is near the edges. But it is a funicular, if we stretch the meaning of the word a little.
The cable of a suspension bridge follows a curve called a funicular, which is essentially the path of the forces in it. The cable has to do this because it is flexible. The funicular is not a fixed mathematical curve with an equation, and it changes with the live load. For more information please see the page about the funicular.
Using a rough calculation we can see how the funicular changes during the building of a suspension bridge. This calculation is unrealistic on two counts. Firstly, the vertical scale has been exaggerated, and secondly, the deck sections have not been joined together. In practice, they would be, for several reasons, and this would reduce the curvature. The diagrams below show three cases - a complete bridge, an incomplete bridge with weightless cables, and a more realistic case in which both cables and deck have weight. suspension bridges 13
We can also start building from the middle, but this is not necessarily a good way to proceed. The unconnected deck is a big mass, waiting to swing in the wind. By starting at the towers we can anchor the deck at that place, and so reduce the amplitude of oscillations. The Severn bridge was built in this way. Its aerodynamic deck reduced its susceptibility to the wind.
A picture of a suspension bridge under construction usually shows a curved deck. As the deck is extended, the forces gradually straighten the deck towards the shape that it would have without the join. The curve may look alarming, but the radius of curvature is very large compared with the depth of the deck, and so the strain is actually very small. The diagrams above are vertically exaggerated.
Tension in suspension cableThe diagram below shows how the tension in the cable varies along its length. The horizontal component (pale blue) cannot vary because the hangers are vertical. The vertical components (green) increase towards the end because more weight is being carried. The total force (red) is, of course, along the cable at every point.
In the second picture the structure has been flattened. Look at the effect on the tension. suspension bridges 14
If the cable is rather flat, the tension will not vary much along the cable, because the horizontal component will dominate.
The curve of a cable obeys the following rule. If the tension is T at any point, and the radius of curvature is R, then the perpendicular force per unit length is f = T / R. So as the tension increases towards the ends of the span, the curve must straighten. In addition, as the cable gets steeper, f decreases, because f is further from parallel to the weight, again requiring R to increase. The cable could never become vertical, because f would be zero, and R would have to be infinite, and the cable would be straight.
How does the tension in a hanging cable vary along the length?
How does the vertical component vary?
How does the horizontal component vary?
Some suspension bridges have side spans that have less than half the length of the main span. What can you deduce about the tension of the cable holding up the three sections of the bridge? How would you achieve this? What can you deduce aout the arrangements at the tops of the towers?
Outside the anchorages the ground is presumably in tension. What is this doing to the ground at opposite ends of the bridge? Can you deduce anything about the ground under the bridge.
An actual suspension bridge
The annotated picture shows a part of a medium sized suspension bridge, across the river Tamar. It was photographed from a train on an approach span of Brunel's railway bridge. In this example, the separate trusses are supported at the towers, but in many other bridges the truss passes between the legs, and in some cases the span is not connected to the towers at all. Most suspension bridge designs have used steel towers, but the use of concrete has been increasing in recent years. The Humber bridge, for example, has concrete towers. Note the two cross beams in the tower, to provide lateral stability.
Suspension bridge details
Truss decks and aerodynamic deck
Cables and chains
Saddles and other attachments
.ppp suspension bridge 15
Proportions of Suspension Bridges
The proportions of bridges vary widely, suspension bridges being no exception. The next diagram shows the ratio of the lengths of suspended side spans to suspended main spans for bridges with main spans of more than 3000 feet. A small fraction of suspension bridges do not have suspended side spans. The Menai Straits bridge and the Clifton bridge are two of the older ones, while the Tsing Ma is a very recent example. The Humber bridge appears twice in the diagram, because its side spans differ greatly - 919 feet and 1739 feet.
The diagram does not seem to show a clear trend, though removal of the two top left points (Mackinac Straits and Tagus River) leaves a remainder showing a small correlation, though that suggestion rests mainly on a single point at the right (Akashi-Kaikyo bridge). At any rate it demonstrates that there isn't a "right way" or a formula for building a suspension bridge, any more than there is for any other type.
Part Two - Actual Suspension Bridges - Click here
Links to other web-sites about suspension bridges
Interesting site about art - http://www.eskimo.com/~ofsound/temp.html
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