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For Cantilevers Part Two - Click Here
Please note that many of the diagrams in this page depict trusses with none of the internal members: only outlines are shown.
Cantilever is a clumsy word, reflecting a basic problem for designers - how to achieve a satisfactory appearance from a bridge which includes two types of structure, cantilever and suspended span. The requirements of these are completely different. Perhaps only on the grand scale of the Forth bridge can the builders get away with it. On a smaller scale there must be compromises.
Just as as a great medieval cathedral can include several different styles, something that would look silly in a small house, so the Forth bridge is big enough that the eye can move from the huge cantilevers to the suspended spans, which are sizable bridges in their own right, and not be too offended. It is in the smaller cantilever bridges, such as the ones over motorways, where subterfuge is needed to get a good shape.
A cantilever is really a large bracket, held rigidly at one end. Here are pictures of some examples.
When we hold something, we know from experience that it is harder to hold it out at arms length than closer to the body. We might refer to this in terms of leverage. In the picture we see a leaning tree that has become unable to support itself, because its position is unstable. The more it leans, the greater the ability of its weight to turn it further, and of course, unlike any bridge, it is still grower, making the effects even worse. The solution was one that bridge designers use whenever possible, make the span or the cantilever arm as short as possible, here achieved with two props. Although bridges do not grow, the traffic on them can certainly do so, in some cases necessitating expensive strengthening or even duplication of the bridge. Strengthening can tax the ingenuity of engineers even more than the original design, especially if the bridge is a very old one which they wish to preserve with as little disfigurement as possible.
|The Forth bridge is unique - its
fame lasts more than fifteen minutes, if only because of the painting. But
of course it would hardly be famous for that if it weren't a magnificent
achievement, which has never been surpassed in cantilever spans, except for the one span of the
bridge in Quebec. To carry a railway locomotive and train up to 260 metres from the
nearest support was a daring project, completely
From the side the structure of the bridge is easy to read.
The great tubes take the compression, held up by the narrow top members, with bracing and strutting to keep the forces from buckling them. The spread legs of the towers suggest great stability, though in fact the outer cantilevers have counterweights at the ends to keep the balance. The outer towers could in fact have been hinged at the bottom, but that would have been out of keeping with the middle one, which has to be totally self supporting. What position of a locomotive would maximize the stresses in a cantilever arm? Not the centre of a span, but at the end of one arm. Why? Draw a graph of the moment at a tower as a locomotive passes from it to the next tower.
|The arch-like lower tubes
of the Forth bridge remind us
that the various types of bridge are not so different as they might
seem. Take an open arch bridge, put a tension member in the top deck, and
you can cut it in the middle and make two cantilevers. Morph the parts and
you can end up with something more like a normal cantilever shape. Morph
the arch a bit more and you have a self-anchoring suspension bridge.
This is illustrated in the right hand column.
The picture shows how two halves of two arches could be held together by a cable, forming a pair of cantilevers, with exactly the same stresses as in the arch condition. But of course this is not an efficient way of making a cantilever, and it does not allow of building without a lot of falsework, which is one advantage of a proper cantilever. The white line represents a cable which holds the half-arches together. Arches such as the Eads bridge and Sydney harbour bridge were built without temporary supports underneath by the use of cables, enabling the channel to remain open.
|One great advantage of a large cantilever bridge is that it can be built outwards from the piers without falsework. Then the suspended span can be lifted into place. Another is that it is inherently rigid - heavy railway trains are no threat if the structure is properly designed. Railways have always been a problem for suspension bridges, one which the cable-stayed type has now overcome, as in the bridge which joins Denmark to Sweden near Copenhagen and Malmo.||The problem with appearance is that the cantilevers need to taper from the supports to the ends, whereas the suspended span ideally needs to be narrow at the ends and thicker in the middle. In practice, for a motorway bridge, the suspended span is often a simple beam, often shaped to continue the line of the cantilevers. There are some interesting asymmetric cantilever bridges on the M1, which take advantage of the sloping terrain. Each span comprises a cantilever supporting the next beam, which itself projects beyond its pier to form the next cantilever.|
Please click here to read a page about moments in cantilevers.
What do we mean by the moment of a force about a point? It is the product of the force and the perpendicular line of the force from the point in question. In a system in equilibrium the total clockwise moment must balance the total anticlockwise moment. If there is an imbalance, the system will accelerate. The diagrams below show some examples.
In the third diagram the whole horizontal bar is considered as a single object acting at a point with a weight W7. This point is called the centre of gravity. We could have considered the bar in two parts, to the left and to the right of the pivot, acting in opposite directions, and we would have obtained the same result. We could even have considered that the bar is like the first two examples, completely balanced, but with a piece missing, and we could have treated this an object with negative mass, or as a positive mass acting on the other side. Tricks like that can sometimes make the calculations simpler.
Note that using the centre of gravity is acceptable for this type of static working, but for rotating objects it will not work. If a skater is spinning with arms outstretched, and the arms are brought in to the body, the rate of spinning increases, although the centre of gravity has not moved.
The next diagrams are intended to emphasise the fact that the distances used in calculating moments are always perpendicular to the forces.
Note that when we calculate moments, the point used need not be an actual pivot; if something has been done wrongly, and the forces are strong enough, they may turn the point into a pivot by bending or breaking something. Note also that although the forces F8 and F9 are balanced in rotation, they both have a horizontal component to the right, and so the fixed part of the mechanism, and its fixtures to the base, have to resist being moved sideways and rotated clockwise.
To summarize - for a structure in to be in equilibrium the forces must add up to zero in all three dimensions, and the moments must add up to zero about axes in all three dimensions. Structures are seldom as static as we might like: live loads, and wind and water, provide transient forces that may induce oscillations, which are usually well damped. Suspension bridges, for example, may as susceptible to torsional oscillations, involving rotation, as linear one, involving only translation.
The next example is more practical, though based on an old object.
These pictures of an old crane illustrate the way that the moment at the base varies with the position of the load. One indicator shows the relationship of the maximum load to the radius, as the jib is : the other shows the variation with angle. The distance of the load from the base is varied by raising or lowering the job; this is called derricking or luffing. In the last picture, if you calculate the products of load and radius to get the moments, you find that these are not constant - 100, 100, 96, 84, 70. Why is this? It is because the jib has weight, and as the radius is increased, the moment of the jib increases. From the figures, we ought to be able to calculate the weight of the jib on the assumption that its centre of gravity is halfway to the load. We also need the horizontal distance from the base of the jib to the ground wheels, but with the number of data available, we can eliminate it. The calculation would only be approximate, because a safety factor would have been applied. Even so, the static safe maximum load might not be as it seems. Suppose you are just below the limit, and you try to suddenly jerk the load off the ground. This might be enough to topple the crane.
In the diagram below, the variables are -
L = weight of load
J = weight of jib
C = weight of remainder of crane.
Then the critical equation is
L x M + J x K > C x D,
for the crane to remain upright with no safety factor.
This picture illustrates the problems of building a cantilever. In order to hold the sunshade out at a distance from the base, both it and the arm have to be as light as possible. The arm is made of plastic tubes, giving rigidity with lightness. The base is filled with heavy material. A look at the base reveals that the turning moment has begun to pull the support out of the vertical where it is fitted to the base. This of course moves the centre of gravity of the system further from the base, thus increasing the moment. In other words, the cantilever does not in this instance possess intrinsic stability. The attachment to the heavy base was supposed to be rigid enough to create the stability. Of course, like the Tour Eiffel, this sunshade would be in its lowest state of energy when lying on its side. One aim of the designer is to ensure that between the upright position and the prone position there is a higher energy state that cannot be reached with any input of energy that could come from sources such as the wind. This is true of most structures - they must be able to withstand any likely application of forces without moving unduly.
The cantilevers of the Forth bridge are designed in the same way. What makes the sunshade and the bridge stable in practice is that if they are tilted, the centre of gravity has to rise before it can fall. The job of the designer is to ensure that for all foreseeable loads, the system remains on the rising part of the force-displacement curve. Should a falling part of the curve be reached, collapse is inevitable.
People carrying heavy suitcases in one hand lean over in order to increase their stability. This is especially necessary when their weight is on the foot further from the suitcase.
"Obviously" a giraffe is less stable than a dog or a person. But if a giraffe should begin to fall over, it has more time than the smaller animals in which to recover. It is very likely that giraffes hardly ever fall over. So we must beware of transferring ideas about static objects to those with active control. Many people "know" that an aerodynamics expert once said that bumble-bees could not possibly fly. It is far more likely that he said that he didn't understand how they could fly - not the same thing at all. Throwing a scale model of a bumble-bee teaches us nothing, because the bee relies on active feedback for control.
Cantilever bridges can often be built out from the supports without blocking the channel below. Then the suspended span can be lifted into place. Because there is no rigid connection through the bridge, small vertical movements of the foundations are not as dangerous as they might be with other bridge types.
Note that the connections at the two towers can all be hinges, but if more than two cantilevers are to be strung together, only the outer two towers can be hinged. The central pier and cantilever of the Forth railway bridge can stand alone. Another solution for multiple spans is a continuous beam, which may look at first sight like a row of shallow arches or cantilevers if it is haunched. Waterloo bridge in London is of this type.
Concrete Cantilever Bridges
Several examples of this bridge are found in Gloucestershire and Wiltshire. More about these later.
The third picture distorts the shape of the image unpleasantly, but it does show the joint where the suspended span is hung. That span does not need to be deeper at the ends than the middle, but the bridge would look rather odd if the suspended span were deeper in the middle, which it ought to be, in response to the bending moments. This is discussed in the page about beams. Many road bridges are treated in the same way. The suspended span is made of seven separate beams for ease of transport and assembly.
Looking at the Forth railway bridge reveals the shape of a suspended span when engineering considerations are allowed to rule, as they have to in large structures. On such a scale, it would be difficult to do otherwise.
This is a very unusual concrete cantilever bridge, the Geelong sewerage aqueduct, in Australia. The pictures are shown here by kind permission of Stewart Beveridge. The aqueduct is continuous, of necessity, which makes this bridge very unusual, as almost all cantilever bridges carry suspended spans which are supported at their ends by attachments that allow rotation. Another unusual feature is the use of a rather complicated design, reminiscent of the Forth railway bridge, on a small scale.
Here is a slender cantilever footbridge made possible by the use of steel stressing wires inside the suspended concrete span. It is a typical type used over a dual carriageway. As the road is in a cutting, the designer did not have a problem connecting it to the footpaths on either side.
|Here are two solutions to the connection problem, long straight ramps and helical ramps, though neither bridge uses cantilevers.|
If you recognize this as a footbridge over the M6 motorway, you may be puzzled by the lack of traffic. The picture was taken near mid-day, as the angle of the shadow shows. It was taken during a normal working day. In 1968. In this example the designer opted for steps on each side for access to the bridge. Sometimes ramps are used in a town, because the bridge will be used by people with push-chairs and wheel chairs. Designing these ramps is no easy matter - they need to be quite long if they are not to be too steep. Making a tidy job of this and integrating the ramps into the design has been attempted using long straight ramps, folded ramps, and helical ramps.
The picture below shows the first Thelwall bridge, over the river Mersey and the Manchester Ship Canal, east of Warrington. Recently it was refurbished, and a new bridge was built alongside to cope with the huge increase in traffic. This bridge has many welded plate girder beams, and a riveted cantilever span of about 335 feet over the canal. The river span is about 180 feet. The total length is about 400 feet, with about 36 spans. The first bridge was completed in 1963, and the second in 1998.