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Cantilever Bridges

In the arch and beam we saw that the bridges were supported at two places - the ends.  In fact, if you want to hold any object in position in two dimensions, you always need two points of attachment. In  three dimensions you need three points. Two-legged animals need feet in order to achieve stability in the three dimensions. This statement is not strictly true, because animals can balance using their muscles to change their position in response to perceived movement. Nevertheless, feet make the process easier.

FrogCantApril.jpg (68616 bytes)FanCantApril.jpg (111079 bytes)CantBooks.jpg (111675 bytes)A cantilever differs from the arch and the beam in that the attachment points are not necessarily at opposite ends. The cantilever is rather like a bracket, projecting out into space. The two forces almost always act in opposite directions. In the lower half of the first photograph, the oscillation in the wind is revealed by the longer exposure. Whenever there is both mass and elasticity, there are natural resonant frequencies. The second photograph shows a vertical cantilever deflecting in a wind, with oscillation in the right hand half of the picture.  

The diagrams below show two basic types of cantilever, though in fact the second includes the first - above the support pier, there are forces like those in the first example, holding the two arms together, as we see in the third diagram. The connection to the pier may be a hinge or a rigid support. The central cantilever pair of the Forth bridge has to be supported rigidly, because both cantilever ends are free. Almost all other cantilever bridges have only two pairs, each of which has a fixed, end, and therefore hinged supports are sufficient, except during construction.

 

As with the beam, the bending stresses and shear stresses vary throughout the structure.

A cantilever is really a large bracket, held rigidly at one end. Here are pictures of some examples.

PylonSmall.jpg (35649 bytes) ForthRailCC.jpg (24245 bytes) QuebecSideC.jpg (40690 bytes) ZoonsCC.jpg (28191 bytes)

PylonCant.jpg (112599 bytes) CraneZCC.jpg (55000 bytes) TVTowerAXC.jpg (52542 bytes) WindCantA.jpg (127978 bytes)

Most cantilever bridges have two cantilevers, with a beam suspended between their free ends, like the example shown below, typical of many motorway bridges, usually built from reinforced concrete or pre-stressed concrete. The largest cantilever bridges are made of steel, though medium sized ones are sometimes in pre-stressed concrete. Ancient ones in Asia were made of wood.

The cantilevers can be maintained in position in two different ways. Firstly, they can be supported by pivots or hinges at the balance point, with the fixed end held in place at the abutment; secondly they can be supported at the balance point by a tower with a base so wide that no practical load can tip the structure. The central part of the Forth railway bridge is of the second type, which is why it has a wider tower than the outer parts. Most cantilevers are of the first type. In the first type there are two ways of holding the structure in position. One is to make the anchored span so heavy that no practical load at the free end can tip the structure. The other is to fix the anchored end to the ground. The outer ends of the Forth railway bridge are high up on masonry piers, which cannot withstand tension. The steel structures therefore have heavy weights attached, which hang down inside the piers. These weights are so heavy that the spans cannot be tipped by any likely load.

QuebecSide2A.jpg (55318 bytes)The longest cantilever span is the centre span of the Quebec bridge, in Canada, part of which is shown here.

Here are some pictures of other cantilever bridges.

Haw1.jpg (38141 bytes) ForthEntireAS.jpg (309037 bytes) Zoons2.jpg (26988 bytes) FBG.jpg (26361 bytes)

For more pictures, please click here.

Advantages of cantilevers

Building out from each end enables construction to be done with little disruption to navigation below. The span can be greater than that of a simple beam, because a beam can be added to the cantilever arms. Cantilever bridges are very common over roads.  Because the beam is resting simply on the arms, thermal expansion and ground movement are fairly simple to sustain. The supports can be simple piers, because there is no horizontal reaction. Cantilever arms are very rigid, because of their depth.

Disadvantages of cantilevers

Like beams, they maintain their shape by the opposition of large tensile and compressive forces, as well as shear, and are therefore relatively massive. Truss construction is used in the larger examples to reduce the weight.

Download a program about building a cantilever.

Click here for a list of the longest cantilever spans.

For more information please see the main pages about cantilevers.

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More notes about stresses in cantilevers

The diagram below represents a rectangular beam which is balanced on a pivot at its centre, so that each half is a cantilever.

The two diagrams below represent the distribution of shear forces and bending forces along the structure. The discontinuity in the shear graph is the result of the vertical force at the pivot. To understand why the shear changes sign at the centre, imagine a vertical cut suddenly being made at any point. The part of the system that is furthest from the pivot will accelerate downwards. Since in reality, with no cut, there is no acceleration, there must be forces to prevent this. We can see that to the right of the pivot the relative motion of the parts will be clockwise, while to the left it will be anti-clockwise. These directions are distinguished by giving them opposite signs. If the cut is nearer the end, the weight of material that has to be held by the shear forces is less: this is way the shear forces fade away uniformly.

The next two diagrams show the distribution of these forces within the structure.

If we ignore live loads it is clear that with weaker forces towards the ends, less material is needed to resist them. This is the reason that so many cantilevers are tapered. The change of shape in fact changes the force distribution, and exaggerates the variation.  

forthcentral.jpg (212874 bytes)TVTowerAX.jpg (58080 bytes)The shape of the Forth railway bridge reflect these force distributions quite well, if we remember that the top chord is in tension, and the bottom chord is in compression. Not only is the overall shape tapered, but the main members taper as well. In any trussed cantilever, there are no members in shear: what happens is that the shear forces in the solid structure are represented in the truss by the forces in the sloping struts and ties. If the top and bottom chords were joined only by vertical members, the structure would tend to sag. In a sense, the purpose of triangulation is to prevent change of shape, which in a solid object is associated with shear. A similar general shape is presented by the transmitter mast, which is a cantilever against wind forces.

Another way to illustrate the stresses is by using three dimensional graphs like those below, where the sideways displacement of the drawing represents the magnitude of the stress. The upper picture represents shear, and the lower represents bending. The discontinuity is the result of the action of the central support. None of these pictures is of any use when we need to look at the stresses in three dimensional objects.  In those cases we either need a mathematical expression or a matrix of numbers.

    

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