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Connecting Things Together

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Looking at a large structure may not reveal all its secrets, especially in respect of the ways in which parts are connected together. Let's take a simple example - a 1 km steel bridge in a region where the temperature can vary from - 30 C to + 30 C, a range of 60 C.

A typical linear coefficient of thermal expansion for steel might be about 10-5 / C, so a 1 km structure subjected to a change of 60 C will expand by 1 km X 60 X 10-5, which is 1000 X 60 X 10-5 m, or 0.6 m.  This is not only a large distance in terms of geometry, it is large in terms of possible stress on the structure and its supports.  The strain is 60 X 10-5, or 6 X 10-4, and if we assume a Young's modulus of about 200 GN / m2, we can see that the stress is large, being about 6 X 10-4 X 100 X 109 = 6 X 107 N / m2. If expansion and contraction are ignored, breaking or buckling may occur, as happens with railway tracks during periods of exceptionally high temperature.

The designer's problem is to allow the expansion to occur, while precisely defining the position of every part of the structure.

What do we need in order to define the position of an object? In three dimensional space, we need to constrain it in all three dimensions. In addition, we need to prevent it from rotating. How many independent axes of rotation are there? The position of a point on the surface of the earth is defined by two angles, latitude and longitude, so we can define the orientation of an object by two angles. Instead of angles, we can define the positions of three points on the object, though we don't need three coordinates in each case.

We must, however, distinguish carefully between geometrical positioning and the requirements of an actual system. For example, the photographer and the surveyor are satisfied to use a tripod to support their apparatus, which is relatively small. The three feet of the rigid tripod exactly determine the position of the apparatus at the top. A snooker table too, could be positioned exactly with only three legs, but this is never done in practice, because it is very heavy, and not sufficiently strong and rigid to cope with a three point support. The practical method is to use six legs, with adjustments such that on the given floor, the playing surface is as close to plane and horizontal as may be obtained. We must always distinguish between geometry and engineering.

On the other hand, the heaviest part of a "grand" piano is a strong steel frame which is very roughly triangular, and three legs are enough.

Like a snooker table, a bridge span is rarely supported at only three points, two at one end and one at the other, for example.  Almost all bridges are supported by at least two points at each end, and sometimes more, especially wide bridges with shallow decks.  On the other hand, at intermediate places on the bridge, single columns may indeed be used, relying on the torsional stiffness of the bridge to take twisting forces back to the the ends.  Such a design is possible with a deep box girder, whether of concrete or steel.

The diagram below shows an example with a roller at each end, the left hand roller being trapped by metal plates.  The right hand roller allows for thermal expansion.  Having allowed for expansion, we then have the problem of the gap at low temperatures.  Even a small gap will produce a noticeable sound as rubber tyres traverse it.  Sometimes metal combs are used to provide some continuity of support.

Beam1RollerA.gif (2665 bytes)

If the structure is curved, as in an arch or a suspension bridge, changes of temperature may be allowed to change the curvature, though the change in length of the deck must still be accommodated.

The next example is a little more subtle.  The diagram represents the way that a suspended span might be attached to the two cantilevers of a large steel bridge.

CantileverLink.gif (3549 bytes)

The arrangement in the top diagram will support the span, but will let it move longitudinally, so we must add one link as in the second diagram.  In practice, for the sake of appearance, three more links are often added on each side of the bridge, but they should be slotted to allow for expansion.  They make the bridge look like a continuous span.  Irrespective of these considerations, the railway track or roadway must continue across the joints, in a way which allows for thermal expansion.

These two examples give only a very simplified insight into a very complex area of design.

A suspension bridge, because of its flexibility, might be thought to present fewer problems than other bridges, but as always, we have to think carefully.  The central part of the deck of one the very longest suspension bridges may vary in vertical position by as much as three metres from summer to winter, as a result of changes in the length of the main cables.  That is not much, compared with the huge span, but we have to consider also the hangers.  As the curve changes, will the deck move in such a way as to share the forces among the hangers in the way that the designers intended?  Luckily the stiffness of the deck renders it relatively rigid over only a small fraction of the span, so it can follow the changes in the main cable rather well.

See also attachments.

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