Moments are central to structures, especially cantilevers. The moment of a force measures its effectiveness for rotating an object around a point. A spanner or a wrench is designed to increase the moment of the force applied by the operator, making it possible to turn the nut. A screwdriver has a fat handle for the same reason. The word torque may be more familiar.
The moment of a force about a point is the product of the magnitude of the force and the perpendicular distance from the point to the line of the force. The spanner in the picture is not being used as effectively as it could be, because the force is not perpendicular to the spanner. If the spanner were held at the end and pulled at right angles, the moment would be greater.
Note we are not obliged to calculate ("take moments") using an actual pivot: we can take moments about any point. For an object that is stationary or rotating uniformly with constant speed, then the sum of the moments of all the forces acting on it, about any point, is zero.
Here is an example of a poorly designed and overloaded bracket. The moment of the weight has been too much for the strength of the metal to withstand, even after someone has stuffed a piece of metal in to try to rectify the situation. This example is only a minor irritation: a bottle of corrosive, toxic or inflammable liquid sliding off would be dangerous. The collapse of a cantilever (a giant bracket) of the Quebec bridge was a human tragedy; the collapse of a large dam could be a human catastrophe. But the principle is the same: design it right, build it right, use it right, and maintain it right.
What has happened in the metal of the bracket is shown in the picture below?
The pressure on the vertical flange was resisted by compressive and tension forces which together created a bending moment that opposed the externally imposed one. But when the external moment became too great, the flange reached its yield point before it had built up enough stress to oppose the imposed ones. And so it bent.
The next two diagrams illustrate the effect of the distance between forces.
Let us work out the conditions for equilibrium.
Taking moments about A, FB X G = W X J, so FB = W X J / G
Taking moments about B, FA X G = W X J, so FA = W X J / G
Taking moments about C, FD X G = W X J, so FD = W X J / H
Taking moments about D, FC X G = W X J, so FC = W X J / H
The horizontal forces FA = FB and FC = FD, as is necessary for horizontal equilibrium.
We can also work out the ratio of FC to FA, as follows.
FC / FA = (W X K / H) / (W X J / G) = (K X G) / (J X H)
Since K is less than J, and G is less than H, we can see that FC is smaller than FA.
These diagrams use vectors, which are lines representing the strength and direction of forces. In the upper diagram a bar is attached to a wall, making a cantilever. The wall must exert an upward force to counteract the weight of the bar, both being drawn green. But this is not enough - these forces act in different places, and there is a torque, or couple, tending to rotate the beam clockwise. In order to stop the bar from turning, other forces need to be applied. If the bar is attached only at top and bottom, the forces will be as shown in red. Both are much bigger than the weight. This is because the bar is narrow and the forces are close together, almost cancelling out. The moment of a force about a point is the product of the size of the force and the closest distance its line makes with that point.At the position of attachment, and indeed elsewhere, the bar is subject to stresses which bend it. The moment of the forces is called the bending moment. The beam bends until the internal stresses just counteract the bending moment.
In the lower diagram, the beam is tapered. This has two beneficial effects - the centre of gravity is moved nearer to the wall, reducing the bending moment, and the greater depth at the wall means that the required moment can be had with smaller forces. Why is the beam tapered and not fatter all the way? Because as we go out from the wall, the remaining beam is both shorter and lighter, and so the bending moment gets less. So less thickness is needed to keep it reasonably rigid. And the taper in turn decreases the weight even further.
Why is there a horizontal thrust in an arch? We can look at this in several ways, of which one will be given here. The diagram below shows a simple arch. In the second part we imagine the two halves of the arch being separated. (Artists imagine things, so why not engineers?)
The weight (green arrow) tends to produce an anti-clockwise rotation about the abutment, so we need a force to counteract it (red arrow). Thus the two halves push each other with just enough force to stabilize the system. But there is a problem. With the two forces shown, the half-arch would accelerate upwards and rightwards: there must be something more, as we see in the next diagram.
Now we have no net force in any direction, and no rotation. But wait: we still have a problem: the arch is pushing on the abutment, but what pushes on the abutment to stop it sliding? The foundations, which themselves are pushed by the ground. Does the series of forces never end? No, it doesn't. The forces just spread out and the stresses become weaker as the forces spread out.
The word "rigid" is technically wrong. In practice no object is rigid - all objects deflect under stress, giving a strain. So the structure must be stiff enough for both static and dynamic effects to be negligible. Dynamic effects must never be neglected - oscillations under wind or load can be very destructive. The non-rigidity of objects is closely related to the ability of structures to provide the required forces. Any object which is fixed at some point, and subjected to a stress, will deform or deflect until it produces forces that oppose the ones that were imposed on it, assuming that it does not break before this point is reached.
Given that each point in the beam experiences both vertical forces (weight and support) and horizontal ones (reducing bending) we can see that the size and direction of the forces in a beam can be quite complicated. In fact in large structures, much of the material can be dispensed with, leaving only narrow members, as in the case of a Truss.
In practice, cantilevers would not normally be built out from a support as shown - much more often a balanced beam is made. The next diagram shows an idealised cantilever bridge, haunched in response to bending moments, and with a curved suspended span for the same reason. Click on the link in the diagram to see a photograph of a real example.
|If the outer arms of the cantilevers
are long and heavy enough, even the maximum live load will not cause the inner
arms to tip downwards. But if the outer arms cannot supply enough moment
by their weight,
they have to be held down to the piers. The outer cantilevers of the Forth
railway bridge have large weights inside the piers to provide the necessary
Many motorway bridges are built using simple cantilevers, but the suspended span is usually shaped to continue the line of the cantilevers, giving a better appearance, and avoiding a hump in the upper road.
A world in which only engineering functionality was considered important would be very dull. That is not to say that ornament should be added to bridges unthinkingly. Many engineers have in fact shown that a "functional" structure can be elegant in its own right.
Cantilevers can often be built out from the supports without blocking the route 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 four piers can all be hinges, but if two or more cantilevers are to be strung together with suspended spans, only the outer two 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.
Here is a cantilever bridge of a type which is found in several places in Gloucestershire and Wiltshire. This one crosses the Barnwood bypass, east of Gloucester. There is another one about a kilometre to the east.
Here is a slender cantilever footbridge made possible by the use of steel wires inside the concrete. See also pre-stressed bridges.
The principle of the haunched beam works just as well for towers. The Eiffel tower has well spread legs to withstand the moments caused by the wind. So do Eiffel's magnificent arch bridges.
If the first Firth of Tay railway bridge had had piers with wide bases, it would have had a better chance of surviving the ferocious wind that blew it down. In fact there were other flaws in the design, and poor construction techniques as well.