Momentsc

Moments are central to structures, especially cantilevers. The moment of a force measures its effectiveness for moving 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.

These diagrams use vectors, which are lines representing the strength and direction of forces. In the upper diagram a beam is attached to a wall. The wall must exert an upward force to counteract the weight of the beam, 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 beam from turning, other forces need to be applied. If the beam 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 beam 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, the beam is trying to bend - the moment of the weight is called 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 thick 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.

The diagram below shows the bending moments along a cantilever caused by point loads at nine different distances from the point of attachment at the left. The free end of the cantilever is at the right. The moment at the attachment clearly increases as the load moves out. This principle is used in the steelyard.

As the moment for any load is greatest at the support, we can see why a cantilever needs to be widest at that point. In fact, if we consider a cantilever of constant depth, we can learn about the moments caused by its dead weight by adding together a lot of these graphs. To do the calculation properly we would use integral calculus, which is essentially doing an infinite number of additions in a finite time.

The diagram below represents the sum of a set of diagrams for point loads. It shows clearly how the bending moment varies along a cantilever which is fixed at the left and free at the right.

This is why a cantilever can be tapered, often strongly, as in the Forth railway bridge.

The tapering of a tree trunk and branches may be thought to correspond well to this effect. But some coniferous trees and palm trees do not taper very much. If all the branches are near the top, the flow of liquids in the trunk needs to be about the same all the way up. This may influence the growth.

More about moments can be found by clicking here to see the moments page, which explains more about cantilever design. A moment is more or less what people mean by leverage.

The wing of an aircraft is a good example of a cantilever.

On the ground, the wing root of a 747 has to support the weight of the wing, two engines, and some fuel. The wings, of course, are not simply bolted to the fuselage - a beam goes right through and forms a strong unit with the two wings.

In the air, the situation is reversed. Now there is upward lift on each wing, with the weight of the fuselage pulling down in the middle. The bending moment is greatly reduced by using the wings to carry the engines and much of the fuel.

Here is an unfinished example of a haunched beam bridge made using I-section steel. This is one of the new bridges near Over, west of Gloucester.

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, 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.

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 forces.

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.

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 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.