Force

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Bridges, like other structures, are designed to exert forces at required places. The Firth of Forth rail bridge, for example, has to exert an upthrust on a train which is exactly equal to the weight of the train. wherever the train may be on the bridge. The balancing of the upthrust and the weight is automatic, because the weight of the train deflects the bridge until the bridge produces the right force. It is of course possible that a structure might break or buckle before it  reached the required force. That is the meaning of "not strong enough".

What are the forces on a bridge?

Among the external forces are -

Vertical - reaction of the supports to the weight of the bridge and live loads

Vertical - weight of traffic

Vertical - lift or down-force caused by wind

Longitudinal - reaction to thrust of an arch

Longitudinal - reaction to the tension of a suspension bridge

Longitudinal - force generated by traffic, especially if accelerating or decelerating

Transverse - wind pressure

Transverse - pressure of river water or tidal water

Transverse - forces generated by traffic on a curved bridge

Many forces are intermittent or erratic, such as turbulence in the air, and vibration from live loads, and of course, simple variation of traffic.

A subtle variation is that which occurs during construction, a period in which forces may be very different from those in the final structure.  This can be a significant effect during the building of long cantilevers or box girders.  Jacking may be necessary during construction.

Simple ideas about forces

If two objects are in contact the forces between them are equal and opposite.

If an object is not accelerating in a given direction, the sum of the forces on it in that direction is zero.

The minimum force needed to support the weight W of an object is W, obtained when the object is supported by a vertical pillar or a vertical cable.

Supporting a weight W by means of a force applied anywhere but at the object requires a total force greater than W.  Even if the supporting forces are parallel with the weight, there is the weight of the intermediate members to consider.

Tension members can be much thinner than compression members.  The rope that held up the bust while it was being lifted and positioned was very much thinner than the thinnest pillar that would safely support it.  This results from the asymmetry of the two cases - a taut rope aside and let go returns to its original position - but a compression member is liable  to bend more when moved

Supporting a weight W by means of forces applied in a direction other than directly opposing the weight requires a total force bigger than W.  The picture below shows one support of the Severn bridge near Aust, before it was painted white.  

 

Severn1.jpg (29757 bytes)The force at the bottom of one leg is the weight of one leg plus the weight of one quarter of the deck and one quarter of the cables.  Each cable at the top has to support the weight of itself plus one quarter of the deck.  Because of the shallow angle of the cable the force in it is very much greater than the force in the leg, yet the leg is huge compared with the cable.  This is because a strut can buckle, whereas a cable remains straight under any applied force, until it breaks.

 

Forces during assembly

The structure has to be built and assembled. This may require as much ingenuity as  the design of the bridge itself.  It will also require special equipment.  While the bridge is incomplete it may be vulnerable to oscillation or deflection  in ways which would not be possible after its completion.  The Pont de Normandie is a good example.  A number of box-girder bridges collapsed during construction until the stresses were well understood.  This type of event is sometimes the price of innovation by extrapolation.

An extreme case is provided by the erection of Sydney Harbour bridge.  During construction the two halves of the arch were held back by cables at the top chord of each half.  So in addition to the normal arch thrust at the abutments, there was extra thrust caused by the pull of the cables.  The arch itself had to withstand the extra thrust.  On the other hand, it was not carrying the deck at that stage.

When a structure is partially completed, the forces may differ substantially from the final ones, resulting in different deflections also.  Progressive jacking may be needed during erection in order to compensate, as in the case of the Tour Eiffel.

The structure has to be maintained.  Painting the Forth bridge is an archetypal symbol for something that can never be finished.

To find out more about the forces in structures try Stresses and Strains.

The easiest case to understand is that of two equal and opposing forces.  The Forth bridge could in principle consist of a ship which carried the trains across the firth.  This is called a ferry.  The great advantage of it is that it produces an upward force just where it is wanted, and no bigger than the weight carried.  The disadvantages are the times needed for loading and unloading, and the slow speed of crossing.  That is a great motivation for building a bridge.

 

Kodaly2.jpg (46571 bytes)In this picture of a bust on a  plinth, the two objects have been separated to show the forces.  The two white vectors represent the weights of the two objects.  The two black vectors represent the reactions needed to support the object.  The plinth must push up against the bust with a force that is equal to the weight of the bust.  The ground must push up against the plinth with a force that is equal to the sum of the two weights.  

A bridge could consist entirely of plinths with no gaps between them, supporting a road.  This would in fact be a wall.  The problem with it is that it would block up any road, railway or river that crosses its path.  The design and construction of bridges amounts to the solution of the problem of getting a road, railway or canal over a road, railway, canal or river, with no unwanted impediment to the lower traffic, for minimum cost of parts and construction, including the costs of disruption of traffic.

The plinth or wall is the simplest possible support, and it contains the simplest possible forces - vertical compressions.  But where it is connected to the ground, things are not so simple.  If we imagine a pillar sitting on a weakly elastic substance like foam plastic or rubber, we see that the substrate will be deformed.  The change of shape suggests that there may be shear stresses, and the curvature of the top surface means that it has lengthened, so tensions must be present as well.  In fact, the stresses in the foundations can be more complex than those in the actual structure.  But they need to be understood if a safe system is to be constructed.  The next two pictures simulate a pier standing on a weak substrate.  In the second picture the yellow line reveals the range over which the deformation occurs.

PushFoam.jpg (25981 bytes) PushFoam2.jpg (26447 bytes)

Real rock or soil deforms much less than this, though the behaviour of soil can be complex.  Nevertheless, we see clearly that the stresses and strains do not stop where a structure reaches the surface.  In any case, the support must deform so that at any point, the forces are completely balanced.  And of course, the elastic energy stored in the medium must be equal to the potential energy lost by the movement of the object.  Is this second statement correct?  If not, why not?  

Let's imagine that everything is perfectly frictionless, and we let go of the structure when it is level with the surface.  It will sink in, but when it has reached what ought to be the equilibrium position, it will have kinetic energy, and will continue to move.  The system will in fact oscillate.  Real systems are not perfectly elastic, and energy will be changed into heat, and the oscillations will die away.  So the final energy is not equal to the initial energy, once the heat has dissipated into the surroundings.  So what fraction do you think remains?

Forces in an arch

The big disadvantage of a bridge is that the supports are not usually where the load is.  This means that there are structures in the bridge that carry the load to the supports.  The forces in these are by no means parallel to the load, and are therefore larger, often much larger, than the load itself.

Here is an example - one pier of a viaduct near the east channel of the River Severn at Gloucester.

SR3Z1.jpg (52966 bytes) SR3Z2.jpg (53549 bytes)

Vectors

In the left picture the arrows represent the direction and magnitude of the static loads.  They are called vectors. The vertical line represents the weight of that part of the bridge that the pier carries, two halves of arches.  The sloping vectors are the thrusts of the arches.  Because they are sloping their total length is more than the weight, which is equal to the sum of their vertical components.  Their horizontal components cancel out.  This means that for static loads the piers need not be very wide, as the transverse loads are carried through the arches only.  

The second picture shows the case of a live load on the right, in this case a locomotive, as this is railway viaduct.  The force in the pier is now not vertical, because the horizontal components of the thrusts do not cancel.  The line of thrust must of course remain inside the pier, right down to the foundations, or it would fall.  In fact the line of thrust should stay well within the pier, to prevent tension at the far side.  This example is exaggerated, but it shows how the line of thrust in the pier would tend to move with the passing of a train.  In fact the next arch would probably provide an increased thrust to counteract some of the live load.

The weight of the masonry above the pier also serves to make the force more vertical.  In medieval cathedrals, large pinnacles or statues were often placed on the piers of flying buttresses, for same reason - they tended to keep the force within the pier by steepening the line of thrust.

It is often convenient to resolve a vector into two components at right angles.  The coordinate system is chosen for utility.  The directions could be horizontal and vertical, or along a component and at right angles to it.  If the force is F and the angle is A, the components are given by 

F cosine(A) and F sine(A), shortened to 

F cos(A) and F sin(A),

as in the diagram below.  Two components at right angles can be added in quadrature, that is, by adding their squares and taking the square root of the sum of the squares.

(F cos(A))2 + (F sin(A))2 = F2

VectorsX.gif (62889 bytes)

WireVectors.jpg (74662 bytes)This picture shows a very simple structure.  Can you work out what is going on in all the parts?  Consider, for example, a plan view.

Roman arches

The ancient Romans built magnificent arches, but they did not often use the possibility of transmitting thrust along a line of arches.  If one arch was swept away by flood or destroyed by an enemy, the others would stand without it, because the piers were so wide.  Looking at the the great aqueducts they built, you wonder whether the structure would survive the loss of one arch.

Components of a force

The next two diagrams show how the rise of an arch affects the outward thrust.  Any force can be considered to behave as the sum of two or more components, as long as these are added in the correct way.  As an example, for an aircraft in steady flight the total of lift, weight, thrust and drag add to zero.

In the diagrams below the outward force exerted by the arches has been resolved into components at right angles.  The vertical component must equal the weight.  The horizontal part is clearly bigger for the flatter arch.   The thrust of an arch has  to be  resisted by the ground.  Soft ground is therefore very unsuitable for an arch, unless the abutments are built to spread the thrust over such a wide area that the ground can resist without significant movement.

 

Arches3XA.gif (5681 bytes)

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