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.  

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. 

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 pu 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 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.

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 

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.

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.