Since there is no means of resisting outward thrust, the object must be a beam.  Suppose we now bring up material from each side, until it just touches the beam on each side, as below.  Do we now have an arch?

If the new abutment only touches the beam, and does not provide any force, then the object remains a beam.  But what happens if we place a load on the span?  The new stresses will push the beam downwards, and the ends will move outwards, and the load, though not the beam, will be supported by arch action.  How can we make the bridge itself act as an arch?  Look at the diagram below.  It shows compression as red, and tension as blue.  Obviously there are stresses throughout the beam, but the diagram only shows where they are strongest.

To create arch action, it’s no good just adding material as we did before: we have to create thrust.  But how much?  Imagine the beam lying on its side on smooth level ground.  Now it will have no bending forces in it.  What we now have to do is measure it, and provide abutments that exactly fit.  How is this different from what we did before?  Look at the diagram below.

We see that when it is subjected to its own weight, it slightly straightens out, and is slightly longer between abutments.  The diagram is of course greatly exaggerated.  So when we make the abutments to fit the weightless arch, they will provide thrust when we add the weight.

But people seldom build arches and then put them in place, apart from tied arches.  So how do these observations affect a real arch? Many arches are built on falsework known as centring, which is removed when the arch is complete, thus turning the structure from  a beam into an arch in a short time.  But we already saw that the required stresses are produced by a slight shortening of the span.  But we aren’t shortening it.  So how does the structure create the right forces?  It does it by sagging slightly.  But when it does that, are the stresses what we want?  Perhaps we should have placed the abutments even closer together.

Some steel trusses, such as the Sydney Harbour bridge, are made by cantilevering them out from each side, and letting the two halves push each other only when complete.  But this raises another question.  In our unrealistic thought experiment we introduced forces to change a beam into an arch.  But suppose we introduce too much thrust.  The result is shown below, along with the original result.

So we can have either the top or the bottom in compression or tension.  This means that it is no simple matter to complete a big truss arch with top and bottom members vertically well separated top and bottom chords.  What if the temperature is outside the expected range.  What if in fact the gaps are not as predicted.  It’s no good just letting the two haves push as they may.  We have to control the forces in the top and bottom chords.  This can be done by inserting suitable spacers before closure.

If we look at the three shapes above, we see that the ends in the three cases are not parallel.  This raises the question, given the deflection that happens when the falsework is released, of just what is the correct shape of the abutments.  In fact, many structures are provided with jacks, which can be adjusted as often as required during construction.  

And what happens if the end of the arch does not exactly fit the abutment?  Something like this –

The obvious thing to do is to make it more accurately.  But what happens when changes of load or temperature cause deflections?  

The next diagram shows very roughly how forces spread from a small area, rather like a rumour spreading among a crowd of people, or water flowing from a small pipe into a trough.

Instead of trying to perfect the alignment, why not accept the situation, and make something like this –

We have a hinge.  Now, when deflections occur, the connection can accommodate them without extra stress.  Note that the lines of force meet the surface at right angles.  Why must this be the case?  Of course, friction between structure and hinge could change the lines.  This looks vaguely like the way you might imagine the gas emerging from a rocket engine, emphasising the apparent similarity of fluid flow and stress flow.   There is, of course, no turbulence in stress – all is laminar.  Nevertheless, there are critical phenomena, such as the onset of buckling.  In fluid flow we have the transitions from laminar to turbulent flow, and from subsonic flow to supersonic flow. 

So we might conclude that it is not sufficient to make a structure in the shape of the funicular – it is necessary to control the stresses in detail.  One virtue of a truss is that to a first approximation the forces are along the members.  This makes calculation simpler.  It is perhaps one reason why trusses are so popular for model bridge competitions.

An example of a connection with a large area is the British and Irish three-pin power plug, which is very large compared with those of many other countries.  These include the USA, where the voltage is less than half as big as in Britain, and currents are therefore much higher.  The large flat pins probably touch the mating conductors over a relatively small area, and so the bulk of the metal is not doing anything useful.

Having referred to the virtues of pins, we might ask about the opposite strategy – using large connections.  Are there advantages?  Yes.  We can, for example, use them to transfer forces from one part of a structure to another, in such a way as to reduce the variation in bending moment or some other effect.  An early example was Robert Stephenson’s tubular bridge over the Menai Straits, in which the connections of the four sections of each tube were made in such a way as to spread the stresses more evenly than they would have been with separate beams.

More information about these topics can be found in arches, beams, funicular and indeterminacy.

Is the object in the next picture an arch or a beam, a mixture of the two, or something else?

The next picture shows a construction that is definitely not an arch.  It is just a pile of planks.  This is related to the corbelled arch.

Here is our original beam, with a tie across the bottom, and vertical ties to take the weight of the main tie.  This is now a tied arch, which can be built off-site and moved into place if required.  The arch is of course much too thick: it could be a lot thinner, as it does not have to produce beam action, except for the live load.

The next picture shows a structure with sloping ties and struts.  Because the triangles introduce rigidity, the top chord can be thinner than before, and does not have to act as an arch or a beam.  Its individual sections act as struts.   This structure would be called a truss rather than a tied arch.  We could imagine intermediate structures.  The point is that although we can create categories of bridges, or indeed anything else, not everything can be placed definitively in one category.  What really matters is to have as complete an understanding of forces and structures as possible, enabling the design and construction of a wide variety of structures to solve a diversity of problems.  This as true in music or poetry as in engineering – knowledge of rules is useful, but insufficient.