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.

The pictures above, one exaggerated vertically, show the lintels over some windows. The rightmost picture has been compressed laterally to show the sag more clearly.  What are these structures? Arches? Beams? Surely a beam has to be in one piece. If we make a cut in a beam we will get something like the result shown below. 

In these two sets of photographs, one of which has been vertically exaggerated in each case, we see that lintels have sagged. In the first example, the central mullions have been removed from some of the windows, allowing more light into the rooms. This has allowed the two-piece lintels to sag, the plastic window frames being insufficiently rigid for the weight of the lintels. In the second case, the window was built without mullions from  the start, with a similar effect. You can see where the plastic frame has bent.

Let’s start with a simple beam and try to understand what is going on in multiple-stone beams.

Next we split the beam in two.

Clearly this isn’t a good idea, so we modify the support, making the "beam" a good fit.

But a heavy enough load will still produce a sag, by deforming all the parts.

So we will go further, and compress the lintel before placing it into a gap that is slightly too small, producing a pre-stressed beam.

This is a very abnormal way of producing a pre-stressed beam. Without jacks in the supports, there is no means of compensating for movement or shrinkage. Still, in principle, it is a solution to the problem of making a beam in two pieces.

Returning to the simple, non-pre-stressed beam, we could have split it into three or more pieces, which is, of course, not a good idea.

But suppose we make the cuts at an angle.

Will this hold up? If so, how? It doesn’t hold up if the cuts are vertical, as we have already seen. What if we make the cuts at a bigger angle?

This clearly won’t survive, as we see in the diagram below. See also the photograph at left. 

The picture at left, compressed horizontally, shows the same lintel as in the previous paragraph. The sharp taper of the keystone means that the two halves of the lintel are acting as cantilevers rather than as parts of an arch. Furthermore, the small steps in the keystone ensure that it pushes its neighbours down, rather than pushing them sideways as a keystone should do. The block on the left side has cracked under the bending stress. It is a fair guess that the designer of this window did not understand the forces in the blocks. The picture shows the angle in the left block, and the cracks at top right, resulting from the right hand block having moved.

So what is the range of angles for the cuts that will enable the structure to hold up? Look at the next picture. Do you agree that the diagram below shows an arch and not a beam? Or is it a beam as well? In the two-piece "beams" that were shown above, the self-weight and the load wedges them together at the top, so that the line of thrust runs from the bottom corners at the supports to the top line in the centre, effectively forcing the system to act as an arch. The two-piece lintels with the plastic window frames were able to sag because there was no means of providing inward thrust with that type of window construction. Modifying a structure is always risky unless you fully understand it. One TV episode of "Some Mothers Do ‘Ave ‘Em", with Frank Spencer, depicts the complete collapse of a house as a result of "do-it-yourself" repairs. Compare this with the events related in "The Destructors", a short story by Graham Greene.

Does this make it easier to decide what angles will work for the cuts? Think about the funicular. The funicular here is very shallow, and as always for a voussoir arch, must remain inside the structure for all loads. The flatter the curve the greater will be the outward thrust that the abutments must resist. Large scale arches are not made like this, but Perronet designed an arch across the Seine that looks rather like this, but in fact the appearance is achieved by the use of cornes de vache.  

Cornes de vache were used in Over Bridge near Gloucester, which was given a low rise to minimise the height of the approach roads. Unfortunately, the ground in that area is poor, and the abutments moved when the centring was taken out. The crown of the bridge sank by about 25 cm. Nevertheless the bridge was used for vehicular traffic well into the 20th century, and was abandoned only when the width of the road became too narrow for modern traffic volumes.

There’s always another way of looking at things.

The pink shapes are imaginary arches inside the bricks. Near the top the arches are deep, but the small rise means large thrust.  Near the bottom we have smaller thrust but thinner arches. If these lintels are built into a wall, this idealised picture is grossly modified because of the distribution of forces in the wall.

Here are some pictures that are not unrelated to this topic, as they illustrate some arches with a small number of straight segments.

1    2    3    4

The sloping ends of the slab in the first picture makes clear that arch action is intended. Vertical ends would require great shear strength in the mortar, while horizontal gaps would denote a beam.

If you don’t believe in the funicular, try these examples.  Will they both work?  Will either of them work?  Why not make a model and find out.

This idea of a funicular within a shape is very old – it was used in the inner shell of Brunelleschi’s octagonal dome in Florence. The internal shell, though octagonal, is thick enough to contain a complete circular shell of significant thickness. Just as these flat arches are sometimes called jack arches, we could call the dome a jack dome.

Returning to an earlier diagram, repeated below, the dotted line shows the highest possible thrust line from the top of the central block. It doesn’t reach the abutment. Hence the collapse. The broken line and the full line are at right-angles.

We can now see what the maximum theoretical angle is for a keystone in this three-block arch-beam.

Here the supports are within the area which includes a funicular within the wood. Cutting all the pieces from one piece of wood is a crude equivalent of match-casting in concrete.

In this example the supports are much too high. Why did the failure occur at the centre?   See Beams to find out.

Here the beam has vertical ends, and wooden wedges were used to pinch it high up at each end. This is a crude form of pre-stressing, and it almost works.  In practice, it would be far better to squeeze at the bottom.  Why?

Here is another example. Can they be acting as arches or as beams? They are probably held up by the window frames.

Here is a more extreme example.

Here again are the lintels we began with.

So we have to very careful in deciding what type of structure we are looking at. The shape may mislead us unless we look at the details. A beam has to be a unified structure, unless the supports are able to provide inward thrust, producing some arch action. When you look around a town or city, don’t just glance at the famous buildings. Look at some details to see if you can find something unusual. And look at some "ordinary" buildings. All sorts of fascinating features can be found in an ordinary street.

This idea of a shallow arch is roughly the converse of the Millennium Bridge in London, in which the eight cables are stretched to a very shallow curve. The tensions are correspondingly high, and are held by enormously deep invisible anchorages.

Are we any more clear about arches and beams, and indeed struts and ties? How would you define each of these unambiguously?

As a start, could we say that a member joining two points is a tie if it is in tension, a strut if it is in compression, and a beam if it exerts neither push nor pull on its fixtures?

It is certainly true that a strut should be in compression throughout, and a tie should be in tension throughout. A beam, as we have seen, is normally in compression along the top and in tension along the bottom. Suppose we prestress a beam with a single steel wire, to the extent that the entire beam is in compression. Could we regard this member as a strut and a tie in opposition?

What about an arch and a strut? Both are supposed to be in compression at all points. So how do they differ? Perhaps the difference is that a strut is intended solely to resist axial forces, while an arch is intended to carry perpendicular ones also, almost always in the form of weight, except for arch dams, which resist the lateral thrust of water. Another way to look at this is to ask whether a strut is a member that has compressive forces that far outweigh its own weight. Even that may not distinguish between an arch and a strut. Think about a very light arch which is stiffened by a very heavy deck. The arch segments would probably be straight, and might be regarded as struts. The heavy deck might be regarded as a beam with many supports.  

What has happened here is that the two functions of a traditional arch, stiffness and strength, have been separated into two different parts of the structure. The advantage is that the deck has to be stiff in any case, to support the live loads, some might as well make it stiffen the arch. Although this model is made of straight members, it is not a truss: a truss uses triangles to obtain rigidity. In this case, a single member, the beam, provides the rigidity for the whole structure. If you look at the page about beams, you will see how a beam is related to a truss, an arch and a suspension bridge. The pictures below illustrate the principle of the deck-stiffened arch by using the inverse model, a suspension bridge..

We can see how this works by considering the example of a suspension bridge. (A cable can be used as a model for a strut as regards the funicular.). The two pictures at left show parts of the Severn suspension bridge and the Clifton suspension bridge respectively. Although the bars of the Clifton bridge are relatively heavy, they are straight, whereas the cables of the Severn bridge are slightly curved between the hangers. The bars must therefore experience some bending moment. Look also at a picture of the chains of the Tower bridge in London.

Actually, we need to think very carefully before we decide that we understand what goes on inside structural members. Consider a very thick cable that was built straight. If we now hang it up, we can surely believe that it now has more tension on the outside of the curve than on the inside. But if we build suspension cables, strand by strand, all parallel, and then we bind them together, we can believe that the tension is uniform throughout. But if we now add the hangers and the deck, the cable will be slightly less curved between the hangers, and rather more curved at the hangers. Does this mean that the tension will vary slightly from top to bottom of the cables? If no slippage occurs between the strands, surely it does.

Many old stone walls cross the paths of streams. Rather than create two ends, which could be sources of weakness, many builders simply bridged the streams with stone beams to carry the walls. Here is a modern example across a moat. Do you think it is an arch, or a beam with fake voussoirs?  Note the unequal lengths of the subdivisions. If it’s an arch, it’s a very flat one, and it will generate a lot of thrust.

Consider a uniform beam, resting on piers. What is the funicular for this? It is a parabola for a uniform distribution of mass, and its height is inversely proportional to the external horizontal thrust. So it must be infinitely high. But if we add pre-stressing wires near the bottom of the beam, and start to increase the tension, the height of the funicular will shrink, and if we create enough tension, we can place the whole curve inside the beam. With enough extra tension, we can ensure that the funicular remains inside the beam for all reasonable live loads. Before pre-stressing was invented, people could achieve a similar effect by curving the beam, and adding a tie between the ends. The curved member was then a tied arch, and the whole structure acted as a beam. In a sense, the pre-stressed beam is a very shallow tied arch enclosed in a solid case. We could even make the channels for the pre-stressing wires in a parabolic shape, and then we would have something like an enclosed version of Brunel’s Royal Albert bridge. See also funicular.

This picture shows the City Union bridge in Glasgow. Is it a beam or an arch? If the lower chords simply rest on the supports, the spans must be beams, with the lower chords in tension, but if the ends are prevented from moving horizontally, the spans could be arches, with the lower chords in compression. In fact, one could imagine that the horizontal reaction at the supports could be adjusted to create an intermediate structure in which some or all of the segments of the lower chord could have zero force in them. In such a case they could be removed. How can this be? The spans are in fact statically indeterminate. The spans could also be supported using only the ends of the upper chords, in which case they could only be beams, and the outer segments of the lower chords would be redundant.

What is the difference between an arch and a beam? Suppose someone told you that a beam is bent, but an arch isn’t. That can’t be true. Or can it? What happens if you build a beam and set it on its supports. It bends. Not much, but it does bend. Beams are often pre-curved the other way, but they still deflect from that shape when placed. They may end up looking straight, but in terms of stresses, they are bent. You can even build a beam that is distinctly curved, but if it rests on flat supports with no horizontal thrust, it’s still a beam. What about an arch? If you design and built it perfectly, it will be purely and uniformly in compression. It is curved, but it hasn’t been bent. A live load will, of course, deflect an arch.

At Whitney-on-Wye there is a bridge that is unusual in several ways. It is a toll bridge, one of only a very few in Britain that are not very large and very new. Three of its five spans are made of wood – greenheart from Guinea, a very hard wearing material. And it is a grade II listed building.

The three wooden spans at Whitney are propped beams, as you can see from the top left corner of this photograph. (Or could they be considered as three-segment four-hinged arches with stabilisers?) The general views are looking upstream, and on the far side of the cutwaters you can see the sheets of metal which protect the timbers from flood water and debris.

In the diagrams above, the deck was assumed to be in segments which spanned the piers. But if the deck were continuous, the compression in the centre of the spans would induce tension above the piers, in order to maintain the same total length, as in the first diagram below. The assumption here is that the deck is fixed at the ends. Such a bridge is somewhat indeterminate in both type and forces. The deck could be post-tensioned or post-compressed to attain a situation with the same polarity of stress throughout.

In the second diagram, we revert to a segmented deck, but with the joints near the props. Now the bridge is based on cantilevers.

We see that what looks like a simple structure allows of several different interpretations, depending on the exact mode of assembly. Unfortunately, it is not possible to get close enough to the bridge to examine the details.