What actually happened was that after the first main beam had been installed, the two on either side were placed with a distinct upward tilt. Then they were joined to the first beam. When these two beams were lowered to the horizontal position, they reduced the sag in the first beam, and themselves ended up with a lesser sag than if they had been separate.

A fourth solution for sagging is to hang the bridge from cables, so that the bridge is no longer a beam but a  Cable stayed bridge or a  Suspension bridge. A fifth solution for sagging is to invert the suspension bridge idea and make an Arch.

That short spans are stiffer than long ones is not the sole criterion, as the picture shows. The stream is very narrow, and a three-span beam would have been easy to build. But the extra complications of two more foundations, two piers and the extra connections would simply not have been worthwhile. And the bridge might not have been much lighter. A much thinner span than this could take the weight, but at this size, stiffness may be the criterion which determines the thickness. People need to feel something solid under their feet. In almost every field of construction, large and small examples are seldom very similar.

Going back to the second solution, the great gain from extra thickness shows that merely turning a plank or a joist on edge is beneficial. Of course the potential for sideways wobble makes such a bridge a precarious crossing. But two vertical planks joined by cross-members begins to look like something good, an inverted trough.

Alternatively a horizontal plank along the top and bottom of a vertical one makes an I-beam, which is rigid in all directions. Two of these joined by cross-members makes a strong bridge, which is used in the chassis of many trucks, to span the distance between the front and rear wheels.

The pictures below show how a piece of thick card behaves as a plate or a beam. Although it is vertically rigid when on edge, it resists transverse bending moments feebly. A much better solution is an I-beam, in which the top and bottom members resist the vertical and horizontal bending forces. The web holds these two members in place. The fifth picture shows the strip wedged between two abutments to make an arch. This is not a true arch, because much of the thrust is caused by the bending of the beam. We can see this because it is more curved than the free curve of the beam in the second picture. The last picture shows two ways of using a beam of expanded polystyrene.

Why is the flat plank so poor? The sagging beam is in compression on top and in tension underneath. Gravity is tending to bend the plank, while these other two forces are tending to straighten it. Equilibrium is reached when the two effects balance. With a thin plank the compression and tension are acting only a few centimetres apart, and are therefore extremely ineffective in resisting the bending moment.

From examples such as bicycle pedals and wheel-braces we know that a pair of forces is much more effective when well separated. That is why making the plank vertical is so much better. In fact, most of the material along the centre-line is doing nothing useful – it is neither stretched or compressed. It might has well have holes to lighten it, leaving only enough material to hold the top and bottom together, and resist the shear action, that would, if unchecked, make the two halves slide upon each other.

Here is a steel rule, which is slightly curved transversely, showing the effect of getting more depth. When the concave surface is facing up, the rule can sustain about a 76 cm cantilever without collapsing. With the concave side down, it collapses at a much lower span. Can you see why? The collapse near the support is typical, because that is where the bending moment is greatest. The first Quebec bridge did the same thing, and so did several early box girder bridges which were constructed as cantilevers, with the intention of joining the ends to make beams.

Try this out with a long narrow piece of paper. Fold it neatly on the long centre line, and give it an L-shaped cross section. Support in at each end in A – a roof configuration, and B – a trough configuration.  If necessary, load it with a small weight.

The bridges shown in the first picture above consist of several I-beams, braced together for greater rigidity. Both are wider than they need to be for the road and footpath, which reduces the tunnel effect. The nearest one carries a slip-road of the M5 at junction 11A, but the second one, which carries a dual-carriageway, the Brockworth bypass, is still rather like a short tunnel. The second picture shows inclined supports, perhaps giving a tendency to arch action, and generating horizontal thrust at the foundations.  

This bridge carries a dual-carriageway, the Hucclecote and Barnwood bypass near Gloucester, and would be almost a tunnel if had not been made so wide.

The resulting object presents a neat appearance to the world, and has no external nooks and crannies for water and corrosion to work upon. See Box Girder Bridge where the advantages of box girders are described.

Here is another tidy beam bridge.

Asymmetric Beams

These are asymmetric beams across roads. This type of construction is very suitable when the road is in a cutting in sloping ground. These bridges all have only one intermediate support, even though they are spanning wide roads. An asymmetric structure is often a response to an asymmetric site.

Suppose that the height of the pier is increased slightly. The force at each end will be reduced, and the distribution of bending stresses will be changed. In a  sense, the left side is acting partially as a cantilever which balances a part of the weight of the longer part.

A bridge is not just a lifeless lump of steel or concrete: it has complex live and static forces within it.

Now look at the diagram below. This is a pretty silly way to build: ignoring the support that the ground could give at one end. But if we deliberately pull down the left hand end so as to reduce, but not eliminate, the weight on the ground at the right hand end, we change the stresses right through the bridge. Could there be benefits in such a strategy? This topic will be mentioned also in the page about cantilevers.

In fact, jacking structures to produce the required distribution of stresses is very common, if only because the stresses in a structure vary considerably during construction, and may need to be adjusted from time to time as the structure grows. A striking example is provided by Sydney Harbour bridge, which was built as a pair of cantilevers, becoming an arch only at the very end of the work.

A Symmetrical Beam

This beam bridge spans the M11 motorway and a joining road, in two almost equal spans (one incompletely shown) with a central support. The spans are tapered like the ones shown above. What is the advantage of the tapered beams?

Bridges with two supports are far more common. Anything that reduces the span is worth considering,  because the cost of a structure rises as a  very strong function of the span. Sloping struts reduce the span still further, and offer the possibility of some arch action if the deck has some rise. The first diagram below shows three examples with straight supports, and the second shows a range of shapes between propped beams and arches. In those cases the depth of the beam would probably be varied along the span to take advantage of the slight arch action, or to control bending moments.  

These are examples of the way in which the boundaries between the classical types of structures are in fact not well defined. Some very interesting structures have been made, especially in recent times, by using intermediate designs. The stresses in straightforward, "pure" arches, beams, cantilevers and so on may be easier to calculate than in more complex designs, but calculating techniques using fast computers can solve immensely complicated problems in reasonable times.

Note the slight taper of the struts. Anything that reduces the monotony in a town or on a road is worth considering. It need not be consciously noticed.  Few people know every detail of the decor their favourite restaurant, pub, or town, but what they do know is that wherever they look, they will probably not be displeased. A bridge does not need to be big, famous, or "original" to do a good job.  Indeed, it may never be noticed.

Here is a picture of the Whitney-On-Wye toll bridge, which is a Grade II listed building.

Here are three bridges for you to compare.

See also propped beams and inclined supports.

The  First  Thelwall  Bridge

The picture below shows the first Thelwall bridge, over the river Mersey and the Manchester Ship Canal, east of Warrington. Recently it was refurbished, and a new bridge was built alongside to cope with the huge increase in traffic. This bridge has many welded plate girder beams, and a riveted cantilever span of about 335 feet over the canal.

The main river span is about 180 feet/55 m long. The total length of the bridge is about 4400 feet/1341 m, including about 36 spans of about 110 feet/34 m. The first bridge was completed in 1963, and the second in 1998.

Other  Beam  Bridges

The picture at left shows five of the bridges which were needed when the Brockworth bypass was built. They are near its junction with the M5 motorway. Building a road with minimal disruption of traffic on existing roads, railways and waterways requires careful planning.

These two bridges take the M5 and a slip road over the Hucclecote and Barnwood bypass quite near the bridges in the previous picture. The slip road bridge is strongly skewed. The large bridge is still equipped with four plastic 30.60.90 set-squares, one of which can just be seen on the right. The concrete abutments have been textured to reduce monotony.

Nearby, the M5 motorway passes over Horsebere Brook. This double-deck tunnel allows works vehicles to go under the motorway on a level above the brook. A cantilevered concrete platform carries a public footpath across the brook.

Further upstream, the brook is crossed by the link road from Gloucester Business Park to the Brockworth bypass and southbound M5 motorway, over an arch based on curved concrete slabs. The slightly non-circular profile increases the headroom over the farm track and the footpath. Although this bridge is seen by a relatively small number of people, the designers have achieved a very pleasant appearance. At the other end of the tunnel, a wooden beam footbridge crosses the brook.

The link dips under the road that connects Brockworth and Hucclecote. The footpath and the road are carried on two bridges based on pre-stressed concrete beams. The whole site has been the subject of attention to detail. The last two pictures show the side walls of the cutting, which use textured and sealed concrete slabs. The exact appearance depends on the angle of the light, and whether it is direct or diffuse. Although the panels are all the same, this is only apparent on close inspection. Attempts in earlier times to disguise concrete by patterning have often failed because the repetition was all too obvious. The entire approach to the site presents a pleasant appearance.

Click here to see a very new beam bridge.

Not far from these bridges, another takes the A40 road over the M5 motorway between Cheltenham and Gloucester. This bridge looks unusual.  That’s because it is. Until 1999 it was a standard beam with two sets of vertical piers. But it was weak, like several others in the region. The engineers repaired it by replacing the vertical piers by the triangular supports, which had the effect of reducing all three spans. Throughout the repair work, traffic flowed continually under and over the bridge, though with lane restrictions. To see more about this bridge, click here.

This picture shows the same bridge, and in front and behind it, the two bridges that carry the linking roundabout over the M5. On the right is one of the two beam bridges that carry the A40 over this roundabout; these also needed repairs. So the junction requires five bridges. In the background we see the scarp of the Cotswolds.

A small portal frame carries an internal road over a drain at Wildlife and Wetland Trust, Slimbridge. The swans are waiting for the man with the barrow-load of grain.

Stresses in a Beam – Bending

The diagrams below represent a simple beam. In the top diagram it rests on the ground, and the only stress in it is a vertical compression due to its own weight. In the second diagrams it rests on two point supports near the ends. The intensity of the colours indicates the magnitude of the stresses, which are compressive at the top and tensile at the bottom. Near the middle of the beam the material is only lightly stressed, and we might ask whether it is needed.

It is indeed a good principle to place material where it is most useful, and to remove it where it does little. That is why the vertical plate with flanges top and bottom is so effective. It is also relatively cheap to make, and easy to integrate into a structure. Note that in these and some other diagrams in this web-site, the stress concentrations at the supports have been ignored in the interests of simplicity. The idea here is to illustrate general beam behaviour.

Some people say that a rigid structure needs to have a large radius of gyration, but this is a technical term related to dynamic rotation. It does not help in understanding static structures. It is surely better to say that material should be kept as far from the neutral axis as possible, to oppose bending moments or torsional forces. The term root mean square distance (rms) might be better. Click here for more about this.

The second diagram above is too simple.  it neglects shear stress, and it ignores the variation of bending moment along the beam. The bending moment is greatest in the middle, and zero at the ends.

The third diagram attempts greater realism. Taking into account the shear stress, we can see that the magnitudes and directions of the stresses in even this simple case vary in a complex way. 

From the diagram we see that cast iron and plain concrete are unsuitable for beams because of their weakness against tension. Click here to find out how concrete can be used in beams.

The contours of the colours suggest that the forces are not parallel to the axes of the beam, which is indeed the case.

Each support takes half the weight of the beam. How is that weight transmitted to the support? Why is the bending moment zero at the support, which is clearly pushing hard on the beam and helping to bend it? Why is the bending moment greatest at the centre of the beam, where you cannot see any forces at all?

This derrick crane includes a number of trussed members, all of which are widest near the middle, where the bending moment of a beam is greatest. And of course the main longitudinal members are away from the neutral axis, so that they have the greatest effect. But we should observe that none of these members is acting as a pure beam: some are struts and some are ties.  If the compression or the tension is large enough that both top and bottom of the member is of the same nature, we could say that there is no beam action. But that is not true: what we have is a mixture of beam and tie or beam and strut. Very often in real structures we find members that are not purely of one type. Click here for a page about the variation of beam depth to take bending moment into account.

We cannot see inside a metal beam, except possibly by something like neutron diffraction, but a transparent material offers the possibility of "seeing" where the stresses are. Using polarised light, and a material composed of asymmetrical molecules, any strains show up because they change the optical properties of the material. The pictures below show a small strip of perspex. In the second picture it has been curved, and the strains can be seen as coloured areas. The neutral region along the axis remains dark, as in the unstrained condition. Compare the picture with the diagram shown earlier, and repeated here. The correspondence is not exact because the plastic strip is being pushed at a few places, not uniformly along its length by its weight. But the difference is not as great as we might expect, because in a heavy beam, the most effective part for producing the stress is around the middle.

So let’s do the calculation again, for a point load in the centre, and ignoring the weight of the beam. The result looks reasonably like the photograph.