Simple beam theory

Asymmetric beams

Propped beams

Thelwall bridge

Other beam bridges

Stresses in beams

Frames

Developing the beam

Beam sections

Beam railway bridges

Over-constraint

Simulator download

Beam architecture

Churn bridge

This is not the smallest beam in this page, but it is one of the simplest – a simple skew bridge over a stream.  In spite of small scale, you can see two features – the netting to give a good grip in wet conditions, and the way the soil has eroded asymmetrically because of the skew.  So even at this scale, construction is non-trivial.

Here is a somewhat larger example.  Already at this scale we see a differentiation of functions; many transverse planks rest on two beams spanning the stream.

Books about bridges often begin by saying that early bridges were possibly tree trunks, later squared off after tools were invented.  What could be simpler?  Compared with a truss the solid beam looks easy to understand.

The opposite is true.  A pinned truss made of narrow ties and struts, if not over-determined, can be worked out using a set of linear equations; tedious, but not difficult.  But a solid beam has an almost infinite number of parts.  Even if it is completely uniform in every way, and supported in a simple manner, calculating the stress and strain at each point is not easy.  Using finite element analysis, the calculation can be made as accurate as we want, at the expense of computing time.  But having done it, we don’t necessarily "understand": all we have is a set of data for the forces at many points.  But then, that is all we need for most purposes.  

Understanding is a strange business.  If we knew the position and velocity, and the chemical bonding and energy levels, of each atom in a living thing, would we understand?  No more than a person viewing a cricket match or a soccer match without knowing the rules.  Such a person views, but someone who knows the rules, sees, and understands.

The difficulty of calculation was demonstrated when several box-girder bridges collapsed during construction around 1970.  These bridges are based on boxes, which are like hollow beams with stiffening diaphragms and flanges.  Before building the Britannia bridge, comprising hollow beams, Robert Stephenson had extensive tests made, using scaled down models.  He knew that current calculating techniques were not adequate as the sole guidance for building such a bridge.

One great advantage of the beam is its very simple appearance.  In a town this can be very important.  Another is the relative simplicity of construction.

This page tries to give some idea of what goes on inside a beam.  Further information can be found in the page about pre-stressing.  The idea of the simple log bridge is a little erroneous in the cases where the tree has generated stressing forces inside itself.  Cutting up such a log and reassembling it produces a structure with different forces in it.

In reality, building a small beam bridge may not require complicated calculations.  Many types of small structures can be designed using codes of practice developed over many years of experience.  But anyone who uses rules outside their areas of validity courts disaster.  A simple decision like using an O-ring or a solder at temperatures where they do not work, can cause catastrophe.

As already stated, one of the simplest ways of spanning a gap is to put a long object across it, such as a flat stone, a log, or a plank.  The first picture here shows a single stone slab over a small stream south of  Naunton.  It has a clear span of only about half a metre.  The second bridge, Keble’s bridge, with four stone slab spans, is at Eastleach.  The other two are at Lower Slaughter. 

All four are in the Cotswolds.

On another island, thousands of miles away in Japan, we can find very similar constructions, this one being in an ancient garden.

And this example reminds us that engineering is not the only consideration: in this garden, the desired appearance is obtained by using beams that are much stronger than they need to be.

The diagrams under the white panel below show what can happen as the span is increased, everything else remaining the same.

The loss of one support of this roof ridge, probably from the effects of age, has doubled the span – see first picture.  The effect is dramatic.  Removal of supports in a building is a bad idea, unless you understand the building and you know that you can safely make the changes.  The pillars at the front are capped with short cantilevers that spread the load into the beams.  The builders of Chinese buildings took this idea to an almost extravagant degree of development, producing complicated, though logical, series of cantilevered brackets to support roofs.  This practice migrated to Japan.  The second picture shows the result of completely rebuilding the roof.  Perhaps as the years go by, the roof will acquire a new population of lichens, mosses and sedums.

These pictures, the second compressed laterally to clarify the effect, shows the effect of removing an intermediate support.  These windows were built with a central mullion.  Removing this and replacing the old frames by plastic ones has increased the area of glass, and improved the illumination in the rooms.  The lintels look thick enough to span the gaps, but in fact they are not continuous: they are in two parts.  The original mullions acted as piers at the joint.  The pictures show how the lintels are sagging.

For a given material and cross-section, increasing the span beyond a certain point makes the sag unacceptable.  Furthermore, adding a load would produce greater sag, and such a structure would be alarmingly flexible.

Galileo showed that for animals and structures, it is impossible to scale something up while keeping the same proportions.  This is because different variables vary linearly, as the square, and as the cube of the dimension.

The crane fly could not be scaled up by even a factor of ten without some changes in relative dimensions.  We instinctively recognize this when we notice the absurdity of the cruder type of science fiction film or horror film which depicts giant insects or other animals.  Small and large aircraft, small and large bridges, small and large mammals, do not resemble each other very closely in proportions, though the anatomy of mice and elephants, and many other mammals is in fact quite similar (homologous) in many ways.  People even have the same number of toes as lizards.  More about insects can be found in the page about tubes.

There is a hidden assumption, namely that the beam does not break.  Stone would break before bending this much.  It requires a force to bend a rigid object, and in applying this force and making a deflection, energy is put in.  Any rigid object will break when a certain amount of energy has been put in, unless it fails by some kind of plastic deformation or creep.

Some natural substances have a great capacity for absorbing energy.  Examples are tendons, spider threads and leather.  

Glass is very rigid, and it breaks very suddenly.  But if glass fibres are embedded in another substance, the composite material can be very strong indeed.  Many gliders are made in this way.

The problems of sagging and of breaking can be solved in several quite different ways.

One way to solve the sagging problem would be to build a beam which is curved in the opposite direction, so that it would sag to a straight line when placed in position.  Prestressed concrete beams are in fact bowed slightly upwards, if only because of the tension in the wires.

But for a large sag such as those shown above, this solution would not work, because beams that flexible would sag further under live load.  The effects of heavy vehicles travelling at speed would be unpleasant, if not dangerous.

We have seen that scaling structures to bigger sizes is not straightforward, and that a longer beam needs to be made stiffer than a shorter one.  But we must be careful not to generalise this idea too much.  If we scale the Severn suspension bridge down to a 10 metre span, we would have a very thin structure.  Indeed, a rope bridge would do the trick.  But who would want to use it to cross a river in a city, with bags of shopping or a push-chair?  So rigidity seems to work the other way in this instance.

Katsushika Hokusai made a picture showing a funicular – Famous Bridges of Various Provinces: The suspended bridge between Hida and Etchu.  He shows clearly the discontinuity in slope at the position of each of the two people, but he has made a bigger change of slope for the person with the smaller load.  He has also assumed zero mass for the bridge.  Most pedestrians would be happier with something more rigid.

What is the curve of a sagging uniform beam?  It is tempting to think that it might be related to the curves of suspension bridges.  But we can easily see that this is not the case.  If we consider a suspended cable, and we imagine a longer cable, and towers further apart, we can superimpose the central part of the new cable on the old cable.

We cannot do that with beams.  At the two supports, the bending of a beam is zero.  We can see that this is the case because there is nothing outside the support to force the beam into a particular shape.  So making a long beam cannot replicate the shape exactly.  The curve contains a parabolic part and a quartic part, giving a point of inflexion at the supports, where the bending moment is zero.

Already we see that the behaviour of a beam is quite complicated,  be also see that some understanding of a system can be obtained from simple principles, just by considering the forces and the boundary conditions.  Just because we can’t work out everything, it doesn’t mean that we can’t work out anything.  The curves below represent a circle, a catenary, a parabola, and a beam, all having the same curvature at the lowest  point.

This picture shows a rack for drying hay, in Slovenia.  Sometimes a wooden beam twists until the ends are relatively displaced by a right angle.  At the centre of the beam, the cross section is at 45 degrees to the horizontal.  Does this make the beam less rigid or more rigid?  Can you work this out by thinking about it, rather than doing some mathematics?  This design is good engineering, because wood is plentiful in Slovenia, and the weather is conducive to drying, without consumption of non-renewable energy.  And little land is required.  Would you place this arrangement along the direction of the prevailing wind or at right angles?

Instead of merely deepening a beam, it is better to split the material into several parallel beams which are deep but narrow; this is a better use of the material.  Later in this page you can find more explanations about stresses in beams.    

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.

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.

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 trying to bend the plank, while these other two forces are trying 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 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.

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 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.

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.

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.

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.

Propped 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.

The  First  Thelwall  Bridge

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, traffic flowed continually under and over the bridge, though with lane restrictions.

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 Slimbridge.  The swans are waiting for the man with the barrow-load of grain.

Stresses in a Beam – Bending

Near the middle of the beam the material is only lightly stresses, and we might ask whether it is needed.  It is indeed a god 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.

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 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 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?

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.