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 problem

Japanese foot-bridge

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 has doubled the span.  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.

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.

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.

What is the difference between the curves of a uniform beam sagging under its weight, and of a uniform beam with three points of pressure?  For small deflections, the first is a parabola, whereas the second comprises two halves, each of which includes a cubic term.  There is a point of inflexion at each outer support, where the bending moment is zero.  In the page about suspension bridges we see that for small deflections, the shape of a parabola is close to the shapes of a circle and of a catenary.

We see here the operation of a simple rule.  At a point support or load, there is a discontinuity in some property of the structure.  In this example the structure itself continues smoothly through, as does the slope, but the curvature, or second derivative of the position, has a discontinuity at the load.  Some graphical examples are given later.

We must always remember that the apparent geometry is the result of the physics, as is shown in the pages about Nature’s maths, elsewhere in this site.  But at the lowest known levels of matter, such as quarks and gluons, nobody knows what material objects, if any, hide behind the maths.  Even at the level of atoms, the rules of quantum mechanics lead to results which are correct but incomprehensible.  If we know enough maths to describe every basic phenomenon, what else is there to know?

Here is a piece of expanded polystyrene, supported at the corners, and loaded in the centre.  This is a two-dimensional analogue of a beam.

These pictures show the result of bending a large square of foam.  The glancing illumination in the left hand picture shows that the surface is not planar.  On the right we see that the outside surface is not cylindrical.  The material takes up the position of least energy.  Apart from springs, not many engineering artefacts are treated like this, but you can sometimes see the effect in ornamental ironwork.  Bending a rectangular eraser will demonstrate it nicely.

Let’s see if we can work out how the shear stresses go in a beam.  In the diagram below we see the same beam as before.  Suppose we make a vertical cut through the beam near the right hand end.  With an iron beam we could hold it together with a magnetic field.  Or we could use explosive bolts.  What will happen if we suddenly break the connection between the two pieces?

It is clear that for a cut at the centre there will be no vertical sliding, because the halves will rotate equally.  So the shear stress is a maximum at each end and zero in the middle.  We can assume that the curve of stress is a smooth function, so it is probably a parabola, that is, the shear stress is proportional to the square of the distance from the centre.  This type of coloured picture is not very useful in practice:  the diagram below is a more practical representation.  The choice of positive and negative is arbitrary.  Diagrams of shear stresses and bending moments are very useful in analyzing complicated structures.  These diagrams are analogous to the Feynman diagrams used by physicists, providing a simple means of representing physical behaviour.

If we now consider horizontal shear, the story is quite different.  Like the tracks on a running circuit, the layers of the beam would be staggered if they were not connected together, and so there must be shear stress along the beam.  The diagram below shows the very simple case of two beams.  The shear stress is maximal along the axis, and zero at the top and bottom.  We can see this by realising that along the axis two strong halves are competing, while near top and bottom, a thick strong part competes with a thin weak part.  The horizontal shear stress varies parabolically from middle to top and bottom.

Another way of illustrating the stresses in a beam is shown below, with the same colour convention as before.  But now we can see the directions of the forces.  The magnitude of stress is indicated by the closeness of the lines, as in a diagram of an electric field or a magnetic field.  The shear stress can be invisaged also from the same diagram.

The same type of diagram enables us to see how a part of a beam is held up by the outer parts.  Look at the green section of the beam below.  Lines of compression and tension lead into it from the outer parts.  The compression lines are sloping upwards into the the green area, while the tension lines are sloping upwards and outwards.  Both sets of forces are counteracting the weight of the green section.

The beam looks as though it is full of arches and suspension cables.  We could in fact remove all the material except the red and blue, and we would have a sort of truss, and looking along the centre line you can see why a truss has diagonal struts and ties.  What has happened is that we have spanned a gap with a shape which is not funicular, and the beam itself has provided the necessary forces.

This happens inside any object that is subjected to outside forces.  It is deflected from its original shape until it sets up a set of forces that are mutually in equilibrium, and in equilibrium with the external forces.  In some cases, the object may break before this condition is reached.  This is called not being strong enough.  

Think about a long railway train crossing a long bridge.  The locomotive at the front provides enough force to pull against the frictional forces produced by the whole train.  The last wagon only requires enough force for its own friction.  How does each part of the train create the forces that are just right for pulling the part behind it?  Each part stretches just enough to create the tension required to pull the rest of the train.  A similar thing occurs when you sit on a chair.  The chair generates the right forces to hold you in place.

What happens in any structure is that a load path is produced from the load (in the beam its weight plus the live load) to the supports (in the beam the piers at the ends).  One job of the engineer is to create a structure which produces an adequate load path safely and economically.  If an architect or an engineer is tempted to use an unsuitable structure, the price will be unnecessary weight and expense.  People who build models from popsicle sticks may usefully try to imagine the load path in the structure they are creating.

We can look at the beam in another way, by considering stored energy.  The energy density (grey) is proportional to the square of the stress (colour), which is why it varies more strongly.  These coloured pictures are only qualitative indicators.

The next pictures show a piece of foam plastic being stressed, the same piece with five cuts made about halfway through with scissors, in two orientations.  Where the cuts are on the compressed side they have little effect on the rigidity of the beam, but where they are on the tensioned side, they open up in proportion to the curvature of the beam, which is itself related to the bending moment.  In fact, the outer two cuts hardly open at all, because the bending moment there is low.

This picture shows four slabs of expanded polystyrene, firstly acting as four unconnected beams, and secondly, acting as a single beam, having been bound together with tape.

These boards on a building site have been bound together, but not in a way that makes them act as a single unit, and they have collapsed under the weight.

This foot-bridge crosses a narrow tidal creek on the south coast.  The two beams comprise five baulks of wood joined together.  The bridge could have been made without camber, but it wouldn’t have looked as good, and the swans and other wildfowl would not have been able to swim under it – an important point in a nature reserve.  A raised flat bridge would have required ramps, making access harder for disabled people.

The final conclusion is – 

There is no compression or tension along the centre line of the beam, and at the middle of that line there is no vertical shear stress either, although the horizontal shear is maximal.  

Therefore you could drill a hole at the centre with very little effect.  In fact you could drill holes along the neutral axis, as long as you left enough material to handle the shear.  In fact some beams are made with quite large holes in them.  An example is shown below.  The compression and tension are now confined to the regions above and below the holes, because the holes cannot transmit the forces.

Here two small trees developed branches.  Subsequent annual growth cylinders of the trunk then enclosed the branches in a manner that suggests the flow of stresses around the "hole".

We have looked at horizontal and vertical stresses, but we could have used any two axes at right angles.  It is a matter of convenience and utility.

If you like to think diagrammatically, consider buying "Building Structures" by Malcolm Millais, E and FN Spon, ISBN 0 419 21970 6.  This book contains a vast number of diagrams, visualising the stresses in many different structures.  It is an excellent complement to a mathematical book.

A more mathematical type of book is "Fundamental Structural Analysis" by W J Spencer, Macmillan Education, ISBN 0 333 43467 6.

It isn’t difficult to sketch diagrams of shear stress.  The graph must be continuous between discontinuities in the structure.  The shear stress must be zero at a plane of symmetry of the forces.  The shear stress must have a discontinuity at the position of an applied force.

The last picture above, made using crossed polaroids and a piece of plastic, reminds us that a simple operation like making a hole can result in unwanted stresses.  There are more notes about this in the pages about indeterminacy and polarizing.  If we enlarge the holes in a beam and change their shape, we can end up with a truss as we shall see in a later section.  But before that, let’s look at the case of a continuous beam resting on piers.  The graphs of shear stress, above, and the bending moment, below, oscillate between positive and negative as the graph shows.  If the beam were in separate sections, the bending moment would be always in the same direction, and it would reach higher levels.  The coloured diagram shows the horizontal stresses.

These calculations were made on the assumption of a perfectly straight beam, picked up and laid on piers of uniform height.  But be shaping the parts a little differently, the bending moments can be changed if required.

The next two pictures give a direct comparison between the effects of separate beams and a continuous beam.  Joining the beams greatly reduces the stresses.

In the case of the continuous beam, there are places with no bending moment.  So if we took one section, and supported it at those two points, it would have vertical ends, just as it has in the continuous case.  These points are called the Airy points.  They are very important in metrology and interferometry, and other fields where exact dimensions are important.

The next picture shows the shear stress in the separated spans, using red and blue for opposite polarities as usual.

These coloured pictures are descriptive, but not very useful for calculation.

One approach is to use the stress tensor, a mathematical construction which describes a three dimensional stress field.  Tensors can be used to describe stresses in solids, the forces in viscous fluids, electromagnetic fields, and gravitational fields.  But for any but the simplest shapes, the calculations can be difficult.

At the opposite extreme, we can abandon any attempt to "understand" in terms of mathematics, and we can just divide the object into little cubes.  For each cube, and for each boundary between cubes, we can write down the internal equations, the equations for the interfaces.  For a complicated object there can be a vast number of equations, and fast computers are needed.  The two approaches could be likened very roughly to chess as played by a grand master, and chess as played by a programmed computer that works by searching through billions of chains of moves.

United we stand, divided we fall

To get more material at the edges of a beam, which is where we have seen the main stresses, it is common to employ flanges.  The picture at left shows a Z-section post supporting a crash barrier with a roughly square U-section.

The rolled steel joist (RSJ) has an I section, and is usually solid in the smaller sizes.  The top and bottom plates take the compression and tension respectively, and the web holds them in place.  Larger sizes can be pierced with circular holes, or with hexagonal ones.  

This picture shows a part of a haunched beam bridge.  The roadway rests on small beams across the gaps between the main beams.  Why make haunches?  The entire weight of the bridge rests on the piers, which transmit vertical forces.  But the effective position of the load is not over the piers, and therefore has a moment about the piers.  At the pier, the load is resisted by tensile forces in the top half of the beam, and compressive forces in the bottom half.  As the moment is the product of distance times force, making the beam deeper decreases the force.  The bridge is in fact beginning to look rather like a cantilever bridge, or even like a flat arch, which is misleading, because an arch should have no tension at any point. 

These pictures show parts of an old swing bridge.  In the first picture, the yellow lines show where the flange is made of more layers near the pivot, marked green.  The pivot is near one end of the bridge.  Near the other end, the beams are lightened by replacing the plates by truss panels.  Two of the pictures  show that the layers of the flange have buckled between the rivets.  Why do you think this happened?  Why were the resulting gaps filled in?  And why was the filling made using a water-resistant material?

Here is a detail from a three-pin arch.  Although this bridge is almost rectangular, it has a hinge at the centre of the span, and a hinge at the bottom of each leg.  The I-beam construction is used throughout.

The three dimensional beam, as in the Z-beam and I-beam, is of course prevalent in nature.  The picture below, of the left hind wing of a dragonfly, Libellula depressa, a fairly typical member of the Anisoptera, illustrates this very well.

The orange lines point to the strongest and most rigid part of the wing, comprising the leading edge (costa) and two other ribs (sub-costal and radius).  This structure is strongly three dimensional  These lead out to the nodus (yellow pointer).  The white lines point to the four "fingers", which are three dimensional, like the leading edge.  The fore-wings show a similar structure, but are significantly different, which is why the sub-order is called Anisoptera.  Compare this wing with the wings of birds and bats, where there are also arm and fingers, and you will see the result of convergent evolution.

These wings would be of no use alone, but they are controlled by a series of nerves and muscles, which in turn control the complicated shell of the thorax of the dragonfly.  Perhaps we can compare this mechanism with that of a helicopter, which may include swash plates and levers in the rotor pivot to produce cyclic pitch and collective pitch variations.

This wing differs markedly from those of butterflies, which use a completely different method of obtaining lift: they clap the wings above the body, and flick them apart, leaving a low pressure volume above.  Vortices probably play a part as well.

Some of the larger dragonflies, like the bird, Apus apus, or common swift, can fly continuously from dawn to dusk, catching prey, finding a mate, and mating, without ever needing to land.

At the other end of the spectrum of Odonata, we have the Zygoptera, or damselfies, in which all the wings are similar.  Here are two zygopteran wings.

Here the structures are far less differentiated, and the Zygoptera are regarded as more primitive than the Anisoptera.  Nevertheless we can see how the wing seems to grow from the leading edge.  The effect of the more rigid leading edge and the more flexible trailing edge is to provide an angle of attack on both upstroke and downstroke.  These creatures may be primitive, but they are still here, like every other living thing.  In that sense, no living thing is more "advanced" or "superior" than any other – the genes of all have made it to the present in spite of all difficulties.  The overall design of dragonfly wings has not changed much in 300 million years, apart from a reduction in venation and in the size of the insects.  So it must be in some sense a good design.  Unfortunately, because of their dependence on water, the numbers of dragonflies are diminishing in many areas.

In a sense, the development of insects has followed that of aircraft: starting with two pairs of wings and evolving to one pair.  In orders such as beetles (Coleoptera), bugs (Hemiptera), butterflies and moths (Lepidoptera), and flies (Diptera), we find that even though there may be four wings (or their remnants), two are either used for other purposes than flapping flight, or are linked to the other two to make effectively a single pair of wings.

The diagram below shows how an I-beam can be cut along a zig-zag line (bottom) and welded together to make a deeper beam, called a castellated beam.  To make use of all the material, if the zig-zags are suitably cut, and one half of the beam is reversed, no material need be cut off at the ends.  Is this last statement correct?  I-beams with circular holes are also sometimes made by welding two halves together.

"United we stand", in this context, does not mean close together.  Moving the material apart, in the form of two flanges, divided by distance, but united by a web, makes the system stronger, because the forces are in opposite directions when the beam experiences a bending action through the dead and live loads.  All the forces are within the beam, which therefore only requires simple supports to take the weight of beam and live load.  Another example of moving forces apart to obtain increased moment is the wheel-brace used to turn the nuts on the wheel of a car.

An electrical analogue is the parallel or twisted-pair transmission line, which is designed to minimise the external electromagnetic field from transmitted signals, and to minimise the effect of ambient fields on received signals.  The coaxial cable can be seen as the analogue of the pre-stressed concrete beam, or the limb of an insect or crustacean, in which one set of forces encloses the other.  The mammalian limb is the other way round – the most rigid parts are inside.

If we imagine enlarging the holes, we can move towards a truss, as in the diagram below.

Making holes is all very well, but the static stresses are not the whole story.  In some materials there is the possibility of metal fatigue.  If a hole has sharp corners, or ragged edges, the resulting stress concentration sits there like an incipient cancer, waiting to spread its effects.  Once a crack starts, a sustained load or a varying load may cause it to get longer.  Ships have been known to break in half, due to cracks which started at square hatches.  Railway tracks and aircraft are subject to the same phenomenon.  The common feature in all these cases is the occurrence of cyclic, or at least varying, loads.  This topic is discussed in the page about cracks.

These pictures show a typical rivetted plate girder bridge carrying a railway over the approach to a town.  The thickening of the horizontal flanges towards the centre of the span is clear in the second picture, as are the vertical stiffening flanges.  The dip in the road is liable to cause flooding during and after a heavy downpour.  The next two pictures have been squeezed horizontally to emphasise the effect.

Many modern bridges do not have variable thickness flanges.  What are the reasons for this?

If we imagine a beam as composed of a set of individual weights, we can imagine the static effect as the sum of a lot of lines like these.  You can see that the resulting curve will be highest in the middle.  Here is the curve for a parallel beam.  We shouldn’t of course, find the curve by adding nine triangles, as in the first picture, though that is what Archimedes might have done.  That’s what integral calculus is for, as we see in the second picture, where the curve is as smooth as the pixels allow.

So in terms of static bending moment a parallel beam is not ideal – it is wasteful of material.  Nevertheless it can be cheaper than a curved one.  We could make the beam deeper towards the middle.  This could cause headroom problems below or a steep deck above.  A good solution is to use two curved beams with a horizontal deck between.  What is the shape of beam whose depth matches the bending moment?  It’s no good copying this curve, because the new shape will give a new curve.  Can the trick even be done?

The coloured shape below represents a beam shaped to support the static bending moment we saw above.  But its own bending moment curve is the one shown above it, still not the same shape as the beam.  However, remembering that a bridge has to support live loads, this form of beam will do nicely.

The diagram above shows approximate contours of compression in red, and of tension in blue.  Along the axis, as elsewhere, the compression and the tension do not cancel.  There is no bending moment on the axis, but that does not mean that there is no stress.  The resultant of the the red and blue forces along the axis is a shear stress.

If a vertical cut were made near one end, with very viscous oil in it, you can see that sliding would take place if the parts were released.  Thought experiments like this are a good way to imagine what will happen in a structure.  In a sense, inside a beam there is a suspension bridge and an arch.  What Brunel did in the Saltash bridge was to let them out, as a sculptor releases a statue from the stone, by removing the inessential material.  Maillart did a similar thing with concrete arches.

If you wanted to make an ideal beam, you could make its outline follow a red contour and a blue one.  In practice, it often costs more to make the ideal structure, and so an approximate solution is used.  If the approximate solution is almost as efficient as the ideal one, and is cheaper, it may be the best one to use.  Some simple shapes are shown in the next set of diagrams.

The Supermarine Spitfire had ideal elliptical wings, but was more expensive than the Hawker Hurricane, which used a slightly less efficient shape, easier to make, and therefore cheaper.  Some small aircraft even have rectangular wings, enabling many of the ribs to be the same.

Nature, however, is not constrained by the same types of economy as people.  a growing plant or animal may have constraints of energy or of time, but it can have far greater freedom  in choice of shapes than people can use.  If you look at a bone or the branch of a tree, you will see very subtle shapes.

Having seen where the stresses go in a beam, we are not surprised that replacing a beam by a truss results in a structure with diagonal struts.

Frames

Isolated beams, except for things like long thin space craft and cabers in mid-air, are uncommon.  Most beams rest on something.  As usual, how this is achieved is quite important.  The four diagrams below illustrate some possibilities.  Which span do you think is the most rigid?  Which could be made with least material?  The pointed parts are to emphasise that they are hinges, not rigid connections.

Actually, this is a bit of a cheat.  Diagram A shows a frame.  The beam, legs and ground are all connected rigidly.  Diagram B shows a beam simply resting on supports.  Diagram C shows a two-hinged portal, while D has inclined legs to hint at the relationship with an arch.

There is an important point about these frames and beams.  For any distribution of weight there is an optimum line for the stresses.  The more the shape of the structure departs from this, the greater the stresses within it.  So D, being slightly more like an arch, will generate slightly smaller stresses, and the structure can be a little lighter.

Maillart, in designing some reinforced concrete buildings, flared the pillars near the ceiling.  This effectively reduced the span, and allowed the stresses to be a little nearer the ideal line, and to flow more smoothly from floor to pillars.  It was a gesture in the direction of the fan vaulting which some medieval builders used to create wide spans in churches.

Here are pictures of a foam plastic frame, with a weight in the middle.  In the first one we can see how forces from the beam are transferred into the legs.  In the second picture, the legs have been cut halfway through, so that tensile forces are not transmitted.  The legs now contribute much more weakly to the stiffness of the structure.

Developing  the  Beam

Both shapes match the curve of bending moment for a uniform weight distribution, which is roughly what they have, so these are good forms.

Beam Sections

These two pictures show curved beams carrying the M5 motorway along the side of a valley south west of Bristol.  The two carriageways are at different levels, to keep the piers to reasonable heights.

Beam Railway Bridges

Let’s spare a thought for the humble railway bridge.  Many British towns have one, even if the railway has long been disused.  Some are brick arches, occasionally of the skew variety which always create interest.  Others are beams, often comprising two vertical plate girders with flanges top and bottom.  Sometimes these flanges are thicker towards the middle.  This is achieved by riveting a series of plates of several different lengths to the main plate.  There are often vertical flanges at intervals along the span.

Spanning the gap between the two main plate girders there will often be a series of I-beams.  And across the spaces between these the older bridges often had a series of small arches, supporting the floor of the bridge.  Later bridges may have some kind of slabs, a construction which is often used when a beam bridge carries a motorway.

In older times, people would not have been able to calculate the stresses at every point in such a bridge.  For short spans they would have used the knowledge gained from earlier experience.

A Footbridge Over the Seine

This footbridge spans the river Seine in Paris.  This is a very long narrow beam indeed.

A Small Beam Over the Thames

The designer of this bridge, with a span of about five metres, only about four miles from the source of the river Thames, was taking no chance with the force of the river when he designed the cutwater.  Definitely over the top, as the water is only a few inches deep, and unlikely to flow very fast, even when flooded, and the fall from the source is very slight.

An elegant Beam

This bridge is built on a skew with the road, but is in fact symmetrical.  Note the shape of the end walls, reducing the tunnel effect, the twin beams, and the large overhang of the deck at each side.  A tidy design.

The diagrams above represent a simple beam bridge, which has been affected by subsidence (exaggerated).   One response is for the beam  to remain so straight that it is only supported in two places, leading to a bigger effective span. Another is for it to bend.  A third would be to break, if either of the first two conditions were unsustainable by the structure.

In practice the designers might include jacks at the base of the piers, to allow for adjustment.

What happens as a result of the movement is that the beam suffers stresses which were not in the design.  In fact the problem exists from the start.

The four support points can never be perfectly aligned, but the alignment is of course made so small that the beam can adjust its shape without absorbing too much energy.

The penalty for a through beam, is the over-determination.  The benefit is the spreading and controlling of loads and stresses.

The diagram above represents the response of a simple cantilever bridge to subsidence.  In this case the joints allow stress-free movement, so nothing is distorted.

Jacking might still be provided.  In a very slender foot-bridge, the slightest error could be noticeable, and so some adjustment may be needed.

After a bridge has been completed, jacks may be concreted over, or they may be left as usable adjusters.  The Eiffel tower is a good example of the jacking requirement.  The stresses, and therefore the strains, at the base, changed markedly during construction.  Jacking enabled the builders to compensate.  This subject is developed further in Indeterminacy.

In a multi-span beam bridge, the deflection between the supports can be reduced by joining the spans, as mentioned earlier.  Better still, by pre-stressing the spans, the deflection can be made even smaller.  Robert Stephenson did something like this in the Britannia bridge across the Menai Strait.

The diagrams below suggest crudely how this works.  At the bottom is a set of separate spans.  At the top is a set that have been joined.  The middle diagram is the result when the temporary supports are removed.

A Multi-span Beam

The A417/A419 from the M4 to the M5 provides a dual carriageway for most of the route.  A tremendous effort has been made throughout in order to create an attractive route.  Some farm land had to be purchased, and some old Cotswold walls had to come down.  Some were already old and crumbling.  In return, the road builders have preserved a well-known wind break near Duntisbourne, they have planted large numbers of saplings, and they have built miles of new Cotswold walls.

The designs of the new bridges are as attractive as possible, given the technical requirements, and the whole route is a good example of modern construction.

Near Baunton in Gloucestershire, an interesting beam bridge crosses the valley of the river Churn, north of Cirencester, carrying the A417(T).  This part of the route is a bypass of Cirencester, which relieved that town of through traffic, and took as much as fifteen minutes off typical journeys.  The pictures below show this bridge from below.  The line of the road is slightly curved in both dimensions, and the bridge is a continuous beam.

Crossing this valley, including the tiny river Churn, presented the problem of appearance.  The actual construction tends to obscure the view of the valley from the road.  Let’s look at this in some detail.

Firstly, the valley is barely visible from the road in most of the route, because of trees and hedgerows.  That it is visible here may be partly due to the effects of construction work.  Within a few years, trees will have grown up to hide the valley from the road.  So it could be argued that the view doesn’t matter very much.

Secondly, what is the intention when building such a bridge?  If we want to make the bridge as inconspicuous as possible, we can build it on very narrow piers.  If this had been done in this case, replacing each pier by two narrow ones, making four at each position, there would have been a good view through the bridge, but the effect of all these piers would have been very untidy.  Obtaining an ordered appearance from all directions is an almost insoluble problem with a two-dimensional array of piers.  The other difficulty would have been the avoidance of a top-heavy appearance.

Another possible solution would have been to have one pier at each location, on the centre line of the road.  The very wide beam would have had to be very stiff indeed to carry torsional forces to the abutments, increasing the cost of the structure.  Alternatively, these single piers could have been splayed at the top to provide paths for the asymmetrical forces.

At the other extreme, solid walls would have produced a very heavy effect.  The actual design balances the mass of the beam and the piers very well, and the hints of gothic curves echo those of the nearby Cirencester church and Gloucester cathedral.  The view across the valley is more or less blocked, but was not especially inspiring in any case.  In a few years’ time, most of the bridge will probably be hidden by the growing hedge and trees.

The river Churn, which lies at the bottom of the valley, is only a few metres wide at this point.  The pictures show it a few miles downstream as it reaches Cirencester.  Fairly soon after Cirencester it is joined by the little river Thames.  The names Churn and Cirencester probably echo the Roman Corinium, an important fortified town.  The A417/A419, apart from the new bypasses of Cirencester and Latton, follows the old Roman road, later called Ermine Street, quite closely.  Comparing a map of Roman roads with a modern map shows that the routes of many Roman roads can still be traced via modern roads, tracks and footpaths.  The influence of the Romans on our language is also enormous.  On the road we may come across words such as bus (omnibus), car, engine, entry, exit, logistics, traffic, trailer, transport, and vehicle, all derived from Latin.  The Corinium museum in Cirencester offers an excellent insight to Roman life in Britain.

Just south east of this bridge, a minor road, the Whiteway, crosses the bypass.  At this point, as on the other side of the Churn valley, the A417 is in a cutting, to reduce the gradient down to the Churn bridge, and to reduce the height of that bridge.  The typical bridge in this situation would have been a simple concrete beam or cantilever type, but what we see is apparently a flat arch, faced attractively with Cotswold stone.  But as we pass underneath, we see that the arch is only a facade, and that the bridge is a concrete beam.  In fact, under the right hand side of the bridge, you can see a part of the sloping haunch.

Is it a fraud?  Well, it looks a lot better in that location than a grey concrete bridge would have done.  It avoids the concrete piers that would have blocked the view of the cutting, and it generates a lot less thrust than a real flat arch would have produced.

This picture, looking north from the small over bridge, shows the deck of the big beam in the distance.  We can see the slight curvature of the bridge.  On each side of the road, on both sides of the bridge, the Jurassic limestone is cut into steps, to improve the appearance of the cutting, and to provide footholds for the growth of vegetation, as the second picture shows.  No doubt the fissures in the rocks will provide shelter for a wide range of invertebrate and vertebrate life.  Kestrels are a common sight over verges, and the occasional buzzard is seen on a fence on days when thermals are not forthcoming.

These are just a few examples of the thinking that lies behind the A417/A419 link road.

Another  Beam  Bridge

This footbridge crosses the A34 at Chilton.  Since we have seen that a beam requires only two hinges as supports, the extra two here can be used to control bending moments.  This bridges exhibits the use of completely unadorned grey concrete to create a simple and elegant structure.  But what if this treatment is applied to buildings, such as apartment blocks or universities (for example the university of Essex)?

The pictures below show some of the possibilities for bridges that look similar to this one, by making cuts in the beam.  In each case, what are the advantages and the disadvantages of the design as compared with the actual one.

 A Japanese Bridge – Take your time

The topography of Japan, and its division into islands, has led to the building of many spectacular bridges, including some of the longest in the world, in order to achieve speedy travel between all regions of the country by rail and road.

But the first bridge depicted here typifies an earlier purpose, when a bridge in a garden would be designed to make sure that people stopped to view the garden at certain points.  This one, in the Manyo Botanical Garden in Nara, has different levels, as well as a zig-zag.

The second picture shows a simple beam bridge in the same garden.

More Japanese Bridges

Many Chinese and Japanese beams are curved quite strongly, but have supports that show them to be beams and not arches.

The Japanese Garden – Islands of Serenity     Photographs by Haruzo Ohashi –

    Graphic-Sha – ISBN 0-87040-731-7

Creating Japanese Gardens –  Phlip Cave – Aurum – ISBN 1-85410-423-3

A Helical Staircase

Each step of this elegant helical stair is a simple beam.  Simplicity so often goes with good design.  This is in an old farm building in Tuscany that has been converted into a holiday apartment.

The helical stair has been a popular way of getting steps into a small space since medieval times.  If you climb the tower of an old cathedral or church you will almost always find yourself inside a narrow cylinder of stone, with stone beams bridging the space between a centre pillar and the wall.  These beams keep the pillar straight, in spite of its height, and the pillar supports the beams in the vertical direction.

Bridges, and other structures too, are made by army ants, using their own bodies, to allow colleagues to cross a gap.  When the Herald of Free Enterprise rolled over near Zeebrugge, a very brave and altruistic passenger lay across a gap to allow others to escape.  A fictitious living bridge is described in "Dr Doolittle’s Post Office", by Hugh Lofting.  In this episode, the good doctor is privileged to see the "bridge of monkeys", which is rarely seen by people.  This bridge was actually a suspension bridge, rather than a beam.

Definition of a Beam

After looking at this page, what do you think a beam is?  This web-site is not intended as a text-book, and is not arranged in the logical fashion of a text-book.  Although structures can be classified broadly into different basic types, in practice, few structures are pure examples.

Let’s look at beams.  How’s this for a description of a "pure" beam? 

A beam is a structure that is able to resist bending, without imposing bending moments or longitudinal forces on its supports.  In other words, it can create within itself all the forces it needs, apart from those provided by the supports, to hold it up.  Many larger beams are constructed in the form of trusses, which have their own page.

Many real structures are far from being "pure", but the ideas like "arch", "beam" and "truss" are useful in learning to understand.  Conversely, many elegant structures have been made by combining features of different types.  Look at some structures and work out what is going on in them.

Conclusion

Often comprising an apparently simple series of rectangles, the beam, as we have seen, conceals many subtleties, its lines of force even simulating both arch and suspension  bridge, or lenticular truss, as Brunel revealed in his Royal Albert bridge.  And we barely looked at the effects of live loads.

This multiplicity of ideas, and the complexity of the forces, is not an uncommon occurrence.  Many ideas in science are in themselves very neat and simple, but not the consequences and the working out.  Examples are Newton’s law of gravity, Darwin’s theory of natural selection Einstein’s relativity theory, and Dirac’s electron theory.  But even today, there is no way of calculating exactly the orbits of three gravitating bodies – only approximations are possible.

The truss looks far more complex, but with narrow members the paths and magnitudes of all the forces are fairly simple to work out.  The penalty is the multitude of attachments, where the forces have to change direction very sharply, as undesirable here as hairpin bends on a road.

You could say, fancifully, that inside a beam there are other types of bridges waiting to be released, just people sometimes think of a sculptor releasing a statue from a block of stone.  Whether we use these other types is a matter of economics.  In a small bridge it just isn’t worth designing all those parts and connections and then joining them all together.  But on a very large scale, we are forced into the more complicated design by sheer weight: as Galileo rightly said, simple scaling does not work.

Since the living world was here before the works of people, we could finish on that note.  Because of the growth of plants from seed, into the air and into the soil, most plants are like cantilevers, but in the animal world the beam comes into its own.  The backbone of almost any quadruped is rather like a beam, perhaps a pre-stressed one, from which much else is suspended.  But when you see the wonderfully elegant economy of the trotting rhinoceros, or the alternate storage and release of energy in the dashing cheetah, you see a glimpse of an incredibly complex system, for which the term beam is utterly inadequate. 

The information processing, chemistry, engineering and physics that are needed are all beautiful integrated into a creature that is a good compromise between the conflicting requirements of its life and surroundings.  But if these surroundings change faster than evolution or migration can cope with, there will eventually be one last member of the species who will find itself in the position of the last glass bead game player.  And if the world changes more slowly, each species will vanish as it evolves into others.  Can any species last for ever?

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