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9th July 2001      Back to Bridges    back to Home Page

Arch   Beam   Box Girder   Cable Stayed   Cantilever   Pre-Stressed   Suspension

Mostly empty space, but very effective.

Compared with a plain beam, a truss can look very complicated, yet when you come to calculate what happens, the truss can be the simpler case, especially if the joints are pure hinges.

For each node, you need three equations for the three dimensions of space, with the forces in the member as unknowns.  Neglecting the weight of the members, the forces at the ends of a member are equal.  If this is a bad approximation, the weight can be added in as a vertical component.

If all is well, there are as many equations as unknowns, and everything can be worked out.  If there are too many members, the system is indeterminate at some level, and the calculation will require further techniques.

With rigid joints, and members that can bend and twist, things get harder to calculate.

Some pictures of insect wings have been included above.  Please note that these are not strictly trusses, as they are not based on triangles.  They obtain stiffness in the plane of the wing by means of the thin membrane.  In this respect they are related to I-beams.  Their inclusion here is to highlight the fact that the best distribution of material is seldom the uniform one.  In fact the main bending forces faced by a wing are out of the plane.  Look at the leading edge of a dragonfly wing, and you will see that along the chord, the section is zig-zagged for greater rigidity, so that the wing is not flat.  And of course, animal wings are not rigid – they can use controlled flexibility in ways that are not at present available to the makers of aircraft.  If the leading edge is stiffer than the rest of the wing, merely flapping the wing up and down will create an angle of attack in the airflow.

The pictures also remind us that nature has preceded people in so many ways.  The shapes that nature makes are not often purely "geometric", because nature is not limited by the cost of design and fabrication in the same way as people, and there has been plenty of time to reach optimal solutions.  Nature is an engineer, not a mathematician.  Perhaps the coming of genetic engineering and evolutionary computing may produce designs that are currently unimaginable.

Like the wing of an insect, the roof of a medieval cathedral is rather like a set of struts which support a membrane.  To get such an airy appearance in stone was a great achievement by both Christian and Moslem builders. 

Triangles

Why are trusses almost always composed of triangles?  Most people know the answer – because as triangle is the only rigid shape they can be made with pinned joints – so there is only one set of angles that will go with a given set of sides.  If the lengths of the sides are A, B and C, and the angles opposite them are a, b and c, then it is possible to prove that 

A sin a = B / sin b = C / sin c,  and a + b + c = 180 degrees.

These equations are only satisfied by one set of angles, for a given set of lengths A, B and C.

Wrong.

You can add or subtract 360 degrees to any angle, and the formula still works.  There are infinitely many solutions, but of course they all look the same.  The really important thing is that you can’t get from one angle to the other because the values in between do not work.  To add 360 degrees to an angle you would have to disconnect, rotate, and reconnect, unless the struts were very rubbery.   Mentioning these extra angles was irrelevant in this case – it was just a mathematical trick.  But it does show that things are not always as simple as they look.  And we will see later that the rubbery struts are not entirely fanciful.

Extra solutions to equations have not always been so empty: in 1928 P A M Dirac discovered a quantum mechanical equation that seemed to describe the behaviour of an electron very well.  There was one problem, a very serious one; it provided solutions over and above those that were needed to account for the known behaviour.  It turned out that these solutions were for anti-electrons – the equation predicted that anti-matter should exist.  And it was found.

What have geometry and quantum mechanics to do with engineering?  Nothing.  The point is that you can’t treat engineering like geometry.  In geometry, you can create two shapes that are very similar, with no possible transition between them.  But in real life, if you have two possible configurations for a mechanism, and they are close enough, you can force the mechanism from one to the other, because the components are not infinitely rigid, hence the earlier reference to rubbery struts.  This is the basis of many kinds of latch, based on over-ride mechanisms.  Engineering is not geometry – there is no such thing as a rigid body in the real world.

In quantum mechanics, the situation is reversed: a system can make a spontaneous transition between two apparently stable states, even though the intermediate configurations are impossible in classical mechanics.

Moments before the collapse of the first Quebec bridge, and of several box girder bridges in the 1960s, the incomplete cantilevers were in just the condition of a latch: technically they were in a stable state, but creep, or the addition of a small extra force, was enough to send them into another stable state, lying on the ground or the bottom of the river.  So one purpose of engineering is to make sure that the stable state we want is separated from other stable states by peaks of energy that are insurmountable.  

The Tacoma narrows bridge was an apparently satisfactory design: to get from the normal state to a pile of wreckage in the water would have required far more energy than was available from traffic loads, o from simple deflection by the wind.  What was unknown to the designers was the ability of the structure to build up its energy by storing it in the form of oscillations of increasing amplitude. 

Why not forget about triangles and just make the joints so strong and rigid that they cannot move?  If you have ever tried to break up old wooden furniture you will have discovered one reason.  Suppose that you have two struts of length L, and width W, joined at right angles.  The longest possible dimension in the joint, even if it is glued or welded, is 1.414 W, and if it is riveted, the most distant rivets will be separated by less than this.

Any moment applied to the ends of the struts will result in forces at the joint of not less than L / (1.414 W).  By adding gusset plates, the effective value of W can be significantly reduced.  In some situations, triangular bracing may not be convenient.  In buildings with windows, it is clearly easier to use horizontal and vertical members only, and if wind pressure is the only source of moment, this solution can suffice.  But numerous very high buildings have adopted other solutions, such as strong central tubes or boxes.  The John Hancock Center in Chicago has diagonal members on the outside.

The ramps are well designed also, incorporating a simple but effective way of joining two box-section beams to make a much longer one.

This foot-bridge connects Les Avanchets, a very large complex of apartment buildings in Geneva, with Balexert, a shopping centre.  The cylindrical tubes produce a tidy appearance, but the curved transparent walkway does not quite suit the trusses.

A similar idea was used to create a covered walkway around the Brunel centre in Swindon.  The truss is rather massive, but it provides long spans between the supports, leading to an open appearance at ground level.

A common sight on railway stations is the footbridge, with ramps parallel with the tracks.  The zigzag edge of the canopy at the top of the picture shows that the structure is an old one, in contrast to the later buildings on the other side of the tracks.

Access ramps at a marina.

Bridges across small streams.  A nice feature of small truss bridges is the free guard rail.

The exciting Wildwalk at Bristol includes a tropical area with a tented roof and trussed ends.  Here we see at the bottom right how the support leans inwards, because the funicular can never become vertical.  Next to Wildwalk is Explore At Bristol, which is a magnificent exploratory of engineering and science.  Unfortunately, the road was too wide for the camera lens, and the camera was tilted, leading to converging verticals in the image.

Both the bridge and the gas-holder frame use a truss construction, one very heavy and one very light.  The frame of the gas-holder is reminiscent of the design of rigid airships in the first half of the 20th century, which often used very light trusses.

Wooden trusses are often used in houses to support the roof.  In larger buildings, steel is much more common.  These trusses comprise two I-beam struts, one I-beam tie, and one vertical tie.  The trusses make a contrast with the concrete walls and galleries.  Many large open spaces are spanned by trusses, in the form of girders or space frames.

How a truss works

Bicycle frames are simple examples of triangulated frames.  The bicycle is a superb machine, which can carry many times its own weight.

Many trusses use triangular arrangements, because this makes for rigidity.  The joints, whether welded or riveted, must not be relied upon to provide rigidity, which is already inherent in the triangles.  In principle, all the joints could be hinged.

One advantage of a truss is that it can be built as a complete unit and then lifted or translated into place.  The absence of falsework removes the need for interruption to traffic below, except at the time of installation.  The truss is therefore ideal as a replacement span over railways and roads.

Here is section of the jib of a crane.  The panels are not precisely composed of triangles, but the difference is unimportant.  What is more interesting is that the pitch of the panels in the two dimensions is different.  Why do you think this was done?  What is the effect?

The point is to get the material as far from the neutral axis as possible in order to oppose the bending moment.  Material near the neutral axis isn’t doing anything useful in this context.  For a tension member, of course, you might as well use a wire as a tube, unless the member is very long and in danger of vibrating.

Trusses do not always taper as much as the example shown above: we have to remember that they carry live loads.  The suspended spans of the Forth rail bridge and the Quebec bridge are good examples of curved trusses.  Both girders would have been sizable bridges in their own right when built.

These shapes are discussed in the page about beams.

The process of lightening a structure can be taken a stage further.  The tension members of the Forth rail bridge are themselves made in the form of trusses.  In principle this process of subdivision could go on for ever, resulting in a fractal structure.  But economic considerations render this absurd.  Biological systems, with their different constraints, can evolve efficient structures that engineers could not consider, provided that for each stage of the process, the organism has a higher probability of producing viable offspring as a result.  Evolution cannot pass through stages of lower efficiency to reach higher ones, just as a river cannot cross a ridge to reach a deeper valley.

The ladder of this rescue vehicle can be a very light truss, because the rigidity is provided by the box section cantilever.  All the ladder has to do is support a few people.  Note also the strong box section outriggers that provide stability.  Together with the chassis, these form a bridge that carries the machinery.

When the Wright brothers built the famous Flyer, they braced the two pairs of wings together with truss-like arrangements of struts and wires, which became the standard structure for many years.

The diagram below shows bridges constructed as separate spans and as a continuous truss.  What are the advantages and disadvantages of each type?

The third diagram shows two higher spans for navigation below.

Some Shapes for Trusses

Here are some possible shapes for trusses.

The middle three are shapes for trussed beams that have been commonly used; they include the bowstring truss and the lenticular truss.  These shapes show how the different types of bridge are not rigidly distinct: the bowstring truss is related to the tied arch, and its inverse is related to the self-anchored suspension bridge.  This shape has lower clearance below the deck than the bowstring, which can be a disadvantage.  The two outside those are not practical shapes, but the lowest type truss has been used many times for two-pinned arches, such as the one in Newcastle and and magnificent ones by Eiffel.  The top shape is weird, but if you look at a picture of the Tower bridge, also rather weird, you will see the similarity.  Note that the details of individual trusses may vary greatly, and many trusses will not include all the cross-members shown here.

More ideas are shown below.  Allowing for the removal of redundant members, and other modifications, you can see resemblances to Telford’s iron arches and Sydney Harbour bridge, for example.

Here are some types of truss which are sometimes used for the main spans of multi-span bridges.  Again, designers will usually omit some of the members that are included here.

The next diagram shows a method of creating a long span with high clearance.

This idea can be simply adapted to create an arch.

The next diagram shows the same design with redundant members removed.  See also Indeterminacy.

All these main spans with through connection to the side spans can in principle be constructed by cantilevering, keeping the navigation channel clear at all times.  The next diagram shows the two sides at different stages of construction.

Now let’s colour some of the members in red for compression and blue for tension.

But what happens when the two halves meet, and we complete the top and bottom chords?  We could arrange over-size rivet holes and join the two parts, retaining the existing forces, but that is not the usual way.  By jacking the bottom chords apart, the span is turned into an arch, relieving the tension at the two outer piers, and creating outward thrust at the main abutments.  But there is still the top chord.  That, too, can be jacked, until it is in compression and not tension.

Can we still confidently assign blue and red to the vertical and sloping members?  What we learn from this thought experiment is that we cannot always, just by looking at a structure, know even qualitatively what all the parts are doing.  In fact, it is quite possible to construct in such a way that we cannot fully understand even by calculation.

These railway arches in Berne are trussed, as the vertical supports and the deck.  Unlike Eiffel’s arches, which taper towards the springing, these taper towards the crown.  What does this tell us about the structures?

Magnificent examples of trussed arches were created by Eiffel.  These had maximum depth at the crown, tapering to a point at the springing, and were therefore two-hinged arches.  But transversely they were wide at the bottom and narrow at the top, to withstand the winds that can blow along the valleys.  One of the weaknesses of the first Tay railway bridge was the use of parallel piers instead of using a straddle.  

The  Royal  Albert  Bridge  at  Saltash,

by  Isambard  Kingdom  Brunel,  Engineer

The penalty was paid when the bridge caught fire in 1870.  The stresses were released, and the bridge was no longer usable.  The spans were replaced by arches.

It is recorded that Stephenson referred to his bridge as a "magnificent blunder" when he heard of Roebling’s design for a suspension bridge at Niagara.  This remark, though generous, was in fact unfair to himself.

Given the state of knowledge, the Britannia bridge was a good solution to the problem.  To have invented the box-girder and pre-stressing could only have been done by a great and imaginative engineer.  The great tubes were far beyond anything that had been done before, but Stephenson’s research and preparation were meticulous.  In fact, Stephenson’s original instinct was right, in that suspension bridges have seldom been successful for railways.

It was shown that rectangular tubes were stronger than round ones.  This may not seem obvious, but we have to remember that the trains were to run inside the tubes, which could therefore not have large flanges or diaphragms.  Systematic tests were made, using several different cross-sections on a reduced scale.  A circular tube has the greatest symmetry, and copes perfectly with axially symmetrical stresses such as torsion and pressure differences.  An airliner and a submarine illustrate this well.

But in a bridge, supported at its ends, the symmetry is broken by the vertical bending moment.  A deep section is needed.  So why are Brunel’s tubes so good?   We must not confuse geometrical symmetry with the symmetry of the stress-flow.  Suppose we create a curved coordinate system that follows the dead-load stress path.  In that space a well-designed arch will appear more-or-less straight.  In the Saltash bridge the bending moments are taken mainly by the truss action, leaving the tubes to deal with the thrust.  A brilliant and beautiful solution.

Let’s compare the Saltash spans with the Britannia spans.

There it is.  Brunel’s lighter trusses saved one third of the material costs, and made lifting into position much easier.  Yet in general, would you normally associate Brunel with economy?  He had in fact toyed with this idea twice before, at Windsor and at Chepstow, so he must have dreamed of it long before starting work at Saltash.

At Windsor the tube had a triangular cross-section.  At Chepstow the tubes were elliptical in cross-section, and some bars were used in a kind of cable-stayed configuration.  On the sides of the deck Brunel embossed curved lines continuing the lines of these stay-bars, hinting at the suspension aspect of the spans.  This bridge lasted until quite late in the 20th century, when it was replaced by a new bridge.

Why did Brunel not build a simple bowstring arch?  The Saltash bridge had to provide clearance for shipping.  It was desirable to have the railway as near as possible to the bottom of the bridge to miminise gradients on the approaches.  The arch had to be entirely above the railway loading gauge.  With horizontal ties, the bridge would still have required hangers to hold the deck, and bracing for rigidity.  By curving the tie of the arch into a suspension chain, Brunel could use it to help the rigidising action.

He could have  provided a tied arch on each side of the track, but any cross-bracing would have had to be above the loading gauge.

The Saltash tubes are elliptical in cross section – wide enough for the bridge to accommodate one train, and deep enough to provide the required stiffness, allowing for the movement of the line of thrust as a heavy train goes over.  Internal flanges, both transverse and longitudinal, increase the stiffness.  A good picture of this is given in "Track Topics", a book for boys of all ages.

An example of the relative weakness of a tube against bending is seen in the Forth railway bridge.  The tubular lower members can give the illusion of being arches.  In fact the individual sections are straight.  

At about the mid-point of each tube a tie connects it to the junction of the strut and tie above it.  The unsupported span of the tube is halved, and its resistance to deflection greatly reduced.

Note that the main tension members of this bridge are trusses, both in the top chord and within the cantilevers.  The railway itself is carried by a truss, which is a bridge within the bridge.  And the suspended spans are large truss bridges in their own right.  The Forth railway bridge is one of those things which when seen are found to be as impressive as their reputation.

Many suspension bridges have trussed decks for rigidity, a practice which was emphasised after the Tacoma Narrows collapse.  Many years later, the aerodynamic deck began to be used instead, with saving in weight and cost.

In the 19th and 20th centuries, many seaside resorts had at least one truss bridge to nowhere, like this one, photographed on a dull day in winter.  In some cases, small ships could moor at them, but in others there was only a pleasure hall at the end.

In the 19th century, a number of designers experimented with a variety of trusses, such as the Howe, Pratt and Whipple trusses.  This link is to a picture of two bridges which combined arch and a truss.

The truss can be expanded into the third dimension to create many kinds of space frames, which are useful for making light roofs, often over wide open areas.  By curving such a structure into the third dimension, Buckminster Fuller created spherical forms that were very light and very rigid.

Some trusses are not among the most elegant of bridges.  This one carries a railway across the river Lune.

Definition of a Truss

After looking at this page, what do you think a truss 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 trusses.  How’s this for a description of a "pure" truss?  

A truss is made entirely of straight members that are in pure compression or tension, all singly pinned at the joints, and constructed in such a way that removal of one member would allow the structure to deform significantly.  Trussed arches and trussed beams exist, so these pages should be looked at for further information.

In practice, joints are usually welded, bolted or riveted in such a way that that the joints confer extra stiffness on the structure.

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.