Trusses - Part Two
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How a truss worksBy building a truss from narrow metal strips a very strong and light structure can be built. Many different types of truss have been used. One advantage of a truss is that it can be assembled in a convenient position and moved into position as a complete unit. This is not always possible. The suspended spans of the Firth of Forth railway bridge were built as cantilevers and joined as trusses when the halves met. The two suspended spans look rather small by comparison with the rest of the bridge, but they are actually 107 m long, which was a sizeable span at the time.
Bicycle frames are simple examples of triangulated frames. The bicycle is a superb machine, which can carry many times its own weight. Note, however, that the frame is a structure, not a mechanism.
How does a truss work? The diagrams below give a simple explanation. The top diagram shows a simple beam jutting out from a support. The green vectors represent the weight of the beam, and the upthrust from the support, which are equal and opposite. The red forces are needed to prevent the green forces from rotating the beam. (See the page about Moments) At the top left, the resultant of green and red is a sloping line, and in fact the beam is in tension at that point. At bottom left the resultant is again sloping, but here the beam is in compression. Throughout the beam the forces are generally not parallel to the axes of the beam.
In the lower pair of diagrams, most of the material has been removed. The loads are carried by specially designed members - thick struts for compression, and narrow for tension. The difference in thickness arises because a strut is inherently unstable - increasing the tends to bend it more. Tension members tend to straighten under load.
These diagrams apply to a trussed cantilever. Now look at the diagram below, which shows one half of a trussed beam.
Again the weight is shown as acting at one point. Again the weight and the support are attempting to rotate the structure, resisted by the forces shown in colour. Ignoring some of the members, we can trace a route (red for compression) from the centre to the support. The three red members are akin to a flat arch. With this arch as a support, we can trace a route for the tension along the blue route. If we now allow for the presence of the other four members, we can see that the forces will be shared out among all the members, but the general idea is the same - a beam includes elements of arch and suspension.
To calculate the forces in all the parts, we could write equations for each connection point. At each point, the sum of the vertical components of the forces must be zero, or it would move. Likewise for the two horizontal directions. So we have three equations per node. If the structure is "perfect" (having no unnecessary parts) the equations can be solved uniquely. Thus, although the truss looks complicated, we can in some ways understand it more easily than we can understand a beam.
You can now see that the imaginary bridge at the top of the page has not been drawn properly: it should have had the thick part at the top, and the thin part at the bottom. Of course, the lower part would have to be thickened to take the local loads of the train.
The directions of the members need not follow the lines of force of the original beam, because the forces are constrained to lie entirely in the struts. The design can be optimised for lightness, strength and economy. For a small structure it may be cheaper to have a repetition of parts, rather than go for a fully optimised design. For the same reason small aircraft often have parallel wings rather than the more efficient tapered ones, and the ideal elliptical shape is very seldom used. The Supermarine Spitfire had elliptical wings, which must have increased the cost significantly. The Hawker Hurricane was cheaper, and also very effective, as was the Bf109.
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.
construction of a gate is in principle a simple matter, but there are
practical difficulties. The gate must be wide enough for large
farm equipment. This has two consequences. Firstly, the gate
requires at least one diagonal member to maintain its rectangular shape,
making it a trussed cantilever, and secondly, it is hard to keep the
supporting post vertical, because of the large moment of the weight. The post will usually be anchored in soil, and a large expenditure on a
foundation is hard to justify. Consequently, these gates often
droop over a period of years, a behaviour which is exacerbated by
people, usually tourists, who lean on them and climb over them, often at
the end furthest from the hinges, where the moment is greatest.
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 Severn bridge at Upton-on-Severn comprises two plate girders joined by cross-bracing. Triangulation ensures rigidity, but is much lighter than a load-bearing truss. Note also the footpath cantilevered out on brackets - well, that's what cantilevers are.
Developing a Beam into a Truss
At the bottom of the picture the diagram represents a simple plate girder. In the next diagram some attempt has been made to shape it to suit the bending moments. In the third diagram this is taken further, and in the fourth picture the structure is greatly lightened by changing it into a truss. Finally, at the top, we see a tied arch or bowstring arch.
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.
Here is another way of developing a beam into a truss.
Some of our examples of trusses are tapered towards the ends. In the pages about beams, we see that this reflects the distribution of bending moments. Trusses do not always taper to almost a point at the ends: 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. In spite of the overall curved shape, all the individual members are usually straight, because they are intended to carry forces along their axes. The example shown here is unusual in having a top chord based on curves.
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.
These two pictures show the cantilever bridge in Quebec. Because of the great size of the bridge, a large proportion of the members are trusses, some of them substantial structures in their own right. The central truss is a bridge in its own right. Although it appears to be continuous with the cantilevers, this is an illusion fostered by short top members which are structurally not really needed. The pictures are about forty years old, and the colours have changed badly. No more cantilever bridges were built on this gigantic scale.
For comparison -
main span 1700 feet/518 m - suspended span 350 feet/107 m
main span 1800 feet/549 m - suspended span 640 feet195 m.
The suspended span of the Quebec bridge weights 5200 tons: remember that its supports are 580 feet/177 m from the feet of the cantilevers. Compare this with the small balsa trusses that are entered into competitions: a load : weight ratio of over 2000 : 1 has been achieved. It's all a matter of scale. As another page shows, there is a limit to bridge spans at which they can only just hold themselves up, with nothing in hand for a load.
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 a rescue is in progress.
the early days of the American railroads, when it was necessary to build
many bridges as cheaply and simply as possible, a vast number of trusses
was built. Many different trusses were invented.
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.
When we see an apparently complicated network of struts, can we work out why each one is there, and why no others are there? Two different questions can be asked. Firstly, what is the simplest arrangement that will make the construction rigid? Secondly, which members are in tension and which in compression?
Looking at the second question, we could ask, thirdly, does each member remain always in tension or always in compression as a live load moves across the bridge?
Let's see if we can answer these questions.
If we want to find out which members are in tension and which in compression, we can often get somewhere by imagining them made of something like stiff rubber, that can change visibly when stressed. A member in tension will tend to stretch. A member in compression will tend to shrink.Or we can imagine a normal rigid bridge, and then imagine removing one member. By working out what would happen we can see whether the member was in tension or in compression.
This has been done in two of the diagrams below. What can you tell about the missing members?
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 in London, 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.
Click here to see a picture of a lenticular truss.
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. This can happen if we include more members than are strictly necessary, so that we cannot know how the forces are shared among the load paths. See also Indeterminacy.
This link - Design a truss - will enable you to design your own truss bridge and find out the forces in it.
This picture shows a typical small tower for antennas. Three is the smallest number of legs that will produce rigidity in all directions, and three is the number provided. These legs are always in compression, unless some catastrophe is about to happen. But all the remaining members may act as either struts or ties, depending on the strength and direction of the wind. The green lines show the position of some of the main cross members. When these are in compression they may fail by buckling, so subsidiary connections are made, shown in red. The third dimension is catered for by the horizontal triangles shown in yellow.
Not many people when travelling would think of electricity pylons as interesting, but in fact they show a remarkable diversity among different countries, although they are solutions, often ingenious and elegant, to the same sets of problems. Some are even two-dimensional, relying on guys for the third dimension. And some antenna towers are very narrow, relying on guys for overall rigidity, leaving the narrow trusses to deal with local rigidity.
These railway arches in Berne are trussed, as are 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
bridge, opened in 1859, carried the Great Western Railway over the wide
and deep river Tamar between Devon and Cornwall. It has two main
spans of 138 m each. These illustrate several principles of
design, as well as Brunel's ability to find unusual solutions for
problems. The design signals very clearly the principles by which
The spans were built on land, floated out on pontoons, and raised into position. Each truss comprises a rather flat tubular arch and a suspension chain, linked by other members to form a rigid whole, with a suspended deck The idea is that the outward pull of the arch is balanced by the inward pull of the chains. This principle has been used in many large domes, though in those, the restraining chain or cable is at right angles to the thrust. Each Saltash span weighs 1190 tonnes. The spans can be regarded as tied arches or as self-anchored suspension bridges. They do look like trusses, in that they have some diagonal bracing for stiffening, but unlike most trusses they have vertical members, not sloping ones, which are dominant, and so they are probably not to be regarded as trusses.
These 138 m spans may not now seem impressive compared with the great feats of the 20th century, but if we compare them with other structures of the age, we see that they were not so small. The suspended trusses of the Forth railway bridge, made of steel, a much superior material, spanned 107 m, though in 1917 the Quebec bridge was completed, with a suspended truss of 195 m.
What is the inscription on the portals of the Saltash bridge?
I. K. BRUNEL
The bridge was Brunel's last design. You can read more about it and see more pictures here.
Click here to see a picture of a lenticular truss.
tube uses material efficiently to make a strong, rigid member. Tubes are used extensively in the natural world as well as in
manufactured articles. The veins of an insect's wing are filled
with fluid under pressure to unfurl the wings after emergence.
Subsequently they provide a stiff network, designed to allow the wings
to bend in just the ways that efficient flight demands. Flow of liquid
in the tubes can be used to convey heat energy to the body on warm days,
as insects do not have an internal regulator of temperature. Many plant
stems and bones are tubular. The principle is to get the material
as far from the axis as possible. This construction resists
torsion very well, and resists bending because the tension and
compression are far apart, providing a large moment. Other
examples in bridge building are the Forth rail bridge and the Menai
Straits tubular bridge.
It is one thing to say that a tube is the ideal shape for a compression member, or strut. It is another thing to implement the idea. Certainly a tube achieves the ideal of getting the material as far as possible from the neutral axis, but a large tube is not a simple or a cheap thing to make. In the Forth railway bridge and the Saltash railway bridge, the tubes were built up by rivetting many curved plates together. This was labour intensive. Furthermore, if a tube is very large it will need internal circular flanges, and possibly longitudinal flanges, to stiffen it. It is not easy to connect these to a tube without introducing unwanted strains. Click here for more about tubes.
is much easier to connect to a plate girder. The same problem arises
when a tube has to be connected to another member. Achieving this it
satisfactorily is not easy. Look at a photograph of the foot of one of
the great towers of the Forth rail bridge. he method of connecting the
tubes of the towers and of the cantilevers is quite complicated, in
order that the stresses could be transmitted satisfactorily from the
cantilevers to the piers, and from the piers to the foundations below.
In fact a structure can even be made weaker by adding
"strengthening", if the additions introduce undue strains, and
therefore stresses, that were not there before. This can happen if
the resulting structure is over-determined and poorly constructed.
A similar type of problem with stress concentrations occurs in designing the fuselage of a pressurised aircraft, or the deck of a ship, when material has to be removed from the ideal tube. The fuselage of an aircraft has to be pierced by various holes for doors, windows, wheels, antennas, and so on. The openings have to be designed carefully, to prevent stress concentrations. The Comet 1 airliner suffered explosive decompression when fatigue, starting at a hole, resulted in catastrophic spreading of cracks. This phenomenon is now much better understood, and all designs would now include measures to reduce the probability of cracks being generated, and also measures to prevent their propagation over long distances in the structure.
cornered hatches in the deck of a ship can result in stress
concentrations which can be the source of cracks, which can propagate if
not stopped. As a result of considerations
about construction, tubes are not employed very often in bridges hat
is beautiful to the engineer, the aesthete, and the financier may differ
quite strikingly. Fritz Leonhardt, in his book "Bridges", explains the desirability of reaching a
satisfactory resolution of these questions.
Nature is not restricted by the same constraints as people. Nature's constraint is that each step in evolution be attainable from the previous one, and that it should be a slight improvement in some way. The improvement need not be one that can be recognised millions of years later, when the use of an organ may be completely different from a previous use. Improvements that require a temporary set-back, however small, in overall probability of reproduction, cannot happen. Evolution has no foresight.
Stephenson completed the Britannia tubular bridge over the Menai Strait
in 1850, by driving the last rivet himself. Like the Saltash
bridge, it has two spans that were floated across the river and raised
into position. Stephenson was able to use an idea that was not
possible with Brunel's design. The spans were connected together with a
small angle of divergence from a straight line. When they were
lowered gently into their final positions, the induced stresses
cancelled some of the unwanted stresses that a straight beam would have
suffered. Stephenson had anticipated Freysinnet by introducing
The penalty was paid when timber on the bridge caught fire in 1970. 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.What are the Saltash spans? Tied arches? Self-anchored suspension bridges? Trusses? Or something unusual by a great individualist? In the USA a number of lenticular trusses have been constructed (see links below).
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. To prevent the tubes from buckling, 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 these pictures we see that that numerous members are themselves trusses. The fifth picture shows clearly that only the strut are treated in this way, and that the bracing is against lateral bending. The Tour Eiffel and the Forth Rail bridge are classic examples of the subdivision of members. In principle the subdivision could go further, but it would be uneconomic. The economics of the natural world are completely different, and many layers of subdivision may be found. In mathematics, where economics has no place, the existence of fractals demonstrates that infinite subdivision is possible, as in the Sierpinski gasket. Such plane curves can have a finite area, but infinite length, and solids can have infinite area with finite volume. And you can read about facts such as the area of a human lung being equal to that of a tennis court. If you make a section through a bone from a bird, which has to be strong and light, you will see the results of millions of years of evolution. Evolutionary "design" is one of numerous ways in which people are beginning to imitate the ways of nature.
fuselage of an airliner is tubular. This gives great strength to
resist many kinds of forces. The pressure difference between the
inside and the outside generates large tensions. The weight tends
to bend the front and rear downwards as they are cantilevers projecting
from the wings. In fact the fuselage is a continuous beam, with
holes for doors and windows.
Most tunnels are tubular, like submarine hulls, so resist the immense pressures around them. Many living structures are tubular, such as plants stems, insect limbs, and bones. In the case of birds, where lightness and strength are in direct conflict, the interior of the bones may be stiffened by innumerable struts, enabling the tube itself to be thinner without buckling.
a very elegant solution using tubes see the section about Hampton
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. The domes of the Eden Project are good examples.
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.
Near the beginning of the page, a problem was mentioned, a problem that looms over the simplicity of the truss. What happens at the places where the parts are joined? All those simple longitudinal forces begin to diverge from simple straight lines: the price we paid to avoid the complexity of wide members like beams is to concentrate our difficulties in a few places. To find out more about this, see Truss Joints.
If you got this far, try a superb game about bridge building - http://firingsquad.gamers.com/games/pontifex/default.asp.
Here you can design your own truss span.
If your question is not answered in these pages, please send an e-mail.