Some pictures of insect wings are included in this page. But if you look for triangles in the wings of insects you will probably look in vain. Why is this? The reason is that the triangles of a truss are to provide stiffness against movements in the plane of the truss: in an insect wing this function is provided by the membrane. The stressed skin of many aircraft is analogous. The veins of the insect wing are there to stiffen it against bending out of the plane, though their resistance to torsion may be weaker. In fact, they do not prevent it – they control it, because some bending and torsion can actually be beneficial or even essential in creating propulsion and lift. The veins may also guard against buckling caused by shear stress: without the veins, although the membrane would be able to resist shear very well in its own plane, it may easily give way by bending into the third dimension.

So we need to be very careful in looking for analogies and general rules. Things that look similar may not be functionally related. The insect wings were included to illustrate the idea of retaining stiffness while minimising weight, but they are not trusses. In the case of the stiffened membrane, a grid can be created from shapes such as triangles, squares and hexagons. Which polygons give the lowest weight for a given performance?  If you look at the pages about beams, you will find pictures of bridges which use I-beams, divided into rectangular panels by vertical flanges.

The wings of the lacewing are built mainly of hexagons. Most cloth is based on rectangular weaving, because it is not required to be rigid in any plane, and because any other construction would be difficult to achieve. But the cut of the cloth, relative to the weave, makes a tremendous difference to the way it hangs. The art and engineering of clothing design requires appreciation that the materials are anisotropic, and that the foundation on which they hang is very mobile.

What is the connection between a damselfly and a B52? The wings of the insect are based on quadrilateral and pentagonal cells, braced against shear by a thin membrane. The fuselage of the B52, like those of many other aircraft, is based on rectangular cells defined by frames and longerons. The shear is handled by a metal skin, which in this case looks inadequate for the job in the cantilevered front section. But what matters is the behaviour when the aircraft is in its natural element, and aerial photographs show clearly that these wrinkles are smoothed out when aerodynamic forces lift the nose of the aicraft. The photograph was taken at the superb Imperial War Museum, home to a large number of aircraft, at Duxford.

Before the introduction of the stressed skin, and even afterwards, in many light aircraft, a triangulated frame was used, and covered with a light but strong fabric which had no load-bearing capability. Another Duxford picture.

In another hangar in the same museum, an aircraft being restored, and also the roof of the hangar, use truss construction for lightness and rigidity.

This hangar at Duxford is based on a series of trusses, providing a large uninterrupted space for the display of aircraft. Why do you think that the trusses at each end of the hangar are on the outside of the cladding and not on the inside?

In contrast to the previous example, here is a particularly ugly example. Unfortunately, it is quite easy to build an unattractive truss, whereas arches and suspension bridges are usually quite attractive. The skew of the railway below, from perpendicular to the road, is about 48 degrees. This truss is in fact alongside a skew arch in brick. The truss was built to convert the stretch of road into a dual carriageway.

This type of truss was frequently used in the 19th century to carry railway tracks. Why do you think it was preferred over a simpler array of triangles? This example is in a bad state.

This bridge employs a different structure, except in the centre of the spans, though it still includes a huge number of members. Both bridges span the Clyde in Glasgow, a good place to see a lot of bridges in a small area.

The roof of the Science Mall in Glasgow is a shell based on quasi-square cells. In spite of the thickness of the tubes, ties are used to triangulate the system.

In the wires that brace the wings of this biplane, we see triangles. Although the air flows over the two pairs of wings tend to interfere with one another, the gain in rigidity is very valuable if the aircraft is intended for aerobatics. Furthermore, two pairs of shorter wings have a much smaller radius of gyration than one longer pair, making the aircraft quicker to bank. Remember that the moment of inertia is proportional to the square of the radius of gyration, for a given mass. The short span and the bracing enable the moment of inertia to be small. A good example of the aerobatic biplane is the Pitts Special. If you have been in both gliders and light powered aircraft, you will know that a glider flies in a more stately manner than the small plane, especially if it is one with wide span. One or two gliders have wing spans approaching that of a Boeing 737. The spars and ribs of the wings are not based on triangles. Some of the rigidity can be obtained by using stresses in the skin.

Is this bridge a truss?  Most of the members look very thin. Can they really take compression? Perhaps this is really a tied arch.  What do you think? Look at the second picture, copied from the page Trusses Three, which illustrates the forces in a uniformly loaded truss.

The simplest truss is the kingpost truss, which has only two triangles. Click here to download a program that simulates the behaviour of a kingpost truss, assumed to have negligible weight compared with the load. The load can be moved, and the height of the truss can be varied.

The ability of triangles to provide rigidity has been known for a long time. Compare these buildings with the Bank of China building, the Hong Kong and Shanghai Bank building and the John Hancock centre.

This part of the page has dwelt on triangles, probably too much. As we saw earlier, geometry is not the same as engineering. Geometry is also not the same as surveying, though triangles are the basis of surveying. Using the properties of triangles, we can measure the position of a point that is inaccessible, just as a triangle of struts can fix a point in a structure.

But there is a difference between the engineer and the surveyor. The engineer uses a triangle to fix a point: once he or she is satisfied that the employed members are strong enough, that is the job done. The surveyor’s philosophy is quite different: he or she will make measurements from a number of different baselines, and will compare all the results. They won’t agree, because no physical measurements except counting objects can be perfectly exact. So the surveyor must average the results to get a "best value". The surveyor can even give different weights to different values, if some are considered more accurate than others.

There is more – the surveyor will have some idea of the potential inaccuracy of the measurements. The spread of the actual results can be compared with the expected spread, to provide a test of the consistency of the data, both with the expected inaccuracy, and with each other. Standard procedures are available for this.

Such problems could not occur in pure geometry.

What if the engineer decided to strengthen something by using more than one triangle to fix a point in space? What is wrong with this? Well, like surveying, engineering is subject to errors. What happens if you try to fix a point in two different ways, and the two ways are not consistent? The surveyor can do some maths to extract and average result. The engineer’s struts and ties have to stretch or contract in order to fit. In other words, unwanted stresses are introduced. If the stress in a member is pushed too near the permissible limit, the structure will fail under a load that is less than the design load. By adding material, we have effectively weakened the structure.

Please see indeterminacy if you need more information.

This is a very neat foot-bridge, over the M5 motorway at Michael Wood services. The plain box-section struts and the simple triangulation give this bridge a pleasant appearance from the motorway. Simple calculations based on this bridge

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 edges of the platform canopies show that the structures are old.  In fact they are the original structures.

Access ramps at marinas.

Bridges across small streams and a river. A nice feature of small truss bridges is the bonus 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 narrow for the available camera lens, and the camera had to be tilted, leading to converging verticals in the image.

Three different ways of making a crane.

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.

The helical guides here eliminate the need for the complicated frames.  Only the stairways are trussed. Why are the metal plates arranged like bricks?

Wooden trusses are often used in houses to support the roof. In larger buildings, steel is much more common. These steel 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.

Steel framed buildings are sometimes provided with a degree of triangulation to increase resistance to distortion. This arrangement is one of three that are present at each end of a long building.

Why are triangles not needed throughout the frame? And why are they needed throughout a truss bridge?

Many buildings in Slovenija use wood, and a large proportion of these employ trusses for stiffening, as we see in the pictures above. The first picture also includes frames that hint at both arch and strutted beam. The fourth picture shows a modern example of a very old Slovenian design. Older examples, like many other Slovenian buildings, had wooden roofs. The modular design can be made in many lengths, with braced frames at intervals. The main area below can be used for the storage of tractors and farm equipment. The parallel bars along the sides can be used for drying hay. Along the centre, seen in the fifth picture, a long narrow storage volume is bounded on the two long sides by trusses. The sixth picture shows a bus-stop shelter which pays tribute to the tradition al designs.

Tourists in England are familiar with the timber framed buildings which were once a standard type. This example looks as though some attempt was made to use triangles for stability.

In Britain you can see early trussed arches, such as these splendid examples by Thomas Telford.