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.

The webs of these beams, which will support the floor of a room, are made largely from recycled chips of wood, bonded by glue, forming a strong composite material.

How wide should we make the two flanges, in relation to the height of the web? The first picture shows some examples. The next picture shows why wide flanges are not useful. A thin slab of transparent plastic has been clipped to a mirror with some coins underneath the slab, to simulate  crudely the bending of a flange along its centre line. The mirror makes it easy to see that the slab does not take up a cylindrical shape: instead, it takes the position of lowest strain energy. If, therefore, we made an I-beam with wide flanges, much of the material would be relatively ineffective. Because the effect of the flanges falls away with distance, they are often tapered in section. The third picture, taken with the mirror covered, shows how the curvature varies. The blue lines were drawn on the plastic slab, while the straight red lines were drawn on the picture. The surface has a very subtle shape.  Along lines at right angles to the imposed bending, the curvature changes polarity along two lines, one each side of the centre.

These pictures illustrate a common way of extending the beam concept into an extra dimension. One steel I-beam (not shown here) is placed along each side of the building. Transverse rolled steel joists (RSJs) are used to span the gap between the main beams. The gaps between the joists may be spanned in a variety of ways.  Here the floor is built from a set of preformed steel sections. A completely different way of extending the beam idea is to build a slab, usually in concrete. You can see that the detailed stress calculations are more difficult for the slab solution than for the RSJ solution, where the forces can be calculated separately for each member.

The plastic slab is provided with an array of dots, which, together with their reflections, makes a Moiré pattern. The pattern is a little clearer in this picture, which has not been sharpened. The Moiré fringes are suggestive of contours of displacement, and indeed Moiré effects are often used to measure small movements.

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 pictures below show some of the I-beams that support the roof of SeaLife at Birmingham. They radiate from a single pin near the ground.

This photograph shows a piece of packaging material made of thin card and paper. It comprises two sheets of card, separated by many strips of paper which have been glued together to form hexagons. Unlike the combs of honey-bees, this structure does not have six-fold rotational symmetry: the glued sides of the hexagons are shorter than the other sides. Being twice as thick, and being stiffened by glue, they need not have the same length as the others, and making them shorter saves on glue. The structure is remarkably stiff, given its very low weight.

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, and its mechanical equivalent, the Bowden 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 riveted 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 the bending moment curve, because the new shape will give a new curve. Can the trick even be done?

Here are some examples of non-parallel beams, in the form of trusses. Don’t forget that if a structure creates no horizontal thrust at the ends, it is acting as a beam.  

Does this mean that a tied arch acts as a beam? Think about the way it derives its rigidity. How does it differ from a truss with a similar shape?

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, compared with a diagram shown earlier. 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.

Lastly, here are two pictures that show clearly the deflection of a span of the Royal Albert bridge at Saltash when loaded by a train. There is no such thing as a rigid body: every body deflects under the action of external forces until it generates within itself exactly the right forces to balance the external ones. These pictures would have been more convincing had they been taken from the same viewpoint with the same equipment, as the movement of the bridge against the background would have been conclusive. One function of the designer is to achieve an acceptably small deflection with the most economical arrangement of material.

Here is a Japanese garden bridge with two spans which taper at the ends, whether for reasons of aesthetics or of structure.

The first diagram below shows how a simple truss can be related to the simple beam shown above. It also hints at the relative crudity of most artificial structures compared with most natural ones, because of the different criteria for efficiency in the two cases. What are the causes of these differences? What we have done in this picture is to remove almost all the material, making the forces pass through narrow struts and ties. The fact is that in the simple beam, a lot of the material is only weakly stressed.  From this diagram we see also why the I-beam is so much more efficient that the plain beam. The two flanges take the compression and the tension, while the web takes the stress. In many bridges, the web has flanges to prevent it buckling into the third dimension. I-beams are discussed elsewhere in these pages.

The thicknesses of the lines in this diagram of a uniformly loaded truss represent the magnitudes of the forces. The variations of tension, compression and shear are strongly reminiscent of those in a solid beam.

If you wanted to make an ideal beam, you could make its outline follow a red contour and a blue one, and the lenticular truss uses the same idea. 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 most of the ribs to be the same. The A-10 Thunderbolt II was designed to maximise the interchangeability of parts. But aircraft such as the Concorde or SR-71A, designed for extreme performance, have far fewer compromises. All four air intakes of the Concorde are different: symmetry is broken because all the engines rotate in the same direction. Apart from the economics of producing clockwise and anti-clockwise versions of an engine, such a method would probably have led to mistakes in assembly or in maintenance.

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. Among artificial constructions, it is perhaps only where extreme performance is required that we see relatively complex shapes, for example in fast aircraft and yachts.

Artists such as Henry Moore and Jean Arp produced some very subtle shapes, but although these were often strongly reminiscent of natural shapes, they were not driven by the requirement to minimise energy or stress. The artists were probably interested in form rather than function, though much of Moore’s work certainly conveys impressions of power and strength.

Having seen how the stresses behave 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.

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