Structural Failure - Bending and Buckling
On 29th August, 1907, one incomplete half of the Quebec bridge fell into the river, leaving a crumpled mass of steel, and eighty five dead men. The wreckage was over a hundred feet/30 m below the original position, and therefore had far less energy than before. Some of the difference had gone into bending the girders: the rest had gone into making waves and sound, eventually being dissipated as heat.
All bridges would have less energy if they collapsed, so why don't they? We all "know" that it is because they are too strong, but it's not quite that simple. We can hang a heavy weight on a length of rope, but we cannot make the rope stand up vertically, even without the weight. This is trivial - it is because the rope is flexible.
What if we use a stiff piece of wire? If we now point it upwards, and gradually increase the length, it will eventually, and suddenly, flop over to one side. The wire is definitely strong enough to support its own weight as a pillar - it is not floppy like the rope - yet it fails.
Why do things not fail if they are designed properly? As we saw, many things have less energy if they fall over or collapse. The Eiffel tower is a good example. If we disconnected two neighbouring legs from the ground and started pushing that side, we would find it very hard to topple the tower. Why? Because tilting the tower would at first raise the centre of gravity, which requires energy, as in the simple example below.
Note that as the object tilts more, the green energy line becomes less steep. This means that the required force decreases, and it falls to zero as the diagonal becomes vertical, and the object becomes balanced on its corner. Any further tilt, and it now needs a force in the opposite direction to stop it falling.
The next diagram shows the variation of the force and energy needed to topple an A-frame, as a function of the distance between the legs, assuming that the weight does not vary with the opening of the legs.
The structure could also fail if a leg were to bend or buckle. Again, this should require more energy than would be gained from lowering the centre of gravity. A large proportion of the material in the Eiffel tower is in the form of bracing, which effectively divides the compression members into shorter sections which are resistant to buckling.
It doesn't matter if a structure is not in the lowest conceivable energy state, as long as it has to acquire more energy before it can reach that lowest state. In other words, the structure must be in a local minimum of energy in an imaginary space that is described by all the variables. That minimum must be deep enough to contain any energy that can be added by external forces such as live loads, wind, temperature effects, and so on. The first Tay bridge collapsed because the wind found it too easy to wrench the piers from their supports and tip it over.
Suppose you have to break a piece off a shrub with your bare hands. You won't push the branch together, you will soon give up pulling it, and you will end up bending and twisting it. Objects can in fact fail under bending, compression (by buckling or crumbling), tension, and torsion.
Bending and Buckling
If you take a piece of piano wire or a thin strip of plastic, and push the ends, it may bend. If you push more and more, it doesn't become very much harder to push. This is in contrast to simple compression, in which the object exerts an increasing force as it shortens. So the bending behaviour is unstable, whereas the compression is stable. If the energy of a strut is less if it bends than if it merely shrinks, the condition is inherently unstable.
We must not confuse this with the stable behaviour of a cantilever, which is pushed or pulled by a lateral force, to which it responds proportionally. Bending a rod by pushing at the end requires more force than bending it by pushing it from the side: but as we have already seen, the significant difference lies in the variation of the force with the distance through which it has acted.
Euler, one of the greatest mathematicians, worked out the conditions for the stability of a strut. The next diagram is a simplified version of what happens when forces are applied into the ends of a strut.
The upper lines represent the variation of energy with the deflection. For small movement, the compression energy is less than the flexural energy, but at a critical load, the flexural energy becomes less than the compression energy. The rod can snap into a new position in which it is curved but not compressed. Looking at the lower graph we see that the variation in force now becomes very small, and great deflection can occur for a small increment in load. The structure is unstable, and will fail unless it is restrained or the load is removed.
Why does a rod behave like this? In a linear system, the energy is usually proportional to the square of the deflection: for a compressed rod, the deflection is the shortening; for a bent rod it is the angle of bend. In the bent rod, the shortening between the ends is not proportional to the angle of bend. You can test this by bending a rod by pushing and by grasping the ends and bending it. You will feel in the second case how the force increases with the bend.
A good way to test this idea is to use a piece of flexible plastic, perhaps about 30 cm in length, and an elastic band of a similar length, which can be made from several shorter ones tied together. If you pull one end of the strip towards the other, you can see and feel how little the elastic band changes as the plastic strip bends.
If you don't believe that massive objects can bend, go to Salisbury cathedral, and stand very close to each of the piers that support the tower and the spire. Some aircraft, such as the B47 and B52, had rather flexible wings, as do some high performance gliders. This behaviour can look alarming if you are in an airliner, but the actual strain at any point is quite small, because the wings are relatively thin and the radius of curvature of the bend is very large.
The nose of the B52 shows the tendency of thin material to bend out of plane when subjected to shear stress. This behaviour occurs only on the ground and at low speeds: at cruising speeds, aerodynamic forces lift the nose. Such effects can be reduced by the use of longitudinal and circumferential flanges.
Incidentally, the Euler strut exhibits behaviour called broken symmetry, which is an important idea in particle physics and other disciplines. Consider a symmetrical strip of material which is pushed by precisely axial forces. When it bends, it must do so either one way or the other. The resulting object has less symmetry than the original, yet nothing in the system was asymmetrical. In the case of a cylindrical strut, the bending can occur in any one of an infinite number of ways, each of which defines a plane, which has less symmetry than a cylinder.
Living things exhibit broken symmetry, in that their helical DNA always twirls in the same direction: it could presumably have gone the other way. Perhaps there are living things elsewhere which have the opposite handedness. Irrespective of this, our bodies could presumably have had the organs arranged the other way around. Somehow, from an apparently symmetrical first cell, an asymmetrical object is created. Is the cause in the genes, or is it physical?
Since the stability of a strut is a function of the length and the resistance to bending, we can stabilise a strut by intermediate supports that divide it into shorter sections.
The graphs were drawn on the assumption of linear, continuous behaviour. In practice, a strut may fail suddenly by bending or folding in a small region, rather than by curving uniformly. In any case, the result is often catastrophic.
Here is a picture of a part of the Firth of Forth rail bridge. The huge sloping tubes that run from the bottom chord to the top meet the latticed ties and are stabilised at the crossing points. The arrow points to a vertical tie that connects a crossing point to one of the great tubes that form the bottom chord. Without the various connections, the bottom chord would have to act as a pure arch, and it would buckle. For it to form a stable arch, it would have to be much wider. It looks like an arch, but it isn't an arch in the usual sense, in that the cantilevers are self supporting. The two halves of a true arch are each prevented from falling by the pressure from the other half.
We cannot dogmatically say that the bottom chords are not half-arches, unless we know something about the stresses in the struts and ties that connect them to the top chords. Are there any circumstances in which you then call it an arch? Suppose we made the central suspended span rigid against strong compression, and we disconnected the top chords from the towers. We would then have a four hinged arch, which would be unstable. But suppose the central span were connected rigidly to the cantilevers, what then?
Had the bridge been built as a series of arches, the piers would have had to absorb any unbalanced longitudinal forces. Compare the rise of the lower chord at the Firth of Forth with the rise of the arches at Sydney and Bayonne. What does this tell us? You can in fact make a cantilever with a completely horizontal lower chord.
Some arches, such as the Sydney harbour bridge, have been constructed without any false-work across the channel, by holding the halves back with cables. These are slowly eased when the halves are complete, until the final pins or connections can be placed. Until that happens, the half-arches are acting as cantilevers.
The span of the Quebec bridge, slightly longer than that of the Forth bridge, has straight bottom chords. Is there any advantage in either straight or curved chords? The chords of the Quebec bridge are built up from flat plates. How is this better than using tubes? How are tubes better than flat plates?
Buckling is different from bending. If you bend something, up to level called the elastic limit, it will return to its original shape when you release it. Some substances, such are not elastic at all. Buckling is what happens when you bend something in a way that not only exceeds the elastic limit, but changes the shape qualitatively. The impact of a car with another object often results in an expensive deformation, sometimes so expensive that the car is a "write-off".
In 1970, during construction, one span of the Milford Haven bridge over the Cleddau collapsed suddenly, with loss of life. This was one of several box girder bridges which collapsed by buckling in the 1960s and 1970s. Prediction of buckling is exceedingly difficult to calculate, and in those days was much less well understood than it is now.
This phenomenon of something changing irreversibly is described in loving detail in Stendhal's book "De l'amour", though the change occurs in an immaterial medium, someone's personality. He uses the word "crystallization" for this process, though in material terms, crystallization is often reversible. An irreversible example of crystallizing can occur in glass, which is normally an extremely viscous supercooled amorphous material. If glass begins to form crystals, its appearance changes drastically. Of course, it can be made amorphous again by heating, but that is not very useful if your original object was a valuable antique glass vase.
Here are tubes that have buckled. From here there is no way back. The onset of buckling can be so fast that little or no warning signs may be observed. In the case of the collapse of the first Quebec bridge, the warning signs would have enabled work to be stopped. The bridge would not have been saved, but lives would not have been lost. As so often when accidents happen, investigation shows that poor communication was one of the causes. In some cases the channels of communication were not well designed. In others, communication occurred, but was ignored or misinterpreted. Good design doesn't just mean stress analysis: it includes every foreseen step from inception to completion, and it includes including plans to deal with unforeseen problems, and that, of course, requires that they be noticed and recognized. "Why didn't they see what was happening?" is a question that can be applied not only to structural failures, but to many human interactions as well.
The Boomkicker TM
The diagram shows the use of curved rods in an application where a roughly constant force is needed for many different positions.
The diagram shows a part of the mast and boom of a sail-boat. The boom is a beam that holds the bottom of the sail, against the tension caused by the wind.
The boom is held down by a vang, the rope which is shown under the boom. Using this, the boom can be set at a required vertical position.
If the wind drops, the boom will not be pulled upwards, and will drop.
To pre-stress the beam against the vang, various arrangements are in use.
A very elegant and simple one is the BoomKickerTM (USA patents 507082 and 6062155). It consists of parallel fibre-glass rods, which are connected to the boom and mast with a bend that produces the pre-stress.
Most structural materials have a high Young's modulus, giving a huge change in force for a small change in length. The curved rods produce a force which varies rather weakly with the distance between the ends, which is the requirement here.
The height of the boom can then be adjusted using only the vang.
Useful Links About Buckling