Oscillation of bridges
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|The behaviour of a bridge is not fully specified by
the static forces within it. Any bridge can move. All the parts have
both mass and elasticity, and can exchange energy between kinetic energy of the
and the strain energy of
bending, stretching or torsion.
The bridge does not sit in a vacuum, doing nothing. It experiences the wind, and it experiences the live loads caused by traffic. These two facts have a profound effect on bridge design. Even the steps of pedestrians can affect a light, flexible foot-bridge.
Consider, for example, a heavy train going on to the bridge. The stable shape of the bridge with the train is very slightly different from the shape without it, because the need to change the stresses means a change of strains as well. After the deflection, the bridge is in a lower state of energy than before. Where has the extra energy gone? Clearly, for the bridge to have changed its shape, it must have moved, so it must have had kinetic energy. This page is concerned with that movement, and the form it takes.
The left hand bar represents the static situation. The right hand shape (exaggerated) reminds us that the tower can oscillate. In principle there could be higher modes, the next mode having a node near the top, but these modes have higher frequencies and smaller amplitudes. The oscillations can be a serious problem before the bridge has been completed, especially when the two legs have not yet been joined by a cross member at the top, or lower down.
Like any other massive elastic object, the tower will have a natural resonant frequency. Energy transferred from the wind will tend to excite this resonance.
Furthermore, because the towers are not streamlined, it is possible for them to shed vortices downstream. These tend to occur on alternate sides of an obstacle, making for an oscillatory situation. This can be observed with a flapping flag or even with the rope slapping against the flag-pole.
Tall metal chimneys are usually provided with helical strakes, which affect the flow in such a way that the vortices do not occur. The helical shape makes the system work well whatever the direction of the wind.
Flutter has been known to affect the wings of high-speed aircraft, though this is now unusual, as the phenomenon is well understood and the technology is mature.
The picture below represents the first five modes of oscillation of a hanging cable, such as an empty clothes line or a cable of an incomplete suspension bridge.
In the fundamental mode at the top, the oscillation would require large changes in the length of the cable, so this mode is strongly suppressed. The main mode is therefore the second harmonic, with a node in the middle. This has important consequences for the behaviour of suspension bridges.
The millennium bridge in London has eight parallel cables, spaced away from the deck. Had some been curved inwards, so that they were closer to the deck at the centre, the fundamental horizontal mode might have been somewhat suppressed. Having little depth, it has little resistance to vertical movement also.
Moving Demonstration Download for Cables
Click here to download (or run in place) a simple movie (48 Kbytes) simulating oscillations of a stretched cable and a hanging cable.
The next picture represents the lowest modes of oscillation of a beam which is constrained in position but not angle, at the ends.
Because the cables of a suspension bridge cannot support the fundamental oscillation with an antinode in the middle, this mode is suppressed in the deck also. In fact the slight upward curvature of the deck, found in most suspension bridges, will also tend to suppress the fundamental mode. The main mode of the deck will be the second harmonic, with a central node. The oscillation can be a simple vertical oscillation, or a twisting motion with opposite sides of the roadway moving in opposite directions, and opposite ends of the bridge tilting in opposite directions.
Suppose we were to connect a huge hydraulic ram to the deck of a bridge, and we slowly swept the frequency of oscillation from zero to perhaps 1 Hz. The response of the bridge might look rather like the diagram below, though the diagram is not intended to be at all realistic in detail. The horizontal scale denotes the harmonics, whose responses are coloured in red, green, blue and black. The result is shown in white.
The second diagram shows results from colliding "elementary" particles. The times involved were of the order of 10-23 seconds, and the masses of the order of 10-27 kg, worlds away from civil engineering. Yet here, too, in these mass spectra, we see evidence for three resonances, two very clear, and one not so clear. Massive objects such as stars and pulsars can also resonate in many different modes. The mass of the sun, the start that warms us, is of the order 2 x 1030 kg. Halfway from the particles to the sun in powers of ten, we get about 50 kg, for us a pretty ordinary sort of quantity.
In practice the frequencies of the various resonances in a bridge might not be so simply related, for example because the side spans are often not equal to half the main span, and because the cable is curved and not straight. Although the details are not realistic, we see that the fundamental is missing, and that the harmonics are lower and wider as the frequency increases. The fourth harmonic is barely visible as a slight bump. A wider resonance means a faster dying away of the oscillation after the excitation has been removed.
This relationship between widths of structures in the time domain and in the frequency domain is fundamental to waves and oscillations. It means, for example, that the location and frequency of a wave cannot both be determined with absolute accuracy. The product of the two uncertainties can be calculated. The behaviour of damped oscillations can be heard in another page - Damping - as if the bridge oscillations had been speeded up by a factor of several hundred.
Railway workers used to walk along the side of a train in a station, tapping the wheels with a hammer. Sound wheels would ring, but a cracked wheel or axle would produce a dull sound. Pottery and glass can be tested in the same way, and a "cracked bell" is a metaphor for someone or something that is not as it should be, whereas "sound as a bell" is a synonym for good quality. Is this how the word "sound" came to be associated with quality?
Click here to download a simple movie (48 Kbytes) simulating a suspension bridge oscillating in its second harmonic mode.
Click here to download simple movie (79 Kbytes) simulating the relationship between travelling waves and standing waves or oscillations.
We have looked at three modes of oscillation - towers, cables and deck. Should any two of these have similar frequencies there could be a serious problem. What could excite a resonance? Any source of energy of about the right frequency could be dangerous. This is why soldiers break step on a small bridge.
But in normal conditions a bridge is not apparently subjected to periodic forces. So thought Moisseiff, the designer of the Tacoma Narrows bridge, and few dissented when he built the most slender long span suspension bridge ever seen. The diagrams below represent the cross-section of the long, narrow and shallow deck of the Tacoma bridge.
The top picture shows the static situation. If there is a wind it can slightly tend to tilt the deck, especially if gusty. If the tilt happens to increase the lift near the leading edge, the tilt may increase. In a strong enough wind the tilt may increase until the airflow breaks away, and a stall occurs. The energy stored in the torsional strain now starts to transform into kinetic energy of rotation. When the bridge reaches the normal position it does not stop because of its inertia, and it will tilt in the opposite direction. In certain conditions a steady wind can sustain such an oscillation. A falling leaf or sheet of card may oscillate in a similar way.
Near the bottom the circles represent a train of vortices downstream of an obstacle in a wind. To see a crude simulation of a vortex street please click here.
The frequency of generation is equal to the speed of the wind divided by the distance between the vortices. If this frequency of a pair of vortices is related to the natural torsional resonance frequency then there could be a problem. The Tacoma narrows bridge was reduced to wreckage in a short time, by a moderate wind. Henry Ford is reputed to have said that history is bunk. Without knowing the context, and what he meant, we cannot evaluate this statement. But if you are building aircraft, bridges, or any other critical structure, it is a good idea to know what has happened before, and to find the right balance between over-cautiousness and rashness. Informed boldness is perhaps a good way to make progress. Suspension structures had been damaged or destroyed before by wind-induced oscillations. The Wheeling bridge and Brighton chain pier were examples.
The immediate response to the Tacoma Narrows crash was to build deep trusses under decks, much as the first aircraft used biplane construction to achieve rigidity.
It may seem strange that a steady wind can produce oscillations in a simple object, but, as we see from the picture above, simple causes do not always have simple effects.
This picture shows the tear in a KitKat wrapper after the back of a knife has been pulled through it. The pull on the knife was uniform: the result was not.
It was not so difficult after the event to see what Moissieff should have done. But let's look at this in more detail. Firstly, consider a situation where you do something numerous times, and nothing bad happens. Can you draw any conclusions. Here are some examples -
A The sun rises thousands of times in a way that seems to allow people to predict
the time of sunrise for centuries to come.
B You stress a piece of steel, well below its breaking stress, many times without
apparent effect, and so you expect to be able to continue to do so.
C You stress a piece of aluminium alloy, well below its breaking stress, many
times without apparent effect, and so you expect to be able to continue to do so.
D You walk across a road many times without looking around, and nothing
E You toss a coin eight times, getting the head side every time.
These cases, though superficially similar, are in fact all different.
Case A is certainly the most reliable, the earth's motion having been very predictable for thousands of year. Yet in a complicated system such as the sun with its planets, the possibility of chaotic motion cannot be ruled out indefinitely.
Case B is reliable, because until the steel becomes rusty or is in some way damaged, there is a stress level below which there is no known limit to the number of safe cycles.
Case C is quite different. Cycling the stresses aluminium alloy can lead to metal fatigue, which makes the material weaker and weaker, until it fails at a stress far below the initial breaking stress. This example refers to the Comet 1 accidents.
Case D is a little more complicated. The road may be out of use. The crossings may have been done during periods of little traffic. Some drivers may have seen you and taken avoiding action. You may simply have been lucky.
Case E resembles Case D, but may be treated mathematically. The coin may have a head on both sides, in which case the probability of a head is 1.0. It could also be a normal coin, and if it is perfectly balanced, the probability of a head is 0.5. The probability of eight heads is 1/256. The probability of the next throw giving a head remains as 0.5.
The Tacoma Narrows result differs from all these examples in that what was repeated was not the same construction, but the continuation of a trend: bridges were made more and more slender. What was being assumed was that known data could be extrapolated to further values, just as the Challenger space shuttle was launched at progressively lower temperatures. Since the theory of suspension bridges was becoming well understood, and the behaviour of steel was extremely well understood, there seemed to be no reason why the Tacoma Narrows bridge should not be built as designed.
Unfortunately, as we now know, one factor had been omitted.
It is, of course, extremely easy to say, in the light of later knowledge, what ought to have been done, and it is true that numerous disasters have been the result of culpable behaviour, such as poor design, poor construction, poor supervision, poor maintenance, poor communication, and even sheer determination to ignore advice and "go for it".
It is also easy to say that people shouldn't do anything or make anything until they understand all the technologies and processes concerned. The reality is quite different: we don't have to understand everything - all we have to know is that a certain set of actions and processes is always followed by a certain set of behaviours. This can be ascertained by thorough and scientifically controlled testing.
Testing may fail to produce the required effects in cases where the results are apparently the same up to a certain number of applications of the forces, after which they differ. A well known example is metal fatigue, as in case C above.
Testing may fail to produce the required effects if we extrapolate the behaviour too far beyond the region we have tested.
Testing may fail in cases where an important variable is not exercised. The Tacoma Narrows structure was not tested for the possibility of oscillations, even though oscillations induced by both wind and pedestrians had wrecked some suspension bridges. This was a case where history certainly wasn't bunk. (As the context of Henry Ford's famous quotation - "History is more or less bunk" - is not known, it tells us nothing about either Henry Ford or about history. We don't even know whether he meant the facts of the past, or history as taught in some schools, or some other meaning.)
Returning to the subject of doing only what we understand, how much of our minds and bodies do we understand? And how much do most of us understand about the way that people interact with each other? Yet somehow we manage to get through life, albeit in some cases with a great deal of pain to ourselves or others, partly because of this ignorance.