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Bicycle Wheel
Suggested viewing: Intermediate Tutorials
DESCRIPTION AND LOCATION
A slightly unusual example today: we’re going to look at how a bicycle wheel works. This is a deceptively si
mple looking structure, though a wheel may have a
reputation for being ludicrously simple, bicycle wheels contain hidden complexity!
APPLIED LOADS
Most directly, a bicycle needs to support the weight of the rider or riders, plus the self-weight of the frame. This could be quite considerable, consider two people on the back of a bike along with the weight of the frame. We could be talking about 200kg, almost 2kN of force.
Because the centre of mass of a bike with a rider is towards the back, the majority of the load is transferred to the back wheel. It’s why pulling a front wheelie is so much easier than a back wheelie; you simply have less vertical force to overcome.
So let’s say we have 1.5kN on the back wheel, being resisted by those tiny spokes, quite a serious prospect.
The wheel is also required to resist dynamic loads while it is in motion. Travelling at a constant speed is effectively no different to a static case. Newton’s second law: F=ma, tells us that with no acceleration there will be no net horizontal force (of course vertically a=9.8m/s2).
Now, strictly speaking the wheel is accelerating around in a circle. However, because it’s almost perfectly symmetrical in the plane of acceleration we can safely ignore this effect.
But there will be a force applied to the wheels during periods of acceleration. This means both ‘normal’ acceleration (speeding up) and braking. In fact, harsh braking is likely to be the critical case.
CONCEPTUAL/QUALITATIVE BEHAVIOUR
Supports:
There are two clearly pinned supports at the hub. If they didn’t allow rotation the bike would be pointless.
However, the connections of the spokes to the wheel are in fact technically rigid connections. It may seem like there isn’t much rotational resistance because of the slenderness of the member, but there is a subtle distinction here.
The rigidity of a support is relative. In basic terms, if the member will fail before the support allows rotation, then it’s rigid. This is simplistic, because both member and support are constantly deforming due to applied loads. In reality, it is the relationship between the stiffness of the two objects that defines how rigid a connection is. There is no black or white, nothing is entirely rigid or entirely pinned.
The connection is rigid then because the support (shown in black) is considerably stiffer than the grey spoke. Note that this logic doesn’t apply when the connection is literally a pin explicitly allowing rotation.
Dead Loads (Permanent Actions):
Let’s look at the load caused by the weight of a rider and try to establish a satisfactory load path to the ground. I would suggest taking a few minutes and trying to
figure it out. Come back to this page when you think you have a solution or if you get totally confused.If you have worked on the principle of trying to find the shortest load path, you may have come up with a solution resembling the one below.
A direct load path with the bottom spokes going into compression
Alternatively, you may have concluded that the spokes of the wheel were too slender to carry the load in compression. They would simply buckle! Similarly, the spokes have insignificant bending resistance and can really only transfer loads in tension. Based on the assumption that the spokes should carry the frame in tension, you might have thought of the top half of the wheel as an arch-like structure, with the axle hanging from the spokes acting as cables. Something like this:
The frame would transfer the reactions at the supports of this ‘arch’ down to the ground through bending and shear forces. The lower spokes may also be useful in restraining the frame in this case.
Both these conclusions are based on logical principles. They both sound convincing in isolation. So which one is right?
The answer is that they both are right, and neither is right. The axle is carried in tension. The force does take the most direct load path. To reconcile this apparent paradox we need to realise one piece of information is wrong in our assumptions, and changes the behaviour of the structure.
The wheel is already pre-stressed in tension!
This effectively prevents the spokes from buckling (a member in tension can’t buckle). Without the problem of buckling we would get something almost identical to the first case. High compression in the bottom few spokes, and relatively small tension elsewhere.
We can superimpose these two cases: the original pre-stressed case, which removes the possibility of buckling, and the direct compression case, which takes the shortest load path.
Adding the prestressed tension case to the direct compression case gives a superposition with net tension in all the spokes
The end result is very surprising. All the spokes are in tension, because the pre-stressing force is designed to be higher than the applied load, but almost all the work is being done by the bottom spokes. The axle is standing on spokes in tension!
To get a feel for why this is possible, think of a bowstring. When you pull it back you would expect compression on one side of the string and tension on the other. That’s what the deformation would indicate. Except compression is impossible in a string.
What’s happening is something similar. The bowstring is pre-tensioned, so although you are compressing one side relative to its original state, the whole cross-section of the string remains in tension.
This underlines a very important principle in design. Your logic and calculations may seem perfectly correct, but if you’ve missed something fundamental in the way the structure you’re analysing behaves, you may get very different results from what you expect. Many good engineers have fallen into this trap.
Visit Ian's Bike Wheel Analysis, an excellent in depth discussion of this effect for more information.
Imposed Loads (Variable Actions)
Now we’ll consider the imposed load discussed early caused by hard braking. We get a force from the ground acting on the tyre. This causes a moment acting around the axle, making it tend to rotate.
The need to resist this moment explains the slightly odd configuration of the spokes around the hub. Imagine a wheel we could call the ‘child’s bike wheel’, shown below, where all the spokes emanate from a single point. The spokes will have to resist the moment by bending, and the result is likely to be deformation shown in the diagram on the right. Not ideal!
What actually happens is that pairs of spokes are offset, and they provide reactions of equal magnitude but in opposite directions. The result of this is that no net force is generated, but the spacing between the members does produce a moment, resisting the external moment. Pairs of forces like this are called a ‘couple’
OTHER THINGS TO THINK ABOUT
When the bike is subjected to dynamic loads, these will act in addition to the variable action caused by the rider. How might this change the tension and compression in the spokes?
The spokes are also offset in the direction normal to the plane of the wheel, so they have a dimension coming ‘out of the page’ as it were. What might the design purpose of this be?
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