Well, this topic is a month old, but as I have just seen it now for the first time, I thought it should be resurrected!
I liked Scott's approach of considering the "ideal" model first - friction being independent of the tyre's "footprint", since downward force is inversely proportional to footprint, and then seeing how the real world differs from that ideal model. I.e., basic principles first, then modify them to represent the more complex scenarios...
I think it's worth noting that in the same vein, a tyre is somewhat like a balloon sitting on a flat surface. Put a given load on the top of it, and it deforms against the flat surface in proportion to its internal air pressure. The footprint of an "ideal" tyre deforms (for any given load) in direct proportion to its pressure, regardless of its width or its height.
Which means that a narrow tyre and wide tyre at the
same pressure will have
identical contact areas, albeit different shapes - the wide tyre will have a short fat footprint, and the narrow one, a long narrow footprint.
At a pressure of 2 Bar (~2kgf/cm2), and a load of 500kgf on the tyre, there will be 250cm2 of footprint. At 1 Bar, there will be 500cm2 of footprint. (I would guess that a 235/85-16 tyre at 1 Bar and a 500kg load would probably have a footprint about 25cm wide and 20cm long. A 185/65-14 at 1 Bar might have a footprint, say, 20cm wide and 25cm long).
This basic principle of pressure illustrates firstly why a narrow tyre doesn't necessarily offer less flotation than a wide one, and secondly, why airing down has such a huge effect. (Airing down to half pressure does in fact roughly double the contact area. Most of this is invisible, as it tends to increase the length of the footprint far more than the width.)
Now, granted, a tyre is not a balloon; it has a degree of rigidity, which decreases the actual footprint (but of course that is true of both wide
and narrow tyres). Also, off-road terrain is not a nice flat hard surface - it gives way, and both wide and narrow tyres sink some distance into it, and then start behaving differently to each other.
But by far the biggest determinant of contact area (for any given load), is tyre pressure, NOT tyre dimensions. And actual tyre construction is (a very distant) second.
That's not to say that tyre dimensions are unimportant - it's just that contact area and flotation are not that dependent on anything other than pressure. Tyre dimensions do play an
indirect part, in that the low-pressure deformation of a small tyre will be much more concentrated than that of a similarly aired-down big tyre. Not only will the small tyre heat up much more (because of the concentrated warping), but it can't shed the heat as quickly either. This makes a big tyre (diameter
or width) more suitable for running at low pressures than a small one. The 185/65-14 example I used above, would have to deform hugely to carry 500kg at 1 Bar, and hence put down 500cm2 of footprint.
And of course, tyre dimensions play a big part in other aspects of handling and performance. Width and aspect ratio affect cornering ability. Smaller, lighter tyres with lower unsprung mass make for better suspension, acceleration, and braking. From an off-road point of view, even without a bigger footprint, tall tyres have huge advantages in terms of vehicle ground clearance, and in terms of how they themselves roll over obstacles more easily than small diameter tyres. But they require more torque to turn them, and hence sometimes put more strain on a vehicle's running gear.
Contrary to some of what's been said in this thread, for thick sand I would choose a tall narrow tyre over a short wide tyre any day, assuming both were able to run at the same pressures. In sand, the main issue is the wall of sand that builds up in front of each tyre. Each tyre is continuously trying to climb this wall, and at the same time, it's having to dig its way through the wall. For any given sand trap, this wall has three characteristics that affect how the tyre (and vehicle) copes with it:
- The height of the wall. This largely depends on the pressure of the tyre, as discussed above. Both tyres should have similar levels of flotation, at the same pressure, so they will sink down to a similar depth into the sand.
- The width of the wall. The narrow tyre has a distinct advantage here, as there is physically less sand to shift out of its way. That's why you need much more horsepower to get through sand with wide tyres.
- The angle of the wall. The tall tyre can "climb" the wall at a significantly smaller angle than the small diameter tyre. This translates into less resistance, and less digging.
This explanation gels with my experience, by the way - I have done a LOT of sand driving in deserts, beaches, and general sand tracks, with vehicles of all kinds, and tyres of all shapes and sizes. In general, I would put the ability to air-down head and shoulders above any other factor when driving in sand. But as a distant second factor, I would rank having tall, narrow tyres as the best shape. (I could be convinced, that under some circumstances, tall wide tyres might be as good or better than tall narrow ones - but I would always choose extra diameter
in preference to extra width).
So why don't tall tyres sell? Simple - other things being equal, a bigger diameter tyre presents a whole raft of disadvantages for the truck designer. Lots of rotational and unsprung mass (worse braking, steering, suspension, acceleration), higher centre of gravity, bigger wheel arches eating into load and engine space, bigger spare wheel to accommodate, more expensive etc. etc. etc. So they design the trucks to take a small diameter tyre, and to hell with optimum off-road performance. Then if the after-market guys want to sell you bigger diameter tyres, they first have to persuade you to raise your suspension, cut away bits of your bodywork, curtail your ability to turn sharply, alter your gearing, and uprate your running gear to cope with the extra torque requirement. Frankly, it's easier to sell you a set of offset wheels and wide tyres. So that's where the volume is.
Enough of a ramble - three cheers for whoever mentioned Michelin XZL 255/100-16's! That's a tyre! (But I really miss the old Michelin XCL in 9.00x16...)
Rgds,
Michael...