Tyre load, area of contact, and pressure

[Shredcheddar]

Hi again. I'm curious as to the relationship between contact area and footprint pressure and how they change with vertical tyre load.

I found a very elementary case study that includes tyre data from Avon, which is the most important part. The person's analysis isn't particularly helpful to me. I'm wondering if any of you brilliant people could dish out some more knowledge and insight of your own.

I'm willing to accept that the relationship F = (P)(A) won't necessarily apply. What I want to explore are the reasons why, and what kind of relationship between footprint area, contact pressure, and inflation pressure does exist in application to the pneumatic tyre.

I intend to do some of my own poking around with that data... I gave that article a once-over at 3:00 AM so I haven't gleaned much. In the meantime, I figured I'd let you all provide your insights or at least mull over the question.

[Jersey Tom]

I'd say that's got to depend entirely on the amount of re-enforcement in the tire, which is going to vary, potentially pretty significantly. If you had an infinitely rigid carcass and tread you'd have a line of contact patch with near-infinite unit pressure. If you had a rubber balloon and pressed it down on a plate, I'd imagine you'd have a pretty linear relationship between load an area.

In any event, you do know that the load on the tire is going to be reacted by the integral of pressure over the entire area of the print.

[Ciro Pabón]

Well, we already know that the contact patch area (the real area where the rubber touches the ground) is very different from the flat area of the tyre (the nominal footprint), because the asphalt is rough, like this:

The "true" contact area is really small... at least for Pirelli tyres.


... and that's why the tyre heats so quickly


As Jersey Tom explains, what they're measuring is more the flexion of the tyre than the true contact area. So, for friction estimation purposes, is more or less a useless exercise. I'd say you're learning more about the flexion of the tyre and its "internal heat" generation than anything else, but I might be wrong.

However, I was curious about the relationship you could find from the figures given in the article linked by Shredcheddar. So, this is what I got in ten minutes, only for the first relationship between contact area and load shown in that article:

The blue line shows the data, the pink line the best fit


The relationship is not lineal, as the article states. The best fit I got is:



The coefficient of correlation squared () is 0.994, which is pretty good.

Conclusion: this is a "very strong" potential relationship (by potential, I mean a relationship of the form , I'm not sure about the english term), because the exponent of X is 0.3. This means that for very high loads the area does not increase significantly, which is reasonable.

How do I visualize it? My "visual" is this: for example, as JerseyTom states, if it is an spheric tyre and there is no way to make it explode , then there is also no way for the contact patch to be larger than the maximum diameter of the tyre, once the tyre is "flattened completely" and it rests on the asphalt, so it's definitely not a lineal relationship. Of course, real tyres burst at the walls long before reaching that state, but, hey, "it's all about the vision"...



I have no time to make the same correlation for the other data given in the article. Just in case, here you have the Excel file I used. Maybe someone wish to try the other data sets to see what results he gets. The only thing to do is to replace the data sets (the first five columns).

Excel file with the correlation between Contact Area and Load

Finally, I find curious (but not conclusive in any way, it's just a correlation) that the patch area is roughly proportional to the cube root of the load. I was expecting an exponent of 2/3, as we discussed in the previous thread.

The exponent I got is 0.3, very close to 0.333 which is the cube root. This could mean that the tyre expands roughly the same way in two of its three axis. I repeat: could. Am I wrong? Has anyone seen pictures of the nominal footprint to discern if it expands the same way in length as in width? It seems it doesn't... perhaps the steel reinforcement restricts somehow the "length", but I'm just guessing here.

[Conceptual]

See, this post by Ciro is exactly why it is not a bad thing to ask the same question again. By taking the time and effort to condense many pages of the previous thread, you have effectively created a very nice FAQ that will no doubt bring much traffic to this website over time.

I thank you for not n00bing this question, it was very informative!

[Shredcheddar]

See, this post by Ciro is exactly why it is not a bad thing to ask the same question again. By taking the time and effort to condense many pages of the previous thread, you have effectively created a very nice FAQ that will no doubt bring much traffic to this website over time.

I thank you for not n00bing this question, it was very informative!

I agree... bravo, Ciro!

One of the reasons I posted this thread was to look more into Ciro's mention of friction force being proportional to the 2/3 power of the load.

As Jersey Tom explains, what they're measuring is more the flexion of the tyre than the true contact area. So, for friction estimation purposes, is more or less a useless exercise. I'd say you're learning more about the flexion of the tyre and its "internal heat" generation than anything else, but I might be wrong.

I agree with you here, certainly. An unscientific method at best, but the trends are still there.

However, I was curious about the relationship you could find from the figures given in the article linked by Shredcheddar. So, this is what I got in ten minutes, only for the first relationship between contact area and load shown in that article:

The relationship is not lineal, as the article states. The best fit I got is:



The coefficient of correlation squared () is 0.994, which is pretty good.

Conclusion: this is a "very strong" potential relationship (by potential, I mean a relationship of the form , I'm not sure about the english term), because the exponent of X is 0.3. This means that for very high loads the area does not increase significantly, which is reasonable.

I intended to look at those trends when I got home today, so perhaps I will try to match your contribution.

Finally, I find curious (but not conclusive in any way, it's just a correlation) that the patch area is roughly proportional to the cube root of the load. I was expecting an exponent of 2/3, as we discussed in the previous thread.

Ciro, something I wondered when you posted that was a more exact meaning of the relationship you stated. What I mean is, you said friction force is proportional to the 2/3 power of load. You explained that the apparent linearity of friction with load (for metals) was attributed to the fact that real contact area changes as a function of load. Based on that statement, I still don't really understand the nature of real contact area's effect on friction. I understand that it changes with load, certainly, but mathematically I don't understand how to interpret "friction force is proportional to the 2/3 power of load" with respect to area, when at the same time metallic friction appears linearly related to load.

Follow me on a little mind-trek that probably doesn't make any sense. I'm operating on some assumptions that are confusing and probably wrong:

I remember thinking that if you meant friction was proportional to the 2/3 power of normal stress, that leaves room for a term expressing real contact area as a function of load; more specifically, proportional to the 1/3 power of load.

Robert Smith, in his book Analyzing Friction in the Design of Rubber Products and Their Paired Surfaces, defines the metallic coefficient of friction as the ratio of the bulk shearing stress of the material divided by the bulk yield stress of the material.

So, do we have something that looks more like:















Which simplifies to:





Where D is a unitless constant.

So, even in spite of that above (and perhaps meaningless) mathematical manipulation, shouldn't we be looking for evidence that area is a function of ? That seems to be a key trend.

All of this makes Ciro's finding very interesting. But wait, there's more of this exponent.

I just read this last night, on page 20 of Robert Smith's Analyzing Friction in the Design of Rubber Products and Their Paired Surfaces. Smith presents Schallamach's findings when testing the friction coefficient of sliding rubber versus the applied pressure:

Soft rubber:

He calls this an "empirical, Hertz-like equation" (for those of you familiar with Hertz's equation - I am not).

So there are some trends here. My mind is too mixed up with the alphabet soup of new information. But I will post this hoping that one of you can correct or guide me, or at the very least to archive my thoughts.

So, as a disclaimer, I probably have everything mixed up. That thought process above resulted from an assumption that Ciro was talking about metallic friction and its apparent linearity with load, and my attempt to figure out how two quantities could vary non-linearly and still appear linear when combined. A very reverse-engineered derivation with a lot of holes, I'm sure.

Phwew. Not exactly something to wind your mind down on after work. I've got a headache... time for some tea.

[Conceptual]

This is gonna be good...

[Shredcheddar]

I hope we're getting somewhere. Waiting on the heavy hitters Mr. Pabón and Jersey Tom.

[Jersey Tom]

F=ma.

[Gecko]

One should first try to make a clear distinction between the "large scale" area and the "small scale" area of the contact patch.

When looking at the whole tyre, the contact area that is being discussed is of a large scale, where it appears that the whole bottom part of a (slick) tyre is in contact with the road. However, on the small scales, as Ciro explains above, the actual contact area is much lower due to the irregularities of the road surface.

It's therefore best to think in terms of fractions, taking the ratio of the small scale area to the total large scale area in a part of the contact patch. This can never be greater than one. This fraction, however, does not depend on the larger dimensions of the tyre directly but only to the local pressure applied, with a highly nonlinear (power law, this is the word you are looking for, Ciro) dependence, as explained in Persson's work. The local pressure in the contact patch, on the other hand, is indeed determined by the larger tyre properties such as radius, pressure, load etc.

[riff_raff]

ShredCheddar,

Tire contact area is a function of applied force and the tire's structural stiffness. The tire's structural stiffness is affected by the construction and materials used in the tire body, and the inflation pressure of the tire (or more precisely, the distributed internal force pre-loading the tire body that must be overcome to create the deflected tire contact patch).

Having said all of that, the area of the tire contact patch is not important with regards to its traction capability. The tire traction capability is solely a function of its traction coefficient (Mu) times the normal force (Fn) applied to it. That's why aero downforce is so important.

F1 tires are primarily a suspension device. If you've ever looked at an F1 tire, you'll have noticed that they have very tall sidewalls. The tire sidewall is where most of the suspension travel in an F1 car takes place. The tire inflation pressure determines its spring rate, so inflation pressures are a very critical part of chassis set up.

[Gecko]

That is not really true. For rubber, the coefficient Mu is very much dependent upon the local pressure applied and therefore directly relates to contact patch area. The bigger this area, the lower the pressure onto the contact patch and the higher the coefficient mu, therefore bigger traction.

[xpensive]

As I can recall, it was rather plausibly concluded in another thread recently, that traction generated between a soft and adheisive tyre and a rough surface is a little too complicated to be explained by a conventional "Mu" factor?

[Shredcheddar]

Riff Raff, allow me to respectfully disagree. I have been enlightened by the findings and research of some brilliant men. Please read on.

I've been doing some reading and I think I have found some answers that tie a lot of these loose ends together.

This stuff comes once again from Robert Smith's Analyzing Friction in the Design of Rubber Products and Their Paired Surfaces.

In 1886, a guy named Hertz derived expressions for the contact radius (r) for a sphere in contact with a flat paired surface.


The expression is as follows:



Where R = radius of the sphere,
, = Young's moduli of the sphere and flat surface materials, and
, = Poisson's ratios for the two materials.

Remember, r is the contact radius between the two materials, the quantity that we're interested in.

Let's look at that equation again! Note the exponent.



So Hertz's expression matches the findings from that simple tire data above (Ciro's post).

So how does real area of contact vary to the of applied load ? Seems simple once it's pointed out: the area of the contact circle (for a sphere) is just pi multiplied by the square of r.

= =

Which is simply equivalent to



Here is the most important part:

"Because adhesive forces in rubber friction arise in the real areas of contact, can be replaced by , the adhesive force developed between the two solids; and is replaced by , the adhesional constant for the two surfaces in contact. - Analyzing Friction

Thus,



Turns out this equation is modified later in the book to account for Van Der Waal's adhesion forces (remember Ciro's post?). Anyways, this post and what I've read answer some immediate questions, but, at least for me, understanding the big picture still looms. For example, justifying that final statement by Smith is somewhat hard for me... Hertz's equation applies to many paired materials. It was originally conceived for glass spheres!

So I want to understand metallic friction in parallel. If this real contact area mechanism affects both metal and rubber friction, and the expression for real contact area is non-linear with load, why does metal friction behave linearly with load while rubber friction does not?! Grr. I feel like the answer is too obvious and right under my nose but it simply does not click.

To state it in more confusing (mathematical) terms, why does the following happen?


For rubber:
For metal:

[Jersey Tom]

Don't think I agree with the comment that the tire is where most of the suspension travel is, either. Unless you're running on packers.. tire rate is probably going to be higher than wheel rate, so you'll still get most of your travel through the linkage.

[Gecko]

The part about wheel rate as compared to the spring rate in an F1 car is actually correct; the two are roughly comparable. This is also the reason why the mass dampers proved such an advantage for Renault; once you have little suspension movement, it is tricky to achieve the right amount of damping in the dampers themselves, and a mass damper can take over that role.