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Bulldozing equation derivation

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I am trying to understand where the terms for the bulldozing equation come from and cannot find mention of it in my searches. The equation is mostly sourced from the book, "Theory of land locomotion - the mechanics of vehicle mobility" by M.G. Bekker (1962)

The first term in the equation is derived from the passive earth pressure equations and only this term is used for tracked vehicles, the additional terms only apply to wheels. These additional terms are what I am unsure of and do not know how they have been derived. It seems as if they have something to do with the geometry of the wheel but beyond that I am unsure.


where bw(m) is the width of the contact area, α(◦) is the angle of approach of the wheel,  z_max(m) is the depth of the rut left by the wheel, γ(gcm−3) is the soil density,  D(m) is the diameter of the wheel, and lr(m) is the distance of rupture. 
Kγ is the modulus of density of soil deformation, c(Pa) is the soil cohesion, Kc is the modulus of cohesion of soil deformation, φ(◦) is the angle of internal friction; all of these are experimental soil parameters. 

The diagram below might give some additional context.

Image preview

Xc = 45◦ - φ /2, 

z_max = h


Any help would be much appreciated.

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Great question, and I really wish I could help. Sadly, this is way outside my own expertise.

That said, I can offer a few suggestions.

1) If you have the book you referenced, study it really carefully, working through it line by line, equation by equation, doing all of the derivations for yourself.

2) If you do not have that book, make a real effort to get it or something else like it. Here the internet can be a huge help. I'm continually amazed at what I find. And what you find tomorrow will likely be different from what you find today. I'm not sure why this is true, but I've found it to be the case a number of times.

3) You might look in some agricultural engineering books. Tractors and other farm machinery operate on soil, and this is definitely a soil oriented problem.

Good luck!1


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After a little bit more thought, I have one more suggestion (probably the best one yet!).

Have you heard of an area of study called "dimensional analysis"? I'm not speaking simply of checking dimensions in your calculations, but a process to help you determine the proper form for an empirical equation. (I would not be at all surprised to learn that the equation you posted is largely empirical.) It is a technique to work from what you think you understand about a process to produce at least the form of the describing equations. One of my own teachers often said that he had made more money as a consultant using dimensional analysis that any other tool.

Dimensional Analysis is closely allied with a subject called "Similitude" or "Theory of Models." It is more than I can present at this point, but it is a very powerful engineering tool. Do some web research on

-- dimensional analysis

-- similitude

-- theory of models

I think this will get you to the answer, but not without a good bit of work.



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