Theory for a towed wheel in soil

Journal of Terramechanics, 1966. Vol. 3, No. 3, pp. 93 to 100. Pergamon Press Ltd. Printed in Great Britain.

DISCUSSION Theory for a Towed Wheel in Soil J. L. MCRAE (Vol. 1, No. 4, 1964)

Author's Reply WILLS* is correct in his assumption that the author is considering the performance of the wheel after equilibrium has been attained. The depth, z, is assumed to be constant, and since l for the wheel is a direct geometrical function of wheel sinkage z, then l is also constant. The justification for the hypothetical situation as represented in Figs. 4 and 5 of the paper resides in the fact that, although the distances z and l are constant, the incremental strain quantities /Xz and A1 must be constantly integrated in the z and l directions, respectively. The author assumes the unit pressure at the m a x i m u m rut depth to be approximated by the total load W divided by the projected area bl, and he considers this to be the 'effective average pressure' since the soil being dealt with is that which has experienced complete displacement for full depth z and length l, represented by volume A B C D for the full width of the rolling wheel (see Fig. 3 of the subject paper). It is apparent, however, that Wills is not aware of the fact that the author is using this assumed 'effective average pressure' at rut depth z. It should also be noted that use of this pressure at rut depth z is at variance with other theories which use the average of the pressure between the ground surface and depth z. This may explain why some predictions using another theoretical formula have been reported to be about one-half the magnitude of the actual measured values.i" It is the author's opinion that if the wheel was not rolling but was continually sinking, or if the wheel was skidding, then the average of the pressure between the surface and depth z would better represent the effective pressure. However, the wheel is considered to have reached equilibrium at sinkage depth z and it is no longer experiencing further sinkage, nor is it skidding, but it is roiling in a horizontal direction at constant rut depth z and all soil that is displaced by the wheel has therefore experienced the full range of pressures from zero at the surface to a maximum at rut depth z. The pressure at depth z is therefore the ultimate pressure experienced by all the soil that has been displaced in the rut beneath and behind the wheel. The volume of soil displaced by the wheel is the volume of soil being dealt with in the equations considered in this paper. The question now arises as to what the magnitude of this pressure is at maximum rut depth z. Referring again to Fig. 3 of the subject paper and the related explanation on pages 33 and 34, it was pointed out that the volume of soil displaced by towing force, TF, acting over area bz through horizontal disance I is equivalent to *Wills, B. M. D., 'Discussion of "Theory for a towed wheel in soil".' Journal o[ Terramechanics, Vol. 2, No. 4, 1965. 1Reece, A. R. Principles o[ Soil-Vehicle Mechanics. Report No. B.R. 186, Fighting Vehicles Research and Development Establishment, Chertsey, Surrey, England, October 1965. 93

94

J . L . McRAE

the volume displaced by load W acting over area bl through vertical distance :. This being the case, then the effective pressure that is acting at depth z is believed to be approximated closely by the average unit pressure acting on the horizontal projection of the wheel contact area. In other words, the author has assumed that the wheel sinkage is the same as that of an area bl, loaded with total load H'. As stated in the paper, the soil is constantly' experiencing a progressive failure phenomenon beneath the rolling wheel, but it is hypothesized that there is no bulldozing, only vertical compression of the soil: therefore, the only' internal soil strain quantity' dealt with is -~z, The strain component 5l does not arise directly from horizontal strain within the soil mass since it is assumed that no bulldozing occurs: the horizontal strain components and the horizontal stress components are both assumed to arise indirectly as a result of the mechanics of the towed wheel which converts externallv applied horizontal force and motion into vertical force and motion within the soft soil mass. It is hypothesized further that the soil strain 5:, ranges from zero at the point of contact at the front of the wheel to a m a x i m u m at the full rut depth and that the corresponding vertical stress ranges from zero at the point of contact at the front of the wheel to a m a x i m u m at the full rut depth. The writer believes that both sides of the equation should be integrated in the fashion indicated in the paper because of the phenomenon of progressive failure in the soil beneath the rolling wheel and because of the use of the vertical pressure at m a x i m u m rut depth in lieu of the average pressure between the surface and depth ~:. This is illustrated in the following derivation in which the physics is the same as in the subject paper, but the material is presented in somewhat greater detail in an effort to achieve greater clarity. Since the load-flow theoretical picture for the wheel deals with volume of soil displaced, it is convenient to utilize some mathematical expression that encompasses a pressure-volume relation in the development of this approach. The equation assumed is

p = k V '~

(1)

where F - p r e s s u r e at depth z V = volume of soil displaced k=coefficient relating to area of penetrating element (mathematically., the p intercept at V = 1 on a plot of p vs. V to log scales) n = e x p o n e n t relating to the load-deformation properties of the soil (mathematically, the slope of the /7 vs. V curve on a log plot) Consider the fact that as an object forces its way through the soil, it causes a volume displacement. The equation for the total a m o u n t of energy expended can be derived as follows :

elementofwork=Fdz.=pA(d~VA) = pd I/ where A = area of penetration F = total force dz, = element of depth p = p r e s s u r e at depth z d V - e l e m e n t of volume displaced

DISCUSSION

95

The total energy expended is then found by the following integration:

total energy = Or = f p d V 0

Substituting from equation (1) l~ O~ = .J k V " d V

k V "+I n+ 1

pV tz+ 1

0

Now derive an expression for the energy expended per unit volume of soil displaced :

0=

k V '~+t n+l V -

pV n+l V

JP- - - n +

0

p n+ 1

1

(2)

The equation p / O = n + 1 was found (see appendix of the paper under discussion) to be a significant general equation relating unit pressure of penetration to energy expended per unit volume of soil displaced.* The term (n + 1) was found to be an important load-sinkage curve shape factor, n, as already stated, being the slope of the curve for pressure versus volume when both are plotted on log scale. With these relations in hand, the analysis for a towed wheel now proceeds as follows : The expression for the total energy expended in displacing material vertically through distance a while the wheel is being pulled forward through horizontal distance 1 is Or-

PY th + 1

Pr lbz bz nL + 1

prl th + 1

where p~=the equivalent of the pressure on the projected area bz, i.e. pr V = volume of soil displaced = lbz n~=slope of the hypothetical loading curve for p, vs. 1 (log plot) Pr = towing force = T F As pointed out earlier, no part of Pr is assumed to arise from a bulldozing or plowing action, but it is assumed rather to arise entirely from the vertical compression of the soil by the wheel. The expression for the total energy expended as the wheel compresses the soil in the vertical direction through distance z is *C. J. Nuttall calls O the average cone index (Clz~l) when the penetration is made with the Waterways Experiment Station ONES) cone penetrometer in his report A Dimensionless Consolidation of WES Data or, the Performance of Sand Under Tire Loads, Contract Report No. 3-130, December 1965.

96

J . L . McRAE

~r-

p:V

b--[ lba

W.

where p: = pressure at depth ,-. V = volume of soil displaced = lbz ~z:= s l o p e of the soil loading curve for p: vs. z (log plot) W = total vertical load on wheel By neglecting minor losses such as bearing friction and assuming no bulldozing, the above two expressions for total energy can be equated. The writer believes that by far the most serious criticism of this step is the neglect of the energy consumed in lateral soil strain. As stated, however, when the above assumptions are accepted, the above expressions for total energy can be equated :

ttz + 1

tt. + I

but -P- =~z+ 1

,p

therefore

itS, ,z,+ 1; ~ - =~z:+ 1 Making substitution"

prl

W:

lh

P:

F r o m which follows the dimensionally correct formula

Pr= ~ \ p : / If the simple equation cited by Wills (vertical work done on the s o i l = T F × l) is used in deriving the formula for towing force when the vertical pressure at maxim u m rut depth is assumed to be the effective pressure, then the factor (~z=- 1) appears in the denominator and the resulting expression must be multiplied by (tz+ 1) in order to check the load-flow formula for towing force. Such an erroneous derivation is illustrated as follows :

p = kz ,~

(4)

z

~=

bl

z

pdz=blk !)

z~d~- - -

.

tz+

1

-

0

blk.~ ~ . .r~ _ n÷

l

W b--T blz

pv n÷

l

n÷

l

m z

tt+

t

DISCUSSION

TFI =

TF =

97

Wz

n+l

Wz

(5)

l (n + 1)

The correct final form of the towing force formula T F = W : / I is the same, regardless of whether the effective vertical pressure is assumed to be that at maximum rut depth z or the average between the surface and rut depth z. If the average pressure is used, then the derivation, as Wills points out, would be as follows: p = kz ~

~r=blfdz=blkfz~dz o

0

blkz '~+~ n+ 1

blkz"z n+ 1

blpz n+ 1

but the average pressure=

W bl

p n+ I

Making substitution : bl t~r= b l W z then TFI = W z from which follows the dimensionally correct formula : TF= Wz

---U

(6)

It is therefore evident that as far as the final form of the correct formula is concerned, it can be the same whether the pressure beneath the wheel is assumed to be the average between zero sinkage and depth z or the average on the horizontal projection of the wheel contact area. While these two approaches are both based upon the general principle of the conservation of energy, they are different in the actual steps of the detailed physics; and in practice, they lead to quite different values for the predicted towing force. This will be illustrated by the following actual case for a wheel test conducted in sand.

98

J.L.

McRAE

Wheel d a t a : L o a d on wheel = W = 877 lb D i a m e t e r of w h e e l = D = 27-3 in. A p p r o x i m a t e width of wheel = b = 9 in. L e n g t h of h o r i z o n t a l projection of wheel contact area = l = 6-84 in. Sand test results: kc=0 k,~=9.8 n = 0"71 T h e a s s u m p t i o n that the average vertical pressure beneath the wheel should a p p l y leads to a towing force f o r m u l a that is often used in the following f o r m : Z/g'~

TF=

(n + 1) (k~ + b k o Y ~:~"

'

I

(1-71) (9-x 9.8) ' ~ '

(3 - n)

877] a~e 3• : ~ = 181 Ib 2-29 ¢ 2 7 . 3 1

Using the l o a d - f l o w f o r m u l a : TF=

Wz 8.77 × 2 . l = 270 lb -1 684

The actual m e a s u r e d value for this test was T F = 242 lb. A s pointed out by Wills, the p a p e r contained no e x p e r i m e n t a l evidence to s u p p o r t the k vs. A relation. Some d a t a can now be supplied. F i g u r e s 1 a n d 2 present test d a t a to substantiate the a s s u m e d straight-line relation between pressure and sinkage, and Figs. 3 and 4 present e x p e r i m e n t a l s u p p o r t for the a s s u m e d relation between k and A.

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