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The prediction of tractive performance on soil surfaces
Pergamon
Journal of Terrarnechanics, Vol. 30, No. 6, pp. 443-459, 1993 Elsevier Science Ltd Copyright © 1994 ISTVS Printed in Great Britain. All rights reserved 0022-4898/93 $6.00+0.00 0022-4898(93)E0004-R
THE
PREDICTION
OF TRACTIVE PERFORMANCE SOIL SURFACES
ON
RAVI GODBOLE,* RALPH ALCOCK* a n d DANIEL HETTIARATCHIt
Summary--A new approach to the traction prediction equation is described. The proposed equation uses the soil deformation modulus and physical properties of agricultural tyres as parameters. The novel features of this approach include the assumption of a non-linear shear stress distribution and change in the value of soil deformation modulus with the normal stress. A model which suggests a relationship between the contact patch area and the soil deformation modulus is also introduced. The prediction equation was compared with the widely used Wismer and Luth equation and measured data obtained by Wittig. The proposed approach results in an improvement over Wismer and Luth in the prediction of traction and it also involves minimal testing.
INTRODUCTION
DRIVE TVRES for agricultural tractors are required to provide traction on agricultural soils, to support the vehicle, and to provide a minimum resistance to movement over the surface in the intended direction of travel. This paper addresses one element of this multifunctional requirement for the tractor drive tyre, that of traction. Different approaches have been suggested for predicting tractive performance, however, they all rely on some soil strength parameter and generally assume a uniform distribution of shear stress between the tyre and the soil. The mathematical definition of the peripheral force exerted by the vehicle is essentially based on experimentally established soil shear diagrams. Bekker [1] was one of the first to suggest such a relationship: r = (c + p t a n ( q 0 ) {exp[(-k2 + ~(k21 - 1))kxAs) - e x p [ ( - k l + V'(k~ - 1 ) ] k z A s ) } Y (1) where r is shear stress, c is cohesion, p is normal stress, ~ is angle of internal friction, kl and k2 are shear moduli, s is the tyre soil displacement (i x j), where i is the slip and j is the distance from initial contact between the tyre and soil, and Y is the maximum value of the expression within the larger numerator bracket. Janosi and Hanamoto [2] proposed a simpler equation to describe the asymptotic curves of shearing stress versus soil deformation: ~" = l~max(1 -- e -ffK)
(2)
*Department of Agricultural Engineering, South Dakota State University, College Station, Brookings, SD 57007, U.S.A. tDepartment of Agriculture and Environmental Science, The University, Newcastle upon Tyne, U.K. 443
444
et al.
R. GODBOLE
where K is a soil deformation modulus which varies depending how rapidly, under deformation, the soil strength reaches its maximum value. Wismer and Luth [3] suggested an equation for traction on the basis of the above equation. P
- 0.75(1 - e-°3c~i).
(3)
W They used cone index (based on cone penetrometer resistance) as a measure of topsoil strength properties. Gee-Clough [4] suggested a similar empirical equation for traction. The principal terms CT and (CT)m~x were defined in terms of the vehicle mobility number. Mobility number is based on cone p e n e t r o m e t e r resistance, tyre dimensions and tyre deflection. Brixius [5] developed several prediction equations for bias ply tyres operating in cohesive-frictional soils. These equations contained more wheel parameters and were a result of testing that was conducted in a wide range of soil types. However, they also depend on cone index which seems to be an insufficient parameter for defining soil strength for traction prediction purposes. Dwyer et al. [6] conducted traction tests in which they used a cone p e n e t r o m e t e r as well as a soil shear meter to measure soil parameters. Prediction using cone index showed poor correlation between measured data and predicted values for traction coefficient at 20% slip. However, a similar comparison using data obtained from plate sinkage tests gave better correlations. The investigation suggested that soil shear strength and soil rubber friction were likely to be important variables in predicting coefficient of traction. Soil strength is, however, known to vary with soil type, bulk density, soil water content, etc. Cone penetrometer resistance also depends greatly on its design and method of use. The cone p e n e t r o m e t e r fails to take surface effects into consideration and bears little or no anology with the mode of operation of the traction tyre. As an alternative to using the cone penetrometer, Wittig [7] proposed the use of a single wheel tester (SWT) for determining a soil strength parameter in predicting tractive performance. The SWT is designed to provide information on the tyre-soil interaction. Different normal loads were applied to the SWT and the maximum torque that could be developed was measured. The slope of the plot of normal load versus wheel torque was used as a soil strength term. The general form of the equation used was: P / W = a(1
-
e -bi)
(4)
where a and b are empirical constants. This method was shown to give more accurate predictions on Vienna loam soils than the cone penetrometer approach of Wismer and Luth. A fully instrumented device capable of measuring soil sinkage and shear parameters was developed by Upadhyaya and Wulfsohn [8]. It was employed to conduct in situ sinkage and shear tests. An 18.4R38 tyre was used for traction tests at different inflation pressures. Soil parameters obtained by the instrumented device were related to terms in an equation for traction prediction. In an earlier study, Wulfsohn and Upadhyaya [9] had conducted field tests for three different types of radial tyre on five different soils with three axle loads. They obtained a prediction equation for different soils and loading conditions and concluded that soil conditions had the greatest effect on the outcomes of their study.
TRACTIVE PERFORMANCE ON SOIL SURFACES
445
There is, however, another interesting approach suggested by Janosi and Hanamoto [2] whereby the thrust is assumed to act similar to the shear force, in a horizontal plane under the traction device:
P= F[l+K(e-i~-l)]il
(5)
where F is the maximum shear force and P is pull. This equation is applicable to soils showing asymptotic behavior on a shear-displacement curve. Different soil types are assumed to give different forms of shear-displacement diagrams. The asymptotic or work hardening nature of the shear diagrams was thought to be for cohesionless soils (normally consolidated or lightly consolidated) whereas cohesive soils (dense or heavily consolidated) tend to give curves of a humped or work softening nature (Fig. 1). Investigations by Komandi [10], however, established that humps in soil shear diagrams are not caused by cohesion exclusively and that friction and cohesion are non-separable phenomena. Equation (5) was therefore considered as forming the basis of this study. This paper proposes a modified form of equation (5) [11]. This can be derived as follows: For a track,
F = bf~rdx where b is the track width and dx is the elemental distance along the contact length, and from equation (2),
F=
bfo(C+ 1
otan40(1 -
e-J/r)dx.
(6)
For uniform normal pressure a is independent of x and is equal to W/bl where the contact area, assuming a rectangular contact patch. The final form of the equation is
bl -- A,
WORK
./
///"
WORK
SOFTENING
HARDENING
CURVE
CURVE
f
Displacement FIG. 1. Typicalshear deformationcurves.
446
R. GODBOLE et al. --
c + tan(p
1 +--(e
~ -
1)
(7)
il
where
A l i K c q0 P W
is the contact area: is the contact length; the slip: is the soil deformation modulus: is soil cohesion; is the angle of internal friction for the soil; is the pull; is the axle load.
The experimental and analytical work that the authors present makes use of the data obtained by Wittig [7]. Note that all the terms involved in the equation (7) can be directly measured and are not empirical. Further, a theory is presented that makes use of an approach suggested by Krick [12] which correlates the term K with tractor tyre properties.
THEORY Contact patch geometry
Figure 2 shows the basic tyre geometry assuming a rigid surface where B is the tyre width, D is the tyre diameter, S is the tyre section height, f is the tyre deflection, l is the contact patch length, b is the contact patch width:
12 Of - f2 _ 4
Since D >> f , l2 = 4Dr
.'. l = 2"~fDf.
(8)
Similarly for the cross section, b
=
2~f-Sf.
(9)
Contact area A, assuming an elliptical patch: lb A =rr-4 .'. a
= rrf~-~.
(lO)
The contact area assuming a rectangular patch is A = Ib
A = 4f~ DF~S.
(11)
In practice, the contact area may be different from the values given by equation (11).
447
T R A C T I V E P E R F O R M A N C E O N SOIL S U R F A C E S
D/2
~b-4 F~
t
I-
I
"l
B
RECTANGULAR
B
PATCH
ELLIPTICAL
PATCH
FIG. 2. Contact patch geometry.
The principal reason for this is the difference in sinkage on various surfaces. This can be accommodated by introducing an appropriate coefficient. Thus for the contact patch length, l = 2Cl'~/Df.
(12)
C1 = 1 for hard soil; C1 > 1 for soft soil. For the contact area: A = 4CAfV~
(13)
for rectangular patch. Ca = 1 for hard soil, Ca > 1 for soft soil. Tyre deflection (f) on a hard surface was determined by Krick [10] to be a function of the section height and the tyre numeric, T. The tyre numeric was defined as follows, T-
pDB W
(14)
where p is inflation pressure. Krick established the following empirical prediction equation for deflection, f: f - 0.67T -°'8. S
(15)
448
R. G O D B O L E et al.
For the purposes of this investigation, Krick's equation was re-evaluated using the section height S instead of tyre section width B is the expression for the tyre numeric. Thus the modified tyre numeric, T', is: T'-
(16)
pDS
W The result of this analysis for different tractor tyres gave a similar relationship to that given by Krick: - 0.54(T') -°79
f
(17)
S which is of the general form f = Q S T '-m where Q is the coefficient of deflection, S is the section height, T' is the modified tyre number and rn is the exponent. Krick proposed his equation based upon the experimental data of the most commonly used agricultural tyres. However, for smaller tyres the coefficient Q and the exponent may be different. In order to investigate this possibility, a similar analysis was conducted for the 6.5 x 15 (17 x 38.1 cm) tyre which had been used previously by Wittig in the design of the single wheel test device (SWT). An experiment was conducted wherein the tyre deflection was measured for a set of loads at different inflation pressures. The resulting load-deflection curves on a hard surface are shown in Fig. 3. A corresponding plot of the dimensionless terms f / S and T' is
T A B L E 1. O B S E R V A T I O N S
FOR T H E L O A D - - D E F L E C T I O N
TESTS ('ONDI_JCTED BY
IHE
AUIHORS
AI
SDSU.
LISTS T H E L O A D A P P L I E D ON T H E T Y R E , T H E C O R R E S P O N D I N G D E F L E C T I O N A N D I N F L A T I O N P R E S S t l R I
Deflection (mm)
Load (kN)
1.27 3.708 6.299 8.89 11.48 I4 16.54 19.15 21.77 24.31 26.75 29.06 31.06 32.84 1.727 4.496 7.29 9.931 12.78 15.6 18.36 21.13 23.83 26.44 29.06 31,52
0.624 0,892 1.382 1.895 2.43 2.898 3.367 3.813 4.147 4.459 5.106 5.708 5.975 6.287 0.691 1.088 1.538 2.007 2.453 2.898 3.246 3.572 4.013 4.281 4.794 5.239
Pressure Deflection (kPa) (ram) 206.8 206.8 206,8 206,8 206.8 206.8 206.8 306.8 206.8 206.8 206.8 206.8 206.8 206.8 172.4 172.4 172.4 172,4 172.4 172.4 172.4 172,4 172.4 172.4 172.4 172.4
34.01 36.3 38.61 1.27 3,988 6,807 8.966 11.79 14.53 17.32 20,19 22.86 25.73 28.55 31.37 34.06 36.83 39.17 41.63 44.04 1,753 4.496 7.315 10.26 13.23 16.1
Load (kN)
Pressure (kPa)
Deflection (mm)
Load (kN)
Pressure (kPa)
5.574 5.841 6.283 0.312 (I.669 I.(17 1.449 1,739 2.029 2.363 2.675 3.032 3.389 3.79 4,147 4.437 4.86 5.039 5.329 5.739 0,491 0.803 1.137 1.427 1,659 2.002
172.4 172.4 172.4 137,9 137,9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 103,4 103.4 103.4 103.4 103,4 11)3.4
21,84 24.79 27.76 30.71 33.5 36.47 39.37 42.16 45.09 1,943 5.055 8.052 11.24 14.5 17.6 20.78 23.9 27.08 30.28 33.38 36.3 39.57 42.7 45,67 48.74 51.82
2,426 2,782 3.121 3.434 3.701 3,978 4.241 4.508 4.771 0,303 0.513 t/.705 0.883 1.106 1.249 1.342 1.65 1.953 2,14 2.319 2.493 2.707 2.947 3.21 ! 3,518 3.768
103.4 103.4 103.4 103.4 1(t3.4 1(13.4 103.4 1(/3.4 1(/3.4 68.94 68,94 68,94 68.94 68.94 68.94 68.94 68.94 68.94 68.94 68.94 68.94 68.94 68.94 68.94 68.94 68.94
I1
TRACTIVE PERFORMANCE ON SOIL SURFACES
449
7000
46000 5000 z
4000
J
3000
x' ,x'
x
.x. ;<'
2000 1000 0 0
10
20 30 40 Deflection rnm
50
60
FiG. 3. Load-Deflection characteristicsfor different tyre inflation pressures.
] -1.5
-2
f/s ___= 1.o5 ~(pOS/W)
-2.5 ~
~
-3 -3.5-
~
-4 -4.5 -5
0.5
~
~
1'.5
2
2~.5
~
315
~4
In[{p/W}*S*D]
FIG. 4. Non-dimensionaltyre deflection parameters for 6.5X15 tyre.
also shown (Fig. 4). The resulting deflection equation for the 6.5 x 15 tyre is as follows [R 2 = 0.97 and d f = 77]:
Sf - 1 ' 0 5 [ - ~ - ] -1'24
(18)
which is of the same general form, but has different Q and m values as might be suspected. Traction equation It is assumed that the normal stress (a), shear stress (r) and soil deformation modulus K will be approximately constant along the contact patch (Fig. 5). The following equation for the pull does, however, provides for quantification of the coefficients and exponents based on soil and tyre parameters. From equation (7),
l'-ni P = A ( c + otanq~) [ 1 + __re nt where n = 1/K and a = normal stress.
- 1) ]
(19)
450
R. G O D B O L E et al.
k
\ "\ '\\
/,
>
l
p~
/ n
J
FIG. 5. Schematic diagram for tyre-soil interface.
Previous work has largely negelected the influence of o on K. There is, however, a tendency for K to decrease with increasing values of o, as the soil becomes stiffer• A simple linear relationship between K and o was found to give a reasonable fit to experimental data for the soils around Brookings. The resulting relationships are plotted in Figs 6 and 7 for two sites. The equations describing this relationship are of the form K = K0 - C~o, where Ck is the slope of the K versus normal stress plot. K Values• Values of the deformation modulus K, were determined from shear stress-displacement curves for different levels of normal stress, o. The value of K was interpreted as the ratio of maximum shear stress (~'max) and the slope of the tangent drawn at the origin of each curve. Thus K = rmax/tan (oi) (Fig. 8). The shear displacement curves were generated by using a direct shear testing machine ( G E O T E S T Digital Model $2215). A schematic diagram of the machine is shown in
0007 .
0.006 ~ i d )t,18!
4 ;'[
b " N yr~
( ; t r . ,:,
.: 0.005 7:
•
£ 0.004.
0.003 ! 0.002
0001 ' 0
*',:i
10
20
30
40
50
)4orm~d $t r,s~
60
70
80
90
100
kP, R
FIG. 6. Plot of K values vs normal stress for soil 1.
-,
T R A C T I V E P E R F O R M A N C E ON SOIL S U R F A C E S
451
0.006
•
0.005
•
.~ 0.004
" "*~'~.~1 [ k= o0514.35E.Sx No~alStress_ ~
]
I :
0002 0.001 -,
"'~
0. . . .
0
10 20
---,
30 4'0 50
- ~
4
6'0 7'0 8'0 90 100
Normal Stress - kPa R~= 0.B1
Fio. 7. Plot of K values vs normal stress for soil 2.
Typical Shear-DisplacementCur~
,
/
b~
Displa[~ment FiG. 8. Definition of K, the soil deformation modulus.
Fig. 9. The shear tests conducted can be categorized as consolidated-undrained tests. Shear rate was held constant at 0.061 mm/s. Soils were collected from two sites. Four samples of each soil were used for testing. Each sample was tested at five levels of normal stress. The resulting relationships for one such sample are shown in Fig. 10. The maximum shear stress was also noted each time. This information was used in generating the Coulomb's [v = c + tan (q~)a] equation for the soils (Figs 11 and 12). In order to find out which factors influence the soil deformation parameter K , the following model is proposed. It was assumed that the parameter K was dependent on shear stress (v), normal stress (a), shear rate (s) and the contact area (A). K = f [ r , a, s, A]
452
R. G O D B O L E et al. s
Dial gauge to measure vertical movement
EED f - Alignment pins Lateral deformation measuring gauge
Loading bar /
'
'
so,;
L
'
'
'
/
Serrated edges Io hold ~ . a m p l e /
Setscrews tO Separate shear box
. ~
F](~. 9. Schematic diagram of the soil testing machine. 80-
c•! x
!6A97 kff~, 50
x
9~1;, kia
CO
30[
•
03
•
20~ . 10 t C 0
•
nn
.
nn
nn
•
•
•
•
•
n~
• 2
4
6 8 10 12 Displacement j - m •
14
16
18
FI(;. 1(). Shear displacement curves for different normal stress levels. 80 i
70~ i
•
GO~
5o] 40'
;.
so, 201 0
i , 10
20
, . : 30 40 50 6Q 7'0 Normal Stress kPa
8r0
• 90
i 100
FK;. 11. Plot of shear stress vs normal stress for soil 1.
T R A C T I V E P E R F O R M A N C E O N SOIL S U R F A C E S
453
100
"i
90 80
4oJ 60
i
/
/~ j
5o1 O3
.-
f
70
!
/ ~11 /
/
i . ~ ~/ ~
~ I
Shear
~:
= 2 9 7 4 + t&n (27 3 9 ) x N o ~ a J
[
1 J
-4 ~
3O 20
0
10
20
30 40 50 60 70 N o r m a l S t r e s s kPa
80
90
100 z
R = ,. ,c,
Flo. 12. Plot of shear stress vs normal stress for soil 2.
where K - [L], r - [ML-1T-2], o - - [ML-1T-2], s - - [LT -1] and A - [ L 2 ] . Using Buckingham PI theorem, K 2 / A = Ccr ('r/a) cl where Ca is an experimental coefficient and Cl is an empirical constant. The K value is, therefore expected to vary with the contact area involved. This rationale was applied in converting the K values determined in the laboratory tests to equivalent K values for the tyre-soil interaction. That factor w a s Kequiv. = Klab. test X N/(Atyrecontactpatch/Alab. test ). The rate of shear was assumed not to affect K based on the above analysis. This assumption was confirmed by Bekker [13] who found that for large range of soil deformation rates (0.127-127 mm/s) the effect upon the soil strength is insignificant. Deflection calculations
Contact length and deflections were calculated at a given axle load from the formulae given earlier. The validity of these calculations was checked with the empirical equation suggested by Painter [14]. This equation is of the form, W = ao + a s f + a6pf [2-(az+a')]
where f is the tyre deflection, p tyre inflation pressure and a0, a2 etc. are empirical constants. Values determined by Painter for the coefficients are as follows:a0 = 61, a5 = 16, a 6
=
448,
(a2
+
a4)
=
0.7.
Using Painter's equation, for a typical load of 21 kN and a standard tyre size of 13.4 x 38, the deflection f = 0.102 m. This compares quite well with the deflection value of 0.0969 m determined from the equation for the deflection given earlier. Contact area calculations
Area calculations were made using the formulae developed in the earlier section assuming a rectangular contact patch. Painter has suggested a similar empirical equation for contact area,
A = ~--~-ala3Da2cazf 2-(a~+a4) 4
R. GODBOLE et al.
454
where D is the outside d i a m e t e r of the tyre, C is the tyre cross section equivalent d i a m e t e r of curvature, f is the tyre deflection and a~, a2 etc. are empirical constants, were ala3 = 5.6, a2 = 0.35, (a2 + a4) = 0.7. Using this e q u a t i o n gives an area A = 0.2252 m 2 which c o m p a r e s reasonably well with o u r calculated value of A = 0.2985 m 2. The difference in these two results could be attributed to the fact that the Painter's e q u a t i o n uses C, the tire cross section, as the equivalent d i a m e t e r of the tyre section which is greater than B, the section width, used in o u r calculations. U p a d h y a y a and W u l f s o h n [15] have p r o p o s e d a new model for calculating the contact areas and lengths which involves measuring the incremental lateral arc lengths ( A S ) of the profile at discrete locations along the contact length and fitting the coefficients of the m o d e l of soil d e f o r m a t i o n at the s o i l - t y r e interface to experimental data. The contact lengths and areas calculated by this m e t h o d are as shown in Table 2. T h e y c o m p a r e favourably with the calculations m a d e by the authors.
n value computations n values were c o m p u t e d as l/K, however, a new a p p r o a c h to the determination of n is suggested in the following:1 n
--
K" Substituting for l = 2CI~/ Df and K = Ko - Cko,
2 C, ~-Df n
--
Ko-
Cko"
Dividing the n u m e r a t o r and d e n o m i n a t o r by a,
(2C,~7/a) n
--
( K o / a ) - Ck Substituting for a = W / A ,
(2CID°Sf°sA/W) Ft =
( K o A / W ) - Ck Substituting for A = 4Caf ~ - ~ and f = (QCfST '-m) gives: (SCICaDQ1.SS2T ' 1"Sin~W) n = [(4CaKoQS 1"5D °5 T ' - m ) / w ] C k TABLE 2.
Load (kN) 18 27 18 27
CONTACT
Inflation pressure (kPa) 85 85 125 125
LENGTHS
AND
AREA
CALCULATIONS
Contact length (mm)
BY DIFFERENT
METHODS
Contact patch area (m:)
W & U*
Author
W & U*
Aut
Au +
522 710 520 630
777.4 912.1 667.3 783.7
0.1883 0.3004 0.1647 0.2795
0.294 0.405 0.2169 0.2987
0.231 0.318 0.171 0.2345
*As computed by Wulfson and Upadhyaya [15]. tAs computed by authors assuming a rectangular patch. :~As computed by authors assuming an elliptical patch.
TRACTIVE PERFORMANCE ON SOIL SURFACES
455
It was found that (4CaKoQSI"SD°'ST'-m/w)>> Ck [Ca = 1.1, K 0 e q u i v ' = 0.058 m, Q = 0.54, S = 0.467 m, D = 1.82m, T' = 3.91, m = - 0 . 8 , W = 21 kN and Ck = 4.2 x 10 -5 m/kPa]; thus Ck will have little influence over the outcome of the n value. Hence upon simplifying,
CDO.SQo.sso.5T,-,n n --
Ko where C is a constant and
SDQT'-2m ]0.5 . '.
n
oc
K~
(20)
which is a dimensionless ratio for n.
RESULTS AND DISCUSSION
Wittig [5] conducted several tests under different soil conditions. Wittig's measured values of dynamic traction ratio were compared with Wismer and Luth predictions and an equation was developed based on a single wheel device to determine soil strength. The first approach was to determine if Wittig's measured data would fit an equation of the following general form: P
[ ( e - B l i - 1)] - B0 1 + .
Bli
(21)
It was found that an equation of this form provides a good fit to the measured data (R 2 = 0.98). Each term in the modified equation was then calculated using theoretical and empirical formulae. Soil samples were collected from the same fields where Wittig had conducted his tests. The following four graphs (Figs 13-16) and Tables (3-6) show a comparison between the P/W values obtained in three ways.
~et 0.! 0.; 0.| 02
~
J
Modlflnd ~l.n IN
ueam~ed Values
0.3 0.2 0.1 ¢
Wlemer-Luth Eqn
-I1.1
-0.2 "-11.3
o
03
o:2
o13
o74
FIG. 13. Graphs showing comparisons between
o15
0.6
P/W values obtained using various methods.
456
R. G O D B O L E
et al.
P,/qq ratio v/s Slip Rough Chisel plowed (partially covered) 0.8 0.7 0.6
S
0.5 0.4 0.3
f
~m
Meas,.u~ Values ~
.
+
W'mmer-Luth
0.2 0.1 0 -.0.1 -0.2 -{3.5 0.1
Flo. 14.
0.2
0.3
Slip i
0,4
0.5
Eqn1
0,6
Graphs showing comparisons between P/W values obtained using various methods.
P/W ratio v/s Slip Bare Soil after slight rain O.g
0.8 0.7 O.fi 0.5 0.4
Values
ll•u•d
W'~nn~-Luth
03 0.2 0.1
/
0 -0.1 -0.2 -0.3 0
Fro. 15.
011
0;2
0:3 0?4 Slip i
Graphs showing comparisons between P / W
015
0.6
values
obtained using various
methods.
(1) by using the modified equation suggested by the authors, (2) from the measured data of Wittig, (3) using the Wismer and Luth equation. The difference between the P/W values obtained by actual measurement and those obtained by using the modified equation can be explained as follows: (1) The testing conducted by Wittig was done in 1990 whereas the authors conducted their soil tests from samples collected in 1992 at the same locations. This has obvious
TRACTIVE PERFORMANCE ON SOIL SURFACES
457
r/w ratio Rough ChiselPlo'~xl o.g 0.8
~
03 0.6
~
~= 0.5
=
w
ilensm~l Valuta
f
:,,3 0.4 ~, 0.3
Irlmn--laall ~la
0.2
j
c
o
,-0.1 -0.2 ,,~....x.-0.3 o
o:1
o:2
o:3
o:~
FIG. 16. Graphs showing comparisons between
o:s
o.6
P/W values obtained
using various methods.
TABLE 3. TABLES SHOWING COMPARISONS BETWEEN P / W VALUES OBTAINED USING VARIOUS METHODS
W (kN)
i
21.9 21.9 22.1 22.2 22.4 22.4 22.85 22.9
0.005 0.04 0.072 0.091 0.117 0.146 0.243 0.298
K(m) 0.0353 0.035 0.035 0.0349 0.0347 0.0347 0.0341 0.0341
l(m) 0.843 0.843 0.843 0.843 0.843 0.843 0.843 0.843
n 23.89 24.09 24.09 24.16 24.3 24.3 24.73 24.73
Pull (kN)
P/W
P/W
P/W
modified
measured
Wismer-Luth
1.142 7.078 10.52 12 13.48 14.81 17.02 17.65
0.052 0.323 0.476 0.54 0.602 0.661 0.745 0.771
0.258 0.316 0.35 0.404 0.41 0.534 0.552
-0.131 -0.083 -0.056 -0.021 0.016 0.123 0.175
TABLE 4. TABLES SHOWING COMPARISONS BETWEEN P / W VALUES OBTAINED USING VARIOUS METHODS
W (kN)
i
21.7 21.9 22.5 22.7 22.8 22.15 22.8
0.048 0.077 0.181 0.242 0.245 0.268 0.334
K(m) 0.0355 0.0353 0.0345 0.0343 0.0342 0.035 0.0342
/(m) 0.843 0.843 0.843 0.843 0.843 0.843 0.843
n 23.74 23.9 24.41 24.58 24.67 24.11 24.67
Pull (kN)
P/W
P/W
P/W
modified
measured
Wismer-Luth
7.971 10.79 15.68 16.9 17 16.92 17.89
0.367 0.493 0.697 0.745 0.746 0.764 0.785
0.187 0.266 0.432 0.495 0.513 0.485 0.546
-0.098 -0.051 0.092 0.16 0.163 0.185 0.244
i m p l i c a t i o n s fo r t h e c o m p a r i s o n o f p r e d i c t e d values with t h o s e m e a s u r e d in t h e field. H o w e v e r , t h e c o m p a r i s o n is i n c l u d e d in t h e p a p e r since soil m o i s t u r e c o n t e n t s w e r e similar for the V i e n n a l o a m soils. F u r t h e r testing is n e e d e d to p r o v i d e a valid comparison. (2) C h a n g e s in t h e c o n d i t i o n s o f th e soil an d c r o p r esi d u e m a y h a v e r e s u l t e d in d i f f e r e n t (c + o tan q~) values.
458
R. G O D B O L E TABLE 5.
W (kN)
i
21.55 21.65 22.2 22.45 22.85 22.85 22.85 23.1 23.3 23.25 23.4
0.042 0.051 0.113 0.132 0.214 0.222 0.257 0.387 0.41 0.448 0.539
TABLES SHOWINQ COMPAR|SONS
K(m)
TABLE 6,
W (kN)
i
21.45 21.95 22.15 22.25 22.65 23.1 23.15 23.3
0.022 0.052 0.067 0.102 0.164 0.312 0.384 0.508
/(m)
0.0357 0.0356 0.0349 0.0346 0.0341 0.0341 0.0341 0.0338 0.0336 0.0336 0.0335
TABLES
[).843 0.843 0.843 0.843 (I.843 0.843 0.843 0.843 (/.843 0.843 (I.843
n
23.62 23.7 24.15 24.36 24.71 24.71 24.71 24.93 25.11 25.07 25.2
BETWEEN
0.0358 0.0352 0.035 0.0349 [).0344 0.0338 0.0338 0.0336
/(m)
0.843 0.843 0.843 0.843 0.843 0.843 0.843 0.843
n
23.54 23.94 24.11 24.19 24.54 24.93 24.98 25.11
P/W
VALUES OBTAINED
USING VARIOUS METHODS
Pull (kN)
P/W
P/W
P//W
modified
measured
Wismer-Luth
7.201 8.285 13.17 14.15 16.55 16.69 17.18 18.39 18.62 18.77 t9,16
0.334 0.383 (I.593 (I.63 0.724 //.73 0.752 (1.796 0.799 0.807 0.819
0.134 0.168 0.352 0.415 0.537 (1.499 0.543 0.605 /I.665 (/.657 (1.7
--0.01 1~.009 ~.145 0.179 0.303 1t.314 [I.353 t~.458 0.476 0.496 IL534
SHOWING COMPARISONS BETWEEN
K(m)
et al.
P/W \ A L U E S
OBTAINED
U S I N G V A R I O t ~ S ME 1 H O D S
Pull (kN)
P/W
P/W
P/W
modified
measured
Wismer-kuth
4.309 8.52 10.09 12.62 15.33 17.88 18.41 19.01
0.201 [/.388 0.455 0.567 0.677 (/.774 (/.795 [/.816
(/.104 0.241 0.325 0.353 0.475 0.611 0.621 0.664
-[~.21 0.17 - 0.16 -(t.12 0.055 0.074 (I.126 0,201
(3) The contact area A is either calculated by formulae or checked using empirical relationships. This may differ from the exact value for contact area. In addition, effects such as soil wall build up around the tyres and soil topography may affect the area term. (4) The contact length term used in the prediction equations may differ from actual values. Despite these provisos, the four graphs suggest that the predictions obtained are reasonably accurate and that with some improvement in the measurement of the relevant terms, traction prediction could be improved. The expression obtained for n; n = C [ S D Q T - Z m / ( K o ) 2 ] °'5 is a valuable result. Researchers need to confirm the validity of the expression experimentally. Once confirmed, for a range of soil types, then given the tyre size and soil type, it will be relatively simple to predict the n value. When the n value is known, the gross traction can be predicted for given axle loads and assumed slip values. It is suggested that this approach will be useful in the design of tracked vehicles and wheeled tractors. CONCLUSIONS
(11 The equation predicts traction coefficient more accurately than some other methods and justifies the use of the soil deformation modulus in traction studies.
TRACTIVE PERFORMANCE ON SOIL SURFACES
459
(2) Soil d e f o r m a t i o n m o d u l u s is a g o o d e s t i m a t o r o f soil s t r e n g t h a n d has r e l a t i o n s h i p with n o r m a l stress levels. (3) E x p r e s s i o n s d e v e l o p e d for c o n t a c t l e n g t h s a n d a r e a p r o v i d e results t h a t a r e c o n s i s t e n t with o t h e r m o d e l s .
FURTHER WORK (1) T h e a u t h o r s w e r e a b l e to establish a r e l a t i o n s h i p b e t w e e n K a n d o which was n o t a v a i l a b l e p r e v i o u s l y , h o w e v e r t h e r e a r e o t h e r factors o n which K m i g h t d e p e n d , such as a r e a , s h e a r r a t e , m o i s t u r e c o n t e n t o f soil a n d b u l k density. R i g o r o u s testing n e e d s to b e c o n d u c t e d in this r e g a r d . (2) T h e single w h e e l t e s t e r ( S W T ) is a r e l a t i v e l y e a s y i n s t r u m e n t with which to c o n d u c t soil tests. It is n e c e s s a r y to e s t a b l i s h a r e l a t i o n s h i p b e t w e e n n for large t y r e a n d n for the small ( S W T ) tyre. Scaling can t h e n b e a p p l i e d in c a l c u l a t i n g n for large tyres.
REFERENCES [1] M. BEKKER, Off the Road Locomotion. The University of Michigan Press (1960). [2] Z. JANOSl and B. HANAMOTO, The analytical determination of drawbar pull as a function of slip for tracked vehicles in deformable soils. First Int. Conf. Mechanics o f Soil-Vehicle Systems, Torino-St. Vincent (1961). [3] R. WISMER and H. LUTH. Off-road traction prediction for wheeled vehicles. Trans. A S A E 17 (1), 8-14 (1974). [4] D. GEE-CLOuGH, Selection of tyres for agricultural vehicles. J. agric. Engng Res. 25,261-278 (1980). [5] W. W. BRIXIUS, Traction prediction equations for bias ply tyres. ASAE Paper No. 87-1622 (1987). [6] M. J. DWYER et al. The field performance of some tractor tyres related to soil mechanical properties. J. agric. Engng Res. 19, 35-50 (1974). [7] V. Wlx'rlG, Prediction of tractor drawbar pull on agricultural soils. M.S. Thesis, Dept of Agricultural Engr, South Dakota State University, Brookings, SD 57007, unpublished (1990). [8] S. UPADHYAYA and D. WULFSOHN, Traction prediction using soil parameters obtained with an instrumented analog device. J. Terramechanics 30 (2), 85-100 (1993). [9] S. UPADHYAYAet al. Traction prediction for radial tyres. J. Terramechanics 311 (2), 149-175 (1989). [10] G. KOMANDI, The determination of soil properties from slip-pull curves. J. Terramechanics $ (2), 9-23 (1968). [11] R. N. YONG et al. Vehicle Traction Mechanics, Elsevier, Amsterdam (1984). [12] G. KRICK, Radial and shear stress distribution under rigid wheels and pneumatic tyres operating on yielding soils with consideration of tyre deformation. J. Terramechanics 6 (3), 73-98 (1969). [13] M. BEKKER Introduction to Terrain-Vehicle Systems. The University of Michigan Press (1968). [14] D. PAINTERA simple deflection model for agricultural tyres. J. agric. Engng Res. 26 (1), 9-20 (1980). [15] D. WULFSOHNand S. UPADHYAYA, Determination of dynamic three-dimensional soil-tyre contact profile. J. Terramechanics 29 (4/5), 433-439 (1992).
Journal of Terrarnechanics, Vol. 30, No. 6, pp. 443-459, 1993 Elsevier Science Ltd Copyright © 1994 ISTVS Printed in Great Britain. All rights reserved 0022-4898/93 $6.00+0.00 0022-4898(93)E0004-R
THE
PREDICTION
OF TRACTIVE PERFORMANCE SOIL SURFACES
ON
RAVI GODBOLE,* RALPH ALCOCK* a n d DANIEL HETTIARATCHIt
Summary--A new approach to the traction prediction equation is described. The proposed equation uses the soil deformation modulus and physical properties of agricultural tyres as parameters. The novel features of this approach include the assumption of a non-linear shear stress distribution and change in the value of soil deformation modulus with the normal stress. A model which suggests a relationship between the contact patch area and the soil deformation modulus is also introduced. The prediction equation was compared with the widely used Wismer and Luth equation and measured data obtained by Wittig. The proposed approach results in an improvement over Wismer and Luth in the prediction of traction and it also involves minimal testing.
INTRODUCTION
DRIVE TVRES for agricultural tractors are required to provide traction on agricultural soils, to support the vehicle, and to provide a minimum resistance to movement over the surface in the intended direction of travel. This paper addresses one element of this multifunctional requirement for the tractor drive tyre, that of traction. Different approaches have been suggested for predicting tractive performance, however, they all rely on some soil strength parameter and generally assume a uniform distribution of shear stress between the tyre and the soil. The mathematical definition of the peripheral force exerted by the vehicle is essentially based on experimentally established soil shear diagrams. Bekker [1] was one of the first to suggest such a relationship: r = (c + p t a n ( q 0 ) {exp[(-k2 + ~(k21 - 1))kxAs) - e x p [ ( - k l + V'(k~ - 1 ) ] k z A s ) } Y (1) where r is shear stress, c is cohesion, p is normal stress, ~ is angle of internal friction, kl and k2 are shear moduli, s is the tyre soil displacement (i x j), where i is the slip and j is the distance from initial contact between the tyre and soil, and Y is the maximum value of the expression within the larger numerator bracket. Janosi and Hanamoto [2] proposed a simpler equation to describe the asymptotic curves of shearing stress versus soil deformation: ~" = l~max(1 -- e -ffK)
(2)
*Department of Agricultural Engineering, South Dakota State University, College Station, Brookings, SD 57007, U.S.A. tDepartment of Agriculture and Environmental Science, The University, Newcastle upon Tyne, U.K. 443
444
et al.
R. GODBOLE
where K is a soil deformation modulus which varies depending how rapidly, under deformation, the soil strength reaches its maximum value. Wismer and Luth [3] suggested an equation for traction on the basis of the above equation. P
- 0.75(1 - e-°3c~i).
(3)
W They used cone index (based on cone penetrometer resistance) as a measure of topsoil strength properties. Gee-Clough [4] suggested a similar empirical equation for traction. The principal terms CT and (CT)m~x were defined in terms of the vehicle mobility number. Mobility number is based on cone p e n e t r o m e t e r resistance, tyre dimensions and tyre deflection. Brixius [5] developed several prediction equations for bias ply tyres operating in cohesive-frictional soils. These equations contained more wheel parameters and were a result of testing that was conducted in a wide range of soil types. However, they also depend on cone index which seems to be an insufficient parameter for defining soil strength for traction prediction purposes. Dwyer et al. [6] conducted traction tests in which they used a cone p e n e t r o m e t e r as well as a soil shear meter to measure soil parameters. Prediction using cone index showed poor correlation between measured data and predicted values for traction coefficient at 20% slip. However, a similar comparison using data obtained from plate sinkage tests gave better correlations. The investigation suggested that soil shear strength and soil rubber friction were likely to be important variables in predicting coefficient of traction. Soil strength is, however, known to vary with soil type, bulk density, soil water content, etc. Cone penetrometer resistance also depends greatly on its design and method of use. The cone p e n e t r o m e t e r fails to take surface effects into consideration and bears little or no anology with the mode of operation of the traction tyre. As an alternative to using the cone penetrometer, Wittig [7] proposed the use of a single wheel tester (SWT) for determining a soil strength parameter in predicting tractive performance. The SWT is designed to provide information on the tyre-soil interaction. Different normal loads were applied to the SWT and the maximum torque that could be developed was measured. The slope of the plot of normal load versus wheel torque was used as a soil strength term. The general form of the equation used was: P / W = a(1
-
e -bi)
(4)
where a and b are empirical constants. This method was shown to give more accurate predictions on Vienna loam soils than the cone penetrometer approach of Wismer and Luth. A fully instrumented device capable of measuring soil sinkage and shear parameters was developed by Upadhyaya and Wulfsohn [8]. It was employed to conduct in situ sinkage and shear tests. An 18.4R38 tyre was used for traction tests at different inflation pressures. Soil parameters obtained by the instrumented device were related to terms in an equation for traction prediction. In an earlier study, Wulfsohn and Upadhyaya [9] had conducted field tests for three different types of radial tyre on five different soils with three axle loads. They obtained a prediction equation for different soils and loading conditions and concluded that soil conditions had the greatest effect on the outcomes of their study.
TRACTIVE PERFORMANCE ON SOIL SURFACES
445
There is, however, another interesting approach suggested by Janosi and Hanamoto [2] whereby the thrust is assumed to act similar to the shear force, in a horizontal plane under the traction device:
P= F[l+K(e-i~-l)]il
(5)
where F is the maximum shear force and P is pull. This equation is applicable to soils showing asymptotic behavior on a shear-displacement curve. Different soil types are assumed to give different forms of shear-displacement diagrams. The asymptotic or work hardening nature of the shear diagrams was thought to be for cohesionless soils (normally consolidated or lightly consolidated) whereas cohesive soils (dense or heavily consolidated) tend to give curves of a humped or work softening nature (Fig. 1). Investigations by Komandi [10], however, established that humps in soil shear diagrams are not caused by cohesion exclusively and that friction and cohesion are non-separable phenomena. Equation (5) was therefore considered as forming the basis of this study. This paper proposes a modified form of equation (5) [11]. This can be derived as follows: For a track,
F = bf~rdx where b is the track width and dx is the elemental distance along the contact length, and from equation (2),
F=
bfo(C+ 1
otan40(1 -
e-J/r)dx.
(6)
For uniform normal pressure a is independent of x and is equal to W/bl where the contact area, assuming a rectangular contact patch. The final form of the equation is
bl -- A,
WORK
./
///"
WORK
SOFTENING
HARDENING
CURVE
CURVE
f
Displacement FIG. 1. Typicalshear deformationcurves.
446
R. GODBOLE et al. --
c + tan(p
1 +--(e
~ -
1)
(7)
il
where
A l i K c q0 P W
is the contact area: is the contact length; the slip: is the soil deformation modulus: is soil cohesion; is the angle of internal friction for the soil; is the pull; is the axle load.
The experimental and analytical work that the authors present makes use of the data obtained by Wittig [7]. Note that all the terms involved in the equation (7) can be directly measured and are not empirical. Further, a theory is presented that makes use of an approach suggested by Krick [12] which correlates the term K with tractor tyre properties.
THEORY Contact patch geometry
Figure 2 shows the basic tyre geometry assuming a rigid surface where B is the tyre width, D is the tyre diameter, S is the tyre section height, f is the tyre deflection, l is the contact patch length, b is the contact patch width:
12 Of - f2 _ 4
Since D >> f , l2 = 4Dr
.'. l = 2"~fDf.
(8)
Similarly for the cross section, b
=
2~f-Sf.
(9)
Contact area A, assuming an elliptical patch: lb A =rr-4 .'. a
= rrf~-~.
(lO)
The contact area assuming a rectangular patch is A = Ib
A = 4f~ DF~S.
(11)
In practice, the contact area may be different from the values given by equation (11).
447
T R A C T I V E P E R F O R M A N C E O N SOIL S U R F A C E S
D/2
~b-4 F~
t
I-
I
"l
B
RECTANGULAR
B
PATCH
ELLIPTICAL
PATCH
FIG. 2. Contact patch geometry.
The principal reason for this is the difference in sinkage on various surfaces. This can be accommodated by introducing an appropriate coefficient. Thus for the contact patch length, l = 2Cl'~/Df.
(12)
C1 = 1 for hard soil; C1 > 1 for soft soil. For the contact area: A = 4CAfV~
(13)
for rectangular patch. Ca = 1 for hard soil, Ca > 1 for soft soil. Tyre deflection (f) on a hard surface was determined by Krick [10] to be a function of the section height and the tyre numeric, T. The tyre numeric was defined as follows, T-
pDB W
(14)
where p is inflation pressure. Krick established the following empirical prediction equation for deflection, f: f - 0.67T -°'8. S
(15)
448
R. G O D B O L E et al.
For the purposes of this investigation, Krick's equation was re-evaluated using the section height S instead of tyre section width B is the expression for the tyre numeric. Thus the modified tyre numeric, T', is: T'-
(16)
pDS
W The result of this analysis for different tractor tyres gave a similar relationship to that given by Krick: - 0.54(T') -°79
f
(17)
S which is of the general form f = Q S T '-m where Q is the coefficient of deflection, S is the section height, T' is the modified tyre number and rn is the exponent. Krick proposed his equation based upon the experimental data of the most commonly used agricultural tyres. However, for smaller tyres the coefficient Q and the exponent may be different. In order to investigate this possibility, a similar analysis was conducted for the 6.5 x 15 (17 x 38.1 cm) tyre which had been used previously by Wittig in the design of the single wheel test device (SWT). An experiment was conducted wherein the tyre deflection was measured for a set of loads at different inflation pressures. The resulting load-deflection curves on a hard surface are shown in Fig. 3. A corresponding plot of the dimensionless terms f / S and T' is
T A B L E 1. O B S E R V A T I O N S
FOR T H E L O A D - - D E F L E C T I O N
TESTS ('ONDI_JCTED BY
IHE
AUIHORS
AI
SDSU.
LISTS T H E L O A D A P P L I E D ON T H E T Y R E , T H E C O R R E S P O N D I N G D E F L E C T I O N A N D I N F L A T I O N P R E S S t l R I
Deflection (mm)
Load (kN)
1.27 3.708 6.299 8.89 11.48 I4 16.54 19.15 21.77 24.31 26.75 29.06 31.06 32.84 1.727 4.496 7.29 9.931 12.78 15.6 18.36 21.13 23.83 26.44 29.06 31,52
0.624 0,892 1.382 1.895 2.43 2.898 3.367 3.813 4.147 4.459 5.106 5.708 5.975 6.287 0.691 1.088 1.538 2.007 2.453 2.898 3.246 3.572 4.013 4.281 4.794 5.239
Pressure Deflection (kPa) (ram) 206.8 206.8 206,8 206,8 206.8 206.8 206.8 306.8 206.8 206.8 206.8 206.8 206.8 206.8 172.4 172.4 172.4 172,4 172.4 172.4 172.4 172,4 172.4 172.4 172.4 172.4
34.01 36.3 38.61 1.27 3,988 6,807 8.966 11.79 14.53 17.32 20,19 22.86 25.73 28.55 31.37 34.06 36.83 39.17 41.63 44.04 1,753 4.496 7.315 10.26 13.23 16.1
Load (kN)
Pressure (kPa)
Deflection (mm)
Load (kN)
Pressure (kPa)
5.574 5.841 6.283 0.312 (I.669 I.(17 1.449 1,739 2.029 2.363 2.675 3.032 3.389 3.79 4,147 4.437 4.86 5.039 5.329 5.739 0,491 0.803 1.137 1.427 1,659 2.002
172.4 172.4 172.4 137,9 137,9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 137.9 103,4 103.4 103.4 103.4 103,4 11)3.4
21,84 24.79 27.76 30.71 33.5 36.47 39.37 42.16 45.09 1,943 5.055 8.052 11.24 14.5 17.6 20.78 23.9 27.08 30.28 33.38 36.3 39.57 42.7 45,67 48.74 51.82
2,426 2,782 3.121 3.434 3.701 3,978 4.241 4.508 4.771 0,303 0.513 t/.705 0.883 1.106 1.249 1.342 1.65 1.953 2,14 2.319 2.493 2.707 2.947 3.21 ! 3,518 3.768
103.4 103.4 103.4 103.4 1(t3.4 1(13.4 103.4 1(/3.4 1(/3.4 68.94 68,94 68,94 68.94 68.94 68.94 68.94 68.94 68.94 68.94 68.94 68.94 68.94 68.94 68.94 68.94 68.94
I1
TRACTIVE PERFORMANCE ON SOIL SURFACES
449
7000
46000 5000 z
4000
J
3000
x' ,x'
x
.x. ;<'
2000 1000 0 0
10
20 30 40 Deflection rnm
50
60
FiG. 3. Load-Deflection characteristicsfor different tyre inflation pressures.
] -1.5
-2
f/s ___= 1.o5 ~(pOS/W)
-2.5 ~
~
-3 -3.5-
~
-4 -4.5 -5
0.5
~
~
1'.5
2
2~.5
~
315
~4
In[{p/W}*S*D]
FIG. 4. Non-dimensionaltyre deflection parameters for 6.5X15 tyre.
also shown (Fig. 4). The resulting deflection equation for the 6.5 x 15 tyre is as follows [R 2 = 0.97 and d f = 77]:
Sf - 1 ' 0 5 [ - ~ - ] -1'24
(18)
which is of the same general form, but has different Q and m values as might be suspected. Traction equation It is assumed that the normal stress (a), shear stress (r) and soil deformation modulus K will be approximately constant along the contact patch (Fig. 5). The following equation for the pull does, however, provides for quantification of the coefficients and exponents based on soil and tyre parameters. From equation (7),
l'-ni P = A ( c + otanq~) [ 1 + __re nt where n = 1/K and a = normal stress.
- 1) ]
(19)
450
R. G O D B O L E et al.
k
\ "\ '\\
/,
>
l
p~
/ n
J
FIG. 5. Schematic diagram for tyre-soil interface.
Previous work has largely negelected the influence of o on K. There is, however, a tendency for K to decrease with increasing values of o, as the soil becomes stiffer• A simple linear relationship between K and o was found to give a reasonable fit to experimental data for the soils around Brookings. The resulting relationships are plotted in Figs 6 and 7 for two sites. The equations describing this relationship are of the form K = K0 - C~o, where Ck is the slope of the K versus normal stress plot. K Values• Values of the deformation modulus K, were determined from shear stress-displacement curves for different levels of normal stress, o. The value of K was interpreted as the ratio of maximum shear stress (~'max) and the slope of the tangent drawn at the origin of each curve. Thus K = rmax/tan (oi) (Fig. 8). The shear displacement curves were generated by using a direct shear testing machine ( G E O T E S T Digital Model $2215). A schematic diagram of the machine is shown in
0007 .
0.006 ~ i d )t,18!
4 ;'[
b " N yr~
( ; t r . ,:,
.: 0.005 7:
•
£ 0.004.
0.003 ! 0.002
0001 ' 0
*',:i
10
20
30
40
50
)4orm~d $t r,s~
60
70
80
90
100
kP, R
FIG. 6. Plot of K values vs normal stress for soil 1.
-,
T R A C T I V E P E R F O R M A N C E ON SOIL S U R F A C E S
451
0.006
•
0.005
•
.~ 0.004
" "*~'~.~1 [ k= o0514.35E.Sx No~alStress_ ~
]
I :
0002 0.001 -,
"'~
0. . . .
0
10 20
---,
30 4'0 50
- ~
4
6'0 7'0 8'0 90 100
Normal Stress - kPa R~= 0.B1
Fio. 7. Plot of K values vs normal stress for soil 2.
Typical Shear-DisplacementCur~
,
/
b~
Displa[~ment FiG. 8. Definition of K, the soil deformation modulus.
Fig. 9. The shear tests conducted can be categorized as consolidated-undrained tests. Shear rate was held constant at 0.061 mm/s. Soils were collected from two sites. Four samples of each soil were used for testing. Each sample was tested at five levels of normal stress. The resulting relationships for one such sample are shown in Fig. 10. The maximum shear stress was also noted each time. This information was used in generating the Coulomb's [v = c + tan (q~)a] equation for the soils (Figs 11 and 12). In order to find out which factors influence the soil deformation parameter K , the following model is proposed. It was assumed that the parameter K was dependent on shear stress (v), normal stress (a), shear rate (s) and the contact area (A). K = f [ r , a, s, A]
452
R. G O D B O L E et al. s
Dial gauge to measure vertical movement
EED f - Alignment pins Lateral deformation measuring gauge
Loading bar /
'
'
so,;
L
'
'
'
/
Serrated edges Io hold ~ . a m p l e /
Setscrews tO Separate shear box
. ~
F](~. 9. Schematic diagram of the soil testing machine. 80-
c•! x
!6A97 kff~, 50
x
9~1;, kia
CO
30[
•
03
•
20~ . 10 t C 0
•
nn
.
nn
nn
•
•
•
•
•
n~
• 2
4
6 8 10 12 Displacement j - m •
14
16
18
FI(;. 1(). Shear displacement curves for different normal stress levels. 80 i
70~ i
•
GO~
5o] 40'
;.
so, 201 0
i , 10
20
, . : 30 40 50 6Q 7'0 Normal Stress kPa
8r0
• 90
i 100
FK;. 11. Plot of shear stress vs normal stress for soil 1.
T R A C T I V E P E R F O R M A N C E O N SOIL S U R F A C E S
453
100
"i
90 80
4oJ 60
i
/
/~ j
5o1 O3
.-
f
70
!
/ ~11 /
/
i . ~ ~/ ~
~ I
Shear
~:
= 2 9 7 4 + t&n (27 3 9 ) x N o ~ a J
[
1 J
-4 ~
3O 20
0
10
20
30 40 50 60 70 N o r m a l S t r e s s kPa
80
90
100 z
R = ,. ,c,
Flo. 12. Plot of shear stress vs normal stress for soil 2.
where K - [L], r - [ML-1T-2], o - - [ML-1T-2], s - - [LT -1] and A - [ L 2 ] . Using Buckingham PI theorem, K 2 / A = Ccr ('r/a) cl where Ca is an experimental coefficient and Cl is an empirical constant. The K value is, therefore expected to vary with the contact area involved. This rationale was applied in converting the K values determined in the laboratory tests to equivalent K values for the tyre-soil interaction. That factor w a s Kequiv. = Klab. test X N/(Atyrecontactpatch/Alab. test ). The rate of shear was assumed not to affect K based on the above analysis. This assumption was confirmed by Bekker [13] who found that for large range of soil deformation rates (0.127-127 mm/s) the effect upon the soil strength is insignificant. Deflection calculations
Contact length and deflections were calculated at a given axle load from the formulae given earlier. The validity of these calculations was checked with the empirical equation suggested by Painter [14]. This equation is of the form, W = ao + a s f + a6pf [2-(az+a')]
where f is the tyre deflection, p tyre inflation pressure and a0, a2 etc. are empirical constants. Values determined by Painter for the coefficients are as follows:a0 = 61, a5 = 16, a 6
=
448,
(a2
+
a4)
=
0.7.
Using Painter's equation, for a typical load of 21 kN and a standard tyre size of 13.4 x 38, the deflection f = 0.102 m. This compares quite well with the deflection value of 0.0969 m determined from the equation for the deflection given earlier. Contact area calculations
Area calculations were made using the formulae developed in the earlier section assuming a rectangular contact patch. Painter has suggested a similar empirical equation for contact area,
A = ~--~-ala3Da2cazf 2-(a~+a4) 4
R. GODBOLE et al.
454
where D is the outside d i a m e t e r of the tyre, C is the tyre cross section equivalent d i a m e t e r of curvature, f is the tyre deflection and a~, a2 etc. are empirical constants, were ala3 = 5.6, a2 = 0.35, (a2 + a4) = 0.7. Using this e q u a t i o n gives an area A = 0.2252 m 2 which c o m p a r e s reasonably well with o u r calculated value of A = 0.2985 m 2. The difference in these two results could be attributed to the fact that the Painter's e q u a t i o n uses C, the tire cross section, as the equivalent d i a m e t e r of the tyre section which is greater than B, the section width, used in o u r calculations. U p a d h y a y a and W u l f s o h n [15] have p r o p o s e d a new model for calculating the contact areas and lengths which involves measuring the incremental lateral arc lengths ( A S ) of the profile at discrete locations along the contact length and fitting the coefficients of the m o d e l of soil d e f o r m a t i o n at the s o i l - t y r e interface to experimental data. The contact lengths and areas calculated by this m e t h o d are as shown in Table 2. T h e y c o m p a r e favourably with the calculations m a d e by the authors.
n value computations n values were c o m p u t e d as l/K, however, a new a p p r o a c h to the determination of n is suggested in the following:1 n
--
K" Substituting for l = 2CI~/ Df and K = Ko - Cko,
2 C, ~-Df n
--
Ko-
Cko"
Dividing the n u m e r a t o r and d e n o m i n a t o r by a,
(2C,~7/a) n
--
( K o / a ) - Ck Substituting for a = W / A ,
(2CID°Sf°sA/W) Ft =
( K o A / W ) - Ck Substituting for A = 4Caf ~ - ~ and f = (QCfST '-m) gives: (SCICaDQ1.SS2T ' 1"Sin~W) n = [(4CaKoQS 1"5D °5 T ' - m ) / w ] C k TABLE 2.
Load (kN) 18 27 18 27
CONTACT
Inflation pressure (kPa) 85 85 125 125
LENGTHS
AND
AREA
CALCULATIONS
Contact length (mm)
BY DIFFERENT
METHODS
Contact patch area (m:)
W & U*
Author
W & U*
Aut
Au +
522 710 520 630
777.4 912.1 667.3 783.7
0.1883 0.3004 0.1647 0.2795
0.294 0.405 0.2169 0.2987
0.231 0.318 0.171 0.2345
*As computed by Wulfson and Upadhyaya [15]. tAs computed by authors assuming a rectangular patch. :~As computed by authors assuming an elliptical patch.
TRACTIVE PERFORMANCE ON SOIL SURFACES
455
It was found that (4CaKoQSI"SD°'ST'-m/w)>> Ck [Ca = 1.1, K 0 e q u i v ' = 0.058 m, Q = 0.54, S = 0.467 m, D = 1.82m, T' = 3.91, m = - 0 . 8 , W = 21 kN and Ck = 4.2 x 10 -5 m/kPa]; thus Ck will have little influence over the outcome of the n value. Hence upon simplifying,
CDO.SQo.sso.5T,-,n n --
Ko where C is a constant and
SDQT'-2m ]0.5 . '.
n
oc
K~
(20)
which is a dimensionless ratio for n.
RESULTS AND DISCUSSION
Wittig [5] conducted several tests under different soil conditions. Wittig's measured values of dynamic traction ratio were compared with Wismer and Luth predictions and an equation was developed based on a single wheel device to determine soil strength. The first approach was to determine if Wittig's measured data would fit an equation of the following general form: P
[ ( e - B l i - 1)] - B0 1 + .
Bli
(21)
It was found that an equation of this form provides a good fit to the measured data (R 2 = 0.98). Each term in the modified equation was then calculated using theoretical and empirical formulae. Soil samples were collected from the same fields where Wittig had conducted his tests. The following four graphs (Figs 13-16) and Tables (3-6) show a comparison between the P/W values obtained in three ways.
~et 0.! 0.; 0.| 02
~
J
Modlflnd ~l.n IN
ueam~ed Values
0.3 0.2 0.1 ¢
Wlemer-Luth Eqn
-I1.1
-0.2 "-11.3
o
03
o:2
o13
o74
FIG. 13. Graphs showing comparisons between
o15
0.6
P/W values obtained using various methods.
456
R. G O D B O L E
et al.
P,/qq ratio v/s Slip Rough Chisel plowed (partially covered) 0.8 0.7 0.6
S
0.5 0.4 0.3
f
~m
Meas,.u~ Values ~
.
+
W'mmer-Luth
0.2 0.1 0 -.0.1 -0.2 -{3.5 0.1
Flo. 14.
0.2
0.3
Slip i
0,4
0.5
Eqn1
0,6
Graphs showing comparisons between P/W values obtained using various methods.
P/W ratio v/s Slip Bare Soil after slight rain O.g
0.8 0.7 O.fi 0.5 0.4
Values
ll•u•d
W'~nn~-Luth
03 0.2 0.1
/
0 -0.1 -0.2 -0.3 0
Fro. 15.
011
0;2
0:3 0?4 Slip i
Graphs showing comparisons between P / W
015
0.6
values
obtained using various
methods.
(1) by using the modified equation suggested by the authors, (2) from the measured data of Wittig, (3) using the Wismer and Luth equation. The difference between the P/W values obtained by actual measurement and those obtained by using the modified equation can be explained as follows: (1) The testing conducted by Wittig was done in 1990 whereas the authors conducted their soil tests from samples collected in 1992 at the same locations. This has obvious
TRACTIVE PERFORMANCE ON SOIL SURFACES
457
r/w ratio Rough ChiselPlo'~xl o.g 0.8
~
03 0.6
~
~= 0.5
=
w
ilensm~l Valuta
f
:,,3 0.4 ~, 0.3
Irlmn--laall ~la
0.2
j
c
o
,-0.1 -0.2 ,,~....x.-0.3 o
o:1
o:2
o:3
o:~
FIG. 16. Graphs showing comparisons between
o:s
o.6
P/W values obtained
using various methods.
TABLE 3. TABLES SHOWING COMPARISONS BETWEEN P / W VALUES OBTAINED USING VARIOUS METHODS
W (kN)
i
21.9 21.9 22.1 22.2 22.4 22.4 22.85 22.9
0.005 0.04 0.072 0.091 0.117 0.146 0.243 0.298
K(m) 0.0353 0.035 0.035 0.0349 0.0347 0.0347 0.0341 0.0341
l(m) 0.843 0.843 0.843 0.843 0.843 0.843 0.843 0.843
n 23.89 24.09 24.09 24.16 24.3 24.3 24.73 24.73
Pull (kN)
P/W
P/W
P/W
modified
measured
Wismer-Luth
1.142 7.078 10.52 12 13.48 14.81 17.02 17.65
0.052 0.323 0.476 0.54 0.602 0.661 0.745 0.771
0.258 0.316 0.35 0.404 0.41 0.534 0.552
-0.131 -0.083 -0.056 -0.021 0.016 0.123 0.175
TABLE 4. TABLES SHOWING COMPARISONS BETWEEN P / W VALUES OBTAINED USING VARIOUS METHODS
W (kN)
i
21.7 21.9 22.5 22.7 22.8 22.15 22.8
0.048 0.077 0.181 0.242 0.245 0.268 0.334
K(m) 0.0355 0.0353 0.0345 0.0343 0.0342 0.035 0.0342
/(m) 0.843 0.843 0.843 0.843 0.843 0.843 0.843
n 23.74 23.9 24.41 24.58 24.67 24.11 24.67
Pull (kN)
P/W
P/W
P/W
modified
measured
Wismer-Luth
7.971 10.79 15.68 16.9 17 16.92 17.89
0.367 0.493 0.697 0.745 0.746 0.764 0.785
0.187 0.266 0.432 0.495 0.513 0.485 0.546
-0.098 -0.051 0.092 0.16 0.163 0.185 0.244
i m p l i c a t i o n s fo r t h e c o m p a r i s o n o f p r e d i c t e d values with t h o s e m e a s u r e d in t h e field. H o w e v e r , t h e c o m p a r i s o n is i n c l u d e d in t h e p a p e r since soil m o i s t u r e c o n t e n t s w e r e similar for the V i e n n a l o a m soils. F u r t h e r testing is n e e d e d to p r o v i d e a valid comparison. (2) C h a n g e s in t h e c o n d i t i o n s o f th e soil an d c r o p r esi d u e m a y h a v e r e s u l t e d in d i f f e r e n t (c + o tan q~) values.
458
R. G O D B O L E TABLE 5.
W (kN)
i
21.55 21.65 22.2 22.45 22.85 22.85 22.85 23.1 23.3 23.25 23.4
0.042 0.051 0.113 0.132 0.214 0.222 0.257 0.387 0.41 0.448 0.539
TABLES SHOWINQ COMPAR|SONS
K(m)
TABLE 6,
W (kN)
i
21.45 21.95 22.15 22.25 22.65 23.1 23.15 23.3
0.022 0.052 0.067 0.102 0.164 0.312 0.384 0.508
/(m)
0.0357 0.0356 0.0349 0.0346 0.0341 0.0341 0.0341 0.0338 0.0336 0.0336 0.0335
TABLES
[).843 0.843 0.843 0.843 (I.843 0.843 0.843 0.843 (/.843 0.843 (I.843
n
23.62 23.7 24.15 24.36 24.71 24.71 24.71 24.93 25.11 25.07 25.2
BETWEEN
0.0358 0.0352 0.035 0.0349 [).0344 0.0338 0.0338 0.0336
/(m)
0.843 0.843 0.843 0.843 0.843 0.843 0.843 0.843
n
23.54 23.94 24.11 24.19 24.54 24.93 24.98 25.11
P/W
VALUES OBTAINED
USING VARIOUS METHODS
Pull (kN)
P/W
P/W
P//W
modified
measured
Wismer-Luth
7.201 8.285 13.17 14.15 16.55 16.69 17.18 18.39 18.62 18.77 t9,16
0.334 0.383 (I.593 (I.63 0.724 //.73 0.752 (1.796 0.799 0.807 0.819
0.134 0.168 0.352 0.415 0.537 (1.499 0.543 0.605 /I.665 (/.657 (1.7
--0.01 1~.009 ~.145 0.179 0.303 1t.314 [I.353 t~.458 0.476 0.496 IL534
SHOWING COMPARISONS BETWEEN
K(m)
et al.
P/W \ A L U E S
OBTAINED
U S I N G V A R I O t ~ S ME 1 H O D S
Pull (kN)
P/W
P/W
P/W
modified
measured
Wismer-kuth
4.309 8.52 10.09 12.62 15.33 17.88 18.41 19.01
0.201 [/.388 0.455 0.567 0.677 (/.774 (/.795 [/.816
(/.104 0.241 0.325 0.353 0.475 0.611 0.621 0.664
-[~.21 0.17 - 0.16 -(t.12 0.055 0.074 (I.126 0,201
(3) The contact area A is either calculated by formulae or checked using empirical relationships. This may differ from the exact value for contact area. In addition, effects such as soil wall build up around the tyres and soil topography may affect the area term. (4) The contact length term used in the prediction equations may differ from actual values. Despite these provisos, the four graphs suggest that the predictions obtained are reasonably accurate and that with some improvement in the measurement of the relevant terms, traction prediction could be improved. The expression obtained for n; n = C [ S D Q T - Z m / ( K o ) 2 ] °'5 is a valuable result. Researchers need to confirm the validity of the expression experimentally. Once confirmed, for a range of soil types, then given the tyre size and soil type, it will be relatively simple to predict the n value. When the n value is known, the gross traction can be predicted for given axle loads and assumed slip values. It is suggested that this approach will be useful in the design of tracked vehicles and wheeled tractors. CONCLUSIONS
(11 The equation predicts traction coefficient more accurately than some other methods and justifies the use of the soil deformation modulus in traction studies.
TRACTIVE PERFORMANCE ON SOIL SURFACES
459
(2) Soil d e f o r m a t i o n m o d u l u s is a g o o d e s t i m a t o r o f soil s t r e n g t h a n d has r e l a t i o n s h i p with n o r m a l stress levels. (3) E x p r e s s i o n s d e v e l o p e d for c o n t a c t l e n g t h s a n d a r e a p r o v i d e results t h a t a r e c o n s i s t e n t with o t h e r m o d e l s .
FURTHER WORK (1) T h e a u t h o r s w e r e a b l e to establish a r e l a t i o n s h i p b e t w e e n K a n d o which was n o t a v a i l a b l e p r e v i o u s l y , h o w e v e r t h e r e a r e o t h e r factors o n which K m i g h t d e p e n d , such as a r e a , s h e a r r a t e , m o i s t u r e c o n t e n t o f soil a n d b u l k density. R i g o r o u s testing n e e d s to b e c o n d u c t e d in this r e g a r d . (2) T h e single w h e e l t e s t e r ( S W T ) is a r e l a t i v e l y e a s y i n s t r u m e n t with which to c o n d u c t soil tests. It is n e c e s s a r y to e s t a b l i s h a r e l a t i o n s h i p b e t w e e n n for large t y r e a n d n for the small ( S W T ) tyre. Scaling can t h e n b e a p p l i e d in c a l c u l a t i n g n for large tyres.
REFERENCES [1] M. BEKKER, Off the Road Locomotion. The University of Michigan Press (1960). [2] Z. JANOSl and B. HANAMOTO, The analytical determination of drawbar pull as a function of slip for tracked vehicles in deformable soils. First Int. Conf. Mechanics o f Soil-Vehicle Systems, Torino-St. Vincent (1961). [3] R. WISMER and H. LUTH. Off-road traction prediction for wheeled vehicles. Trans. A S A E 17 (1), 8-14 (1974). [4] D. GEE-CLOuGH, Selection of tyres for agricultural vehicles. J. agric. Engng Res. 25,261-278 (1980). [5] W. W. BRIXIUS, Traction prediction equations for bias ply tyres. ASAE Paper No. 87-1622 (1987). [6] M. J. DWYER et al. The field performance of some tractor tyres related to soil mechanical properties. J. agric. Engng Res. 19, 35-50 (1974). [7] V. Wlx'rlG, Prediction of tractor drawbar pull on agricultural soils. M.S. Thesis, Dept of Agricultural Engr, South Dakota State University, Brookings, SD 57007, unpublished (1990). [8] S. UPADHYAYA and D. WULFSOHN, Traction prediction using soil parameters obtained with an instrumented analog device. J. Terramechanics 30 (2), 85-100 (1993). [9] S. UPADHYAYAet al. Traction prediction for radial tyres. J. Terramechanics 311 (2), 149-175 (1989). [10] G. KOMANDI, The determination of soil properties from slip-pull curves. J. Terramechanics $ (2), 9-23 (1968). [11] R. N. YONG et al. Vehicle Traction Mechanics, Elsevier, Amsterdam (1984). [12] G. KRICK, Radial and shear stress distribution under rigid wheels and pneumatic tyres operating on yielding soils with consideration of tyre deformation. J. Terramechanics 6 (3), 73-98 (1969). [13] M. BEKKER Introduction to Terrain-Vehicle Systems. The University of Michigan Press (1968). [14] D. PAINTERA simple deflection model for agricultural tyres. J. agric. Engng Res. 26 (1), 9-20 (1980). [15] D. WULFSOHNand S. UPADHYAYA, Determination of dynamic three-dimensional soil-tyre contact profile. J. Terramechanics 29 (4/5), 433-439 (1992).