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The role of displacement in a simple traction system
Journal of Terramechanics, 1966, Vol. 3, No. 1, pp. 47 tO 62. PergamOn Press Ltd.
Printed in Great Britain.
THE ROLE OF DISPLACEMENT IN A SIMPLE TRACTION SYSTEM* J. H. TAYLOR and G. E. VANDEN BERGt INTRODUCTION INSUFFICIENT information is available to analytically determine the thrust a vehiclo can generate in a given soil condition. Empirical methods based on observation and measurement are presently used but they can be greatly improved. If only the maximum pull of a traction device is of interest, then only the pulling force need be measured. However, if the slip at which the maximum pull occurs is of interest, then it becomes necessary to measure relative displacements. The degree of sophistication necessary in describing any phenomena depends upon the level or intensity of one's interest. In traction research we are seldom interested only in the "point" of maximum pull, but rather the entire slip-pull "curve" is desired for analysis of performance. The Coulomb failure criterion expresses the maximum shearing stress of a soil for any given normal load. If shear stress-displacement curves for each normal load are plotted, the parameters of the Coulomb equation are evaluated by using only one point from each curve. The point used is the maximum shearing stress and this, of course, is based on the assumption that the maximum shearing stress of the soil is reached within the range of displacement used. The maximum shearing stress expressed by the Coulomb equation is closely related to the maximum pull of a traction device. A similar relationship exists between slip-pull curves and shear stress-displacement curves. The complexity increases as our level of interest increases just as it does in statistics when we progress from the arithmetic mean, a "number", to the regression "line" as our basis for evaluation. The validity of the analogy ultimately depends upon how effectively the soilmeasuring device system represents the soil-track or soil-wheel system. In any case there is little likelihood that a slip-pull curve can be predicted until an accurate description of a shear stress--displacement curve can be made. Considerable effort has been made to use soil shearing stress equations to predict thrust. Micklethwait [1] used a very simple approach in 1944 when he multiplied both sides of the Coulomb equation, S = C + P tan $, (S is the maximum soil shear stress obtained for a normal load P) by the ground contact area A to give SA = C A + PA tan $. This reduces to H = C A + W tan $, where PA is the total weight W and H is thrust. *Paper presented at the 1965 Annual Meeting of the American Society of Agricultural Engineers. tNational Tillage ,Machinery Laboratory, Auburn, Alabama, U.S.A. Communicated by A. R. Reece.
47
48
J.H.
T A Y L O R and G. E. V A N D E N BERG
Bekker [2] observed that the form of some shear stress-displacement curves is similar to that of displacement-natural frequency diagrams of an aperiodic damped vibration. He replaced the damping constant and the spring constant by soil parameters and developed his well known modification of the Coulomb equation. He then developed an equation for thrust by designating track dimensions and assuming a uniform normal pressure.
f
DEFORM&~'|ON
~1 ( I N )
FIG. 1. Typical soil shear stress-strain curve.
Janosi and Hanamoto [3] found that in the majority of cases the soil does not exhibit a hump and a decay in the shear stress-displacement curve but is of the type shown in Fig. 1. They developed a simpler modification of the Coulomb equation and proceeded from there to the development of an equation for thrust in the manner of Micklethwait and Bekker. All of these modifications attempted to account for the effect of displacement on soil shear strength. ANALYSTS
In the Coulomb failure criterion the normal stress is considered acting on the failure surface, and a distinct failure surface is assumed to occur. The horizontal line along the bottom of the cleats in Fig. 2(a) represents this failure surface as it might apply to a track. If vertical distribution of forces is ignored, the normal force is assumed to be acting on the failure surface. The only shearing strain that has occurred in Fig. 2(a) was within the plane of failure so that once failure was reached all other displacement was as a rigid body. Soehne [4], Bekker [5] and others have used various methods to show soil deformation beneath a traction device. Figure 2(b) is a composite concept drawn from personal observation and a study of the literature. In addition to the displacement along the plane of failure, Fig. 2(b) shows compaction occurring from the surface to some depth well below the plane of failure. Horizontal shearing begins at some undetermined depth with the accumulated horizontal displacements increasing to a maximum at the plane of failure. At sufficiently low slip values, no actual failure plane would develop. The system in Fig. 2(b) may be separated into units or elements for closer study. Stress is defined as force on a unit area. The stress measured is on a unit area
THE ROLE OF DISPLACEMENT IN A SIMPLE TRACTION SYSTEM
49
_~l i i I I I I I~~~_. __ :.:.:.'..: .....
"~' I:_I._L_L_I_.L_I,
: -" ~ "':::::
i o e o o
:::::
iii:
e e e e e
(a)
e o o o o
__~ ~ ~ :.:___ e l o t l
:::::
e o o o o
(b)
FIo. 2. (a) Soil displacement only in the plane of failure. Co)Accumulating soil displacement and compaction. which composes the topmost layer of a column of soil extending to some unknown depth. The column of soil shown in Fig. 3, which is originally vertical, may be used as the element which represents the basic unit of the system. According to Reece [6] and others, the depth of this column of soil under stress increases with increasing normal load. A logical representation of the soil strain in Fig. 3 would be the ratio of the accumulated horizontal displacements 3 to the vertical depth of soil under stress d. The horizontal displacement of the unit area on top of the column now becomes relatively meaningless unless we know the depth of the soil column affected. P2>PI
J_L= J_.22 dr d 2
, ,!///
FIG. 3. The column of soil which represents the basic unit of the system.
50
J . H . TAYLOR and G. E. VANDEN BERG
Neither the plane of failure nor the volume of soil stressed is usually controlled under a traction device. Therefore, when additional increments of load P are added to a unit area in a traction system as represented in Fig. 2(b), either the depth of the soil column in shear beneath the unit area will increase, or the density of the column will increase, or both. Further, when the depth of the stressed column d increases, the total accumulated horizontal displacement J at the surface must increase if the strain, defined as the J/d ratio, is to remain constant. Measuring the depth of soil stressed or the change in density as P is increased would be difficult, but since a change in P is the only action taken to cause these changes then both must be a function of P for a given soil condition. Now if P somehow represents the depth of the soil column under stress and J represents the total accumulated horizontal displacement at the top of the column, then some function of J/P should represent the soil strain as well as or better than the J/d ratio.
OBJE,CI'IVE The Coulomb failure criterion, which is a form of limit analysis, involves assumptions that seem to be incompatible with conditions that frequently exist in a soil-traction device system. The slip of a tire or track is known to be important, and it is equally well established that soil shearing stress varies with displacement. Since the Coulomb equation ignores displacement, it seems unlikely that it could ever provide a satisfactory mathematical representation of the soil-traction device system. Therefore, instead of attempting to modify the Coulomb equation, an experiment was designed to determine empirically the relation of soil shearing stress to normal pressure and displacement on a unit area. The objective was to establish a stress-strain relationship based on the deformation problem as exhibited in S-1 curves rather than a limit analysis based on a failure criterion. Concisely, the objective was to determine the general form of f where S = f (P, J).
APPARATUS AND PROCEDURE The requirements for a special device to simulate a unit area in traction are that neither the plane of failure nor the volume of soil stressed is controlled. Displacement J and normal stress P should be controlled and programmed and the resulting shearing stress S measured. Thus , / a n d P would completely determine S for a given soil. An annulus shear apparatus, called a Desometer, was designed and built with the added capability of measuring sinkage. It is shown in Fig. 4. The shearing head was an annulus with 6 in. i.d. and 8 in. o.d. Two annuli were used; one with a smooth rubber face and the other with ¼ in. lugs spaced radially 20 ° apart. When making the measurements the normal load was applied first using an air cylinder, and then torque was applied by an electric motor. The rate of rotation was I. 1 rev / min. Two soils were prepared and tested in the small circular soil bin. Hiwassee sandy loam was prepared in three different moisture-density conditions and Lloyd clay
THE ROLE OF DISPLACEMENT IN A SIMPLE TRACTION SYSTEM
51
FIG. 4. The annulus shear apparatus.
in two. Penetrometer measurements were made for each test. A summary of the soil data is contained in Tables I and 2. RESULTS AND DISCUSSION
Ten sets of S--J curves were secured from the combination of five soil conditions and two annuli used in the circular soil bin. In addition, S-1 curves were obtained from the Land Locomotion Laboratory, Warren, Michigan, and from soil prepared in the outdoor bins at the N T M L . Since only three basic variables are involved, one way to look at S as a function of ] and P is to make a three dimensional graph. Figure 5 is a perspective of one surface generated by the three variables. The S - ] curves in Fig. 5 resemble the ones called typical by Janosi and Hanamoto [3]. In this type curve S continues to increase as J increases through the range of ./ used. Although the value of S rises very slowly after the initial surge, S does continue to increase and the maximum S becomes a function of the magnitude of ]. These circumstances prevent the
J. H. T A Y L O R and G. E. V A N D E N BERG
52
TABLE 1.
SUMMARY OF SOIL MOISTURE-DENSITY CONDITIONS
Soil type and Test No.
Moisture content
Bulk density
Soil depth (in.)
By depth (%)
Test av. (% )
By depth (g / cm s)
Test av. (g / cm 3)
0"0-2'5 2.5-5"0 5-0-7'5
9"3 9.3 9.1
9.2
1-31 1-28 1.35
1.31
H-2
0.0-2"5 2" 5-5'0 5.0-7.5
12-9 15-0 14.9
14"3
1.63 1.68 1'77
1.69
H-3
0-0-2"5 2"5-5"0 5"0-7"5
8-2 8.3 8"2
8-2
1'38 1-41 1.49
1-43
L-I
0.0-2"5 2"5-5'0 5"0-7"5
22.4 21 "9 22"6
22'3
0.95 0"99 I'05
1"00
L-2
0"0-2"5 2"5-5"0 5"0-7"5
24' I 24-2 24"8
I"23 !'24 1"30
l"26
Hiwassee H-1
Lloyd
TABLE 2. Soil test No. Depth (in.)
24"4
PmcgT1tOM~T~t M]~SUREME1CrSFOR T~s'r CONDITIONs Cone index (psi)
I 2 3 4 5 6
'
3
H-1
Hiwassee H-2
H-3
L-I
Lloyd L-2
20 25 25 30 35 45
25 35 40 40 45 45
50 60 70 75 8O 90
15 30 35 40 45 50
55 70 80 85 9O I00
12'
6 8
Flo. 5.
Perspective of the surface generated by shearing stress S, normal stress P, and displacement ./.
THE ROLE OF DISPLACEMENT IN A SIMPLE TRACTION SYSTEM
53
determination of the maximum value of S needed for plotting against P in order to establish the Coulomb parameters C and ~b. Any representation of shearing stress which is adequate for traction must consider the relative displacement of the soil. 16
H - I-- STEEL M. C. •9.2 •
.
/
J'&O
o
J-4~
o') (/)
d -2.0
~s
J • l'O
n.
J -(>5
~4 -r
I
I
1
I
4 8 12 NORMAL STRESS P
16 (PSi)
Flo. 6. Shear stress vs. normal stress for constant displacement. Figures 6 and 7 illustrate how the shear stress-normal stress relationship varies with displacement. This shows that shearing stress is not a linear function of normal stress when displacement is held constant. When P is constant. S increases as J increases. 16 H - 2 - STEEL M.C. • 14-3
u)
~t2
•
J,
.
8-0
J - 4.0 u~ 03
J , 2'0
~e
J , I'0 J " 0'5
w •! -
4
I
0
4
I
I
8 12 NORMAL STRESS P
/
Ie (PSi)
FIG. 7. Shear stress vs. normal stress for constant displacement. To check the effectiveness of I/P in representing the changing strain as suggested in the analysis section, values of S and P were read from the S--J curves and plotted for convenient whole number ratios of J to P. This is shown in Fig. 8. After all
54
J . H . T A Y L O R and G. E. V A N D E N BERG
tests used in this study were checked in this manner, the relationship between S and P was found to be very close to linear when the value of J/P was held constant. The three dimensional plot of S, P, J (Fig. 5) helps to clarify the full significance of the relationship described in the preceding paragraph. The C-intercept will be ignored for the present discussion. The locus of a point described by J/P=a, where a is any constant, is a straight line in the J-P plane, or floor, of the perspective view in Fig. 5. In Fig. 8 this same line is shown to be straight in the S-P plane, A little geometrical reasoning will show that the line must also be straight when projected in the third perpendicular plane (S-J). Therefore the line OA, which lies on the surface, must be a straight line. dip = 1/4---~
8
¢n 6 u~ uJ ¢n 4
<= 2
~i
r
FIG. 8.
J/p=l/8
I
I
I
I
2
4
6 NORMAL
8 STRESS
I
P
LO (PSi}
I
I
12
14
,
16
Shear stress vs. normal stress when J/P is constant.
The surface in Fig. 5, which represents the relationship between S, P, and J, is generated by the path of a straight line OA as its projection in the J-P plane (l/P=a) is rotated about point O throughout the J-P plane. The actual path of the line will have to be determined by parameters not yet established. The specific case of S-J curves with well defined maxima fits well into this generalization. The conditions require that maximum S for each P be reached at the same J/P value. This l I P line is the projection of a line on the surface which corresponds with the Coulomb equation for curves with well defined maxima. Rotation of the straight line still generates the S-P-J surface. A quick and simple check can be made on S-1 curves to determine if constant J/P ratios give a linear relationship between S and P. The technique used is shown in Fig. 9. A straight-edge was used to draw lines at convenient J to P ratios with all lines converging to a single point near the origin. This point is an estimator of the C-intercept. The compliance is quickly noted by using whole number ratios. If J/P=¼, then when 1 = 1 , P = 4 ; when J = 2 , P = 8 etc. Figures 9, 10, and 11 represent curves of basically different shapes yet the S-P relationship in each case is essentially linear for all values of 1/P. The steps necessary in order to establish the mathematical model relating shearing stress to normal stress and displacement can perhaps be visualized best
THE ROLE OF DISPLACEMENT IN A SIMPLE TRACTION SYSTEM
P,3
L
;,/
/.
.... F H- i-- STEEL
J/P= 118 '5
/
J/P'I/4
1 J/P,I/2
L/..k"/ li:l~
= <
--
'
'<
"/
¢/'~"
t
L.
0
zP'8
I
,
2
,
3 4 DISPLACEMENT
5 d (IN)
6
7
FiG. 9. The method used in checking original data for agreement with the derived equation.
./ /
J/P,I/2
L
,,P., P : S _ _ ~ . - -
L-..-411---'t'~"~ ~
, /./
r~dT- / I ,,P., ~, vr t i ~ _ z ~ / ' I : ~,'l ,~'M-/ X_ l! I/ ' ~ 2 ''~ •I I I
I/
.~
J/P'8
~Ui ,i
I
2
6 8 DISPLACEMENT
IO (IN)
J
12
i I I
-
14
16--
FiG. I0. The method used in checking original data for agreement with the derived equation.
15 I0
JfP=t ,+
,,
-,..7_
/
P.s
gl'/
//
../ ~
]L:~o. 11.
4
J
i
p.4
_f
~P" I
P'Z
~
/_~ 2
NTML TRACK T E S T NORFOLK S L ~ 18 JULY 1963
~ JIP=4
, 6 DISPLACEMENT
J
I0 (IN)
12
14
16
method used in chewking original data for agreement with the derived equation.
55
56
J.H.
T A Y L O R and G. E. V A N D E N
BERG
by reviewing Fig. 8. The problem is to find the function relating the J-P relation to the S-P relation and this is equivalent to determining how tan ~b in the Coulomb equation varies as the value of ,I/P varies. It is unnecessary to plot S-P graphs such Fig. 8. The Coulomb tan ~ = S - - C / P can be computed from S - C / P = IS - C /.1] [] / P ]. Both S - C / J and J/P can be determined from the original S-J graphs as shown in Figs. 9, 10, and 11. Tan ~b was found to be related to J/P by a power function such that In tan ~b---n In ( l / P ) + l n (T), where n and T are system parameters. The parameters n and T are constant for a
1
2'0 H-I- STEEL
(iNTERCEPT) T,O.BT- - ~
I'O (SLOPE) n -0"191
.-..'-"'~
~"'--' ~
,, 0-5
-e-
O'1 "OI
•O5
O't
0"5
t-o
2-0
J/P
FIG. 12.
The method used for determining the parameters n and T.
20
H-t-
STEEL
S • 0 " 0 + O'e7 pO.eOt j o ' l ~ a. co
w
~5 •
c,
P,2
8 i
Q I
•
t
L
O
~
2
FIG. 13.
Computed values of S plotted on measured S-] curves.
3 a DISPLACEMENT
i
,5 d (IN)
i
I
i
6
7
8
specified soil condition and shearing device. They are not true soil parameters since the annulus with the rubber surface and the annulus with lugs gave different values for n and T in the same soil condition. A convenient method for determining n and T is illustrated in Fig. 12. Tan ~b is plotted vs. J/P on full-logarithmic graph paper, and the resulting straight line has the slope n and the intercept T. The intercept is read at J / P = 1.0 since In 1.0=0.
THE ROLE OF DISPLACEMENT IN A SIMPLE TRACTION SYSTEM
57
If tan ~b (the S - P relationship) is replaced by ( I / P ) ' T (the expression relating tan ¢k to I / P ) , the shearing stress may then be expressed as S = C + P (]/P)'T; or when simplified S = O + pI-,,1,,T. The parameters n and T were determined for each set of S - J curves in this study. Then the equation S = C + PI-"J"T was used to calculate S for several points on each S-J curve, and these points were superimposed on the original curves. These data are presented in Figs. 13-26. The 20
H- 2 - STEEL S" 1'0+0"77 pO.eOIjo-m o o
IO P'8 v 5
~..
P'4
i*l
8
P,Z Q O
I
i
I
2
/
I
o I
3 4 5 DISPLACEMENT J (IN]
I
I
I
6
7
8
RG. 14. Computed values of S plotted on measured S-/curves.
o --'-v)IS
~
'~
H'-3- STEEL S • I'0+ 0-83 pOq93~ . .
e
t~
il
I 2
P'2~I I i | 3 4 .5 DISPLACEMENT J {IN)
0 l 6
i 7
I 8
F]o. 15. Computed values of S plotted on measured S-,I curves. soil type, soil fitting, and sheafing device used (designated as rubber or steel) are listed in each figure along with the shearing stress equation used for that set of curves. In examining these figures remember that the continuous curve is original data and the points are calculated from the derived equation. The calculated values fit the measured curve very well except when P = 32 lb/in j where the results were erratic. There is ample evidence in the literature to indicate
58
J.H.
T A Y L O R and G. E. V A N D E N
BERG
that a normal load of 32 lb/in ~ will affect the soil to a greater depth than the 10 in. of soil used in these tests. The derived equation's description of the S-J curve is limited in one case. If the shearing stress reaches a maximum value and declines then only those values of S up to the maximum are in agreement with the equation. In the one set of curves from the outdoor bins (Figs. 11 and 23) the small initial peak was ignored and, since J was of no effect after the initial peak, these curves fit the general equation. The broken ] / P line in Fig. 11 shows that three of the four maximum
2O L - - ~-- S T E E L 0
/
S = 0 " 4 + 0 , 7 2 pO.e25 dO,;%
P".L2L-- o
P=8
n
P=4
I*1 n
o
co 0 I
FIG. 16.
P=:) ~ 2
I 1 3 4 DISPLACEMENT
o 0
J
L 5 (IN)
i 6
l 7
I 8
Computed values of S plotted on measured S-.I curves.
7 20 L- 2- STEEL S = 0 " 0 + 0 " 9 7 pO.Si7 jo.,33
=~IS c/)
wlO
--
i I
I 2
0=8
..gg--
P=4
8
i 3
I 4
DISPLACEMENT
FIG. 17.
•
I 5
d
i 6
i 7
J 8
(IN)
Computed values of S plotted on measured S-1 curves.
values of S occur at the same ,I/P value thus following the same pattern as all the other data. More data will be needed to expand the initial part of such curves to accurately determine the relationship existing before maximum S is reached. Further insight into the relationship of the variables in this study may be gained by expressing the shearing stress in the following manner: S = C + T X , where X = P1-"1".
THE ROLE OF DISPLACEMENT IN A SIMPLE TRACTION SYSTEM
59
This form of the equation emphasizes that S is a linear function of the combination of P and ]; that T is the parameter relating S to the combined variables, while n distributes the combined effect between P and J. From this it follows that the range of n is restricted to 0 ~ n ~ 1. If the value of the C-intercept is ignored, the actual value of n represents the per cent of shearing stress that comes from displacement. A look at Figs. 13-26 reveals that the effect of displacement varies from zero to 22 per cent under the test conditions used. 20
P= 32 0
H-t-RUBBER S =O'0+ 0 ' 7 8 pO.e3l jO'=e2
tn O io
,5 = ,~0 ' ' ' ~ . . . . e ~ .3 i
~ O i
I
0
P=2 L
2
3
O
o I
I
4
.DISPLACEMENT FIG. 18.
I
5 J
I
6
7
/
s
(IN)
Computed values of $ plotted on measured S-J curves.
20 H--2--RUBBER S " 0 " 4 0 + 0 ' 6 5 pO.,gO jO'ZZO 0 P-32
u~
~o
0
~
i
6 0
P'8 _
~
P,4
I
I
I DISPLACEMENT
FIG. 19.
J (IN)
Computed values of S plotted on measured $-$ curves.
2or
o
J
~
~
15
H- 3-RUBBER O
S • OeO + 0 , 7 9 5 pC-loo jo.zoO O
~'oI-I
~
¢
P=8
O
r"
,
I
,
2
Flo. 20. Computed
,P.z
0
.
3 4 DISPLACEMENT
.
J
.
5 (IN~
.
6
.
7
8
values of S plotted on measured $-J curves.
60
J.H.
T A Y L O R and G. E. V A N D E N
2°I
BERG
L-I-RUBBER
o
S " O'ZO + 0 " 6 5 pO44O jo.~6o
u~ P.16
~o
O 0
O
~B P-40
0
:, 1
,
,
o ,, I
o ,
.L .....
2
~
4
DISPLACEMENT
Fzo. 21,
e
P-2
1
i
7
•
J (IN)
Computed values of S plotted on measured S - J curves.
20
L-
0
2-- RUBBER
S " 0 " 6 0 + 0 " 7 5 pO4M)4 jo-,e
0
P~32
~
pffi8
0
Pffi4
0
t~
S
<
t 5 n
._.
0
.
pffi2 -
J I
1 2
I
,I
I
3
4
6
DISPLACEMENT
FIo. 22.
J
....
I
6
{IN)
Computed values of S plotted on measured 3 - 1 curves.
NTML
IO
TRACK TEST
NORFOLK S L 18 JULY 1963 S = 0 . 4 0 + 0.70 P
ul
Q
tn
vt
XC
-:,
"--
0
0
P=4
P.2
F[o. 23.
I
I
2
4
6 8 DISPLACEMENT
I0 d (tN)
';3
12
14
16
Computed values of S plotted on measured S-] curves.
THE ROLE OF DISPLACEMENT IN A SIMPLE TRACTION SYSTEM P,6
~~ l
~/
L,NoLOtL,.
,fO
MUNSON TEST
s I <,<,,4,o.,soP 0"1240"0711
./i" ~
i
It F
_
o
2 FIG. 24.
,
~
,
13
........
6
s to 12 14 ,s DISPLACEMENT J (IN) Computed values of S plotted on measured $-1 curves. u
"P'S P=4
,N LAND LOC. LAB. MUNSON TEST
~2 pt3
0 EAST ON CENTER 28 AUGUST ~)63 S,0,0~ 0 ~ ) 0 po'~s4j ¢'O~s u
P,2
O~
B
C
I;1
2
4
. . . . . .
6
Pi
la
,.
P,I
8 ~O DISPLACEMENT J (IN)
12
I4
_:
IG
.....
is
FIG. 25. Computed values of S plotted on measured S-J curves. 0 f
--
P~S
~z
LAND LOC. LAB. MUNSON TEST O
P~5
........ O
u
.....
CENTER 2B AUGUST 1963 S,O'OS 4.0'512 p~S~ j o ' ~ s
0
C.
i
2
O
,
, I
4
...... P-i
! ,
~
~
I
s
.....
....
O.
I
!
L
I
,0
,2
i4
is
DISPLACEMENT J (IN)
Flo. 26,
C o m p u t e d v a ~ e s of S plotted on measured S-J curves.
18
61
62
J . H . TAYLOR and G. E. VANDEN BERG
CONCLUSIONS The most important conclusions of this investigation are condensed and listed below : 1. The Coulomb equation is valid only for soil shear displacement curves which have a well-defined maximum value for shearing stress and then only for this maximum value. 2. No representation of shearing stress which is adequate for traction investigations can ignore the soil displacement. 3. The shearing stress is not a linear functiton of normal load when displacement is held constant. 4. While J / P is not soil strain, it is a logical and adequate representation of the deformation in the soil-traction system which results from soil strain. 5. The shearing stress is accurately described by the equation S = C + TX, where X=PI-~J n. This form of the equation emphasizes that S is a linear function of the combination of P and J, that T is the parameter relating S to the combined variables, while n distributes the effect of the combination between P and J. REFERENCES [1] E.W.E. MICKLETHWAIT.Soil Mechanics in Relation to Fighting Vehicles. Military College of Science, Chertsey (1944). [2] M.G. BEKKER. Theory o/ Land Locomotion. The University of Michigan Press (1956). [3] F. JANOSt and B. HANAMOTO.The Analytical Determination of Drawbar Pull as a Function of Slip for Tracked Vehicles in Deformable Soils. Paper No. 44, Proc. of 1st Int. Conf. on Mech. of Soil-Vehicle Systems, Turin, Italy (1961). [4] W. SOEHNE.Stress Transmission Between Tractor Tires and Arable Soils. Grundl. Landtech. 3, 75 (1952). [5] M.G. BEKKER. Off-the-Road Locommion. The University of Michigan Press (1960). [6] A. R. REECE. Discussion of Paper No. 44. Proc. of the 1st Int. Conf. on the Mech. of Soil-Vehicle Systems, Turin, Italy (1961).
Printed in Great Britain.
THE ROLE OF DISPLACEMENT IN A SIMPLE TRACTION SYSTEM* J. H. TAYLOR and G. E. VANDEN BERGt INTRODUCTION INSUFFICIENT information is available to analytically determine the thrust a vehiclo can generate in a given soil condition. Empirical methods based on observation and measurement are presently used but they can be greatly improved. If only the maximum pull of a traction device is of interest, then only the pulling force need be measured. However, if the slip at which the maximum pull occurs is of interest, then it becomes necessary to measure relative displacements. The degree of sophistication necessary in describing any phenomena depends upon the level or intensity of one's interest. In traction research we are seldom interested only in the "point" of maximum pull, but rather the entire slip-pull "curve" is desired for analysis of performance. The Coulomb failure criterion expresses the maximum shearing stress of a soil for any given normal load. If shear stress-displacement curves for each normal load are plotted, the parameters of the Coulomb equation are evaluated by using only one point from each curve. The point used is the maximum shearing stress and this, of course, is based on the assumption that the maximum shearing stress of the soil is reached within the range of displacement used. The maximum shearing stress expressed by the Coulomb equation is closely related to the maximum pull of a traction device. A similar relationship exists between slip-pull curves and shear stress-displacement curves. The complexity increases as our level of interest increases just as it does in statistics when we progress from the arithmetic mean, a "number", to the regression "line" as our basis for evaluation. The validity of the analogy ultimately depends upon how effectively the soilmeasuring device system represents the soil-track or soil-wheel system. In any case there is little likelihood that a slip-pull curve can be predicted until an accurate description of a shear stress--displacement curve can be made. Considerable effort has been made to use soil shearing stress equations to predict thrust. Micklethwait [1] used a very simple approach in 1944 when he multiplied both sides of the Coulomb equation, S = C + P tan $, (S is the maximum soil shear stress obtained for a normal load P) by the ground contact area A to give SA = C A + PA tan $. This reduces to H = C A + W tan $, where PA is the total weight W and H is thrust. *Paper presented at the 1965 Annual Meeting of the American Society of Agricultural Engineers. tNational Tillage ,Machinery Laboratory, Auburn, Alabama, U.S.A. Communicated by A. R. Reece.
47
48
J.H.
T A Y L O R and G. E. V A N D E N BERG
Bekker [2] observed that the form of some shear stress-displacement curves is similar to that of displacement-natural frequency diagrams of an aperiodic damped vibration. He replaced the damping constant and the spring constant by soil parameters and developed his well known modification of the Coulomb equation. He then developed an equation for thrust by designating track dimensions and assuming a uniform normal pressure.
f
DEFORM&~'|ON
~1 ( I N )
FIG. 1. Typical soil shear stress-strain curve.
Janosi and Hanamoto [3] found that in the majority of cases the soil does not exhibit a hump and a decay in the shear stress-displacement curve but is of the type shown in Fig. 1. They developed a simpler modification of the Coulomb equation and proceeded from there to the development of an equation for thrust in the manner of Micklethwait and Bekker. All of these modifications attempted to account for the effect of displacement on soil shear strength. ANALYSTS
In the Coulomb failure criterion the normal stress is considered acting on the failure surface, and a distinct failure surface is assumed to occur. The horizontal line along the bottom of the cleats in Fig. 2(a) represents this failure surface as it might apply to a track. If vertical distribution of forces is ignored, the normal force is assumed to be acting on the failure surface. The only shearing strain that has occurred in Fig. 2(a) was within the plane of failure so that once failure was reached all other displacement was as a rigid body. Soehne [4], Bekker [5] and others have used various methods to show soil deformation beneath a traction device. Figure 2(b) is a composite concept drawn from personal observation and a study of the literature. In addition to the displacement along the plane of failure, Fig. 2(b) shows compaction occurring from the surface to some depth well below the plane of failure. Horizontal shearing begins at some undetermined depth with the accumulated horizontal displacements increasing to a maximum at the plane of failure. At sufficiently low slip values, no actual failure plane would develop. The system in Fig. 2(b) may be separated into units or elements for closer study. Stress is defined as force on a unit area. The stress measured is on a unit area
THE ROLE OF DISPLACEMENT IN A SIMPLE TRACTION SYSTEM
49
_~l i i I I I I I~~~_. __ :.:.:.'..: .....
"~' I:_I._L_L_I_.L_I,
: -" ~ "':::::
i o e o o
:::::
iii:
e e e e e
(a)
e o o o o
__~ ~ ~ :.:___ e l o t l
:::::
e o o o o
(b)
FIo. 2. (a) Soil displacement only in the plane of failure. Co)Accumulating soil displacement and compaction. which composes the topmost layer of a column of soil extending to some unknown depth. The column of soil shown in Fig. 3, which is originally vertical, may be used as the element which represents the basic unit of the system. According to Reece [6] and others, the depth of this column of soil under stress increases with increasing normal load. A logical representation of the soil strain in Fig. 3 would be the ratio of the accumulated horizontal displacements 3 to the vertical depth of soil under stress d. The horizontal displacement of the unit area on top of the column now becomes relatively meaningless unless we know the depth of the soil column affected. P2>PI
J_L= J_.22 dr d 2
, ,!///
FIG. 3. The column of soil which represents the basic unit of the system.
50
J . H . TAYLOR and G. E. VANDEN BERG
Neither the plane of failure nor the volume of soil stressed is usually controlled under a traction device. Therefore, when additional increments of load P are added to a unit area in a traction system as represented in Fig. 2(b), either the depth of the soil column in shear beneath the unit area will increase, or the density of the column will increase, or both. Further, when the depth of the stressed column d increases, the total accumulated horizontal displacement J at the surface must increase if the strain, defined as the J/d ratio, is to remain constant. Measuring the depth of soil stressed or the change in density as P is increased would be difficult, but since a change in P is the only action taken to cause these changes then both must be a function of P for a given soil condition. Now if P somehow represents the depth of the soil column under stress and J represents the total accumulated horizontal displacement at the top of the column, then some function of J/P should represent the soil strain as well as or better than the J/d ratio.
OBJE,CI'IVE The Coulomb failure criterion, which is a form of limit analysis, involves assumptions that seem to be incompatible with conditions that frequently exist in a soil-traction device system. The slip of a tire or track is known to be important, and it is equally well established that soil shearing stress varies with displacement. Since the Coulomb equation ignores displacement, it seems unlikely that it could ever provide a satisfactory mathematical representation of the soil-traction device system. Therefore, instead of attempting to modify the Coulomb equation, an experiment was designed to determine empirically the relation of soil shearing stress to normal pressure and displacement on a unit area. The objective was to establish a stress-strain relationship based on the deformation problem as exhibited in S-1 curves rather than a limit analysis based on a failure criterion. Concisely, the objective was to determine the general form of f where S = f (P, J).
APPARATUS AND PROCEDURE The requirements for a special device to simulate a unit area in traction are that neither the plane of failure nor the volume of soil stressed is controlled. Displacement J and normal stress P should be controlled and programmed and the resulting shearing stress S measured. Thus , / a n d P would completely determine S for a given soil. An annulus shear apparatus, called a Desometer, was designed and built with the added capability of measuring sinkage. It is shown in Fig. 4. The shearing head was an annulus with 6 in. i.d. and 8 in. o.d. Two annuli were used; one with a smooth rubber face and the other with ¼ in. lugs spaced radially 20 ° apart. When making the measurements the normal load was applied first using an air cylinder, and then torque was applied by an electric motor. The rate of rotation was I. 1 rev / min. Two soils were prepared and tested in the small circular soil bin. Hiwassee sandy loam was prepared in three different moisture-density conditions and Lloyd clay
THE ROLE OF DISPLACEMENT IN A SIMPLE TRACTION SYSTEM
51
FIG. 4. The annulus shear apparatus.
in two. Penetrometer measurements were made for each test. A summary of the soil data is contained in Tables I and 2. RESULTS AND DISCUSSION
Ten sets of S--J curves were secured from the combination of five soil conditions and two annuli used in the circular soil bin. In addition, S-1 curves were obtained from the Land Locomotion Laboratory, Warren, Michigan, and from soil prepared in the outdoor bins at the N T M L . Since only three basic variables are involved, one way to look at S as a function of ] and P is to make a three dimensional graph. Figure 5 is a perspective of one surface generated by the three variables. The S - ] curves in Fig. 5 resemble the ones called typical by Janosi and Hanamoto [3]. In this type curve S continues to increase as J increases through the range of ./ used. Although the value of S rises very slowly after the initial surge, S does continue to increase and the maximum S becomes a function of the magnitude of ]. These circumstances prevent the
J. H. T A Y L O R and G. E. V A N D E N BERG
52
TABLE 1.
SUMMARY OF SOIL MOISTURE-DENSITY CONDITIONS
Soil type and Test No.
Moisture content
Bulk density
Soil depth (in.)
By depth (%)
Test av. (% )
By depth (g / cm s)
Test av. (g / cm 3)
0"0-2'5 2.5-5"0 5-0-7'5
9"3 9.3 9.1
9.2
1-31 1-28 1.35
1.31
H-2
0.0-2"5 2" 5-5'0 5.0-7.5
12-9 15-0 14.9
14"3
1.63 1.68 1'77
1.69
H-3
0-0-2"5 2"5-5"0 5"0-7"5
8-2 8.3 8"2
8-2
1'38 1-41 1.49
1-43
L-I
0.0-2"5 2"5-5'0 5"0-7"5
22.4 21 "9 22"6
22'3
0.95 0"99 I'05
1"00
L-2
0"0-2"5 2"5-5"0 5"0-7"5
24' I 24-2 24"8
I"23 !'24 1"30
l"26
Hiwassee H-1
Lloyd
TABLE 2. Soil test No. Depth (in.)
24"4
PmcgT1tOM~T~t M]~SUREME1CrSFOR T~s'r CONDITIONs Cone index (psi)
I 2 3 4 5 6
'
3
H-1
Hiwassee H-2
H-3
L-I
Lloyd L-2
20 25 25 30 35 45
25 35 40 40 45 45
50 60 70 75 8O 90
15 30 35 40 45 50
55 70 80 85 9O I00
12'
6 8
Flo. 5.
Perspective of the surface generated by shearing stress S, normal stress P, and displacement ./.
THE ROLE OF DISPLACEMENT IN A SIMPLE TRACTION SYSTEM
53
determination of the maximum value of S needed for plotting against P in order to establish the Coulomb parameters C and ~b. Any representation of shearing stress which is adequate for traction must consider the relative displacement of the soil. 16
H - I-- STEEL M. C. •9.2 •
.
/
J'&O
o
J-4~
o') (/)
d -2.0
~s
J • l'O
n.
J -(>5
~4 -r
I
I
1
I
4 8 12 NORMAL STRESS P
16 (PSi)
Flo. 6. Shear stress vs. normal stress for constant displacement. Figures 6 and 7 illustrate how the shear stress-normal stress relationship varies with displacement. This shows that shearing stress is not a linear function of normal stress when displacement is held constant. When P is constant. S increases as J increases. 16 H - 2 - STEEL M.C. • 14-3
u)
~t2
•
J,
.
8-0
J - 4.0 u~ 03
J , 2'0
~e
J , I'0 J " 0'5
w •! -
4
I
0
4
I
I
8 12 NORMAL STRESS P
/
Ie (PSi)
FIG. 7. Shear stress vs. normal stress for constant displacement. To check the effectiveness of I/P in representing the changing strain as suggested in the analysis section, values of S and P were read from the S--J curves and plotted for convenient whole number ratios of J to P. This is shown in Fig. 8. After all
54
J . H . T A Y L O R and G. E. V A N D E N BERG
tests used in this study were checked in this manner, the relationship between S and P was found to be very close to linear when the value of J/P was held constant. The three dimensional plot of S, P, J (Fig. 5) helps to clarify the full significance of the relationship described in the preceding paragraph. The C-intercept will be ignored for the present discussion. The locus of a point described by J/P=a, where a is any constant, is a straight line in the J-P plane, or floor, of the perspective view in Fig. 5. In Fig. 8 this same line is shown to be straight in the S-P plane, A little geometrical reasoning will show that the line must also be straight when projected in the third perpendicular plane (S-J). Therefore the line OA, which lies on the surface, must be a straight line. dip = 1/4---~
8
¢n 6 u~ uJ ¢n 4
<= 2
~i
r
FIG. 8.
J/p=l/8
I
I
I
I
2
4
6 NORMAL
8 STRESS
I
P
LO (PSi}
I
I
12
14
,
16
Shear stress vs. normal stress when J/P is constant.
The surface in Fig. 5, which represents the relationship between S, P, and J, is generated by the path of a straight line OA as its projection in the J-P plane (l/P=a) is rotated about point O throughout the J-P plane. The actual path of the line will have to be determined by parameters not yet established. The specific case of S-J curves with well defined maxima fits well into this generalization. The conditions require that maximum S for each P be reached at the same J/P value. This l I P line is the projection of a line on the surface which corresponds with the Coulomb equation for curves with well defined maxima. Rotation of the straight line still generates the S-P-J surface. A quick and simple check can be made on S-1 curves to determine if constant J/P ratios give a linear relationship between S and P. The technique used is shown in Fig. 9. A straight-edge was used to draw lines at convenient J to P ratios with all lines converging to a single point near the origin. This point is an estimator of the C-intercept. The compliance is quickly noted by using whole number ratios. If J/P=¼, then when 1 = 1 , P = 4 ; when J = 2 , P = 8 etc. Figures 9, 10, and 11 represent curves of basically different shapes yet the S-P relationship in each case is essentially linear for all values of 1/P. The steps necessary in order to establish the mathematical model relating shearing stress to normal stress and displacement can perhaps be visualized best
THE ROLE OF DISPLACEMENT IN A SIMPLE TRACTION SYSTEM
P,3
L
;,/
/.
.... F H- i-- STEEL
J/P= 118 '5
/
J/P'I/4
1 J/P,I/2
L/..k"/ li:l~
= <
--
'
'<
"/
¢/'~"
t
L.
0
zP'8
I
,
2
,
3 4 DISPLACEMENT
5 d (IN)
6
7
FiG. 9. The method used in checking original data for agreement with the derived equation.
./ /
J/P,I/2
L
,,P., P : S _ _ ~ . - -
L-..-411---'t'~"~ ~
, /./
r~dT- / I ,,P., ~, vr t i ~ _ z ~ / ' I : ~,'l ,~'M-/ X_ l! I/ ' ~ 2 ''~ •I I I
I/
.~
J/P'8
~Ui ,i
I
2
6 8 DISPLACEMENT
IO (IN)
J
12
i I I
-
14
16--
FiG. I0. The method used in checking original data for agreement with the derived equation.
15 I0
JfP=t ,+
,,
-,..7_
/
P.s
gl'/
//
../ ~
]L:~o. 11.
4
J
i
p.4
_f
~P" I
P'Z
~
/_~ 2
NTML TRACK T E S T NORFOLK S L ~ 18 JULY 1963
~ JIP=4
, 6 DISPLACEMENT
J
I0 (IN)
12
14
16
method used in chewking original data for agreement with the derived equation.
55
56
J.H.
T A Y L O R and G. E. V A N D E N
BERG
by reviewing Fig. 8. The problem is to find the function relating the J-P relation to the S-P relation and this is equivalent to determining how tan ~b in the Coulomb equation varies as the value of ,I/P varies. It is unnecessary to plot S-P graphs such Fig. 8. The Coulomb tan ~ = S - - C / P can be computed from S - C / P = IS - C /.1] [] / P ]. Both S - C / J and J/P can be determined from the original S-J graphs as shown in Figs. 9, 10, and 11. Tan ~b was found to be related to J/P by a power function such that In tan ~b---n In ( l / P ) + l n (T), where n and T are system parameters. The parameters n and T are constant for a
1
2'0 H-I- STEEL
(iNTERCEPT) T,O.BT- - ~
I'O (SLOPE) n -0"191
.-..'-"'~
~"'--' ~
,, 0-5
-e-
O'1 "OI
•O5
O't
0"5
t-o
2-0
J/P
FIG. 12.
The method used for determining the parameters n and T.
20
H-t-
STEEL
S • 0 " 0 + O'e7 pO.eOt j o ' l ~ a. co
w
~5 •
c,
P,2
8 i
Q I
•
t
L
O
~
2
FIG. 13.
Computed values of S plotted on measured S-] curves.
3 a DISPLACEMENT
i
,5 d (IN)
i
I
i
6
7
8
specified soil condition and shearing device. They are not true soil parameters since the annulus with the rubber surface and the annulus with lugs gave different values for n and T in the same soil condition. A convenient method for determining n and T is illustrated in Fig. 12. Tan ~b is plotted vs. J/P on full-logarithmic graph paper, and the resulting straight line has the slope n and the intercept T. The intercept is read at J / P = 1.0 since In 1.0=0.
THE ROLE OF DISPLACEMENT IN A SIMPLE TRACTION SYSTEM
57
If tan ~b (the S - P relationship) is replaced by ( I / P ) ' T (the expression relating tan ¢k to I / P ) , the shearing stress may then be expressed as S = C + P (]/P)'T; or when simplified S = O + pI-,,1,,T. The parameters n and T were determined for each set of S - J curves in this study. Then the equation S = C + PI-"J"T was used to calculate S for several points on each S-J curve, and these points were superimposed on the original curves. These data are presented in Figs. 13-26. The 20
H- 2 - STEEL S" 1'0+0"77 pO.eOIjo-m o o
IO P'8 v 5
~..
P'4
i*l
8
P,Z Q O
I
i
I
2
/
I
o I
3 4 5 DISPLACEMENT J (IN]
I
I
I
6
7
8
RG. 14. Computed values of S plotted on measured S-/curves.
o --'-v)IS
~
'~
H'-3- STEEL S • I'0+ 0-83 pOq93~ . .
e
t~
il
I 2
P'2~I I i | 3 4 .5 DISPLACEMENT J {IN)
0 l 6
i 7
I 8
F]o. 15. Computed values of S plotted on measured S-,I curves. soil type, soil fitting, and sheafing device used (designated as rubber or steel) are listed in each figure along with the shearing stress equation used for that set of curves. In examining these figures remember that the continuous curve is original data and the points are calculated from the derived equation. The calculated values fit the measured curve very well except when P = 32 lb/in j where the results were erratic. There is ample evidence in the literature to indicate
58
J.H.
T A Y L O R and G. E. V A N D E N
BERG
that a normal load of 32 lb/in ~ will affect the soil to a greater depth than the 10 in. of soil used in these tests. The derived equation's description of the S-J curve is limited in one case. If the shearing stress reaches a maximum value and declines then only those values of S up to the maximum are in agreement with the equation. In the one set of curves from the outdoor bins (Figs. 11 and 23) the small initial peak was ignored and, since J was of no effect after the initial peak, these curves fit the general equation. The broken ] / P line in Fig. 11 shows that three of the four maximum
2O L - - ~-- S T E E L 0
/
S = 0 " 4 + 0 , 7 2 pO.e25 dO,;%
P".L2L-- o
P=8
n
P=4
I*1 n
o
co 0 I
FIG. 16.
P=:) ~ 2
I 1 3 4 DISPLACEMENT
o 0
J
L 5 (IN)
i 6
l 7
I 8
Computed values of S plotted on measured S-.I curves.
7 20 L- 2- STEEL S = 0 " 0 + 0 " 9 7 pO.Si7 jo.,33
=~IS c/)
wlO
--
i I
I 2
0=8
..gg--
P=4
8
i 3
I 4
DISPLACEMENT
FIG. 17.
•
I 5
d
i 6
i 7
J 8
(IN)
Computed values of S plotted on measured S-1 curves.
values of S occur at the same ,I/P value thus following the same pattern as all the other data. More data will be needed to expand the initial part of such curves to accurately determine the relationship existing before maximum S is reached. Further insight into the relationship of the variables in this study may be gained by expressing the shearing stress in the following manner: S = C + T X , where X = P1-"1".
THE ROLE OF DISPLACEMENT IN A SIMPLE TRACTION SYSTEM
59
This form of the equation emphasizes that S is a linear function of the combination of P and ]; that T is the parameter relating S to the combined variables, while n distributes the combined effect between P and J. From this it follows that the range of n is restricted to 0 ~ n ~ 1. If the value of the C-intercept is ignored, the actual value of n represents the per cent of shearing stress that comes from displacement. A look at Figs. 13-26 reveals that the effect of displacement varies from zero to 22 per cent under the test conditions used. 20
P= 32 0
H-t-RUBBER S =O'0+ 0 ' 7 8 pO.e3l jO'=e2
tn O io
,5 = ,~0 ' ' ' ~ . . . . e ~ .3 i
~ O i
I
0
P=2 L
2
3
O
o I
I
4
.DISPLACEMENT FIG. 18.
I
5 J
I
6
7
/
s
(IN)
Computed values of $ plotted on measured S-J curves.
20 H--2--RUBBER S " 0 " 4 0 + 0 ' 6 5 pO.,gO jO'ZZO 0 P-32
u~
~o
0
~
i
6 0
P'8 _
~
P,4
I
I
I DISPLACEMENT
FIG. 19.
J (IN)
Computed values of S plotted on measured $-$ curves.
2or
o
J
~
~
15
H- 3-RUBBER O
S • OeO + 0 , 7 9 5 pC-loo jo.zoO O
~'oI-I
~
¢
P=8
O
r"
,
I
,
2
Flo. 20. Computed
,P.z
0
.
3 4 DISPLACEMENT
.
J
.
5 (IN~
.
6
.
7
8
values of S plotted on measured $-J curves.
60
J.H.
T A Y L O R and G. E. V A N D E N
2°I
BERG
L-I-RUBBER
o
S " O'ZO + 0 " 6 5 pO44O jo.~6o
u~ P.16
~o
O 0
O
~B P-40
0
:, 1
,
,
o ,, I
o ,
.L .....
2
~
4
DISPLACEMENT
Fzo. 21,
e
P-2
1
i
7
•
J (IN)
Computed values of S plotted on measured S - J curves.
20
L-
0
2-- RUBBER
S " 0 " 6 0 + 0 " 7 5 pO4M)4 jo-,e
0
P~32
~
pffi8
0
Pffi4
0
t~
S
<
t 5 n
._.
0
.
pffi2 -
J I
1 2
I
,I
I
3
4
6
DISPLACEMENT
FIo. 22.
J
....
I
6
{IN)
Computed values of S plotted on measured 3 - 1 curves.
NTML
IO
TRACK TEST
NORFOLK S L 18 JULY 1963 S = 0 . 4 0 + 0.70 P
ul
Q
tn
vt
XC
-:,
"--
0
0
P=4
P.2
F[o. 23.
I
I
2
4
6 8 DISPLACEMENT
I0 d (tN)
';3
12
14
16
Computed values of S plotted on measured S-] curves.
THE ROLE OF DISPLACEMENT IN A SIMPLE TRACTION SYSTEM P,6
~~ l
~/
L,NoLOtL,.
,fO
MUNSON TEST
s I <,<,,4,o.,soP 0"1240"0711
./i" ~
i
It F
_
o
2 FIG. 24.
,
~
,
13
........
6
s to 12 14 ,s DISPLACEMENT J (IN) Computed values of S plotted on measured $-1 curves. u
"P'S P=4
,N LAND LOC. LAB. MUNSON TEST
~2 pt3
0 EAST ON CENTER 28 AUGUST ~)63 S,0,0~ 0 ~ ) 0 po'~s4j ¢'O~s u
P,2
O~
B
C
I;1
2
4
. . . . . .
6
Pi
la
,.
P,I
8 ~O DISPLACEMENT J (IN)
12
I4
_:
IG
.....
is
FIG. 25. Computed values of S plotted on measured S-J curves. 0 f
--
P~S
~z
LAND LOC. LAB. MUNSON TEST O
P~5
........ O
u
.....
CENTER 2B AUGUST 1963 S,O'OS 4.0'512 p~S~ j o ' ~ s
0
C.
i
2
O
,
, I
4
...... P-i
! ,
~
~
I
s
.....
....
O.
I
!
L
I
,0
,2
i4
is
DISPLACEMENT J (IN)
Flo. 26,
C o m p u t e d v a ~ e s of S plotted on measured S-J curves.
18
61
62
J . H . TAYLOR and G. E. VANDEN BERG
CONCLUSIONS The most important conclusions of this investigation are condensed and listed below : 1. The Coulomb equation is valid only for soil shear displacement curves which have a well-defined maximum value for shearing stress and then only for this maximum value. 2. No representation of shearing stress which is adequate for traction investigations can ignore the soil displacement. 3. The shearing stress is not a linear functiton of normal load when displacement is held constant. 4. While J / P is not soil strain, it is a logical and adequate representation of the deformation in the soil-traction system which results from soil strain. 5. The shearing stress is accurately described by the equation S = C + TX, where X=PI-~J n. This form of the equation emphasizes that S is a linear function of the combination of P and J, that T is the parameter relating S to the combined variables, while n distributes the effect of the combination between P and J. REFERENCES [1] E.W.E. MICKLETHWAIT.Soil Mechanics in Relation to Fighting Vehicles. Military College of Science, Chertsey (1944). [2] M.G. BEKKER. Theory o/ Land Locomotion. The University of Michigan Press (1956). [3] F. JANOSt and B. HANAMOTO.The Analytical Determination of Drawbar Pull as a Function of Slip for Tracked Vehicles in Deformable Soils. Paper No. 44, Proc. of 1st Int. Conf. on Mech. of Soil-Vehicle Systems, Turin, Italy (1961). [4] W. SOEHNE.Stress Transmission Between Tractor Tires and Arable Soils. Grundl. Landtech. 3, 75 (1952). [5] M.G. BEKKER. Off-the-Road Locommion. The University of Michigan Press (1960). [6] A. R. REECE. Discussion of Paper No. 44. Proc. of the 1st Int. Conf. on the Mech. of Soil-Vehicle Systems, Turin, Italy (1961).