Performance of elastic wheels on yielding cohesive soils

Journal of Terramechanics, 1977, Vol. 14, No. 4, pp. 191 to 210. Pergamon Press Printed in Great Britain.

PERFORMANCE

OF

ELASTIC

COHESIVE

WHEELS

ON

YIELDING

SOILS

Y. FUJIMOTO* Summary--Conventional methods for predicting the performance of wheels on soft soils consider the wheels to be rigid. It is difficult to describe the performance of elastic wheels such as pneumatic rubber tires due to the lack of an adequate expression of their deformation on yielding soils. In this paper a simple method to describe the configurations of elastic wheels moving in deformable soils is proposed. It treats elastic wheels as rigid wheels of larger diameters. Also a theoretical method that can predict the rolling resistances, depth of ruts and driving torques of pneumatic rubber tires on yielding cohesive soils is developed. Exl~rimental resultsshow that the method is good enough to predictthe performance of pneumatic rubber tiresfor practicaluse. PERFORMANCE OF RIGID WHEEL ON YIELDING SOILS A THEOItE'I1CALmethod to predict the performance of rigid wheels on yielding soils was introduced by Wong and Reeee in 1967 [I]. Rolling resistances, depth of ruts, draw-bar pulls and driving torques, can be predicted if we know the magnitudes and distributions of radial and tangential stresses acting on the periphery of rigid wheels. However, because of the complexity of the stress distributions proposed by Wong and Reece, it is rather impractical to calculate the performance of rigid wheels by using their formulas. According to reports by Uffelmann [2] and Krick [3], we can consider that the distributions of radial and tangential stresses acting on the periphery of pneumatic rubber tires are flatter specially in the case of cohesive soils. So we can assume as follows: (a) The radial stress acting on the contact surface between wheel and soil is constant over the contact surface, and equals a certain value q that is related to the bearing strength of cohesive soil. (b) The tangential stress ~" is given as a function of shearing displacement by the equation [4], T - - 3..(1 - e "-'~x)

(1)

where

• is the shearing displacement of soil, • ,~ is the max. value of tangential stress, K is a constant shown in Fig. 1. (c) The elastic deformation of soil is considered to be negligible, so the depth of rut can be regarded as the sinkage of wheel.

*Construction Method and Machinery R.eseareh Institute, 3154, Ohbuchi, Fuji, Shizuoka, Japan. 191

192

Y. FUJIMOTO

i. K

lm Shearindigsplecemen'S"

FIG. 1. The assumed relationship between shearing stress and displacement. (d)

In the case of a towed wheel, frequently the tangential stress becomes negative in some part of contact area; in such a case we assume that the resultant force of the horizontal components of tangential stresses is always zero. Therefore, the rolling resistance of a towed wheel is regarded as the same as for a nonslipping wheel. Assuming the above, we can get the following relations (Fig. 2), w

~-~

%---

FIG. 2.

eo

~e

w='b{f.qc°sede+f

(2)

eQ

P = rb

v,,(1 - e-e/

el

cosOdO -

qsin0d0 ,

J

(3)

eo

R =rb

fq

sin O d 0

(4)

P E R F O R M A N C E O F ELASTIC W H E E L S O N Y I E L D I N G C O H E S I V E SOILS

193

Ot

f

T =r2b

rm(l - e-elK) d 0

(5)

where

W is the vertical wheel load, P is the d r a w - b a r pull, R is the rolling resistance, T is the driving torque, r is the radius of wheel, b is the width of wheel. The angential shearing strain ¢ is expressed as follows, • = r (0 o - O) - r (1 -- s ) ( s i n Oo - s i n O)

where s is the slip of the wheel. If the sinkage of the wheel is relatively small, we can consider approximately s i n 0 ~ 0, and • .~ r(Oo - O) - r ( l

(6)

- s)(O o - O) = $ r ( O o - O)

Substituting (6) in equation (2), we get

ry

e.

~o

W=rb

tq

cosOdO

e,_

f

+ • ,, s i n O d O

o

-r,,,

I

f* e

sr

( Oo - - O)

sinOdO

e

} sr o,

=rb

Tm

q sin Oo + r,,,(l - cos 0o) -

sr

sr "~

(--~--)+ 1 or W = r / r r m { ~ sin 0o + (l - cos 0o)

(-'-~" sin 0 o -- cos 0o + e

1

K

)l 3

1 + ~z (~ sin Oo - cos Oo + e

_~e. 1

)~ (7)

where ~ = q/rm ~ = s r/K. Similarly, f

1

P = r b x m ~ sin 0o + ~. (cos 0o - I) - 1 + ~' (~ cos 0o + sin 0 - ; R =rbq(l

2brmt

(9)

- cos0o) r

T=r

L

1 0o - - 7 " ( 1 ~

e-r~0 °) } (8)

-~0 °) - e

(10) J

194

Y. F U J I M O T O

Also, we get the sinkage of wheel Zo

Zo=r(1 -co;Oo)

(ll)

Therefore, if we know the values of r, b, r~.. ~ and ~ we can determine O0from equation (7), and substituting 0o in equations (8), (9), (I0) and (I I), we can get the value of draw-bar pull P, roiling resistance R, driving torque T and wheel sinkage Zo of a rigid wheel on yielding cohesive soil. R O L L I N G RESISTANCE O F A P N E U M A T I C R U B B E R TIRE ON R I G I D G R O U N D

Instead of being rigid, wheels usually have pneumatic tires. If we increase the static vertical load on a pneumatic rubber tire resting on rigid ground such as concrete pavement, the rubber tire will deflect vertically and its contact length, contact width and contact area will increase accordingly. The relationships among vertical deflection A, contact length L, contact area A and vertical load W of typical pneumatic rubber tires for construction machinery are shown in Figs. 3-5. Also, other examples I00

.

~

D

;

I0.00 -20,)10 PR

eO

=2.4 I~g/cm z p ; : 3 , 0 kg/cm 2

4060 f

•;:3,4

kg/cm z

zo I

1,0

O,S

F[o. 3.

I

1,5

I

Z.O w(t)

I

2.5

I

3.0

3.5

The relations between tire deflection A and wheel load IV.

p; : 2.4 kq/cm 2

SO

E 40

~p =3.0 kg/cm 1 \

~O,OO- 20, IOPR

p; : 3.4 kg/cm 2

30

'~ o .d

p; =2.4 I~g/=m z i);=3, 0 kg/cm 2 p;: 3.4 kg/cm /

20

Io o.~ FIG. 4.

; ~.o

I ).s

; z.o w (t)

I z.',

I 3.0

3.s

The relations between contact length L, contact width b and wheel load W.

P E R F O R M A N C E OF ELASTIC WHEELS ON Y I E L D I N G COHESIVE SOILS

p;:2.4 I000

195

k ~ l / ¢ rn 2 ;~; : 3 . 0

kg /(:m 2

p :3.4

KIJ / ¢ m z

000

~a

600

<~

400

200

0.5

;

I

I

I

I

i.O

1.5

Z.O

Z. '~

3.0

S.S

W{t

FrG. 5. The relations between contact area A and wheel load IV. are shown in Figs. 6-8 which describe the relations among vertical deflection, contact length, contact area and inflation pressure while the tire load remains constant. 90

Si.O

X

23.5-25~12 PR

I1.0

23.5-2.~1

IO

"t.O

2PR

E w

1.0

.

60

5,0

P I.0

1.4

I.I

pi,

kg /¢m

2.2 Z

2.6

E.4

Pi '

FIG. 6 FIG. 6. FxG. 7.

1.0

J

t

1.8

Z.Z

~.g

kQ / e r a z

FIG. 7

The relations between tire deflection A and inflation pressure p~. The relations b=tween contact length L and inflation pressure p~.

In these cases we can consider that those relationships are approximately linear especially in the case of the relation between vertical deflection and tire load. Then we can express vertical deflection A as, 5 = 5 o -4- k l W

(12)

where A o is a constant depending on the kind of rubber tire. The other constant kz we can consider to be a function of the inflation pressure of tire. In fact, according to our experimental results on various kind of tires for construction machinery,

196

Y. F U J I M O T O

3500 23.5-25~12 PR 3000~

~

2SO0L~

~

,c~ 2 ° 0

:

01--

~'~°°~,0 / FIG. 8.

"

L4

I

I I.8

1

Oi,

k c J / c mz

2.2

2.6

The relations between contact area A and inflation pressure pt.

such as 10.00-20, IOPR, 14.00-24, 12PR, 17.5-25, 12PR, 18.00-25, 12PR, 23.5-25, 12PR, 64 × 31.0-25, 12PR etc., shown in Fig. 9, we can express the coefficient/c o as, ki --

(13)

at (P0 + Pl) a

where as and ~, are constants depending upon the kind o f tire, and Po is a value to express the stiffness o f tire carcass as an equivalent inflation pressure. llO

I

O0 !0 80 70 80 ~o 40

kI 30 15

20 15

Io 1

3

( go

4

5

I

7

el

I0

"" P : )

FiG. 9. Substituting (13) in equation (12), we get

A = Ao +

a~

(.Do + P,)~

. W

(14)

P E R F O R M A N C E O F ELASTIC WHEELS ON Y I E L D I N G COHESIVE SOILS

197

Generally speaking, the rolling resistance of a pneumatic rubber tire at a constant speed is due to various energy losses as follows: (a) Energy loss due to the internal friction of rubber as the tire deflects periodically. (b) Energy loss due to local slip within the ground contact area. (c) Air resistance and losses due to generation of noise and electricity. In the case of off-the-road tires, travel speeds usually are not large, so the effect of item (c) can be neglected. It is still very difficult to analyze the remaining phenomena theoretically, so we are obliged to treat them approximately as follows. We consider that both items (a) and (b) must be mainly related to the amount of tire deflection A. We can assume that the rolling resistance of a pneumatic rubber tire on rigid ground Ro is proportional to the energy E required to get the vertical deflection &. Then, we get Ro = aa E =- ~ ( ~ - -~o)W -- kx2al W2

(15)

where aa is a constant of proportionality. Substituting (13) in equation (15), we get Ro _

a l a2

• Wz

(16)

2(Po + pi) a DEFORMATION OF A PNEUMATIC RUBBER TIRE ON YIELDING SOIL It is very difficult to predict analytically the configuration of a pneumatic rubber tire rolling on soft soil, because the deformations might take place in both the tire and the soil. In his analysis Bekker [5] assumed that pneumatic rubber tire rolling on the soft soil usually forms a rut of depth zo, and at the same time deforms as if it were on solid ground located at the depth of zo as shown in Fig. 10. Using this assumption w

FIG. 10

198

Y. FUJIMOTO

it becomes rather simple to predict the performance of rubber tire on soft ground. But in fact, the deformations as assumed by Bekker seldom take place especially in case of the plastic soils and rubber tires for construction machinery. As a rule, the degree of deformation of a pneumatic rubber tire rolling on yielding soil is governed by the stiffness of the tire as well as the strength of the soil if the vertical load is constant. When we keep the inflation pressure of a rubber tire constant and increase the strength of soil concerned, the deflection of the tire will increase accordingly. On the other hand, if the strength of soil decreases the tire deflection will decrease. Also, if the inflation pressure of a tire is decreased in the same soil condition, the deflection of the tire will increase, thus, the degree of tire deformation varies with both the soil strength and the tire stiffness, i.e. the inflation pressure of the tire. At one extreme on rigid ground the tire only deforms, and at the other extreme of the rigid wheel on yielding soil only the soil will deform. Figure 11 is a schematic i

fioll $1"renql" h 'trlflotiofl pressure

m- d e c r e a s e ---------~,flcte~$e

FiG. 11

model illustrating these relationships, showing the degree of tire deformation according to the variation of soil strength and inflation pressure. (From left to right, the soil strength varies from large to small, and the inflation pressure varies from low to high.) Freitag and Smith [6] have published the results of their research on the center line deflections of pneumatic tires moving in dry sand. Some of the typical results are shown in Fig. 12 and Fig. 13.

I:}~ 0.88 ~(j / c m 2

~

"

\\

k\'X

/ . C I - ' a 2 ~g/cr~2

]

CI"I'8 ~"/cm2 '/// FIG, 12

I

~ '\ \

I~. = 2 . 8 3 x g / ¢ m

/C,=5.7 kglcm'

<

2

/ //

/ /

FIG. 13

Using these experimental results, we can say, at least qualitatively, that for pneumatic rubber tires rolling on yielding soils,

PERFORMANCE

OF ELASTIC W H E E L S O N YIELDING COHESIVE SOILS

199

(a) The form and degree of tire deformation is mainly decided by the relation between the stiffness of tire and soil strength, (b) if the strength of soil becomes smaller compared with tire stiffness, the deflection of tire becomes smaller, and beyond some point we can regard it as a rigid wheel, (c) in contrast with the above, if the strength of soil becomes greater than tire stiffness, the deflection of tire becomes larger and approaches the case of a tire on rigid ground, and (d) between (b) and (c) there must be a intermediate state as shown in Fig. I I. A PROPOSED METHOD

TO PREDICT T H E P E R F O R M A N C E OF A P N E U M A T I C RUBBER TIRE ON YIELDING SOIL To derive a method for predicting the performance of a pneumatic rubber tire on yielding cohesive soil, first of all we adopt the same assumptions as for the case of a rigid wheel. In addition, for a pneumatic rubber tire, we assume the following. An elastic wheel rolling on yielding cohesive soil forms a rut of depth Z'o due to the plastic displacement of the soil, but at the same time is continuously deformed itself. The form and degree of deformation varies according to the relation between the stiffness of tire and the strength of soil, and can be expressed as shown in Fig. 14.

\

/I / \

8

I,'~'vI~v/:'o

F1o. 14. Assumed configuration of elastic wheel on soft soil. The contact surface between soil and wheel is regarded as a part of a cylinder and is approximated by arc AB, whose center O' is located at the intersection of the vertical line through the point A and the right bisector of chord A-B. The central angle 0~ which determines the location of point A where the tire leaves the soil is given by the following equation as a function of the ratio of tire stiffness and soil strength.

(17)

200

Y. F U J I M O T O

where 0misthe central angle subtending the contact length of the tire on rigid ground. Pi is the inflation pressure of tire, Po is the value of the stiffness of tire carcass converted to an inflation pressure, q is the radial stress acting on the tire-soil interface and is assumed to be constant. k is an experimental constant. The above assumption is based on the fact that the value of e or 0z decrease gradually going from left to right in Fig. 1 t. r' sin /_ A O ' C = r s i n / A O C

r' sin

O1"

= r sin

Therefore,

r' = r

(18)

Within the range dealt with, we can consider that 01 + 02 < g and 01 > 02 So

or

r p >1. r

This means that, when an elastic wheel having a radius r rolls on yielding soil, it can b¢ regarded as a rigid wheel having a larger radius r' as the result of tire deformation. The performance of a rigid wheel of radius r , can be predicted by equations (7)-(1 l). So, if we know the values of b, W, s, q, rm and K, we can get a relation between r' and O, as a modification of equation (7). f

W=

r ' b r , , t ~ sin 00 + (1 - cos 0,)

1 -+- ~'= (~' sin 00 - cos 0 o + e -¢'e')

(19)

where, ~' = s r ' / K and, from Fig. 14 we get 0 o as follows: 0 o = /__ A O ' B = 0 1 - 02

(20)

Using equations (17)-(20), we can finally determine the value of 0o and r', and then we can predict the depth of rut Z ' o, rolling resistance due to the rut formation

P E R F O R M A N C E OF ELASTIC WHEELS ON Y I E L D I N G COHESIVE SOILS

201

R '~, and the draw-bar pull P' of an elastic wheel having radius of r rolling on yielding soil by following equations.

P' =r'brm ~sin0 o + ~ ( c O S 0 o - 1)

(~,'cos 0o + sin 0o - ~'e-;'°') t (21)

1 +

L

R'I = r' b q (I - cos 0o) t

zo =r'(1

- cosOo)

(22) (23)

However, we cannot get the driving torque T' by simple modification of equation (I), because the driving axle is not located at point 0'. So, we must calculate T' as follows. In Fig. 15, if we designate the tangential stress working at a point D on the tiresoil contact surface A D B as 3, /_..A O ' D as 0 and Z O D O ' as ~, then the driving torque T' is O,

T' ----f b r' T cos ~,.O'-D.de

(24)

0

0

-o A

I

FK}. 15

From the relation of the sides and their opposite angles of triangle O0'D, we get n

m

00' sin

OD

r'

sin (0 - 00/2)

sin (n - 0 + 0 , / 2 - =)

Also, from triangle OO'A, O0 ~

sin 0~ - -

--

r

0o

sin --~

or 0 0 '

= r.

sin 02

sin

0o

(25)

202

Y. FUJIMOTO

Substituting OO' in equation (25), we can get

OD=r.

sin 02.sin(0 - 00/2)

(26)

O0

sin ~-. sin Then, from the first and the third term of (25), we get r'

sin 02 sin ~, = r . - - - - f f , sin (n - 0 + 00/2 - :0 sin

--

2

sin O: isin (0

0o ) cos~ + c o s

=r---~o sin ~-

O0

cos • or

sin ~

~.r' sin 02

) sin~t 3

O. } /sin(O

rr' sin-~

J

(0 -

0o - ~)

cos (0 - ~ )

(27)

Substituting (26) and (27) to equation (24), we get 0o

rsin02

T ' ~--- b r ' 0

cos(0-

?)}d0.

2

While, the tangential stress r is given by equation (1), [ ~" = "~'nl ( ' - - e - " ~ ) = ~ m ~

$r"

"~

'-e-K-(°'-A'f

Therefore, 0e

r 0

L

J L O.

'/

= br'v,, ~ r'

sin 0~. cot

dO - r sin 0~.. cot 0

?f

8,.

cos 0 dO - r sin 0~

0

e~"

f

0

0,

0

- r sin 0=.sin 0 ~ dO

O.,,

k

- r'e--F

boos 0 2

0,

dO + r s i n 0 2 . c o t " ° . e - T

e~e'.cos0d0

2

0

sin 0 d 0

PERFORMANCE OF ELASTIC WHEELS ON YIELDING COHESIVE SOILS

203

On

sr' O~ f

-~-rsinO2.e-'g

sr' 0

!eT

"]

.sinOdO t

.

J 0

t" r Oo + r sin 02 (cos 0 o - 1 ) - ~-7 (1 - e -;'°°) == b r ' r m l r ' O o - r sin Oo.sin O~.cot

r sin 02 cot oo ~- (~, 1 ~r'2

rsinO,~ ~', . cos Oo + sin 0 o -- ~'e -7°' ) + -~ T~-r2(~ sm Oo-cOs Oo+e-;'°')j (28)

ROLLING RESISTANCE DUE TO THE DEFORMATION OF A PNEUMAT|C RUBBER TIRE ON YIELDING SOIL The rolling resistance of a pneumatic rubber tire caused by the formation of a rut is given by the equation (22). But, when we discuss the total resistance of rubber tire, we cannot neglect the effect of energy loss due to tire deformation. It was very difficult to analyze theoretically the rolling resistance of pneumatic tire in case of rigid ground, it is even more difficult for yielding soils even if we assume the deformation of the tire to be as shown ~n F~g. 14. So we have to simplify as follows, combining the concept that the deformation of a pneumatic rubber tire on yielding soil is a function of the ratio of tire-stiffness (P0 + Pi) and soil strength q, with that of the rolling resistance of pneumatic tire on rigid ground. al a2 W. •f' - - 2 ( p o + P i) ~

~r l - "P°--+ Pi) l (--q J

(29)

Where, f ' is the coefficient of rolling resistance of a pneumatic rubber tire on yielding soil due to the tire deformation. In equation (29), if q ~ P0 + P; or Pi ~ q - P0, that is to say, if the inflation pressure of tire is greater than the so-called critical pressure, the tire would not show any deformation. In such a case we need not consider rolling resistance due to tire deformation. Also, in the case o f q ~ P0 + Pi, equation (29) approaches the equation of rolling resistance on rigid ground. Assuming the above, we can get the equation expressing the total rolling resistance of pneumatic tire on yielding soils by combining (22) and (29). R' = R' 1 +f'W

= r'bq(1

- cosOo) +

2(po + P i ) a W2

1 - Po k q

/

(30)

204

Y. FUJIMOTO EXPERIMENTS

Tests were performed to examine the theoretical results using 13 full size pneumatic rubber tires in a 24 m long × 3.5 m wide x 1.0 m deep soil-bin in our laboratory. (Fig. 16).

I_ 3.5~

"•..._._. ...............

J

I;.... L.... ,/ _ . _ . . . . . . . . - . - - - . . . . . - . - . -

.

.

.

.

.

.

.

.

.

.

...........

........

i .........

I

!ij ........

FIG. 16. Soil bin used for experiments.

Tire testing machine The tire testing machine was designed to measure the performance of pneumatic rubber tires used for construction machinery. It is capable of measuring rolling resistance, draw-bar pull, driving torque and slip ratio of towed or driven tires on concrete pavement and on soils of various conditions. The test machine and measuring apparatus are shown in Fig. 17 and Fig. 18 respectively. Tested tires The pneumatic rubber tires used for this experiment are described in Table 1. Soil conditions A sandy loam and a volcanic clay were used for these tests. The grain size distribution of each soils is shown in Fig. 19, and the test conditions i.e., water contents w, densities ),, cohesions c, angles of internal friction ~, average bearing strength of soil q and average cone indexes CI are shown in Table 2.

FIG. 17. Tire testing machine.

P E R F O R M A N C E OF ELASTIC WHEELS ON YIELDING COHESIVE SOILS

FIG. 18.

Measuring apparatus of tire testing machine.

TABLE 1.

Tire size 10.00-20, IOPR 10.00-20, 14PR 13.5-20, IOPR 18-20, IOPR 14.00-24, 10PR 14.00-24, 12PR 17.5-25, 12PR 18.00-25, 12PR 23.5-25, 12PR 23.5-25, 12PR 64 x 31-25, 12PR 64 x 31-25, 12PR 66 x 43-25, 6PR

SPECIFICATIONS OF TESTED TIRES

Diameter D (mm)

Section width b (mm)

1,080 1,032 1,074 1,107 1,326 1,360 1,352 1,643 1,617 1,608 1,636 1,626 1,665

286 262 360 463 367 384 434 537 587 604 757 750 1,068

TABLE 2.

Kind of soil Sandy loam Volcanic clay

D/b 3.8 3.9 3.0 2.4 3.6 3.5 3.1 3.1 2.8 2.7 2.2 2.2 1.6

Water

Density

of soil Soil & soil y(g/cm a) c(kg/cm 2) ¢° 1.78 1.57 1.14 1.45 1.45 1.44 1.38 1.33

Inflation Tire pressure load p;(kg/cm 2) W(kg) 3.0 3.0 2.4 1.8 1.6 1.3 1.1 1.4 1.5 1.4 0.7 0.9 1.1

Tread pattern

2,600 ,, ,, ,, ,, ,,

FG ST FG FG FG Y67A Y67A ,, 5,000 PL ,, YI03 ,, G15 ,, G29 ,, GI5W ,, Sup TG

Makes Bridge Stone ,, ,, ,, ))

Yokohama ,)

Toyo Tire Yokohama Toyo Tire ,, Good Year

SOIL C O N D I T I O N S

content w(~) 6.9 12.9 75.5 98.2 101.7 103.7 115.5 123.6

205

Soil properties

0.00 0.30 0.33 0.16 0.16 0.13 0.32 0.48

39.8 31.0 29.0 30.0 36.5 27.0 20.0 25.2

Soil & rubber c'(kg/cm 2) ¢'° 0.00 0.23 0.10 0.04 0.01 0.00 0.20 0.34

34.5 32.0 37.5 29.0 30.0 26.8 23.5 24.2

q (kg/cm 2) 2.55 1.78 1.41 2.10 2.70 2.00 1.02 1.39

C1 (kg/cm s) 4.7 3.6 3.1 4.0 4.4 3.2 2.5 2.8

206

Y. FUJIMOTO

tOO 90 80 70 60 50

/

I

- -

eTx. /

/

/

\oo ~ o6"~/

40 30 2O

-

]0 I 1 ~ 1 i ; I 0 0.001 0.002 0.005 0.010.02 0.05 0.1 0.2 Groin s i z e ,

i

I

J

0.5 1.0 2.0 mrn

1

l

i

5.0 I0.0 20.0 50.0

FIG. 19. Grain size distribution of soils used for experiments. Here, the most important requirement is to predict the value of q assumed to be the mean contact pressure between the wheel and the soil. The soil properties, c, % K can be obtained by appropriate in situ direct-shear tests such as a shear annulus. The values shown in Table 2 were measured by using a shear annulus with an outer diameter of 150 m m and inner diameter of 110 mm. The value q may be predicted from a plate penetration test, but unfortunately the bearing strength of soil varies with the shape and size of footings even on the same soil condition. In principle we cannot recognize the existence of a law of similarity a m o n g them except for special cases. This seems to be one of the most difficult problems of Terramechanics. Studies or discussions have been conducted on this subject by many researchers, but up to the present we have not reached a final conclusion. But nevertheless we must predict the value of q, so tentatively we have adopted the idea that the radial stresses acting on the contact surface between soil and wheel are approximately equal to the mean value of the penetrating pressure of a rectangular plate having a width equal to the width of tire-lug over the distance from the soil surface to the rut depth. Actually, 64 × 198 mm rectangular plate was used, and the results are also shown in Table 2. From our experiments, it was found that there was a close relationship between the average penetrating pressure q of this rectangular plate and cone index CI, which could be approximated by following equation (Fig. 20). q = 0.7 C I - 0.6

RESULTS

AND

(31)

DISCUSSIONS

A rolling resistance test and a draw-bar pull test are shown in Figs. 21 and 22, respectively. Typical records of such tests are shown in Figs. 23 and 24. According to equation (17) which provides the configuration of pneumatic rubber tires rolling on yielding soil, if the sum of carcass stiffness P0 and inflation pressure Pi equals the contact pressure q 02 ~ 0 .

PERFORMANCE OF ELASTIC WHEELS ON YIELDING COHESIVE SOILS

207

/

3.0

2.5 0:0-7

CI-0.6

,, /

2.0

/ .

1,5

1.0--

I

2

3

,4

5

CI FIG. 20.

The relationship between average plate penetrating pressure q and cone index C1.

FIG. 21. Viewof rolling resistance test. Then, from equation (18) we get r ~ r

t

This means that in case of P0 q- P; >/ q or Pi >/ q - Po the pneumatic tire can be regarded as a rigid wheel, so we can call this inflation pressure the critical pressure. A m o n g our experiments are some that can be regarded as having a rigid wheel. We have predicted performances for these tests using equations (7)-(11). The relationships between measured and predicted values of rut depth, rolling resistance, maximum draw-bar pull and maximum driving torque are shown in Figs. 25-28.

203

Y. F U J I M O T O

FIG. 24,

FIG. 22.

View of draw-bar pull test.

FIG. 23.

Record of rolling resistance.

Record of draw-bar pull and driving torque.

PERFORMANCE OF ELASTIC WHEELS ON YIELDING COHESIVE SOILS

~a

30

3OO0 /

20

200o r

E E

E u ~0

I000

r

I

IO Z,

20

/ tO00

30

cm ; predlcl"ed

2000

5000

Kg:pred~cted

R,

Fro. 25 F I G . 25. F I G . 26.

209

FIG. 26

Relationship between rmmsuredand predicted wheel sinkage Z of rigid J,heel. Relationship I:etween measured and predicted rolling resistance R of rigid wheel.

3000

/

3000

|

2000

2000

:.

E

E E ~ooo

I000

C - - ~ 1

e IOOO P,

2OOO kq ; predicte~

FI~. 27 F I G . 27. F I G . 28.

I 3000

~000 T,

Kg-m

2000

3000

: predicted

l::lG. 28

Relationship between measured and predicted draw-bar pull P of rigid wheel. Relationship between measured and predicted driving torque 7"of rigid wheel.

For cases that should be treated as elastic wheels, we have calculated performance by equations (21), (23), (28) and (30) assuming ~. ---- 1.5. The results are shown in Figs. 29-32 as the relationships between measured and predicted values. In every case the measured results are more or less scattered but the mean values agree approximately with the predicted ones in case of rut depth and rolling resistance. On the other hand the measured values of maximum draw-bar pull and driving torque at 100 % slip generally exceed predicted ones. The reason why this is so is not clear at present, and we can only point out that the phenomenon of slip-sinkage of driving wheels was observed especially for high slips.

210

Y. F U J I M O T O 30

3000

20 b

=

200c

m

E

/

S

IO00

I

I

~0

20

Z,

cm;

30

,000

R ,

predtc?ed

2000

3000

K(J: p r e d , c - t ' e d

FiG. 29

FIG. 30

Fro. 29. Relationship between measured and predicted wheel sinkage Z of elastic wheel. FtG. 30. Relationship between measured and predicted rolling resistance R of elastic wheel. 3000

3000

2000

2000

E

ql

E °.

IOO0

I000

I000 P,

2000

kg;

•

•

I000

3000

T,

ored ic~*ed

~p e

2000

3000

K g - m : predicted

FiG. 32

FIG. 31

Relationship between measured and predicted draw-bar pull P of elastic wheel. FIG. 32. Relationship between measured and predicted driving torque T of elastic wheel. FiG. 31.

REFERENCES

[I] JO-JUNG W o N o and A. R. RateCE, Prediction of rigid wheel performance based on the analysis of soil-wheel stresses,J. Terramechanics, 4, (I) (1967). [2] F. L. UFrELMANN,The performance of rigid cylindrical wheels on clay soil, Proc. 1st Int. Conf. I S T V S (1961). [3] G. K&IcK, Radial and shear stress distribution under rigid wheels and pneumatic tires operating on yielding soils with consideration of tire deformation, J. Terramechanics, 6 (3) (1969). [4] Z. JANOSXand B. HANAMOTO,The analytical determination of draw-bar pull as a function of slip for tracked vehicles in deformable soil, Proc. 1st Int. Conf. I S T V S (1961). [5] M. G. BeK]CER, O~'-The-Road Locomotion. University of Michigan Press (1960). [6] D. R. FRirrAo and M. E. SMITH, Center line deflection of pneumatic tires moving in dry sand, Y. Terramechanics, 3 (1) (1966).