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USA TAC ltr, 10 Apr 1974

THIS PAGE IS UNCLASSIFIED

STECHNICAL REPORT NO. 10278

| ANALYSIS OF SOIL INDENTATION BY A TRANSLATING

S~GROUSER PLATEI coe -CAL LIBRARY'TEHIC RREFPENCE NCop

*~~E GRUE LT

II ,Each transmittal of thisBes/•mabe C pydocument outside the DepartmentBet+•fbeC p of Defense must have prior

S~approval ofU.S. Army Tank-Automotive Command

S~ATTN: AMSTA-BSL

1 -i I II

LAND LOCOMOTION DIVISION

..MOBILITY SYSTEMS LABORATORY."

ilU.S. Army Tank-AutOmOtAWa , ivc an d

* ATTN: AI'4STA-BSL

DISCLAIMER NOTICE

THIS DOCUMENT IS BEST QUALITYPRACTICABLE. THE COPY FURNISHEDTO DTIC CONTAINED A SIGNIFICANTNUMBER OF PAGES WHICH DO NOTREPRODUCE LEGIBLY.

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Th~ ~i+ Aic~ '~ n RcIl productsThe t" _,.t O' O ': s on-0ot corrtitflte

an ~'~C-~--*~-n ot prOor -iOP - ro C.

TECNIUAL REPORT NO. 10278

ANALSIS OF SOIL INDENTATION BY A TRANSIATING

GROUSER PLATE

By

J. L. Dais

October 1968

A•MS CODE: 5535.12.426

ACKNOWLEDGEMENT

The results reported herein were in part obtained during the courseof summer employment with the Land Locomotion Division of the U.S. ArmyTank-Automotive Command

ii

"ABSTRMCT

Soil is idealized as a weightless isotropic frictional materialwith cohesion, and quasi-static solutions are obtained to thi two-dimensional problems of indentation by a plate with a single spud andan infinitely long plate with equally spaced spuds of uniform depth.

*iii

TABLE .OF CONTENTS

Page No.

Acknowledgement.. ... ....•-•• . . .ii

Abstract . . . .... 0 0 0 q

List of Figures ". .0 ' V

Object - " ,W.

Introduction ' I a' -

Governing Equations o'a o s e o v e 2

Analysis of the Single Spud Problem . . . ... . . . . . . . 4'

Plate Performance Curves ..... . .. .... ..... 9

Analysis of Infinite Plate Problems. .. ...... a 0 9

Conclusions .. . . . . . .. . . . .0 . . 0 12

References . . . . . ...... . . ....... . . . 23

Distrbutron List .on. is.......... . 24

DD Form 1473, . .. 0 . .a a. .. . . . . . . 29

LIST OF FIGURESFigur Page No

1. Configuration of Plate and Soil Before and AfterSmall Identation .n.ati.n. . . 13

2. Plates with Curved or Inclined Grousers ... . . . 13

'"3. Coulomb Yield Condition. . 14

4. The or Line and P Line Characteristic Curves, . . . 14

5a. Stress Characteristic Field 0 = 306 =-250 * @ 15

5b. Stress Characteristic Field $ = 30b Q = 100 ... 15

5c. Stress Characteristic Field 0 =300 9 = 620 . . . 16

5d. Stress Characteristic Field 0 = 300 9 = 450 . . . 16

6a. Configuration of Initially Square Grid After SmallPlate Displacement 0=300 Q-25 . . . . . .. 17

6b. Configuration of Initially Square Grid After SmallPlate Displacement $ = 300 9 = 100 ....... 17.

6c. Configuration of Initially Square Grid After SmallPlate Displacement =300 = 75 0 ... . . 18

6d. Configuration of Initially Square Grid After SmallPlate Displacement $=300 = 750. . . . . . 18

7a. Plate Load and Stress Characteristic Field. . . . . 19

7b. Mohr Diagram. 19

8. Plate Performance Curves, . 30 0 .... .. . . 20

S59. Section. of Infinitely Long Plate with Equally SpacedSpuds .of Uniform Depth Before and After SmallIdentation, .o. ................... 21

10. Alternative Characteristic Fields for Infinite PlateProblem , . 0 0 0 0 a 0 a 0 0 0 . * a 0 . . 0 0 . 0 21

11. Configurations of Grids of Initially Vertical Lines. 21

12. Collapse Loads for Infinite Plate . . . . . . ... 22

v

OBJECT:

The object here is to analytically investigate the action ofgrouser plates under applied load. The grouser plate of Figure 1simulates a weapon spade. and that in Figure 9 a bulldozer track. It willbe attempted to obtain solutions in a form which can be interpreted by adesigner in selecting an optimum plate geometry.

INTRODUCTION:

The problem of initial indentation of a half-space of soil by arigid translating grouser plate will be considered as a two dimensionalproblem as shown in Figure 1. The plate can be taken to translate atangle C to the horizontal; the essential analytical result is then themagnitude P and the line of action of the corresponding collapse load.Alternatively, the problem can be considered as having the inclination dof the collapse load specified; P and 8 are then the essential results.For definiteness in limiting the scope of the paper, solutions will notbe considered for? .

The specification of descriptive and yet analytically simple plate-.soil interface conditions for this problem is tricky and reflects judgmentbased to some extent on experimental and field experience. The interfaceis here taken to be perfectly rough unless the solution involves separationthere (as in Figure 6(a)), in which case the traction vector is taken tovanish. Thus, the solution will in part determine the interface condition.

Bekker (1) stressed the practical importance of developing groundfailures rather than grip failures in land locomotion. If 9 is largein Figure 1, then the situation corresponds to what Bekker termed "ground"failure. If 0 is sufficiently negative, then the situation correspondsto what Bekker termed "grip" failure. Deformed configurations for9 = - 250 and +100 corresponding to equal amounts of plate displace-

ment are shown respectively in Figures 6(a) and 6(b). In Figure 6 (a) theplate has lifted, and thus it is a smaller section which remains to be.failed. Furthermore, the deformation is concentrated on a plane leading"to the free surface. It is likely that failure (i.e. the completesplitting away of a soil mass translating rigidly with the plate) wouldoccur at a much smaller plate displacement in an experiment or a fieldsituation described by'Figure 6(a) as contrasted with Figure 6(b.).

This analysis takes the soil parameters cohesion and internalfriction to be constants and neglects soil weight, end effects of thegrouser plate, and soil flow out of the plane. Plate end effects becomeimportant in situations where the grouser width/depth ratio. is not large.Soil weight is important in this problem if the soil is cohesionless,and also if the soil has cohesion and the grouser depth is large.

Inclusion of these effects would lead to larger loads so their neglecthere could be thought of as lending an analysis on the safe side. Soilflow out of the* plne becomes important in situations where the platelength/width ratio is substantial and flow can occur at an-almost verticalload. A practical example of this is the tracked vehicle problem wheresoil bearing~capacity does not substantially exceed the vehicle weight.The inclusion of this effect would lead to a smaller collapse load.

Plates are frequently designed with curved or inclined grousers asshown in Figure 2. Provided that the back face of the grouser does notapproach the soil behind the plate, the present analysis is applicable.The essential plate parameters here are OD and IE 0 Db as shown inthe figure.

Two dimensional indentation by an infinitely long rigid plate withequally spaced grousers is considered as in Figure 9. For :this problem.material incompressibility prevents plate displacements with - > 0.Corresponding to a given value of d , distinct values Q of collapseload/unit plate length will be found. Until experiment dictates other-wise, it seems reasonable to conjecture that the solution to which therecorresponds the smallest value of Q would be most descriptive of fieldsituations.

THE GOVERNING EQUATIONS:

For a frictional material (2), the stress state is taken to obey theCoulomb limit condition; i.e. on no surface is the magnitude '1Y of theshear stress vector allowed to exceed C- Gw , where. c iscohesion and tan cP and 'To are respectively the friction coefficientand normal stress on a surface. Expressed alternatively, the Mohrls circledefined by G, and qs , respectively the algebraically greatest andleast principal stress components, must lie inside of or just-touch theCoulomb line of Figure 3(a). If the circle touches the envelope, then asindicated-by Figures 3(a) and 3(b), the equation

(ax+'ay)sin p + -y)2 + 4T 2 2c cos * (

holds, where x and y axes are taken in the plane of q and "3.In the absence of body force, the equations of plane equilibrium are

2

Bax + • 0ox •y

(2)

The equations (1) and (2) can be transformed into a system of two firstorder. quasilinear hyperbolic differential equations with characteristics,"usually termed c lines and 0 lines, inclined at -(2'-r ) and

+ to the direction of ..

If V denotes the angle of inclination of the direction of 6 tothe x-axis as shown in Figure 4., and - + is denoted by

S.S 3)

then the state of stress is determined by p and V whereeverequation (1) holds, and the characteristic equations

cot q dp+ 2(p + c cot p)dYn 0 (4)

and

cot c dp - 2(p + c. cot cp)dY - 0 (5)

must hold on sL and P lines respectively.

The material is rigid if the circle in-Figure 3(a) does not touchthe envelope. If the circle touches the envelope, then deformation canoccur by plane strain in the plane df Gi and Ts either by a tangentialjump of velocity across an cL or P line or with zero volume change bya continuous velocity field in which the direction of algebraicallygreatest principal strain rate, f , is inclined at either - or

+ 40,.Y to the direction of g, . The velocity equations correspondingto either of these inclinations are a system of two first order linearhyperbolic equations; for the former inclination the % line is onecharacteristic and the line orthogonal to the c line, termed a I line,is a second characteristic; for the latter the a line is one character-istic and the line orthogonal to the • line, termed a p line', is a

3

second characteristic. The characteristic curves are shown in Figure 4.If vw. , Va, Vp , and Vy denote the projection of "the velocityvector on the corresponding characteristic line, then in the former case

d vadY

S( )

must be satisfied on • and a lines respectively; in the latter case

.dv + VdY=O0p (8)

and dv -vd 0P (9-)

must be satisfied on • and t lines respectively.

ANALYSIS OF THE SIGLE SPUD PROBLEM:

Solutions exhibited will consist of the stress characteristic fieldsof Figures 5*(a)-(ld), solutions of equations (4) and (5), and velocityfields which satisfy equations (6)-(9) where they are continuous and takea tangential jump across and c or 0 line where they are discontinuous.The velocity solutions, if maintained during a small plate displacement,will distort initially square grids as shown in Figures '6(a)-(d). Theregions outside of the characteristic field Of Figures'. 5(a)-(d) are takento be rigid. A collapse load will be associated with the. stress fields.

The introduction of the variables f , , and 4A of Figure 5will facilitate the analysis. The inclination to the horizontal of the

cc line which intersects the point D is denoted by Q . 4 EOD> inFigures 5(b)-(d) is denoted by p ; in Figure 5(a) the inclination tothe plate of the c. line which intersects the point -D is denoted by

S; the point of intersection with the plate is labelled as A . ODin Figures 5(b)-(d) is denoted by ' , in Figure 5(a) the length of thecc* which intersects the point D is denoted by e . An expression

*.The stress characteristic field of Figure 5(b) was proposed byHaythornthwaite (3).

4

for the collapse load will be obtained which will equilibrate the uniformstress states along the lines AD of Figure 5(a) and OD of Figures5(?))-(c).

For plates with < EOD . sufficiently small (if @ = 3009 then< EOD > cannot exceed about 360) the characteristic field of Figure

e 5(c) can be constructed with <BOD> =" <CDG> . For < EOD> in'this range, one of the fields of Figures 5(a), (b), or (c) will beappropriate depending on OF and <EOD > . Otherwise one of the fields"of Figures 5(b), (c), or (d) is appropriate depending on i and . EOD>

An equation which relates 0 to < EOD > in-Figure 5(c) can beobtained by equating the length of ED to C sin < EOD > and also tothe length of FD times s;n (2 0 -- -X ) . There results the expression

sin(-Iy -qp)sin< EOD > -

2sLn20q, ..)os-'- -- sin(..+p- EOD >-e)e

It follows from trigonometric identities that

- .~- +e)tantan(e - cos S e cot < EOD >. (10)

'"' 2cos(28-qp)cos( - - 7r)coS(O-q,)

For sufficiently small <. EOD>, equation (10) will have a solution for 9.

A pressure distribution which satisfies equations (4) and (5) in thefields of Figures 5(b)-(d) is

P c Cos 0P "(-in• •7 in OAB,

~ 2+8-0 tan CPp c cot [ (1- sin ." in OBC,

I- j.+e)ta n cpand p c cbt . ( -1 in OCDE.. (11)[ (1-sin ]

.1€

Equation (11) holds -also in EAD of Figure 5(a).

If the chara'cteristic field of Figure 5(b) applies, then a velocitysolutidn for which ,the perfectly rough plate translates with' speed V atangle 0 = 0 to the x-axis is

vV Vin OCDE, (12)

V~r- 0

v~ -V~ --~+6) tanp." vO Ve ''

}in OAB,

V -Y

and v -V sin c e p tan

v•in OBC.v V Vcos cp e"- tan c

P

Equation (12) also applies to region EAD of Figure 5(a). The velocityvector experiences a tangential jump across the or line'-intersecting thepoint D.

A velocity solution valid for . _. • 7t corresponding to thefield of Figure 5(c) is

v V cos(O -e) O Ein, OCDE,

v .= -V sin(e -6)

V - -V tan cp cos ( -e - O+ )e1-V tan cp

v... , in OBC,

vp V eo e -8 + CP) e"1p tan .

.. .. 6

V = cos(e- +)tanCos Cos (8 +,)

in OAB,v -O0V

v - -V tan • sin(e Qe)e4'1tan .e

in DCG2v w V sin(e -e)e'•'tan nD

and v= - V sin(@ -e)e"- + +)tan q

"V in FEDG.

The velocity vector undergoes a tangential jump across the curves FGC,CBAt, DC and CO, and the region translates with the zone FGD.

If the stress field of Figure 5(d) applies, then a velocitysolution valid'for <EDO>+

v = V cos(@ - <EDO> -eP) 'iaJ in OCE,

v -V sin(8 -< EDO >-CP)Y)- -V tan ( cos(- < < EDo >)e" ttan

P,

T- V+cos( - < EDO >)e"tan } A

an cos( < EDO >)e +<E O>Tand v•"COS CP

in OBC.

V7Y

Tangential jumps in :.velocity occur across CBU and CO.

Figure 7 w-ill facilitate the determination of the .magnitude P andinclination (f of the corresponding plate collapse load. 'The inclination

(If-f ) of the plate collapse load to. a is the same as the ihclinationof the stress vector on OD to Z and the magnitude P of the load ise times the magnitude of the stress vector. It follows from Figure 7(b)

and equation (il) that

¶cn R cos(20,+2-cp) - P/t sin(6-C) , (13)

E Lp + R sin(2e +29-tp) - P/t, Cos(6-g) (14)

where-- +e) tan .

R C C Cos CP{ ( -s I.. (15)

and

P/R - U-(i. sin co)e'2 - ,,,,,+0)tan (16)sin cp

From equations (13), (14), and (16) there follows

coCt(6-9) M pin [ 29t29-qp+. 1- U(-sibqC)e -

cos(2 0e+ 2 )L s Si-nl J(17)

+2-)Cs 2(w OT~ +e) tan cpP/$ "cos(-281+29-4~cos e

PA C0( 2 e if. 6# 0 C (18)s!,(.6-9) (1-sin CO

So'. 8

and ..

2 l+G) ~ tan.~ += cos C- e (

cos(8Q L1si COJ][

1-01-sin ,e C ) + (19)... sin cp

PLATE PRFORMANCE CURVES:

For applications, equations (10), (17), (18), and (19) can beconveniently represented by plate performance curves. For 4 = 300(a reasonable value of # ) Figure 8 shows such curves for = 00,

150, 300, 45O, 600, 750, and 9006 and their use in performance predictionof plates with < EOD> = 00, 15 , 300, 450 600, 750, or 900 will beexplained. Plates with other values of < EOD> cbn be treated by inter-polating between the curves.

if f is given, then ordinarily the curve for f = <'EOD > willassociate values of49 and P/Ic, with Cf . If tF is so small thatthe curve for < = EOD > is not ouched by a vertical line from W.,then the plate will translate with 9> • , the situation thus beingoutside the scope of the present paper. If d is so large that thecurve for ( = (EOD> is not reached, then 9 and P/n, can beobtained using the dotted curves; ( EAD > of Figure 6(a) is then equaltoG.

If 9 -is given, then ordinarily the curve for t= <•EOD > willassociate values of ( with 0 . If 9 is so small that the curvefor 0 = <EOD7 is not reached by a horizontal line from .9 , thenthe dotted line will associate a value of 6 with 9. ; e. EAD> ofFigure 6(a) is then equal to S . In either case, P/%ee can then bedetermined from d as in the preceding paragraph.

ANALYSIS OF TBE INFINITE PLATE PROBLEM:

In Figure 9, if 4. is sufficiently small (less than about 25O if4= 300), the plate will not displace. For larger values of (

three alternative solutions are exhibited. The stress characteristicfields for these solutions are shown in-Figures 10(a)-(c)., where t inFigures 10(a) and 10(b) is arbitrary. If the pressure is an arbitrary

9

constant in these fields, then equations (4) and (5) are satisfied. Thesolution corresponding to Figure 10(c) requires.- *> 4 EOD> by anamount sufficient so that the region DAOB can support the requiredtraction along -AD . A collapse load will be ýassigned to each of the'stress fields.

Take the plate to translate with speed. V. Then-a velocity solutioncorresponding to Figure 10(a) for. which the region above M, translateswith speed V at =0 is

V =0Y

VsV

between Vt. and ' , where s denotes distance'albng a Y line fromThe region below oto is taken .rigid. A velocity solution ,-

corresponding to Figure 10(b) for which the region above P, translateswith speed V at = 0 is

V =0 ...

V VS~P t

between P. and - , where s, denotes distance along a , line from'.. 'The region below Po is taken rigid. These velocity. solutions

maintained through a small displacement would distort a grid of. initiallyvertical lines as is shown in Figure 11(a). A velocity solution.-corresponding to Figure 10(c) for which the region above. •. translateswith speed V. at 9 =Qis

V V

Vo =0

above 'le. The velocity vector experiences a tangential jump acrosscc and the region below of, remains rigid. This velocity solution

10

maintained through a small displacement would distort a grid of initiallyvertical lines as is shown in Figure 11(b).

A collapse load can quite simply be assigned to the characteristicfield of Figure 10(a) by noting that 1 = C -' N tan cp on a lines."It then follows from ¶ Q sin 6 and aN = - Q cos 6 that

1

Q =sin 6 -cos 6 tan (20)

For the case - 300 this equation is exhibited graphically in Figure12. It follows readily from the Mohr diagram of Figure 3(a) that

(I+ 2 lan )r-c- -6;jo, on P lines. Then from I = Q sinedand = -Q cos( it follows that

Q/c 1 (21)sin 6(1 + 2 tan2c)- cos 6 tan c .

corresponds to Figure 10(b). This equation is also exhibited graphicallyin Figure 12 for the case of -= 300. It follows from the geometry ofFigure 10(c) that '/L = tan 4 EOD > /sin .0. The results of theprevious sections are applicable to find 9 and P/c- ; the collapseload is then given by

Q/c " sin < EOD > [pkc) (22)sin

"

For the case = 300 and plates with E ROD> = 40, 70, and 100 theplate performance curves of Figure 8 were used to exhibit equation (22)graphically in Figure 12.

Bekker (1) analyzed this problem previously and obtained a collapseload equivalent to the one obtained here corresponding to Figure 10(a).As was discussed by Bekker.(3), (4), this solution implies that thedrawbar pull of tracked vehicles should be independent of spud depth and:spud spacing. The solutions obtained here have the following implications,for track design: As long as < EOD 7 is greater than roughly 80 of 10othen drawbar pull is independent of < EOD > . However, for lesservalues of < EOD 7 significant reductions of drawbar pull can resultfrom decreasing - EOD > . Such reductions would occur only in terrainsof reasonably high strength soil,

11

CONCLUSIONS:

It is the opinion of the author that the solutions obtained for theplate with the single spud are sufficiently descriptive of fieldsituations involving weapon spades that useful implications for, spadedesign can be deduced.

The infinitely long plate problem, however, involves a troublesomenonuniqueness of solution, as is indicated in Figure 12. For sufficientlysmall inclinations of the collapse load to the horizontal, three solutionshave bean obtained, each having a distince collapse load magnitude. Forsteeper inclinations, two solutions have been obtained; each with adistinct collapse load magnitude. It appears likely to the author thatthe solutions with the lowest load magnitude will best correlate withfield situations. This, however, is a matter which would be moresatisfactorily approached by experiment than conjecture.

12

OD

D- "r -

= -

Figure 1. Configuration of plate and soil before and after

small indentation. Soil below the dashed curve

has remained rigid. OD and < EOD > are the

essential plate parameters.

,E 0 60

050 OV

0 0

Fig. a . .. Fig. b

Figure 2. Plates with Curved or Inclined Grousers

13

e3 -,,-

Fig. a Fig. b

Figure 3. The Coulomb Yield Condition

03-

Figure 4. The a line and B line are characteristic curves

of the stress equations. Either the o and y lines.or the p and O lines are the characteristic curves

of the velocity equations.

i4 I I

D

Figure 5(a). Stress Characteristic Field c = 300, : -.250

"Figure 5(b). Stress Characteristic Field cp = 300, .6 - 100

15

'67 7

Figure 5'(c).. Stress Characterist ic Field c = 300, e = 620

Figure 5(d). Stress Characteristic Field c - 300, 8 450

16

Figure 6(a). Configuration of Initially Square Grid after.Small Plate Displacement . s 30O, 0 - -25°

r • -- - I II I ...1 - - -1

Figure 6(b). Configuration of Initially Square Grid afterSmall Plate Displacement • - 300, 8 = 100.

17

Figure 6(c). Configuration of Initially Square Grid. after

Small-Plate Displacement t = 30 0, 75 0

Figure 6(d). Configuration of Initially Square Grid after-

Small Plate Displacement -30, 8) 750

18

• " .

•~ 4.

Figure. 7(a). Plate Load and Stress Characteristic Field.The line of action of the plate load intersectsthe "line OD at its midpoint.provided A liesto the right of 0 . If A lies to the left of0 as in fig. 5(a), then the line AD is inter-sected at its midpoint by the plate load.

ft

Figure 7(b). Mohr'Diagrani. '::-.(:.*: + ! •), ' :Tr:.

-,0

0~ o :O o-••+0 P-4T-++

30. 0<I 07." '9

a)U

v- OA Inlntio toVri o 0t~ad ere

0 va

o .. _ .. o

i. s, 60..,.0

LL,.4I-w

60.i

=,-44e 15 T~

Ir 60

0 0,-

15 30.5 . 7 •

o

4jU

4-j r-

oI'30 S 60 7.+Sq

8 - Inclination to Vertical of Plate Load - Degrees'

600

Figure 8. Plate Performance Curves -+-30°20

o 45 ||• ,

Figure 9. Section of Infinitely Long Plate with Equally Spaced.Spuds of Uniform Depth Before and After Small Indentation

m.6

_- i _.. ... _-_ ' . . .. '• '

A-

WAD0

Lt ~(b) 1

L_

Figure 10.. Alternative Characteristic Fields for Infinite PlateProblem. - 300

• . • .. (a)

(b)

Figure 11. Configurations of Grids of Initially Vertical Lines.

"Fig. 11(a) corresponds to figs. 10(a)"and 10(b).Fig. 11(b) corresponds to fig. 10(c).

21

CU

S8Collapse Load Corresponding to fig., 10(a)

oCollapse Load Corresponding to fig. 10(b) '

$4-

S6

COr-4

93

U2

0

4-''Claps ColpeLodCrosodigtaigd0b

"w -

t...

a,

o

* a

• .

- -. ?

0 /0.• 2, ..•/ ,'••7

•- i• "Collapse Load0' / i " corresponching.. .

S.to fig, •lQ(c) ••-

3Z0 0 4•'0 6"0 60 * 70 80 * O.

6 - Inclination to Vertical of Collapse Load

Figure 12. Collapse Loads.for. Infinite Plate.. qc- 300'

22

REFERENCES

I.~ Bekker, M. G., "Introduction to Research on Vehicle Nobility,

.Part I - Stability Problem", Ordnance Tank Automotive Command,Land Locomotion Research Branch Report No. 22.

2. Dais, J. L., "An Isotropic Frictional Theory for a Granular MediumWith or Without Cohesion", Brown University, Technical ReportNo. NSF-GKlO13/6, 1967.

3. Haythornthwaite, R. M., "Methods of Plasticity in Land LocomotionStudies", in "Rheologie et Mechanique des Sols", Proc. IUTAMSymposium, •Grenoble, April 1964, edited by J. Kravtchenko andP. J. Sirieys, Springer, 1966, pp. 28-44.

4. Bekker, M. G., "Theory of Land Locomotion", The University ofMichigan Press, 1956.

23

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C ominanding General1U.S. 'Armyr Supply & Maintenance Cotunxd-Washington., 3). C. 2Q310ATTN: AMSM-Mf

C omnmanding General 118th-Airborne Corps

* Fort *Bragg., ýN(?th Carolina 28307

* Commanding GeneralU.S* -Army AlaskaAPO 4,09

* Seattle., Washington

Office, Chief of Research & Development 2'Department of the Army,

*Washington, D.: C.

U.S. Army Deputy Chief of S8taff for'Logistics .2Wasbington, D.' C.

No. of

.Commander 1U. S. Marine CorpsWashington., Do C.

*Attention: AO-rH

'Commanding OfficerU-S. Army Aviation Material LabsFort Eustis., VirginiaAttention: TCREC -SDL

Commanding General .U.S. Army General Eq~uipment Test Activity.-

* Fort Lee., Virginia 23801*Attn: Transportation Logistics.Test Directorate

Commanding General .. 2U.S. Army Medical Services Combat Deve 'lopment Agency.Fort Sam Houston, Texas 78234

Commanding Officer .2Signal Corps. .

Fort Monmouth., NewJer sey 07703ATTN:. CSRDL

Commanding Officer1Yuma Proving Ground.Yuma,- Arizona 85364,ATTN: -STEYP-TE

Corps of Engineers1* U.S. Army Engineer Research &-Development Labs

* Fort Belvoir., Virginia 22060..

President 1U.S. Army Maintenance Board

* Fort Knox.,*Kentucky 401-21,

President1U.S. Army Armor Boar~1Fort-Knox., Kentucky 40121-

* - President1U.S. Army.Artillery BoardFort Sill., Oklahoma .73503.

*President *. . 1*U.S, Army, Infantry Board

*; Fort BonningGeorgia ,31905

::U... rmy Aibore Eectoid & pec al fr WýT4Board

Director, Marine.Corps' CneLanding Forces Development 6neQ~uantico, Virginia 223-34

Commanding -General2Headquarters USARAL

.APO 949Seattle, WashingtonATTN: ARAC)D

* Commanding OfficerAberdeen Proving Grouzid, Maryland. 210O5* Attention: STEAP-TL

C . manin General.Sho *

Office- of the-LibrarianFort Rucker., Al-abamaATTN: . AASPI-L

* Plans Officer (Psy~chologist)PP&A DiV, G-Hqs, tISACDCBCFort ORD., California 939i4l

Commanding General .

Hq., U.S. Army Materiel CommandResearch-flvisionResearch-and Development Directorate;.,.Washington., D. C. 20025.

* Canadian Army Staff. * .2450 Massachusetts'Avenue4

Washington, D. C.

commanding General.U.S. Army materiel Command'ATTN:; 'A1RD-Rv~z, mr. Paul Carlton.

* Wasbington,- D., C.23$

No. ofCopies

Director: 3U.S. Army Engineer Waterways Experiment Station"Corps of EngineersP.O. B6x 631.Vicksburg, Mississippi 39181

Unit XDocuments Expediting ProjectLibrary .of CongressWashington., D, C.Stop 303.

Exchange and Gift Division:Library of CongressWashington,D. C. 2.025"

United States Navy 10Industrial College of the Armed FrocesWashington, D. C.ATTN: *Vice Deputy Commandant

d

Continental Army CommandFort Monroe, Virginia

7

Department of National Defense ..Dr. N. W. MortonScientific AdvisorChief of General StaffArmy HeadquartersOttawa, Ontario, Canada

Chief 'IOffice of Naval ResearchWashington, D. C.

SuperintendentU.S. Military AcademyWest Point, New YorkATTN: Prof. of Ordnance

SuperintendentU.S. Naval AcademyAnnapolis, Maryland

Chief, Research Office.-Mechanical Engineering DivisionQuartermaster Research & Engineering Command"Natick, Massachusetts

' .; ": ' • : ~ ~28 i . . .

Security ClassificationDOCUMENT CONTROL DATA - R&D

(Security claeelfication of title. bo0 of obetmct awd tdwebwd annmtelatin tuet be enteed when me 0v'ell neport to cheeoled)

1. ORIGINATING ACTIVITY (Co*;ree aum-o') 2*. REPORT SECURITY C LASSIFICATION

,Land Locomotion Division, AMSTA-ULY 2b. GROUP

3. REPORT TITLE

Analysis of Soil Indentation by a Translating Grouser Plate

4. DESCRIPTIVE NOTES (Type of.mport ad Inchaelve dotes)

S. AUTHOR(S) (Last awe, Brat name, Intia)

Dais, J. L.

6. REPORT DATE 7a. TOTAL NO. OF PAGES i& NO. OF MKFS

October 1968 3O.. CONTRACT OR GRANT NO. $a. ORGINATORS REPORT NUMlt*)

b. PROJEcT No. 10278

c. A1.MS CODE: 5535.12.426 Sb. O•M PORT NOMS) (Any oetr u.o"mMa ma.tmybe&eetsig.d

d.

10. A

• iocument outside the agencies• " te U.S. Government must

;=A 0a± o .. .... .Or 'M 12. SPONSORING ELTARY ACTIVIT*

U.S. Army Tank-Automotive Comman I Land Locomotion Divisionl.obility Sstems LaboratoATTN:, AMSTA-BSL -- .S. AmTank-Automotive Command-Warren,. *ichizan 4±8090.

IS. ABSTRACT

Soil is idealized as a weightless isotropic frictional materialwith cohesion, and quasi-static solutions are obtained to the two-dimensional problems of indentation by a plate with a single spud andan infinitely long plate with equally spaced spuds of uniform depth.

4/

D oD IJAN6.1473 29 UNCIASSI=Security Classification

.. B est.Available Copy :Security Classification

I4. LINK A LINK 8 LINK CKEY WORDS -

ROLE WT •ILF* 'WT ROLE WT

Spade indentationSpade holding7 abilityGrousers

INSTMJCTIONS

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should follow normal pagination procedures, Le., enter the 13. ABSTRACT: Enter -an abstract giving a brief and factualnumber of pages containing information. summary of the document indicative of the report, even though

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key words but will be followed by an indication of technicalcontext. The assignment of links, rules, and weights isoptional.

30, UNCASSFl'2DSecurity Classificgtion