Fundamentals of soil action under vehicles (Part two)

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Fundamentals of soil action under vehicles (Part two)

NATIONAL RESEARCH COUNCIL OF CANADA

ASSOCIATE COMMITTEE ON SOIL

AND

SNOW MECHANICS.

TECFmICAL MEMORANDUM NO. 8

FUNDAMENTALS OF SOIL ACTION UNDER VEHICLES

(PART TWO)

ANALYZED

By

M. G. BEKKER

(Directorate of Vehicle Development

Department of National Detence.)

OTrAWA, CANADA.

ABsrRACT

This paper presents a mathematical analysis of the stability of

a model representing a wheel or track of a moving vehicle.

The proposed method of determining a trafficability curve is

based on accepted theories of Soil Mechanics and represents a

con-tinuation of the work described in Technical Memorandum No.6.

A

cohesionless medium and the stability of a single wheel or

a

single track shoe with or without grouser is alone considered.

The proposed method, however, is general in scope, and may be easily

extended over cohesive soils and several wheels and shoes.

TECHNICAL

l1ENORANDUM NO. 8

Fundamentals

of

Soil

Action Under Vehicles

o

(part Two)

Contents

Introduction

Co • • u • • • • • • • • • • • • • • • • • •• ·••• · . " o .

page

3 Prob 1em •••••. It • • • • • •e I • • • • • • 8 G • 101• " o • ｾ • ID • • •e 3

Propo sed

Solution

Q • • • • • • • • • • • • • • • O• • • ｦｬＮＬｾｗｾ 4

Numerical Example ••••••••••••...••....•.. 10

Conclusions .•••••••••••••••••••••.•

0• • • • • • •

11

TIlts work presents further

development

of

ｾｨ･

theory on the

relBtion-al.ip ｲｊＺｴ［ｶＬｾ･ｮ the Soil and Vehicle 1'101618 as presented in thereclL"'li;:!81 l'Iemoran-:h:; L.:, 6 by iLF. ri!:.f'F.iT and H.G" BEYJCER, published in October l'.H6 by the

The 1.11181Y818 outLined in the present work is based on well estBblished ＺｮＧｾｌＧＮｃＧ､Ｒ ｾＧＩｩＧ ［ｩｯＮｾＬｊ J"Iecba:lics. ｾＳｴｵ､･ｮｴｳ of this paper would be a ast et ed in

follow-1\,1;' ｖｩ･ｾＬＺｲＮＺｊＮＨ｜ cf t:;Ci;lr:;lt by reeding "Theoretical Soil 11echenics" by Karl 'rERZAGHI

21:h\ Fall It.d,, Loudon, and John 'w11 ey and &,:m» New York - 1944) from

v,', j.G'" ｾｾＬｴｬｰｴＺＢｲＺＳ ;:03, VI T and VIII

are

especially

r-ecommended : "Pa asr s e Barth

.i::-rnc."u'c" (;\''-l:,-;efl ｬｏｏｾＡｴｙ［Ｌ 108-110) and "Beuri ng Capacity" (pages 120-129)0

.P R

O,JLL.!11

It 15 ｫｾｬｯｷｮ that the stability of a ｴｷｯｾ､ｩｲｮ･ｮｳＱｯｄ｟ＶＩＮ ｲｮｯｾ･ｬ ｲ･ｰｲ･ｳｮｴｾ

ｬＺｾＬ (;<):2' '.1C:'lJ ＧｾＬ･ｭｯｮｾｲｴｲＸｴＮ･､ already tb8t the stabilJ.T,y nf such a. l;\odel

\i

ｾＬ［ｾ

'T>U, f:;oiJ deper,ds on the aile"Ie of internal friet:Lon

ｾ

:Ul(1 Uw

¥

of'thf, soiL. It also depends on the aho e le,gtl.

.s

gr or ser

"crt,' e·n lond

V

and horizontal pull

H

&ppli!::'d:'o

ｾＢｌｦ［［ＧｵＩＮｊ･ｬ

0

aLso that the relatioZl3hip between

H

£iEd

V

. L '\ 'cfi;5.t'or,5' of' tw(::ilt'!'erent. types of veru.cLe failure in rlr.'gotiating t:le

H-

V

h

Ｋｾｴ｡ｮｾ

pull

H

angle

ｾ

The above equation enables one to calculate 1nunediately tte drawbar which causes the grip failure, once the vertical load

V,

friction

and model dinensions

ｾ

and

h

are given.

l'he second function which defines condttion s of ground failure was Js-t.;,ern:ined ersl,hically by applying the Logari t.hmtc Spiral method. Thi s rune-tien was traced in a few partfcu.Lar cases and resenbled a hyperbola of an unspecified equation.

The graphic method adopted, as well as any other similar met.ho d applicable in this field requires considerable amount of work and involves a grcD.t, deal of cumber-some computation. It appears advisable therefore, that

s. mathemat.I caj analysis of the stability of 8 model be made with the purpose of determining an approximate eq,uation of ground failureo

TIlis would not only shorten the work required for determining both types of TRAFFIGABILITY CUHVES, but 81so will form a broader basis for future discussion of the stability of vehicles and their performenceo

PROPOSED SOLu'rrON

which is sloped to the horizon at an aDele o

Take an indefinitely long grouser plate which is shown in Fig.

1.

'I'he action of this plate loaded with a vertical load V Lbs, and a horizontal pull h Ibs. may be replaced by the action of an imaginary continouB footing

aPr

-'(H.)

e

=

ta.n

vI

Ie this case trajectories of principal stresses have their ｦｯｾＱ in points £! 'lnd b. If the ang l.e enclosed between the shoe plate, oS and the hy;'.:.:ter,uce ;;; 1 S

f3

and if the angle of internal friction is

f>

,the:. the

ｩＮＮ｜ｮｦｾｾｬ･ｳＬ r ef'e r-r ed to the dimensions and to the location of the imaginary fcot-inf : at a r e t,ose shown in Fif,o L

;..s in the cLa asfca l case which determines the bearing capacity of a st.rip ':ond t.nat lw s e. "ldth ab , the task is to dt;termine ｾＬｨ･ pa s si v e earth resistance on retaining walls dB and db.

Since wall db is the weaker element of our structure, it may be

｡ｓｐｷｾ･､ that its failure determines the stability limit of our model. In

:(?RRATA

-

TEC?NICAL _{ｾｅｍｏｐＮＮａｂｬＲｾ｟ｌｾ}

-fAGE

LINE No.

_-

_-,

_..｟ｾＭＭ

-_

..

_-5

10

Equation should be numbered

1

5

23

11

"

"

n ₂

6

2

11 11 11 11

₃

6

₅

_"

11

l'

11

₄

6

9 11 11 11 11

₅

6

12

11 11 11 I) ₆

6

15

."

_"

11 _{" 7}

6

20

!I Il 11 Il ₈

6

25

_"

_"

_"

11 ₉ 7

20

11 11 11

"

11

7

21

11

"

11

"

12

8

Last line

_"

_"

_"

_"

₁₃

aLd do !JOt appear ;".' ,h £'t.o1't t:LE: fin':)] result ,}ILl.climB)" be checked by exp e

r-rmente,

I

COlisirlE'I', fi r st , c; conponerrt ;;: of eLE: passive ec r-tu pr-e

saure

due to the 8u:-charC'; e xe r-ci sed by t.he soil layer which has the thickness

k,

(Fig. 1).

TLe mCJc.ni tude of "hi s component has been. shown by Terzaghi. Res-pectiv61y modified, it is.

where

hi

is the deptL of the retaining wall, end

lS'

or on thE. d ept.h

ｾ･Ｍ

is a pure number

hi .

hence and but lind finally

h

I mav 'be de t.e rmt ncd fromｾＮ Fir'.c , 1

h -

cj

ｾ

1 , . ) .

I -

2cas

rp

reos

e

+

3"

;fIne; .fII7(¢

«

e)

In a similar way:

orI after having exp re ase o

J"/rj!3

J

COjiJ

by respLcti ve f'un ction s of /, and

CS

k,

=

-s

cos

e

ＨｾＮＬ

COS

e -

s-/o

t? )

,

The doub Le '.;or::pcnc;n:..:)f

ｾ

frof,''; equation 1 may be then determined

22'==

<j2.

0C

()J 8

K

(coJer

h

.si

n

8) ( !;

",.sB-Slne)

p C0,[ 2.

f;

P

8-

"J

If the model ls

｢｣［ＡＬＡＨＺｾ

01, d ,,):;:t:::..':' deptb

k2.

tFit:' 2) which does not change

with the anrle

e

as 1;"; •)·t rye;iou s case, ｾ［ｾｬｦＳＬＧ nil; ad dt tiona} component of the passive eur-t.h pn;,:;;"lrc ,1'lU to ';;hc abov e const.ant surcl.arge is (Terzoghi):

If?.')

=

hi

k

s

K

( I r -

Ji,,(¢t-6)COJ

¢

2

PO-.ti. doubled value of' Ll.t s emilj'cDcr:t. nay be obtained ,;'.·;sily by ｣ｯｮｴｩｲＺｩＺｲＺＬｾＺ equations

5 end 2:

By sub stt tutinf."

hi

obtained:

of our model 13 thE' &.lJ'1 ＮＺ［ｾ ＬＬＺｾｵ､ｌｩｇｮ 4, band

e.

To simplify this expression Let :

. 1 ; 1 1 . 1 . 0 "• • • • • • •10

Values

q,6S

and

1119

depend on watto

his

ang.l e s

¢

and & ,

ｾ

0"

as well as

K

_pff

depend only on

¢;

and

e

,therefore values of

flJ6e

and

'Ih8

may

be determined erapllicfJlly once and for 211, for any Civen

hh.1

f

4nq

e.

Value s ll1.rn19 and

ｾＶＧ

were computed by tile ..;rite r by mean c of a logari thmic spiral metr.oo fc,r

rj>

=

35°

ｾ

various ur:g1ss

e

and for

h/-::s

=

0,0025.00 5 , 0 . 7 5 , L i{esrective figures are p l otte d in t;raph shown in Figo

4

and

5.

The above graph and fOl'mul& 10 e nab I e one ';0 tre ce the traffic9bility curve for any model ( I f tho."

｡ＧｯＧｾＧ

.specified

0

raU.os in cor.e sn oru e es sand,

the friction of whi ch i.s

cf

=

35°. Tti s may be de)):", tn the. t'c l Low l ng W8_{J' :}

(

2 Z ｾ

2

F;e)

=

H

+

Y

and hence

H

=

('11?.fhe

e:Y..--J

kz

+

ｾｨ･

ｾ

z..)

Jill

e

V= (

t?:l"3IJ(!;

o-:sk

2

+

17-:J6e

O-;J

l)

(as

o

It then be ccme s evt derrt that the requirt3d t'uncti on wln ch (]",t'irJ€s the

ground failure may be eas!ly determined in polar coordinates, namely in an angle

e

and in a radius

)==

2

ｾ

For this purpose it is sufficient to for various

ceLcuLat.e a series of radii

Z':;9

By

connecting ths ends of these radii the trafficabil i ty curve wt.ict. def'Lnes the

at the moment of groand failure for given

h

and

-$ and

t6

ney be r-eadily

v

obtained. Loud s

II

and

V

v,il1 b6 '3"tsrminf-i:.1 from e quation s 11 and 12, or merely by projecting the respective '";;(':,01"08

2

P

p e on

Ii

and

V

axis of Caresian coordinates which have ｴｨｾ r zero point at th( beginninc of our polar coordinates (Fig. 3)0

Attempts to eliminate parame ter

e

from equations 11 and 12 in order to obtain a single squat.Lon in Carte si an coordinates

/I

and

V

seems to be futile as the form of this latter equation

ｈＺｆＨｖｾｙｾｨＩ

is too cum-bersome for practical applicBtion,

1quation 10 may be u sed 8S a basis for general consideration. For instance it may be Lnter-e st ing to know wheri the Grip Failure ceLiSE<J and when Ground Failure begins under given conditions. In other words the position of point ｾ (Fig. 3) is to be ､ｴｇｮｾｩｮ･､Ｎ

In order to do this , it should be noticed tllat the required angle

B

of radius

OA

may be deterr:11ned from the formula. £) .::

ta

-I (

h

I--S

t;qn

¢ )

V

17

-;J -

I,

tLl"

9)

The corresponding values

ｾｨＤ

rind

ｮｾｊＬｾ

nay be Immed i at e ly ascertained from graphs shown in Fig. 4 and

Ｕｾ

and the required radius

2P ::'

ｏｾ

may be

ca1-P9

cul.at.ed e e aily t'ron 6;.'.:.<\':1')1' I C.

Another point of tnt.er-e st nay be the mextnum obtainable drawba r pul l (H max.), with reference to the vertical load

V

I u7Gileble for given

con-di tions (

¢

oJ

I,

arid

--S ).

SinCE t.he differeIitial

<I¥

V

cannot ce attained directly. equBtions

By di viding

､ｾ･

G

may

oe

found.

11 'c1nd 12 .nust be differenti,':) ted wt th reference to

e

｢ｹ､ｾ･｡ｮ､

by assumi ng the re su Lt e.que I to zero a value

To do this i t is sufficient to solve the : : : :

0

,,;quetion:

dfl (

dm

k

s-» {

d&

=

dB

2.

+-

dB

--S

JIIJB

+

ｭｾ

-» --$)

cos

e :::::

0

e

₌

23°

e

₌

230 459

B

_•

24° 309

o

: 26° 0' (;

₌

27° 20r

In order to find

9

from thi s fOI'Ill'11aI functions

m

=

Ife)

and

n:=

fie)

must be known. ii2 these functions have 'been determined graphically

(Fig. 4 and 5) equation 13 may also be solved graphically by designeting

trg

and;:: values as tangents of angles which are enclosed between

8

axis and

respective lines tangent to

rn

and

n

curves as shown in Fig. 4 and

5.

The graph plotted in Fig. 6 contains equation 13 solved for

kz..==-J.

The results indicate that for a surcharge

1<2,

e que I to tl16 width of the

shoet the maximum dr-awbai- pull

h'

may be obtained if the angle fj which is

determined by the corresponding

H/V

ratio takes the following values: For a plate: his

=

0 For grouser plates:

hi

5 :. ,,25' his

=

.5 hiS r : .75 his

=

LO

If there is no surcharge;re-

dm -0

and

m.:

0

.

Equation 13 is much simplified and may be solved as shown in Fig. 7.

In this case the maximum obtainable drawbar pull takes place at the following

&

values:

For a plate: his

₌

a

For grouser plates:

his

_: .25

his

₌

.5 his

_-

.75

his :=

LO

s::

19° 30' (} - 200 309 (): 22° 20':

e..

24° 15i

For any of the above quoted

ｾ

angles the respective

I?}hs

may be found from graphs shovm in Figo 4 and 5"

and

By substituting these values in equations 11 and 12 the maximum drawbar pUll (H max.) and the corresponding

V

load may easily be obtainedo

Hence the location of the maximum point ｾ (Fig.3) may be plotted.

A third matter of interest may be the point B \Figo 3) in which soil fails due to the vertical load only. This point may be found by substituting in equation 11 and 12

ｾｨＹ

and.

ns.t.,e

values for

e-

0

(Fig. 4 and 5)0

It will be noted that in tLi e case fomulae 10 and 12 reduce them-selves to an equation which :Is identica} to the equation given by TERZAGHI for the determination of the henrinr'::qac1 ty of a ct.r-Lp load0

and

n-S

_h

9' become identical wi t.L tile TErIZAGHI! s va] ue s

ｾ

in his "Theor-eti ca I Soil MechEllt1cS(I.

N U M

ｾ

RIC ALE X A

ｾ

F L E

:;o;::fficients

In

she

and

ｾ

as quoted

Determine polar coo rdi nat e s and values of

If

and

V

in points A,

M and B of the Trafficability C1.11"V6 _{(1"i.(;o 3)} for the following grouser plate: S

=

5". h

=

20 5 " . The plate who se "iath is w ':: 2011, is acting without a

sur-charge upon a homogeneous eoh eSJOnLESS sand (frietion

rf ::

350 and specific weight

¥'

=

0006 Ib

s/cu ,

ino).

Solution

(a) Coordinates and loads of point

A:

o

= 'fa.

- I (

Z·$

+St-q;,n.J,j-Q ) ::::

6/(>

ｾ

11 .) _ 2.S

ea»

,J,J"Q

The respective value of

n

_Sf, f9

(for

%::::

z.yrs= .

ｾ

4). Hence the required radius OA i B;

is about 3 {Fig.

and the polar co-ordinates C poi nt A are:

e

=

610•

Y

=

40 5 lbsu/ino

Corresponding dr-ewbar pun H and the ve.rti ca 1 load V for the total width

w :

20" of the plate are:

H

.::3

ＴＮｾ［

X

20 )(

<$Ii-}

6/

0

s:

78.

3

-U.r

V

=

4. '""

;<

20

X UJJ

ｾ

/0 -::

4J.

S-

tJ...r

(b) Co-ordinates of point 11 and maxrmum obtainable drawbar pull;

From data quoted in tbis paper,

ｨｾＮ

005 and

ｾ

=

350 give the maximum drawbar pul I at (9 lJf 20° 30! (no sur-char-ge}, The corresponding

169

value is about 20 (Fig" 4). Hence

f

max, is:

@ :=:

20

J' •

00

;<

6'

2.

=

30

0

-a

/r,.,.

.) mQ.)( . it'f "

Loads which can be safely supported

are:

Ii

='

30x

2()--;)/,,(zooJo/j

:::

210

(bt/.

f11tJ

r

/

V -

JOXtOCOf(tOJQ/;

5-62. £t.f.

(c) Co-ordinetes of point B and the vertical Load '/;

r

f

0=

0°, d b

From the de inition 0 point B, an' t.e ｣Ｐｲｲ･ｾ｜ｯｮ､ｩｮｧ

n

ShB

wG

x

20

X 'Of()o:::

/32 (/

.as.

£..9.

N C L U S I 0 ｾ

f=

V=

and

value is about

44 (Figo 4)

hence;

44

x .

0'

X

5'

z:;;

6'(;

ｾｾｩＬ

The above analytio

method

enables

one to find

all

or three

basic

points ＨａｾｍＬｂ - Fig. 3) of the 'l'rafficab1l1ty curve which determines the

Ground Failure and thus makes it easy to trace

this

curve for any model, In order to apply tbi s mett.od to all cohe si onLe as soil s , value s

mSh(;J

and

ｾｨＶＧ

should be comput ed in the same way as was done in this

paper for

¢> •

35°.

The

deduced formula appears to have the same meaning and limits of application e snther Similar fonnulae in Soil Mechanics

H

- - - - t _

v

ｾ

ＭｾＭＭ

k•

•

I

hi

____l

F\ G. ,

... _._._ r

-e

l\.

F\G.2

B

v

Gt2IP FAILUQE

h+,.o Tan

d>

A -

h

Tan

¢)

GQOUND FAlLUr2E..

'2

PP8::

m s

1S k'l

+

n

ｾ

..6

2

H

=

'2

PP9

Sin.

e

V=

'2

Pp9

Cos.

e

H

H

ｍａｾＮ

F'G.3

_res.

70

ＭＭＭＭＭＭＭｾ ｾ

nhse -

VALUES AS COMPUTED FOQ

68

!\

¢

=

35° AND

FOR MODELS

WITH,

\

- - -

VARIOUS his QATIOS

6..

_1---

5---,

\

Ｖｾ

\

r

\

-C

56

L

\

\

S1

1\

\

｜Ｍ｜ｾＭＭ

48

--.---._._- - _... - - - -- --- ＭｾＮＭ .--- - _..

_--44

1\

-- _.._ - - -e--.

\",

\

__ L . _

40

"

ｾ｜

\-- \---_._---, - - _ . -- - . _ - - - . - - -- '

36

-

-\

1.-32

ｾ

-.

- --. ｾ - f - - -

'--,

\?

ｾ｜

ｾＸ

,

...

ｾＭ

1

-\ .

\?

ｾｾ

ｾＴ

1\

ｾ

\

･ＭＮ｟ｾｾｾ

ｾｓＭ

1\ \

'lD

ｾ

ｾＢ

.K

ｾ

'", '"

I'

''l

--

ｾ

'"

",,--

ｾ

""-

-,

ｾ

'"

ｾＭＭＭＭ

8

_____

<,

...

ｾｾ

<,

ＢＢＭｾ

- - -- ---"""";;;c- - - -1

-ＢＭＮＲｲＭＭＭｴＧｲＭ］ＺＺｾｾ

.

4

r---

1'----__

-0

5

10

15

20

'25

30

35

40

45

50

55

60

6!>

70

eO

..

c--7d

l

l

CD

en

.J.:

C

FIG.4

mshe-VALUES AS CALCULATED

FOI2.

f-40

Q> ::

35

0

AND

VA1210US

his

QATIOS

ｾ

ｾ

<,

i s - j

-f!

ｾ ｾ

r-,

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