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1 11 UBLIC ROADS ’ > > A lourna11111 i
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p} .- J . I Highway Research}
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5. ‘ ’2}; Issued by the 1‘ AN A
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; FEDERAL WORKS AGENCY 1
1.. 1} 1 1 PUBLIC ROADS ADMINISTRATION
1 i
f; , D. M. BEACH. Editor HIS R
5 1 1 1 11 Volume 20’ N0. 8 OCtOber 19311 TleSICh
l} . _ 1 \l tions.1 .2 1
E1} ‘1 .3 ' The reports of research published in this magazine are necessarily qualified by the conditions of the tests from which the data are obtainell analytlcill .
'\ } i 131‘ Whenever it is deemed possible to do so. generalizations are drawn from the results of the tests; and, unless this is done. the conclusion}; ,tlorllstf lglli
} Q 1‘ f ‘ formulated must be considered as specifically pertinent only to described conditions. hlbcwirtifi a.
1 .1. I fill when 131
‘ 1 1 1.3 1 As Show
1 21'}, . ‘ 3 problem ill
'1 .l ' } ing two _12
1 1:} In This Issue ‘ mew-m1.
»‘ ‘ 1 1 PM“ 1 analytlcal I
i. } 1 1 j . . . '
l - gilt DeSIgn of a F111 Supported by Clay Underlaid by Rock . . . . . . . . . . . 157 OIulglfttfilgl
‘1} } ‘ 1; 1 _ ; nxls '
1 1} _ _ _ . . _ , 3 1s con31dere
1,1 1‘ Slgmficant Trends In Motor-Vehicle Registratlons and Receipts . . . . . . . . . 163 in this dir
.‘1 1 1:1; occur are ii
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‘ ,1 ‘ 178157—
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DESIGN OF A F ILL SUPPORTED BY CLAY , s
”ll UNDERLAID BY ROCK ”I
team}; I I
e LICATION OF SOIL MECHANICS IN SOLVING A HIGHWAY FILL PROBLEM II I
i BY THE DIVISION OF TESTS, PUBLIC ROADS ADMINISTRATION -, i
‘ L: Reported by LJA. PALMER,_Associate Chemist I I ‘. I
"f HIS REPORT is a continuation of the theoretical b fit» :I I ,
Oberwii Tconsiderations contained in two previous publica- I‘. UNIFORM LOAD , i
Mons}? Its purpose is to present in usable form the - J ,
Bobtail analytical methods based on the assumption of condi— l a. I
, f, tions of plane strain 2 and to extend. these analyses. to 'X o +X 3 ; I
""‘Im'll include the problem of d etermining the supporting 2 I '5 “‘
I power of a clay stratum supporting a symmetrical earth CLAY STRATUM 7 w" I I
s: fill When the clay stratum is underlaid by rock. ROCK sumcs x '3‘
:5" . As shown in one of the'preVious publications2 a ”WM/Wmmmmmmammmmmmmmmmm ail,-
‘ problem involving Plane strain condlhons is one involv- / “"H ” W i M ” ”M ” ”” ""/ M ”’ H W ”7”" W "W” "/ ' III I
‘ ing two dimensions. The load is distributed over an ;i I
{I ' area that is quite long as compared to its Width and the . II I
is: 3;; analytical procedure is applied to a vertical cross section I 3;
I7 i of unit thickness in the direction of the longitudinal +2 I “I
,5? axis of theload. This is taken-as the Y direction. It I I '
i , is considered that there is no displacement of material FIGURE 1-—UNIF0RM LOAD ON A LONG STRIP SUPPORTED BY III
’3 in this direction and that whatever SOll movoments 0“” UNDERLAID BY SOLID ROCK‘ IIII
" occur are in the Z direction which is toward the center - - _ I j
g; of the earth, and in the X or horizontal direction, that tilrlfiltsccilggkggc’h boundary below the clay constitute a ; II I ‘
is is; perpendicuIar to bOth the Y and 7' directions. . Probably the fourth assumption is the least valid of , I I
f2; The analytical procedures used in the theoretical the four I ‘: I;
“’ch solution 0f PI“? present problem involve two_theories, Since it is assumed that there is no displacement I I? I
Go (Edi that Of .elastic1ty and that Of plastic ‘fthbrlufin’ and either in the fill or in the supporting soil in the direction I
’ 2‘ four principal assumptions ai‘le 11.1“) vedh T e first of the longitudinal (Y) axis of the fill, the problem is i I
isii three are common to bOI’h t 601163.. T e_ fourth is one of plane strain. One vertical cross sectionperpendi- I II
Ii made onlyrwhen the theory Of plastic equilibrium ls cular to the Y axis is the same as any other insofar as j' I II
a, applied. hese fire-f h 1 d d stresses and deformations are concerned, assuming, of ;‘ I I
Q . II The strengt‘ Q tT‘JLC ay strafiulm 61361111 slessen- course, that both the fill material and the supporting Ii I
lemme" Ilfaf Y 0.11 “1300116510” . 1 e Strefilgt ue to t e element clay are, in themselves, homogeneous. Since the rock "I I?
‘E°'“"".% % rietionhis comparative yisma land mayhbe neg. ected. is supposedly rigid, it follows that there is no vertical ,- , ‘I
Ishire’lI ence, w enever and wheiever tie unit s earing stress displacement of soil at this boundary. ‘ l I
“L % becomes equal to the unit coheSion, c, the s01] becomes 1 I
,bmwN-‘I plastic and undergoes plastic flow; that is, the soil fails. STRESSES IN THE CLAYACOMPUTED FROM THEORY or ELASTICITY [I . I
I d Dinii‘ ,, 2'fTI:€,adII\IIeSK1’.n Of the clay to film rOka surface is Carothers 3 has shown that'for a uniform load 1) per , ‘? I
g: fig“ 60 ‘ b (IS ipplage occurs at_t 15} sur ace although unit area on a long strip of Width 2b (see fig. 1) at_the ‘ ,, ,
“smug veiyenldaaryth:rdIi-Jlgrgurlgggement m t 16 clay at pomts surface, the shearing stress, 3", at the rock surface is I I
if . , I I,
mag: 3. The soil deformations considered in this paper are _ ? I,
3 those that occur at an assumed constant volume. It s,z=g[sech—725£2Tb —sechg§2+Tb]___ _. . (1) ‘3 ,
km,“ seemsd ieasonablle to assume that the deformations ,; I i
3,- cause. y atera yield in the X direction occur during - - . - ‘vi t
and ‘; aperiod of time that is brief in comparison with the Wlifrhi’szgxfirzgsifilfggss fidfr ihiifgfligiifgioggfidafi 3:1 I I vi
‘ tim ' ' ' ' - I: b I I I
‘““'F':% of tfi: Sgiégsgfagagtigfiif: 1abiliiiflilegldaefgfrfiiirtlisdifilsdddéoI; Other expressmns for stresses for. other types of surface , II I]
d' y i ' e e u loading (see for exampleequation 12) are developed PII i
of]; at co t ~ s - - , 1 from thetheory of elast1c1ty. When these express10ns 3 ,I
:1: --, HS: ant volume, P0185011 S ratio ls talxen as 2 (the are used it is cons1dered that the clay mass has not been 5 III»
aJPPI‘OXImate value). stressed to its ultimate supportingpower and is there- _ If
,5 0ch 4..In applying the method of plastic equilibrium it is fore not reduced to aplastic condition throughout. , III,
:16 °°PI FOIESldered that the fill acts like an absolutely rigid body In the followmg discuss1on equations 2, 3, 4, and 8 {psi
s in its production of stresses in the clay stratum when are those frequently seen in texts on the theory of III
.1 _ Win the plastic state Thus the fill above and elastimty.‘1 II I
.P- -1 - - . . . T— u - I ~ . . ‘4:
,_ E, 3,: gig: °§sis$3e§f$g°§ §I$i§§°ii7§$alfcif 11110333523333? Mitedfiglifée'gir ind Ps03$?5523i:iéi‘lfon‘i‘iti‘i’iifiisfifiiiiihg5,33???ifoiifiigfifiiif’bis.gagiif’the’s’ I s
‘2‘": Ab t D es of $011 Mechanics Involved in the Design of Retaining Walls and Bridge ‘ See, for example, pp. 8—20, incluSive, of Theory of Elastimty, by S. Timoshenko. -,‘ ,
u ments, 11- A. Palmer, PUBLIC Roms, vol. 19, No. 10, December 1938. MeGmw-Hill Book 00., lst. ed., 1934. y, I) s
I, 178157‘39 157 ”‘3‘
a.“ I
Ii ’ - “II
:11? ‘ I ,1, I
- "‘II i
'3 ' ri‘ ‘
"' . 1*

 11777
311 1.111 ‘
W i 11':
‘ f .“s i :1 '
1 - l .
.7 7 *7
,1 _111 158 P UB L I C R 0A D S VOLNNM October.
11 1 1 1‘1 1 11 m g;—
77 7 '1 7 2 E7 The fundamental strain relations are — _“‘—— as ori
1:1 1 771—“; (bd—B :1 problé.
1 P k from
. 7 77 €2=EV[PI—fl(py+Pa)]———————————~(2) i1. ‘- d '
7 1 '1 17 ‘ BS
3 7 ‘ 7 7:77 1 1 1 giltlhe
._77 717i e'I=E‘[5p”—7”(PI—740271“"""""‘(3> ‘ k a clay
‘1 ‘ 1 117.11 ' 1— 7X 0 77 7151371717775
11 17111 171 €1=ELPz—H HUEmMgMMmHK
711.137.11.11. , . aai, . craw-i oo o.
1 19 12.11 .
1 "
51111 .7‘ _
77177737772711 1

 ' I fit“:
1 1 11
Vol. ”No.1 October 1939 1. 1'1
a A 1 ill
as originally devised and a ' 159 '1
' pplled b Pr 7 l -
Frofilfigtlllustrated by figure 3’ the 131% {13(11th fit; the E . f1.
10 S 3.111316% tgvotriglcl plates. Certain equati O 8011 - 11“ l .
s res es e erived 1n the application Th Ons for H c 1 11
efill‘lessmnfilfor the stresses will be used in the enlth'ese ‘1, )1
O 1 6 pr: 1? m of the fill: ABOD: figure 4 su $01; 1111011 A i 1.1
—++x a my; 111 um underlald by rock. Fir’st gfp 01116‘ b-y 111‘; 1 1
25121115174 ( Es. 3 and 41) that h is either equgfl t g 11 is _X «fireh \‘ .1 1
. _ n no case 111 the following dev 1 or eSS aw _—’ —* —>—._.D I
h be conmdered as greater than b /4 eqopment may CW 2 new .X 5
follows. ' - he SOlutlon ._”°W newly/1c: h 111'
—* 2b ~fiemi~1 “‘" 1'11
P _ z 1
1 1 AREA AED~ AREA ABC‘D q=suppunrme POWER :3}? mama or SAFETY- q 1 1 1
. FIGURE 4.—Sma ’ 5 1
_ RIGID PORTING P 1 1 ,. .
SUPPORTED- ‘ FRAEAE r . BY ROCK, M23131) OOFF (EILAY LAYER UNDERLAID 1
4— “. .‘ ’ v” FLow OF ENCKY' ‘1 "
ring Stre —-FI7{5TIC MATERIAL 5:60' b 1 i 1
, ' SS =60' 1r 1‘ :1
‘ ' 1: ’. 1
dlng 15m" SLIP LINES B ‘E 1 131
10 and 12 H=26%' ' ‘ C ‘j, , 1:
ofhandb. pzwfl 111
smax~ de Z / 1 30' . . 11 11
1 FOI,‘ ex- ‘Ifl... —__AA ' 30 _’T‘_40' \‘ 1. 11 1
l- .--_, D—->X 111
.S l b then 217"“—Evl.«_fi+li_oX z QFLOW I‘- SEA; W'T” A UNIT COHESION 1‘ 11
2 i -X LL FLOW4__ ~h —>FT_o_v1/-—+X 0° POUNDS PER SQUARE FOOT 2h=%b=30' . 11 1‘
« fl. 7 ROCK SURFACE " 1
he Clay has 7&1) _ 5x1 : c 1 Wmémemwam ma'm 1 4 .
~~ __ I / 4 A ,, M I , , , 1. .
913.131 -- Z b — .w ,, ,I/W1 11
x=gggd511 FIG DISTRIBUTION OF STRESS / / m 1 11
Uli’ingss—ngggifgnogg litvEAIIiLURE IN A PLASTIC MATERIAL FIG 2 1 ll
W11 OUGH PARALLEL PLATES' sgnggivi—si?§ficmli OF EHE SUPPORTING POWER 0F A C 1 51 l
. E 'I 1
load Igllslaleefigmgrbq pressed between the plates b ROCK. D ETWEEN A FILL: ABC'D, AND Selig: 1 1
I ' ’ eCO ( ' y a: . _ 1 j],
tcohesmn. Wlth _a constant maxififis a pfiastic mass, flow occurs Equatlon 17 is now solved b a ‘ 11 11 1
equation, . m s ear expressed by the Pends. on 2 alone and not ony Ssummg that-8,2 de_ 1' 11
5 and the “ equatlon 17 reduces to 1'- When thls IS true 11‘ -l
smax =\/l:pz;p95:r 2 62 11
' r = ' . 8 1 ‘
'2 +833 the unlt’ 0011881011 6 0r #OZ:Z=0""“-——— _‘_-~__— (18) l“ 211
__ _ — Which is re (1'11 ' ‘1 1:1
P: p —i2 02_ 2 a y Integrable anl - -
1‘” aficiording to theorv 8 ;CO 3/ 8“ "-2-03) 8 _I’{ ‘ there Is Obtalned 1 1
18m”, may CPHdltlons_ There aré,t‘:flgzbtl IlS 51.1113th under these _ 22— 1_l—‘K:‘Z~Z————___________ 19 1 , C
rlum 1er e uat The h '
@112 The ,namely’ q Ions of equ111b_ the uni: earing stress 8“ cannot anywhere 3 1 1
.mg_ stress, 0 b t ‘ _COhesion, If K1 be taken as exceed 6, ‘1 .-
btamed iii 8:52 W0 Stlalght llnes (th zero, there are 1 1
l and 02: +E20” ‘ ‘ ’ " - - > — , — — — _ _ (14) filg‘ 3), the equatioDS (ff $13311; filed 210W“ bOUIldaries, i "I 11
'DAMENTAL SiggngEmh £0,116 Shearing stress Szzbebgrrltegd sZ=—h 11
quilibrium g—Z)‘+%=0 In theyprtlaggntoissf sum=sxz at the reek (rigid) 53:13:: I :11
tions illus- Z 533 ‘ " ’ " ‘ ‘ ‘ ' * - - — — — (15) 2: ih, Which forme t ere are . two rigid Surfaces at 1 i
,0 the 0011- The stresses 1; The si 11 of K . na ural llmltS for the plastic m, . 1 ,. .
Id , equat. :57 P2, and 8“ ma b d . g 2 111 equatIOn 19 d 3155. 1 .: 1
’ the (he Ions 13, 14 and 15 Y e Qtermmed from Szz=+c or s =—c f0 epends on whether ‘ 1 '
clay layer respect to x and 114 With - Dlflerentlatlng 15 with then for Klzfzo e u t_I‘ Z=h- If for z=+hy 8“: +0 1 1111
I‘face area. there is Obtained I'espect to z and subtracting- , q a 1011 19 becomes , 1 1
2 . i .
LteS whese Szz=smx = +021“ 3 1 1
andtwhlch 02 528 2 0r - 2 1 j .1 .
‘3) IS 031:. met—17,):VZL %s_:_ (16) K C :11
Slon an 1 Sub - . _ at "”"n— 2+1 ,3:
0f 11113611181 Stltutmg equation 13 in equation 16 and th 2 h 11 i2
be shown 52 __ 62 ’ erefore for any Value of 2 between +h and l :11.-
ition Problem ig—ch—S 2=i§_ 328" —— L, 1 1'
,vol. 21,No.11 —\ 51352 ”z a 2 —2-______ (17) c? 1.1“
en Karma 11. 1 71“ mud“. Zeitschrm m: mg \r th 2 bx _ _ 8“: +711 - — — - — — — _ _ _ - (20) 111
g, p. 29], p.141» -. 1 a ematik und Mechanik, 1. 6, ' . . . _ _ _ _ 1- 1, 1
:00. v0 1923 by substitution 1n equation 19, ‘11111
:1111
x11
11" ‘ " 11
7'» if. ‘11

 (213171 ; ‘ ~
‘ 12‘ 2 ‘22 i2 . :
I 12 .2 2; 2 2
1 5 22- E
1 22 2 160 P UB L I C R 0A D S Vol.20N92 0220b,
.2 ‘22 ‘2 2 2 My ;
2 222 22 2 Now, from equation 14, Therefore or
;2 22 22 ‘ __C(b—_x) __ 2 2
3 :1 banana ire __g (21, er 22021 ”‘W (23
2 2:2“ 23 2 Or be — 02 h — h“”"
‘11.: 1 . $21) The
2 ‘ 222 12 and from equation 15, 292— h ""”"_"""‘ (292 Stm
1 2.11 1
2 21 23 . d -
‘ 2 112. a . as a an In”
2112 .1: 1 1/ _—_'_~___—
.1 2 12 "
2 2 22 1 . . . . 0]] l
2 2 2 By integration, equations 21 and 22 yield At the boundaries, 222—12 and z: _}22 othé
‘5 ‘ .11 2
2, f2: 1 ex
1‘5 " 2 1 = — — —‘ that
22 2 2222 ‘22 2” h +f1(2)“"“"“""(23) 1022;55:167—96) and su=smx.=ic.
5 1 ”'22 32
12. 11311 1 and 25531
22 2 222 22 272=f2(9c) ____________- _ __ (24) HENCKY’S METHOD APPLICABLE IN FILL DESIGN 22222
22 :2 2122122 1 , . , ‘
322 2 2.2222 3 .2 2 . . From equation 29 it is seen that p, is a maximum
21 1 “952‘: 2 2518885020“ 3132’ where f 1f (z) 115 a fugctllon 0f 2 alone and when 23:0, and diminishes as 90 increases (6 is positive
22 2 2 2 .2 s a unction o a: aone. _Ot1f1 (2) and f2 (ac) th "crht d t‘ tl If f OZ
1 2' "2‘ 2' must be so determined that equation 13 on - e “b an nega we on - 16 et_ 0 )‘ The
22 2 2 ’2 2! ’ loadipg on tlhe surfaces of plasitic clay is therefore tIi- whei
1 ‘ 2_ 222 — angu ar as s own in figure 3 a though the load applied
2 22.222 2: Pz—Pz: iQW/Cz—szZ—un—"u (13) toff‘he rigid frames is uniform. 2
22 22: 212.1 . . . . H . . he problem illustrated in figure 4, a fill, [MOD
2 22 2.2222 2 “7le 1be 532;:Sfiffd' S liquation 132115 0511219411 the condition supported by a clay stratum underlaid by rock, is con- Hen
222 :22: 222 2 0 P_ as 191 3’2 2n stituting t e V3 ues for P2 andpx Sidered next. The computation of the supporting 1-
. 22 .2222 2 as given in equations 23 and 24 and for s“ from equation power, q, of the soil layer, figure 42 is based on 2222 whe
2 222122 22 20 m equation 13 there 195111“: assumption that the structure, ABOD, is absolutely 2-
22‘ 2 222 2 22 rig2c2. Thisbassumption isequifvalent to saying that the hem
2'23 252 222 5‘ I2 f2(x)+c~x—'f1(2)= i2cJ1iéTlfi______ (25) $01 ayer is etween two rigid rames, the fill above andl T'
22 ‘2 212 1 h the rock below. But in order to use equations 28, 29. ‘
2 22122 2 and 20, derived for soil between two plates, there must regi
22 22 .2222 22 Putting x=0 in equation 25. Then be made another _ Simplifying assumption for th1 3214
2 ‘22 22222 22 problem fillustigftteddiil: figlilire 4, which is that the mast T
232 22 f (Z):K:2:20_\/T:2—h2‘ where K=f (0) ance to OW o ere V t e soil in the clay layer to the
123 2 :1 1 2/’ 2 ' 1f “ ' -
22 2 . 21 2 2 e t of A and to the right of D (figure 4) is small enough pow
22 22 522232 22 Putting 2:0 in equation 25. Then (reéapifielyfi tphbe neglecitqd: t' t'
2,2 2:2 22 1 a ese simp 1 ying. assump 110118 equaions
22 22 222: 2 2 f 2 ) K cx h K f (O) 2 28, 2f9, wild 2O laplply in cfpmputing the supporting power, star
2 222. 1222 22 2 x = ——, W ere = 1 i c. » q, o t .e soi ayer gure 4. Since the structure:
22 2.22 2:22 i. h AfiOD, 1s 21gid, thélan’ according to equation 29 theddis- 0f t
222 2222 222222 2 2 . __ tri ution o Vertica pressure, p2, at the upper boun 1112
2 225 ‘222 22 11135211320261??? Séfiglgeflelfiigggéiic—f2(0)‘ Hence (figure 4) is triangular. The same vertical stress dist12- S-
2 122' 22212 2" By substitutionyin equations 23 and 24 there results bution at this boundary WOUld be realized in fwd safe
2221 222| 2 ‘ ‘ ’ 7 the load diagram, ABCD, becomes triangular, AER belo
22 12222 2 ex the area of ABC’D and that of AED being identical com
222 2-2 2222 2 pz=K—— 272F202/1_z2/h2,________ (26) since the total load of the fill cross section (1 foot th102 1,
222 2 22 2 and H111 the direction perpendicular to the plane of fig. 4)L‘ 2,
2 2 2222 22 t e same. 3.
22 . 2 3 2 P _K_c_x (27) ( The i20t1'112vertical florce, P, on a strip of unit Width B0
.2 2, :22 :2 2_ h “"”"“"“ y=1 g.4 on thepane boundary z=h is 4
, 1 122 7 J 7 .
22 22 2522? 22 where K_is a constant. (I b peg
2 222 222} 22 Equations 26 and 27, together with equation 20, 13:22 Pzd$=2[ M3 he -'
2 222 ‘22“ 21-1 completely determine the stresses at any point in the 0 - 0 h in?
22 2 2 2 plastic mas