The overall stability of free andpropped embedded cantileverretaining wallsby J. B. BURLAND*, MSc(Eng), PhD, CEng, MICE, MIStructE, MSAICE, FGS,D. M. POTTSf, BSc, PhD, & N. M. WALSH

10—

7H/Pl

6

3510 15 20degrees

(0)

25 30

~1 Fp= 23

C0

2 3 Fd =1.7C0II

1

2 Fry=2

I

35I II

10I

50

302515 20degrees

(b)'nd (a) H/h and (b) factorFig. 3. Relationship between rh'nd afor various design methods

of safety on shear strength

h

It H

QL >

, o.o-Q4 F, =1.5

H/Il

1 I

0.4 0.5I

0.6 0.7 0.8 0.9 1.02Cu

yhc h and H/h for various design methodsFig. 4. Relationship between 2c„/yh an

form rictionaI I. Line (

f'

I soil with the water tableat excavation (or dredge) level. ineis of course commcommon to all methods andcorrresponds to a factor of safety of one.Line (4) corresponds to a factor o yof safeton strengt h F, = 1.5(=Ff) and is easilyderived from line (L).

Line (1) is obtained using method 1for a load factor F = 2. For values of

y'bovea outb t 30'ine (1) gives similarvalues of H/h to line (4). This probablyaccounts orf r the success of the method

larwhen use ord for embedded walls in granu arlow 30'oils,However, as rf'ecreases be ow

a value oof F = 2 gives increasingly con-servative va ueslues of H/h in comparisonwith F„= 1.5. Indeed it can be shownthat at y' 10'he value of / ap-Hha-

2 comparedproaches infinity5 for Fpwith a value of 6.5 for F„= 1.5.

Line is(2) is given by method 2 withF„„=2. It can be seen to lie closline (L) implying a very low factor of

3 ihf n shear strength. In contrast line

(3) hich is given by methodF„= 1.7, nearly coincides with inI'e 4im ying an a mI 'lmost constant factor of safe-ty on strength of 1.5.

F' 3(b) the above lines have beenreplotted giving a more direct comp ari-

b F and the three other met-1d . I 'lear that both methods

of Fand 2 are unsatisfactory. A value o= 2 ives very variable levels of safetywith respect to shear strengt an vconservative values of H/h for values offh'ess than about 25'. A value of Fnp2 gives a constant level of safety but thef ctor of safety on strength is always lessa

n 1.1. Surprisingly method 3 w'ththan .. uost favour-1.7 appears to compare mos

ably with method 4. The main objectionthe use of method 3 would be for vari-to t e use

able ground conditions becauseh d does not take account of ground conditions below the limiting dept o p

o oes

tration d„Undrained clays

In Fig. 4 line (L) shows the limitingrelationship between H/h and 2c / hfor a propped embedded cantileverwall in a clay with c„constant withdepth (ignoring tensile stresses). t

h Id be noted that there is acritical depth of excavation h,„l, (=below which instability occurs irrespec-tive of the depth of penetration of the

II. L'4) represents the relationshiprF = 15between H/h and 2ca/yh for

/ ). Lines (1) are given by(= Ca Ca ummethod 1 and correspond to F = 1.5 and2.0. I be seen that they differ drama-t can

I start-tica y roII from line (4). A particular y s a.5 ling feature is that when 2c„/y Isconstant the value of F decreases withincreasing wall penetration-—a resultwhich is clearly not logical. The lines cor-responding to Fn = 2 and Fa ——= 2, thoughnot identical, are so similar that they areshown as a single line in Fig. 4 labelled 2and 3 respectively. It is evident that met-hods 2 and 3 both give very low factorsof safety with respect to strength.General remarks

The use of both F and F„can lead tovery ury unsatisfactory results. There appears

b no logical or consistent relations 'phibetween F„and the factor o sa e yf safet onstrength and its use can lead to very con-servative values of wall penetration for

It follows from method 1 that as ~h IK /K, i.e. F is finite for zero heig t op~ K

retained material.

July, 1981 29

(a)

qf

lllHlllll

(a)

Passivezone CI c

Active zone

PA

J.Y .d. Kp ~ qf. KA Y .d. KA

rigid wall

PROP

PN

LAt

, PAti

LA2

HI I

q

Jlllllllll

.d .(Kp—KA) A. cI.K

net resisting pressure8 net activating pressureIIFr = PpN. LpN "/2 .Ys d '- A) pN

PAt.LA1 + PA2. LA2 /2 ~ Ys . h . KA. LA1+Ys.h.KA.d.LA2

S ys d (KP KA) qf.KAnet resisting pressure I net activating pressureFb= /2. (s.d . ( Kp —KA)1 2

q. d. KA(c)

(b)

Fig. 5 (left). Bearing capacity of a strip load(a) classical failure mechanism(b) simplified failure mechanism(c) definition of factor of safety

Fig. 6 (above). Revised factor of safety for an embedded wallin a frictional soil

drained conditions with q'ess than25'nd

for undrained conditions. With regardto F„„its use with currently recommendedvalues of about 2 leads to factors of safe-ty on shear strength generally less than1.1 for both undrained and drained condi-tions. It should therefore only be usedwith great caution and with much highervalues which are compatible with accept-ed values of F, For drained conditionsin uniform ground the use of the simplescale factor Fd appears to be entirelysatisfactory. However, it should not beused for undrained conditions or wherethe strength properties of the ground varysignificantly with depth.

The concept of factoring passive resis-tance of the toe of an embedded retain-ing wall is attractive in principle as theoverall stability can be expressed as asingle number, This is not the case forshear strength where, for all but the sim-plest ground conditions, the engineer maybe faced with the possibility of a numberof factors for strength and wall friction.Hence there are considerable benefits indeveloping a definition of factor of safetyon passive resistance which can be shownto be logical and consistent.

3. Simple bearing capacity analogyThere is a reasonably close analogy be-

tween the overall stability of a retainedheight of ground and the bearing capacityof a strip load. It was decided to pursuethis analogy to see if a more logical defi-nition of factor of safety on passive re-sistance for an embedded retaining wallcould be developed,

30 Ground Engineering

Consider a uniform surcharge q actingon the surface of a granular material hav-ing an angle of shearing resistance if'' andno cohesion. At limiting equilibrium clas-sical bearing capacity theory (see Fig.5a) gives an active failure zone I immed-iately beneath the loaded area coupledby a zone of radial shear, zone II, to apassive failure zone III outside the loadedarea. For present purposes this can besimplified to an active and passive zoneseparated by a rigid wall, Fig. Sb. Atlimiting equilibrium the passive resistance

Pp ———,'.7, da. Kp

and the active force on the wall

Pt=qf.d.Ki+ i /..da.KtFor equilibrium P,, must equal P, andhence solving for q< givest

—,'.7, da. (Kp —K,)ql ——

d. Kt(3)

where F> is a load factor. The quantity (K,,

$ Althouoh this is a very simple approach, pro-wded the values of K, and K„ take account offull wall friction a surprisingly accurate resultis obtained,

Now the factor of safety with respectto bearing capacity is given by:

qf —,'.7, da. (K~ —Kt)F„=—= " (

10

7H/I E

2 c0

1c0

vi

r 05 10 15 20 25 30 35

I)) —degrees

(a)

y .d. Kp - 2.c.Kp~

(a)

KA.(q ~ y .d ) —2,c. Kac

2520'i V ii ii 1I 11 tr 1I 1I)Ib e c

a = (s.d.( Kp- KA)b= Z.c.Kpcc = q ~ Ka 2.c.Kac

e = 2.c'.Kac

I,et resisting pressure II net activating pressure82 3

H/ Il

Fig, 7(a) Comparison between revised factor of safety F„andthe factor of safety of shear strength F, for a frictional soil.(b) Relationship between F„and H/h for various values of It,'

= '/g .'(s.d (Kp- K„) ~ 2.c.d ( KAC+ Kpc)bq. Ka.d

(b)

Fig. 8. Definition of factor of safety for a strip load on a c' 1tr'oil

(4) are taken from Fig. 3a. Also shownin Fig. 7a are the relationships betweenF, (= tan If'/tan if'. ) and III'or F„=1.5 and 2.0. It is eviden™t that a given valueof F„corresponds to a reasonably con-stant factor of safety with respect toshear strength. For example, when F, =2.0 the values of F, vary from 1.4 to 1.52as (tr'aries from 30'o 20'. When F„=1.5 the corresponding variation in F, isfrom 1.22 to 1.27. It can be seen fromFig. 7b that there is an approximately linearrelationship between F„and H/h for agiven value of rfi'. This is a feature thatcan be useful in design.

It appears that the revised factor ofsafety on passive resistance F„providesa satisfactory method of obtaining thedepth of penetration d in a purely fric-tional soil. The application of F, to soilspossessing cohesion will now be con-sidered.

p, = K, . Ir.' 2. c'. Kec

pp = Kp . Ir ' 2 . c' Kppwhere

PROP,IL ii

P ROP.

IIH H

ii

Kp .ys.d 2.c .Kpc

( a )

KA ~ (s.H —2, c.Kac ( a )

IL IL

PROP.IL

PROP.

il ii

PN1

PNI PN2

Lal

PA I:tl A2

s(s. d.(KP-Ka )I

b = 2.c.( Kpc 'Kac)

LAI

— r.c,

Ii

c e c = Ka " Ys —2.c.Kacf

e = 2.c.Kactivating pressurenet resisting pressure II net acII

4Cu s(s. h-2.C„

(b)

Fr = pN pN

Pal . Lal ~ Pa2. LA2

Fr — PNI 'NI PN2'N2PA;'LAI 'A2 LA2

Fig. 9 (left). Revised factor of safety for an embedded wall in ac'ct 'oil

Fig. 10 (above). Revised factor of safety for undrained conditions

citly assumed that the active pressure atexcavation level is positive, that is to say2. c„/ys h < 1. If this is not the case and2. c„/ys h ) 1 it follows from the con-ventional practice of ignoring tensilestresses that there is no active thrustacting on the wall above excavation level.The activating moment on the wall thenconsists solely of the moment derivedfrom the surcharge effect of the soil aboveexcavation level. The revised factor ofsafety is then evaluated by substitutingP~, = 0 in the above expression for F„which then reduces to eqn. 6 with q re-placed by y, h.

In Fig. 11 the revised factor of safetyF„ is compared with the factor of safetyon strength F, (= c„/c„u ) for a prop-ped embedded wall in a clay with c„constant with depth and no wall adhesion.It can be seen that as H/h increases fora given value of 2. c„/yh both factors ofsafety tend rapidly to the same limitingvalue (= 4 . c„/yh). Both methods showthat there is normally little advantage indeepening the wall beyond about H/h =1.5 to 2.0 as thereafter there is little gainin factor of safety.

For a given value of 2. c„/yh the re-vised method tends to the limiting valueof factor of safety more rapidly than thestrength factor method. The reason forthis difference can be explained by refer-ring to Fig. 10b. The value of F„ is obtainedusing the actual strength c„of the soil.However, Fs is derived by factoring c„to obtain c„u . In doing this not onlyare the forces P~,, P„, and Pp,. changed,but also the lever arm LLI is altered, i.e.the geometry of the loading as well asthe magnitude is changed,

The case when 2. c„/yh = 1 is of par-ticular interest. The retained material ex-

32 Ground Engineering

erts no pressure (P„, = 0 in Fig. 10b)and the revised method treats the prob-lem as purely one of bearing capacitywith no benefit derived from wall pene-tration. However, if c„ is factored, wallpressures develop and hence the strengthfactor approach requires some wall pene-tration and is intrinsically more conserva-tive.

The above example is of course artificialas in practice the wall is always design-ed to resist earth pressures. These canbe derived either by reducing the valueof c„above dredge level to allow forsoftening or by assuming partial or com-plete drainage and carrying out an effect-ive stress earth pressure calculation. In

either case the revised method providesa logical approach by separating the acti-vating moment of the earth pressure ex-erted by the retained material from theactivating moment generated by itsbearing pressure.

For comparison the dotted line in Fig.11 shows the variation of F, (method 1in Section 2) for 2 . c,/yh = 0.7. As H/hincreases F increases to a maximum of1.53 and then decreases tending to oneas H/h becomes very large. Such varia-tions in factor of safety are clearly notlogical,

The results given in Section 4, 5 and 6demonstrate that the revised factor ofsafety F„gives logical and consistent re-

2Cu/Yh = 1

0.9

Iu 1,50

If)0.7

0.8

0.7

0.6

F5FrFp

0.51

Fig. 11. Comparison of the revisedconditions

I I

2H/h

factor of safety F„with F, and F„ for undrained

P

JJJJJJJ Il Il Il Il II Il

cc Pa d

(o)

P

J J J J J J il li ii ii li Ii ii Ii ii

net resisting pressureS net activating pressureIIIy, d (Kp Ka) p(Kp Ka) (q-p) ~ Ka

suits for both frictional and cohesive soils,It is pertinent to examine how F„may beused to take account of surcharge andseepage pressures and this will be donein the following Sections.

7. SurchargesIf a surcharge is placed on the soil sur-

face behind the wall it causes an increasein the active thrust and reduces the sta-bility of the wall. When calculating therevised factor of safety F„ the moments ofthe lateral earth pressures due to this sur-charge are added to the activatingmoment.

Surcharges acting at the bottom of theexcavation tend to increase the stabilityof the wall by increasing the passive re-sistance of the soil. The evaluation of F„for such a situation is not immediately ap-parent. However, the situation can beclarified by analogy once again with thebearing capacity problem. Considering thesimplified failure condition discussed insection 3, (see also Fig. 5b) with a sur-charge loading p we arrive at the situationdepicted in Fig. 12a, where, at limitingequilibrium,

Fb- p. d. ( Kz —Ka) + ~a .y,.d . (Kp —Ka)(q-p) ~ Ka-d

(b)Fig. 12. Definition of factor of safety for a strip load with sur-charge acting

s

lllllllll,ll

Pn p ~ d Kp + z i/ d' Kpand

PK=ql.d.K,+-',. f .ds.K~

As these forces must be in equilibrium,we obtain

p . d . Kp + —,' f„.dd . (K,. —K,)ql

d.KK(9)

Now the net ultimate pressure that maybe applied, q„l is given by

LPN1

PN21i

PN1 V

L pN2

P

I I I I I

A2

PA1 1iand noting that it is customary to define the

qsrallowable bearing pressure q = —+ p

F„we obtain the following expression for theload factor F„:

p. d . (Kp —Kx) +. /s. ds.(Kp —K,)

F(, = " (10)(q —p) .d.K,

net resisting pressure

PA1 = s. KA. h1~2 ~ Ys. h . KA

net activating pressure

pN1= 2 ~ Ys.d ( Kp —KA)

1 2

pN2— p . d ( Kp- KA)

The forces contributing to the numera-tor and denominator of the above expres-sion are shown shaded in Fig. 12b.

Applying the above thinking to the re-taining wall situation, see Fig. 13, we ob-tain the following expression for F,

4sI + PesF„= (11)Px. L:I:

PA3—

( s ~ Ys.h —p ).d. KA

Fr = PpN1 . LpN1 + PpN2. LpN2A1'1 A2'2 A3 '3

d ~ ( K KA ) ( "K'2 ~ Ys ~ d . L P, + p . LPN2 )Ks [s. h . Ls, ~ 'ls . ys. h . Lss ~ I s ~ y, h —pi.d . Lss]

Fig. 13. Revised factor of safety for embedded wall with surcharge

8. SeepageTo illustrate how seepage pressures may

be taken into account, consider the situa-tion shown in Fig. 14 where we have aretaining wall embedded in a cohesionlesssoil with a water table at a distance jbelow the ground surface behind the walland at excavation level in front of thewall. As there is a difference in hydraulichead across the wall seepage will occurinto the excavation and hydrostatic condi-tions will not prevail. The exact nature ofthe pore pressure distribution in the vicinity of the wall may be determined usingone of several seepage calculation proce-dures. For illustrative purposes it is as-

July, 1981 35

PROP.

(a)

h

trated force is assumed to represent thepassive resistance acting on the back ofthe wall for a short distance above thetoe. The limiting depth of penetrationd„can be obtained from the conditionthat

Pp. Lp ——P,. L,

hydrostatic

II

PWTP~= 2 Yw (H I)'d

(2H-h —j )

A more rigorous approach is outlinedby Terzaghi, 1943 (page 357, 8th Edition)to take account of the length over whichQ acts.

The revised design method requires thatthe moment activated by the retainedmaterial should not exceed the fraction1/F„of the moment of the net availablepassive resistance (given by the net pas-sive pressure coefficient K„—K,), Refer-ring to Fig. 15b it can be seen that

LPN

(b)

PROP.

-hkPA1 4 LA1C~ fs tf LA2

1

a PA2 1 rLA3

PROP.

(c)

——/~nl

w1 LW2

Pwt

- Pwr.(h-j)(H-j )

Pw2

F. =rPp, Lax

.L,, + Paa L„s( 13)

Water pressures, surcharges and co-hesion may be included by following theprocedures outlined previously.

It is of interest to note that Krey (1936)appears to have proposed essentially thesame approach for the design of embeddedcantilever walls with the added refinementof including the length over which Q acts(the passive resistance at the heel of thewall). Similarly the revised approach isessentially the same as that proposed byTerzaghi (1943) for the design of embed-ded rigid walls subject to lateral loads ator above ground level.

a = (Kp Ka) d Ys 2 Yw CI [( Kp KA) ~ (H j)+ Ka.( h-j )](2H-h-j )

b = KA.Ys.[h 2 'Y (h j) dYs.(2H —h —j )

c = I'Ys ~ KA

PPN . LPNFr =

PA1. Lat+ PA2. LA2+PA3. LA3+Pwt.Lwt+Pw2 Lw2

Fig. 14. Embedded wall with seepage pressures(a) pore water pressures around the wall(b) effective earth pressures(c) net water pressures

Adopting this pore pressure distributionthe effective horizontal soil pressure andnet water pressure diagrams can be con-structed as shown in Figs. 14b and 14c.The expression for F, is then

sumed that the hydraulic head varieslinearly down the back and up the frontof the wall which in turn gives the porepressure distribution shown in Fig. 14a".Taking the datum for the hydraulic headcoincident with the water table behind thewall, the hydraulic head at any pointaround the wall is given by:

Ppx 4xF„= (12)+ Pul Lll,l+ Plfia LtrHead = —x. (h —j) . y /(2H —h —j)

where the forces Pf,„, P„, etc. are againevaluated from the shaded part of the pres-sure diagram shown in Fig. 14b.

where x is the distance around the wallfrom the datum level. Knowing the magni-tude- of the hydraulic and elevation headsthe pore pressures may be evaluated, Forexample at the toe of the wall x = H —jand the pore pressure is given by:

9. Embedded cantilever wallsIn this section the application of the re-

vised load factor to unpropped cantileverwalls will be described briefly. It is com-monly assumed that at limiting equilibriumpassive and active pressure distributionsexist down the front and back of the walland that a concentrated force acts at thetoe as shown in Fig. 15a for a wall em-bedded in a frictional soil, This concen-

pn t = (H-i) ym —(H-i) (h-i) ./fc/(2H —h —j)

= 2. y„.. (H —j) . d/(2H —h —j)

*For many practical problems it will be neces-sary to make a more rigorous assessment of thedistribution of pore pressure around the wall.

36 Ground Engineering

10. Conduding remarksIt is important to emphasise that the

definition of factor of safety is a matterof convention which varies from one classof problem to another. It should, how-ever, be both logical and consistent forthe full range of conditions likely to beencountered. In this Paper the customarydefinitions of factor of safety with res-pect to passive failure of the toe of anembedded retaining wall have been shownto be inconsistent, illogical and at timesunsafe. A revised definition has been pro-posed which is based on an analogy withthe bearing capacity of a strip load. Inessence the method identifies the limit-ing horizontal forces on the back of thewall that are activated by the presence ofthe retained material. The maximum re-sultant force that the underlying groundcan offer to resist these activating forcesis derived from the net passive pressurecoefficients (Kf, —K„) for friction and K„c+ K „,) for cohesion. The method requiresthat the moment of the activating forcesshould not exceed a factor 1/F„of themoment of the net available passive re-sistance of the underlying ground.

The revised definition leads to resultswhich are both logical and are consistentwith the factor of safety on shear strengthfor soils ranging from purely frictional tocohesive. Procedures for including theeffects of seepage pressure and surchargeare presented.

The main benefit of the revised defini-tion of factor of safety is that it leads tomore rational values which are consis-tent with bearing capacity problems andare free from the inconsistencies inher-

elient in the conventional definitions. It is,also worth noting that in many problems,

; particularly those for which the cohesionjIIj is small or negligible, the revised defini-Itition leads to smaller depths of penetra-fl

I

PP

(o)

PA

il

LA

PA1

Q

II ~md Referencescaquet, A. 8 Kerisei, J. (1948): Tab'es for thecalculation of passive pressure, active pressureand bearing capacity or foundations. (trans atedfrom French by Maurice A. Bec; rev. trans. byMinistry of Works, Chief Scientific Adviser'sDwision, London), Gauthier-Villars, Paris,Civil Engineering Code of Practice No 2 (1951).Earth Retaining Structures.Hansen, J. B. (1968): "A revised and extendedformula for bearing capacity". Danish Geotech-nical Institute, Bulletin No. 28.Krey, H. (1936): Erddruck, Erdwiderstand undTragfahigkeit des Baugrundes, 5th Ed„Bar in, W.Ernst u. Sohn,NAVFAC DM-7 (1971) Design Manual —soil mec-hanics, foundations and earth structures, U S.Naval Facilities Engineering Command, Washing-ton D.C.Piling Handbook (1979). British Steel Corporation.Taylor. D. W. (1948); Fundamentals of Soil Mec-hanics, Wiley, New York.Terzaghi K, (1943): Theoretical Soil Mechanics,Wiley, New York.Tschebotarioff. G, P. (1973); Foundations, retain-ing and earth structures, McGraw-Hill.USS Steel Sheet Piling Design Manual (1975).United States Steel International, Pittsburgh.

PPN

PN

net resisting pressure net octivoting pressure

APPENDIX

Example 1: Propped wall retainingcohesionless soil(Fig. 16)

Soil properties:

(I>

' —30', Fi ' —rb '/2, 7,. ——19.6kN/m"-,7ia = 9,8kN/m"-, K, = 0.3, Kp = 4.9Net water pressures:

Fig. 15. Free cantilever wall

(b)

tion for a given value of factor of safetythan method 1 (described in Section 2)which is widely used. This may lead tosignificant savings particularly in situationswhere the angle of friction is low andthe groundwater pressures are high —asituation that is not uncommon for cut-tings and excavations in clay soils.

Regarding the magnitude of F, to usein design it would appear from the workdescribed in this Paper that values of F„between about 1.5 and 2.0 would normallybe appropriate. These values are less thanthe load factors normally used in the de-sign of footings where uncertainty of load-ing is usually greater and the consequen-ces of excessive settlement are often sev-ere. When selecting a value or F, not onlyshould the uncertainty of the soil strengthand its variation with depth be consideredbut also the uncertainty attached to theestimation of seepage pressures and theirinfluence both on the water pressuresagainst the wall and the effective stresseswhich control the earth pressures.

Two examples which incorporate themajority of the work discussed in thisPaper are presented in the Appendix andserve to illustrate the calculation proced-ure in detail. In the first example a proppedwall retaining a cohesionless soil with dif-ferent water levels on either side has beenconsidered and in the second a short-term analysis for a propped wall retaininga clay overlain by a granular fill and a sur-charge has been investigated. For bothexamPles the values of Fp, Fe„and Fxhave also been calculated for comparisonpurposes. Whereas F, and F, are reason-ably consistent it can be .- 'n that F isgenerally conservative and F„ is veryoptimistic.

As stated previously, it has not been

A

B

6m

E

GEOMETRY

Fig. 16. Example 1

NETWATERPRESSURE

the purpose of this Paper to debate therelative merits of load factors and strengthfactors. It will, however, have been notedthat the consistency of F, (which is es-sentially a load factor) has been assessedlargely by comparing it with F,. It canbe argued that for this type of problemthe use of load factors should be discard-ed in favour of factors of safety with re-spect to strength. Alternatively the use ofpartial factors on all the major variablesmay be thought to be more rational. Sucha debate is outside the scope of thisPaper. At present the concept of factorof safety with respect to passive failureof the toe of an embedded retaining wallis widely used in design and has the at-traction that the overall stability can be-expressed by a single factor. The reviseddefinition is logical and consistent for awide range of conditions and overcomesthe major deficiencies that are inherent incurrent definitions,

Position Net water pressure (kN/ms)C 2.67 x 9.8—2.67 x 2.67 x 9.8/

17.33 = 22.1D 5.33 x 9.8—5.33 x 2.67 x 9.8/

17.33—2,67 x 9.8 = 18.0

Net lateral effective earth pressures:

PASSIVE SIDE

Position(Passive Epressureat E)

(Activepressureat E)

Earth pres-ure (kN/ms)4.9(2.67x 9.8 + 6 x 19.6—(11,33x 9.8-11.33x 2.67 x 9.8/17.33) )

—0.3 (14 x 19.6—(11.33x 9.8—11.33x 2.67 x9.8/17.33) )

(Activepressureat D)

+ 0.3 (8x 19.6—(5.33x9.8 —5.33 x 2.67 x 9.8/17.33)) = 223,9

A

T

B

D

=-V4--r II

11

ELATERAL EFFECTIVE

EARTH PRESSURE

ACTIVE SIDE

Position Earth Pressure (kN/ms)B 0.3x 2.67 x 19.6 15.7D 0.3 (8 x 19.6—(5.33 x 9.8—

5.33x 2.67 x 9.8/17.33) ) = 33.8

July, 1981 37

-~- i i l i i t I i i(FILL

II5m

3mC LAY

A

T

B

Dy

5m

'5'4'EOMETRY

Fig. 17. Example 2

ELATERAL EARTH PRESSURES

Moments about tie position

Moment due to net available passive re-sistance

Zone Moment (kNm/m run)Earth pressure 5 0.5 x 223.9 x

6x (4+ 6) = 6717.0

Moments activated by retained material

Zone Moment (kNm/m run)Earth pressure 1 0.5 x 15,7 x 2.67

x (0.67 x 2.67 —2)-4.4

Earth pressure

Earth pressure

Earth pressure

Water pressure

Water pressure

Water pressure

Water pressure

2 15.7x 5.33 x (0.67j 2.67) = 2793 0.5 x (33.8—15.7)

x 5.33 x (0.67 +0.67 x 5.33) = 205

4 33.8x6x(6 + 3) =1825

6 0.5 x 22.1 x 2.67x (0.67 + 0.67 x2.67) = 72

7 18 x 2.67 x (0.67+ 1.5x 2.67) = 224

8 0.5 (22.1 —18)x 2.67 x (0.67+ 2.67 + 0.33x 2.67) = 23

9 05x18x6x(6+ 2) = 432

(ii) Clay: cs = 35kN/m"-, y', =15kN/m'et

lateral earth pressures:

ACTIVE SIDE

Position

A

—D

+D

Earth Pressure (kN/m')10x 027 = 270.27 (10 + 20x 2) 13.510+ 2x 20+3 x 15 —2 x 35 = 2510+ 2x20+3x15 95

PASSIVE SIDE

D 4x35

Moments about tie position:

25I ength y = —= 1.67m

15

Moment due to net available passive re-sistance

Zone Moment (kNm/m run)Clay 5 140 x 5 x 5.5 =3850

Moments activated by retained material

Factor of safety F,:

moment of net availablepassive resistance 6 717

F„ =—= 2.2moment activated by 3 056retained material

Total 2 651For comparison the factors of safety cor-responding to Methods 1, 2 and 4 respec-tively are: F = 1.5

F „=4.7F, = 1.4

Example 2: Short-term analysis of apropped wall retainingclay overlain by a gran-ular fill and a surcharge

(Fig. 17)Soil properties:

(i) Fill: y' 35', )I' 0,.'. K, = 0.27, y', = 20kN/ms38 Ground Engineering

Factor of safety F„:

moment of net availablepassive resistance 3 850

F„= =—= 1.45moment activated by 2 651retained material

For comparison the factors of safety cor-responding to Methods 1, 2 and 4 res-pectively are:—F = 1.63

F „=32.1F, = 1.41

Zone Moment (kNm/m run)Total 3 056.0Fill 1 2 x 2.7x (-1) = -5.4Fill 2 0.5 x (13.5—2.7) x

2 x (-0.67) = -7.2Clay 3 0.5 x 25 x 1.67x (1.33

+ 0.67 x 1.67) = 51.12Clay 4 95 x 5 x 5.5 =2612.5

Soil mechanicsand tunnelling

(continued from page 26)

used, and the method and rate of exca-vation. It is recognised that it is oftendifficult to assess the influence of each ofthese factors in isolation, but it has beenthe aim of this Paper to show how it ispossible to separate some of the moreimportant parameters and interpret thebehaviour of a tunnel in soft ground interms of well-established principles of soilmechanics. With this approach to softground tunnelling, a more rational under-standing of tunnel deformation behaviourcan be developed and a framework provi-ded for tunnel design and interpretationof field data.

AcknowledgemetTtsMuch of the work referred to in this

Paper was carried out at Cambridge Uni-versity as part of a programme of inves-tigations into the behaviour of tunnels insoft ground, supported by the Transportand Road Research Laboratory (TRRL).The authors are grateful to Mr. M. P.O'Reilly, Head of the Tunnels Division atthe TRRL, for his continual interest andconstructive criticisms. The authors areindebted to Professors A. N. Schofield andC, P. Wroth for their overall direction ofthe research work,

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