19_Geotechnique_No38_Is2_167_189

5/28/2018 19_Geotechnique_No38_Is2_167_189

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Bolton, M. D. & Powrie, W. (1988) GLmechnique 38, No. 2, 167-189

Behaviour of diaphragm walls in clay prior to collapseM. D. BOLTON* and W. POWRIEt

Centrifuge model tests have heen used in anattempt to gain a coherent view of the soil-structure interaction behaviour following the exca-vation of soil in front of a pre-constructed wall.Excavation was simulated by the removal of a suit-ably heavy fluid from a preformed cavity. Thebroad replication of stress magnitudes and stresspaths permitted the full representation of walldeformation, soil strain and swelling, completing a50 year full-scale lifespan in under 24 hours of con-tinuous centrifuging. Measurements were made ofsoil displacement vectors, pore water pressures,wall displacements and bending moments togetherwith forces in props when they were present. Thesehave made possible the validation of simplifiedgeostructural mechanisms which offer the samedegree of advantage to the designer as does theidealization of heam behaviour encapsulated inengineers heam theory. A serviceability criterionfor soil or wail displacements can he entered intosimplified admissible strain fields appropriate tothe kinematic constraints so that the effectivemohilized soil strain in the major zones of soildeformation can he deduced. This can lead,through triaxial or plane strain test data, to theselection of a mobilized soil strength and thence toan equilibrium analysis of the wall, from whichunknowns such as the required wall penetrationand bending strength and the required prop forcecan he determined. This approach leads to theevaluation of a design in terms of the chosen dis-placement criterion and avoids the question ofdefining or calculating a factor of safety. Safetycan he judged against separate collapse criteria,linked to the establishment of severe but realisticcombinations of influences.KEYWORDS: diaphragm walls; deformation; centri-fuge modelling; soil-structure interaction; design; timedependence.

Des essais sur mod&s en centrifugeuse furentemploy& pour une etude approfondie du com-portement dinteraction sol/construction apr&sIexcavation du sol devant un mur pr&onstruit.Lexcavation fut simulCe par Ie&vement dunfluide de pesanteur convenable P partir dune cavitiprCform8e. La reproduction essentielle des valeursde contrainte et des chemins de contrainte permiten moins de 24 heures de centrifugeage continu lareprbentation compBte de la deformation du muret de la contrainte et du gonflement du sol quiauraient eu lieu en vraie grandeur pendant une p&-iode de 50 ans. Des mesures furent effectuees desvecteurs de diplacement du sol, des pressions deIeau interstitielle, des d&placements du mur, desmoments de flexion et des forces dans des supportsBventuels. Des mkanismes g&structuraux furentainsi valid&s qui offrent le m@me avantage aux con-structeurs que Iid(?alisation du comportement despoutres qui forme partie de la thborie des ingi?n-ieurs concernant les poutres. On peut introduire uncrit&re dapplication pour les dbplacements du solou du mur dans des champs de contrainte admis-sibles simplifiBs qui conviennent aux contraintescinematiques de sorte quien puisse en d&duire lacontrainte effective du sol mobili& dans les zonesprincipales de la dbformation du sol. Ceci peut con-duire par moyen des donn&s dessais triaxiux oude contrainte plane B une analyse dbquilibre dumur, $ partir de laquelle des quantitb inconnuespeuvent se determiner comme, par exemple,lencastrement necessaire du mur, le moment deflexion et les efforts dans les buttons. Cette mCth-ode conduit $ Pitvaluation dun projet en fonctiondu critkre de dbplacement choisi et Cvite le proh-l&me de dhfinir ou de calculer un facteur de skuri-tC. La s&curiti peut s&aluer en fonction descriti?res choisis deffondrement combinb avec ladbfinition de comhinaisons strictes mais rCalistesdes influences.

Factors of safety may serve two purposes: to dis-tance work ing states of a structure from condi-tions which would lead to collapse; and to ensurethat working deflections are tolerable. In thedesign of rigid-plastic structures the formerpurpose would predominate, while the latterwould be more important in the design of very

Discussion on this Paper closes on 1 October 1988. For flexible structures. As soil is particularly flexiblefurther details, see p. ii. and compressible in comparison with other con-* University of Cam bridge. struction materials, it may be that the avoidancet Kings College London. of strain should supersede the avoidance of col-167


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168 BOLTON ND POWRIElapse in geotechnical design evaluations and thatfactor of serviceability should supplant factor ofsafety.It is recognised that the allowanc e of anundrained bearing pressure on clay of only 2c,,compared with an ultimate capacity of 5-6~arises mainly from the need to avoid significantplastic yielding in the soil under the edges of thefooting, an d settlements which wo uld be very dif-ficult to predict. Instead of referring to thisproblem as requiring a factor of safety of about 3,it would be more meaning ful to speak of a ser-viceability factor of about 3 which is a pre-requisite for simple elastic-type settlementcalculations.No such recognition exists in the design ofretaining walls, which are conventionallydesigned using plastic analyses based on soilstrength, without the benefit of any soil stiffnessanalysis. Serviceability is then guarantee d(hopefully) through the introduction of a factor ofserviceability which is labelled as a factor ofsafety. It is sometimes applied in the form

F, = stabilizing forces/de-stabilizing forces (1)This form is inherently unsatisfactory, because thestabilizing and de-stabilizing forces must alwaysbe exactly equal and opposite if the structure is tobe in equilibrium . Furtherm ore, the frictionalstrength of soil increases if the applied load isincreased, so it is unclear how one is to segregatethe stabilizing and de-stabilizing effects.

Burland, Potts & Walsh (19 81) discussed theproblems of defining F, for cantilever retainingwalls and expressed the opinion that the mostfundam ental definition would beF, = maximum soil strengthmobilized soil strength (2)

Considering that this would, in engineering prac-tice, lead to unacceptab le complications in designanalysis, they created a nother definition using ananalogy with bearing capacity calculations. Thiswas held to be more acc eptable than other formu-lations because it accorded most closely with theresults from Eqn 2. How ever, the selection of avalue for F, in relation to the desired limitationson wall deformations is not obvious.The objective of this Paper is to explore thepotential of mobilized soil strength as the para-meter for the control of deformations in cantile-ver retaining walls. The results of a numb er ofcentrifuge model tests on both propped andunpropped walls are introduced. Back analysesare made which take approximate account ofeach of the three compone nts of a complete solu-tion in solid mecha nics-quilibrium, compat-ibility and a constitutive relationship. A method

is then proposed by which a performance cri-terion in terms of wall or soil displacement can beconverted into a soil strain lim itation and therebyinto a limitation on the soil strength which can bemobilized. Calcula tions identical in form to con-ventional collapse analyses can then be performedusing only that mobilized soil strength which willnot transgress the performance criterion for ser-viceability.

The tests were conducted on the Cam bridgegeotechnical centrifuge. Equivalent full-scaleprototypes were stiff walls in overconsolidatedkaolin, with a retained height of 10 m. The wallswere either unpropped, or propped at the crest.Grou ndwa ter levels were initially at ground leveland were generally kept high in the retained clayas the process of excavation was simulated . Dis-placements, pore water pressures, bendingmom ents and prop forces were then monitored inthe subsequent phase of softening following therelief of total stresses. Results are presented atprototype scale using a dimensionless time factorT, to relate elapsed time to the timescale of porewater pressure equilibration. Values of TV werecalculated using the average length of the drain-age path in the centrifuge model, and a coefficientof consolidation and swelling was deduced frommeasurem ents made during reconsolidation of theclay sample in the centrifuge.

A typical centrifuge model is illustrated in Fig.1 and represents a section of a long retainingwall. The length of the model wall section was150 mm, corresponding to 18.75 m of a prototypewall at a scale of 1: 125 . For ease of back-analysis,the deformation took place under conditions ofplane strain. The plane vertical boundaries per-pendicular to the face of the model wall shouldtherefore have been rigid and frictionless. Thecentrifuge strong-box designed for the model testshad an aluminium alloy back-plate 16 mm thickwith two horizontal stiffening beams, and aPerspex front window 80 mm thick. To reducefriction to a minim um, the inside of the backpla tewas well lubricated with silicone grease and theinside of the Perspex window was sprayed with amould release agent so that the view of the modelwas not obscured. It was estimated that the totalrestraining force due to friction from all sourceswould be less than 10% of the typical fully activeforce (including porewater pressure) on theretained side of the model wall above e xcavationlevel (Powrie, 1986).The clay used in the model tests w as speswhitekaolin, chosen principally because of its relativelyhigh permeability k = 0.8 x 10e9 m/s (Al-Tabbaa, 1987). Kaolin powder was mixed withde-ionized water under a partial vacuum to aslurry with a moisture content of 120% (abouttwice the liquid limit). The slurry was then poured


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170 BOLTON AND POWRIEheight of 80 mm in the model represented 10 m atprototype scale. The model w alls were intendedto be imperm eable to groundw ater and effectivelyrigid in bending. They were made of either 9.5mm or 4.7 mm alum inium alloy plate, givingequivalent bending stiffnesses EI at prototypescale of approximately 10 and 1.2 x lo6kNm /m. The faces of the model walls werecovered with a coating of resin 2 mm thick toprotect the strain gauges and wires and to achievea uniform and repeatable surface finish.In most tests, a full height groundw ater levelon the retained side of the wall was modelled andspecial silicone rubber wiper seals were used toprevent water from leaking between the edges ofthe wall and the sides of the strong-box. Stand-pipes with overflow outlets at fixed elevationswere supplied with water from hydraulic sliprings in order to create con stant head devices. Byadjusting the supply flow rate, the elevation ofwater above the stand-pipe outlet could be finelyadjusted. D uring the initial reconsolidation, waterwas supplied at the elevation of the groundsurface to the ground surface, a base drainagesheet, and the floor of the excavation. After exca-vation, solenoid valves were used to switch d rain-age lines so as to isolate the base drain and keepthe water level in the excavation drawn down tothe floor. It can be shown that the presence of theisolated drainage sheet at the base of the modelcaused the steady seep age solution to mimic thatof a much deeper soil stratum. The general prin-ciples of centrifuge modelling are discussed inmore detail b y Schofield (198 0), and the design ofthe model diaphragm wall tests is detailed in fullby Powrie (1986). A companion paper (Bolton &Powrie, 198 7) was concerned specifically with col-lapse limit states. Back-analyse s were based onequilibrium stress fields which form the point ofdeparture for the present study of deformationsprior to collapse.

E Q U I L I B R I U MStatically admissible stress fields are simplifiedstress distributions which are in equilibrium withgravity and any other applied loads, and whichnowhere violate soil strength limitations usuallyembodied either by an effective angle of shearingresistance 4 or by an undrained strength c, at aparticular void ratio. Figure 3 shows one suchstress field used (Bolton & Powrie, 198 7) to back-analyse the collapse of unpropped cantileverwalls. Frictionless surfaces are invoked on thewall surfaces and on the horizontal planesthrough the base of the wall and through anassumed pivot point about wh ich active andpassive soil zones w ere taken to switch over.Similar distributions were found to be reasonably

Settlementn,__----_ n- rIII I=,,-, - n,Heaved,

K, and K are earth pressure COeffiClentSdo is the Initial depth of embedmen tho= 10mFig. 3. Admissible stress field for unpropped cantile-ver wall

accurate in back-analysis of cases of limiting sta-bility.Milligan & Bransby (1976) described the failureof sand retained by rigid model walls which wereconstrained to rotate about a point in theirlength. These authors investigated whether thestress discontinuity in the retained soil at the level

of the pivot might be treated with a plastic fanzone, thereby enhancing the passive pressures byone or two orders of magn itude. They concludedthat no theoretical stress distribution couldconvey the true situation, which depended on themobilization of strain in the soil around andbeneath the pivot. In their experiments, highlylocalized strains developed from the bottom edgeof the wall, correlating with extremely strongpassive stresses at the same point. The simplerstress field shown in Fig. 3 is preferred, as therequired depth of wall below the pivot will bevery small. It remains to be investigated whetherthis can lead to a useful estimate of mobilizedstrength and mobilized strain.The focus of these approximate analyses is theequilibrium of the wall. Although points in thesoil well awa y from the wall are taken to be in thesame stress state as points on the same elevationnear the wall, this is unrealistic. The plane divid-ing active and passive states would, if it were notfrictionless, permit each zone to develop supportfrom the other so that the state of stress in thesoil would become safer further from the wall.This is ignored because it is not important for thestability of the wall; all that is desired from alimit-equilibrium standpoint is that a safe stressdistribution be found.

It is assume d that stress distributions of thetype shown in Fig. 3 are of relevance not only atcollapse, but also in the phase of increasing strainprior to collapse. If a particular wall would be inlimiting equilibrium with earth pressure coefli-cients corresponding to 4 = 20 in certain zones


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BEHAVIOUR OF DIAPHRAGM WALLS 171it is assumed that the soil in these zones will uni-formly mobilize 20 of its available angle of shear-ing even if that w ere to be somew hat larger, sothat the wall was not on the point of failure. Suchan assumption might seem unpromising.How ever, equilibrium is the essential conditionleading to the selection of a value of mobilizedstrength. Only if the shape of the stress distribu-tion prior to collapse were substantially differentto that at failure would the assump tion lead tosignificant error. This contingency can be investi-gated experimentally.Non-linearity of the lateral stress distributionbehind a wall propped near the crest was investi-gated by Rowe (1952, 1955). He defined anempirical soil stiffness constant m by the relation

Aa = m z AyJD (3)for the local increase in lateral stress due to a dis-placement Ay at depth z following the rotation ortranslation into the soil of a wall of embedm entD. The bending stiffness of the wall was charac-terized by EIJH4, or its inverse p = H4/EI.Rowes relative soil/structure stiffness was there-fore

R = mp (4)It is not possible to relate the apparent soilstiffness m to any fundame ntal soil modulu s.However, if in Eqn (3) Ay/D can be treated as anapproxim ate meas ure of local strain A E it followsthat

Ao=mzAs (5)so that an equivalent Youngs Modulus is

E = mz (6)

To an order of magnitud e therefore, m can betreated as the rate of increase of soil modu luswith depth.For the kaolin used in the models m = 2000kN/m 3 at an axial strain of about 0.2% falling toabout 1000 kN/m 3 at 0.5% strain. T he more flex-ible retaining walls had a prototype bendingstiffness of 1.2 x lo6 kNm /m and the stifferwalls, 10 kNm /m; this range brackets typicalstiffnesses of existing walls. Prototype wall height(crest to foot) varied from 15 m to 30 m leadingto a wall flexibility p between 0.7 and 0.005. Thelargest applicable soil/structure stiffness ratiowould then be R = 1350 for the longest of themore flexibile walls generating small soil strains,but value s a factor of ten or a hundred timessmaller would be more typical of the tests as awhole. Rowe suggested that arching seriously

affects the linearity of lateral stress diagrams onlywhen R 1000 in the case of such cantileverwalls. Diaph ragm walls propped near their crestmay therefore be described as rigid rather thanflexible, in relation to the clays which theysupport. Howe ver, relative soil/structure stiffnessdepends not only on the modu lus of rigidity ofthe wall but also on its support conditions. Wa llswith extensible anchors, or multiple levels ofanchors, or walls supported at the level of theexcavation must each be considered separately.

A numerical analysis by Potts & Fourie (1984)based on elastic-frictional soil properties investi-gated the sensitivity of the pressure distributionbehind a propped diaphragm wall to variations inthe effective earth p ressure coeflicient K, prior toexcavation. In particular, they showed that for ahigh initial K, of 2, the post-excavation earthpressures behind a typical diaphragm wall werefar from linearly distributed. This was in strongcontrast to their analysis for K, = 0.5, which fea-tured linear distributions of stress. How ever, theprecise significance of this result is difficult toassess, as the analysis took no account of ground-water. An initial earth pressure coefficient of 2corresponded to a rate of increase of effectivelateral stress w ith depth of 40 kN /m3. It has beenargued (Powrie, 198 5) that the process of con-struction of a diaphragm wall in clay would inevi-tably reduce K , towards unity prior toexcavation. Furtherm ore, the presence of a highgroundw ater table would generally reduce theeffective submerged unit weight of clay to lessthan 10 kN /m3. A rate of increase of effectivelateral stress of about 10 kN/m 3, correspondingto K, = 0.5 in the Potts & Fourie analysis wouldappear to be more typical of real conditions.Their analysis therefore leaves open the questionof whether stress distributions in practice mightbe approxim ated as linear for the purposes ofdesign.

Equilibrium stress-field calculations based onlinear stress distributions for walls proppedrigidly at the crest are shown in Fig. 4. If theprops do not fail, rotation of the wall must tak eplace about the position of the props. Therequired depth of penetration follows from thecondition of mom ent equilibrium about this axis,together with earth pressure coefficients based onthe mobilized angle(s) of soil shearing.The condition of moment equilibrium aboutthe prop in Fig. 4 provides one equation in the

two unknowns K, , K, (see Appendix 1). Oneother equation is necessary to complete the solu-tion. (The same applies to the unpropped wall inFig. 3 which possesses both a n extra unk nown zplocating the pivot point, and an extra equation ofhorizontal force equilibrium .) The missing rela-tion could be provided using the assump tion that


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172 BOLTON AND POWRIE

Fig. 4. Admissible stress field for wall propped at crestmobilized angles of shearing in the two zones areequal

&lob 1 = d&b 2 (7)This might be justifiable for initial states K, = 1if it was thought that equal strains would beinduced in both regions, tending separately andsymm etrically towards active and passive states.If the effects of wall friction were ignored, this ele-mentary assump tion would lead to the condition

K = l/K, (8)from which a solution to the mobilized earthpressures can be obtained by iteration.Figure 5 shows the results of this elementarysolution for 4kob in the case of a wall retaining 10

m of kaolin with full groundw ater seepage. While,for example, an embedm ent of 20 m is vulnerableto drained failure with &, = 22 , an embedmentof 30 m reduces 4ko,, to 1 5 . This result is basedon Eqn (8) which is as yet unsubstantiated . Inorder to substantiate statements about the degreeof soil strength mobilized in various soil zones itis necessary to consider the mobilization of soilstrain.

Fig. 6. Dilatant strain field, admissible for wall rotationabout toe

COMPATIBILITYBransby & Milligan (1975) introduced a kine-matically admissible soil strain field which wascompatible with the outwa rd rotation of aretaining wall. Figure 6 illustrates such a strainfield in the simplest case of a soil shearing with aconstant angle of dilatancy Ic, behind a rigid wallpivoting at its base, 0. A feature of these admis-sible strain fields is that the boundaries of thedeforming regions are either zero-extension linessuch as OZ in Fig. 6, or principal planes such asOV . As OV is the interface on the retaining wall,it is necessary to stipulate that the wall is friction-less. It was shown, however, that w all roughnesshad a negligible effect on the strain fields whichemerged in model experiments. It was also shownthat the increment in shear strain 6y due to a wallrotation 68 is given by

6y = 2 set 60 (9)This is insensitive to the dilatancy angle ;over the us ual range, 0 for loose soil to 25 fordense soil, set varies only between 1.0 and 1.1.Milligan & Bransby (1976) reported modeltests in which a rigid wall retaining dense sand

Fig. 5. Mobilized angle of shearing for frictionless walls, propped at crest, withfull-height groundwater


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BEH VIOUR OF DI PHR GM W LLS 173

YI: IlL(b)

.Major rmc~palstramdirect~on

Fig. 7. After Milligan Bransby (1976): (a) strain contours; (b) major principal strain direc-tions in dense sand, following 1 wall rotation about x

was rotated about an axis contained within itsown length. Figure 7(a) is their contour map ofshear strain at a wall rotation 0 = 1 = 1.74%.Definition beneath the pivot was poor at smallrotations, but the general shear strain of 4% in atriangular region above the pivot can be com-pared with F ig. 6 and the prediction 6y = 3.8%(Eqn 9). In Fig. 7(b) the directions of principalcompressive strain are seen to be vertical abovethe pivot and rotate towards the horizontal belowthe pivot, through a fan zone.James, Smith & Bransby (1972) computedforce/deformation solutions for the particularproblem of the translation of a wall into soil, bystepping through consecutive plastic stress andstrain field solutions portrayed using character-istics. This method could, in principle, be used tosolve for any kinematic boundary conditions, butproblems arise in compu tation where strong dis-continuities or singularities have to be insertedinto the net of characteristics.

The present objective is to enhance the stressfields shown in Figs 3 and 4 with equivalentadmissible strain fields. A simpler approach is tosubdivide the active and passive soil zones intotriangles, the verticals and horizontals of whichare frictionless displacement discontinuities while

the hypotenuse of each is a zero extension line.It is necessary to select an angle of dilation for theclay. Zero dilation satisfies the undrainedbehaviour of clay. When permitted to drain, over-consolidated clay will initially shear quasi-elastically, but at higher stress ratios it will dilateuntil it ruptures. Shear softening will then occuruntil sufficient soil has reached a critical state. Nofurther dilation will take place unless the changeof geometry forces new rupture bands to develop.Since dilation is significant only in determiningthe size of the shear zone rather than the magni-tude of strain within it, it was decided to imposethe simplification = 0 throughout.Figure 8(a) depicts the simplified admissiblestrain field compatible with a frictionless rigidwall rotating outwards by a small angle 60 aboutits base. In the absence of dilation, zero-extensionlines such a s OA are at 45 to the principal direc-tions, which must be vertical an d horizontal. Theuniform increment in horizontal strain 8~~ insidetriangle OVA can be calculated by the extensionh60 in AV

bE = - hdO/h = -60 10)taking compression positive. The vertical strainincrement &, can be calculated in this plane


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BOLTON ND POWRIE

Freedomto slide

I (b)Fig. 8. Adm issible strains for wall OV ro tating aboutbase: shear at constant volume

strain problem by invoking constancy of volume(zero rate of dilation) so thatBE+ 6&h= 0 (11)

8E = +%I (12)Inward rotation would simply cause the signs

of the vertical and horizontal strain increments toreverse. Figure 8(b) shows the correspondingMo hr d iagram of strain increments. It will beseen that the increment of shear strain

6y = 2 se (13)is in accordance with Eqn (9).Figure 9 shows a similar strain field compatiblewith rotation of a wall abou t its crest V . The fric-tionless discontinuities have been moved to theopposite sides of an imaginary square ABO Vdrawn in the soil. Hypotenuse OA remains a zeroextension line. That triangle AOV can remainrigid is demons trable by considering the move-ment (6u, &I) of a point P(x, y) on AO . Taking thestrain increments in ABO from Eqns (10) and (12)

6u= +x606v = -y 60

Fig. 9. Admissible strain field for wall rotating aboutcrest

Substitutingy=h-x

it is found that6U (h - Y)-= - = _-6V (h - 4 (14)

This is compatible with rigid body rotation of tri-angle AOV about V. The imposed rigidity ofAO V and the artificial step created in thedeformed soil surface at A, are compromises forthe purpose of characterizing the actua l mecha-nism. Once again, reversed passive rotationwould simply lead to reversed strains.Superposition of Figs 8 and 9 results in Fig. 10for uniform strain within the square ABO V dueto translation 6u of the wall. Equ al and oppositewall rotations 68 = 6u/h about crest and baserespectively give strain increments

SE,, = -6u/h (15)6~ = f /h (16)

and a shear strain increment6y = 2( /h) (17)

everywhere within the square.A V

---

Fig. 10. Admissible strain field for wall in translation


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Fig. 11. Admissible strain field for embedded wall rotating about crest

The result of these admissible strain fields isthat a wall rotation 60 can be seen to be consis-tent with an increment of soil shear strain6y = 260 in a well defined zone of deformationirrespective of whether that rotation is inward oroutwa rd or whether it is about the crest or thebase of the wall, and that translation of the wallcan be synthesized by superposition.

Superposition of unequ al comp onents of Figs 8& 9 results in a strain field compatible with arotation about any desired point alo ng the heightof the wall. Although the two straining trianglesdo not conflict, the magnitude s of strain withinthem differ. Howev er, if the objective is solely toextract a reasonable value for the ratio betweenmobilized shear strain and wall rotation, thelarger of the two strains should s&ice. Thiswould be

Sy = 2 6,Jh (18)where a,.,,,, is the larger of the deflections of thewall at the top and bottom of the adjacent soillayer. For exam ple, for the passive block in Fig.11 restraining a rigid cantilever wall from rota-tion abo ut its crest, E qn (18 ) gives

6y = 2(h + d/d)? (19)With a typical ratio d/h = 2 this gives 6y = 3 60.More complicated modes of wall deflexion canbe derived either by the superposition of existingstrain fields for the whole wall, or by the creationof a set of smaller compatible fields which su it aparticular case and which can be assembledwithout violating the compa tibility condition.For example, Fig. 12 shows an admissible strainfield for an embedded cantilever wall VW oflength L rotating by 68 about a point 0, distanceb from its base. Above 0, an active triangle AO V

and a passive triangle POQ are compatible witha frictionless wall element OV rotating about 0.Below 0, four triangles have been assembledwhich are compatible with a circulatory motionof the soil about the base W . For consistency,frictionless discontinuities have been insertedalong both OS and RT to provide displacementdiscontinuities on these principal planes. Thequadran ts undergo a cyclic variation of state(passive-active-passive-active) from OR W anti-clockwise to OTW . Again, the shear strainincrement at any point is characterized as beingeither zero or 2 68.

In comp aring the strain increments in Fig. 12with the stresses in Fig. 3, it can be seen that theimposed frictionless discontinuities coincide, andthe directions of major principal stress and majorprincipal strain increment also coincide for allzones of implied soil deformation in the vicinityof the wall. In all such zones a unique state ofmobilized strength could be deduced from a givenconstitutive relation. In zones remote from thewall, where stresses were unrealistically unsafe,the soil is taken to be unrealistically rigid so as tocompensate.

CONSTITUTIVE REL TIONSHIPThe strength and stiffness of a soil element isknow n to be a function of its effective stresshistory (particularly its maxim um previous effec-tive stress state), its present effective stress state,and its future state path (pa rticularly in relationto any intended reversal or rotation of the prin-ciple strain direction). Soil is known to be aniso-tropic after one-dimensional consolidation duringdeposition and burial, and to be relatively stiffafter each subsequent stress reversal. T hat thisbehaviou r creates problems for the back analysis(or design) of soil constructions is clear (Ward,


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176 BOLTON AND POWRIE

Fig. 12. Admissible strain field for unpropped wall

1961; Burland & Hancock, 1977; and Burland,Simpson & St John, 1979).It could be considered that the typical rangesof stress arou nd the walls in the present studycorrespond to &2 5 m of active soil and (X 15 m ofpassive soil. The range of minor effective stresswould be O-100 kN/m on the active side and&120 kN/m* on the passive side, accountingapproxim ately for pore pressures due to fullydeveloped seepage around the wall. The range ofeffective cell pressure for testing should thereforehave been of the order of cl20 kN/m *. At anyparticular stress level, two samples should havebeen tested in imitation of each of the active andpassive strain regimes. Ideally, a variety of statepaths wo uld also have been employed, includingundrained and drained tests and the increase andreduction of the mean effective stress as the devi-atoric stress w as increased in drained tests. Allstrain paths would approximate to plane strain.The principal advantag e in the production of soiltest data for the back analysis of laboratorymodels is that in every case the maxim um effec-tive vertical precompression of every soil elementwas known to be 1250 kN/m.Conventional practice would demand onlyroutine undrained triaxial compression tests withpore pressure m easureme nt carried out onsamples trimmed with vertical axe s. It wasdecided to use this class of data and to enhance itwith various exploratory tests (for example, at dif-ferent OC R, or drained rather than undrained, orin plane strain rather than triaxial strain). Stressreversal was investigated only in terms of hyster-esis loops derived from compression tests. The

plane strain tests were carried out using a simpleadapter fitted onto the base of a standardWy keham Farrance triaxial test cell, as shown inFig. 13.A dimensionless constitutive relation wouldafford the best opportunity for generalization.Accordingly, a graph of secant mobilised qYagainst shear strain y (both defined in terms of aplane in the soil containing major and minorprincipal axes) was selected, so that

4 = sin- [(or - e,)/(c, + a,)1 (20)y= ,-Ej (21)

Figure 14(a) shows curves of q%kobagainst yobtained from undrained triaxial tests on threesamples, with initial overconsolidation ratios

Side platenStrut

Top cap

Sample

Porousd~scswith skirts

Pedestal

Fig. 13. Plane strain adaptation for triaxial cel l


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BEHA V IOUR OF D IA PHRA GM W A LLS

24r177

0 I 1 I I I I I I0 2 4 6 6 1 12 14 16Shear st rai n:

24

I I I I I I I2 4 6 8 10 12 14 16

Shear str ain :@Fig. 14. Mobilized angle of shearing as a function of shear strain: (a)

undrained triaxial compression tests from various OCRs; (b) undrainedcompression tests in plane and triaxial strain

(based on cr) of 11.0, 5.9 and 3.2. The secantstiffness & ,Jy, at a mobilized angle of shearingof 18 varied by less than 30% . In contrast, Fig.14(b) indicates that at &,,, = 18 , the secantstiffness of similar samples tested in plane s trainand under triaxial conditions differed by a factorof three. Apparen tly, triaxial data errs on the safeside rather more tha n sho uld be tolerated.Figure 15(a ) shows the effective stress paths inq-p space followed in drained and undrained tri-axial tests on two samples with initially similarstress histories. The relationship between the cor-responding kob-y curves shown in Fig. 15(b ) isas expected, the undrained path leading to some-what larger strains as it traverses through statesof high stress ratio at a reduced mean effectivestress. Soil elements, in either the model or the

field, initially suffer undrained shear strains onexcavation, followed by the drained phase of porepressure equilibration which can lead to addi-tional shear strains. The lowest credible stiffnessshould be used in design. It was therefore con-sidered appropriate to use the &,,ob-y curveobtained from the undrained plane strain test(Fig. 14(b)) with a relatively low initial OC R of3.3, to calculate the soil displacements projectedby the proposed design methodology.

UNPROPPED WALLSIt is possible for a wall which is initially stableto collapse gradua lly as the excess pore watersuctions induced o n excavation dissipate an d


12/23

178 BOLTON ND POWRIE

120-

I OO-

60 -

P,kN/ m2(a)

' 1'I

OCR Pon CT,,1: 11:4

' kN/ m110

2: 11. 0 114

I I I I I I I I2 4 6 6 10 12 14 16

Shear str afn :WFig. 15. Comparison between drained and undrained triaxial tests: a)

stress paths; h) mohilized angle o f shearing as a function of shear strain

steady-state seepage is approach ed. This class ofbehavio ur was observed in two of the tests onunpropped model walls reported by Bolton &Powrie (1987 ): test DW C07 (10 m retained height,15 m penetration) and test DW C08 (10 m

retained height, 20 m penetration). In the case ofthe centrifuge model test, the pore w ater pressureswere measu red and the effective stress equilibriumanalysis shown in Fig. 3 can be applied at anystage to calculate the mobilized angle of soil fric-


13/23

BEHAVIOUR OF DIAPHRAGM WALLS 179

\ \ \Fig. 16. (a) Soil movements during excavation, test DWCO8; (b) calculated displacements during excavation, testDWCOS

tion. A corresponding shear strain can then bededuced and displacements would follow fromthe shear strain mecha nism shown in Fig. 12.These stress and strain fields are applicableonly to a smooth wall, and may overpredict thedisplacements of a rough wall. In this case anentirely empirical adjustme nt can be made, using

the modified earth pressure coefficients given byCaquot & Kerisel (1948) to derive 4kob, buttaking the relationship between w all rotation 60and characteristic shear strain 6y to remain unaf-fected, i.e. 6y = 2 60.In the strain field shown in Fig. 12, deforma-tion is supposed to take place at constant volume.

A first order correction for volumetric strainswould be to superimpose one-dimensional con-solidation or swelling effects due to the changes inpore water pressure. A more thorough treatmentwould involve the careful segregation of elasticand plastic strain increments, and the use of adilatancy relationship between plastic com-ponents of volumetric and shear strains. Th eadmissible strain fields would then have to bemodified to take dilation into account. This morerigorous approach was considered not to be justi-fied unless the model test data indicated that thesimpler analysis was in gross error.

An equilibrium calculation following F ig. 3


14/23

180 BOLTON AND POWRIEPore water pressures kN/m

100 00 100

Fig. 17. Pore water pressures around the wall, 12.3 years after excavation,test DWC22

based on the undeformed wall geometry and the The long-term deformations which occurredpore water pressures meas ured at the end of exca- during centrifuge test DW COS would probably bevation in test DW C 08, indicates that a mobilized unacceptab le in a prototype structure. The behav-angle of shearing of approxim ately 17.5 is iour of an unpropped wall of the same geo metryrequired with 6 = 4kob, o r a mobilized angle of (10 m retained height, 20 m depth of embedment)shearing of just over 23 if the effect of wall fric- but with a lower groundw ater level, was investi-tion is neglected (6 = 0). The corresponding shear gated in test DW C22. The pore water pressuresstrains from Fig. 14(b) are 1.1% and 3.5% respec- measured 12.3 years after excavation at prototypetively, and the depth to the pivot zP is approx- scale (TV= 1.3) are shown in Fig. 17. Under theseimately 18.8 m in both cases, leading to crest groundw ater conditions, the equilibrium calcu-deflexions of 158 mm and 5 04 mm. F igure 16 lation shown in Fig. 3 indicates a mobilized angleshows that the displacements measured from of shearing of 13.5 (6 = &J or 17.8 (6 = 0).photographs of the marke rs visible in Fig. 1 are The depth to the pivot zP is approximately 19 msimilar to those calculated using the strain field in both cases and the corresponding shear strainsshown in Fig. 12 with a characteristic shear strain of 0.6% (6 = & ,,,,,,) and 1.2% (6 = 0) yield calcu-of 1.1%. Displacements calculated using a char- lated tip deflexions of 86 mm and 172 mm respec-acteristic shear strain of 3.5% would be much tively. The observed deflexion was 126 mm atlarger than those measu red, which implies that in prototype scale, which lies within the calculatedthis case there was sufficient movem ent to mobi- range. These calculations are summarized inlize full friction 6 = +aob at the soil-wall interface. Table 1.

Table 1. Summary of equilibrium calculations for unpropped walls of 20 m embedmentTime factor: Input parame ters Calcula ted paramete rs Crest deflexion:

T, mmPore water pressures wf 4kb Y Predicted Measured

0 Measured, DWCOS 0 23.0 3.5% 5041 17.5 1.1% 158 1701.3 Measured, DWC22 0 17.8 1.2% 172(Fig. 17) 1 13.5 0.6% 86 126


15/23

BEH VIOUR OF DI PHR GM W LLS 181BMT 2

0 2 4 6 8 10 12Time Followng excavation: years

(a)

PPl

(b)Fig. 18. Transient response scaled-up), test DWC 16

WALLS PRO PPED AT THE CRESTThe model wall used in test DW C16 waspropped at the crest, and represented a retainedheight of 10 m, a depth of embedment of 15 mand a bending stiffness EZ of 1.2 x lo6 kNm /mat prototype scale. Under the prevailing ground -water cond itions measu red at the end of the test,the mobilized angle of shearing for a smooth wall(6 = 0) according to the calculation in Fig. 4 is21.3. If the wall is rough (6 = d), the mobilizedangle of shearing can be estimated from the earthpressure coefficients given by Caqu ot & Kerisel(194 8) and reduced to 15.8. Thus the wall wouldcertainly not be expected to have collapsed out-right.

The bending m oments measured during testDW C16 are shown as functions of time in Fig.18(a). The gradual increase in the bendingmom ents (and prop forces) is directly attributableto the readjustmen t of the pore water pressures totheir long-term equilibrium values, and in partic-ular to the dissipation of the excess pore w atersuctions induced in the soil on excavation (Fig.18(b)).The pore water pressures measured near themodel w all at two separate instants during thetest are recorded in Fig. 1 9(a) . The first instantwas shortly after excavation, and the second wasnear the end of the test after 12 .3 years wouldhave elapsed at prototype scale (T, = 1.3) and

steady seepage established. The idealized linearpore water pressure distributions shown in Fig.19(a) were used in the back analysis of the modeltest. In Fig. 19(b ), the pore water pressures mea-sured after 12 .3 years at prototype scale are com-pared with the values obtained from the idealizedsteady-state flow-net for seepage from a fullheight grou ndwa ter level behind the wall to awetted su rface within the excavation. It may benoted that the idealized gro undw ater conditionswere not replicated exactly in the model test, butwere closer to those w hich migh t be expected inpractice with a reduced groun dwater level on theexcavated side of the wall.

The bending moments measured in the modelwall at these two instants are shown (at prototypescale) in Fig. 20, together with the correspondingprop forces. Computed bending moment dia-grams are also shown. These were calculated onthe assump tion that the effective lateral earthpressure is propo rtional to the depth below thesoil surface, and that the wall is perfectly rough ,i.e. 6 = &,,,,,. For a wall propped at the crest, theadmissible strain field (Fig. 11) indicates that for agiven wall rotation 8, the maximum shear strainon the retained side is 28, and that on the exca-vated side is 20 (1 + h/d). In this case, h = 10 mand d = 15 m and the maximum shear strain onthe excavated side is S/3 greater than that o n theretained side. Accordingly, the mom ent equi-


16/23

182 BOLTON AND POWRIE200 100 0 0 100 200I I I I I I

0 Before excavation

/ ./ 1/ \. - Hydrostatic0 Shorilyatierexcavat~on-- - - ldeallzed\ A After 12.3 years--- ldealrzed.Jy/ \ -..//,/ / \ Y/ \,-QD\ 1, \. 0\d ,Y , \

/\ \.\/ \.I/. \ 1.

\.

,/

\\./ ,I \\ \/ /

N\

/ /,/ -.,

0 o/ \./ .jA0 , /. A a

I

i I I I Plezometrlc levels: m above excavation6-7 0 MeasuredI I I 1 8.6 7 -4- Flow-net

Fig. 19. (a) Pore water pressures measured around wall, test DWC16; (b) piezometric levels, test DWC lQ after 12.3years

librium calculation for the earth pressure coefli-cients K , (on the retained side) and K, (on theexcavated side) was repeated until a pair of valueswas found corresponding to mobilized angles ofshearing on each side of the wall which w ould beconsistent (Fig. 14(b )) with this difference in char-acteristic shear strain. The measured pore waterpressures were introduced into the calculation bymeans of the linear idealizations shown in Fig.19(a). The prop loa d was then obtained from thecondition of horizontal equilibrium, and thebending moments were calculated from the loadson the wall in the normal way.Figure 20 shows that the bending momentsmeasured just after excavation in test DW C16 areclose to those calculated using this method withK, = 0.57 and K, = 2.22 , which correspond to6 = 4 = 12.5 and y = 0.5% on the retained side,and 6 = 4 = 15.9 and y = 0.8% on the exca-vated side of the wall. After 12.3 years at proto-

type scale (T, = 1.3) the measured bendingmoments are close to those calculated usingK, = 0.55 (6 = 4 = 13.5, y = 0.55%) and K, =2.35 (6 = 4 = 16.8 , y = 0.95%). The measuredand computed prop loads are also in agreement.These calculations are summarized in Table 2.If the analysis had been simplified by makingthe assumption K, = l/K,, the calculatedbending moments would not have differed bymore than 4 %. In some cases, this assumptioncould usefully be made without introducing anysignificant error.

Figure 21 shows the soil displacements(measured from photographs) which occurredduring excavation. The horizontal movement ofthe retained soil near the top of the wall waslarger than would be expected for a wall proppedrigidly at its crest. This was due to a lack ofcontact between the wall and the props at thestart of the excavation process. While every care


17/23

BEHAVIOUR OF DIAPHRAGM WALLS 183Bendmg moment. kNm/m

1000 2000 3000

Prop loads kN/mCalculated Measured

Fig. 20. Bending moments and prop forces, test DWC16,at 12.3 years prototype scale

was taken to ensure that the props were correctlyplaced, no control could be exercised over relativemovem ents between the wall and the propsduring reconsolidation in the centrifuge. Dur ingexcavation, the top of the wall moved forwardsby just un der 0.6 mm, which corresponds toabou t 70 mm at prototype scale. The effect of thislack of fit is not tak en into accoun t by the admis-sible strain field shown in Fig. 11. Therefore, thedisplacements measu red in centrifuge testDW C14, on a wall of similar geometry but with abending stiffness a t prototype scale of 10 kNm /m which was rigidly propped at the crest,will be comp ared with those calculated using theequilibrium calculation and admissible strain fieldalready described.

According to the stress field analysis shown inFig. 4, the wall of test DW C14 would be in equi-librium with the pore water pressures measuredimm ediately after excavation and soil stressesgiven by earth pressure coefficients K, = 0.4 7and K, = 2.85 . These coefficients correspond to6 = 4 = 17.25 and y = 1.05% on the retainedside, and S = 4 = 20.1 and y = 1.72% on theexcavated side of the wall. If the effects of wallfriction are neglected, the equilibrium earth pres-sure coefficients become K, = 0.44 (4 = 23 ,y = 3.1%) and K, = 2.46(@ = 25 , y = 5%).

The measu red and calculated soil settlementsare compared in Fig. 22. As with the unproppedwall used in test DWC 08, the actual settlementsare closer to those calcu lated using the smallercharacteristic shear strains, on the assum ption6 = &nab. It is unlikely that long-term displace-ments of the magn itude observed in the centrifugetests would be acceptable in a prototype struc-ture, which would probably be designed to alower mobilized soil strength.

The methods used for the back-analy sis ofevents which have already occurred can be

--_------------ ro-. . . . . ---%~~\\I\\~ space Scale 15 mm.*. , -a ~\.\~\\\\I~ Dlsplacemenf scale 15mm H. . . . . , .\r*h,L\\Illr...,..,, b \ h * * \\[I. . . . . , ,.., ~~~~~~~------------* . . . \ . . . . . r\\\\\,L*/P C-I... . . . .. . . . (.. I.,. \.b\.,.+ I Pr,T>rrP.*.._+.r. . . . . . , _ , (I,,,,P.,* . . , . . . . . . * wmhII , ,. . s. _,, ?**\:,l;::::::::: .:::a.... . . . . . . . . ..I. . - . . . . , . \ . . . . . . III.Ia . . . ..I. I:,... , . 1 . . . . . . . . f . . . . 11. I.. . . . . . .LJ-,.*............ r* . . ..-. e . . . a*..*.,,,.,. _ . . . . ...* -.... . . . . . . . -.-*( . , . . . . . . . . . n - . * . . . . . . . , . . .

Fig. 21. Soil movements during excavation, test DWC16


18/23

Te2SmyoebumccuaooteDWC1memmpoathce

Tm

Inpame

Ccaepame

Pofoc

Mmmbn

fao

kNm

mmkNm

5 z

T

Pewe

6

Oh

Ransd

Eesd

Pece

Mue

Pece

Mue

peue

eL

Y

d&

Y

5

0

TDW

1

Y=5Y

15

05%

19

08%

4

44

2

2

(Fg1

K=1K

11

08%

14

06%

3

2

9

13

TDW

1

5

Y=5Y3

15

05%

18

09%

5

6

3

3

(Fg1

K=K

11

10%

12

07%

5

3


19/23


Me asurementsx LVDT. photographic(sub-surface)

PredictIons-.-6-0--o=q5mob

I Prop

.-.--_./.---

-5k-S-wt-r zj , . . . . . .

H 2 Ill spaceH 200 mm D6placement

Fig. 22. Measured and calculated settlements, DWC14

applied to design, but the approach would differslightly in two respects. First, depending on thepre-excavation soil stresses and the proposedmethod of construction, it may be appropriate toinvoke a different relationship between the anglesof soil friction on each side of the wall. Thiswould be based on in situ test data for K, andlaboratory test data for 4 as a function of shearstrain.Second, the pore water pressures must be pre-dicted, perhaps using the methods suggested pre-viously and in Appendix 2. Alternatively, ashort-term analysis might proceed in terms of amobilized shear strength c,,,,~ as described byBolton et al. (1987). The transient pore w ater suc-tions induced on excavation are a function of the

structural stiffness. If the soil is assume d to be iso-tropic and reversibly elastic, the transient porewater pressu res can be estimated from the condi-tion that p = constant. The long-term pore waterpressures are dominated by seepage, an d aretherefore a function of geometry and not of struc-tural stiffness; they can be estimated easilyenough from a flow-net.

The side walls of the M2 5 motorway tunnel atBell Com mon, south of Epping, were constructedin situ using the secant pile technique, and act asembedded cantilevers, propped near the crest bythe roof slab. The performance of one side wall iscurrently monitored by the Transport and RoadResearch Laboratory (TRR L) and the BuildingResearch Establishment (BRE).

Fig. 23. Ideal d geometry of the Bell Common wall, Hubbard et al. (1984)


20/23

186 BOLTON AND POWRIERetained sidepore water pressures: kN/m150 100 50I I I0 Measured

- ldeallzed

- Proo

ccavated sidewe water pressures kN/m

0

/Fig. 24. Measured and ideal d pore water pressures atBell Common, Tedd et al. (1984)

The design of the Bell Common tunnel wallwas described by Hubbard, Potts, Miller &Burland (1984), and the instrumentation by Tedd,Chard, Charles & Symons (1984) who alsoreported on the behaviou r of the wall between thestart of construction (April 1982) and May 19 83.The idealized excavation geometry is shown inFig. 23.

Analysis of the Bell Comm on wall is compli-cated by the presence of three distinct soil typeson the retained side. Given appropriate labor-atory test data, allow ance could be made for thedifferent behaviou r of the three soils. How ever, inthe following calculations any such difference hasbeen neglected.

The lateral earth pressures were calculated onthe assum ption that their variation was linearwith depth (Fig. 4) using the linear idealizationsto the Stage VI pore water pressures measu red byTedd et al. (1984) shown in Fig. 24. In the calcu-lation of the effective earth pressure coefficientsfrom the condition of mom ent equilibrium aboutthe prop, it was assum ed that K, = l/K,. Figure25 shows that the prop load an d the total lateralearth pressures thus calculated are close to thevalues measured by Tedd et al.The pattern of soil movem ents observed at BellCommon was similar to that observed during thecentrifuge test DW C16, the secondary excava-tions at Bell Common presumably having asimilar effect to the slight lack of fit between theprops and the wall in test DW Cld It would seemthat, given appropriate laboratory test data, the

Retained sidetotal lateral stress kN/m

300 200 100I I IL

20

c Prop load:measured 440 kN/mcalculated 502 kN/m

Excavated sidetotal lateral stress: kN/m*100 200 3000

00

0 Measured (Tedd eta/)- Calculated (mobilized angle of friction method)

Fig. 25. Measured and calculated total lateral stressesacting on the Bell Common wall, Tedd ef al. (1984)

Bell Common wall could be analysed quite fullyusing the procedures described. The effect of theconstruction sequence (including secondary exca-vations such as those made at Bell Common)could also be taken into consideration.

CONCLUSIONS AND SUMMARIZINGREMARKSKinem atically admissible strain fields werederived which idealize soil behaviou r in terms ofuniformly deforming triangles. A ctive and passivetriangles were defined which are free to slide onvertical and horizontal surfaces but which can beattached to rigid zones through zero extensionlines. In the absence of dilation, zero extensionlines are at 45 to the principle axes of strain. Anydeformation of a rigid retaining wall can beaccomm odated in this idealized fashion. Forexample, a field comprising six triangles was usedto represent the rotation of an embedded wallabout a point above its base. Discontinuities ofdisplacement were seen to have been invoked onthe same surfaces as those required in an elemen-tary approach to equilibrium invoking simpleactive and passive zones.


21/23

BEH VIOUR OF DI PHR GM W LLS 87The strain fields indicated that wall rotation 68

could m obilize a shear strain increment 6y = 2 60within the neighbou ring zones of deformation,whether the wall rotated abou t the top or thebase of the adjacent soil layer. An embedded can-tilever was also shown to possess an admissiblestrain field within which the shear strain alsoincreased everywhere by 2 60. Only in the case ofthe passive zone ben eath the level of the excava-tion of a wall propped at its crest did it appearlikely that the shear strain incremen t wou ld beslightly larger.

It has been demo nstrated that centrifuge modeltests can form the basis of research into soil-structure interaction. It is necessary to simplifythe complex of data by imposing upon it certainidealizations which represent the essential behav-iour patterns. Such idealizations can lead to theenrichment of engineering design calculations.

How ever, experimen tal techniques do need tobe refined. Problems of misfit of stiff soil againstrelatively stiff structures were found to limit theexercise of fine control over kinem atic boun daryconditions. Methods of grouting and excavationin-flight will have to be developed if more difficultproblems are to be researched.

A constant mob ilized strength approa ch wasused to back-an alyse a group of centrifugal modeltests on rigid walls, unprop ped or propped. For aparticular geom etry and groundwater condition,a pair of earth pressure coefficients could beinferred from a simple equilibrium analysis.Curv es of 4 against y were insensitive to smallchanges of overconso lidation ratio and initialeffective stress, but plane strain was found toevoke a stiffer response than triaxial strain.Anisotropy was not explicitly d ealt with.How ever, the assumed mobilized shear strain ywas measured and displacements were calculatedusing an approp riate idealized strain field. Wh encomp ared with centrifuge test data of soil andwall deformations, the concordance was found tobe suniciently accurate, in the sense that a designbased on this approach would have performed ator slightly within the specificied deform ationenvelope. In this respect the mobilized 4approa ch can be seen to be more useful, reliableand logical than the safety factor approach,without creating any serious difficulties for thedesigner.

The adoption of geostructural mechanisms inthis Paper has followed the spirit of engineersbeam theory. Beam theory neglects shear defor-mations in favour of a simple mech anistic treat-ment of bending in terms of plane sectionsremaining plane. Althoug h wrong , engineersbeam theory proves more useful to the designerthan more complete approach es, since it charac-terizes stress and strain in a consistent fashion

which is geom etrically simple. Experim ent provesit to be acceptably accurate for a particular classof beams which are slender.

Designers facing difficulties with soil-structureinteraction have suffered from a lack of such sim-plified treatments. Code formu lae have often beenbased on poorly digested inform ation peculiar toa given site which should not have been ap pliedin other contexts. Howev er, finite elementanalyses produce such a volum e of detailed pre-diction that patterns are dillicult to discern andassump tions (especially of the material stress-strain laws) difficult to evaluate.

Geostructural mechanisms which are based onlower boun d stress fields, but which incorp orateconsistent strain fields, are felt to be a suitabledesign tool. It has been demon strated that anydesired stress-strain relation could have beenused as the basis of prediction, though the tech-nique should lead to the adoption of real datafrom stress paths app ropriate to the problem .Such proposed mechanisms must, of course, betested. The centrifuge model technique appearssuited to this task.

CKNOWLEDGEMENTThe work reported was carried out under con-tract to the Transport and Road Research Labor-atory (TR RL). The views expressed are notnecessarily those of the Departm ent of Trans port.W. Powrie received suppor t from the Science andEngineering Research Council.

The Authors are grateful to Mr I. F. Symons(TR RL ) for his advice and constructive criticism.We also recognise the suppo rt by colleagues inthe Soil Mechanics Group at Cambridge, and inparticular the assistance of Mr C. H. Collison, MrJ. Doherty and Mr D . I. Stewart.

PPENDIX 1: EQUILIBRIUM C LCUL TION FORW LL PROPPED T THE CRESTFigure 26 shows the distributions of pore water pres-sure and horizontal effective stress assumed to be actingon each side of the wall. Taking moments about theprop yields

K, fin h + q2- 2+ % (h + d)?r,,(h + 4 - us, - U,,)1


22/23

188 BOLTON ND POWRIE

h

0: suctionu: pressure

Fig. 26. Pore pressure and effective stress distributions

If K, = K = l/K,, this equation reduces to a quadraticin K (the values of iir,, us , I&, uB2 y . h and d beingknown). A different relatlonship between K, and K,might be assumed (for example, to take account of theeffects of wall friction) in which case an iterative methodof solution would probably be necessary.

The total lateral stress is negative (tensile) on theretained side of the wall above a depth given by

z. = i&,(1 - KJh + d)[K ,(Y,,,(~ + d) - us, - u,,)l + Q,, + k, 23)

For a surface suction UT, = 20 kN/m*, z0 is likely to beof the order of 0.8 m, and if Ur, = 100 kN/m*, z,, =3 m. For the purpose of back-analysis, these figureswere not considered to be unreasonable and a no-tension cut-off was not applied.

APPENDIX 2: C LCUL TION OF PORE W TERPRESSURES IMMEDI TELY FTER EXC V TION

If the soil is assumed to behave as an isotropic andreversibly elastic material, the changes in pore waterpressure induced on excavation may also be calculated,on the basis that the average effective stress p remainsthe same. Before excavation, it is supposed that thelateral earth pressure coefficient is unity and that thepore water pressure at a depth z below the retained soilsurface is given by 1(1 = Y, z - &, where tr,, is the porewater suction at the surface. Then, initially at a depth z,P = (Y,,, - y,)z + ti,, After excavation, it is supposedthat the pore water pressure at a depth z is u2, the coef-ficient of lateral earth pressure in the direction perpen-dicular to the face of the wall is K , and that the effective

stress in the longitudinal direction is equal to the meaneffective stress u,(K + 1)/2.

Then, at a depth z on the retained side of a friction-less wall, the average effective stress is given by p = (K+ l)(y,,, z - Q/2 so that

2(Y,,, - Y,)u2 = Ysat - K+l2l-l2-2

K+l (24)

Similarly, at a depth z below the soil surface on theexcavated side of the wall

2(Ysa, Y,)u2 = Ysat - K + 1 1 [z+ uK + 1 - Oyla)(25)

where u,, is the pre-excavation pore water pressure atthe level of the excavation, and the depth of the excava-tion is 10 m. If the wall is perfectly rough, so that 6 =&,ob these expressions becomemzKnob cd 4kob _uz =

[Yra - ~ K (Ysat Y,)

1z - 7 u,,

(26)On the retained side of the wall, and

uz = [ Ysat -f +L+ (y,,, - y,)

12

cos2 kOb+ ~ (u,, - lOY,,JKon the excavated side, where K = o,,/(y _, z - u2)

REFERENCESAl-Tabbaa, A. (1987). Permeability and stress-strain

response of speswhite kaolin. PhD thesis, Cam-bridge University.


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BEHAVIOUR OF DIAPHRAGM WALLS 89Bolton, M. D. & Powrie, W. (1987). Collapse of dia-

phragm walls retaining clay. Geotechni que 37 No. 3335-353.

Bolton, M. D., Powrie, W. & Stewart, D. I. (1987).Effects on diaphragm walls of groundwater pressurerising in clavs. Proc. 9th Eur. Conf Soil Mech. FdnEngig 2,759-762.

Bransby, P. L. & Milligan, G. W. E. (1975). Soil defor-mations near cantilever sheet-pile walls. Geotech-nique 25, No. 2, 75-195.

Burland, J. B. & Hancock, R. J. R. (1977). Undergroundcar nark at the House of Commons-geotechnicalaspects. St ruct . Engr 55, 87-100.

Burland. J. B.. Potts. D. M.&Walsh. N. M. (1981). Theoverall stability of free and propped embedded can-tilever retaining walls. Ground Engng 14, No. 5,28-38.

Burland, J. B., Simpson, B. & St. John, H. D. (1979).Movements around excavations in London clay.Proc. 7t h Eur. Con/I Soi l Mech.. Bri aht on 1, 13-29.

Caquot, A. & Kerisel, J. (1948). Tabies for the calcu-lation of passive pressure, etc. Paris: GauthierVillars.

Hubbard, H. W., Potts, D. M., Miller, D. & Burland, J.B. (1984). Design of the retaining walls for the M25cut and cover tunnel at Bell Common. Gtotechnique34, No. 4,495512.

James, R. G., Smith, I. A. A. & Bransby, P. L. (1972).Prediction of stresses and deformations in a sand

mass adjacent to a retaining wall. Proc. 5th Eur.Confi Soil M ech., M adrid 1 3946.

Milligan, G. W. E. & Bransby, P. L. (1976). Combinedactive and passive rotational failure of a retainingwall in sand. Geotechni aue 26. No. 3. 473-494.

Potts, D. M. & Fourie, A.B. (1984). The behaviour of apropped retaining wall: results of a numerical inves-tigation Geotechni que 34, No. 3, 383404.

Powrie, W. (1985). Discussion on Fifth GeotechniqueSymposium in Print. The performance of proppedand cantilevered rigid walls. Geotechni que 35, No. 4,54&548.

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