supporting excavation in clay keynote bolton et al

8/2/2019 Supporting Excavation in Clay Keynote Bolton Et Al

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1 INTRODUCTION

The Mobilizable Strength Design (MSD) methodhas developed following various advances in the useof plastic deformation mechanisms to predict grounddisplacements: (Milligan and Bransby, 1975; Boltonand Powrie, 1988; Bolton et al. 1989, 1990a,1990b). MSD is a general, unified design methodol-ogy, which aims to satisfy both safety and service-ability requirements in a single calculation proce-dure, contrasting with conventional designmethodology which treats stability problems andserviceability problems separately. In the MSDmethod, actual stress-strain data is used to select a

design strength that limits ground deformations, andthis is used in plastic soil analyses that satisfy equi-librium conditions without the use of empiricalsafety factors.

Simple plastic mechanisms are used to representthe working state of the geotechnical system. Themechanisms represent both the equilibrium and de-formation of the various soil bodies, especially attheir junction with the superstructure. Then, rawstress-strain data from soil tests on undisturbed sam-ples, taken from representative locations, are useddirectly to link stresses and strains under working

conditions. Constitutive laws and soil parameters areunnecessary.

The MSD approach has been successfully im-plemented for shallow foundations (Osman and Bol-

ton, 2005), cantilever retaining walls (Osman andBolton, 2005), tunneling-induced ground displace-

ments (Osman et al, 2006) and also the sequentialconstruction of braced excavations which inducewall displacements and ground deformations (Os-man and Bolton, 2006).

Consider the imposition of certain actions on asoil body, due to construction activities such asstress relief accompanying excavation, or to loadsapplied in service. The MSD method permits theengineer to use simple hand calculations to estimatethe consequential ground displacements accountingfor non-linear soil behavior obtained from a singlewell-chosen test of the undisturbed soil.

The MSD approach firstly requires the engineerto represent the working states of the geotechnicalsystem by a generic mechanism which conveys thekinematics (i.e. the pattern of displacements) of thesoil due to the proposed actions. Analysis of the de-formation mechanism leads to a compatibility rela-tionship between the average strain mobilized in thesoil and the boundary displacements.

The average shear strength mobilized in the soildue to the imposed actions is then calculated, eitherfrom an independent equilibrium analysis using apermissible stress field (equivalent to a lower bound

plastic analysis), or from an equation balancingwork and energy for the chosen mechanism (equiva-lent to an upper bound plastic analysis).

Supporting excavations in clay - from analysis to decision-making

Bolton, M.D. and Lam, S.Y.University of Cambridge

Osman, A.S.Durham University

Keynote Lecture 6th International Symposium on Geotechnical Aspects of UndergroundConstruction in Soft Ground, IS-Shanghai, 2008, Vol. 1, 12-25.

ABSTRACT: Finite Element Analysis (FEA) is used to calibrate a decision-making tool based on an ex-tension of the Mobilized Strength Design (MSD) method which permits the designer an extremely simplemethod of predicting ground displacements during construction. This newly extended MSD approach ac-commodates a number of issues which are important in underground construction between in-situ walls, in-cluding: alternative base heave mechanisms suitable either for wide excavations in relatively shallow soft claystrata, or narrow excavations in relatively deep soft strata; the influence of support system stiffness in relationto the sequence of propping of the wall; and the capability of dealing with stratified ground. These develop-ments should make it possible for a design engineer to take informed decisions on the relationship betweenprop spacing and ground movements, or the influence of wall stiffness, or on the need for and influence of ajet-grouted base slab, for example, without having to conduct project-specific FEA.


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The location of one or more representative soilelements is then selected, basing this judgment onthe soil profile in relation to the location and shapeof the selected mechanism. The centroid of themechanism can serve as a default location if a singlelocation is to be employed. Stress-strain relation-ships are then obtained from appropriate laboratorytests on undisturbed soil samples taken from the se-

lected locations and carried out with precise strainmeasurements. Equivalent in-situ tests such as self-boring pressuremeter tests can alternatively be car-ried out. The mode of deformation in the soil testsshould correspond as closely as possible to the modeof shearing in the MSD mechanism. Otherwise, ani-sotropy should somehow be allowed for.

Finally, the mobilized shear strength requiredfor equilibrium under working loads is set againstthe representative shear stress-strain curve in orderto obtain the mobilized soil strain, and thereby the

boundary displacements of the simplified MSDmechanism.

2 MSD FOR DEEP EXCAVATION PROBLEM

Osman and Bolton (2006) showed for an in-situ wallsupporting a deep excavation that the total deforma-tion could be approximated as the sum of the canti-lever movement prior to propping, and the subse-quent bulging movement that accretes incrementallywith every sequence of propping and excavation.

A method for estimating the cantilever movementhad been suggested earlier in Osman and Bolton(2004). It begins by considering the lateral earthpressure distribution for a smooth, rigid, cantileverwall rotating about a point some way above its toe,in undrained conditions. A simple mobilizedstrength ratio is introduced to characterize the aver-age degree of mobilization of undrained shearstrength throughout the soil. By using horizontalforce and moment equilibrium equations, the twounknowns the position of the pivot point and themobilized strength ratio are obtained. Then, a mo-

bilized strain value is read off from the shear stress-strain curve of a soil element appropriate to the rep-resentative depth of the mechanism at the mid-depthof the wall. Simple kinematics for a cantilever wallrotating about its base suggests that the shear strainmobilized in the adjacent soil is double the angle ofwall rotation. Accordingly, for the initial cantileverphase, the wall rotation is estimated as one half ofthe shear strain required to induce the degree of mo-bilization of shear strength necessary to hold thewall in equilibrium. Osman and Bolton (2004) usedFEA to show that correction factors up to about 2.0

could be applied to the MSD estimates of the wallcrest displacement, depending on a variety of non-dimensional groups of parameters ignored in the

simple MSD theory, such as wall flexibility and ini-tial earth pressure coefficient prior to excavation.

A typical increment of bulging, on the otherhand, was calculated in Osman and Bolton (2006)by considering an admissible plastic mechanism forbase heave. In this case, the mobilized shear strengthwas deduced from the kinematically admissiblemechanism itself, using virtual work principles. The

energy dissipated by shearing was said to balancethe virtual loss of potential energy due to the simul-taneous formation of a subsidence trough on the re-tained soil surface and a matching volume of heaveinside the excavation. The mobilized strength ratiocould then be calculated, and the mobilized shearstrain read off from the stress-strain curve of a rep-resentative element, as before. The deformation isestimated using the relationship between the bound-ary displacements and the average mobilized shearstrain, in accordance with the original mechanism.

The MSD solutions of Osman and Bolton (2006)compared quite well with some numerical simula-tions using the realistic non-linear MIT-E3 model,and various case studies that provided field data.However, these initial solutions are capable of im-provement in three ways that will contribute to theirapplicability in engineering practice.

(i) The original mechanism assumed a relativelywide excavation, whereas cut-and-cover tunnel andsubway constructions are likely to be much deeperthan their width. The MSD mechanism thereforeneeds to be adapted for the case in which the plastic

deformation fields for the side walls interfere witheach other beneath the excavation.

(ii) The structural strain energy of the supportsystem can be incorporated. This could be signifi-cant when the soil is weak, and when measures aretaken to limit base heave in the excavation, such asby base grouting between the supporting walls. Inthis case, the reduction of lateral earth pressure dueto ground deformation may be relatively small, andit is principally the stiffness of the structural systemitself that limits external ground displacements.

(iii) Progressively incorporating elastic strain en-

ergy requires the calculation procedure to be fullyincremental, whereas Osman and Bolton (2006) hadbeen able to use total energy flows to calculate theresults of each stage of excavation separately. Afully incremental solution, admitting ground layer-ing, will permit the accumulation of different mobi-lized shear strengths, and shear strains, at differentdepths in the ground, thereby improving accuracy.

It is the aim of this paper to introduce an en-hanced MSD solution that includes these three fea-tures. This is then compared with existing FEA ofbraced excavations which featured a range of ge-

ometries and stiffnesses. It will be suggested thatMSD provides the ideal means of harvesting FEAsimulations for use in design and decision-making.


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3 PLASTIC FAILURE MECHANISMS

Limit equilibrium methods are routinely used in sta-bility calculations for soft clay which is idealised,unrealistically, as rigid-plastic. Slip surfaces are se-lected as the assumed focus of all plastic deforma-tions. Failure mechanisms should be kinematicallyadmissible, meaning that unwanted gaps and over-

laps should not be produced. Furthermore, in thecase of undrained shearing of clays, a constant-volume condition should be respected at every point.A consequence is that undrained plane-strain failuremechanisms must comprise only slip planes and slipcircles. The soil on such failure surfaces is taken tomobilize its undrained shear strength divided by asafety factor, to maintain the mechanism in limitingequilibrium under the action of gravity, and anyother applied loads. Calculated in this way, thesafety factor literally offers an estimate of the factor

by which the strength of the soil would have to dropbefore the soil construction would collapse. Such es-timates might err either on the high side or the lowside, depending on the particular assumptions thatwere made.

In the case of base heave in braced excavations,plastic solutions were derived from slip-line fieldsbased on the method of characteristics. Such solu-tions comprise both slip surfaces, as before, andplastic fans which distribute plastic strains over a fi-nite zone in the shape of a sector of a circle. Not-withstanding these zones of finite strain, the addi-

tional presence of slip surfaces still restricts theapplication of these solutions to the prediction offailure. Furthermore, no such solution can be re-garded automatically as an accurate predictor offailure, notwithstanding their apparent sophistica-tion. All that can be said is that they will lead to anunsafe estimate of stability. Their use in practice canonly be justified following back-analysis of actualfailures, whether in the field or the laboratory.

Two typical failure mechanisms as suggested byTerzaghi (1943) and Bjerrum and Eide (1956) areshown in Figure 1. They have each been widely used

for the design of multi-propped excavations. Ter-zaghi (1943) suggested a mechanism consisting of asoil column outside the excavation which creates abearing capacity failure. The failure is resisted bythe weight of a corresponding soil column inside theexcavation and also by adhesion acting along thevertical edges of the mechanism. Bjerrum and Eide(1956) assumed that the base of the excavation couldbe treated as a negatively loaded perfectly smoothfooting. The bearing capacity factors proposed bySkempton (1951) are used directly in the stabilitycalculations and are taken as stability numbers,

N = H/cu. Eide et al. (1972) modified this approachto account for the increase in basal stability owing tomobilized shear strength along the embedded lengthof the rigid wall.

Figure 1. Conventional basal stability mechanism and notation(after Ukritchon et al., 2003)

ORourke (1993) further modified the basal sta-bility calculations of Bjerrum and Eide (1956) to in-clude flexure of the wall below the excavation level.It was assumed that the embedded depth of the walldoes not change the geometry of the basal failure

mechanism. However, an increase in stability wasanticipated due to the elastic strain energy stored inflexure. This gave stability numbers that were func-tions of the yield moment and assumed boundaryconditions at the base of the wall.

Ukritchon et al. (2003) used numerical limitanalysis to calculate the stability of braced excava-tions. Upper and lower bound formulations are pre-sented based on Sloan and Kleeman (1995) andSloan (1988), respectively. The technique calculatesupper bound and lower bound estimates of collapseloads numerically, by linear programming, while

spatial discretization and interpolation of the fieldvariables are calculated using the finite elementmethod. No failure mechanism need be assumed andfailure both of the soil and the wall are taken care of.


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However, both soil and wall are again assumed to berigid perfectly plastic so the failure mechanism in-cludes a plastic hinge at the lowest level of support.

All these collapse limit analyses provide usefulguidance on the possible geometry of plastic defor-mation mechanisms for service conditions. But thekey requirement for MSD mechanisms is that dis-placement discontinuities (slip surfaces) must be

avoided entirely. In that way, small but finite grounddisplacements are associated at every internal pointwith small but finite strains.

4 WALL DEFORMATIONS

Consider now the deformations of a multi-proppedwall supporting a deep excavation in soft, undrainedclay. At each stage of excavation the incrementaldisplacement profile (Figure 2) of the ground and

the wall below the lowest prop can be assumed to bea cosine function (ORourke,1993) as follows:

= )2

cos(12

max

l

yww

(1)

Here w is the incremental wall displacement at anydistance y below the lowest support, wmax is itsmaximum value, and l is the wavelength of the de-formation, regarded as proportional to the lengths ofthe wall below the lowest level of current support:

l= s (2)

ORourke (1993) defined the wavelength of the de-formation as the distance from the lowest supportlevel to the fixed base of the wall. Osman and Bol-ton (2006) suggested a definition for the wavelengthof the deformation based on wall end fixity. Forwalls embedded into a stiff layer beneath the softclay, such that the wall tip is fully fixed in positionand direction, the wavelength was set equal to thewall length ( = 1). For short walls embedded indeep soft clay, the maximum wall displacement oc-curs at the tip of the wall so the wavelength wastaken as twice the projecting wall length (= 2). In-termediate cases might be described as restrained-end walls (1


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virtual change of gravitational potential energy ofthe soil. Afactor can then be found so that a corre-sponding mobilized shear strain can be read off fromthe chosen stress-strain curve. The incremental dis-placement can then be calculated by the correlationbetween the average shear strain increment and theincremental wall displacement.

This approach offered a straightforward way to

estimate the bulging displacement of the retainingwall. However, the approach requires refinement inorder to include some additional features that maybe significant in deep excavations. Firstly, the ap-proach did not consider the elastic strain energystored in the support system. Secondly, it is commonto find a non-uniform soil stratum with undrainedshear strength varying irregularly with depth. Fur-thermore, the geometry of the deformation mecha-nism changes as the construction proceeds, so therepresentation of mobilization of shear strength

through the whole depth, using a single mobilizationratio, is only a rough approximation. In reality therewill be differences in mobilization of shear strengthat different depths for calculating incremental soildisplacement. Lastly, the original mechanism ofOsman and Bolton (2006) shown in Figure 3(a) onlyapplied to wide excavations; narrow excavationscalled for the development of the alternative mecha-nism of Figure 3(b).

In view of these issues, a new fully incrementalcalculation method has been introduced, allowingfor the storage of elastic strain energy in the wall

and the support system, and respecting the possibleconstriction of the plastic deformations due to thenarrowness of an excavation.

5.1 Deformation pattern in different zones

From Figure 3, the soil is assumed to flow parallel tothe wall at the retained side above the level of thelowest support (zone ABDC) and the incrementaldisplacement at any distance x from the wall isgiven by the cosine function of Equation 1, replac-ing y by x.

By taking the origin as the top of the wall, the de-formation pattern of retained soil ABDC is given inrectangular coordinates as follows:

))2

cos(1(2

max

l

xdwdwy

= (3)

0=xdw (4)

In fan zone, CDE, by taking the apex of the fanzone as the origin

+

+= 22

22

max 2cos12 yx

xl

yxdwdwy (5)

t

H

h

s

B

L

Excavation

depth

Hard stratum

l

dwmax

maxdw

dwmax

maxdw

A B

C D

F

H

I

E

(a) Incremental displacement field for wide excavation

t

H

h

s

L

Hard stratum

l

dwmax

maxdw

dwmax

A B

C D

F

E

I

H

Excavation depth

l

B

0 1 2 3 4 5 6 7 8 9 10

-5

-4

-3

-2

-1

0

1

2

3

4

5

(b) Incremental displacement field for narrow excavation

Figure 3 Incremental displacement fields

F I

E H

l

wave

max2

=

l

wave

max2.2

=

l

wave

max2

=


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+

+=

22

22

max2

cos12 yx

y

l

yxdwdwx

(6)

For fan zone EFH in very wide excavations as in-dicated in Figure 3(a), by taking the junction of thewall and the current excavation level as the origin:

[ ]

+

++

= 22

22

max2

cos12 yx

x

l

yxhdw

dwy (7)

[ ]

+

++=

22

22

max2

cos12 yx

y

l

yxhdwdwx

(8)

For the triangular zone FHI in very wide excava-tions, again taking the junction of the excavation andthe wall as the origin:

+

== l

yxhdw

dwdw xy

)(2

22

cos14

2 max

(9)

For narrow excavations as shown in Figure 3(b),a rectangular zone EFHI of 2D shearing is now pro-posed. The origin is taken as the mid-point of FE,mid-wavelength in the excavation, at the wall.

++

=B

x

l

y

l

y

B

dwldwy

sin

2sin

2

4

max (10)

+= lx

B

ydw

dwx

cos

2

cos12

max (11)

In order to get more accurate solutions, it is sup-posed that the soil stratum is divided into n layers ofuniform thickness t (Figure 4). The average shearstrain d(m,n) is calculated forn layers in m excava-tion stages. The incremental engineering shear strainin each layer is calculated as follows:

+

= dVol

dVoly

w

x

w

x

w

y

w

nmd

yxyx 4

),(

2

(12)

In order to get a better idea of the deformationmechanism, the relationship between the maximumincremental wall displacement and the average shearstrain mobilized in each zone of deformation shouldbe obtained. On the active side of the excavation, thespatial scale is fixed by the wavelength of deforma-tion l, and all strain components are proportional todwmax/l. The average engineering shear strain incre-ment mobmobilized in the deformed soil can be cal-

culated from the spatial average of the shear strainincrements in the whole volume of the deformationzone. For a wide excavation i.e. Figure 3(a), the av-erage shear strain is equal to 2dwmax/l. For a narrow

excavation, the average shear strain (ave) of activezone ABCD and fan zone CDE is 2dwmax/l and2.23dwmax/l, respectively, while ave in the passivezone EFHI depends both on the wavelength lof thedeformation and the width B of the excavation. Therelationship between the normalized average shearstrain in EFHI and the excavation geometry isshown in Figure 5. The correlations are as follows:

61.02

38.0/2 max

+

=B

l

Bw

ave for 1B

l (13)

83.32

16.32

98.0/2

2

max

+

=B

l

B

l

Bw

ave for 1B

l (14)

l

A B

C D

F

E

H

I

1

1

1

1

DSS

DSS

PSAPSP

Layer 1

Layer 2

Layer 3

Layer 4

Layer n

Layer (n-1)

Figure 4 Mobilizable shear strength profile of an excavationstage in an layered soil

2l/B

0 2 4 6 8 10

mob

/2w

max

1.0

1.5

2.0

2.5

3.0

3.5

4.0

61.02

38.0/2 max

+

=B

l

Bw

ave

83.32

16.32

98.0/2

2

max

+

=B

l

B

l

Bw

ave

Figure 5 Correlation between normalized average shear strainand excavation geometry for a narrow excavation


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Apart from the first excavation stage, all subse-quent deformation mechanisms must partially over-lay the previous ones (Figure 6). Due to the non-linearity of soil it is important to calculate the accu-mulated mobilized shear strain in each particularlayer of soil in order to correctly deduce the mobi-lized shear strength of that layer. This is done by anarea average method described as follows:

),(

),1(),1(),(),(),(

nmA

nmAnmnmAnmdnm

+=

(15)

where ),( nm is the total shear strain of the nth

layerin the m

thexcavation stage, and ),( nmA is the area of

deformation in the nth

layer in the mth

excavationstage.

With the help of some suitable stress-strain rela-tion for the soil (discussed in the following section),the mobilized strength ratio(m,n) at excavationstage m for soil layer n can be found (Figure 7).

5.2 Shear strength mobilized in mechanism

In soft clay, the undrained shear strength generallyvaries with depth, and with orientation of shear di-rection. The strength matrix cmob(m,n) mobilized forexcavation stage m for layern can be expressed us-ing a matrix (m,n) on the appropriate undrainedshear strength profile. Regarding orientation, anisot-ropy of soft soil can be a significant factor for exca-vation stability. For example, Clough & Hansen(1981) show an empirical factor based on the obser-

vation that triaxial extension tests give roughly onehalf the undrained shear strength of triaxial com-pression, with simple shear roughly half way be-tween. Figure 4 shows the orientation of the majorprincipal stress direction within the various zones ofshearing in the assumed plastic mechanism for wideexcavations, and indicates with a code the soil testconfiguration that would correctly represent theundrained shear strength of at the specific orienta-tion. For locations marked DSS the assume direc-tions of shearing are either vertical or horizontal, sothe ideal test on a vertical core is a direct simple

shear test. In areas marked PSA and PSP, shearingtakes place at 45 degrees to the horizontal and thesezones are best represented by plain strain active andpassive tests, respectively. Since the undrained shearstrength of the direct simple shear test is roughly theaverage of that of PSA and PSP, the relative influ-ence of the PSA and PSP zones is roughly neutralwith respect to direct simple shear. As a result, thedesign method for braced excavation can best bebased on the undrained shear strength of a directsimple shear test. A similar decision was made byO'Rourke (1993).

A B

C D

F

E

H

I

C" D"

F"

E"

H"

I"

n th Layer

Deformation

mechanism of

Excavation stagem-1

Deformation

mechanism ofExcavation stagem

Enlarged strip of the nth layer

A(m-1,n)

A(m,n)

Figure 6 Overlapping of deformation field

Figure 7 Typical stress strain relationship of soft clay

The equilibrium of the unbalanced weight of soilinside the mechanism is achieved by mobilization ofshear strength. For each excavation stage, mobiliza-tion of shear strength of each layer is considered bythe following:

(n)cn)(m,n)(m,cumobu, = (16)

where cu,mob(m,n) is the mobilized undrained shearstrength for layer n in excavation stage m; cu(n) isthe undrained shear strength for layer n; and (m,n)

is the mobilized strength ratio for excavation stage mand soil layern.

d(3,n)d(1,n)

(1,n)

(3,n)


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5.3 Incremental energy balance

By conservation of energy, the total loss of potentialenergy of the soil (P) must balance the total dissi-pated energy due to plastic shearing of the soil (D)and the total stored elastic strain energy in bendingthe wall (U).

UDP += (17)

The potential energy loss on the active side of thewall and the potential energy gain of soil on the pas-sive side can be estimated easily. The net change ofgravitational potential energy (P) is given by thesum of the potential energy changes in each layer:

=

=

n

i volume

ysat dVolimdwimP1

),(),( (18)

where dwy (m,i) is the vertical component of dis-placement of soil in the i

thlayer for the m

thconstruc-

tion; sat (m,i) is the saturated unit weight of soil inthe ith layer for the mth construction.

Since there are no displacement discontinuities, thetotal plastic work done by shearing of soil is givenby summing the internal dissipation in each layer:

=

=

n

i Volume

u dVolimdimcimD1

),(),(),( (19)

where cu(m,i) is the undrained shear strength of soilin the i

thlayer for the m

thconstruction; d(m,i) is the

shear strain increment of soil in the ith

layer for the

mth construction; and the corresponding mobilizedstrength ratio is given by:

(m,i) =),(

),(,

imc

imc

u

mobu

The total elastic strain energy stored in the wall, U,can be evaluated by repeatedly updating the de-flected shape of the wall. It is necessary to do thissince U is a quadratic function of displacement:

=

s

x dx

dy

wdEIU

0

2

2

2

2

(20)

whereEandIare the elastic modulus and the secondmoment of area per unit length of wall, and s is thelength of the wall in bending.L can be the length ofwalls below the lowest prop.

By assuming the cosine waveform equation(Equation 1), the strain energy term can be shown tobe as follows:

+=4

)4

sin(

3

2

max

3

l

s

l

s

l

EIdwU

(21)

where l is the wavelength of deformation, dwmax isthe maximum deflection of the wall in each excava-tion increment.

5.4 Calculation procedure

The following calculation procedure is programmedin Matlab 2006b.

1. At each stage of excavation, a maximum de-formation wmax, which is bounded by an up-per and a lower bound, is assumed. The soilstratum is divided into n layers. The areas on

both the active side and the passive side ineach layer are calculated.2. For each layer, with the help of the numeri-

cal integration procedure in Matlab, the mo-bilized shear strain and the change in PE onboth active and passive sides in differentzones is calculated. (Equation 18) The totalmobilized shear strain is updated accordingEquation 15.

3. With the use of a suitable stress-strain curve(Figure 7), the mobilizable strength ratio can be found.

4. Total change in PE and total energy dissipa-tion and elastic bending energy in the wallcan be calculated by Equations 18, 19 & 21,respectively.

5. By considering the conservation of energy ofa structure in statical equilibrium, the sum oftotal energy dissipation and elastic strain en-ergy in the wall balances the total change inPE. To facilitate solving the solution, an er-ror term is introduced as follows:

Error = D + U- P (22)6. When the error is smaller than a specified

convergence limit, the assumed deformationis accepted as the solution; otherwise, themethod of bisection is employed to assumeanother maximum displacement and the errorterm is calculated again using steps 1 to 5.

7. Then, the incremental wall movement profileis plotted using the cosine function of equa-tion

8. The cumulative displacement profile is ob-tained by accumulating the incrementalmovement profiles.

6 VALIDATION BY NUMERICAL ANALYSIS

The finite element method can provide a frameworkfor performing numerical simulations to validate theextended MSD method in evaluating the perform-ance of braced excavations. However, finite elementanalysis of retaining walls is potentially problematic.One the most difficult problems is the constitutivemodel used for the soil. The stress-strain relationshipcan be very complicated when considering stress

history and anisotropy of soil (Whittle, 1993).The validation of the extended MSD method isexamined by comparing its predictions with resultsof comprehensive FE analyses of a plane strain


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braced excavation in Boston Blue Clay carried outby Jen (1998). In these analyses, the MIT-E3 consti-tutive model is used (Whittle, 1987). The model isbased on Modified Cam clay (Roscoe and Burland1968). However, several modifications had beenmade to improve the basic critical state framework.The model can simulate small strain non-linearity,soil anisotropy and the hysteretic behaviour associ-ated with reversal of load paths. Whittle (1993) alsodemonstrated the ability of the model to accuratelyrepresent the behaviour of different clays when sub-jected to a variety of loading paths.

Jen (1998) extended the use of the MIT-E3 modelfor analyzing cases of deep excavation in a great va-riety of situations. She considered the effect of ex-cavation geometry such as wall length, excavationwidth and depth of bed rock, the effect of soil profilesuch as cu/OCR ratio and layered soil, and the effectof structural stiffness such as wall stiffness and strut

stiffness. This provides a valuable database for vali-dation of the extended MSD method.

Retaining wall(EI=1440,77 and 19MNm /m)

OCR=1BBCproperties

C=37.5, 50 and 100m

L=20, 25, 30, 35 and 40m

B/2=15,20,25 & 30mLC

h=2.5m

s=2.5m

2

Figure 8 Scope of parametric study to examine excavationwidth effect

Figure 9. Stress-strain response for Ko consolidatedundrained DSS tests on Boston blue clay (After Whittle, 1993)

6.1 An example of MSD calculation

The following example shows the extended MSDcalculation of wall deflections for a 40m wall retain-ing 17.5m deep and 40m wide excavation (Figure 8).The construction sequence comprises the followingsteps:

1 The soil is excavated initially to an un-

supported depth (h) of 2.5m.2 The first support is installed at theground surface.

3 The second level of props is installed at avertical spacing of 2.5m, and 2.5m of soilis excavated.

The undrained shear strength of the soil is expressedby the relationship suggested by Hashash and Whit-

tle (1996) for Boston Blue Clay (BBC) as follows:

[ ]kPazcu 5.2419.821.0 += (23)

The cantilever mode of deformation and the bulg-ing movements are calculated separately using themechanism of Osman & Bolton (2006) and the ex-tended MSD method as described above. The totalwall movements are then obtained by adding thebulging movements to the cantilever movements tothe cantilever movement according to Clough et al.(1989).

Cantilever movementBy solving for horizontal force equilibrium and

moment equilibrium about the top of the wall, themobilized shear stress (cmob) is found to be11.43kPa. The mobilized strength ratio is 0.2886.With the help of direct simple shear stress-straindata for Boston blue clay by Whittle (1993) (Figure9), the mobilized strain is read off for an appropriatepreconsolidation pressure p and an appropriateOCR. The mobilized shear strain (mob) is found tobe 0.2%. By considering the geometrical relation-ship, the wall rotation is found to be 0.1%. The dis-placement at the top of the wall is found to be39mm.

Bulging movementThe first support is installed at the top of the wall.The length of the wall below the support is 40m. Byadopting the iterative calculation procedure, and us-ing the deformation mechanism for a narrow exca-vation, the bulging movement at each stage of exca-vation can be obtained. Then, the incrementalbulging movement profile in each stage is plottedusing the cosine function, using the maximum in-cremental displacement in each stage together withthe corresponding wavelengths. The total wall

movement is obtained by accumulating cantilevermovement and the total bulging movement. Figure10 shows the final deformation profile of the accu-mulated wall movement of an excavation with a


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width of 40m. The maximum wall deflection at anexcavation depth of 17.5m is 115mm. The positionof the maximum wall displacement is located at0.75L, where L is the length of the wall.

Wall deflection, w, (m)

0.00 0.04 0.08 0.12 0.16

Depth(m)

0

5

10

15

20

25

30

35

40

45

50

H=2.5

H=5.0

H=7.5H=10.0

H=12.5

H=15.0

H=17.5

Figure 10 Wall deflections from MSD with different excava-tion depths

6.2 Effect of excavation width

The effect of excavation width on predicted groundmovements is the focus of this section. Undergroundtransportation systems may have excavation widthsranging from 25m (a subway station) to 60m (an un-derground highway). The most widely used designcharts generally incorporate the effect of excavationwidth in estimation of factor of safety against baseheave (Bjerrum and Eide, 1956) or as a multiplica-tion factor in estimating the maximum settlement(Mana and Clough, 1981).

The scope of the excavation analyses are shownin Figure 8. In the analyses, the excavation was car-ried out in undrained conditions in a deposit of nor-mally consolidated Boston Blue Clay with depth C

taken to be 100m. A concrete diaphragm wall ofdepth L = 40m, and thickness 0.9m, supported byrigid props spaced at h = 2.5m, was used for sup-porting the simulated excavation. The excavationwidth varies from 20m to 60m. The wavelength ofdeformation is chosen according to the sl = rule,where was taken to be 1.5 and s is the length ofwall below the lowest prop. Computed results by Jen(1998) are used for comparison. Full details of theanalysis procedures, assumptions and parameters aregiven in Jen (1998). In the following section, onlyresults of wall deformation will be taken for com-

parison.Figure 11 (a) and (b) show the wall deflection

profile with different excavation widths at an exca-vation depth of 17.5m, as calculated by the extended

MSD method and the MIT-E3 model. Figure 11 (a)shows that the excavation width does not have anyeffect on the deflected shape of the wall as calcu-lated by the extended MSD method. Figure 11 (b),simularly, shows a limited effect on the deflectedshape of the wall by the MIT-E3 model. While theMSD-predicted maximum wall deflection increasesby a factor of 1.5 as the width is increased from 30m

to 60m, the MIT-E3 computed maximum wall de-formation increased by a factor of 1.6 with the sameincrease in excavation width.


0.00 0.04 0.08 0.12 0.16

Depth(m)

0

10

20

30

40

50

B=60

B=50

B=40

B=30

(a) Prediction by Extended MSD


0.00 0.04 0.08 0.12 0.16

Depth(m)

0

10

20

30

40

50

B=60

B=50

B=40

B=30

(b) Prediction by MIT E3 model (After Jen,1998)

Figure 11 Wall deflection profile of different excavationwidths at H = 17.5m


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6.3 Effect of bending stiffness of the wall

In general, structural support to excavations is pro-vided by a wall and bracing system. Soldier pilesand lagging, sheet piles, soil mix and soldier piles,drilled piers (secant piles), and reinforced concretediaphragm walls are examples of wall types thathave been used to support excavations. The various

types of wall exhibit a significant range of bendingstiffness and allowable moment. Support walls com-posed of soldier piles and sheet piles are generallymore flexible and capable of sustaining smallerloads than the more rigid drilled piers and reinforceddiaphragm walls.

Wall Deflection, w, (m)

0.000.040.080.120.160.200.24

Depth(m)

0

10

20

30

40

50

0.9m Diaphragm wall (EI=1442.0 MNm2/m)

PZ-38 (EI=76.7 MNm2/m)

PWZ-27 (EI=19.1MNm2/m)

H=12.5m


(b) Prediction by MIT E3 model (After Jen,1998)Figure 12 Deflection profiles of walls with various bendingstiffnesses

The preceding sections have all assumed a 0.9mthick concrete diaphragm wall with elastic bending

stiffness EI = 1440MNm2/m. Although it is possibleto increase this bending stiffness by increasing thewall thickness and reinforcement, or by using T-panels (barettes), most of the walls used in practice

have lower bending stiffnesses. For example, thetypical bending stiffness of sheet pile walls is in therange of 50 to 80 MNm

2/m. This section assesses the

effect of wall bending stiffness on the excavation-induced displacements.

Excavation in soft clay with a width of 40m sup-ported by a wall of length 25m and of various bend-ing stiffness (EI = 1440, 70 and 20 MNm2/m) are

studied. Results generated by the MSD method andFEA are compared. Figure 12 (a) and (b) presentsthe deflection profiles of the excavations predictedby extended MSD and the MIT-E3 model, respec-tively. As the bending stiffness of the wall de-creases, there is no pronounced change in the overallshape of the wall; the maximum wall deflection in-creases and its location migrates towards the exca-vated grade. At H = 12.5m, the maximum wall dis-placement is 47mm for the concrete diaphragm wallwith the maximum deflection located at 7.5 m below

the excavation level, while the result for the mostflexible sheet pile wall shows 197mm of maximumwall deflection occurring at 5.5m below the excava-tion level. In additional to this, changes in wall stiff-ness also affect the transition from a sub-grade bend-ing mode to a toe kicking-out mode. As the wallstiffness decreases, the influence of embedmentdepth reduces, and hence the tendency for toe kick-out to occur is less. Again, a fairly good agreementcan be seen when comparing extended MSD resultsand numerical results by the MIT-E3 model, thoughkinks are usually found at the wall toe in the nu-

merical predictions, which implies localization oflarge shear strains developed beneath the wall toe.

6.4 Effect of wall length

Wall length is one of the geometrical factors affect-ing the behaviour of a supported excavation. Previ-ous analyses were done by Osman and Bolton(2006). The studies showed that the wall end condi-tion should be assumed to be free for short walls(L = 12.5m) since the clay is very soft at the baseand the embedded length is not long enough to re-

strain the movement at the tip of the wall (kick-outmode of deformation). For long walls (L = 40m), theembedded depth was assumed to be sufficient to re-strain the movement at the wall base (bulging modelof deformation). However, the effect of structuralstiffness was not considered in the old MSD method,though similar observations were made by Hashashand Whittle (1996) in their numerical analyses.

In this section, the effect of wall length will be

considered. Excavations with widths of 40m sup-ported by a 0.9m thick concrete diaphragm wall with

varying length (L = 20, 25, 30 and 40m) are studied.

Figure 13 shows the wall displacement profilesagainst depth with different lengths of wall. For

H 7.5m, the deflected wall shapes are virtually


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identical for all four wall cases of wall length. Thisagrees with the conclusion made by Hashash (1992)

that wall length had a minimal effect on pre-failure

deformations. As H increases to 10m, the toe of the20m long wall begins to kick out with maximum in-

cremental deformations occurring at the toe of the

wall. The movements of the 25, 30 and 40 m long

walls are quite similar. At H = 12.5m, the toe of the20m and 25m long wall kick out, while the two

longer walls (L = 30 and 40m) continue to deform ina bulging mode. The difference in deformation mode

shape demonstrates that the wall length has a sig-

nificant influence on the failure mechanism for abraced excavation.

Wall Deflection, w, (m)

0.000.050.100.150.200.25

Depth(m)

0

10

20

30

40

2.55.0

7.5

10.0

12.5

40

3025

20

12.5

20

H

L

Figure 13 Wall deflection profile of excavation with differentsupport wall lengths, by Extended MSD method

B / L

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8

wmax

/H

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

EI = 1.68 x 103

MN m/m2

EI = 1.68 x 102

MN m/m2

EI = 1.23 x 10 MN m/m2

Figure 14 Variation of maximum wall deflection with widthto length ratio of wall

Figure 15 shows a similar set of analyses by us-ing the MIT-E3 model. Similar observations aboutthe wall shape can be made.

Figure 14 summarizes the variation of the nor-malized excavation-induced deflection (wmax/H) withthe width to length ratio (B/L) for different bendingstiffnesses of the support wall, for H=17.5m.

Figure 15 Wall deflection profile of excavation with differentsupport wall lengths, by MIT-E3 method (After Jen (1998))

For a flexible wall (EI = 12.3 MN m2/m), the nor-

malized maximum wall deflection increases linearlyas the B/L ratio increases from 1 to 2. The gradientchanges and wmax/H increases in a gentle fashion as

the B/L ratio increases from 2 to 2.8. For a rigidlysupported wall, the increase in wmax/H ratio is lesssignificant as the B/L ratio increases. In other words,the maximum wall deflection is less sensitive to achange of B/L ratio for a rigid wall.

6.5 Effect of the depth of bearing stratum

The depth to bedrock, C, is an important componentof the excavation geometry. The preceding analyseshave assumed a deep clay layer with bedrock locatedat C = 100m which represents a practical upper limiton C. In practice, however, the clay layer is usuallyless than 100m deep. The following results focus onthe discussion of the geometrical parameter C. Theanalysis involves plane strain excavation in normallyconsolidated Boston blue clay supported by a 0.9mthick concrete diaphragm wall with rigid strut sup-ports spaced at 2.5m.

The wall deflection profiles for excavations pre-dicted by both MSD and MIT-E3 with two depths ofthe clay stratum (C = 35m and 50m) are compared inFigure 16(a) and (b).

In general, the depth of the firm stratum wouldonly affect wall deformations below the excavatedgrade, hence the largest effects can be seen at the toeof the wall. For situations where the wavelength of


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deformation is restricted less by excavation widththan by the depth of the firm stratum (B > C), themagnitude of maximum wall deflection increaseswith the depth of the firm stratum (C). The MSDmethod predicts that the kick-out displacement ofthe wall toe is limited by the restriction of develop-ing a large deformation mechanism. As a result, themaximum wall deflection is also limited. The in-

crease in incremental wall deflection decreases inlater stages of excavation when H increases from12.5m to 17.5m due to the reduction of wavelengthof deformation. On the other hand, when the depthof the firm stratum is much larger than the width ofthe excavation (B < C), the depth of the bed rock hasa minimal effect on the magnitude of wall deflec-tion. Results by FEA by Jen (1998) (Figure 16(b))also showed the same observation. Despite theshortcoming of MSD not being able to model thecorrect shape of wall, the maximum wall deflection

is predicted reasonably well. The net difference inmaximum wall displacement between MSD and thefull FEA is generally less than 20%.

7 CONCLUDING REMARKS

An extended MSD method is introduced to calculatethe maximum wall displacement profile of a multi-propped wall retaining an excavation in soft clay. Aswith the earlier MSD approach, each increment ofwall bulging generated by excavation of soil beneath

the current lowest level of support is approximatedby a cosine function. The soil is divided into layersin each of which the average shear strain incrementsare compounded so that the mobilized strength ratioin each layer can be tracked separately as excavationproceeds, using stress-strain data from a representa-tive element test matched to the soil properties atmid-depth of the wall. The incremental loss in po-tential energy associated with the formation of a set-tlement trough, due to wall deformation and baseheave, can be expressed as a function of thoseground movements at any stage. By conservation of

energy, this must always balance the incrementaldissipation of energy through shearing and the in-cremental storage of elastic energy in bending thewall. By an iterative procedure, the developing pro-file of wall displacements can be found.

A reasonable agreement is found between predic-tions made using this extended MSD method and theFEA results of Jen (1998) who created full numeri-cal solutions using the MIT-E3 soil model. In par-ticular, the effects of excavation width, wall bendingstiffness, wall length, and the depth of the clay stra-tum, were all quite closely reproduced.

It is important to draw the right lessons from this.The excellent work at MIT over many years, on soilelement testing, soil constitutive models, and FiniteElement Analysis, have provided us with the means

to calibrate a very simple MSD prediction method.This was based on an undrained strength profile, asingle stress-strain test, and a plastic deformationmechanism. Were it not for the multiple level ofprops the calculation of ground displacements couldbe carried out in hardly more time than is currentlyrequired to calculate a stability number or factor ofsafety. Allowing for the need to represent various

levels of props, the calculations then call for a Mat-lab script or a spreadsheet, and the whole processmight take half a day to complete.

An engineer can therefore anticipate that impor-tant questions will be capable of approximate butreasonably robust answers in a sensible industrialtimescale. For example: Will a prop spacing of 3m be sufficient for a

wall of limited stiffness and strength? Should the base of the wall be fixed by jet-

grouting prior to excavation?

Will a particular construction sequence causethe soil to strain so much that it indulges inpost-peak softening?

Is it feasible to prop the wall at sufficientlyclose spacings to restrict strains in the retainedground to values that will prevent damage toburied services?

This may lead engineers to take soil stiffness moreseriously, and to request accurate stress-strain data.If so, in a decade perhaps, our Codes of Practicemight be updated to note that MSD for deep excava-tions provides a practical way of checking for the

avoidance of serviceability limit states.

Wall deflection, w,(m)

-0.050.000.050.100.150.20

Depth(m)

0

10

20

30

40

50

H=17.5

H=15.0

H=12.5H=10.0

H=7.5H=5.0

H=2.5

C = 35.0 m

C = 50.0 m



14/14

(b) Prediction by MIT E3 model (After Jen,1998)

Figure 16 Wall deflection profiles of excavation with differ-ent depths to the firm stratum

ACKNOWLEDGEMENT

The authors would like to acknowledge the ear-marked research grant # 618006 provided by the Re-search Grants Council of the HKSAR Government,and also the Platform Grant (GR/T18660/01)awarded by the UK Engineering and Physical Sci-ences Research Council.

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