A Qexibility number for thedisplacement controlled design ofmulti propped retaining wallsby TI Addenbrooke, Imperial College, Universityof London. Cooling Prize winner 1994.

IntroductionTo support excavations in urban areas where constructionspace is limited and excavation depths large (deep buildingsubstructures, or cut and cover transportation tunnels) it iscommon practice to use vertical retaining walls supportedinternally by multi level propping. Construction is usuallycarried out in close proximity to existing roads, buildings andservices.

The analysis of such an excavation should therefore examine:(1) The possibility of collapse of the retaining structure, eitheras a result of excessive prop loads or failure of the soil at thebottom of the excavation.(2) The predicted movements caused by excavation anddewatering, particularly with regard to their effects on nearbystructures (Clayton & Milititsky, 1986).(3) The displaced shape of the wall, as large bending momentsmay be induced (Potts & Fourie, 1985).

This paper considers points (2) and (3), movements andinduced bending moments. It is rare for a multi proppedexcavation to fail due to structural problems. More commonly'failure'f a support system is because of unacceptablemovements (Clough et al, 1989).Buildings sited adjacent todeep cuts are generally less tolerant of the subsequentexcavation —induced differential settlements than similarstructures settling under their own weight (Boscardin &Cording, 1989).

Potts & Day (1990) state that both experimental work (egRowe, 1952) and more recent numerical work (eg Potts &Fourie, 1985) indicate that under the same operatingconditions stiffer walls attract larger bending moments thanmore flexible walls. The stresses imposed by the soil are &ee toredistribute through a more flexible structure thus reducing thestructural forces imposed on the wall. This in itself is beneficial,but occurs at the expense of larger wall and soil movements.They observe that there is therefore a compromise betweenreduced bending moments and increased movements as theflexibility of the wall increases.

If greater movements cannot be tolerated, then more propsare required if a more flexible wall is to be employed. Theengineer needs a framework within which bending momentreduction can still be considered, balanced against the requiredincrease in the number of propping levels. This paperintroduces a new flexibility number for multi propped retainingwall design, the 'displacement flexibility'. The displacementflexibility number permits the engineer to confidently considercost and construction variations within a displacementcontrolled design &amework. Its applicability is justified usingfinite element predictions of movements for various specific,but representative support systems.

Two sets of analyses are presented. Set A models differentretaining wall systems supporting an excavation in dry soil,while Set B models systems supporting a rapid excavationfollowed by consolidation in a water bearing soil.

ReviewPotts & Fourie (1985) presented the results of numericalpredictions for a propped retaining wall. It was demonstratedthat by increasing the flexibility of a wall in stiff clay, theinduced bending moment can be reduced significantly. It maybe possible therefore to provide a range of design solutions&om a stiff to a flexible wall, noting that while a more flexiblewall may be cheaper, it will allow greater soil and wallmovements.

Potts & Day (1990)went on to investigate the complexinteraction between bending moment, soil and wallmovements, and cost, as the flexibility of the wall changes. Byconsidering three cases (Bell Common tunnel, George Greentunnel, and the House of Commons car park) they showed thata five fold reduction in maximum wall bending moment couldbe achieved by using a sheet pile wall in place of a lm concretesection wall. It was concluded that if the increased movementsassociated with more flexible walls can be tolerated, orreduced by extra propping, such walls provide a viablealternative.

Flexibility nuxnbersThere are a number of existing ways of representing a retainingwall's stiflness —material flexibility, Rowe's flexibility number(Rowe, 1952) for single prop retaining walls, and systemflexibility (Clough et al, 1989) for multi propped retainingwalls.~Material flexibility.The flexibility of the material &om which a wall is composed iscommonly defined by:

ln (EI)The natural logarithm of EI, the Young's modulus, multipliedby the second moment of area of its section.~Wall flexibility.The flexibility of a wall can be defined by the logarithm ofRowe's flexibility number, log p:

log ( —)

where H is the wall height. Appendix A shows the theoreticalderivation of the flexibility number. It was devised by Rowe(1952) to define a single propped model wall whichdeformed to the same deflected shape as the prototype itrepresented.~System flexibility.Clough et al (1989) introduced the idea of systemflexibility, p,:

(EI )y„h4

where h is the average vertical prop spacing of a multi proppedsupport system and y„ is the bulk unit weight of water. Thiswas related to wall displacements, and hence groundmovements for a given factor of safety against basal heave and agiven support system.~Displacement flexibility.A logical extension to Rowe's flexibility number is to derive anumber, the constancy of which defines support systems withthe same absolute displacement (refer to Appendix B forderivation). The 'displacement flexibility', A, of a supportsystem is defined thus:

A=(- )h'I

GROUND ENGINEERING ~ SEPTEMBER ~ 1994

Analysis of a multi propped excavationin clayThe finite element program ICFEP (Imperial College finiteelement program) was used to perform different analyses of a20m and 23m wall in a clay material with a large factor ofsafety against base heave. This is representative of typical deepexcavations in London.~Details of analysesThe retaining wall modelled was 20m in height, supporting a40m wide excavation of depth 16m. One analysis was carriedout with a 23m wall, an increase in embedment depth of 75%.

Various elastic wall types were analyzed and are listed inTable 1. Five different rigid prop spacings (h) were employed,corresponding to the use of 2, 3, 4, 5, and 10 props. The propswere evenly spaced over the 16m excavation depth with the firstprop at the original ground surface on each occasion, and noprop at the final reduced level of excavation.

The soil was assumed to be linearly elastic —perfectly plasticwith a Mohr-Coulomb yield surface. A non associatedcondition in which the plastic potential is defined by an angleof dilation, v, was adopted. The soil properties were as follows:

xmvIov

+KOav

oem ~

r/Lvv

zom

——L

CA

toom

CAvv //IXP

Strength parameters

Angle of dilationPoisson's ratioYoung's modulus

Bulk unit weightPermeabilityInitial efFective stress ratio

c' 025'

=12.5',=02

E' 6000z kN/m'herez is the depth below the ground surface.

7 = 20kN/m'= 1.0 x 10-"m/s

Ko = 1.0

MOP AT ONOUND EUNPACE EXCAVATE TO NEXT MOPPINO LEVEL

Table 1:Support systems.Description Prep E In (EO(Potts & Day, 1990) spso919 (GN/msi

1m Concrete 8m 28 14.70

Diaphragm

4m

High Mod. 1BXN sheet 5.33m

Frodingham 4N Sheet 8m

5.33m

4m

Frodingham 1N sheet 3.2m

'Soft'all 1.6m

28 14.70

28 14.70

210 'l2;85

210 11,29,

210 11.29

210 11,29

210 10.19210 6.65

iog p

-1.18 60.27

-1.18 305.66-1.18 964.31-0.29 39,38

0.30 1.990.30 10.11

0.30 31.860.78 25.89

2.32 12.02

log a Rsl'e

-1.87 1

-2.75 2

-3.37 3-1.86 4-0.39 5-1.27 6-1.89 7

-1.89 8-1.87 9

~Finite element simulationOne of the three finite element meshes used for this study isshown in Figure 1. Symmetry has been assumed, and so themesh represents only one half of the excavation, with A-B beingthe centre line. The soil was modelled using eight nodedisoparametric plane strain quadrilateral elements. The wall wasmodelled using three noded isoparametric plane strain beamelements.

Construction was simulated by removal of the elements infront of the wall row by row to a depth of -16m. Threedifferent meshes were required to ensure that all the desiredprop positions matched with the top of a row of elements to be

42 excavated. The initial displacement boundary conditions

CONTINUE TO ~INAL NEDUCED LEVEL

Figure 1 (TOP): Typical 300 element mesh.Figure 2 (BOTTOM): Example construction sequence.

applied were the same for all three meshes and are shown onFigure .1.No displacements were permitted along the bottomboundary, and only vertical displacements were permittedalong the two side boundaries. The boundaries were remote tothe excavation.

The following stages of construction and unloading weremodelled (see Figure 2):ANALYSES SET A:The soil was dry. Systems 1-9 (Table 1) were modelled.(1) Wall installation.A wished in place wall was adopted. The assumption thereforewas that the initial state of stress in the ground and the soilstrength were unaffected by wall installation.(2) Supported excavation.A prop was simulated at the ground surface by the specificationof zero horizontal displacement of the beam element node atthat point as excavation proceeded. When the next proppingposition coincided with the reduced level of excavation a propwas again simulated by the prevention of horizontaldisplacement at that point on further excavation. This processcontinued until the reduced level reached -16m.ANALYSES SET B:A water table at 2m below the ground surface was modelled,

GROUND ENGINEERING SEPTEMBER ~ 1994

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Figure 3a (TOP): Maximum lateral wall displacementand horizontal ground surface movement (as a % ofexcavation depth) against ln (EI) and (EVE+4) for Set Aon completion of excavation.FIGURE 3b (BOTTOM) 8 Maximum lateral walldisplacement (as a % of excavation depth) against log A.

Figure 4 (TOP): Maximum horizontal groundsurface movement (as a % of excavation depth) againstlog A.Figure 5 (BOTTOM): Maximum vertical groundsurface movement (as a % of excavation depth) againstlog A.

with an initial hydrostatic pore water pressure distribution. Theboundary representing the line of symmetry was a no flowboundary, the other boundaries were fully drained. Systems 1,3, 4, 5, and 7 were modelled.(1) Wall installation.As for Set A.(2) Supported excavation.As for Set A. The excavation was carried out over a simulatedperiod of four months with the excavated surface having a porewater pressure of zero at all times.(3) Consolidation.The pore pressures were permitted to equilibrate over a periodof 90 years. At the end of this period steady state conditionshad been achieved.

Support system behaviourThe maximum lateral wall displacement and the maximumhorizontal ground surface movement (a negative displacementis towards the excavation), on completion of excavation fromanalysis Set A are plotted as percentages of the excavated depthagainst both ln (EI) and (EV7+4) in Figure 3a. It is evidentthat there is very little variation in displacement for supportsystems 1, 4, 7, 8, and 9 which cover a wide range of material

flexibilities, and a complete log cycle of system flexibilities.These five systems do however have the same displacementflexibility. Figures 3b and 4 plot the same displacementsagainst log A. The sign convention adopted for the horizontalground movements in Figure 4 is such that a positivedisplacement is a movement towards the excavation, this resultsin the same trend for all three situations. There is clearly notrend between displacements and ln (EI). The system flexibilityand the displacement flexibility however both relate tomovements. The reduced scatter apparent irom usingdisplacement flexibility, A, and its theoretical justification(Appendix B) suggest it to be the best parameter to plotdisplacements against. The predictions from analysis Set Bshow similar results.~Wall movementsFigure 3b shows maximum lateral wall displacement plottedagainst log A. The results fall onto curves between the twoextreme cases of a four prop diaphragm wall and a two propsheet pile wall. The smaller the displacement flexibility, the lessthe maximum lateral wall displacement.~Ground movementsFigures 4, 5 and 6 consider the movement of the groundsurface behind the completed excavation. It is these movements

GROUND ENGINEERING 'EPTEMBER '994

Wall type ifo of propi logs Ref No line ttyie

(fable I)

Oiaphragm 4 -3.37 I

that cause damage in adjacent structures.Figure 4 is a plot of maximum horizontal ground surface

movement as a percentage of excavation depth against log A.Again the results fall onto curves between the two extremecases of a four prop diaphragm wall and a two prop sheetpilewall, with systems 1, 4, 7, 8, and 9 showing very littlespread.

Figure 5 is a plot of maximum vertical ground surfacemovement as a percentage of excavation depth against log t5,.

The excavations modelled in dry soil produce heave behind thewall (plotted as positive displacement). The analyses indicatethat this heave is controlled by the unloading infront of thewall, lifting the support system upwards. All the supportsystems modelled gave rise to the same maximum heaveadjacent to the back of the wall. Excavation in a water bearingsoil followed by consolidation produces settlement profilesbehind the wall (as shown in Figure 6). Plotting the maximumvertical ground surface movement as a percentage of excavationdepth against log A reveals that the lower the system'sdisplacement flexibility, the less the maximum ground surfacesettlement.

Figure 6 shows the marked similarity in settlement profilesbehind support systems with the same displacement flexibility(systems 1, 4, and 7), bounded by a stiffer system (3) and asofter system (5).~Prop forces and wall bending momentsFigure 7 is a plot of the sum of prop forces against log A for allthe support systems. A linear relationship is evident. The lowerthe displacement flexibility, the greater the sum of the propforces.

The bending moments induced in a wall are dependent onthe deflected shape. Bending moments are not thereforeuniquely related to the displacement flexibility which isrepresentative of a support system's absolute displacementpotential. Figure 8 shows that for a given A value (analyses 1,4, 7, 8, and 9), the use of a more flexible wall with acorresponding increase in the number of propping levels(analysis 9 compared to 1) reduces the maximum bendingmoment.

The single analysis performed with a four propped 23msheet pile wall with the same prop spacing as the four propped20m wall, representing an increase in embedment depth of75% showed no significant change in the prop forces (0.96%),the wall and ground movements (2.5%),or the maximumbending moment (3.5%).The average prop spacing thereforeappears to be an acceptable characteristic dimension.

Engineering applicationThe displacement flexibility, A, as defined here, where h is theaverage prop spacing can be used as part of a displacementcontrolled design.

The prescribing of a maximum allowable displacement as adesign parameter does not restrict the engineer in the selectionof a support system. By keeping A constant the bendingmoment reducing benefits &om the use of a more flexible wallmaterial, lower EI, can be accurately balanced against theconstruction requirement of more propping levels, smaller h(Figure 8). Several different support systems can therefore be

44 considered in the knowledge that the maximum allowable

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Figure 6 (TOP): Settlement profiles —Set B afterexcavation.Figure 7 (BOTTOM) t Sum ofprop loads (nondixnensionalised) against log A.Figure 8 (RIGHT) i Magnitude of the maximum bendingmoment (non dimensionalised) against ln EI for supportsystems with the same log A.

displacement will not be exceeded and the profile of groundsettlement behind the excavation will be unchanged.

ConclusionsIt has been shown that existing flexibility numbers for retainingwalls do not provide a complete &amework for displacementcontrolled design of deep excavations supported by multipropped systems.

A new flexibility number, the displacement flexibilityh, (= h'/EI), has been defined to enable the engineer to considerdifferent support options which will meet the samedisplacement criteria.

Support systems with the same displacement flexibility Lec, oncompletion of excavation, give rise to the same maximumlateral wall displacement, the same ground surface response(vertical and horizontal displacements), and the same sum ofprop loads.

GROUND ENGINEERING SEPTEMBER '994

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E.I E.Iwhere z is the depth below the original ground surface level.

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and,

(—modd (—pmroryprM M.

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E.IWalls with the same value of p are likely to yield the samedeflected shape.

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0.08

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Appendix BThe displacement flexibility nuxnber.

For walls to have similar displacements;

(3 ) sysrem I = (3 ) sysrem 2

5 6 7 8 9 1011 12131415In El

This can be expressed as:

(f~dz),~rm(f ~dz), 2

ReferencesBoscardin, MD & Cording, EJ (1989).'Building response to excavation—induced settlement'. Journal of Geotechnical Engineering, ASCE, 115,No 1, 1-21.Clayton, CRI & Milititsky, J (1986). 'Earth pressures and earth retainingstructures'. Surrey University Press, London.Clough, GW, Smith, EM & Sweeney, BP (1989). 'Movement control ofexcavation support systems by iterative design'. Foundation engineering, currentprinciples and practices, Pmc ASCE, Vo12, 869-884.Potts, DM 8t Fourie, AB (1985). 'The effect of wall stiffness on the behaviour ofa propped retaining wall'. Geotechnique, 35, 347-352.Potts, DM 8t Day, RA (1990).'Use of sheet pile retaining walls for deepexcavations in stiff clay'. Proceedings, ICE, Part I, 88, 899-927.Rowe, P (1952). 'Anchored sheet pile walls'. Proceedings, ICE, I, 27-70.

Appendix ARowe's flexibility number (Rowe, 1952).

Rowe derived this number to enable him to perform scaledmodel retaining wall experiments, the deflected shape of whichwere similar to the prototype, ie

(dg)modd —(g~)roororJvse

where y is in the horizontal plane and z in the vertical.

Now,

(f ~d dz) (fMz«)

where M is the bending moment which is assumedproportional to (z'), and EI is the material flexibility.

If we have the same requirements as for Rowe's derivation (seeAppendix A), with D as depth,

(D)modd (D )prororype 'q

and,

(~)~m (~)M M

Then we have,

(f~~ .vi dD )o,, m (f~~D .vl dD ),Integrated over the average prop spacing, h (the characteristicdimension of a multi propped retaining system), this leads to asolution of the form

1 2 hs 1 2 h'EI ' EI

Thus the 'displacement flexibility's defined:

From structural analysis,

(@)=(—')dz E.I

Walls with the same value of 1S are likely to have the samemagnitudes of displacements. 45

GROUND ENGINEERING SEPTEMBER 1994