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19 vues17 pagesModelling of a 3D Exca

Paper 05 Modelling of a 3D Exca in FE Analysis

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Modelling of a 3D Exca

© All Rights Reserved

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19 vues17 pagesPaper 05 Modelling of a 3D Exca in FE Analysis

Modelling of a 3D Exca

© All Rights Reserved

Vous êtes sur la page 1sur 17

7, 497513

L . Z D R AV KOV I C * , D. M . P OT T S * a n d H . D. S T J O H N

Cet expose etudie plusieurs questions liees a` la modelisation dune structure de soute`nement utilisee pour soutenir une excavation dans les analyses delements finis en

3D. Nous examinons plus particulie`rement les effets de la

rigidite du mur dans diverses directions coordonnees et

la fixite rotationnelle dans langle de lexcavation. Des

excavations carrees et rectangulaires sont analysees et

comparees avec les analyses de deformations axisymetriques et planes equivalentes normalement utilisees comme

approximations a` des fins de modelisation. La geometrie,

la sequence de construction et les conditions de sol

choisies sont basees sur une excavation profonde proposee

a` Moorgate a` Londres (pre`s du developpement de Moor

House), qui fera partie dune station souterraine du

projet Crossrail. Lobjectif de cette etude est de donner

une evaluation detaillee des mouvements du mur et du

sol ainsi que des forces structurales dans le mur a` la

lumie`re de differentes hypothe`ses de modelisation. Cette

etude peut sappliquer plus largement a` une variete de

projets qui incluent le developpement infrastructural et

la construction de parkings en sous-sol et de structures

enterrees ; elle pourra sappliquer aussi a` leffet que ces

structures ont sur les zones environnantes.

modelling of a retaining structure used to support an

excavation in 3D finite element analyses. In particular,

the effects of wall stiffness in different coordinate directions and the rotational fixity in the corner of the excavation are examined. Both square and rectangular

excavations are analysed and compared with the equivalent axisymmetric and plane strain analyses, normally

used as approximations for modelling purposes. The

chosen geometry, construction sequence and soil conditions are based on a proposed deep excavation at Moorgate in London (next to the Moor House development),

which will form part of an underground station for the

Crossrail project. The objective of the study is to provide

a detailed assessment of wall and ground movements and

structural forces in the wall in the light of different

modelling assumptions. The study has wider application

to a variety of projects that include the development of

infrastructure, the construction of deep basement car

parks and buried structures, and the effect that these

have on the surrounding areas.

KEYWORDS: deep excavation; numerical modelling and analysis; retaining walls

INTRODUCTION

The construction of tunnels and station boxes in urban areas,

such as London, requires a detailed assessment of the effects

that such construction might have on existing structures.

Sometimes, if there is enough information about previous

similar undertakings, it is possible to make this assessment

on the basis of experience. However, if this is not the case,

then it is necessary to use numerical techniques to make the

necessary predictions.

Current design practice suggests that, in a general rectangular excavation, plane strain two-dimensional (2D) analysis

should be applied to assess the wall and ground movements

in the centre of the excavation (along its longer side),

whereas an axisymmetric analysis should be applied to

assess conditions in the corner and the shorter side of the

excavation (see Fig. 1). To date, full three-dimensional (3D)

analyses have rarely been carried out because of time and

cost constraints.

St John (1975) compared the predictions of ground movements for plane strain, axisymmetric and square excavations

modelled assuming a uniform linear-elastic soil and no wall,

in an attempt to explain the variation of surface ground

movements measured at the Houses of Parliament in London. A number of recent publications describe the 3D

modelling of deep strutted excavations in a variety of soil

conditions and compare the results with those from 2D

Katzenbach, 2002). These analyses have gone further in

modelling the soil as an elasto-plastic material, but the

retaining walls are still assumed to be isotropic elastic.

In reality, however, a concrete retaining wall, for example,

is not an isotropic solid. Whether it is a diaphragm wall, a

contiguous wall, a secant pile wall, or even a sheet pile wall,

it has continuous vertical elements (e.g. diaphragm panels,

piles), but is discontinuous in the horizontal direction, along

the sides of the excavation; see Fig. 2. Consequently, it

cannot sustain any significant out-of-plane bending, and also

the horizontal axial stiffness of the wall is much smaller

than the stiffness of the solid concrete, as a consequence of

joints between the vertical elements. The assumption of

isotropic stiffness (i.e. the same stiffness in all coordinate

directions) therefore introduces a significant limitation to

any analysis. An axisymmetric analysis with isotropic wall

stiffness predicts small wall and ground movements and

shows that the support is provided by hoop stresses, rather

than bending resistance along the vertical direction as would

be the case for a circular shaft sunk using traditional

techniques (Cabarkapa et al., 2003). Even for a truly circular

shaft constructed with an in situ retaining wall this assumption is unrealistic, as the behaviour of the wall will be

dominated by the compression of the joints between the

elements of the wall (e.g. panels or piles). Consequently, if

realistic predictions of wall and ground movements and

structural forces are to be achieved when modelling either

axisymmetric or full 3D excavations, it is necessary to

reduce the out-of-plane wall stiffness (both axial and bending) to an appropriate value.

This paper first investigates the effects of wall stiffness in

a square excavation based on the geometry and ground

conditions for a proposed deep excavation on the Crossrail

accepted 3 June 2005.

Discussion on this paper closes on 1 March 2006, for further

details see p. ii.

* Department of Civil and Environmental Engineering, Imperial

College, London, UK.

Geotechnical Consulting Group, London, UK.

497

t

Cross-section for

plane strain analysis

1137

P1

165

Axisymmetric simulation

for the corner and short

side of excavation

CL

125 P2

P3

230

P4

275

P5

2125

P6

2175

2225 P7

GWT

1137

1100

165

Excavation level

Prop level

Made ground

Terrace gravel

London clay

2220

2270

2330

No out-of-plane

bending stiffness

498

175 m

Lambeth Group

clay

12 m

2400

Thanet sand

Limited horizontal

axial stiffness

Joint

Panel

2530

(a)

Chalk

No out-of-plane

bending stiffness

Pile

Limited horizontal

axial stiffness

Joint

(b)

No out-of-plane

bending stiffness

Pile

No horizontal

axial stiffness

Joint

(c)

This sequence of propping and excavation then continues

until the final excavation level of 27.0 mOD is reached,

making the total excavation depth 40.7 m. Such a large

depth of excavation is required because of the necessity for

the new tunnels to run below the existing London Underground tunnels, and at this location they must be at 30 m

depth. Although this is an exceptionally deep excavation

compared with usual excavation depths for developments in

urban areas, it will be shown in this paper that the results

from the analyses presented here can be used to assess the

behaviour at shallower excavation depths.

The pore water pressure and K0 profiles adopted in the

analyses are shown in Fig. 4. The clay layers are modelled

as undrained, but the remaining layers are drained.

wall; (b) secant pile wall; (c) contiguous pile wall

aims to connect the east and west ends of London via 19 km

long tunnels running beneath central London. This particular

excavation is to serve as a launching platform for the tunnel

boring machine for one of the Crossrail tunnels and also as

a part of a future station with escalators. Some preliminary

results from this study are described in Potts (2003) and

Torp-Petersen et al. (2003). The effects of different moment

connections in the corner of such an excavation are also

examined. Having in this way established the most appropriate approach for modelling a 3D excavation, the study is

then extended to the analyses of rectangular excavations,

with length, L, to width, B, ratios of 2:1 and 4:1. All the

results are compared with the appropriate axisymmetric and

plane strain predictions.

FINITE ELEMENT ANALYSES

Soil conditions and construction sequence

The ground conditions adopted in the analyses reflect a

typical soil profile in central London (see Fig. 3), with the

groundwater table at the top of the London clay. Also shown

in the figure is the construction sequence that is envisaged

for the site at Moorgate station. The dashed lines represent

ground level at different stages of excavation, and the arrows

represent props. The wall behaves as an embedded cantilever

up to the excavation level of +6.5 mOD. Prop P1 is then

The non-linear elasto-plastic MohrCoulomb model (Potts

& Zdravkovic, 1999) is used to model all soil units, apart

from the made ground, which is modelled with a linear

elastic MohrCoulomb model. The non-linearity below yield

is simulated with the Jardine et al. (1986) small-strain

stiffness model. Model parameters for all soil units are

summarised in Tables 1 and 2, and the variation of normalised shear (3G/p9) and bulk (K/p9) stiffness with deviatoric

(Ed ) and volumetric (v ) strain respectively is shown in

Fig. 5.

Geometry

Both 2D and 3D finite element analyses are performed in

this study, using the Imperial College Finite Element Program (ICFEP; Potts and Zdravkovic, 1999). The results from

axisymmetric and plane strain analyses are used as a reference for comparison with those from 3D analyses. The

Moorgate excavation geometry is square in plan (35 m 3

35 m outer dimensions), and therefore only half of the

central cross-section is modelled in 2D analyses. The mesh

used for the 2D analyses is shown in Fig. 6; it consists of

800 eight-noded quadrilateral isoparametric elements. The

props are modelled as two-noded bar elements that can

transmit only axial force, and the wall is modelled with

either solid or beam elements (Potts & Zdravkovic, 1999).

In the 3D analyses advantage is taken of a fourfold symmetry, and this mesh is shown in Fig. 7. The soil here is

15

15

1137

1100

10

5

25

25

210

Elevation: m

1137

1100

10

165

165

210

Hydrostatic

215

215

220

220

2220

225

225

230

230

235

235

2400

240

2220

2400

240

Underdrained

245

245

250

255

499

250

2530

0

100

200

300

400

Pore water pressure: kPa

(a)

500

255

600

2530

0

02

04

06

K0

(b)

10

08

12

Layer

Made ground

Terrace gravel

London clay

Lambeth clay

Thanet sand

Angle of shearing

resistance, 9: deg

Cohesion,

c9: kPa

Angle of dilation,

: deg

25

35

22

22

32

0

0

0

0

0

12.5

17.5

11

11

16

elements, whereas the props are modelled using eight-noded

membrane elements that can transmit only in-plane axial

forces. The wall is modelled with either 20-noded solid or

eight-noded shell elements (Schroeder, 2002). In all the

analyses, structural elements are modelled as elastic, and the

adopted wall thickness is equivalent to a 1.2 m thick

diaphragm wall. The Youngs modulus of 3 3 106 kPa for

the props was estimated from the equivalent stiffness of

tubular steel pipes that would normally be used in such an

excavation. The elastic properties of the walls are summarised in Table 3. It should be noted that full interface

friction was assumed between the soil and the wall: consequently no interface elements were used in the analyses.

REFERENCE ANALYSES

The plane strain and axisymmetric analyses that serve as

a reference for comparison with the 3D results are performed by modelling the retaining wall with either solid or

beam elements. A circular shape is inscribed in a square for

the axisymmetric analysis, similar to the sketch in Fig. 1. In

Youngs modulus,

E: kPa

Small

Small

Small

Small

10 000

strains (see

strains (see

strains (see

strains (see

Table

Table

Table

Table

Poissons

ratio,

2)

2)

2)

2)

0.2

0.2

0.3

0.3

0.2

the plane strain analysis the wall stiffness (in the vertical zdirection) is specified as Ez 28 3 106 kPa, to simulate

properties of concrete. In the axisymmetric analysis the same

value is specified for the axial wall stiffness (Ez ), but zero

stiffness is prescribed in the circumferential direction (E ),

to account for a discontinuous wall in this direction.

The analyses with the wall modelled with solid elements

are performed first, and the horizontal wall movements after

the complete construction sequence (i.e. excavation to a

depth of 40.7 m) are shown in Fig. 8. As expected, the

axisymmetric analysis predicts smaller movements, and for

this case the maximum value (at 21.0 mOD) is about 70%

of that predicted in the plane strain analysis. The two

analyses are also repeated with the wall modelled with beam

elements, placed on the excavation side of the solid elements

(see Fig. 6). The relative difference between the two wall

deflections is similar to the analyses with solid elements;

however, in each of the analyses the wall deflection is larger

than when the wall is modelled with solid elements. This is

a consequence of the lack of the beneficial action of shear

stresses mobilised on the back of the wall. In the case of

solid elements this shear stress acts downward at a certain

500

Table 2(a). Small-strain soil properties: coefficients for elastic shear modulus

Layer

Terrace gravel

London clay

Lambeth clay

Thanet sand

C 3 104 : %

Ed,min 3 104 : %

1104

1400

1400

930

1035

1270

1270

1120

5

1

1

2

0.974

1.335

1.335

1.100

0.940

0.617

0.617

0.700

8.83346

8.66025

8.66025

3.63731

Ed,max : %

0.3464

0.6928

0.6928

0.1645

Gmin : kPa

2000

2667

2667

2000

Table 2(b). Small-strain soil properties: coefficients for elastic bulk modulus

Layer

T 3 103 : %

Terrace gravel

London clay

Lambeth clay

Thanet sand

275

686

686

190

225

633

633

110

2

1

1

1

0.998

2.069

2.069

0.975

1.044

0.420

0.420

1.010

v,min 3 103 : %

2.1

5.0

5.0

1.1

v,max : %

Kmin : kPa

0.20

0.15

0.15

0.20

5000

5000

5000

5000

Coefficients in Tables 2(a) and 2(b) are material constants used in the following equations to give a variation of tangent shear and bulk

stiffness with both stress and strain level:

1

Ed

(

(

p

) B log10

)

3G

Ed

Ed

C 3

p

p

A B cos log10

sin

log

10

p9

2:303

C 3

C 3

(

K

jv j

R S cos log10

p9

T

)

jv j

T

2:303

1

S log10

(

sin log10

jv j

T

)

2500

500

Terrace gravel

London clay &

Lambeth clay

Thanet sand

2000

400

1500

K/p

3G/p

300

1000

200

500

100

0

00001

0001

001

01

Deviatoric strain, Ed: %

(a)

0001

001

01

Volumetric strain, v: %

(b)

Fig. 5. Non-linear stiffness used in the analyses: (a) shear stiffness; (b) bulk stiffness

produces a clockwise moment about this axis that reduces

the anticlockwise moment generated by the horizontal stresses acting on the back of the wall. When the wall is

modelled with beam elements, although its properties take

account of the wall thickness, the actual beam elements do

not have thickness in the finite element mesh, and therefore

there is no clockwise moment from the shear stresses on the

back of the wall, thus resulting in larger horizontal movements.

A parametric study in which the ratio E /Ez in the

order of magnitude (i.e. 1.0, 0.1, 0.01, etc.) was also

performed and showed that if E /Ez < 0.001 there is no

difference in predicted wall deformation between the analyses performed with different E /Ez values. Fig. 8 also

shows the wall deflection for the case of isotropic wall

stiffness (i.e. E /Ez 1.0, termed stiff wall on the figure),

which demonstrates that such a simulation is clearly unrealistic as it predicts negligible wall movements. A similar

relationship between the analyses predictions is observed

for the surface settlement behind the wall. This is an

175 m

501

12 m

1137

Made ground

Terrace gravel

165

125

230

London clay

275

2125

2175

2225

2270

Lambeth group

clay

Wall:solid elements

2330

Wall:beam elements

z

Thanet sand

x

2530

1000

35 m

35 m

Plane of symmetry

y

x

Corner

1137

1000

Wall: solid elements

Wall: shell elements

z

y

2530

0

1000

Plane of symmetry

wall, because the inclusion of any significant circumferential

stiffness, E , results in the resistance to soil pressure on the

back of the wall being provided by the hoop (i.e. circumferential) stresses, rather than by the bending of the wall in the

vertical plane, which is unrealistic.

3D ANALYSES OF SQUARE EXCAVATION: SOLID

ELEMENT WALL

In the first set of 3D analyses the wall is modelled using

20-noded hexahedral solid elements. Two analyses are per-

Ez 28 3 106 kPa) and the other with an anisotropic wall

stiffness (Ex Ez 28 3 106 kPa, E y /Ez 105 ; see Fig.

7 for coordinate directions). As the chosen ratio of E y /Ez is

smaller than the minimum threshold of 103 established in

the axisymmetric analyses, the movements of the wall will

be the maximum possible. Because of the very low stiffness

in the y-direction, the latter analysis broadly simulates the

conditions in a contiguous pile wall. The results are presented in comparison with the equivalent plane strain and

axisymmetric analyses from Fig. 8 (i.e. solid wall simulations).

502

Wall type

Ex : kPa

Plane strain

Axisymmetric

3D isotropic

3D anisotropic

28

28

28

28

3

3

3

3

106

106

106

106

E y : kPa

Ez : kPa

N/A

28 3 101

28 3 106

28 3 101

28

28

28

28

3

3

3

3

Poissons ratio:

Wall thickness: m

0.2

0.2

0.2

0.2

1.2

1.2

1.2

1.2

106

106

106

106

Wall type

E: kPa

Plane strain

Axisymmetric

3D isotropic

3D anisotropic

28

28

28

28

3

3

3

3

106

106

106

106

t: m

Vertical axial

stiffness: %(EA)

Vertical bending

stiffness: %(EI)

Horizontal axial

stiffness: %(EA)

Horizontal bending

stiffness: %(EI)

0.2

0.2

0.2

0.2

1.2

1.2

1.2

1.2

100

100

100

100

100

100

100

100

N/A

0.01

100

20

N/A

0.01

100

1

A is the cross-sectional area of the wall per metre length of wall; I is the second moment of inertia of the wall per metre length of wall.

Plane strain, beam elements

Axisymmetric, solid elements

Axisymmetric, beam elements

Axisymmetric, stiff wall

15

10

Elevation: m

25

210

215

220

225

230

235

2010

2008

2006

2004

2002

Horizontal wall displacement: m

axisymmetric analyses for different wall models

at the end of the complete construction sequence are shown

in Fig. 9. At the centre, modelling the wall as an isotropic

solid predicts about 20% smaller maximum horizontal wall

movement, whereas the movement of the top of the wall is

nearly three times smaller, when compared with the anisotropic wall analysis. At the corner, the wall movement from

movement is seen in the anisotropic analysis. As explained

earlier, this is a consequence of essentially modelling the

wall as a continuous stiff membrane in the ground.

Surface settlement troughs behind the wall, in the centre

and corner, are shown in Fig. 10. They follow a relationship

similar to that of the wall deflections, with the isotropic wall

having the smallest settlement in both cross-sections. It is

interesting to note that, although the maximum horizontal

wall movement in the centre of the excavation of an

anisotropic wall (Fig. 9(a)) is only about 13% smaller than

that of the plane strain analysis (thus suggesting that plane

strain may not be an unreasonable simplification), the maximum surface settlement in the same central cross-section is

significantly overpredicted by the plane strain analysis, being

1.6 times larger than that of the anisotropic wall analysis.

This clearly demonstrates the effects of a 3D geometry on

ground movements.

The vertical wall bending moments M1 , corresponding to

rotation about the y-axis of the wall, at the centre of the

excavation (Fig. 11(a)) are broadly similar for all analyses,

because of the similarity of the curvatures of the deformed

wall. In the corner of the excavation (Fig. 11(b)) the

anisotropic wall gives bending moments M1 that are generally half of those predicted at the centre. This indicates, for

uniform walls, that the corners of the excavation are safe, as

the reinforcement necessary for the centre is sufficient to

cover the bending moments in the corner. However, the

isotropic wall, which essentially simulates a full moment

connection, shows the opposite trend to the other analyses.

This implies that, for a diaphragm wall for example, the

corner panels would have to be reinforced differently from

the central panels, which is not normally done in practice.

A further drawback in modelling the wall with an isotropic stiffness is shown in Fig. 12. Fig. 12(a) shows the

distribution of the out-of-plane horizontal bending moment

M2 , corresponding to rotation about the z-axis of the wall, at

a level of 24.0 mOD (which is the level of the maximum

vertical bending moment, M1 , in Fig. 11). The anisotropic

wall cannot transmit any moment in this direction, but the

magnitude of this moment in the isotropic wall is similar to

the magnitude of the moment M1 , and it also changes sign

towards the corner of the excavation. In a similar way the

horizontal axial force in an isotropic wall is more than five

times larger than that in an anisotropic wall (Fig. 12(b)). As

a result of the jointed nature of any wall type (as sketched

in Fig. 2), such high structural forces in this direction are

considered unlikely.

15

503

15

Plane strain

Axisymmetric

10

10

3D, isotropic wall

25

25

Elevation: m

Elevation: m

210

210

215

215

220

220

225

225

230

230

235

235

2010

2008

2006

2004

2002

Horizontal wall displacement: m

(a)

2010

2008

2006

2004

2002

Horizontal wall displacement: m

(b)

Fig. 9. Horizontal wall movements in square excavation (solid element wall): (a) in centre; (b) in corner

0

2001

2002

2003

Plane strain

Axisymmetric

3D, anisotropic wall

3D, isotropic wall

2004

2005

0

10

20

30

40

50

60

70

Horizontal distance from wall: m

(a)

80

90

10

20

30

40

50

60

70

Horizontal distance from wall: m

(b)

80

90

0

2001

2002

2003

2004

2005

Fig. 10. Surface settlements behind the wall in square excavation (solid element wall): (a) in centre; (b) in corner

ELEMENT WALL

General

In the following study the same square excavation is

analysed, but this time with the wall modelled using shell

elements (Schroeder, 2002). One advantage of using shell

elements is their formulation in terms of structural forces,

rather than stresses, so that the magnitudes of these come as

a direct result from the analyses. In the case of solid

elements in the previous section, structural forces have to be

calculated from the stresses at element integration points,

which makes the whole process slightly cumbersome.

In addition to this, apart from displacement degrees of

freedom, shell elements also have rotational degrees of freedom, which gives greater choice for modelling the moment

conditions in the corner of the excavation. In this study the

shell elements are modelled as elastic, but with the freedom

of having different axial and bending stiffness in the vertical

and horizontal directions.

Five analyses are performed that, because of the properties

assigned to the shell elements, are considered to simulate

the conditions in a diaphragm wall. In analysis 1 (a1), the

shell wall is modelled as isotropic, with Ez E y 28 3

106 kPa, and the rotational degrees of freedom in the corner

are fixed (i.e. full moment connection). This scenario is

similar to that of the isotropic solid element wall in the

previous section. Analysis 2 (a2) also models the wall as

isotropic, but releases the rotational degrees of freedom in

the corner (i.e. moment-free connection). The purpose of

this analysis is to investigate whether just this change in

modelling is sufficient to provide more realistic results.

Analysis 3 (a3) introduces an anisotropic shell wall (i.e.

smaller axial and bending stiffness in the horizontal ydirection), with fixed rotational degrees of freedom in the

504

15

15

10

M1

M1

25

25

Elevation: m

210

215

210

215

(a)

1500

1000

500

22500

1500

1000

500

2500

21000

235

21500

235

22000

230

22500

230

2500

225

21000

225

220

21500

Plane strain

Axisymmetric

3D, anisotropic wall

3D, isotropic wall

220

22000

Elevation: m

10

(b)

Fig. 11. Wall bending moments in square excavation (solid element wall): (a) in centre; (b) in corner

6000

3D, anisotropic wall

3D, isotropic wall

4500

3000

M2

1500

0

21500

Corner

23000

18

Centre

16

14

12

10

8

6

4

Horizontal distance along wall: m

(a)

0

21000

M1

22000

23000

M2

A2

24000

A1

y

x

A2

25000

Corner

26000

18

16

Centre

14

12

10

8

6

4

Horizontal distance along wall: m

(b)

element wall): (a) bending moment; (b) axial force

introduction of anisotropy, but still with full moment connection in the corner, provides more realistic results than analysis (a1). Finally, analysis 4 (a4) introduces anisotropy in

both the axial and bending stiffness of the wall (the same as

a3), and releases the rotational degrees of freedom in the

corner. This is thought to represent the most realistic model

of a diaphragm wall in a 3D excavation. Additional analysis

5 (a5) investigates the effect of a capping beam that is

normally constructed on the top of the wall to connect all

the structural elements. This is achieved by modelling the

top 1.7 m of shell elements as isotropic and with full

moment connection in the corner, whereas the rest of the

wall is anisotropic and with a moment-free connection in the

corner, as in a4.

In the analyses where the shell wall is anisotropic, this is

achieved by assigning the shell elements a full axial and

bending stiffness in the vertical z-direction (Ez 28 3

106 kPa), whereas in the horizontal y-direction the axial

stiffness is 20% of the vertical value, and the bending

stiffness is only a nominal 1% of the vertical value. The

horizontal axial stiffness is estimated on the basis that the

joints between the panels of a typical diaphragm wall may

close by an assumed 1 mm, taking the axial shortening of

each panel at maximum excavation depth from an isotropic

analysis, and reducing the stiffness so that the total shortening (panel shortening plus gap closure) would be produced

under the same load conditions. Although this clearly underestimates the movements where the axial loads are lower,

short of modelling each panel individually, it is a reasonable

approximation.

Figures 1316 compare the predictions of wall and

ground movements, as well as the structural forces in the

wall, for the five analyses described above.

15

15

10

10

25

25

505

Elevation: m

Elevation: m

210

215

220

220

225

225

230

230

235

235

2008

2006

2004

2002

Horizontal wall movement: m

(a)

Anisotropic wall

1 capping beam - (a5)

210

215

2010

Anisotropic wall - (a3)

2010

2008

2006

2004

2002

Horizontal wall movement: m

(b)

Fig. 13. Effect of modelling assumptions in square excavation on horizontal wall movements (shell element wall): (a) in

centre; (b) in corner

0

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Isotropic wall - (a2)

Anisotropic wall - (a3)

Anisotropic wall - (a4)

Anisotropic wall

1 capping beam - (a5)

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Horizontal distance from wall: m

(b)

80

90

surface settlements behind wall (shell element wall): (a) in

centre; (b) in corner

Movements

Figure 13 shows the horizontal wall movements in the

centre and corner of the excavation. In the centre, the

isotropic shell wall with the full moment connection (a1)

results in the smallest deflection, as expected from the

similar analyses with the solid element wall in Fig. 9(a).

Comparison with this figure also shows that the shell wall

predicts slightly larger horizontal movements than the solid

element wall, which was explained earlier as a consequence

of the zero thickness of the shell wall in the finite element

mesh. A similar result will be seen when comparing surface

settlement behind the wall for these two analyses (Figs 10

and 14).

The remaining four analyses predict almost identical

maximum horizontal wall displacement. It appears that the

release of the full moment connection in the corner of the

isotropic wall (a2) is sufficient to give a reasonable prediction of wall deflection in the centre of the excavation, and in

particular the maximum value. The addition of the capping

beam (a5) only restricts the movement of the top part of the

wall; it doesnt affect the rest of it. All analyses, apart from

a1, also predict almost identical maximum horizontal wall

displacement to that of the anisotropic solid element wall in

Fig. 9(a). Although the conditions in the corner for this

analysis are similar to those of no moment connection, this

wall also has negligible horizontal axial stiffness (compared

with the shell element wall for which this stiffness is 20%

of the vertical axial stiffness). This difference in the magnitude of the horizontal axial stiffness in shell and solid

element wall analyses does not appear to affect the maximum wall deformation in the centre. However, wall movements in the corner of the excavation (Fig. 13(b)) are all

negligibly small compared with that of the anisotropic solid

element wall in Fig. 9(b). This appears to be the conse-

506

15

15

10

10

M1

M1

5

25

25

Elevation: m

Elevation: m

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1500

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235

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230

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Anisotropic wall

1 capping beam - (a5)

225

220

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21000

220

21500

Isotropic wall - (a2)

22000

215

(b)

(a)

Fig. 15. Effect of modelling assumptions in square excavation on wall bending moments (shell element wall): (a) in centre;

(b) in corner

(at 224 mOD)

10000

8000

M2

6000

smaller than in the (a4) wall. Consequently, not taking the

capping beam into account gives a slightly conservative

prediction of wall and ground movements, which justifies its

omission in other analyses. The presence of the capping

beam does not appear to influence the surface settlement in

the corner of the excavation.

Isotropic wall - (a2)

Anisotropic wall - (a3)

Anisotropic wall - (a4)

Anisotropic wall

1 capping beam - (a5)

4000

2000

0

22000

Corner

24000

18

Centre

16

14

12

10

8

6

4

2

Distance along wall from centre to corner: m

out-of-plane bending moment; shell element wall

larger axial force, due to larger stiffness, in the horizontal ydirection.

Similar conclusions can be drawn for the surface settlements behind the wall in Fig. 14, in the centre and corner of

the excavation. Again, it is of interest to note that the

maximum surface settlement behind the isotropic wall (a2)

is on average only about 12% smaller than that of the

anisotropic wall (a4), which is considered as the most

appropriate model of the wall. For practical purposes this

would normally be considered acceptable. The inclusion of

the capping beam (a5) reduces significantly the settlement

immediately behind the wall in the central section, because

it is modelled as an isotropic element, with the same vertical

and horizontal stiffness and full moment connection in the

Structural forces

Figure 15 presents the bending moments M1 in the centre

and corner of the excavation. Whereas all five analyses give

similar predictions in the centre of the wall (Fig. 15(a)), the

picture is quite different in the corner (Fig. 15(b)). Analyses

(a4) and (a5), apart from the top part of the wall, give almost

identical bending moment diagrams, whose magnitude is

almost half of that in the centre. This result is also similar to

that of the anisotropic solid element wall in Fig. 11(b).

Analysis (a2) gives smaller bending moments, but of the

same sign as the previous two analyses. This again demonstrates that, although the wall is isotropic, the moment-free

connection in the corner is sufficient to give a more realistic

prediction of the bending moment M1 . In analysis (a3),

although the wall has appropriate anisotropic properties (the

same as in the (a4) analysis), the full moment connection in

the corner causes a change of sign of the bending moment

M1 , similar to that in (a1). These are comparable to the

isotropic solid element wall analysis in Fig. 11(b).

Figure 16 shows the distribution of the out-of-plane

moment M2 along the horizontal y-axis, at 24 mOD, which

is the same level as in the solid element wall analyses. All

three of the anisotropic shell walls (a3, a4 and a5) show that

a negligible M2 moment is transmitted in this direction. Both

of the isotropic shell walls (a1 and a2) transmit a significant

507

M2 moment (with a magnitude similar to that of the M1

(a) stage 1: excavation to +6.5 mOD (i.e. top of London

bending moment), which is unrealistic for any wall that is

clay), at which the wall acts as an embedded cantilever

discontinuous in the y-direction. The only difference between

(b) stage 2: excavation to 7.5 mOD (i.e. depth of

the two is that, whereas in wall (a1) the bending moment

excavation 21.2 m, which is a more usual depth for

M2 switches sign in the corner owing to the full moment

developments in London), at which stage the wall is

connection (similar to the isotropic solid element wall in

propped at three levels (props P1 to P3)

Fig. 12(a)), the corner moment in the wall (a2) goes to zero

(c) stage 3: full excavation to 27 mOD (i.e. 40.7 m

because of the moment-free connection.

excavation depth), with all seven propping levels.

General

In the remainder of this study further analyses are

performed to investigate the behaviour of rectangular excavations. Two geometries are considered, one with a width, B,

to length, L, ratio of 1:2, and the other with B:L 1:4. The

width B is kept the same as in the square excavation,

whereas the length L is changed accordingly. Because of

symmetry only a quarter of the geometry is analysed; see

Fig. 17. The depth of the excavation and the construction

sequence, as well as the soil profile and material properties,

are the same as in the previous analyses. The wall is

represented with shell elements, with the most appropriate

wall model that resulted from the square analyses. This was

considered to be analysis (a4), and consequently in the

rectangular analyses the same anisotropic properties are

assigned to the wall. However, whereas in (a4) it was

possible to release the rotational degrees of freedom of the

shell elements in the corner (because it was on the boundary

of the mesh; see Fig. 7), this is not possible in the analysis

of rectangular excavations, and consequently this corner has

a full moment connection. This is not considered to be a

serious drawback in the rectangular analyses, as the results

from (a3) for the square excavation, which has the same

wall model, showed that the effect of the full moment

connection is only in predicting a high, and of opposite sign,

bending moment M1 in the corner of the excavation (see

Fig. 15(b)). Consequently, this bending moment will not be

presented for these analyses.

In the following, the results from the two rectangular

analyses are compared with the square and appropriate plane

strain and axisymmetric analyses (i.e. in which the wall was

modelled using beam elements; see Fig. 8), in order to

assess at which B/L ratio the plane strain conditions are met

along the longer side of the wall. Also, the results are

compared at three different stages of excavation, in order to

assess whether these conditions are met at earlier stages of

excavation. The stages considered are the following (see

Fig. 3):

Movements

The horizontal wall movements in the centre of the long

and short wall sides, together with those from the square,

axisymmetric and plane strain analyses, are shown in Fig.

18. The movements in all three stages are bounded by the

axisymmetric prediction on the lower side, and the plane

strain prediction on the upper side. In this, the maximum

movements of the short side of the wall and that of the

square excavation are grouped towards the axisymmetric

value, whereas those of the long side are grouped towards

the plane strain value. The maximum deflection of the long

side of the wall, for the first two stages of excavation, is

smaller than the deflection of the wall in plane strain

conditions by 8% and 3% for L/B 2 and 4 respectively.

This difference increases with depth of excavation, but at

stage 3 it is still only 12% and 5% for L/B 2 and 4

respectively. In all stages the maximum movement of the

long side of the wall is for L/B 4, followed by L/B 2

and then L/B 1 (i.e. square excavation). However, the

excavation depth appears to have a greater effect on the

maximum movement of the short side of the wall, which

does not show a clear pattern of deformation dependence on

plan geometry.

Comparing the maximum horizontal movements from

each analysis at the three stages it can be seen that, in the

first 20 m of excavation (stages 1 and 2), although the

position of the maximum deflection moves down with

excavation and propping, its magnitude increases only marginally (by less than 10%). However, with further excavation

the magnitude of the maximum deflection increases dramatically, such that after another 20 m of excavation (stage 3) it

is 70% higher than in stage 2.

The horizontal wall movements in the corner of the

excavation are very small, and similar to those presented in

Fig. 13(b): consequently they are not shown here.

The settlement troughs in the central sections behind the

short and long sides of the walls in the rectangular excavations, together with the plane strain, axisymmetric and

square predictions, are shown in Fig. 19 for all three stages

Plane of symmetry

1000

1137

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2530

0

4000

0

Plane of symmetry

y

x

508

15

15

10

10

10

25

25

25

210

Elevation: m

Elevation: m

L/B 5 2, shortside

L/B 5 2, long side

Axisymmetric

Plane strain

L/B 5 4, short side

L/B 5 4, long side

Square

Elevation: m

15

210

210

215

215

215

220

220

220

225

225

225

230

230

230

235

235

2010 2008 2006 2004 2002

Horizontal wall movement: m

(a)

235

Horizontal wall movement: m

(b)

Horizontal wall movement: m

(c)

Fig. 18. Comparison of wall movements at different stages of excavation (shell element wall): (a) stage 1; (b) stage 2; (c) stage 3

of excavation. Similar to the wall deflections, the axisymmetric prediction provides a lower bound, and the plane

strain prediction an upper bound to the results. Even for L/B

4, the maximum surface settlement behind the long side

of the wall at the end of excavation (stage 3) is about 10%

smaller than that of the plane strain prediction, whereas for

L/B 2 it is some 30% smaller. The L/B 4 prediction on

the long side appears to be closer to the plane strain

prediction at shallower depths of excavation.

Contrary to the wall deflections in Fig. 18, the changes in

the maximum surface settlement with depth of excavation

are more pronounced. For each analysis the maximum settlement at stage 2 is about 35% larger than that in stage 1,

whereas in stage 3 it is about 70% larger than in stage 2.

Surface settlements in the corner of the excavation are

shown in Fig. 20, for all three stages. Note that for clarity

the plane strain prediction for stage 3 is not presented, as its

magnitude is higher than the adopted scale. These settlements also increase with depth of excavation, but for shallow

depths (stage 2) the maximum settlements are close to the

axisymmetric prediction. However, the shapes of settlement

troughs, especially in the first 10 m from the wall, are

different from the axisymmetric prediction, as the corner of

the excavation does not appear to be affected by the

propping system in the same way as the centre of the

excavation (Fig. 19), or plane strain and axisymmetric

geometries.

Figure 21 shows the contours of ground surface settlements at the end of excavation (i.e. stage 3) for the 3D

analyses with L/B 1, 2 and 4. The L/B ratio has a

significant effect on the displacements adjacent to the long

side of the excavation, but it has a much smaller effect on

the short side. In addition, as noted above for the wall

movements, whereas there is a clear dependence of surface

settlements behind the long side of the wall on plan geometry (i.e. L/B 4 is the maximum, followed by L/B 2 and

then L/B 1) for any excavation depth, this pattern is not

so clear for surface settlements behind the short side of the

wall.

Bending moments

Bending moments M1 in the centre of both short and long

sides of the wall are shown in Fig. 22, for all three stages of

excavation. Again, because of the similar curvatures of the

very similar at all three stages of excavation, with the plane

strain prediction being an upper bound for all results.

The out-of-plane bending moment M2 along the horizontal

axis of either the short or long side of the wall is always

negligible (similar to that shown in Fig. 16 for anisotropic

walls) and, for brevity, it is not shown here.

THE EFFECT OF WALL DEPTH

At the beginning of this study it was recognised that the

excavation at Moorgate station was exceptionally deep, with

a significant number of props. This poses the question as to

whether the results presented so far in this paper can be

used to assess the behaviour of a wall and the surrounding

soil at smaller excavation depths (and therefore shallower

walls), which are more common in building construction in

urban areas. For this purpose, an additional analysis is

performed with the L/B 2 rectangular geometry, in which

the maximum excavation depth is 21.2 m (7.5 mOD), and

the depth of the wall is only another 7 m below the maximum excavation depth (14.5 mOD). The excavation stages

up to this level are the same as in the previous analyses, and

the wall is propped by props P1 to P3 (see Fig. 3). The

embedded depth of this wall is similar to that of the wall in

the deep excavation.

The results from this analysis are compared with those of

stage 2 for the analysis with the same L/B 2 ratio but

with the deep wall, as this stage has the same excavation

depth of 21.2 m. Fig. 23 compares horizontal wall movements in the central sections of the excavation, and shows

that the longer embedment depth of the wall reduces the

horizontal movement mainly below the excavation level, the

effect being larger on the short side of the rectangular

excavation. However, this reduction is a maximum of 10%.

Surface settlements in both the central and corner sections

of the wall are compared in Fig. 24. These show negligible

differences (less than 5%) between the shallow and deep

wall excavations. In a similar way, the bending moments M1

are very close in the two analyses, as shown in Fig. 25.

Consequently, the embedment depth of the wall does not

appear to have a significant effect on the behaviour of the

wall and the surrounding soil, and the results from the

analysis of a deep wall can be used to assess the behaviour

of shallow excavations retained by shallower walls.

0005

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L/B 5 2, long side

Axisymmetric

Plane strain

L/B 5 4, short side

L/B 5 4, long side

Square

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Horizontal distance from wall: m

(a)

80

L/B 5 2, long side

Axisymmetric

Plane strain

L/B 5 4, short side

L/B 5 4, long side

Square

20015

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0

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Horizontal distance from wall: m

(a)

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90

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Horizontal distance from wall: m

(b)

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90

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Horizontal distance from wall: m

(c)

80

90

0005

Vertical movement behind wall: m

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Vertical movement behind wall: m

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(b)

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90

0005

Vertical movement behind wall: m

0005

Vertical movement behind wall: m

509

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40

50

60

70

Horizontal distance from wall: m

(c)

80

90

different stages of excavation (shell element wall): (a) stage 1;

(b) stage 2; (c) stage 3

Fig. 20. Comparison of surface settlements in corner at different stages of excavation (shell element wall): (a) stage 1; (b)

stage 2; (c) stage 3

CONCLUSIONS

The objective of this paper is to investigate possible ways

of modelling a retaining wall in square and rectangular

excavations, and provide guidance for the most appropriate

approach to be used in any 3D finite element analysis. The

paper also shows how the 3D predictions compare with

those obtained from equivalent plane strain and axisymmetric analyses, and gives guidance for practical use of

these results.

The following main conclusions can be drawn from the

study.

beneficial reducing moment generated by the downward-acting shear stresses on the back of the wall

(Fig. 8).

(b) Any retaining wall is unlikely to be a continuous

membrane along its perimeter, as it is made from a

number of vertical elements (diaphragms or piles) that

are not fully connected in this direction. Therefore, to

obtain realistic results in axisymmetric and 3D

analyses, the axial and bending stiffness of the wall

along its perimeter must be reduced.

(c) In practice, the realistic conditions in the corner of the

excavation are such that the full moment is not

transmitted. In the analysis, this can be achieved either

by modelling a wall with anisotropic solid elements, or

with anisotropic shell elements that have released

rotational degrees of freedom in the corner of the

excavation.

(a) The retaining wall can be modelled in finite element

analysis using either solid or beam/shell elements. The

latter type of element predicts larger wall and ground

movements because they do not have a thickness in the

510

Settlement: mm

25

50

75

100

125

150

175

200

225

100

75

125

150

175

200

225

(a)

Settlement: mm

75

50

25

100

125

150

175

200

225

250

275

300

100

75

125

150

175

(b)

Settlement: mm

125

100

75

50

25

150

175

200

225

250

275

300

325

125

100

75

150

175

200

225

(c)

Fig. 21. Comparison of surface settlement contours at end of excavation (shell element

wall): (a) L/B

1; (b) L/B

2; (c) L/B

4

15

10

10

10

25

25

25

(a)

(b)

1500

500

1000

22500

1500

500

1000

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235

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235

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230

1500

230

500

230

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225

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220

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220

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21000

215

210

511

L/B 5 2, long side

Axisymmetric

Plane strain

L/B 5 4, short side

L/B 5 4, long side

22000

210

Elevation: m

15

Elevation: m

15

22500

Elevation: m

(c)

Fig. 22. Comparison of wall bending moments in centre at different stages of excavation (shell element wall): (a) stage 1; (b)

stage 2; (c) stage 3

0010

15

10

Shallow wall: short side

Shallow wall: long side

0

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Horizontal distance from wall: m

(a)

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Horizontal distance from wall: m

(b)

80

90

210

0010

215

220

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Deep wall: long side

Shallow wall: short side

230

235

Elevation: m

25

0005

0005

0

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20020

20025

2010

2008

2006

2004

2002

0

Horizontal wall movement for same excavation depth: m

(centre of excavation)

movement (shell element wall)

(shell element wall): (a) in centre; (b) in corner

anisotropic wall properties (i.e. it has to be modelled as

isotropic), but if the corner of the excavation can be

modelled as a moment-free connection, then predictions

of wall deflections and surface settlements (Figs 13 and

14), as well as bending moment M1 , that are reasonable, although on the lower side, can be obtained.

unrealistically high (Fig. 16).

(e) On the other hand, if the wall can be modelled as

anisotropic, but the condition in the corner has to be

that of a full moment connection, then the only

unrealistic prediction will be that of the bending

moment M1 in the corner of the excavation (Fig. 15(b)).

512

15

10

5

0

Deep wall: long side

Shallow wall: short side

Shallow wall: long side

Elevation: m

25

210

215

220

225

230

235

22000

21000

0

1000

Bending moment: kNm/m

(centre of excavation)

2000

moment (shell element wall)

wall has negligible (reducing) effects on movements

and structural forces, and it is therefore reasonable to

ignore it (Figs 13 to 16).

(g) Wall movements and surface settlements behind the

wall in the centre of a square excavation are closer to

202

(Figs 18 and 19).

(h) In rectangular excavations, even for a length-to-width

ratio L/B of 4, the conditions in the centre of the longer

side of the excavation are not fully plane strain in

terms of wall movements and surface settlements

behind the wall, although they are at most 10% smaller

than plane strain predictions, even for the full depth of

excavation. For shallower depths of excavation the

difference is negligible (Figs 18 and 19). For L/B of 2

the influence of depth of excavation is more significant.

(i) The maximum surface settlements behind the wall in

the corner of square/rectangular excavations are on

average about 3050% smaller than the maximum

values in the central sections of the excavation. At

shallower depths (around 20 m) they appear to be well

represented by the predictions of an equivalent

axisymmetric analysis.

( j) The effect of the embedment depth of a wall on

movements and structural forces in the excavations

analysed is negligible. Therefore the results from the

intermediate stages of an analysis of a deep excavation

can be used to assess the wall and ground movements

and structural forces at shallower depths, without

having to repeat the analysis with a shallower wall.

Although the presented study is based on a particular soil

profile and a particular excavation geometry and construction

sequence, the conclusions are general in a sense that they

result from analyses in which only the boundary conditions

on the wall and the wall properties are varied, while the soil

properties, the remaining boundary conditions in the mesh

and the construction sequence are always the same. In this

respect the ground surface settlements predicted in the

analyses can be compared with the observations of wellmonitored excavations and wall systems in stiff clay, summarised in Gaba et al. (2003) and reproduced in Fig. 26.

Also shown in this figure are the normalised maximum

settlements and settlements at a distance equal to two

excavation depths away from the wall (the maximum excava-

1

2

3

At distance of two

excavation depths

201

0

01

02

03

High stif

Low

fness

nes

stiff

At distance of maximum

surface settlement

04

05

06

07

Plane strain

L/B 5 4, long side

L/B 5 2, long side

Average for square and short sides

Axisymmetric

Bell Common | SPW

British Library Euston | SPW

Brittanic House | DW

Churchill Square | CPW

Columbia Center | KP

East of Falloden Way (1) | CPW

East of Falloden Way (2) | DW

Houston Bldgs | KP

Lion Yard | DW

Neasden | DW

New Palace Yard | DW

Rayleigh Weir BP | BPW

Reading | DW

State Street | DW

Walthamstow (1) | CPW

Walthamstow (2) | DW

YMCA | DW

08

Fig. 26. Ground surface settlements due to excavation in front of wall in stiff clay (from Gaba

et al., 2003) (BP: bored piles; BPW: bored pile wall; CPW: contiguous bored pile wall; DW:

diaphragm wall; KP: king post wall; SPW: secant bored pile wall)

tion depth being 40.7 m, i.e. settlement troughs in Fig.

19(c)), for the analyses with little or no out-of-plane flexural

stiffness. The results are clearly in agreement with the

empirical data for walls with high in-plane stiffness.

The analyses also indicate that the current design practice

for rectangular excavations (of using plane strain analysis to

assess the long side and axisymmetric analysis to assess

corners and the short side of the excavation) is broadly

appropriate.

NOTATION

A1

A2

B

c9

D

Ed

Ex , Ey , Ez

E

G

K

K0

L

M1

M2

p9

t

v

j9

axial force in horizontal direction (along y axis)

excavation width

cohesion

excavation diameter

deviatoric strain

Youngs modulus in x, y and z directions respectively

Youngs modulus in circumferential direction

shear modulus

bulk modulus

coefficient of earth pressure at rest

excavation length

bending moment in vertical plane (rotation about y

axis)

out of plane bending moment (rotation about z axis)

mean effective stress

wall thickness

volumetric strain

Poissons ratio

angle of dilation

angle of shearing resistance

513

REFERENCES

Cabarkapa, Z., Milligan, G. W. B., Menkiti, C. O., Murphy, J. &

Potts, D. M. (2003). Design and performance of a large diameter

shaft in Dublin Boulder Clay. Foundations: innovations, observations, design and practice: Proc. BGA Int. Conf. (ed. T. A.

Newson), pp. 176185. London: Thomas Telford

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