three-dimensional modelling of tunnel excavation and lining

Computers and Geotechnics 31 (2004) 171–183

www.elsevier.com/locate/compgeo

Three-dimensional modelling of tunnel excavation and lining

G. Galli *, A. Grimaldi, A. Leonardi

Department of Civil Engineering, University of Rome,‘‘ Tor Vergata’’, Viale del politecnico n.1, Rome, Italy

Abstract

In this study, a 3D finite element model is applied to simulate the conventional procedure of tunnel excavation and lining. Both

shallow and deep tunnels are considered in soils modelled with Mohr–Coulomb elasto-plastic constitutive equation.

A polycentric tunnel cross-section with temporary lining and soil-nailing of the face excavation is studied.

The numerical results show the influence of the soil properties and excavation procedures on face deformation and ground

settlements.

The model allows to evaluate the lining–soil interaction and the stress distribution in both the lining and the reinforcing

structural elements.

� 2004 Published by Elsevier Ltd.

1. Introduction

Tunnel design and construction sets relevant issues,

especially for shallow tunnels in urban environment.

The main problems are the evaluation and the control ofground settlements, deformations and stability of the

excavation front, loads and stresses in the lining. A great

variety of excavation techniques has been developed

[1–3], which employ different methods to reinforce and

support the excavation front. It is therefore important to

evaluate and compare the effect of these methods.

Usually the excavation process is simulated step by

step with FEM modelling. The numerical modellingoften relies on a 2D analysis, implementing elasto-

plastic constitutive models, which are supposed to cap-

ture the limit-state behaviour of drained and undrained

soils. Complete reviews of numerical analyses of tunnels

have been presented in [4,5]. These reviews make it ap-

parent how popular 2D modelling is with respect to 3D

modelling [6].

However, the use of 3D modelling is almost manda-tory if one wants to correctly evaluate the effects of the

excavation process, so that 3D models are under con-

tinuous development, and are being applied to increas-

ingly complex problems [7].

* Corresponding author.

E-mail address: [email protected] (A. Grimaldi).

0266-352X/$ - see front matter � 2004 Published by Elsevier Ltd.

doi:10.1016/j.compgeo.2004.02.003

Specifically, 3D schemes have been used to model

shallow excavations with TBM tunnelling, where the soil

has been modelled in order to simulate time-dependent

consolidation effects [9]. Three-dimensional analysis at

different stages of tunnel excavation has been developed,for instance, in [10] for the Heathrow Express Trial

Tunnel (the first tunnel excavated by the New Austrian

Tunnelling Method in the London Clay, with a poly-

centric section) and in [11] for the case of circular

shallow tunnels, where the effects of sub-horizontal

pipes and umbrella of pipes have been analysed, with

special attention to displacements at the excavation face.

In [12], excavation-induced displacements for shallowtunnels in sandy soils have been analysed both with

experiments and with 3D numerical simulations.

Moreover, a 3D analysis for shallow tunnels is presented

in [13], which investigates K0-influence both on the ini-

tial deformation and on the lining stresses.

In this paper we are concerned with 3D analysis of

shallow and deep tunnels. More specifically, we will

study a full-section-excavated tunnel with polycentricsection, the excavation process being performed with the

use of soil nails and temporary lining.

The Mohr–Coulomb elasto-plastic constitutive model

has been employed; this model is used to describe both

drained and undrained limit conditions. Simulations

have been implemented with the commercial finite ele-

ments code-LUSAS 13.5 [16].

mail to: [email protected]

172 G. Galli et al. / Computers and Geotechnics 31 (2004) 171–183

The main aim of the numerical investigations is to

evaluate:

1. Influence of the protection measures (lining and face

reinforcing) on the ground settlements and face

deformations.2. Soil–structure interaction and stresses in the lining

elements.

In the FEM simulation, standard eight-node volume

elements have been used. Therefore, limit conditions of

stability at the excavation face [14] and localized plastic

deformations are excluded (this case has been investi-

gated, for instance, in [8]).

In the following, the modelling of the excavation andlining phases is illustrated.

The results proposed are referred to various soil

mechanical properties and different procedures of ex-

cavation and lining with or without face-protective

measures.

Finally, a comparison with the simpler 2D model is

given [8,15].

Cross section at the excavation face

Step 1

1

1

Step 2

2

2

Step 3

33

Step 4

44

Step 5

5

5

Step 6

6

Fig. 1. Excavation and lining

2. Modelling of excavation and lining

Excavation and lining cases studied in this work are

referred to the highway and railway tunnel typologies,

with polycentric cross-section (surface 100–150 m2, dia-meter 10–15 m), excavated with conventional methods

(open face, hand mined) and subsequent lining phases.

The first numerical model refers to the shallow tunnel

case with cover height equal about the medium diameter

of the cross-section. A simple numerical procedure is de-

veloped, suitable to evaluate the excavation effects on the

ground settlements and on the stresses in temporary and

final lining. The excavation procedure includes reinforc-ing elements at the excavation face to assure face stability.

The protective measures for the excavation face are

longitudinal fibber-glass nails grouted in the tunnel face

and umbrellas of longitudinal fibber-glass pipes (circular

crone with internal diameter 0.04 m and external di-

ameter 0.06 m), designed to prevent localized break-ups

and front extrusion.

Longitudinal profile

Protective measures forface stability

Temporary lining

6

Final concrete lining

procedure (see Table 1).

Table 1

Protective measures and lining elements

Excavation and lining procedure

1. Umbrella of pipes

2. Soil nailing at face

3. Steel ring beam

4. Shotcrete support

5. Invert arch

6. Final concrete lining

G. Galli et al. / Computers and Geotechnics 31 (2004) 171–183 173

The temporary lining is given by steel ring beams with

standard double-T section (two coupled IPE-180),

shotcrete support (thickness 0.3 m) and concrete invert

arch (thickness 0.8 m).

The arrangement of these elements in the typicalcross-section is illustrated in Fig. 1 and Table 1.

3. Three-dimensional final element modelling

The 3D model of Fig. 2 shows a shallow tunnel case

with a polycentric section (D ¼ 11 m) assuming sym-

metric loading conditions.Excavation and lining phases are simulated through

72 different analysis steps.

In the first step, all the representative elements of the

lining and soil reinforcing are deactivated; in the second

step, the soil-weight is applied to the ground elements; in

the third step, the soil reinforcing elements of the front

(umbrella of pipes and soil nails at face) are activated

and tunnel excavation starts at the end cross-section ofthe model.

70 m

Fig. 2. Three-dimen

Table 2

Lining elements and protective measure parameters

Lining material properties

Ring steel beams E ¼ 2:1Eþ 08 kPa, v ¼ 0:3

Shotcrete support E ¼ 2:5Eþ 07 kPa, v ¼ 0:1

Invert arch E ¼ 2:85Eþ 07 kPa, v ¼ 0:2

Umbrella of pipes E ¼ 1:3Eþ 07 kPa, v ¼ 0:3

Soil nails at face E ¼ 1:5Eþ 07 kPa, v ¼ 0:3

In the subsequent phases, the tunnel excavation is

developed and the lining elements are activated (steel

beams, shotcrete support and invert arch).

The constitutive law used for the soil elements is the

elasto-plastic associated Mohr–Coulomb model with thefollowing material parameters: E ¼ 40; 000 kPa,

m ¼ 0:334, / ¼ 26� (friction angle), cohesion c ¼ 20 kPa,

density c ¼ 20 kN/m3. The lining elements and the

protective measures are assumed to have a linearly

elastic behaviour.

The lining geometrical material parameters are given

in Table 2.

The excavation sequence are:• Construction of soil nails and umbrella of pipes.

• Excavation of 1 m of soil and subsequent installation

of the temporary lining, which includes a steel ring

beam at 1 m distance from the excavation face, a

shotcrete support at 2 m distance from the excavation

face and an invert arch at a variable distance (4–16 m)

from the excavation face.

The numerical procedure simulates an excavationstarting from the end section of the soil model and

stops at the middle section. The excavation crosses 15

soil layers with different thicknesses. The element

thickness (1 m) in the central layers of the model is

reduced with respect to the mesh thickness (4 m) of

the lateral first layers, in order to increase the nu-

merical accuracy in the central part of the model,

where the simulation is assumed representative of theactual excavation phase.

The details of the subsequent analysis steps (load

cases in Lusas 13.5) are as follows:

∼4D

∼3D

D

∼D

sional model.

A ¼ 47:8 cm2, l ¼ 2634 cm4/m

Thickness¼ 0.3 m

Thickness¼ 0.8 m

Circular crown section 0.06/0.04 m

Circular crown section 0.06/0.04 m


Load case 1. Initialisation of the procedure: deactiva-

tion of all lining and reinforcing elements.

Load case 2. Soil weight; lithostatic condition.

Load case 3. Activation of the steel ring beams and

the shotcrete support elements (3D beams) locatedin the first soil layer. Activation of the soil pipes um-

brella and soil nails at the face (3D-bars).

Load case 4. Deactivation of the soil elements in-

cluded in the first layer (release of 100% of the bound-

ary nodal forces).

Load case 5. Activation of the steel ring beams and

the shotcrete support elements located on the sec-

ond layer. Activation of the invert arch in the firstlayer.

Load case 6. Deactivation of the soil elements in-

cluded in the second layer (release of 100% of the

boundary nodal forces).

The procedure (load cases 5–6) is repeated to simulate

the first 20 m of tunnel excavation and lining.

Fig. 3. Stresses and displacement (excavatio

In this first part, the length of the excavation step is 4

m corresponding to the thickness of the soil elements

(load cases 7–13).

In the central part of the model the length of the

excavation step is reduced to 2 m between 20 and 30 m,and to 1 m between 30 and 35 m (middle section of the

model). In this part of the excavation (20–35 m), a more

precise lining procedure is adopted with the subsequent

activation of steel ring beams and shotcrete support

(load cases 13–72).

We assume that final concrete lining is placed at large

distance from the excavation face, hence it does not

appear in the numerical simulations.

3.1. Soil stresses and displacements

The results obtained from the 3D model are relative

to the final situation after completing the excavation and

lining (Fig. 3).

n and lining completed, load case 72).


In this example all the protective measures have been

considered. The numerical results show the spreading of

a plastic zone near the excavation face and under the

invert arch.

3.2. Stresses in lining

The numerical results show the interaction between

lining elements and surrounding soil, and allow to val-

uate the stress resultant in the steel ring beams (Fig. 4),

shotcrete support (Fig. 5) and invert arch (Fig. 6).

The small bending stiffness of the ring steel beams

and shotcrete support implies that the stresses produced

Fig. 6. Stress resultants in the invert arch

Fig. 5. Stress resultants in the shotcrete supp

Fig. 4. Stress resultants in the steel ring beams (doub

by axial force are prevailing on the bending moment

effects. More precisely, the compression stress in the

steel beams is almost constant with the maximum value

r ¼ 142; 600 kPa. In the shotcrete, the maximum com-

pression stress is r ¼ 2600 kPa. In the invert arch, bothaxial force and bending moment are relevant, and the

maximum compression stress in the concrete is r ¼ 1800

kPa.

3.3. Soil nails

The model gives the distribution of the tensile axial

force in the nails at the excavation face (Fig. 7). The

(1 m of lining, thickness¼ 0.8 m).

ort (1 m of lining, thickness¼ 0.3 m).

le T section; A ¼ 47:8 cm2/m, I ¼ 2634 cm4/m).


tensile force in the nails is extinguished at a distance of

about two diameters (D ¼ 11 m) from the excavation

front. The maximum tensile stress in the fibber glass

nails is about r ¼ 17; 707 kPa.

3.4. Ground settlements and face deformation

The numerical simulation has been repeated for three

different procedures of excavation and lining:

Fig. 8. Face mo

Fig. 9. Settlements at the tunn

50

10

15

20

25

0

0

0

0

0

300

0 10 20 30 4

EExxccaavvaattiioonn ffaaccee

Distan

Axial Force in the soilnails (kN

)A

xial Force in the soilnails (kN)

Fig. 7. Axial force in s

• Face with soil nailing at face and construction of the

invert arch at 4 m distance from the excavation face.

• Face without nails at face and construction of the in-

vert arch at 4 m distance from the excavation face.

• Face with soil nailing at face and construction of theinvert arch at 16 m distance from the excavation face.

The excavation face movements are plotted in Fig. 8.

These results show that face movements are reduced

by the soil nails at face (Fig. 8); the analysis shows also a

vements.

el cross-section heading.

0 50 60 70 80

N=278 kN

ce (m)

oil nails at face.

Fig. 11. Vertical displacements at ground surface.

Table 3

Mechanical properties of soil

Soil E (kPa) v c (kPa) u (�)

Case 1 40,000 0.334 20 26

Case 2 40,000 0.334 5 38

Case 3 20,000 0.334 40 26

Fig. 10. Heading displacements at the excavation face cross-section.


reduction of the total displacements near the excavation

face. In Figs. 9 and 10, the effects of the excavation andlining procedure on the tunnel heading settlements are

examined. The strong reduction of the settlements due

to soil nails and invert arch is emphasized.

The distribution of the ground settlements is given in

Fig. 11.

4. Influence of soil parameters

With the subsequent model cases, the influence of

different mechanical properties of the soil (Table 3),

corresponding to the Mohr–Coulomb criterion with

associated flow, has been analysed. Both a shallow

tunnel (Fig. 2) example (with an additional surface load

of 40 kN/m2 that simulates the influence of a four-stories

building) and a deep tunnel example (Fig. 14) are ex-amined.

The excavation and lining procedures are the same

used for the first model.

The numerical results for the shallow tunnel exampleare given in Figs. 12 and 13, and Table 4.

The influence of the soil mechanical properties on the

maximum values of the stress resultants in the lining

elements is shown in Table 4.

Similarly in Fig. 13, the influence of soil parameters

on the settlements (at the heading and at the ground

surface), face displacements and on the maximum values

of the axial force in the soil nails at face is investigated.

Fig. 12. Stress and settlements in the model before excavation and lining procedure and after excavation and lining procedure (case 2).


The next example is a deep tunnelling excavation andlining, corresponding to the three soil cases (Table 3) but

without the asymmetrical surface load in the previous

case.

The mesh is similar to that of the shallow case, but

the tunnel location is about 53 m below the soil plane

(Fig. 14).

The comparison of soil movements and stress in the

soil nails are reproduced in Fig. 15.In these cases, the ground settlements are negligible.

The stress resultant in the lining elements are given in

Table 5.

The previous results have been obtained assuming the

simple Mohr–Coulomb model with associated flow. The

use of this criterion is diffused in practice, and the cor-

responding finite element numerical iterative solution is

generally stable. Some numerical results using the non-associated Mohr–Coulomb model are also shown in

Figs. 16 and 17, where the vertical displacements are

plotted corresponding to different values of the dilatancy

(Table 6).

The comparison shows the increment of the groundsettlements for the cases of non-associative flux.

5. Two-dimensional modelling

The most frequent modelling used for tunnelling ex-

cavation is the 2D finite element analysis [4].

In this case a soil layer, orthogonal to the tunnel andsufficiently far from the excavation front, is considered

(Fig. 18). The excavation and lining phases are analysed

with subsequent load cases corresponding to deactiva-

tion of soil elements and activation of lining elements.

The nodal forces acting on the tunnel cross-section

boundary are gradually relaxed to simulate the interac-

tion between the soil and the lining elements. However,

for the 2D model it is necessary to assume the fraction ofnodal forces release in the subsequent load cases.

On the contrary, the 3D model can automatically

simulate the real procedure of excavation and lining. A

comparison between 2D model predictions and the re-

Fig. 13. Soil displacements and axial force in the soil nails.

Table 4

Axial force (kN) and bending moment (kNm) in the lining elements

Case Steel ring Shotcrete support Invert arch

Nmax Mmax Nmax Mmax Nmax Mmax

1 )634 )29 )188 )167 )509 )1612 )591 )28 )188 )151 )424 )1633 )354 )30 )208 )195 )358 )134

Fig. 14. Three-dimensioinal model of deep excavation.


sults of the corresponding 3D model can be useful to

define the correct nodal forces release in 2D model.

This comparison has been developed for the shallow

tunnel case of Fig. 18.

The excavation and lining procedure (Fig. 18), is

simulated with nine load cases:

Load case 1. Deactivation of all beams elements.

Load case 2. Soil weight application.Load case 3. Deactivation of ground elements and

relaxation of 30% of the nodal forces.

Load case 4. Activation of steel beams.

Load case 5. 20% further relaxation of the nodal

forces.

Load case 6. Activation of shotcrete support.

Fig. 15. Soil displacements and axial force in the soil nails (deep excavation).

Table 5

Axial force (kN) and bending moment (kNm) in the lining elements (deep excavation)

Case Steel ring Shotcrete support Invert arch

Nmax Mmax Nmax Mmax Nmax Mmax

1 )447 )24 )172 )167 )509 )1612 )520 )33 )188 )151 )424 )1633 )745 )49 )208 )195 )358 )134

Fig. 16. Settlements at the tunnel cross-section heading.


Fig. 17. Vertical displacements at ground surface.

Table 6

Mechanical properties of soil (different values of dilatancy)

Soil w (�) c (kPa) u (�)

Case 1 26 20 26

Case 2 12 20 26

Case 3 6 20 26

Fig. 19. Stresses and displacement (excavati

~4D

~3D

D

~D

Fig. 18. Two-dimensional model, cross-section dimension.



forces.

Load case 8. Activation of concrete invert arch.

Load case 9. Relaxation of 40% of the nodal forces.

The 2D model results are given in Figs. 19 and 20,

and show that the stress and displacement distribution

in the soil is similar to the corresponding 3D distribu-

tion.Specifically, with the assumed sequence of nodal force

release, the settlement values at the tunnel cross-section

heading are almost coincident for 2D and 3D model.

In this sense the 3D and 2D model results can be

correlated.

However, the results of the 2D model generally

strongly dependent on the choice of the fraction of no-

dal forces release, as shown in Fig. 21, where the exca-vation and lining procedure, is simulated with the

following load cases:

Load case 1. Deactivation of all beams elements.

Load case 2. Soil weight application.

Load case 3. Deactivation of ground elements and

relaxation of 40% of the nodal forces.

Load case 4. Activation of steel beams.

on and lining completed, load case 9).

Fig. 20. Heading displacements in 2D model.

Fig. 21. Heading displacements in 2D model, second procedure.



forces.

Load case 6. Activation of shotcrete support.


forces.

Load case 8. Activation of concrete invert arch.

Load case 9. Relaxation of 10% of the nodal forces.

This numerical simulation gives values of the settle-ments at the tunnel cross-section heading very different

with respect to the values of the 3D model.

6. Conclusions

The numerical investigation developed in this study

has shown the possibility to simulate the tunnelling ex-cavation and lining phases using a standard FEM

commercial software.

The use of 3D models can be useful to analyse the real

sequence of soil excavation, face reinforcing and tunnel

lining.

The FEM technique of activation/deactivation of the

structural elements is helpful to develop a simple pro-

cedure for the excavation phases.The numerical results show the efficiency of 3D model

to analyse the face deformation and the ground settle-

ments in the soil, and to evaluate the stress in the lining

elements in the subsequent construction phases.

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three-dimensional modelling of tunnel excavation and lining

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