Tunnel Stressess

Stress paths around a circular tunnel Percorsi di sollecitazione attorno ad una galleria circolare Marco Barla∗ Summary The history of stress around a circular tunnel during excavation in a homogeneous, isotropic, linear elastic or plastic medium with a strain softening behaviour, subjected to either isotropic (Ko = 1) or anisotropic (Ko = 2) state of stress, in 3D conditions are considered by using a 3D finite difference model. The results are presented by means of the stress path method with attention paid to the zones located in the crown/invert and in the springlines. It is shown that the stress path in these zones exhibits a more complex trend of behavior with respect to those typical of 2D simplified analyses. The study is to be used in connection with triaxial testing of marly-clay and clay-shales, with the purpose to investigate the behavior of these rocks in short term and long term conditions. Sommario Nella presente nota si affronta lo studio della storia tensionale a cui sono soggetti gli elementi di terreno sul contorno di una galleria circolare durante lo scavo. Tali percorsi tensionali sono definiti utilizzando un modello alle differenze finite tridimensionale, per un mezzo omogeneo lineare elastico o elasto-plastico con condizioni tensionali isotrope (Ko = 1) e anisotrope (Ko = 2). I risultati sono presentati in termini di stress path ponendo l’attenzione sulle zone in prossimità dell’arco rovescio o dei piedritti. Si mostra come i percorsi di sollecitazione in queste regioni mostrino un comportamento più complesso rispetto a quelli di corrispondenti analisi bidimensionali. Lo studio è finalizzato alla simulazione in cella triassiale di tali percorsi con l’intento di studiare il comportamento nel tempo di argille consistenti e argille scagliose.

∗

Department of Structural and Geotechnical Engineering.

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1. Introduction Swelling of soils and rocks is a complex phenomenon with a number of important factors influencing it. One of such factors is the stress history at points in the tunnel surround as face advancement takes place. This can be well described by the use of the stress path representation, as proposed by LAMBE [1967] for a number of applications to Geotechnical Engineering. Therefore, it is of interest to develop in the present paper typical stress paths as created during tunnel excavation, which can be adopted as appropriate input to laboratory testing in three dimensional (3D) conditions. This line of thoughts, which agrees with work previously performed by NG & LO [1985], STEINER [1992], BELLWALD [1990] and ARISTORENAS [1992], is appealing with reference to engineering applications in tunnels, when consideration is to be given to 3D conditions and to the influence of the advancing face. 2. Problem under study The numerical study has simulated the intrinsic behaviour of a deep circular tunnel in homogeneous ground during excavation. The modelled phenomenon is illustrated in Figure 1, where shown is a 10 m diameter circular tunnel, with the surrounding elements where the stress path is computed. Attention has been posed on the behaviour of the elements at the sidewall (S = sidewall) and crown (C = crown), that due to the symmetry conditions is behaving as the invert arch.

A

2. 2D and 3D numerical analyses A number of numerical analyses have been performed using the finite difference element codes Flac and Flac3D [ITASCA , 1996] and the boundary element code Examine3D [ROCSCIENCE, 1998]. Due to the symmetry conditions it has been possible to create a mesh of a ¼ of the real problem in order to optimise computation time. In 2D analysis excavation has been simulated by gradually reducing to zero the forces due to excavation on the tunnel contour. For 3D analysis tunnel excavation has been simulated by removing elements in sequence, for steps of 0.5 m length in the longitudinal direction. The mesh adopted for the analyses is plotted in Figure 2. For the 3D analyses, it is assumed that the excavation has reached the A-A section (where stress paths are computed), which is located at half distance from the vertical limit faces of the model along the longitudinal axis.

C 1m

EXCAVATION DIRECTION

S 5m

1m

A

Fig. 1 - Longitudinal and cross section A-A of the circular tunnel. Fig. 1 – Sezione longitudinale e trasversale della galleria circolare.

The simulation of tunnel excavation proceeds from left to right (Figure 1). Before excavation, the stress state at points C and S depends on the depth of cover and the stress ratio (minimum to

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maximum principal stress, Ko) considered. During excavation, the tunnel face advances, passes trough the A-A section and continues in the opposite direction. Finally, a new equilibrium condition is reached corresponding to the excavation completed with no support installed.

Fig. 2 - Mesh in two and in three dimensions. Fig. 2 – Modello in due e tre dimensioni.

Two different stress conditions, depending on the Ko ratio, have been simulated in the models. For each case a two dimensional and a three dimensional analysis have been performed and the results compared with the closed form solutions available. The ground around the tunnel is assumed to behave according to a linearly elastic isotropic model (e, ILE), with E (elastic modulus) = 400 MPa and ν (Poisson’s ratio) = 0.3, or an elasto-plastic model with strain-softening (p, ELPLA). In the

latter case a limit value (ε 1) for the axial strain (ε) is defined below which the peak strength parameters (cp = cohesion, φ p = friction angle) apply. Then, for values of ε greater than ε 1, the strength parameters are taken to change linearly down to the residual strength parameters (cr, φ r), which hold true for ε greater than the limit value ε 2. A summary of the analyses performed is shown in Table 1.

3D analyses are compared in the same picture. For purpose of comparison of the stress path computed by numerical methods, the closed form solution for a circular hole in a linearly elastic isotropic plate, subjected to an isotropic or anisotropic state of stress is considered. Also plotted are the vertical (σv) and horizontal (σh) stresses as excavation takes place, versus the face position along the longitudinal axis of the tunnel.

Analysis Ko Initial σv 2D-1e 1 1 MPa 3D-1e 1 1 MPa 2D-1p 1 1 MPa 3D-1p 1 1 MPa 2D-2e 2 1 MPa 3D-2e 2 1 MPa 2D-2p 2 1,5 MPa 3D-2p 2 1,5 MPa Tab. 1 - Numerical analysis performed. Ko ratio and σv is the vertical stress. Tab. 1 - Analisi numeriche eseguite. Ko è il spinta a riposo e σv la tensione verticale.

3.1 Elastic analyses

Model ILE ILE ELPLA ELPLA ILE ILE ELPLA ELPLA is the stress

As shown in Figure 3, the results obtained for the 2D elastic Ko = 1 analyses exhibit a stress path which leads to the same state of stress as given by the closed form solution: as the mean normal stress remains constant, the maximum shear stress at the tunnel crown/invert and sidewall is shown to change accordingly. 0.8

coefficiente di

Analytical

0.6

S

Flac 2D

0.4

ARRIVAL OF THE FACE

It is noted that the deformability properties assumed in the calculations are those of BELLWALD [1990] and ARISTORENAS [1992]. For the elastoplastic analyses (for Ko = 2), reference is made instead to the parameters given by G.3S – Ecole Polytechnique [BERNAUD et al., 1993] for the highly fracturated and tectonised clay-shale of the Chaotic Complex, as met during the excavation of the Raticosa tunnel in the Appennines. 3. Results The numerical results obtained in all the analyses performed are described below by depicting the stress path during excavation, which is drawn on the t-s plane, where: σv − σh 2 σ + σh s= v 2

t=

σv and σh are the vertical stress and the horizontal stress respectively. In case the horizontal stress becomes larger than the vertical one this results in a negative t. The different stress paths obtained from 2D and

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t [MPa]

Flac 3D

0.2 Examine3D

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

-0.2 -0.4 -0.6

E = 400 MPa ν = 0,3

HUDER-AMBERG OEDOMETER TEST

ARRIVAL OF THE FACE

C

-0.8

s [MPa]

Fig. 3 - Stress paths for points S (sidewall) and C (crown/invert) for the elastic Ko = 1 analyses. Fig. 3 - Percorsi di sollecitazione per i punti S (piedritto) e C (calotta/arco rovescio) per l’analisi elastica Ko = 1.

The results of the 3D computations, which appear to be in good agreement when comparing the Flac3D and Examine3D stress values, exhibit a different trend of behaviour. As the tunnel face approaches the monitored section A-A the mean normal stress increases. An arrow along the 3D stress path shows the state of stress obtained when the face of the excavation crosses the A-A section. As soon as the face of the excavation overpasses the A-A section, the mean normal stress suddenly decreases and then goes back to the initial value. As shown in Figure 4, this takes place because of an abrupt decrease in the horizontal stress (σh). It is of interest to note that between the highest and the lowest value of s, the excavation proceeds for 2-3 meters only. The

s [MPa]

behaviour is similar, however with an opposite sign for the stresses at the crown. 1.6

Flac3D

1.4

Examine3D

σv

1.2 1 0.8 0.6

AA section

Stresses [MPa]

1.8

0.4 0.2

σh

0 0

20

40

60

80

100

Fig. 6 - Stresses at point S (sidewall) for the 3D elastic Ko = 2 analyses. Fig. 6 – Tensioni nel punto S (piedritto) per l’analisi 3D elastica Ko = 2.

Distance from the beginning of the model [m]

Stresses [MPa]

The results of the elastic Ko = 2 analyses show instead a different trend of behaviour between the sidewall and the crown (Figure 5). In the first case the mean normal stress decreases, in the second it increases. The 3D behaviour is non linear also in this case. As shown in Figure 6 the horizontal stress at the sidewall experiences a sudden decrease when the face of the tunnel reaches the monitored section while the vertical stress slightly increases during all the excavation process. In Figure 7 the corresponding stresses at the crown are reported. It is possible to notice that the changes are less abrupt than for the sidewalls and that the change of s is limited to a few meters of excavation (when the face is near to the A-A section) and probably due to mesh discretization.

3.5

Flac3D Examine3D

3.0

AA Section

4.0

Fig. 4 - Stresses at point S (sidewall) for the 3D elastic Ko = 1 analyses. Fig. 4 – Tensioni nel punto S (piedritto) per l’analisi 3D elastica Ko = 1.

σh

2.5 2.0 1.5

σv

1.0 0.5 0.0 0

20 40 60 80 Distance from the beginning of the model [m]

100

Fig. 7 - Stresses at point C (crown/invert) for the 3D elastic Ko = 2 analyses. Fig. 7 – Tensioni nel punto C (calotta/arco rovescio) per l’analisi 3D elastica Ko = 2.

The numerical results obtained allow one to notice a significant difference between the stresses computed in three dimensional and two dimensional conditions, with a clear influence on the stress path experienced around the tunnel. In order to emphasise this, Figures 3 and 5 also show the typical stress path as obtained in the modified HUDER & AMBERG [1970] oedometer test which is generally used to characterise the swelling ground behaviour. 3.2 Elasto-plastic analyses

Fig. 5 - Stress paths for points S (sidewall) and C (crown/invert) for the elastic Ko = 2 analyses. Fig. 5 – Percorsi di sollecitazione per i punti S (piedritto) e C (calotta/arco rovescio) per l’analisi elastica Ko = 2.

For the failure envelopes used in the computations, strength is exceeded and plastic deformation around the tunnel takes place. In the plastic Ko = 1 analyses the s value decreases strongly with an initial increase for both points S and C (Figure 8). 0,5

2.5

0,4 Flac3D Examine3D

1.5

4

1.0 0.5

0,3 0,2

t [MPa]

σv

ε1= 0,01%

0,1

cr = 20 kPa

φr = 23°

0 -0,1 0

0,5

-0,2

Section

Stresses [MPa]

2.0

σ

E = 400 Mpa ν = 0,3 cp = 30 kPa φp = 25°

S

-0,3

1

1,5 Analytical

C

Flac 2D

ε2= 1%

2

1,0 E = 500 Mpa ν = 0,45 c p = 1 MPa

S

t [MPa]

0,5

φ p = 7° ε 1= 3,5%

0,0 0,0 -0,5

0,5

1,0

1,5

2,0

2,5

3,0

c r = 0,4 MPa 3,5 4,0 φ r = 7°

ε 2= 6%

Ko line Flac 2D

-1,0

Fig. 8 – Stress paths at points S (sidewall) and C (crown/invert) for the plastic Ko = 1 analyses. Fig. 8 – Percorsi di sollecitazione per i punti S (piedritto) e C (calotta/arco rovescio) per l’analisi plastica Ko = 1.

The decrease of s starts when the face of the excavation is still 5-6 m behind. The matter of fact is that a plastic zone is created around the tunnel during excavation (Figure 9). When the elements where stresses are computed change from elastic to plastic behaviour, as soon as the plastic zone (black zone in Figure 10) gets through the A-A section, both the vertical and the horizontal stresses decrease to small values and determine the decrease of s (Figure 11). The change in the state of stress after the tunnel face crossing is small and when the face is just 2-3 m ahead the stresses have reached a new final equilibrium.

Flac 3D

C -1,5

s [MPa]

Fig. 11 – Stress paths at points S (sidewall) and C (crown/invert) for the plastic Ko = 2 analyses. Fig. 11 – Percorsi di sollecitazione per i punti S (piedritto) e C (calotta/arco rovescio) per l’analisi plastica Ko = 2.

In the plastic Ko = 2 analyses performed yielding takes place only at the crown/invert, while the walls are experiencing mainly an elastic behaviour, as well shown by the stress path which is nearly the same as for the elastic analysis (Figure 11).

Fig.

12

-

Plastic zones around the tunnel. Fig. 12 – Zone di plasticizzazione attorno alla galleria. 3,5

2,5

σ

2,0

v

1,5 AA Section

Stresses [MPa]

3,0

Fig. 9 - Stresses at point S (sidewall) for the 3D plastic Ko = 1 analyses. Fig. 9 – Tensioni nel punto S (piedritto) per l’analisi 3D plastica Ko = 1.

1,0 0,5

σ

h

0,0 0

20

40

60

80

Distance from the beginning of the model [m]

Fig. 10 - Plastic zones around the tunnel. Fig. 10 – Zone di plasticizzazione attorno alla galleria.

Fig. 13 - Stresses at point S (sidewall) for the 3D plastic Ko = 2 analyses. Fig. 13 – Tensioni nel punto S (piedritto) per l’analisi 3D plastica Ko = 2. 4,5 4,0

3,0 2,5 2,0

AA Section

Stresses [MPa]

3,5

5

100

σh

1,5 1,0

σ

Fig. 14 - Stresses at point C (crown/invert) for the 3D plastic Ko = 2 analyses. Fig. 14 – Tensioni nel punto C (calotta/arco rovescio) per l’analisi 3D plastica Ko = 2.

The plastic zone around the tunnel has the typical ear shape section as can be seen in Figure 12. By paying attention to point C (crown/invert), it is possible to see that the stress path has two changes in direction during excavation. The first change, where s increases, is due to the plastic zone that intercepts the A-A section, the second one, where s decreases is due to the crossing of the tunnel face (Figures 13 and 14). Also for this case the final equilibrium stress state is reached as soon as the face of the excavation is only a few meters ahead. 4. Conclusions The numerical results obtained allow one to notice a significant difference between the stresses computed in three dimensional and two dimensional conditions, with a clear influence on the stress path experienced around the tunnel. With the 3D analyses a change of the mean normal stress s is evidenced for all the cases under study. For the results pertaining to the elastic Ko = 1 case, the 3D stress path for the sidewall and the crown/invert evidences a variation in the mean normal stress during excavation, which is not shown by the corresponding theoretical solution and 2D results. In the elastic Ko = 2 case, both elements around the tunnel give again a change in the s value: a decrease of s for the sidewall simulation and an increase for the crown/invert respectively. In this case, the change is shown both by the two dimensional and the three dimensional analyses, even though the two dimensional stress path is linear. On the basis of these results, if the swelling behaviour of the tunnel is correlated to a decrease of the mean normal stress [WITTKE,

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1976], this is much more likely for Ko = 2 analyses (at the sidewalls) but it can still occur for the Ko = 1 condition. As the simulation comes near to the most likely ground behaviour, as represented by an elastoplastic constitutive law, the stress paths increase in complexity. For all the cases considered, a decrease of the first stress invariant is evidenced. As shown in Figures 3 and 5 the modified HUDER & AMBERG [1970] oedometer test, which is generally used to characterise the swelling ground behaviour, is not able to reproduce the correct stress history experienced in the near vicinity of the tunnel. It becomes evident that this stress history, in particular near the face of the excavation, can be properly described only by simulating three dimensional conditions, which is possible in a triaxial apparatus. 4. References ARISTORENAS, G. V. 1992. Time-dependent behaviour of tunnels excavated in shale. PhD Thesis. Massachusetts Institute of Technology. Boston, USA. BELLWALD, P. 1990. A contribution to the design of tunnels in argillaceous rock. PhD Thesis. Massachusetts Institute of technology. Boston, USA. BERNAUD, D., H. COLINA , G. ROUSSET 1993. Calculs de dimensionnement du soutenement du tunnel “Linea Alta Velocità” dans les argiles chaotiques. Techincal report, Groupement pour l’Etude des Structures Souterraines de Stockage. Palaiseau Cedex: Ecole Polytechnique. HUDER, J., G. AMBERG 1970. Quellung in Mergel, Opalinuston und anydrit. Schweizerische Bauzeitung. Vol. 88, No. 43, pp. 975980. ITASCA Inc., 1996. Flac2D Ver. 3.3. User’s Manual. Minneapolis, USA. ITASCA Inc., 1996. Flac3D Ver. 1.1. User’s Manual. Minneapolis, USA. LAMBE, T.W. 1967. The stress path method. JSMFD, ASCE, Nov., pp. 309-331. NG, R.M.C., K.Y. LO 1985. The measurements of soil parameters relevant to tunnelling in clays. Can. Geotech, J. Vol. 22, pp. 375-391. ROCSCIENCE Inc., Univesity of Toronto 1998. Examine3D, User’s Manual.

STEINER, W. 1992. Swelling rocks in tunnels: characterisation and effect of horizontal stresses. Eurock ’92. pp. 163-168. Thomas Telford. London, U.K. WITTKE, W., P. RISSLER 1976. Dimensioning of the lining of underground openings in swelling rock applying the finite element method. Pubblications of the Institute for Foundation Engineering, Soil Mechanics, Rock mechanics and Water Ways Construction. RWTH (University) Aachen. Vol. 2, pp. 7-48.

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