insitu testing

193

The influence of in situ stress state on tunnel design

Antiga Andrea & Coppola Pietro

Soil S.r.l., Milano, Italy

SYNOPSIS: The knowledge of in situ stress has to be considered one of the key input parameter in tunnel design. Several approaches have been developed to analyze the behaviour of a rock mass around a tunnel excavation and to estimate the support pressure required to control the extent of the plastic zone and the resulting tunnel convergence. The development of numerical analysis has provided engineers with an extremely powerful analysis tool; it allows simulating complex in situ conditions and an accurate representation of the soil-structure interactions. Analyses carried out by mean of numerical models reveal that rock behaviour is influenced in a decisive way by the state of in situ stress, in particular the horizontal to vertical stress ratio. Since stresses in rock masses are a fundamental concern in the design of underground excavations, it is very important to measure the stress components and to acquire such information before the design. The purpose of this paper is to investigate the influence of the state of in situ stress on tunnel design. It illustrates and sums up the results of a great number of numerical analyses carried out varying the horizontal to vertical stress ratio k and considering different geomechanical conditions.

1. INTRODUCTION

Stresses in the subsurface are commonly divided into primary and secondary; the primary stress, or in situ stress, is the cumulative product of events in its geological history, while the secondary rock stress is man made by e.g. excavations. Therefore, rock or soil, in natural state, is an uncommon engineering material because it is preload, i.e. there is a pre-existing state of stress in the rock; these loading forces are of unknown magnitude and orientation. The tunnel designer is often inclined to ignore specification and determination of the state of stress. It is generally considered that the behaviour of an underground structure is above all influenced by the relationship between the rock strength and the weight of the overburden; fairly often the ratio of the uniaxial compressive strength of the rock mass to the weight of the overlying strata, i.e. *z where is the unit weight of the overlying material and z is the depth below surface, is the only parameter considered in tunnel design. The state of stress is usually represented with vertical stress component valued equal to the horizontal component (hydrostatic condition), identified with (*z). The same goes for the support pressure required to control the convergences and the extent of the plastic zone.

This common approach ignores that in the majority of stress states measured throughout the world the horizontal component of the stress field has greater magnitude than the vertical component and that the stability of the underground structures is often compromised by mechanisms, for instance bending stress, that are influenced by the state of in situ stress, in particular the horizontal to vertical stress ratio, in a way much greater than the rock strength parameters. The purpose of this paper is to investigate, by means of numerical analyses performed using the finite difference method and the FLAC 2D code, the influence of the state of in situ stress on tunnel design; in particular it analyzes the influence of in situ state of stress on tunnel convergences, on shape and extension of the failure zone and on choice of the most appropriate support. At the start, it is presented an overview of the possible way to predict the magnitudes of the principal stresses. Later on, we illustrate and sum up the results of a great number of numerical analyses carried out varying the horizontal to vertical stress ratio k and considering different geomechanical situations.

2. IN SITU STRESS STATE

The tunnel engineer has always to consider that the rock medium is subject to initial stress prior to

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excavation; so the final, i.e. post excavation, state of stress in any underground structure is the resultant of the initial state of stress and of the stresses induced by excavation. Since induced stresses are directly related to the initial stresses, it is clear that it is a necessary precursor to any design analysis. Measuring the in situ stress is demanding and time-consuming but, since stresses in rock masses are a fundamental concern, it is very important to measure the pre-existing stress components and to acquire such information before the design.

2.1 Stress condition

The in situ stress state is generally described by the orientations and the magnitudes of the three principal stresses assuming an approximation that they are one vertical component and two horizontal components. Following this assumption concerning orientations, it becomes possible to predict the magnitudes of these principal stresses through the use of elasticity theory. The in situ principal stresses are in general different and are connected to the geological history. Changes in the state of stress in a rock mass may be related to temperature changes and thermal stress, and chemical and physicochemical processes such as leaching, precipitation and re-crystallisation of constituent minerals. Mechanical processes such as fracture generation slip on fracture surfaces and viscoplastic flow throughout the medium can be expected to produce both complex and heterogeneous states of stress. The vertical stress is mostly based on the depth and density of rock; we might expect that the vertical component increases in magnitude as the depth below the ground surface increases due to the weight of the overburden; so this stress is estimated from this relationship:

zv = (1) where: is the unit weight of the overlying material, z is the depth below surface. In areas of uniform bedrock structure, for example, sedimentary basins, the vertical force at a known depth is dependent on the weight of overlying rock according to hydrostatic pressure. In areas of more complex geology, for example crystalline, hard rock, the vertical stress does not follow this rule with such accuracy.

Measurements made of the in situ stress, in various mining and civil engineering sites around the world, confirm that the estimate (1) of the vertical stress component is basically correct although there is a significant amount of cases where the predicted component is different to the measured component; there are cases at depth less than 500m where the measured value is about 45 times the predicted value. The horizontal stress is much more difficult to estimate. The main sources for horizontal forces are continental plate tectonics and vertical movements of less dense areas of bedrock. It is globally dominant near the surface. Usually, the ratio of the average horizontal stress to the vertical stress is denoted by the letter k:

vh k = (2) The measurements made of the horizontal in situ stress allow determining two formulae as envelopes for all data (Hoek [3]):

+

195

vertical stress component, in particular at depths typical of civil engineering. We can at last observe that, the previous equations and the existing measurements provide a good predictive estimation of the in situ stresses, particularly for the vertical component; however they are not reliable to give an adequate estimation in a specific location (Figure 1). The previous comments also point out that the in situ state of stress in a rock mass is not amenable to calculation by any known method and must be determined experimentally.

2.2 Systems of stress estimate

The direct experimentally determination of in situ stress presents some difficulty. In particular, the spatial variability of the stress tensor suggests that any single experimental determination may bear little relation to volume averages of the tensor components. In the design of an underground structure, it is the average state of stress in the zone of influence of the structure which exerts a primary control on the rock behaviour and on the tunnel stability. A satisfactory determination of a representative solution of the in situ state of stress is not possible

with a small number of random stress measurements. The solution is to develop a site-specific strategy to sample the stress tensor at a number of points in the mass, taking account of the rock structure. It may then be necessary to average the results obtained, in away consistent with the distribution of measurements, to obtain a site representative value. Moreover, stress measurements faces a basic obstacle in that stress is not a physical phenomenon that can be measured directly but it is a concept defined within the framework of continuum mechanics. It is possible to determine the mean stress provided a relationship between the mean stress and a measurable effect that this stress produces over that region (Faihrust [2]): = (5) where is an operator that translates the measured quantity into the stress. The simplest example of operator is the elastic modulus E of an isotropic elastic specimen loaded in uniaxial compression generating the measured uniaxial strain . All stress determination techniques require a relation of the equation (5) and the validity of the stress determination depends of the degree of reliability of this relation.

Eh(GPa) 10

25

50

75

100

Figure 1. Vertical stress and ratio horizontal to vertical stress k (after Hoek 1998)

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The stress state in a rock, or soil, mass is described by six parameters; it is generally presented in terms of the magnitude and orientation of the principal stresses (Figure 2). The stress state is completely described by six parameters and so any system utilized for estimating the in situ stress state must involve a minimum of six independent measurements. It is possible to distinguish between direct and indirect methods of estimating the in situ stresses. The ISRM recommends four direct methods [5]: (1) flat jack test, (2) hydraulic fracturing test, (3) USBM overcoring torpedo and (4) CSIRO overcoring gauge. In Figure 3 the stress tensor obtainable by a single application of each of the cited ISRM direct methods is represented. Among the indirect methods we mention: borehole breakouts-damage to a borehole indicating principal stress orientations, acoustic emission-the rock emits low-intensity noise when it is stressed, observations of discontinuity states, e.g. open discontinuities are not transmitting stress across the gap. An accurate description of methods used to determine the in situ stress is reported in Hudson [4].

3. NUMERICAL ANALYSES: GEOMECHANICAL CHARACTERIZATION

In order to estimate the influence of the state of stress on convergences, shape and extension of the failure zone and actions eccentricity a great number of numerical analyses have been carried out. The numerical analyses have been carried out using

invariable rock mechanics characteristics and varying overburden so to investigate different geomechanical behaviour according to Hoeks squeeze theory. Hoek [7] published details of an analysis showed that the ratio of the uniaxial compressive strength cm of the rock mass to the in situ stress P0 can be used as an indicator of potential tunnel squeezing problems. One of the most interesting aspects of this approach is that potential tunnel squeezing problems are predicted in terms of dimensionless parameters (the ratio of the uniaxial compressive strength cm of the rock mass to the in situ stress P0 and the ratio of convergence to the tunnel radius i.e. strain ). Figure 4 shows Hoek categories and the relationship between the strain and the potential tunnel squeezing problems associated with tunnelling through squeezing rock. In order to analyze the influence of the in situ stress on the analyses results, we have made reference to a "very poor quality rock mass", as defined by Hoek [3], varying the value of the tunnel depth so to cover, with the numerical analyses, the entire range of deformations and conditions of squeeze proposed by Hoek (Figure 5). After all, the rock mass chosen for discussion is characterized by the parameters summarized in Table 1. We observe that the significant depths are: 140 m (case A), 223 m (case B), 315 m (case C) and 443 m (case D) (Figure 6). Online general, we think that the brought back analyses and the results obtained can be extended also to soil or rock mass with mechanics characteristic different from those indicated, conserving the assumption of perfectly plastic behaviour, having like reference the value of squeeze .

23

1

00

0 302001

Figure 2.

197

xx xy xzyy yz

zz

One normal stress component determined, say parallel to x-axsisSymm.

1. Flatjack

1 0 02 0

3

Principal stresses assumed parallel to axes i.e. plane of the fracture, two determined, say 1 and 3, one estimated, say 2.Symm.

2. Hydraulic fracturing

xx xy xzyy yz

zz

Three components in 2-D determined from three measurements of borehole diameter change.Symm.

3. USBM overcoring torpedo

All six components determined from six (or more) measurements of strain at one time.

3. CSIRO overcoring gauge

xx xy xzyy yz

zzSymm.

Figure 3. (after Hudson1997)

Figure 4. Hoek categories of tunnel squeezing (after Hoek 2000)

Table 1. Parameters of the rock mass chosen for discussion

Intact rock strength ci 20 MPa Poissons ratio 0.3

Hoek-Brown constant mi 8 Dilation angle 0

Geological Strength Index GSI 30 Friction angle 24

Rock mass compressive strength cm 1.7 MPa Cohesive strength c 0.55MPa

Deformation modulus Em 1400 MPa

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0.00

2.00

4.00

6.00

8.00

10.00

12.00

50 100 150 200 250 300 350 400 450

DEPTH (m)

STR

AIN

(%)

C

A

B

D

E

H= 315 m

= 10%

= 5%

= 2.5%

= 1%

CASE C

H= 223 mCASE B

H= 140 mCASE A

H= 445 mCASE D

Figure 5. Tunnel depths chosen for discussion

4. INFLUENCE OF IN SITU STRESS ON

CONVERGENCE AND FAILURE ZONE

In a previous paper [1] Antiga, Chiorboli & Coppola analysed, for a circular tunnel, the influence of a not hydrostatic state of stress on convergence and on the shape and extension of the failure zone. The results of the studied cases show that:

(a) for horizontal to vertical stress ratio k 1 the hydrostatic conditions provide valid results. Some corrective coefficients were defined for convergence (CFconvergence) and failure zone dimensions (CFfailure zone), useful to evaluate the effects of k 1 starting from results obtained by hydrostatic conditions.

(b) for horizontal to vertical stress ratio k 1.5 the results obtained with hydrostatic solutions solution differ in a substantial way from those obtained from the not hydrostatic solutions; the

differences are greater when geomechanical conditions become worse. In these cases not hydrostatic conditions methods can be useful only to make qualitative parametric studies and it will be necessary to use numerical models also in the preliminary stages of design, taking into account real tunnel geometry and rock mass conditions.

In Table 2 the values of convergence to the equilibrium (i.e. p=0) for all the analyzed cases are brought back; the convergence is normalised to the value find by hydrostatic conditions (i.e. ratio of not hydrostatic solutions convergence to hydrostatic conditions convergence is represented). In Figure 6 a prudent corrective factor (CFconvergence), for circular shape cross section, is fixed to calculate the correct convergence value from hydrostatic solutions value.

Table 2. Ratio of numerical convergence to analytic convergence

Tunnel Depth (m) k =0.5 k =0.75 k =1 k =1.5 k =2

A - 140 0,81 0,85 1 2,09 3,94B - 223 0,78 0,84 1 2,26 4,06C - 315 0,78 0,83 1 2,14 3,69D - 445 0,81 0,83 1 1,94

average 0,79 0,84 1 2,11 3,90

= MAX / THEORETICAL

Table of differences of displacements

199

Figure 6. Corrective factor of convergence - (CFconvergence)

The Table 3 reassumes the results regarding

the shape and the extension of the failure zone for circular shape cross sections. The dimensions of the failure zone are normalised to the failure zone radius of the hydrostatic conditions solutions. It is observed, as already seen for convergences, that the results obtained with the hydrostatic conditions solution are valid until the coefficient k reach a limit value. In particular, up to k=1.5 it is possible to use for the definition of the shape and of the extension of failure zone the results of the hydrostatic conditions formulation adopting the corrective coefficients defined in table 3. For k > 1.5 the failure zone shape becomes strongly irregular and it is not possible to define a corrective coefficient.

5. INFLUENCE OF IN SITU STATE OF STRESS ON SUPPORT ACTIONS

In order to estimate the influence of in situ stress state on eccentricity of support actions, n. 80 new numerical analyses have been carried out. We have considered a circular tunnel section with excavation area of 100 square metres (R=5.64 m), we have varied horizontal to vertical stress ratio k (0.50, 0,75, 1.00, 1,50 and 2.00) and tunnel lining thickness (0.25 m, 0.50 m, 0.75 m and 1.00 m). The numerical analyses have been developed using the finite difference element codes Flac 2D (Itasca). The ground around the tunnel has been assumed to behave according to a perfectly plastic, homogeneous and isotropic medium. Several different geomechanical conditions have been considered according to hypothesis of geomechanical characterization defined in chapter

n.3 (case A, B, C and D). The numerical study is based on the Mohr-Coulomb failure criterion which gives a very simple solution for the progressive failure of the rock mass surrounding the tunnel; moreover the FLAC code is built to work directly with Mohr-Coulomb failure criterion and generates a Hoek-Brown failure surface by manipulating the Mohr-Coulomb model. The analyses have been performed varying the tunnel depth (z) and the ratio of the average horizontal stress to the vertical stress k. In Table 4 the maximum eccentricity (e=M/N) values, determined for various lining thickness (25, 50, 75, 100) cm are resumed. The value e in each table is the medium eccentricity value. We observe that, for a definite thickness and k, the eccentricity maximum value for internal actions is basically a constant, also varying the geomechanical conditions. Moreover, the value of the eccentricity is almost the same for values of k equidistant from k=1, i.e. for internal actions there are not appreciable differences inverting horizontal with vertical stress. From now on eccentricity is the medium eccentricity value e. In Figure 7a the trend of e value is represented depending on k (horizontal to vertical stress ratio) for each of the analyzed thickness. We observe that the eccentricity value increases in a considerable way increasing lining thickness (eccentricity e is plotted in logarithmic scale). In Figure 7b the trend of e* value is represented depending on k for the four analyzed lining thickness (e* is the value of e normalised to lining thickness, i.e. ratio of eccentricity to lining thickness. e* = e / t in percentage).

200

Table 3. Corrective coefficients for failure zone - (CFfailure zone)

Table 4. Eccentricity (e=M/N) values

Tunnel Depth (m) k =0.5 k =0.75 k =1 k =1.5 k =2A - 140 0,004 0,001 0 0,002 0,003B - 223 0,004 0,001 0 0,002 0,003C - 315 0,003 0,002 0 0,002 0,004D - 445 0,003 0,002 0 0,002 0,004

e (m) 0,004 0,002 0 0,002 0,004

for a liner of thickness 0.25 mEccentricity of internal actions e = M/N


e (m) 0,024 0,008 0 0,012 0,021

Eccentricity of internal actions e = M/Nfor a liner of thickness 0.50 m


e (m) 0,067 0,023 0 0,035 0,063



e (m) 0,142 0,049 0 0,072 0,142


Eccentricity of internal actions e=M/N

0,000

0,001

0,010

0,100

1,0000,5 1 1,5 2

K (horizontal to vertical stress ratio)

e (m

)

0.25 m (t/R=0.04)

0.50 m (t/R=0.09)

0.75 m (t/R=0.13)

1.00 m (t/R=0.18)

a

Percentual eccentricity e* of internal actions (e/thickness)

0,000

0,001

0,010

0,100

1,000

10,000

100,0000,5 1 1,5 2


e* (%

)

0.25 m (t/R=0.04)

0.50 m(t/R=0.09)

0.75 m (t/R=0.13)

1.00 m (t/R=0.18)

b

Figure 7. Medium eccentricity value e

0.44

4.971.113.791.06

6.271.08

1.500.861.570.871.470.851.470.79

1.00

1.00

1.00

1.00

1.17

0.701.040.661.040.671.070.651.08

1.110.441.16

0.451.110.42

A

B

C

D

K0 = 2K0 = 0.5 K0 = 0.75 K0 = 1 K0 = 1.5

Rp Rpmin

Rpm

ax

RpmaxRpmax RpminRpm

in

Rpm

inRpK1Rp=

Rpmin

Rpmax

Rpmin

Rpmax

Rpmin

Rpmax

Rpmin

Rpmax

RpK=1 Rpma

x

201

In Table 5 the value of e* are summarized. It should be noted that, if for the same k we normalize the value of e* to the value of e* obtained for t = 0.25m, we obtain a progression of values on average equal to 1, 3, 6, 9; this observation highlights that the influence of not hydrostatic state of stress is higher when the lining thickness increases and the internal actions eccentricity increase more than proportionally compared to basic value t of lining thickness. In Figure 8 the minimum and the maximum stresses for analyzed cases are summarized; the stresses are normalised to hydrostatic conditions stress, i.e. ratio of (k 1) for to hydrostatic conditions stress (k = 1). Stress acting in a general situation (k 1) (unitary length):

( )*2,1 e61tN = (6) Stress acting in hydrostatic condition (k = 1) (unitary length):

tN

1K = = (7)

Stress acting in a general situation (k 1) normalised to stress acting in hydrostatic conditions (k = 1) (unitary length):

( )*2,1* e61 = (8) For analyzed situations, circular tunnel and horizontal to vertical stress ratio (0.5 k 2.0), the lining cross-section remains entirely in compression (i.e. there is no traction stresses). For not hydrostatic conditions, we observe that increasing lining thickness the maximum normalised stresses rises while the minimum normalised stresses decreases; so the growth of lining thickness handicaps the behaviour of the resistant lining cross section that resists in a heavily not homogeneous way (maximum stress much greater than minimum stress). This effect is greater for values of k farther from k=1 (for k=1 maximum stress equal minimum

stress). Further developing analyses, that will be theme of a future paper, reveal that for not-circular cross section these effects turn much more critical and marked choking of the resistant lining cross section happens.

6. CONCLUSIONS

In case the real in situ stress state it is not defined properly and rough analyses in hydrostatic conditions are carried out, the obtained results diverge from the real ones as a function of the value of the horizontal to vertical stress ratio k. The analyses carried out, for circular tunnel, enable to draw some general conclusions useful to weigh the reliability of the design of an underground structure.

(1) Convergence of an unlined tunnel

- (1a) for k < 1.5 the hydrostatic conditions provide valid results; we observe values of convergence that differ from real values at the most of 2030 %; it is possible to correct the hydrostatic convergence values by mean of corrective factors (CFconvergence) defined at chapter n.4;

- (1b) for horizontal to vertical stress ratio k 1.5 the results obtained with hydrostatic solution differ in a substantial way from those obtained from the not hydrostatic solutions; it is not possible to define corrective factors; the results can be useful only to make qualitative studies.

(2) Shape and extension of the failure zone of an unlined tunnel

- (2a) up to k=1.5 it is possible to use for the definition of the shape and of the extension of failure zone the results of the hydrostatic conditions formulation adopting the corrective coefficients defined in Table 3;

Table 5

t (m) k=0.5 k=0.75 k=1 k=1.5 k=20.25 1.4 0.6 0 0.8 1.40.5 4.75 1.6 0 2.4 4.250.75 8.867 3.067 0 4.667 8.4

1 14.2 4.9 0 7.2 14.2

e* = (e/t)t (m) k=0.5 k=0.75 k=1 k=1.5 k=20.25 1 1 0 1 10.5 3 3 0 3 30.75 6 5 0 6 6

1 10 8 0 9 10

e*(t)/e*(t=0.25)

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Thickness=0.25m (t/R=0.04)*=(16e*)

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

20,5 1 1,5 2


*

1

2

Thickness=0.50m (t/R=0.09)*=(16e*)

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

20,5 1 1,5 2


*

1

2

Thickness=0.75m (t/R=0.13)

*=(16e*)

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

20,5 1 1,5 2


*

1

2

Thickness=1.00m (t/R=0.18)*=(16e*)

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

20,5 1 1,5 2


*

1

2

Figure 8. Medium eccentricity value e - (2b) for k > 1.5 the failure zone shape becomes

strongly irregular and it is not possible to define a corrective coefficient.

(3) Eccentricity of support actions - the influence on action eccentricity of not

hydrostatic state of stress is higher when the lining thickness increases;

- for all analysed cases (0.5 k 2.0), even if the eccentricity increases for higher k, the cross-section remains completely in compression stress;

- for k 1 we observe an increasing in lining stresses that became more and more significant for higher thickness and for value of k more different from k=1.

All the cited influences, of the k value on structural and geomechanical behaviour of a tunnel, are evaluated for a circular cross section. It is evident that, the effects of a not hydrostatic state of stress will be greater for a not circular cross section; in that case an erroneous evaluation of the in situ state of stress can easily lead to an inadequate design. This will be subject of further studies.

REFERENCES

1. Antiga A., Chiorboli M., Coppola P. - Convergence-confinement method: limit of application of the closed form solutions compared with numerical models. ECCOMAS Thematic Conference on Computational Methods in Tunnelling (EURO:TUN 2007). Vienna, 2007.

2. Fairhurst C: Stress estimation in rock: a brief history and review. International Journal of Rock Mechanics and Mining Sciences 40 (2003) 957-973. Elsevier Ltd.

3. Hoek E, Brown ET: Underground Excavations in Rock. Champan & Hall: London, 1980.

4. Hudson J A: Engineering Rock Mechanics -Elsevier Science Ltd 1997.

5. ISRM: Suggested Methods for Rock Stress Estimation Part 1, 2, 3, 4; October 2003.

6. Sheorey, P.R.: A theory for in situ stresses in isotropic and transversely isotropic rock. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 311:23-34, 1994.

7. Hoek E, Marinos P: Predicting tunnel squeezing problems in weak heterogeneous rock masses. Tunnels and Tunnelling International Part 1 November 2000, Part 2 - December 2000.

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BIOGRAPHICAL DETAILS OF THE AUTHORS

Andrea Antiga obtained the Doctorate in Civil Engineering at the University of Padua (Italy) in 1989.

From 1991 he has been involved in some of the most important tunnel project in Italy.

He began to work at Rocksoil S.p.A. (Milan - Italy) specializing in tunnelling and underground constructions.

Since 2002 he has been worked at Soil S.r.l. (Milan Italy), specializing in geotechnic and tunnelling, of which

he is Technical Director. At present, he is working at the design of some important geotechnical work and tunnels, included a stretch of a subway line in Milan.

Pietro Coppola graduated in Civil Engineering in 2002 at the State University of Bologna ( Italy ).

He specialized in seismic engineering and in numerical modelling of FEM and FDM structures and soil.

He mainly deals with civil engineering design, geotechnical and underground works.

He has been collaborating with Soil s.r.l. (Milan-Italy) since 2003.

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