A probabilistic design approach for tunnel supports

Computers and Geotechnics 32 (2005) 520–534 www.elsevier.com/locate/compgeo

A probabilistic design approach for tunnel supports Pierpaolo Oreste

*

Department of Georesources and Land, Politecnico di Torino, Italy Received 15 November 2004; received in revised form 13 September 2005; accepted 14 September 2005 Available online 9 November 2005

Abstract Probabilistic calculation and design tools are not at present widely used for tunnel supports. A probabilistic approach, when it is possible to have sufficient data on the quality of the rock mass, leads to:
a better understanding of the project risks;
a more efficient geomechanical zoning;
a more reliable estimation of the costs. This paper presents a probabilistic numerical approach for the design of primary tunnel supports, according to the hyperstatic reaction method. This approach considers the probabilistic distributions of the geomechanical index of the rock mass and of the mechanical parameters of the support material, using a Monte-Carlo technique. From the calculation results it is possible to obtain the probabilistic distribution of the support safety factors and then to design the supports on the basis of this probabilistic curve.
2005 Elsevier Ltd. All rights reserved. Keywords: Probabilistic approach; Quality index of the rock mass; Probability density function; Cumulative distribution function; Monte-Carlo procedure; Geomechanical zoning of tunnel section; Rock mass–support interaction; Safety factor; Probabilistic distribution

1. Introduction The rock mass in a tunnel section can be homogeneous from the geological and structural points of view, while the geomechanical quality can vary considerably in detail. This variation has consequences on both the method of excavation and on the supports. The quality of the rock mass influences the geomechanical characteristics and therefore also the complex process of interaction between the rock and the supports. As the variability of the physical characteristics of the rock mass that define the geomechanical quality is unknown and unforeseeable prior to tunnel construction, it is possible to deal with these unknowns in a probabilistic manner. Where there is sufficient data, it is possible to relate these unknowns to a probabilistic density function (PDF) of the quality index. In this way it is possi*

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2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2005.09.003

ble to study the interaction process between the rock and the supports in a probabilistic manner allowing the support to be designed as a structure with a safety factor that is lower than that established using a deterministic approach. It is also possible to evaluate the relative merits of different types of support by considering the probabilistic distribution of the support costs for a section of the tunnel and therefore for the whole tunnel. Some types of support, such as shotcrete, have time dependent mechanical parameters that vary in time and space, and which depend on many factors. Some of these factors relate to the details of the installation method. It is necessary to use probabilistic methods for this type of support, to allow an appropriate characterisation to be made. A probabilistic design approach of the supports, based on the probabilistic density function (PDF) of the quality of the rock mass in each section of the tunnel, also allows an efficient geomechanical zoning of the profile to be made.

P. Oreste / Computers and Geotechnics 32 (2005) 520–534

2. Evaluation of the uncertainty of the geomechanical rock quality GSI and RMR are the most commonly used rock mass characterisation indexes. RMR is defined as the sum of the following five parameters: uniaxial compression strength of the intact rock (rci), rock quality designation (RQD), discontinuity spacing, discontinuity condition, and the presence of water [1–4]. A probabilistic density function of each of the five parameter values can be made for data collected along a scan-line on a rock outcrop, from pilot tunnels, site investigation boreholes, laboratory or in situ tests. The probability is obtained first of all for each previously defined interval of values. This is then compared with standard distribution curves (Gaussian, lognormal and negative exponential curves are frequently used for this type of problem), to obtain the cumulative distribution function (CDF) of each parameter. Starting from the CDF of the constitutive parameters, the probabilistic distribution of the RMR (Fig. 1) can be obtained through, for example, the Monte-Carlo method, if it is possible to assume that the parameters are independent of each other. This method consists in randomly extracting the parameters (on the

a

0.1 0.08 0.06

f 0.04 0.02

.5 41 .5 44 .5 47 .5 50 .5 53 .5 56 .5 59 .5 62 .5 65 .5 68 .5 71 .5 74 .5

0

38

Zoning is currently almost exclusively based on either Bieniawski
s geomechanical classification (RMR) [1–4] or Hoek
s GSI method [5,6]. An initial examination of the distribution of the GSI values around the mean value in each tunnel section allows an evaluation to be made of the necessity to modify the tunnel into zones, each with a separate GSI design value, in order to avoid over-designing the support. The geomechanical zoning should always allow the maximum possible simplification of the project, and a reduction of the number of support types and the number of sections that are considered homogeneous along the length of the tunnel. Structures subject to uncertain load conditions are designed in many fields of engineering and science. One branch of rock mechanics in which the probabilistic approach is used to analyse the distribution of the safety factor is the rock slope stability field [6], but it is only used in some particular cases in tunnelling. This is basically due to the complexity of the interaction between the structure and the rock mass and to the lack of suitable and efficient calculation methods. The design of tunnel supports is therefore usually performed using deterministic methods that incorporate all the uncertain knowledge of the rock mass, the support materials and the adopted calculation methodology in one single safety factor. After having defined the geomechanical characteristics of the rock mass that influence the behaviour of the primary supports, a new procedure for the design of the primary supports is presented in this paper on a probabilistic basis, taking into account the uncertainties of the rock mass quality index and the mechanical characteristics of the materials that make up the support.

521

RMR QUALITY INDEX

b

F

Fig. 1. Probabilistic distributions of the RMR quality index, referring to a 1500 m long section of gneiss and granite gneiss in a 12.3 m wide and 8.9 m high Italian road tunnel (Verbania province). Key: (a) probability density function (PDF); (b) cumulative distribution function (CDF).

basis of their individual probabilistic distributions) and selecting an RMR value that belongs to a gradually increasing sample of data (the calculation is stopped when the statistical tests lead to a stable sample). As the five parameters are generally, at least in part, correlated, it is necessary to prepare special filters [7] that exclude those situations which have no physical meaning during the MonteCarlo calculation (such as those with a high RQD and low mean discontinuity spacing or a high UCS and a high level of discontinuity alteration). For the same reason, it would be better to evaluate the RQD (the second parameter in the RMR system) using a direct correlation with the mean volumetric spacing of the discontinuities, e.g., according to the well-known Palstrom relationship [8] RQD ¼ 115  3:3  J v

ð1Þ

for J v > 4:5; 3

where Jv is the number of discontinuities per m , calculated using the following expression:  n
X 1 Jv ¼ ; ð2Þ dm;j j¼1 where dm,j being the mean spacing value of the jth discontinuity family and n the number of discontinuity families.

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As an alternative to the use of special filters to exclude those situations that have no physical meaning during the Monte-Carlo calculation, it is possible to construct the covariance matrix C for the five parameters that contribute to the RMR evaluation:

ð3Þ

Not only the variances of each probabilistic variable r2xi are present in the C matrix, but also the joined covariances of each couple of probabilistic variable rxi xj . In this way, the statistical dependency among the probabilistic variables (parameters that make up the RMR index) is also described. The variances r2xi and covariances rxi xj in the C matrix are calculated starting from the sample measures carried out in the rock mass. Forming the hypothesis that n measures for each of the five parameters that constitute the RMR index are available, we have: n X 1 2  r2xi ¼ ðxi;k  xi;m Þ ; ð4Þ n  1 k¼1 n X 1  rxi xj ði6¼jÞ ¼ ½ðxi;k  xi;m Þ  ðxj;k  xj;m Þ; ð5Þ n  1 k¼1 where xi or xj is one of the five parameters that contribute to the RMR evaluation and xi,m is its mean value. The RMR theoretical probabilistic distribution can be derived by simply calculating the mean value (RMRm) and the variance ðr2RMR Þ of RMR, in the following way: RMRm ¼

5 X

ðxi;m Þ;

ð6Þ

i¼1

r2RMR ¼

5 5 X X X ðr2xi Þ þ ðrxi xj Þ. i¼1

ð7Þ

i¼1 j:j6¼i

The Gaussian CDF of the RMR can therefore be described using the following equation: 0  2 1 Z RMR m 12 vRMR 1 rRMR @pffiffiffiffiffiffiffiffiffi  e A  dv. CDF ¼ ð8Þ 2pr 1 This theoretical curve can be used for the generation of a sample of random RMR values, necessary to the simulation with the Monte-Carlo method (Fig. 2). A random number among 1 and 1000 is extracted by the calculator using a programming language, and it is subsequently divided by 1000. The value obtained is set on the ordinate axis: from it departs a horizontal straight line that intersects the theoretical curve CDF before defined. The corresponding value of the RMR is the output of the carried out throw and goes to constitute an element of the RMR sample that will be formed with the Monte-Carlo procedure.

Fig. 2. Generation of the sample of RMR values in the Monte-Carlo simulation, using the theoretical curve CDF of the RMR index. In the example shown in this figure is extracted by the calculator the number 652, that corresponds to the number 0.652 on the ordinate axis and to the RMR=60 value.

The procedure proposed in this paper uses the MonteCarlo method together with special filters to exclude those situations that have no physical meaning. This approach has resulted to be quick and the final solution has proved to be close to the more rigorous one that is obtained when using the covariance matrix C. In order to derive a correct evaluation of the RMR index, all the parameters have to be calculated at the same points. If the sources of information for the RMR evaluation come from different points (e.g., the rock strength measured on samples from deep boreholes and the discontinuity spacings from outcrops), the regionalised probabilistic variable approach has to be used. In this case, the distance from the surveying points also influence the probabilistic distribution function of the RMR index, but the mathematical procedure results to be more complex and therefore very slow. The GSI index can be easily estimated from the RMR value [5]. A large diffusion of the GSI and RMR values around the modal value indicates that the tunnel section under consideration, which was considered homogeneous in the zoning process, is in fact not homogeneous, from the geomechanical point of view. This would lead, as a consequence, to an over-design of the support in a large proportion of the tunnel. This problem could be contained by redefining the geomechanical zoning into shorter tunnel sections, to make the support system more adherent to the quality of the rock along the axis of the tunnel. The zoning should be able to discretise the variability of the rock mass quality in detail, providing this is economically and technically viable.

P. Oreste / Computers and Geotechnics 32 (2005) 520–534

Conversely, a small distribution of values around the modal RMR value denotes a tunnel section with a homogeneous geomechanical quality. Where there are adjacent homogeneous sections with a similar average quality, these sections can be linked in such a way that the geomechanical zoning of the tunnel is simplified, without affecting the total cost of the support system.

523

Normal spring Support element Support node Support element

Shear spring

3. The design of supports through the analysis of rock mass interaction The analysis of a support structure should consider the interaction that occurs between the structure itself and the rock mass, this being fundamental to the development of the stress on the inside of the support. A simple but effective way of doing this is to use the hyperstatic reaction method (HRM) [9–15]. This requires the definition of active loads that are applied directly to the support structure by the rock mass; further passive loads are due to the reaction of the rock mass to the displacement of the support structure (interaction between the support structure and the rock mass). The HRM assumes the hypothetical existence of normal and shear springs all around the support (Winkler
s approach) (Figs. 3 and 4). These develop forces that linearly depend on the relative displacements between the structure and the rock mass; having reached a limit stress condition, which is a function of the rock mass strength, any increment in the relative displacements do not produce any further increment in the forces (plastic stage in the rock mass–structure interaction) (Figs. 5 and 6). The normal springs disappear in zones where the support structure moves towards the tunnel: this is generally the case of the roof, but when the horizontal active loads are greater than the vertical ones, it occurs at the sidewalls. Therefore, only compressive loads are possible in the normal direction, where the tunnel support moves towards the rock mass: normal springs only work in compression.

Fig. 3. Scheme of the hyperstatic reaction method (Winkler
s ‘‘independent spring criterion’’) used to analyse the static behaviour of a tunnel support. The tunnel support (S) is shared in uni-dimensional finite elements and nodes. These are linked to the rock mass through normal and shear springs (K), which simulate the rock support interaction. Active loads are applied to the tunnel support by the rock mass in the roof (vertical loads, qv) and on the lateral sides (horizontal loads, qh). The active loads are independent of the displacements that develop in the support and at the rock–support interface.

Fig. 4. Details of the rock–support interaction through Winkler
s springs criterion in the hyperstatic reaction method.

A specific code, named FEMSUP, was developed using a FEM framework, to perform calculations with the HRM. This code is able to consider the effective geometry of the support and horizontal active loads that are different from the vertical ones; it is therefore able to analyse the rock mass–structure interaction in detail. The unknown parameters of the problem are the components of the nodal displacements of the support structure. The evaluation of the unknown components of the nodal displacements is obtained through the definition of the global stiffness matrix of the support structure and of its connections with the surrounding rock mass. The global stiffness matrix is derived from the local stiffness matrixes of each element after they are assembled. The components of the nodal displacements are therefore obtained using the following equation: 1 0 1 0 q1 F1 B q C B F C B 2 C B 2 C C B C B B q3 C B F 3 C C¼B C K B ð9Þ B . . . C B . . . C; C B C B C B C B @ ... A @ ... A qnþ1

F nþ1

where: K is the global stiffness matrix of the support structure, including the connections with the surrounding rock mass; (q1, q2, q3, . . . ,qn + 1)T the unknown components of the nodal displacements of the support structure; and (F1, F2, F3, . . . ,Fn + 1)T the external forces applied to the node of the support structure (active loads). From the components of the nodal displacements it is possible to evaluate, through the local stiffness matrices of each element, the bending moments, the axial and shear forces along the support structure and therefore the stress state inside it. The numerical methods that discretise the continuum around the tunnel perimeter (FEM, DEM) do not result to be suitable for the dimensioning of the support structures (steel sets, shotcrete lining), especially when, as in the case of a probabilistic approach, a large number of calculation cycles are necessary. The computational costs are very high because a great number of elements are necessary to obtain a detailed trend of the bending moments, and of the axial and shear forces along the tunnel support.

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Rock mass

Interface detail

Rock-support interface

p τ

u δ

Support structure Fig. 5. The rock–support interaction along the tunnel profile. Key: p is the normal pressure (pressure acting perpendicularly to the support) due to a displacement u of the support structure towards the rock mass; s the shear stress (stress acting parallel to the support) due to a shear displacement d between the support structure and the tunnel profile; u the radial displacement of the tunnel support (positive if towards the rock mass); d the shear displacement on the rock–support interface.

Fig. 6. Stress–displacement relationship at the rock–support interface, both for normal and for parallel interaction. Key: Kn and Ks are the stiffness of the normal and shear Winkler
s springs that simulate the rock mass response perpendicularly and parallel to the support, respectively; pmax the compressive strength of the rock mass; smax the shear strength of the rock mass.

The HRM, on the other hand, allows a great deal of analyses to be performed in a very short time and for this reason it results to be very suitable for the design of support structure using a probabilistic approach. The HRM has two drawbacks: the exact evaluation of the active loads qv and qh and the definition of the stiffness of the normal and shear Winkler
s springs are necessary. The active vertical load can be estimated using the convergence-confinement method [16], by intersecting the ground reaction curve of the tunnel and the reaction line of the support structure. The convergence-confinement method cannot be directly used to dimension the support structures because it is not able to evaluate the bending moments and shear forces along the structures (only axial forces can be evaluated for the case of a closed support). Very interesting direct correlations were found [4,17,18] (see Section 4, Eqs. (19) and (20)) between the rock mass quality index and the vertical loads on the roof of the tunnel supports. The horizontal loads that are applied to the side walls are usually considered to be some percentage of the vertical ones. Generally, the ratio between the horizontal and vertical loads on the support structure is lower than the in situ stress ratio K0 for a high quality rock mass, while it can be higher than K0 for very crushed rock masses

Fig. 7. Terzaghi
s Rock Load Classification as modified by Deer et al. [19]. Key: C = B + Ht; B the tunnel width; Ht the tunnel height.

and for squeezing and swelling ground. For the Terzaghi
s Rock Load Classification, as modified by Deer et al. [19], the horizontal load (side pressure) on the support structure (Fig. 7) is nil or low when the geomechanical quality of the rock mass is medium or good (for RQD greater than 40%). The horizontal load becomes considerable when the quality of the rock mass is very poor (completely crushed rock). This same conclusion can be reached by analysing the potential instability of the rock mass on the tunnel sides and hypothesing a planar sliding surface (Fig. 8). In the incipient movement condition, the unstabilising forces parallel to the sliding surface are equal to the resistant forces due to the cohesion and friction angle along the sliding surface

P. Oreste / Computers and Geotechnics 32 (2005) 520–534

qv

H qh

qh α=π /4+ϕ /2

α=π /4+ϕ /2 B

Fig. 8. Evaluation of the horizontal loads on the support structures, hypothesing a incipient instability of the rock mass on the tunnel sides along planar surfaces.


 qv  H þ W  sin a  qh  H  cos a tan a
 q H þW ¼ qh  H  sin a  tan u þ v tan a cH  cos a  tan u þ ; sin a

where: H is the tunnel height; qv and qh the vertical and horizontal loads acting on the support structure, respectively; a the inclination of the sliding surface, a ¼ p4 þ u2 ; c and u are the cohesion and the friction angle of the rock mass, respectively; and W is the weight of the potentially H2 instable rock mass block, W ¼ 2tan  c. a Once the previous equation is solved, it is possible to evaluate the qh/qv ratio in function of the vertical load qv, the tunnel height H, the specific weight c and the strength parameters c and u of the rock mass

 u 1  tan qh cH tan a ¼ 1þ  2  qv qv 1 þ tan u  tan a   c 1   . ð11Þ qv sin a  ðcos a þ sin a  tan uÞ It is possible to note that for c = clim, where   c clim ¼ qv þ  H  sin a  ðcos a þ sin a  tan uÞ 2
 u 1  tan tan a  ; 1 þ tan u  tan a

has the shape of a parallelepipedon with a width that is equal to the distance between the springs (c), a thickness equal to the unitary study depth (d) (along the tunnel axis) and a height equal to the depth of influence (L) inside the rock mass. L is the distance from the tunnel perimeter at which the tensional effects produced by the action of the support have ceased to exist: therefore, we can assume that, at a distance of L in the rock mass, the displacements induced by the action of the support structure are nil and that a fixed point can be foreseen for the normal springs in this location. With the hypothesis that the stress effect produced by the support action on the rock mass exhaust completely for a distance L, the following relationship [20] can be obtained by placing the strain in the normal spring equal to the strain in the volume of ground that is represented by it Kn ¼

ð10Þ

ð12Þ

the qh/qv ratio is equal to 0 and the horizontal load is nil. The horizontal load obviously also results to be nil for c > clim. Due to the fact that c, u and qv (see Eq. (8)) depend on the geomechanical quality of the rock mass, we can observe, for a tunnel with an height of 8–10 m, that the active horizontal load is negligible when the RMR is greater than 30. This assumption can only be considered valid for K0 < 1, that is, when the in situ horizontal stress is lower than the vertical one. The evaluation of the stiffness of the normal Winkler
s springs (bedding stiffness) requires a ‘‘volume of influence’’ to be identified in the rock mass for each spring [20]. This

525

Ed  c  d ; L=2

ð13Þ

Kn depends on the elastic modulus of the rock mass E, the distance c between two successive springs and on the depth of influence inside the rock mass L. L is equal to (1.25 ‚ 1.35) Æ Deq (Deq is the equivalent diameter of the tunnel) [15] and the stiffness Kn can be estimated using the following simplified expression: K n ffi 1:5 

E  c  d; Deq

ð14Þ

where c is usually automatically evaluated and d is placed equal to 1 in bi-dimensional calculation methods that analyse a transversal section of the tunnel. The definition of the stiffness of the shear Winkler
s springs is obtained in the same way considering a volume of influence for each spring K s ffi 1:5 

G  c  d; Deq

ð15Þ

where G is the shear stiffness of the rock mass. The ratio between Ks and Kn is therefore equal to G/E and to [0.5/(1 + m)], where m is the Poisson ratio of the rock mass. Due to the fact that m is close to 0.25 for the rock mass, it is generally assumed that Ks/Kn is almost equal to 0.4. Apart from the interaction with the rock mass, the behaviour of the support is influenced by the bending and normal stiffness, the restraining conditions, the magnitude and the type of active load applied by the rock mass to the support. The bending and normal stiffnesses depend on the elastic modulus of the support material and on the geometry of the support section. The type of constraint at the foot can influence the support behaviour to a great extent and depends on the type of foundation and on the geometry of the support. Having determined the parameters that influence the problem, it is possible to obtain the trends of the bending moments, of the normal force and of the shear force along

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P. Oreste / Computers and Geotechnics 32 (2005) 520–534

the support profile from the calculation of the rock mass– structure interaction. These parameters and the geometry of the section of the support being known, the stresses induced inside the support structures can finally be determined. 4. Definition of the parameters that influence the behaviour of a support The main parameters that influence the behaviour of the support are those relative to the interaction with the rock mass. These parameters can be written as functions of the geomechanical quality of the rock mass, thanks to empirical relations obtained from previous experience. For example, for the deformation modulus Ed it is possible to use the Hoek relation [21] which expresses Ed as a function of the GSI
 rffiffiffiffiffiffiffiffi D rci GSI10 Ed ¼ 1    10 40 ½GPa for rci < 100 MPa; 2 100 ð16Þ where rci is the unconfined compression strength of the intact rock material in MPa, and D is a factor that depends on the degree of disturbance to which the rock mass has been subjected due to blast damage and stress relaxation. The Hoek–Brown failure criteria is normally used to determine the limit pressure that induces plastic behaviour in the rock mass at the support–rock interface [21,22]. For a zero confining stress (r3 = 0), the Hoek–Brown criterion is reduced to the following expression: r1;lim ¼ rci  sa ;

ð17Þ

where r1,lim is the main stress limit at the start of plastic behaviour;  1 1  GSI 20 a ¼ þ  e 15  e 3 ; 2 6 s is the Hoek–Brown strength parameter, which can be expressed as a function of the GSI through the following expression: s ¼ eð

GSI100 93D

Þ.

ð18Þ

For a deep tunnel, Unal [18] estimated the vertical load of the rock on the supports as the weight of a ‘‘solid load’’ of height HP, this too being obtained from the RMR:
 100  RMR HP ¼  B; ð19Þ 100 where B is the width of the tunnel. The vertical pressure p (force/length2) acting on the support system is therefore given by the product of the specific weight of the rock c and the height of the ‘‘solid load’’ p ¼ c  H P.

ð20Þ

It can be noted that, thanks to empirical expressions obtained from in situ monitoring results, all the parameters that govern the structural interaction between a support

and the rock mass can be evaluated on the basis of the quality index of the rock mass (RMR or GSI) and on the UCS of the intact rock, both of which can be known in probabilistic terms. These empirical relations obviously also present some degree of uncertainty since they try to describe numerous results of in situ measurements in mathematical terms. The available in situ measurements of the deformation modulus of the rock mass, related to the RMR quality index and three different empirical relations that are able to represent them [23] are reported in Fig. 9. It can be seen that, although Eq. (16) seems to approximate the experimental data rather well, a certain error can exist. This error can be considered in the calculations that evaluate the specific SD re.r. of the generic _ empirical relation y ¼ f ðxÞ, as follows: Pn hðy i _y i Þi2 r2e.r. ¼

i¼1

_

yi

n1

;

ð21Þ

where: xi and yi are the results of generic in situ measure_ ments; y i the value obtained by the empirical relation for an x value equal to xi; and n is the number of the available measurement results. For a defined value of x, it is therefore possible to obtain _ a probabilistic density function of y considering y ¼ f ðxÞ _ as the mean value and ðre.r.  y Þ as the SD. In this way, empirical relations, such as Eqs. (16) and (19), can be correctly used in a probabilistic approach considering their effective level of approximation of real data. The re.r. value calculated for Eq. (16) (Fig. 9) results to be 0.35. This means that for a rock mass with GSI = 60 (D = 0 and rci = 100 MPa), it is possible to define a probabilistic distribution function for the deformation modulus Ed with a mean value of 10((60  10)/40) = 17.78 GPa and a SD of 0.35 Æ 17.78 = 6.22 GPa. The re.r. value calculated for Eq. (19) results to be very low (0.2), thus, denoting its very good approximation of real data. The strength and stiffness characteristics of the support material can also be obtained in probabilistic terms, particularly in the case of shotcrete. Studies on the probabilistic distribution of the strength and stiffness parameters of shotcrete (Oggeri et al. [24] and Oggeri and Peila [25]) have shown that the variability of their values can be described by the Gaussian PDF and that the SD can be consistent (the SD can reach 20% of the mean value). It has also been demonstrated that the Mohr–Coulomb strength criterion can be considered valid for a typical shotcrete, in the case of the absence of reinforcing fibres and after 3 days from its casting. For this type of support, it is therefore possible to derive the probability density function of the elastic modulus, the cohesion and the friction angle from a statistically viable sample of laboratory test results. To obtain the cohesion (ccon) and friction angle (ucon) value of the shotcrete, tri-axial tests are necessary or, as an alternative, the uniaxial compression test together with a shear box test (Fig. 10): the cohesion is initially evaluated from the shear box test and then the UCS (unconfined compression

P. Oreste / Computers and Geotechnics 32 (2005) 520–534

527

Fig. 9. Deformation modulus of the rock mass in function of the quality index RMR: 62 in situ measurements were used to define the most widespread empirical relations [23].

strength) rc,con is obtained from the uniaxial compression test. The friction angle is thus calculated by the following equation: ! 4  ccon  rc;con ucon ¼ arccos . ð22Þ r2c;con þ 4  c2con The tensile strength of the shotcrete (rt,con) is evaluated from the uniaxial compression strength (rc,con), knowing the friction angle (ucon) rt;con ¼ rc;con 

1  sinucon . 1 þ sinucon

ð23Þ

However, is important to emphasise that the mechanical parameters of shotcrete are particularly sensitive to small differences in the casting method and in the mix design.

τ ϕ

c

5. Calculation of the stress state in the support The calculations required to design a support on a probabilistic basis are performed iteratively using a MonteCarlo procedure [26]. A vector of values composed of the GSI, the UCS of the intact rock rci, the elastic modulus and the strength parameters of the material that makes up the support (UCS, tensile strength, cohesion and friction angle) is extracted at each calculation step (throw). Some of these probabilistic parameters are correlated among them: GSI and rci; Econ, rc,con and ccon. ucon is obtained directly from rc,con and ccon; rt,con is obtained directly from rc,con and ucon. The correlated parameters are determined one for time to every throw: the first parameter on the basis of the theoretical curve CDF shown in Section 2, the following ones on the basis of the value of the previous parameter just extracted, of the variances of the two correlated parameters and of the covariance, that describes the correlation level between two probabilistic parameters. For example, the uniaxial compression strength of the intact rock rci is evaluated after having extracted the GSI parameter: in order to obtain a value of rci conditioned by the GSI value just extracted, we can use the normal probabilistic joined distribution of two (x and y) probabilistic variables (Fig. 11), whose expression is the following: fX ;Y ðx; yÞ ¼

σ

UCS

Fig. 10. Mohr–Coulomb strength criterion adopted for the shotcrete.

1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  q2XY "
2
2 1 x  xm y  ym   exp  þ rX rY 2  ð1  q2XY Þ #) x  xm y  y m  2  qXY   ; ð24Þ rX rY 2  p  r X  rY  (

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P. Oreste / Computers and Geotechnics 32 (2005) 520–534

a

k=0.3 - hinge k=0.3 - fixed joint k=0.15 - hinge k=0.15 - fixed joint

bending moment (MN x m / m) .

0.06

0.02 0 0.00

0.96

1.93

3.15

5.41

7.68

9.94

12.2

-0.02 -0.04 -0.06

Fig. 11. Normal probabilistic joined distribution of two (x and y) probabilistic variables.

distance (m)

b

1.2 1

axial force (MN / m) .

where qXY is the correlation coefficient, qXY = rXY/ (rX Æ rY), and rXY the covariance of the probabilistic parameters x and y. The CDF of rci conditioned from the extraction of the correlated parameter GSI is obtained replacing in the preceding expression GSI and rci to the x and y variables and attributing to the GSI parameter the value GSI0 just extracted Z rci CDFrci ¼ ½fGSI;rci ðGSI ¼ GSI0 Þ  dy. ð25Þ

0.04

0.8 0.6 k=0.3 - hinge k=0.3 - fixed joint k=0.15 - hinge k=0.15 - fixed joint

0.4 0.2

1

In the same way we can do for Econ, rc,con and ccon. rc,con is obtained on the basis of Econ value just extracted and ccon on the basis of rc,con value just extracted. The trend of the bending moment and the normal and shear forces along the support profile is then determined for each extracted vector using the hyperstatic reaction method (an example of the trend of the bending moment and the normal force along the support profile is given in Fig. 12). From Fig. 12, it can be noted that, for the analysed example, the k factor and the boundary condition at the base of the pillar only influence the bending moments in the pillar. No considerable influence is shown for the axial forces in the support structure. When the support is only made up of a shotcrete lining, the normal stresses acting at the extrados and intrados are evaluated along its profile, the bending moment and the normal force being known, thanks to the simple static pressure-bending scheme rex ¼

N tcon

þ

6M ; t2con

rin ¼

N tcon



6M ; t2con

ð26Þ

where: N and M are the normal force and the bending moment in the shotcrete lining per metre of length along the tunnel axis: N is positive for compression and M is positive, if the tensile stress is produced in the intrados fibre; tcon is the thickness of the shotcrete lining; and rex and rin are the normal stresses acting inside the support at the extrados and at the intrados, respectively (positive when in compression).

0 0.00 0.77 1.54 2.31 3.60 5.41 7.22 9.03 10.84

distance (m)

Fig. 12. Typical trend of the bending moment (a) and axial force (b) in a half of the support structure obtained using the hyperstatic reaction method. The results refer to the example of Section 7 for different hypothesis: k (ph/pv) equal to 0.30 and 0.15; boundary condition at the base of the pillar constituted by a fixed joint or a hinge. Key: distance: measured along the support profile starting from the base of the pillar and ending in the centre of the roof.

Of all the intrados and extrados stresses, only the maximum tensile value (if there are tensile stresses) and the maximum compression value along the whole support profile are recorded. There is also a maximum value for the shear stresses smax, in correspondence to the barycentric axis of the section, that equals smax ¼ 1:5 

T ; tcon

ð27Þ

where T is the shear force per metre of length along the axis of the tunnel. The maximum shear stress along the support profile is also recorded, together with the normal stress N/tcon that acts in correspondence to the barycentric axis in the same point. The UCS, tensile strength and shear strength of the support material, which are present in the vector of values initially extracted with the Monte-Carlo method, are divided into the three maximum values of the just evaluated stresses

P. Oreste / Computers and Geotechnics 32 (2005) 520–534

along the support profile, in such a way as to obtain three different safety factors concerning yielding due to compression, traction and shear, respectively: F s;com ¼

rc;supp ; rmax;com

F s;shear ¼

slim;supp smax

rt;supp ; rmax;trac N ccon þ tcon  tgucon ¼ ; T 1:5  tcon

(c)

F s;trac ¼

ð28Þ

where: rc,supp and rt,supp are the uniaxial compression strength and tensile strength of the material that makes up the support, extracted with the Monte-Carlo method, their probabilistic functions being known; rmax,com and rmax,trac are the maximum normal stresses due to compression and to traction (in absolute values) acting along the lining profile; slim, supp is the shear strength of the material that makes up the support, which, in the case of shotcrete linings, is a function of the cohesion and the friction angle (whose values can be extracted from a probabilistic function using the Monte-Carlo method), and of the normal force N at the point in which the maximum shear force T develops; and Fs,com, Fs,trac and Fs,shear are the safety factors of the shotcrete lining concerning yielding due to compression, traction and shear, respectively. Further ‘‘throws’’ with the Monte-Carlo method would lead to the forming of three samples for the three previously mentioned safety factors; the procedure is stopped when the samples are stabilised. It is then possible to define a cumulative distribution function for each of the three samples. In some cases, in highly stressed grounds, it is very difficult to impose safety factors of a shotcrete lining that does not yield in some points. The formation of local plastic hinges could in fact be very useful to improve the overall stress distribution in the shotcrete lining. This is the case of the connection between the lining and the temporary invert of the top heading. For this reason, when the formation of local plastic hinges is acceptable, the safety factor evaluation is carried out with reference to the strain state induced inside the lining (reference to the rupture), instead of with reference to the stress state (reference to the yielding), as reported in Eq. (28). In this way the concepts of the local yielding and the local rupture of the shotcrete lining are separated. For steel sets the formation of plastic hinges is never considered admissible and therefore the safety factors are calculated with reference to the yielding of the support. The calculation procedure can be summarised in the following stages: (a) definition of the cumulative density functions (CDF curves) of the parameters that are necessary for the calculation, considering also the correlations between some of the probabilistic variables; (b) extraction of the RMR or GSI, the UCS of the intact rock rci, the elastic modulus and strength parameters for the support material, using the Monte-Carlo

(d)

(e) (f) (g)

529

method (in the case of a shotcrete lining these can be represented by the UCS rc,con, tensile strength rt,con, cohesion ccon and friction angle ucon); definition of the bedding stiffness and of the limit pressure of the rock in the rock mass–structure interaction, and of the vertical load acting on the support, on the basis of the extracted RMR value, considering the grade of uncertainty of the used empirical relations; evaluation of the bending moment, the normal and shear forces along the support profile using a hyperstatic reaction method that is able to correctly analyse the interaction between the support structure and the rock mass; determination of the maximum normal and shear stresses induced inside the support structure; definition of the safety factors of the support; repetition of steps (b) to (f) until the obtained safety factor sample is stabilised and verified through opportune statistical tests (v2 test, for example).

6. Design of the supports using the probabilistic approach The design of supports with the probabilistic approach, as previously illustrated, is made through the analysis of the distribution of the cumulative distribution function (CDF) of the safety factors for the different hypothesised geometrical configurations (in the case of shotcrete, for the different thicknesses). For each considered thickness, having identified the most critical distribution of the three safety factors, it is possible to determine the limit value F
s that is related to a predetermined probability
y (Fig. 13). The safety factor will therefore have the probability
y of being lower than F
s .

F

y FS

Fig. 13. Evaluation of the limit value of F
s , for which the safety factor shows a defined probability
y of being lower.

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value = 74.1). The SD of the measure sample was considered as the better estimate of the SD of the population n X 1  ðRMRk  RMRm Þ2 ; ð29Þ r2RMR ffi n  1 k¼1

Fig. 14. Design of the support structure on the basis of the F
s trend, as a function of the lining thickness or of another geometric parameter that characterises the support.

Taking the values of F
s thus obtained as functions of the thickness or of other geometrical parameters that characterise the support structure, the design is performed by  imposing that F
s is equal to a precise minimum value F
s (Fig. 14). In this way the support is designed, imposing that the safety factor has a probability
y of being lower than a pre  determined value F
s . The values of
y and F
s are estimated on the basis of the representativeness of the samples used to describe the probabilistic functions of the parameters that influence the behaviour of the support and on the basis of the type of calculation method used to evaluate the stresses induced inside the support. It would also be better to make a second condition on the probabilistic function of the safety factors. The minimum value of the distribution should be higher than 1, in such a way that it guarantees that the yielding conditions of the support have not been reached, even in the case of  a safety factor that is lower than F
s .

where n is the number of measures of RMR index (n = 38) and RMRm the mean value of RMR. As the mean value and the SD of the RMR index are known, the probabilistic distribution was obtained, hypothesising valid the Gaussian curve. In the same way, in situ and laboratory tests permitted also an evaluation of the probability density function (PDF) of the UCS of the intact rock to be made. The correlation coefficient between these two probabilistic variables qGSI;rci is equal to 0.36. The deformation modulus of the rock mass (and therefore also the bedding stiffness), the limit pressure in the rock–support interaction and the vertical load acting on the support, were all evaluated using the empirical correlations described in Section 4. The active horizontal loads acting on the tunnel sides were considered to be zero, because the rock mass quality was considered fair-good (see Section 3). Horizontal loads were applied by the rock mass to the support due to the structure deformation (passive loads). The boundary condition assumed at the base of the pillar was a mobile hinge (carriage) (Fig. 15). The mechanical characteristics of the shotcrete were obtained both in a deterministic and probabilistic way (in this second case a Gaussian PDF was considered valid, see Table 1).

7. Calculation example An example of the probabilistic calculation procedure proposed for the design of a shotcrete lining is presented in this section. Here, a 1500 m long section of a 12.3 m wide and 8.9 m high Italian road tunnel (Verbania province) was full-face excavated through gneiss and granite gneiss using the ‘‘drill and blast’’ method. The ground was homogeneous, from the geological point of view. A total of 38 geo-structural surveys provided the data for a probabilistic function of the RMR (Fig. 1, mean l = 57.6; sample SD r = 5.6; (r/l) = 0.097; minimum value = 38.9; maximum

Fig. 15. Static scheme used in the calculation for the considered problem and the position of the 16 nodes of the structure connected to the rock mass through normal and shear springs.

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The output samples of the Monte-Carlo simulation reach the stabilisation after 76,836 throws. The test adopted to verify the reached stabilisation of each sample is that named v2 test. The test requested that the unknown SD of the population falls inside an interval of confidence with an amplitude of 1% of the SD of the sample created by the Monte-Carlo simulation, with a confidence level of 95%. The calculation furnished the probability density function, for a thickness of the shotcrete lining of 20 cm, of the maximum normal stresses (compression and traction). The descriptive parameters of these distributions are given in Table 2. The calculation has permitted to obtain the samples of the safety factors concerning yielding due to compression and traction (the safety factor concerning yielding due to shear is neglected because it resulted to be always greater than the other two). This two samples of data show a certain level of correlation and for this reason the problem has to be considered as a system of probabilistic correlated variables. The descriptive parameters of the probability density function of the safety factors are given in Table 3. The correlation coefficient q between the two safety factors is equal to 0.57 for the case 1 and to 0.59 for the case 2. Knowing, then, the mean values, the SDs and the correlation coefficient of the two samples, it is possible to derive a joined normal distribution (Eq. (24)). Imposing a limit value for each of the two safety factors, it is possible to calculate the probability that at least one of the two safety factors is less than the pre-established limit value. Assuming the limit value of 1.2 for both of them, we obtain a probability of 0.0075 for the case 1 and of 0.0083 for the case 2. Table 1 Mechanical parameters of the shotcrete considered in the calculation example, on the basis of the analysis of laboratory test results Mechanical parameter

Mean value ± SD

Elastic modulus, Econ (MPa) Uniaxial compressive strength, rc,con (MPa) Uniaxial tensile strength, rt,con (MPa) Cohesion, ccon (MPa) Friction angle, ucon () Correlation coefficient, qEcon ;rc;con (–) Correlation coefficient, qrc;con ;ccon (–)

27,000 ± 600 20 ± 1.67 2.3 ± 0.23 3.39 ± 0.26 52.5 ± 2.2 0.43 0.52

531

Table 3 Descriptive parameters of the probability density functions of the safety factors, for a shotcrete lining thickness of 20 cm Distribution parameters

Mean value, l SD, r r/l ratio Median, m Minimum value Maximum value

Yielding due to compression

Yielding due to traction

Case 1

Case 2

Case 1

Case 2

2.41 0.35 0.15 2.36 1.70 4.00

2.42 0.40 0.17 2.37 1.40 4.55

1.95 0.38 0.19 1.87 1.35 5.65

1.96 0.43 0.22 1.88 1.04 6.01

Key: case 1 – strength and deformability parameters of the shotcrete considered in a deterministic manner; case 2 – strength and deformability parameters of the shotcrete considered in a probabilistic manner.

F

a

F

Table 2 Descriptive parameters of the probability density function of the maximum compressive and tensile stresses in the shotcrete lining Distribution parameters

Compression stresses (MPa)

Tensile stresses (MPa)

Mean value, l SD, r r/l ratio Median, m Minimum value Maximum value

8.45 1.14 0.13 8.50 5.04 11.78

1.21 0.18 0.15 1.23 0.47 1.68

b Fig. 16. Cumulative distribution function (CDF) of the safety factors, with reference to yielding, due to compression (a) and to traction (b).

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In the following a less rigorous approach but more simplified, that results to be precautionary (pessimistic and, then, in favour of safety), is presented: the two samples of data of the safety factors are analysed separately, without then consider their reciprocal correlation. Subsequently, in the support dimensioning, the only safety factor that requires a structure more massive and so also more resistant, will be considered. The cumulative distribution functions of the safety factors, with reference to yielding due to compression and to traction, are shown in Fig. 16, for an evaluation of the mechanical parameters of the shotcrete both in a deterministic (case 1) and probabilistic way (case 2). The deterministic case was computed using the mean values of the shotcrete parameters. From an examination of the probability density functions of the safety factors and their descriptive parameters, it is possible to make the following considerations:
The probability density functions show a trend that is similar to the Gaussian curve (see Table 4).
The intervals of existence of the safety factors are greater in case 2 than in the case 1. In case 2, the SDs increased by 15% compared to case 1, while the mean values are almost equal.
The (r/l) ratios for the safety factor concerning yielding due to traction are slightly higher than those concerning yielding due to compression.
The (r/l) ratio for the safety factors is higher than that of the maximum compressive and tensile stresses in the lining (0.13 and 0.15, respectively) and also than that of the RMR (r/l = 0.097).
An analysis carried out considering the mean value of the probabilistic distributions as characteristic parameters of the rock and of shotcrete furnishes a safety factor that coincides with the median value of the probabilistic distributions.

Table 4 Comparison between the safety factor values obtained from the calculation and the theoretical values of the Gaussian curve, for different values of the cumulative frequency F Cumulative frequency, F

Yielding due to compression

Yielding due to traction

Case 1

Case 2

Case 1

Case 2

Obtained results F = 0.1587 F = 0.3 F = 0.5 F = 0.7 F = 0.8413

2.07 2.21 2.38 2.57 2.73

2.01 2.18 2.40 2.60 2.82

1.60 1.75 1.90 2.06 2.28

1.55 1.71 1.92 2.11 2.33

Theoretical Gaussian curve F = 0.1587 F = 0.3 F = 0.5 F = 0.7 F = 0.8413

2.06 2.23 2.41 2.59 2.76

2.02 2.21 2.42 2.63 2.82

1.57 1.75 1.95 2.15 2.33

1.53 1.73 1.96 2.19 2.39

Table 5 Descriptive parameters of the probability density functions of the safety factors, for a shotcrete lining thickness of 10 cm Distribution parameters

Mean value, l SD, r r/l ratio Median, m Minimum value Maximum value

Yielding due to compression

Yielding due to traction

Case 1

Case 2

Case 1

Case 2

1.27 0.23 0.18 1.21 0.88 2.33

1.28 0.26 0.21 1.23 0.73 2.76

2.32 – – – 1.05 –

2.17 – – – 0.84 –

The calculation was also carried out for other values of the lining thickness. Three samples of the safety factors, which were analysed in a statistical way, were obtained for each considered thickness, in the same way as before. The descriptive parameters of the probability density functions of the safety factors with reference to yielding due to

F

a

F

b Fig. 17. Cumulative distribution function of the safety factors, with reference to yielding, due to compression and to traction, for thicknesses of the shotcrete lining equal to 10 and 20 cm. Key: (a) global view; (b) zoom for low values of the cumulative frequency.

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Fig. 18. Trend of the safety factors obtained for a probability
y of 0.01, as a function of the lining thickness, with reference to yielding due to compression and to traction.

compression and due to traction are given in Table 5 for a thickness of 10 cm. No descriptive parameters are given for the safety factors for yielding due to traction as the distribution was rendered invalid as some of the values of the tensile stresses were close to zero. Even for a lining thickness of 10 cm, the interval of existence of the safety factors is higher in case 2 than in case 1; the (r/l) ratios are now greater than in the case of the lining with a thickness of 20 cm. By following the criteria given in Section 6, it is possible to identify the values of F
s with reference to a certain probability
y , with a variation of the thickness. The cumulative distribution function (CDF) of the safety factors and the values of F
s obtained for a probability
y of 0.01 are given in Fig. 17, for the two examined thicknesses (referring to case 2). The values of F
s are given in diagram form in Fig. 18 as a function of the lining thickness. By imposing  F
s ¼ 1:2 as the minimum admissible safety factor, the diagram allows the definition of the thickness (about 18 cm in the illustrated example) that provides safety factors which are higher than the admissible one. The second design criterion that requires a minimum value of the safety factor of the cumulative distribution function higher than 1, imposes the choice of a lining thickness of 20 cm, a value that is slightly higher than that which was identified simply through the diagram in Fig. 18 (first design criterion). 8. Conclusions While there is an enormous quantity of data available from primary and secondary site investigations and tunnel

533

construction, this has not led to a correspondingly widespread use of probabilistic calculation and design techniques for tunnel supports. These techniques are widely used in structural engineering, where it is easier to refer to probabilistic distributions of the parameters that characterise the construction materials. The design of tunnel supports based on a probabilistic approach allows a better understanding of the risks and a more efficient geomechanical zoning of the tunnel profile to be made. This in turn allows an estimation of the costs of the different support systems to be made in relation to the reliability level of the economic evaluations (that is, to the probability that such an estimation is right). A probabilistic approach for the design of preliminary tunnel supports has been presented in this paper (with particular attention being paid to shotcrete and the NATM method), which permits an evaluation of the rock mass– support structure interaction to be made, according to Winkler
s theory. The method considers the probability density function of the rock mass quality and of the mechanical parameters of the support material as calculation parameters, using the Monte-Carlo method for the extraction of the data. This method permits a succession of the safety factors to be obtained. In this paper, the safety factors inside the support have been evaluated with reference to yielding (calculated as the ratio of stresses), but in some circumstances, when local plastic hinges are accepted, safety factors have to be evaluated with reference to rupture (calculated as the ratio of strains). The calculation is stopped when opportune statistical tests verify the stability of the succession. It is then possible to describe the obtained succession of the safety factors through the cumulative distribution function (CDF) and to determine the trend of the minimum safety factor value associated to a certain predefined probability, as a function of one of the geometrical parameters of the support system (the thickness, in the case of a shotcrete lining). The design can be performed by ensuring that the minimum safety factor that is associated to a predefined probability is greater than a predetermined value (first condition). A calculation example has been provided for a shotcrete lining installed in an Italian road tunnel. The probabilistic distribution of the RMR was obtained in a section of the tunnel. This example provided the following conclusions:
the probability density function of the safety factors shows a trend that is very similar to the Gaussian curve;
the interval of existence of the safety factors are larger in the case where the shotcrete strength parameters are known in a probabilistic manner than in the case in which they are known in a deterministic manner;
the (r/l) ratios of the safety factors concerning yielding due to traction are slightly higher than those concerning yielding due to compression;
the (r/l) ratios of the safety factors are higher than those of the maximum stresses in the lining and also than those of the RMR, denoting a propagation of the

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uncertainty in the calculations and, therefore, an increase in the uncertainty during the design of the structures. This phenomenon is greater for smaller lining thicknesses and for the tensile stress state than the compression stress state;
an analysis using the mean value of the probabilistic distributions as representative parameters of the rock and of the concrete furnishes a safety factor that coincides with the median values of its probabilistic distribution. The design, on a probabilistic basis, in the illustrated example, shows that the first condition reached a minimum safety factor that was verified for a lining thickness of 18 cm. Only with a thickness greater than or equal to 20 cm is the second condition also verified, that is, the field of existence of the safety factor succession, obtained from the calculation, is entirely above unity. Acknowledgements The author thanks Professor S. Pelizza, Politecnico di Torino (Italy), Professor E. Hoek (Vancouver, Canada), Dr. P. Varley and Eng. B. Marrai, Knight Piesold Consultant, Ashford, Kent (United Kingdom), for their useful suggestions during the preparation of this work. References [1] Bieniawski ZT. Engineering classification of jointed rock masses. Trans S Afr Inst Civ Eng 1973;15:335–44. [2] Bieniawski ZT. Geomechanics classification of rock masses and its application to tunnelling. Advances in rock mechanics, vol. 2A. National Academy of Sciences: Washington (DC); 1974. p. 27–32. [3] Bieniawski ZT. Rock mechanics design in mining and tunnelling. Rotterdam: Bulkema; 1984. [4] Bieniawski ZT. Engineering rock mass classification. New York: Wiley; 1989. [5] Hoek E, Brown ET. Practical estimates or rock mass strength. Int J Rock Mech Mining Sci Geomech Abstract 1997;34(8):1165–86. [6] Hoek E. Practical rock engineering; 2000. Electronic format: http:// www.rocscience.com/ [link ‘‘Hoek
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