Analysis of the Tunnel-Support Interaction Through a Probabilistic Approach

Article history Received: 04-12-2014 Revised: 26-02-2015 Accepted: 02-03-2015 Abstract: The analysis of the behavior of a tunnel requires the evaluation of numerous parameters, relative to the ground and the support structure and which are generally only known with a certain degree of approximation. The uncertainty of these parameters is reflected on the results of the calculation, that is, on the state of stress in the support structure. In order to be able to manage this uncertainty on the representative ground and support structure parameters in a rational way, it is useful to resort to a probabilistic type analysis, making use of the MonteCarlo method and to simple analytical methods for an analysis of the tunnel-support structure interaction. A calculation procedure of probabilistic type that is able to establish representative values of the maximum moment and of the associated normal force in the support structure is presented in this study. This procedure can be used to verify the suitability of the support structure to sustain the induced loads.


Introduction
The analysis of the behavior of a tunnel and the dimensioning of the support structures necessary to guarantee its stability require the knowledge of numerous parameters that are essential for the calculation. These parameters make it possible to describe the ground in which the tunnel is excavated, the entity of the pre-existing stress state at the depth of the tunnel and the geometrical and mechanical characteristics of the support structure that is to be installed. Unfortunately, many of these parameters are only known with a certain approximation, also because they can vary along the tunnel stretch (Karakostas and Manolis, 2000;Oreste, 2005a;Špacková et al., 2013).
For this reason, reference to calculation parameters established according to a deterministic approach could lead to serious problems. Instead, a probabilistic approach allows one to have precise indications on the uncertainty of the calculation parameters and to obtain results associated to the requested level of reliability (Lü and Low, 2011;Fellin et al., 2010;Oreste, 2005b;Kalamaras, 1997;Guarascio et al., 2007a;2007b;2013;Lombardi et al., 2013). In other words, the probabilistic approach is able to deal with the in situ uncertainty of the calculation parameters and of the obtained results in a rational way.
In order to be able to conduct a probabilistic type analysis, it is generally necessary to refer to the Monte-Carlo method, which allows the single uncertain calculation parameters to be extracted randomly, once their probabilistic distribution has been defined (type of distribution, mean value and standard deviation for each parameter considered uncertain) (Oreste, 2006;Tonon et al., 2000).
Moreover, as numerous calculations are necessary for each series of extracted parameters (random vector), it is necessary to adopt simple calculation methods, of an analytical type, that lead to a closed solution which can be reached in a reasonable time (Osgoui and Oreste, 2007;Oreste, 2009a;2013). Numerical type calculation methods (Do et al., 2013;2014a;2014b), which are usually slower and more complex, do not in fact lend themselves to probabilistic type analyses in the tunneling sector.
The convergence-confinement method (Rechsteiner and Lombardi, 1974;Ribacchi and Riccioni, 1977;Lembo-Fazio and Ribacchi, 1986;AFTES, 1993;Panet, 1995;Panet et al., 2001) and the Einstein and Schwartz (1979) method are two of the most commonly used analytical methods in the field of tunneling. The former allows the stresses and strains that develop around a deep circular tunnel to be analyzed as well as the interaction between the tunnel and the support structure (Oreste, 2003a;2003b;2009b): The value of the radial stress acting at the extrados of the support structure is an important result that can be obtained. The latter method allows the maximum moment acting within the support and the normal force associated to it to be simply obtained through the application of a certain load to the support structure.
Once the probabilistic distribution of the maximum moment values and the associated normal force have been obtained, it is possible to proceed with a verification of the capacity of the hypothesized support to absorb its internal induced stresses.
In this study, the convergence-confinement method and the techniques necessary to proceed with a study of the interaction between a tunnel and a support structure are presented first, in order to evaluate the load acting on the support structure. Then, a technique is indicated that can be used to be able to determine the maximum moment and the associated normal force, once the load acting on the support structure is known, through the Einstein and Schwartz method.
Finally, the modalities that can be used to treat the analyses of the behavior of a support structure in probabilistic terms, through the use of the Monte-Carlo method, are given. In particular, the modalities that can be used, at the end of the probabilistic analyses, to obtain the values of the maximum moment and the normal force, from which the maximum stresses in the support structure are obtained, are indicated. These values could then be compared with the strength of the material that makes up the support in order to have indications on whether the hypothesized support structure is adequate, lacking or excessive.

Materials and Methods
The convergence-confinement method allows an analysis to be made of the interaction between a tunnel and a support, on the basis of the following hypotheses: Circular and deep tunnel (depth of the tunnel axis from the surface greater than 10-12 times the tunnel radius R); homogeneous and isotropic ground; constant and isotropic lithostatic stress p 0 (K 0 = 1) around the tunnel (Ribacchi and Riccioni, 1977;Lembo-Fazio and Ribacchi, 1986;Panet, 1995;Oreste, 2009b).
The convergence-confinement curve is the relation that connects the radial displacements of the tunnel wall u R to the pressure applied inside the tunnel σ R . For elastic behavior of the ground around the tunnel, the following is obtained (Rechsteiner and Lombardi, 1974;Ribacchi and Riccioni, 1977;Panet, 1995): Where: E = The elastic modulus of the ground ν = The Poisson ratio of the ground In the case of elastic-plastic behavior, a plastic zone develops around the tunnel when the internal applied pressure σ R is below σ Rpl (radial stress at the border between the plastic zone and the zone with elastic behavior) and when the latter is greater than 0.
Therefore, for σ R ≤σ Rpl and σ Rpl ≥0, the relation between u R and σ R is no longer obtained using Equation 1, but from the following Equation 2, considering the Mohr-Coulomb strength criterion as being valid (Ribacchi and Riccioni, 1977): Where: c p and c r = The ground peak and residual cohesion ϕ p and ϕ r = The ground peak and residual friction angle Ψ = The dilatancy (dilatancy is an angle that can vary between 0 and the residual friction angle of the ground) ( ) Therefore, if σ Rpl <0, the convergence-confinement curve can be obtained from Equation 1, which expresses a linear trend of u R as σ R varies from p 0 to 0. Instead, if σ Rpl ≥0, the convergence-confinement curve is expressed by Equation 1 for σ R varying from p 0 to σ Rpl and from Equation 2, for σ R varying from σ Rpl to 0.
The interaction between the tunnel and the support structure can be studied by overlapping the support reaction line onto the convergence-confinement curve of the tunnel. This reaction line is expressed by the following σ R -u R relation (Panet, 1995;Oreste, 2003a): where, k sup is the stiffness of the support; in the case of a continuous shotcrete lining, the stiffness of the support is given, for example, by the following relation (Hoek and Brown, 1980): E sup = The elastic modulus of the lining material ν sup = The Poisson ratio of the lining material t sup = The thickness of the lining u Ro = The radial displacement of the tunnel that has already developed at the point in which the support has been installed According to Vlachopoulos and Diederichs (2009), this value can be estimated in function of the maximum displacement u Rmax that occurs at a long distance from the excavation face, in function of the distance d from the excavation face at which the support is installed and in function of the value of the plastic radius R pl (σ Req ) for a pressureσ R equal to σ Req (the final pressure acting on the support structure): As both u Rmax and σ Req can be obtained from the intersection of the convergence-confinement curve with the reaction line of the support and they therefore depend on the u Ro , it is necessary to have an iterative procedure that quickly converges to a final value, starting from the initial condition u Ro = 0 ( Fig. 1). At the end of the iterative procedure, a value of the maximum displacement of the tunnel wall u Rmax (at a long distance from the excavation face) and a value of the final load acting on the support structure σ Req are obtained. This load is very useful for verifying whether the initially hypothesized support structure can be considered adequate.
In order to proceed with this verification, it is necessary to refer to the maximum bending moment M max and to the associated normal force N induced inside the support. It is possible to estimate the maximum compression stress and the maximum tensile stress (if it exists) inside the support structure using this previous couple of values (M max and N).
M max and the associated N that act inside the tunnel supports, can be obtained adopting the Einstein and Schwartz (1979). This method belongs to the analytical method category that does not require a numerical solution and it is relatively simple to carry out the calculations. It considers the medium around the lining as being elastic, homogeneous and isotropic and of being of infinite extension, with an initial stress (a different value of the vertical stress and of the horizontal stress) equal to the lithostatic stress. The tunnel lining is treated as an elastic ring, with a normal stiffness and a bending stiffness, which are described by the following dimensionless parameters C * and F * , respectively (Einstein and Schwartz, 1979): Where: A sup = The area of the lining section: A sup = t sup ·1 I sup = The inertia moment of the lining section: I sup = t sup 3

·1/12
Let us consider the support-tunnel wall interface case, which supposes relative displacement without any limitation. It is possible to obtain M max and the associated N through the following simple relation (Einstein and Schwartz, 1979), under the conservative hypothesis that the load σ Req obtained from the convergence-confinement method represents the radial stress acting at the extrados of the support in correspondence to the crown zone: Where: K 0 = The lateral thrust coefficient in litho static conditions (ratio between the horizontal stresses and the vertical ones) a * 0 , a * 2 and α = Dimensionless coefficients: a C F C F 1 2 F 1 6 5 6 1 1 2 K a K 1 3 a K 1 2 In order to be able to obtain the two fundamental values of M max and the associated N, it is necessary to refer to a series of parameters, which are usually known with a certain approximation. In particular, the following characteristic parameters of the ground (c p , c r , ϕ p , ϕ r , E), the lithostatic vertical stress p 0 , the coefficient K 0 and the characteristic parameters of the support structure (E sup , t sup ) can present a certain variability and it is often difficult to attribute a representative parameter for all of the conditions that can be encountered during the excavation of a tunnel.
It would be more appropriate to proceed with a probabilistic study, in which the parameters of the problem are represented by their intrinsic uncertain value; a normal distribution is generally adopted. This distribution is generally interrupted at ± 3·σ of the mean value µ of the distribution (where µ is the mean value and σ the standard deviation of the population). In this way, 0.27% of the values outside the µ±3·σ interval are excluded. Therefore, if a variability interval of an uncertain parameter (β±β·µ, with β expressed in percentage terms) is identified, it is possible to evaluate the standard deviation σ of the probabilistic distribution of the population in the following way: σ = (β·µ)/3. Once the probabilistic distributions of each parameter of the problem have been determined (through µ and σ), it is possible to proceed with a random extraction of these parameters using the Monte-Carlo method (Karakostas and Manolis, 2000;Oreste, 2005a;Fellin et al., 2010;Oreste, 2006;Tonon et al., 2000).
The Monte-Carlo procedure, applied to the convergence-confinement method, allows a value of the σ Req load to be obtained for each extracted vector; each vector is represented by the set of the single values extracted for each probabilistic parameter that characterizes the problem. The procedure therefore leads to the establishment of a sample of σ Req values; the number of values of the sample is considered sufficient when the stabilization of the mean value and/or standard deviation of the sample is verified through an evaluation of the variability of the mean value and/or standard deviation of the σ Req population. For this purpose, the Student probabilistic distribution should be considered to evaluate the variability interval of the mean value and the distribution of X 2 for an evaluation of the variability interval of the standard deviation: Where: As the number of values of σ Req in the sample grows, the estimation interval of the mean value and of the standard deviation of the population of σ Req decreases. A sample that allows the semi-amplitude of the variability interval of the mean value below 1% of the mean of the sample to be obtained with a reliability of 99% and the semi-amplitude of the variability interval of the standard deviation of the population below 2% of the standard deviation of the sample to be obtained with a reliability of 95%, may be considered sufficient.
The thus obtained σ Req sample can then be used to obtain a set of couples of M max and N values though the Einstein and Schwartz (1979) method. Together with the σ Req sample, other samples of uncertain parameters that intervene in the calculation with the Einstein and Schwartz method can be extracted with the Monte-Carlo method; for example, the elastic modulus of the ground E, the elastic modulus of the material that constitutes the support E sup , the thickness of the support t sup and the value of the coefficient K 0 . Again in this case, each single vector of the uncertain parameters produces a couple of values of M max and N. Therefore, the final result is a sample of couples of values of M max and N. This sample can be treated in probabilistic terms in order to identify the value of M max that corresponds to a probability, for example, equal to 99%, that no values of the maximum moment will occur in the support structure above this value. Then, by analyzing the associated N values of all the values of M max close to that probability, it is possible to obtain a probabilistic distribution of the values of N. In this case, the values of N at the ends of the extracted cumulative probabilistic distribution are of particular interest; for example, the values of N relative to a probability of 5 and of 95%, which can be used to identify the values of the maximum compression stress and of the maximum tensile stress (if it exists), in order to compare them with the relative compression and tensile strengths of the material that makes up the support structure.

Results
The probabilistic procedure explained in the previous section has been applied to the case of a circular tunnel, with a 4.5 m radius, excavated in a rock mass with a GSI = 35 (Marinos et al., 2005;Hoek et al., 2013;Marinos and Hoek, 2000;Cai et al., 2004), anuniaxial compression strength of the intact rock σ c equal to 80 MPa and a Hoek and Brown strength parameter of the intact rock m i equal to 18 (Hoek et al., 2002;Hoek and Brown, 1997;Hoek, 2007). On the basis of these parameters, it has been possible to make an estimation of the cohesion c p (0.312 MPa), of the friction angle ϕ p (33°) and of the elastic modulus E of the rock mass (3772 MPa), through the linearization of the Hoek and Brown criteria, as illustrated by (Hoek, 2006). The residual strength parameters have been made equal to the peak values (c r = c p ; ϕ r = ϕ p ). The dilatancy angle has been assumed equal to 50% of the friction angle and the Poisson ratio has been made equal to 0.3. The litho static stress p 0 has been taken as 5 MPa. The support structure is made up of shotcrete lining of a thickness of 0.3 m. A mean elastic modulus of the shotcrete E con of 12000 MPa has been hypothesized, as well as a Poisson ratio ν con of 0.15. The stiffness k sup has been calculated equal to 190.2 MPa/m (Oreste, 2003a;Hoek and Brown, 1980). The distance from the excavation face d at which the support is positioned in the tunnel is equal to 1.5 m.
Given the uncertainty concerning many of the above reported parameters (c p , ϕ p , E, p 0 , k sup ), it was decided to assume a normal probabilistic distribution for each one of them with a mean value equal to the estimated value and the standard deviation obtained on the basis of the probable variability interval (semi-amplitude of the interval equal to 5% of the estimated value, reliability of the estimation equal to 99.73%). According to this hypothesis, the standard deviation of the distribution is equal to a third of the semi-amplitude of the assumed variability interval. The normal distributions were interrupted at a distance of ± 3·σ from the mean value.
The probabilistic parameters were then extracted randomly, according to the Monte-Carlo procedure, in order to obtain a succession of random vectors of dimension 5, in which each component is made up of a random value of one of the five parameters considered variable. The set of the other deterministic parameters, which were instead considered fixed, was then associated to each vector. A load value σ Req , obtained from the calculation according to the modality illustrated in section 2 (through the intersection of the convergenceconfinement curve with the intersection line of the support), was then associated to each random vector. The random vector generation procedure comes to an end when the sample of σ Req is considered stable, in relation to the estimation of the mean value and standard deviation of the population (see section 2). In this specific case, 4539 extractions of the uncertain parameters were necessary and the generated sample of σ Req therefore presents the same number of values. The cumulative distribution of the values of σ Req that were obtained are reported in Fig. 2; values of σ Req varying between about 0.79 MPa and about 0.96 MPa can be observed.
The thus obtained sample of σ Req was then used to proceed with the determination of the maximum bending moment M max and of the associated normal force N induced inside the support structure, through the Einstein and Schwartz (1997) calculation method. Together with the probabilistic variables σ Req , the following parameters were considered uncertain and therefore variable, from the probabilistic point of view: The elastic modulus of the ground E, the elastic modulus of the material constituting the support structure E sup , the area of the section of the support structure A sup , the lateral thrust coefficient K 0 in litho static conditions. A normal type probabilistic distribution was also considered for these parameters and interrupted at ± 3·σ from the mean value of the distribution. The mean value of each distribution is represented by the estimation value of the parameter (E = 3772 MPa, E sup = 12000 MPa, A sup = 0.3 m 2 , K 0 = 0.5); the standard deviation was considered a third of the semi-amplitude of the interval of variability, which was assumed equal to 5% of the mean value.  (Rechsteiner and Lombardi, 1974;Panet, 1995;Oreste, 2003a;Hoek and Brown, 1980) Fig. 2. Cumulative probabilistic distribution of σ Req , obtained from the sample generated using the Monte-Carlo procedure, starting from the normal probabilistic distribution of the parameters considered uncertain Proceeding with the Monte-Carlo method in the same way as previously seen for the convergence-confinement method for each random vector extracted, which was composed of random values of the five uncertain parameters (σ Req , E, E sup , A sup , K 0 ), it is possible to calculate the maximum moment M max on the support structure and the associated normal force N. The procedure continues until the available sample of values of σ Req has been completed.

Discussion
The final result is a set of couples of values of M max -N, which can be well represented as a cloud of points in a M max -N diagram (Fig. 3). The cumulative distribution of the maximum moment M max can also be obtained and as a consequence, it is possible to identify the maximum moment referring to a probability of 99% (0.0168 MN·m/m) (Fig. 4): This value has the probability of being overcome inside the support structure in only 1% of the cases.
It is possible to extract the couples of values M max -N for which M max falls within a relatively restricted interval around the cumulated frequency of 99% from the sample of M max -N couples, for example, between 98.5 and 99.5% (Fig. 3), that is, between 0.01672 and 0.01700 MN·m/M. It is possible to analyze the cumulative distribution of only force N for these couples of values (Fig. 5). By calculating the mean value and the standard deviation of the N of this sub-set of the sample, it is possible to trace the normal cumulative distribution, which is very useful to identify two extreme values of N, for example the force N associated to a cumulative percentage of 5% (N = 3.813 MN/m) and a cumulative percentage of 95% (N = 4.189 MN/m). These two extreme values of N, together with the moment M max relative to a cumulative percentage of 99% (0.0168 MN·m/m) render it possible to obtain a maximum compression strength and a maximum tensile stress (if it exists) which can be compared with the strength of the material that makes up the support structure (which, in this case, is shotcrete).

Conclusion
The parameters that characterize the ground and support structure of a tunnel are usually only known to a certain extent. In order to be able to analyze the statistic conditions of a tunnel, when there is a support structure, it is useful to consider a probabilistic approach that is able to furnish the probabilistic distribution of the stress actions inside the support structure.
The probabilistic approach generally requires resort to the Monte-Carlo method, which allows the uncertain parameters to be extracted randomly, once the probabilistic distribution has been defined for each of them. Moreover, the probabilistic approach should be associated with an analytical calculation method, with closed-form solution, that can manage the remarkable number of analyses that are necessary in contained times.
A probabilistic type of analysis technique that can be used to analyze the behavior of a tunnel and verify the stability conditions of the support structure is presented in this study. This technique, which makes use of the convergence-confinement method and the Einstein and Schwartz method, allows a sample of couples of values of the maximum moment and the associated normal force to be obtained and these values can then be treated from the probabilistic point of view. In particular, it has been possible to determine the maximum moment of the support structure and two values of the normal force, from which it is possible to obtain the representative values of the maximum compression and tensile stresses. These values can then be compared with the strength of the material that constitutes the support structure.
A calculation example, referring to a real tunnel, has been given. This example has made it possible to follow the proposed procedure step by step.