POSTER PAPER PROCEEDINGS

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1 ITA - AITES WORLD TUNNEL CONGRESS April 2018 Dubai International Convention & Exhibition Centre, UAE POSTER PAPER PROCEEDINGS

2 The influence of over consolidation ratio on the mechanical behavior of conventional shallow over burden tunnel Muhammad Shehzad Khalid 1, Mamoru Kikumoto 2, Ying Cui 3, and Kiyoshi Kishida 4 1 Department of Urban Management, Kyoto University, Nishikyo-Ku, Kyoto, , Japan, khalid.shehzad.25e@st.kyoto-u.ac.jp 2 Department of Civil Engineering, Yokohama National University, Yokohama, Kanagawa, , Japan, kikumoto@ynu.ac.jp 3 Department of Civil Engineering, Yokohama National University, Yokohama, Kanagawa, , Japan, sai-ei-mx@ynu.ac.jp 4 Department of Urban Management, Kyoto University, Nishikyo-Ku, Kyoto, , Japan, kishida.kiyoshi.3r@kyoto-u.ac.jp ABSTRACT Until the recent past, the open-cut method had usually been applied for excavating shallow overburden tunnels in unconsolidated grounds. The conventional tunneling method [the New Austrian Tunneling Method (NATM)] had been thought to be suitable for excavating tunnels in mountainous terrains. However, recent advancements in construction and measurement techniques have enlarged the range in application of the NATM to loose grounds, and this method has become popular for shallow overburden tunnel excavations in soft grounds for economic reasons. In the current study, the mechanical behavior of the ground during shallow circular tunneling by NATM is explored under varied overburden heights (H/D = 0.5, 1.0, 1.5, and 2, where H is the depth from the free ground surface to the tunnel crown and D is the diameter of the tunnel) and various over-consolidation ratios (i.e. normally consolidated, lightly over consolidated, and highly over consolidated soils) through various two-dimensional finite difference analyses. The ground is modeled by famous and widely used Modified Cam Clay (MCC) model which obeys an associated flow rule. The MCC model is an incremental hardening/softening elastoplastic model which includes a particular form of nonlinear elasticity, and a hardening/softening behavior governed by volumetric plastic strain. The ground reaction curve concept in conjunction with the stress path is applied as a conceptual tool to interpret the mechanical response of the ground to tunneling. Few consistent behaviors were observed for normally, lightly, and heavily over consolidated soils irrespective to the various overburden ratios. For normally consolidated soils the behavior at the crown follows the hardening trend while at the spring line the behavior follows the initial hardening followed by the softening of the yield surface. In case of lightly over consolidated soils the behavior at the crown remains within elastic limit while at the spring line the behavior follows the softening trend after reaching to the existing yield surface whilst for the heavily over consolidated soils the behavior at the crown and spring line remains within elastic range. In the end, relaxation values for different over consolidated soils are suggested from practical design point of view. Key Words: Shallow overburden tunnel, Convergence confinement method, Over consolidated soils, Ground reaction curves, Stress path 1. INTRODUCTION Due to the recent advancement in construction technologies the scope of the New Austrian Tunneling Method (NATM) has extended to the shallow overburden tunnels. The design of NATM tunnels is currently based on empirical methods (Barton et al., 1974; Bieniawski, 1974). Meanwhile, analytical techniques and two-dimensional or 1

3 three-dimensional numerical methods are collectively known as the Convergence- Confinement Method (CCM), which can account for elasto-plastic ground-support interaction by providing equations for calculating the stress-strain curve of the ground as well as the stiffness of the lining (Panet and Guenot, 1982). The CCM is a simplified approach used to analyze the interaction between the ground and the support. Using an axisymmetric hypothesis, the method provides simplified knowledge on the process of the ground-support interaction that takes place close to the working face. The input is two-dimensional, but the results of the analysis can also be applied to three-dimensional problems (Panet, 1995). The application of the CCM to other types of models, that do not present axisymmetry, requires the use of numerical simulation programs. The CCM relies on three components: the longitudinal displacement profile (LDP), the support characteristic curve (SCC), and the ground reaction curve (GRC). The LDP shows the radial displacement of the tunnel cross-section in the longitudinal direction from the tunnel face under the assumption of no supports. The SCC describes the increasing pressure that acts on the supports as the radial displacement increases. Lastly, the GRC shows the increasing trend of the radial displacement as the internal pressure of the tunnel decreases; it plays a crucial role in deciding when the supports are to be installed and how stiff they should be. The determination of a reasonable GRC is the major part of any study related to the CCM. The interdependency of the tunnel-wall/crown displacement and the support pressure has long been recognized by many, e.g., Fenner (1938) and Pacher (1964). Deere et al. (1969) commented on the idea of a GRC. The GRC concept allows for taking account of the sequential installation of a liner within the context of the deformation of the tunnel opening as earth is gradually removed. Various aspects, such as the stress-strain behavior, the constitutive model of the ground, the primary tunnel supports, and the geometry of the tunnel, have been considered in order to obtain the proper ground response in terms of the GRC. The Mohr-Coulomb and Hoek-Brown models were the main constitutive models of the ground (Brown et al., 1983; Carranza-Torres and Fairhurst, 1999, 2000; Sharan, 2003; Oreste, 2003; Alejano et al., 2009; Shin et al., 2010). However, most of these studies were carried out for deep tunnels (overburden height H > 2.0D; D is the diameter of the tunnel) and mostly under rock mass conditions. As most shallow overburden (H < 2D) tunnels often encounter relatively loose soil masses, it is important to study the behavior of shallow overburden tunnels in soft grounds by utilizing the CCM concept. Due to these poor ground conditions, surface and tunnel settlements frequently occurred mainly due to the low stiffness of the unconsolidated grounds. The typical approaches to movements due to the tunneling in soft ground varies from well-known empirical methods to advance numerical analyses. Due to the lack of accuracy of former, a lot of effort has been put in development of techniques like finite element and finite difference methods. Beside this there is a considerable progress in development of constitutive models over the last few decades but research institutes and designers are still using the basic constitutive models due to the fact that theory behind the basic constitutive models is widely understood and the parameters required for the most advance constitutive models are not easily obtained through conventional in-situ and laboratory testing. 2 The actual stress path in ground during tunnel excavation is very complex. To capture the correct tunnel excavation response, it is important to correctly resemble

4 the in-situ stress in the numerical tool. Stress changes due to mining activity and related ground relaxation can drastically influence the stability of underground opening. For a given ground and in-situ stress conditions, excavation induced failure and damage to the ground depends on the stress path. The essence of this dependency is that the mechanical response of elasto-plastic materials is stress path dependent. Almost half of the studies discussed only one tunneling aspect, commonly the surface settlements (Negro and Queiroz, 2000). Only a few studies have been carried out on stress paths that are experienced by the ground during the construction of tunnels. The results are generally presented in terms of strains. This has been related to the difficulties that exist in measuring stresses on site during construction. Nevertheless, it is important to deeply understand the stress path during tunneling process specifically considering shallow over burden conditions which results in low confining stresses. In the present study, an attempt has been made to study the stress path experienced during tunneling for four different overburden conditions by using very famous and widely used constitutive model i.e. Modified Cam Clay model. Tunnel excavation is carried out by using the CCM. By utilizing the concept of progressive relaxation the GRC were obtained to understand the mechanical behavior of the ground subjected to the tunneling. Three different sets of numerical analyses were carried out in order to account for the various over consolidation states of soils i.e. normally consolidated, lightly over consolidated and heavily over consolidated soils. In addition, the stress paths during the subsequent ground relaxation stages are also presented in order to investigate the state of the ground subjected to the tunneling. During the tunneling process, the relaxation of the ground yields displacement and in the mean while the stress path changes as well. This stress path during subsequent relaxation stages helps in identifying the material yielding state. 2. OUTLINE OF NUMERICAL ANALYSES Panet and Guenot (1982) suggested that the tunneling process could be analyzed under plane-strain conditions, in a plane perpendicular to the tunnel axis, provided that some reasonable representation of the conditions at the face were included in the model. Thus, we carried out a series of simulations of shallow tunneling in sandy ground using a two-dimensional finite difference continuum analysis under plane-strain condition. It is assumed here that the excavated face remains intact and that no kinematic collapse mode due to gravity has been induced in the longitudinal direction. Overburden heights (H = 0.5D, 1.0D, 1.5D and 2.0D) and over consolidation ratios (OCR= 1, 2, and 8) are varied to investigate the effect of OCR on the shallow tunneling. OCR = 1, 2, and 8 represent the normally consolidated, lightly over consolidated and heavily over consolidated soils respectively. The model for the unlined circular tunnel in cohesion-less soil is shown in Figure. 1 below. 3

5 Figure 1. Model chosen for stability of shallow overburden (H=0.5D, 1.0D, 1.5D, and 2.0D) unlined circular tunnel 2.1. Analyses domain and boundary conditions The adequate selections of the analysis domain, the mesh size, and the boundary conditions are essential to obtain accurate numerical solution. Some tentative test runs have been done in elastic materials and show that the obtained errors are negligible. According to Varas et al. (2005), for perfectly elastoplastic materials, the error due to the mesh size selection used to be less than 5%, whereas in the case of strain-softening, the results are more mesh-dependent due to the fact that localization phenomena might take place. Basically, the meshes must be fine enough in the region where significant deformation is expected to occur. A circularshaped tunnel with a diameter D of 10 m is considered in all the analyses, and a finer quadrilateral mesh (size of 0.5 m) was used. Bottom and side boundaries of the analysis domain should be sufficiently far away from the tunnel so that the impact of the excavation on the boundaries can be ignored. The guidelines by the Japan Society of Civil Engineers (2006), entitled The Model Experiments and Numerical Analyses in Mountain Tunnels suggests that the minimum vertical and horizontal extensions of the model are 3D and 5D respectively (where D is the diameter of the tunnel). The vertical and horizontal boundary extensions are set following the above recommendation in this study. The associated boundary conditions are also included in Figure. 1, where smooth, roller boundary conditions are applied at bottom and side boundaries respectively Ground properties 4 The model ground is assumed to be loose sandy ground in a drained condition. For this, we applied the Modified Cam Clay (MCC) model. MCC is an incremental hardening/softening elastoplastic model. Its features include a particular form of non-linear elasticity, and a hardening/softening behavior governed by volumetric plastic strain ( density driven). The failure envelopes are similar in shape, and correspond to ellipsoids of rotation about the mean stress axis in the principal stress space. The shear flow rule is associated; no resistance to tensile mean stress is offered in this model (Roscoe and Burland, 1968; and Wood, 1990). The MCC model is expressed in terms of three variables: the mean effective pressure, p ; the deviator stress, q; and the specific volume, v. The generalized stress components p and q may be expressed in terms of principal stresses:

6 p =! σ!! + σ! + σ! (1) q =!! (2) σ! σ!! + σ! σ!! + σ! σ!! (Note that q = 3 J!, where J 2 is the second invariant of the effective stress deviator tensor. The yie function correspond to a particular vale p c of the pre consolidation pressure has the form: f = q! + M! p p p! (3) where M is a material constant. The yield condition f = 0 is represented by an ellipse with horizontal axis, pc, and vertical axis, Mpc, in the (q, p) plane (cf. Figure. 2). As the ellipse passes through the origin. Hence, the material in this model is not able to support an all-around tensile stress. The failure criterion is represented in the principal stress space by an ellipsoid of rotation about the mean stress axis (any section through the yield surface at constant mean effective stress p is a circle). The potential function g corresponds to an associated flow rule, and we have: corresponds to an associated flow rule, and we have: g = q! + M! p p p! (4) A set of soil parameters was chosen from previous research done by Cui at al., 2010, to investigate th effect of the variations in overburden ratio (H/D) and over consolidation ratio (OCR). The parameters fo the ground material are summarized in Table 1. In MCC model the stiffness is stress dependent and th bulk modulus (K) is defined as:! =!!, where ʋ is the current specific volume and p is the mean! effective stress and k is the slope! = of!!! the swelling line. A higher value of K (i.e. 1.0E9 Pa) was specified the numerical analyses. Poisson ratio (ν) was kept constant while shear modulus (G) varies with respe to K. Table 1. Soil parameters for Modified Cam Clay failure criteria and employing an associated flow rule (Cui et al., 2010) 2010) Model Parameters Value Unit Modified Cam Clay Frictional constant (M) 1.2 Slope of swelling line (k) Slope of normal consolidation line (λ) 0.03 Reference pressure (p 1 ) 100 kpa Specific volume at reference pressure (v λ ) 1.43 Over consolidation ratio (OCR) 1, 2, and 8 Sandy ground with Density (ρ) = 1805 Kg/m 3, K 0 = 0.5, and Poisson s ratio (ν) = Figure 2. Modified Cam Clay failure criterion Figure 3. Grid geometry and boundary conditio 5

7 2.3. Modeling of excavation process The tunneling excavation simulations are carried out by the CCM. This method relies on the GRC, developed by progressively reducing the internal pressure within the excavation region and plotting this support pressure against the tunnel closure. The progressive reduction of internal pressure is controlled by the stress relaxation factor (λ) and using equation. 5. σ = (1-λ) (5) σ _ o The forces on the tunnel periphery at zero relaxation are measured and then reapplied at an incrementally decreasing amount (depending upon the assigned λ) of this force as traction. In the current analyses, the tractions along the entire tunnel boundary were reduced in small increments of 10% from the zero relaxation state (i.e., λ=0.9). Drained isotropic tri-axial compression test Prior to the tunnel simulations, drained isotropic triaxial compression tests were carried out using the simulation. The soil parameters listed in Table. 1 are applied here. The basic purpose of these tests was to show the fundamental material behavior under relatively low confining stresses, which essentially is the condition expected during the excavation of shallow overburden tunnels. Three simulation analyses were conducted by using a confining stress (Po) of 100 kpa and with variable pre consolidation pressure (pc) of 100 kpa, 200 kpa, and 800 kpa. These variable pc values are actually representing the state of normally, lightly and heavily over consolidated soils (i.e. OCR = 1, 2, and 8 respectively as OCR=pc/po). The analyses were carried out using a single element in an axisymmetric configuration. The zone has unit dimensions in the x- and y-directions. Figure. 3 shows the reference axes and boundary conditions in the simulation. The grid was fixed in the y-direction; in-situ isotropic compressive stresses of 100 kpa was prescribed, and constant lateral confining pressures (Po), of 100 kpa was imposed. To perform strain-controlled tests, axial displacement was applied at the top of the model and axial strain of 15 percent was finally given in 15,000 calculation steps. 6 The results of the simulation of tri-axial tests are presented in Figure. 4a, 4b, and 4c, where a, b, and c represent the OCR = 1, 2, and 8 respectively. As the test progresses, Figures. 4a and b show typical effective stress paths with inclination of 3 in p -q relationship for drained CD tests with constant radial effective stress. The stress paths finally reach the critical states at which the deviator stress intersects the critical state line and form a new yield surface. Which means in case of OCR= 1 and 2 there is a hardening of the yield surface. In contrast to this, for OCR = 8, (Figure. 4c) there is an initial hardening with a slope of 1:3, once the stress path intersects the original yield surface there is a softening trend and it continues until the stress path intersects with the critical state line. The corresponding evolution of specific volume (i.e. v =1+e, where e is the void ratio) are also shown for their respective cases. In case of OCR = 1 the starting point lies on the normal consolidation line (NCL). When the test progress the material undergoes compression which is evident as decrease in specific volume and finally it touches the critical state line (CSL). The behavior is same in case of OCR = 2 except the change in slope of specific volume reduction when the stress path cross over the original yield surface. Whilst for the case of OCR = 8 there is an initial decrease in specific volume followed by

8 the increase in specific volume which is align to the fact that there is a softening trend. Figure. 5 show the evolution of volumetric strain with axial strain. In the beginning stage of shearing, all the samples exhibited linear elastic responses and negative volumetric strain was seen due to the increase in mean effective stress. Which means there is a compression state except for the OCR = 8 case, where initial compression is followed by the positive volumetric strain which correspond to the dilation of the soil. 4. Mechanical behavior of shallow overburden tunnel When carrying out the excavation of a tunnel, a redistribution of the stresses and the deformation takes place in the area surrounding the working face of the excavation. This redistribution translates as a displacement of the tunnel wall and crown, which tends to close the new cavity. Due to the relief of the initial in situ stresses, the disturbed ground mass will displace until it reaches a new state of equilibrium that is largely controlled by the manner of the support activation. In this study, four separate analyses for different values of H/D ratios, i.e., 0.5, 1.0, 1.5, and 2.0, are carried out. In the MCC analysis for each individual case, three different values of over consolidation ratios (OCR = 1, 2, and 8) were introduced to evaluate the OCR effect. The mechanical behavior of the shallow overburden is assessed by GRCs and the stress evolution at the crown and the spring line, as shown by the circular markers in Figure. 1, during the progressive relaxation of the ground, which is carried out to develop the GRC. 7

9 4.1. Ground reaction curve During the excavation, the fictitious support pressure (Pi) decreases. This pressure corresponds to the equivalent support pressure required to attain an equilibrium state. The ground responses at the crown and spring line, expressed in the GRCs of the normalized support pressure (Pi/Po) versus the normalized displacement (Ui/a), are presented in Figure 6a, 6b, and 6c (a, b, and c correspond to the previously mentioned three cases of OCR i.e., OCR = 1, 2, and 8). Ui is the vertical displacement at the tunnel crown in case of crown GRC whilst in the case of spring line/wall it represents the horizontal displacement at spring line/wall and D (D = 10.0 m) is the diameter of the tunnel. Each marker in the figures shows the 10% 8

10 relaxation. It is important to notice that in case of OCR = 1, the normalized support pressure cannot be reduced to zero as at the normalized support pressure of approx the failure zone reached to the ground surface as often is the case in shallow overburden tunnel and excessive large displacement occurs and due to this numerical simulation tend not to converge (follows an asymptomatic path). Whilst in case of OCR = 2, only H = 0.5D is the case where numerical solution does not remain stable below the normalized support--pressure of In rest of the cases the normalized support pressure can be reduced to zero. In case of OCR = 8, the normalized support pressure can be reduced to zero for all the overburden cases. In the above Figure. 6 only the results of normalized support pressure until 0.08 are shown that correspond to the 92% relaxation. It is quite evident that in all the OCRs cases the vertical displacement is higher as compare to the spring line which is primarily due to the anisotropic stress condition (K0 = 0.5). With the increase in OCR value the magnitude of normalized displacement reduces as well. In the case of OCR = 1, the maximum displacement occurs for H = 1.0D case followed by 1.5D and 2.0D case and least displacement was observed in case of shallowest overburden case i.e. H=0.5D. In contrast to this for the cases of OCR = 2, and 8; there is a unique sequence in displacement i.e. maximum displacement was observed in case of H = 0.5D followed by H = 1.0D, H = 1.5D, and H = 2.0D cases. Figure 4. Ground reaction curves for (a) OCR = 1; (b) OCR = 2 and (c) OCR = 8. 9

11 4.2. Stress distribution around the tunnel Stress variations with the progress of the tunnel excavation modeled by the decrease in support pressure (Pi) at the crown and spring line are shown in Figures. 8 and 10 respectively. At the tunnel crown and spring line, stress redistribution involves a decrease in radial stress σr and an increase in tangential stress σt. The relaxation values correspond to the λ (i.e., λ=1-pi/po), in percentage form. As K0 is equal to 0.5, the initial radial stress (σyy) is greater than the initial tangential stress (σxx) at the crown, and vice versa at the spring line (cf. Figure. 10). In case of OCR = 1 (Figure. 8a), as the internal pressure decreases (relaxation increases), the stress difference (σt σr) becomes smaller at the crown as the vertical stresses tends to decrease and the horizontal stresses tend to increase. At a certain stage of relaxation, the horizontal stresses bisect the vertical stresses. At this point, the arching above the tunnel crown occurs in the tangential (horizontal) direction, with the tangential stresses as the major principal stresses. If the support pressure is reduced further, the (σt σr) difference increases. Due to this there is an increase in mean stress and at certain relaxation stage the stress path crosses the original yield surface and hardening of the yield surface starts and this continuous until the stress path intersect with the critical state line. After that there is a softening trend and this continuous until the 92% relaxation with the exception in case of H = 1.0D where stress path intersects the critical state line at dry side. In case of OCR = 2 and 8 the stress path remains within elastic region. A selective case for OCR = 1 and 2 is shown in Figure. 7. At the spring line, in the beginning stage of stress relaxation, σr (horizontal) is decreased and σt (vertical) is increased as shown in Figure. 10. For OCR = 1 (Figure 10a), the stress difference (σt σr) increases at the spring line in the beginning stage and approx. at 10% relaxation the hardening of the yield surface initiates and this continuous until the stress path intersect the critical state line at 72% relaxation. After that there is a softening of the yield surface until the final relaxation value and this behavior is consistent in all the overburden cases. In case of OCR = 2 (Figure 10b), there is initial elastic behavior until the stress path intersect the initial yield surface which occurs at the 41% relaxation for all the overburden cases. After that relaxation there is a softening of the yield surface and plastic volumetric dilation took place. Whilst in case of OCR = 8, the stress path remains within elastic region. A selective case for OCR = 1 and 2 is shown in Figure.9. Figure 5. Stress path in p, q space for (a) H=1.0D, OCR=1 and (b) H=0.5D, OCR=2 10

12 Figure 6. Stress evolution during the relaxation at Crown for (a) OCR =1; (b) OCR=2 and (c) OCR=8 Figure 7. Stress path in p, q space for (a) H=1.0D, OCR=1 and (b) H=0.5D, OCR=2 11

13 Figure 8. Stress evolution during the relaxation at Spring line for (a) OCR =1; (b) OCR=2 and (c) OCR=8 5. CONCLUSIONS Four different over burden heights (i.e. H/D = 0.5, 1.0, 1.5, and 2.0) within shallow over burden domain were analyzed for three different OCR values (i.e. OCR = 1, 2, and 8) which correspond to the normally, lightly and heavily over consolidated soils by assuming an idealized ground profile which obeys the MCC behavior. The GRC concept in conjunction to the stress evolution during progressive relaxation and subsequent stress path in p,q space was applied to understand and evaluate the influence of various OCR with respect to various H/D ratios. Following conclusion were drawn: In all the analyzed cases, it is evident that the displacement at the tunnel crown is larger as compare to the spring line which is primarily due to the fact that K0 = 0.5. Moreover, with an increase in OCR value the magnitude of displacement reduces. The displacement pattern within OCR = 2 and 8 follows a unique pattern i.e. with the increase in overburden displacement reduces as well. During the tunnel excavation, at the tunnel crown σyy reduces while σxx increases and at certain stage σxx takes over the σyy and become the major principal stress while opposite occurs at the spring line. At the tunnel crown, for OCR = 1, the hardening of the yield surface took place whilst in case of relatively high over burden i.e. 1.5D and 2.0D there is softening trend as well. Due to the decrease in mean stress, positive dilation was observed for all the overburden cases. Whilst in case of OCR = 2 and 8, the ground remains within elastic range 12

14 for all the cases. Contrary to this at the spring line, For OCR = 1, the hardening of the yield surface starts at 10% relaxation which continues until the 72% relaxation. During this stage material remain in compression state. After this hardening trend material undergoes softening behavior and it continuous until the full relaxation. Whilst in case of OCR = 2, after the initial elastic behavior there is a softening trend which start exhibiting at the 41% relaxation and this relaxation value is consistent for all the overburden cases. For OCR = 8 case, the material remains within elastic range for all the cases. Stress reduction factor (λ) is a dimensionless coefficient in 2D numerical analyses based on the CCM and an important parameter for the support activation and tunnel design. In the typical design procedures, the ground is allowed to relax until a certain relaxation value which satisfies the displacement limitation criteria i.e. either the surface settlement or based on the distance of support activation from the tunnel working face (LDP). After that a desired support system is prescribed which satisfy the safety limits and ultimately 100% relaxation is achieved. As with K0 = 0.5, the yielding always initiates from the spring line and propagates toward the free ground surface and this continues until a proper support system is introduced. Hence the behavior at the tunnel spring line will be the governing factor in the design. From the current research, a relaxation value between 10% to 72% is suggested for the normally consolidated soils (i.e. OCR = 1) for all the overburden cases (i.e. H < 2D) before the support activation as within this relaxation limit, the grounds remain within compression state and there is a hardening of the yield surface. Whilst for the lightly over consolidated soils (i.e. OCR = 2) a relaxation value of 10% to 41% is suggested for all the overburden cases (i.e. H < 2D) before the support activation as within this range the ground remains within elastic range and after that it follows the softening trend. It is worth noticing that the suggested relaxation values have not considered the other important factors like surface settlement and are only valid for the given ground conditions. 6. ACKNOWLEDGEMENT The first author would like to express his gratitude to the generous supported by Grant in-aid for doctoral studies from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) Japan. 7. REFERENCE Alejano, L. R., Alonso, E., Rodríguez-Dono, A., Alonso, E., Fdez-Manín, G., Ground reaction curves for tunnels excavated in different quality rock masses showing several types of post-peak failure behavior. Tunn. Undergr. Space Technol. 24(6), Barton, N.R., Lien, R., Lunde, J., Engineering classification of rock masses for the design of tunnel support. Rock Mechanics. 6(4), Bieniawski, Z.T., Geomechanics classification of rock masses and its application in tunneling. In: Proceeding of the Third Congress ISRM, Denver, 2, Part A Brown, E.T., Bray, J.W., Ladanyi, B., Hoek, E., Ground response curves for rock tunnels. Journal of Geotechnical Engineering. 109(1),

15 Carranza-Torres, C., Fairhurst, C., Application of convergence-confinement method of tunnel design to rock masses that satisfy the Hoek-Brown failure criterion. Tunn. Undergr. Space Technol. 15(2), Cui, Y., Kishida, K., Kimura, M., Analytical study on the control of ground subsidence arising from the phenomenon of accompanied settlement using footing reinforcement pile, Deep and Underground Excavation. ASCE Geotechnical Special Publication, pp Deere, D. U., Peck, R. B., Monsees, J. E., Schmidt, B., Design of tunnel liners and support systems. PB Report for the U.S. Department of Commerce, University of Illinois, Urbana, 111. Fenner, R., Untersuchungen zur Erkenntnis des Gebirgsdruckes. Gluckauf. 74, [ ]. Japan Society of Civil Engineers, Model experiments and numerical analyses in mountain tunnels Maruzen Co., Ltd., Tokyo, ISBN (in Japanese). Negro, A., Queiroz, P, B., Prediction and preformance: A review of numerical analyses for tunnels. International Symposium on Geotechnical Aspects of Underground Construction in Soft Ground. Fujita, Kusakabe and Miyazaki (eds.). Balkema. pp Oreste, P. P., Analysis of structural interaction in tunnels using the convergence-confinement approach. Tunn. Undergr. Space Technol. 18(4), Pacher, F., Measurements of deformations in a test gallery as a means of investigating the behavior of the rock mass and specifying lining requirements. Rock Mech. and Engineering Geology. 1, Panet, M., Le calcul des tunnels par la méthode des curves convergenceconfinement. Presses de l École Nationals des Ponts et Chaussées, Paris. Panet, M., Guenot, A., Analysis of convergence behind the face of a tunnel. In: Proceedings of the Int. Symp. Tunneling Sharan, S. K., Elastic-brittle-plastic analysis of circular openings in Hoek- Brown media. International Journal of Rock Mechanics and Mining Science. 40(6), Shin, Y. J., Kim, B. K., Shin, J. H., Lee, I. M., The ground reaction curve of underwater tunnels considering seepage forces. Tunn. Undergr. Space Technol. 25(4), Varas, F., Alonso, E., Alejano, L. R., Fdez-Manín, G., Study of bifurcation in the problem of unloading a circular excavation in a strain-softening material. Tunn. Undergr. Space Technol. 20(4),

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