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Introductory Chapter: Tunnel Engineering – Rock Load Estimation and Support Design Methods

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Hasan Tosun

Submitted: 27 May 2021 Published: 25 May 2022

DOI: 10.5772/intechopen.104515

From the Edited Volume

Theory and Practice of Tunnel Engineering

Edited by Hasan Tosun

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1. Introduction

The design of an underground opening differs from that of other engineering structures constructed on the ground surface. For subsurface structures, loads caused by overburden pressure and surcharge loadings are taken into account, and support system/systems are designed to meet these total loads, while in surface structures such as dams, bridges, and buildings the designer has to transfer loads caused by the relevant structures to the foundation elements beneath ground. In other words, underground structures are constructed within the ground with many unknown parameters. However, surface structures are designed with materials whose properties are known very well and structure dimensions can be controlled according to the bearing capacity of the ground below the relevant structures. Therefore, rock load estimation is a much more important issue in the design of underground structures. Additionally, typical problems can arise due to the type of rock, structural features in the rock mass, and age of rock formation. For example; older rocks with Precambrian and Paleozoic age can result in huge squeezing pressure while the arching effect cannot be formed in young sedimentary rocks.

In the design of underground structures, as well as rock load estimation, the depth of losing zone, the arching effect, and rock-support interaction also emerge as an important design criteria. For this purpose, so many methods based on country and region facts have been suggested. For example, in North American practice, support design with steel ribs is envisaged depending on the defined simple rock classes, while in Central European tunneling, the design of flexible support systems such as rock bolts, shotcrete, and wire mesh was adopted to form a self-carrying zone around the opening depending on physical and mechanical properties of excavated material, discontinuities of rock mass and mechanical characteristics of support system. In the methods that envisage flexible support design, it is aimed to balance the stresses formed around the underground opening by self-bearing the rock load and to achieve more economical solutions. Numerous studies have been conducted on the reliability and economy of these proposed methods [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15].

This book includes four excellent contributions on the special issues of Tunnel Engineering. The overall aim of the collection is to improve the theory and practice of underground structures. The articles cover chapters on analytical and numerical methods for rock load estimation and design support systems and advance in the measurement system for underground structures.

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2. Empirical methods

There are mainly three separate methods, which are empirically used in rock load estimation and support systems for tunnel structures: (1) Conventional analysis, (2) Geomechanics classification, and (3) Q-system. Some researchers introduced so many studies on the review and comparison of empirical methods [16, 17, 18, 19, 20, 21, 22, 23].

2.1 Conventional analysis

The method was originally developed by Terzaghi [24, 25] for a steel rib support system. In the method, it is assumed that the load magnitude for the analytical analysis depends on the rock height, and is supposedly available that rock mass can be dimensioned as a wedge or inclined block. In the Conventional Method external support system, such as steel ribs are selected depending on the recommendation of Terzaghi’s rock load concept (nine ground categories). For the conventional analysis, rock load is determined as a function of the height of the loosened rock. In this case, the height of loosened rock referred to Terzaghi’s rock load concept was estimated rock at the range of 4.5(B + Ht) to 0, in which B is width and height of the tunnel, respectively and a special definition, which is irrespective of the value of (B + Ht) is given in the method [24]. The Conventional Analysis does not assume a construction sequence.

The model used for the Conventional analysis is mainly based on a graphical solution. In the model of Conventional Analysis, it is assumed that load is radially transferred to the support system and radial deformation does not occur during the loading stage. By this analysis method, the interaction between rock and support system is considered only in developing passive resistance. According to [2], the effects of the relative stiffness of the rock and support system, and the boundary condition between the support and rock are not included in analytical solutions of the conventional analysis.

2.2 Geomechanics classification

Bieniawski [26] suggests a classification system depending on an index (RMR-Rock Mass Rating) for mainly underground openings. The author completed new studies to increase its reliability and to provide optimization on the support system used in tunnels [27, 28, 29, 30, 31]. This classification poses two main sections. In the first section, there are five parameters: (1) strength of intact rock material, (2) rock quality designation, (3) spacing of joints, (4) condition of joints, and (5) groundwater conditions. As a strength criterion, compressive strength is utilized. For weak rocks, the index value on point load can be considered instead of uniaxial compressive strength. The aspect of Rock Quality Designation is considered to evaluate the drill core of rock mass. The term “joint” means all discontinuities of rock masses surrounding the opening.

The first section of Geomechanics Classification takes into account the presence of fillings in joints and wall conditions. It also considers continuity and the separation of joints as well as surface roughness. The method introduces a ratio for defining water pressure in joints for 10 m tunnel length or a qualitative criterion for representing groundwater flow around the opening. Ratings are allocated for all ranges of the related parameter. The summation of all ratings introduces an overall rating that represents the crude RMR value of the rock mass for the selected section of the tunnel. The second section considers the joint orientation impact. The crude RMR value is adjusted for considering the influence of joint orientation. The adjusted value, which was called RMR concept in short term, changes within a wide range (from 20 to 100).

Geomechanics Classification emphasizes the orientation of the structural features to the rock mass while taking no account of the rock stress. It has been found that the Geomechanics Classification has difficulties applying in the extremely weak ground, which results in squeezing, swelling, or flowing conditions. However, it introduces a rating obtained from the detailed geological investigation. An empirical equation (ht = [(100-RMR)/100] x B) has been developed to estimate the rock load acting on the support system, based on the RMR of the Geomechanics Classification System [32].

Geomechanical Classification of Bieniawski suggests temporary support systems depending on the RMR value, tunnel characteristics, and excavation method. For very good rock conditions (the RMR value is between 81 and 100) no support is recommended while locally and systematic bolts with shotcrete are considered for good rock (the RMR value is between 61 and 80) and fair rock (the RMR value is between 41 and 60), respectively. For poor rock (the RMR is between 41 and 60) and very poor rock (the RMR value is between 21 and 40), the method suggests the use of steel ribs with the combination of rock bolts and shotcrete. The Geomechanics Classification suggests construction sequences a full face, top heading-beach, and multiple drifts depending on rock mass classes categorized according to RMR values.

2.3 Q-system

Barton et al. [33] empirically introduced a design tool for underground openings, namely the Q-system, which is a geomechanical aspect based on six separate parameters. These are (1) Rock Quality Designation (RQD), (2) joint set number (Jn), (3) joint roughness number (Jr), (4) joint alteration number (Ja), (5) joint water reduction factor (Jw) and (6) stress reduction factor (SRF). The authors developed the system to optimize support requirements without stability problems [34, 35, 36, 37, 38]. Recently, NGI [39] introduced a manual for using the Q-system.

The Q-value has been formulated as being three quotients [(RQD/Jn), (Jr/Ja), and (Jw/SRF)] depending on six separate parameters mentioned above. The quotient (RQD/Jn) is defined as a parameter for the block or particle size within a wide range (200 and 0.5). The quotient (Jr/Ja), which is also another parameter that measures inter-block shear strength, introduces valuable data about the roughness and alteration degree of discontinuities. The last quotient (Jw/SRF) consisting of two stress parameters (Jw- joint water reduction factor and SRF- the stress reduction factor) considers water pressure which adversely affects the shear strength of joints and evaluates the loosening load resulting by unloading case through discontinuities and very weak rock.

The equations on support pressure introduced in the Q-system provide a convenient means for developing classification rules for dynamic as well as static loading of underground excavations. The dynamic stresses resulting from the passage of seismic waves may presumably exceed the static stresses by some unknown factors [33, 35]. Q-system does not include the joint orientation as a separate parameter. However, the properties of the most unfavorable joint sets are considered in the assessment of the joint alteration number and the joint roughness.

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3. Rational methods based on rock-structure interaction

3.1 Convergence-confinement method (CCM)

The CCM, as a rational method based on rock-support interaction, was first suggested by Ladanyi [40] and then developed by Hoek and Brown [41]. A technical committee approves its recommendation on the CCM [42]. Valuable studies have been realized on the convergence behavior of tunnels by various researchers [43, 44, 45, 46, 47, 48].

It considers rock mass behavior to be a tendency to close the excavation. The excavating of the tunnel changes equilibrium conditions in rock mass as well disturbs original stresses. The unloading caused by excavation results in displacements throughout the rock mass. The support system is installed while a change in the original stress occurs and displacements develop. In other words, the temporary support system resists displacements in the surrounding rock during the excavation process. In fact, stresses redistribution and displacement development are controlled by rock-support interaction. This phenomenon is the fundamental principle of the CCM which recognizes the behavior of rock mass during processes of rock excavation and support installation. The convergence behavior of rock mass is represented by a curve that correlates pressure with displacement. For constructing this curve, the strength criterion of rock mass such as primary stress condition, elastic moduli, uniaxial compressive strength, etc. is needed. The ground curve with support characteristic curve provides an excellent design tool to illustrate the geomechanical problems of the project.

In the Convergence-Confinement method, rock load is defined as a function of primary stress conditions, not depending on the height of loosed rock directly. The stress-deformation curve of the surrounding is drawn for estimating the limit pressure to be supported. The CCM introduces an analysis that widely utilizes different support systems. It is available to select all kinds of support systems. However, the selection of supports is based on the ground classes as given in the New Austrian Tunneling Method.

3.2 The new Australian Tunneling method (NATM)

The NATM is also a rational method based on ground-structure interaction as the CCM. It contains design and construction concepts with contractual improvements and poses different items on technical and operational processes. Numerous researchers studied in the NATM to clarify some items on support systems [49, 50, 51, 52].

The NATM depends on the principle to reduce support requirements by ground resistance mobilization to optimum case without resulting in any instability. The NATM generally recommends two support systems (outer and inner arch). The outer one has a function as a shell zone having more flexibility to provide stability to the surrounding rock (protective support). The supports suggested for this arch are mainly shotcrete combined with a reinforcement mesh and rock bolts. For the unfavorable ground conditions, the flexible support system mentioned above can be combined with light steel sets. The inner arch generally consists of concrete lining which should be installed after providing equilibrium conditions for outer arch. However, the concrete lining is not installed as a permanent support system prior to the outer arch has reached equilibrium. Rabcewicz and Golser [1] state that this application increases the factor safety if it is needed.

The ground classes empirically relate the geological conditions with the excavation procedure and initial support. The classes used in the preliminary design describe the excavation procedure and the support system in detail and include the quantitative information for designing and construction, although the geological conditions described are qualitative.

The behavior of protective support and surrounding rock during the stress redistribution caused by excavation is controlled by a sophisticated measuring system. Observation of ground and support system provides valuable data for well stabilization of tunnel and optimization on support cost in the NATM [3].

The NATM suggests the utilization of technically advanced support and excavation systems to mobilize the ground resistance to its optimum extent, to redistribute the stresses from the heavily stressed to the less stressed zones, and to improve the material properties of the ground. The flexible supports applied for a relatively short time after excavation accomplish the optimum mobilization of ground resistance, and also provide the redistribution of stresses by a flexible cylinder action as described by Peck [53]. The improvement of ground material is achieved by bolts and shotcrete. The reinforcing action of bolts increases the ground shearing resistance.

The total support capacity is the summation of three components including the resistance of lining and rock bolts, and the resistance of rock arch. The total support capacity should exceed the limit support pressure, which was obtained from the ground-support analysis. Otherwise, the structure will not be appropriately stable and safe. A good example for the NATM is Sanliurfa Tunnels, which consists of two tunnels, each having 26 km long and 7.62 m internal diameter (Figure 1).

Figure 1.

The Sanliurfa tunnel: (a) cross-section of tunnel and (b) a general view from inlet portion of the tunnel.

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4. Numerical methods

There are so many numerical approximations for modeling and designing underground structures. The Finite Element Method (FEM), as a most sophisticated numerical analysis, is widely used in analyzing stresses and deformations around an underground opening. In FEM analysis, the structure and the surrounding rock masses are restricted by appropriate boundary conditions and divided into discrete elements, which are triangles and quadrangles connected to each other only at nodes or points of knots. An underground opening can design by the FEM analysis. However, an existing structure can be evaluated as post-failure analysis. It is possible to use it in a wide variety of analyses for sequential construction, control, monitoring, and instrumentation. Recently it is regarded as a common tool for modeling laboratory testing. Numerous studies have been realized on the use of the finite element method for tunneling [54, 55, 56, 57, 58, 59, 60]. Examples of the deformation analyses and design of external support system by FEM is given in Figures 2 and 3, respectively.

Figure 2.

The deformation (vertical) analyses for the construction second tunnel (unloading) of double tube system of Sanliurfa tunnel by FEM [9, 13, 14, 15].

Figure 3.

Design of the external support system (shotcrete+wire mesh+rock bolt) of Sanliurfa tunnel by FEM [9, 13, 14, 15].

Mathematically it is a numerical technique used for solving differential equations. Stresses and strains for defined elements within the model can be determined by the constitutive equations of stress and strain. Physically it is defined as a method for determining element stiffness. In the FEM analyses, a number of alternatives of loading and geometries can be evaluated by two or three-dimensional models. Especially three-dimensional models of underground openings an effectively used for analyzing sequential excavation and support installation. However, the researcher state that the FEM should not be used alone for designing an underground opening [9, 61]. It assists project engineers and consultants in having rational decisions. It poses advantages defining on complex geometry and non-linear nature of geological features as well as providing simplicity for inhomogeneous and discontinuous material [62, 63].

References

  1. 1. Rabcewicz L, Golser J. Principal of dimensioning the supporting system for the new Austrian Tunneling method. Water Power. 1973:88-93
  2. 2. Agbabian Assoc. Improved Design Procedures for Underground Structural Support Systems in Rock. First Technical Progress Report, R-7814-4803, El Segundo, CA. 1979
  3. 3. Einstein HH, Schwartz CW, Steiner W, Baligh MM, Levitt RE. Improved Design for Tunnel Supports-Analysis Method and Ground Structure Behavior. US Department of Transportation, Report no. DOT/RSPA/DPB-50/79/10.1980
  4. 4. Tosun H. Parameters considered for tunnel design and geotechnical approaches. In: Proceedings Book. National Seminar on Geotechnics, State Hydraulic Works. Gümüldür-İzmir; 1984. pp. 45-56
  5. 5. Tosun H. Comparison of the tunnel design methods and their application to Urfa aqueduct tunnel. In: MS Thesis in the METU. Ankara; 1985. p. 141
  6. 6. Tosun H. Finite element method for tunnel design. In: Proceedings Book, National Seminar on Geotechnics, General Directorate of State Hydraulic Works. Dragos-İstanbul; 1986. pp. 33-56
  7. 7. Doyuran V, Tosun H. Comparison of tunnel design methods and their application to Urfa tunnel. Middle East Technical University Journal of Pure and Applied Sciences. 1986;19(1):23-61
  8. 8. Doyuran V, Tosun H. Shield Tunneling for soft grounds. In: Proceedings Book of National Seminar on Geotechnics, General Directorate of State Hydraulic Works. Dragos-İstanbul; 1987. pp. 9-40
  9. 9. Tosun H. Use of finite element method on tunnel design. In: Seminar on Use of Software for Hydraulic Engineering. General Directorate of State Hydraulic Works, Sanlıurfa, Proceedings Book. 1991. pp. 1-28
  10. 10. Tosun H. Comparison on design methods of temporary supports using Sanlıurfa tunnels. In: Proceeding of the 2nd Int. Conference on Hydropower. Lillehammer, Norway; 1992. pp. 201-208
  11. 11. Turfan M, Tosun H. Impacts of Geomechanical characteristic of rock masses on support system for pressurized tunnels. In: National Conference on Development of Water and Earth Resources, Proceedings Book. Vol. 2. 1994. pp. 719-729
  12. 12. Tosun H. Design principles for pressurized tunnels and shafts-case studies. Turkish Structure World. 2001;67:23-28
  13. 13. Tosun H. Geotechnical experience obtained from the pressurized tunnel and shafts in Turkey. In: Int. Conference on Tunneling and Underground Space Use, Proceedings Book. Istanbul; 2002. pp. 145-155
  14. 14. Tosun H. Re-analysis of Sanliurfa tunnels by finite element method. In: Proceedings Book, AITES-ITA world tunnel congress- Underground space use: analysis of the past and lessons for the future. Istanbul, Turkey; 2005. pp. 1085-1090
  15. 15. Tosun H. Re-analysis of Sanlıurfa tunnels. International Water Power & Dam Construction. 2007;2007:16-20
  16. 16. Aksoy CO. Review of rock mass rating classification: Historical developments, applications and restrictions. Journal of Mining Science. 2008;44(1):51-63
  17. 17. Rahmani N, Nikbakhtan B, Ahangari K, Apel D. Comparison of empirical and numerical methods in tunnel stability analysis. International Journal of Mining, Reclamation and Environment. 2012;26(3):261-270
  18. 18. Dehkordi MS, Shahriar K, Maarefvand P, Nik MG. Application of the strain energy to estimate the rock load in non-squeezing ground condition. Archives of Mining Sciences. 2011;56(3):551-566
  19. 19. Franza A, Marshall AM. Empirical and semi-analytical methods for evaluating tunnelling-induced ground movements in sands. Tunnelling and Underground Space Technology. 2019;88:47-62
  20. 20. Khan B, Jamil SM, Jafri TH, Akhtar K. Effects of different empirical tunnel design approaches on rock mass behaviour during tunnel widening. Heliyon. 2019;5:e02944
  21. 21. Soufi A, Bahi L, Ouadif L, Kissai JE. Correlation between rock mass rating, Q-system and rock mass index based on field data. In: MATEC Web of Conferences. Vol. 149. 2018. p. 02030
  22. 22. Farhadian H, Nikvar-Hassani A. Development of a new empirical method for tunnel squeezing classification (TSC). Quarterly Journal of Engineering Geology and Hydrogeology. 2020;53:655-660
  23. 23. Maleki Z, Farhadian H, Nikvar-Hassani A. Geological hazards in tunnelling: The example of Gelas water conveyance tunnel, Iran. Quarterly Journal of Engineering Geology and Hydrogeology. 2020;54(1):qjegh2019-114
  24. 24. Terzaghi K. Introduction to tunnel geology. In: Proctor RV, White TL, editors. Rock Tunnelling with Steel Supports. Youngstown, OH: Commercial Shearing & Stamping Co.; 1946. pp. 17-99
  25. 25. Proctor RV, White TL. Rock Tunneling with Steel Support. Ohio: Commercial Shearing and Stamping Co.; 1968
  26. 26. Bieniawski ZT. Engineering classification of jointed rock masses. The Transactions of the South African Institution of Civil Engineers. 1973;15:335-344
  27. 27. Bieniawski ZT. Rock mass classification in rock engineering. In: Bieniawski ZT, editor. Symposium on Exploration for Rock Engineering. Balkema: Rotterdam; 1976. pp. 97-106
  28. 28. Bieniawski ZT. Determining rock mass deformability: Experience from case histories. International Journal of Rock Mechanics and Mining Science and Geomechanics Abstracts. 1978;15:237-247
  29. 29. Bieniawski ZT. The geomechanics classification in rock engineering applications. In: Proceedings of the 4th International Congress on Rock Mechanics. Montreux; Vol. 2. 1979. pp. 41-48
  30. 30. Bieniawski ZT. Engineering Rock Mass Classifications: A Complete Manual for Engineers and Geologists in Mining, Civil, and Petroleum Engineering. Vol. xii. New York: Wiley; 1989. p. 251
  31. 31. Bieniawski ZT. Classification of rock masses for engineering: The RMR system and future trends. In: Hudson JA, editor. Comprehensive Rock Engineering, Volume 3. Oxford; New York: Pergamon Press; 1993. pp. 553-573
  32. 32. Singh B, Goel RK. Rock Mass Classification: A Practical Approach in Civil Engineering. Elsevier; 1999
  33. 33. Barton N, Lien R, Lunde J. Engineering classification of rock masses for the design of tunnel support. Rock Mechanics. 1974;6:189-239
  34. 34. Barton N. Rock Mass Classification and Tunnel Reinforcement Selection Using the Q-System, Rock Classification for Engineering Purpose. Vol. 984. Philadelphia: ASTM Special Technical Publication; 1988. pp. 59-88
  35. 35. Barton N. The influence of joint properties in modeling jointed rock masses. In: Proceedings of the 8th International Congress on Rock Mechanics. Tokyo, Japan; 1995. pp. 1023-1032
  36. 36. Barton N. Some new Q-value correlations to assist in site characterization and tunnel design. International Journal of Rock Mechanics and Mining Sciences. 2002;39:185-216
  37. 37. Barton N, Bandis SC. Review of predictive capability of JRC-JCS model in engineering practice. In: Barton NR, Stephansson O, editors. International Symposium on Rock Joints. Rotterdam: Balkema; 1990. pp. 603-610
  38. 38. Barton N, Løset F, Lien R, Lunde J. Application of the Q-system in design decisions. In: Bergman M, editor. Subsurface space. Vol. 2. New York: Pergamon; 1980. pp. 553-561
  39. 39. NGI. Using the Q-System- Rock Mass Classification and Support Design. Oslo: Norwegian Geotechnical Instıtute; 2015. p. 54
  40. 40. Ladanyi B. Use of the log-term strength concept in the determination of ground pressure on tunnel lining. In: Proc. 3rd Cong. On Rock Mech., ISRM, v.11, Part B. Denver, Coloroda; 1974
  41. 41. Hoek E, Brown ET. Underground Excavation in Rock: The Institution of Mining and Metallurgy. London, England; 1980
  42. 42. AFTES. In: Panet M, editor. Recommendation on Convergence-Confinement Method. 2001. p. 11
  43. 43. Sinha RS. Underground Structures—Design and Instrumentation. Oxford: Elsevier Science; 1989
  44. 44. Mahdevari S, Torabi SR. Prediction of tunnel convergence using artificial neural networks. Tunnelling and Underground Space Technology. 2012;28:218-228
  45. 45. Oreste P. The convergence-confinement method: Roles and limits in modern Geomech. Tunnel design. American Journal of Applied Sciences. 2009;6(4):757-771
  46. 46. Alonso E, Alejeno LR, Manin GF, Basance FG. Influence of post-peak properties in the application of the convergence-confinement method for designing underground excavations. In: 5th International Conference and Exhibition on Mass Mining. Luleå, Sweden; 2008
  47. 47. Sandrone F, Labiouse V. Analysis of the evolution of road tunnels equilibrium conditions with a convergence–confinement approach. Rock Mechanics and Rock Engineering. 2010;43:201-218
  48. 48. Massinas S. Designing a Tunnel. In: Sakellariou M, editor. Tunnel Engineering. Intechopen; 2019. DOI: 10.5772/intechopen.90182
  49. 49. Rabcewicz L. The New Austrian Tunnelling Method, Part one, Water Power. 1964. pp. 453-457, Part two, Water Power. 1964. pp. 511-515
  50. 50. Muller L. The reasons for unsuccessful applications of the new Austrian tunnelling method, tunnelling under difficult conditions. In: Proceedings of the International Tunnel Symposium. Tokyo: Pergamon Press; 1978. pp. 67-72
  51. 51. Brown ET. Putting the NATM into perspective. Tunnels and Tunneling. November 1981;13(10):13-17
  52. 52. Zhang Y, Zuang X, Lackner R. Stability analysis of shotcrete supported crown of NATM tunnels with discontinuity layout optimization. International Journal for Numerical and Analytical Methods in Geomechanics. 2017;42(11):1199-1216
  53. 53. Peck RB. Deep excavation and tunneling in soft ground. State of the art report. In: Proc. 7th Int. Conf. on Soil Mech. and Found. Eng. Mexico; 1969. pp. 225-290
  54. 54. Mroueh H, Shahrour I. A full 3-D finite element analysis of tunneling–adjacent structures interaction. Computers and Geotechnics. 2003;30:245-253
  55. 55. Roatesi S. Finite element analysis for the problem of tunnel excavation successive phases and lining mounting. In: Chapter in Intechopen Book of Finite Element Analysis-New Trends and Developments. 2012. pp. 301-332
  56. 56. Tang SB, Tang CA. Numerical studies on tunnel floor heave in swelling ground under humid conditions. International Journal of Rock Mechanics and Mining Sciences. 2012;55:139-150
  57. 57. Elarabi H, Mustafa A. Comparison of numerical and analytical methods of analysis of tunnels. In: Eighth International Symposium on Geotechnical Aspects of Underground Construction in Soft Ground. IS-Seoul; 2014. pp. 1-8
  58. 58. Zhang ZX, Liu C, Huang X, Kwok CY, Teng L. Three-dimensional finite-element analysis on ground responses during twin-tunnel construction using the URUP method. Tunnelling and Underground Space Technology. 2016;58:133-146
  59. 59. Akram MS, Ahmed L, Ullah MF, Rehman F, Ali M. Numerical verification of empirically designed support for a headrace tunnel. Civil Engineering Journal. 2018;4(11):2575-2587
  60. 60. Zaid M, Mishra S, Rao KS. Finite element analysis of static loading on urban tunnels. In: Latha Gali M, Raghuveer Rao P, editors. Geotechnical Characterization and Modelling. Lecture Notes in Civil Engineering. Vol. 85. Singapore: Springer; 2020
  61. 61. Kripakov NP. Finite Element Method of Design. Denver, Colorado: U.S. Department of Interior, Bureau of Mines; 1983. p. 17
  62. 62. Swoboda G. Finite element analysis of the new Austrian tunnelling method (NATM). In: Proceedings of the 3rd International Conference on Numerical Methods in Geomechanics. Vol. 2. Aachen; 1979. pp. 581-586
  63. 63. Swoboda G, Marence M, Mader I. Finite element modelling of tunnel excavation. International Journal for Engineering Modelling. 1994;6:51-63

Written By

Hasan Tosun

Submitted: 27 May 2021 Published: 25 May 2022