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Theoretical solutions for NATM excavation in soft rock with non-hydrostatic in-situ stresses Nagasaki University Z. Guan Y. Jiang Y.Tanabasi 1. Philosophy and construction process 2. Key problem: convergence released before and after supporting installation 1. Vertical in-situ stress Pv and horizontal in-situ stress Ph are apparently different from each other in most occasions 1. Constitutive law: strain-softening model 2. Three zones: elastic zone, strain-softening zone and plastic-flow zone 1. Introducing some assumption 2. Relatively simple without numerical method involved and useful for primary design

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Background--NATM Philosophy of NATM Construction process Key problem in the design of supporting Philosophy of the research Analytical model for cross section Take face effect (longitudinal effect) into account Figure1 Schematic representation of NATM Back

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Analytical model for cross section Plane strain problem Strain-softening deformation characteristic Non-hydrostatic in-situ stresses Figure2 Plane strain analytical model for cross section

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Constitutive law for soft rock Figure3 Typical stress and strain curves under triaxial tests Relationship between 1 and 3 Mohr-Coulomb Criterion Plastic Poisson Ratio h Relationship between 1 and 3 Back

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Constitutive law for soft rock Figure3 Typical stress and strain curves under triaxial tests Relationship between 1 and 3 Mohr-Coulomb Criterion Plastic Poisson Ratio Relationship between 1 and 3 Back

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Angle-wise approximation assumption So that the stress state at the inner boundary could verify Mohr-coulomb criterion exactly. Figure4 Classical problem in elasticity loadings Vertical far field stress Pv horizontal far field stress Ph Inner pressure Pi( ) varying with azimuth

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Angle-wise approximation assumption approximate its solution in elastic zone to the classical one mentioned above The essence of this assumption is to neglect shear deformation in rock mass Figure5 Approximation for an infinitesimal azimuth At elastic boundary (r=Re)

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Analytical solutions in strain-softening zone Equilibrium equation Geometry equation Displacement governing equation Stress governing equation

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Analytical solutions in plastic-flow zone Displacement governing equation Stress governing equation u, and in all three zones could be expressed as the functions of radius r, with two parameters Re and Rf unknown Equilibrium equation Geometry equation

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Determination of Re and Rf Continuum condition of tangent stress t at Rf boundary Continuum condition of radial stress r at tunnel wall boundary Kc is Radial stiffness of lining r a is the interaction force between rock mass and lining ua is the tunnel wall convergence Set up an analytical solutions for cross section model u, and in all three zones are totally determined

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Equivalent series stiffness hypothesis Before supporting The face carry the loading partly Pre-released displacement occurs After supporting and face advancing away The supporting together with rock mass carry the full load Displacement release goes on, until to the ultimate convergence KcKc Lining stiffness in reality Back-analyze K ini initial stiffness due to face Pre-released displacement Ultimate convergence K equ ua Equivalent series stiffness Forward-analyze Equivalent series stiffness Figure6 Physical significance of Kequ

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Summary of theoretical solutions Introduce angle-wise approximation assumption to simplify non-hydrostatic in-situ stresses Introduce equivalent series stiffness hypothesis to take pre-released displacement into account For every infinitesimal azimuth , search for proper Re and Rf that verify all the boundary and continuum conditions To determine all the state variables (u, and ) in three zones, especially ultimate convergence (ua)

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Basic case: Pv=2.5, Ph=1.5 Both of two zones connected Solution implementation Parameters employed in the basic case Calculation results E (Mpa) ma hf w (8)w (8) s c (Mpa) s c* (Mpa) Pv (Mpa)Ph (Mpa)Ra (m)Ec (Mpa) mcmc tc (m) h Pv=2.75, Ph=1.25 Only s-s zones connected Pv=3.0, Ph=1.0 Both of two zones separated

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Case studies The object Reveal the influence of different parameters on the supporting effect in NATM Provide primary design and suggestion for NATM The evaluation indices Re (the range of strain-softening zone), ua (the ultimate convergence of tunnel wall) and Eng (energy stored in equivalent lining) Dimensionless indices, Re/Re 0, ua/ua 0 and Eng/Eng 0 are employed in case studies to standardize and highlight the variation of them

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Influence of rock mass properties c and c* influence both Re and ua greatly c and c* determine the energy storage capability of rock mass E influences ua drastically, whereas takes little effect on Re E only change the energy storage proportion between elastic zone and lining

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Influence of supporting properties In theory Kc play identical role to Kini In practice Kini vary hundred times according to It is difficult to control Suggestion Pay more attention to and Kini It is better that make Kini equal to Kc

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Conclusions Establish a set of solutions and implementation for NATM excavation in soft rock with non-hydrostatic in-situ stresses After case studies, it is clarified that these solutions could predict the state of NATM excavation well, and useful for primary design of supporting

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