Theoretical solutions for NATM excavation in soft rock with non-hydrostatic in-situ stresses

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

2. 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

3. 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

4. Constitutive law for soft rock Back Figure3 Typical stress and strain curves under triaxial tests Relationship between s1 and s3 Relationship between e1 and e3 Mohr-Coulomb Criterion Plastic Poisson Ratio h

5. Constitutive law for soft rock Back Figure3 Typical stress and strain curves under triaxial tests Relationship between s1 and s3 Relationship between e1 and e3 Mohr-Coulomb Criterion Plastic Poisson Ratio

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

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

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

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

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

11. Kc Lining stiffness in reality Figure6 Physical significance of Kequ 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 Back-analyze h Kini Pre-released displacement initial stiffness due to face Equivalent series stiffness Forward-analyze Kequ ua Equivalent series stiffness Ultimate convergence

12. 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 q, search for proper Re and Rf that verify all the boundary and continuum conditions To determine all the state variables (u, e and s) in three zones, especially ultimate convergence (ua)

13. E (Mpa) m a h f w(8) sc (Mpa) sc* (Mpa) 2000 0.3 1.33 1.88 0.41 25 1 0.65 Pv (Mpa) Ph (Mpa) Ra (m) Ec (Mpa) mc tc (m) h 2.5 1.5 5 20000 0.25 0.1 0.3 Pv=2.75, Ph=1.25 Only s-s zones connected Pv=3.0, Ph=1.0 Both of two zones separated Basic case: Pv=2.5, Ph=1.5 Both of two zones connected Solution implementation Parameters employed in the basic case Calculation results

14. 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/Re0, ua/ua0 and Eng/Eng0 are employed in case studies to standardize and highlight the variation of them

15. Influence of rock mass properties sc and sc* influence both Re and ua greatly sc and sc* 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

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

17. 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

18. Thank You ! End