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Direct Calculation Method for the Analysis of Non-linear Behavior of Ground-Support Interaction of a Circular Tunnel Using Convergence Confinement Approach

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Abstract

The convergence-confinement method is widely used in conventional tunneling at a preliminary stage of the support design. A circular tunnel through the ground in an initially isotropic stress state and the behavior of ground-support interaction simplified utilizing a two-dimensional plane-strain are postulated. From a point of view of practical application, the direct algorithm process so-called the direct calculation method (DCM) is proposed in this paper to deal with solving the solution of stresses/displacements between the ground reaction curve (GRC) and the support confining curve (SCC) in the final equilibrium state by applying the simultaneous equations in the elastic region and using the numerical analysis known as the Newton recursion method for finding roots of the non-linear equations in the plastic region. This explicit procedure also can realize the analytical solution to an executable computation that can be stepwise estimated by using a simple calculation spreadsheet. The validity of the developed method for the analytical solution was examined by the finite element analysis (FEM) to investigate the influence of mechanical properties of the ground and the time-dependent effects of the shotcrete lining-support on the GRC, SCC and stress path at the intrados of the tunnel. The agreement of results between DCM and FEM was found to be excellent in the elastic and elastic-perfectly plastic media.

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Abbreviations

CCM:

Convergence-confinement method

CLC:

Confinement loss curve

c :

Cohesion of the ground

DCM:

Direct calculation method

E :

Elastic modulus of the ground

E shot :

Elastic modulus of the shotcrete lining

G :

Shear modulus of the ground

GRC:

Ground reaction curve

K o :

In-situ initial stress ratio

k s :

Stiffness of the support

K p :

Coefficient \(K_{p} = \tan^{2} \left( {45^{ \circ } + \frac{\varphi }{2}} \right)\)

K ψ :

Coefficient \(K_{\psi } = \tan^{2} \left( {45^{ \circ } + \frac{\psi }{2}} \right)\)

LDP:

Longitudinal displacement profile

N :

Stability number \(N = \frac{{\sigma_{c} }}{{2\sigma_{v} }}\)

p s :

Support pressure

r :

Distance in the polar coordinate

R :

Radius of tunnel excavation

R p :

Plastic radius

\(R_{p}^{d}\) :

Plastic radius at a certain distance d to the working face

\(R_{p}^{s}\) :

Plastic radius in the equilibrium state

SCC:

Support confining curve

t shot :

Shotcrete thickness

u R :

Radial displacement at the intrados of tunnel

\(u_{R}^{d}\) :

Radial displacement at the supported distances d to the working face

\(u_{R}^{s}\) :

Radial displacement in the equilibrium state

\(u_{R}^{z}\) :

Radial displacement at the distances z to the working face

\(u_{R}^{\infty }\) :

Radial displacement at a great distance to the working face

Z :

Longitudinal axis of the tunnel

ϕ :

Friction angle of the ground

η :

Parameter of hyperbolic function

λ d :

Confinement loss λ at a certain distance d to the working face

λ z :

Confinement loss λ at a certain distance z to the working face

λ e :

Confinement loss in the elastic limit state

λ s :

Confinement loss in the equilibrium state

ν :

Poisson’s ratio of the ground

ν shot :

Poisson’s ratio of the shotcrete lining

θ :

Angle in polar coordinate

σ c :

Uniaxial compression strength (UCS) of the ground

σ R :

Radial stresses at the intrados of tunnel

\(\sigma_{R}^{d}\) :

Radial stresses at the intrados of tunnel at a certain distance d to the working face

\(\sigma_{R}^{s}\) :

Radial stresses at the intrados of tunnel in the equilibrium state

σ θ :

Tangential stresses at the intrados of tunnel

\(\sigma_{\theta }^{d}\) :

Tangential stresses at the intrados of the tunnel at a certain distance d to the working face

\(\sigma_{\theta }^{s}\) :

Tangential stresses at the intrados of the tunnel in the equilibrium state

σ v :

Overburden pressure

ψ :

Dilation angle of the ground

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Authors and Affiliations

  1. Department of Civil Engineering, Chung Hua University, No. 707, Sec.2, Wufu Rd., Hsinchu, 30012, Taiwan, ROC

    Yu-Lin Lee, Yi-Xian Xin & Bo-Yu Zhou

  2. Department of Civil and Disaster Prevention Engineering, National United University, No. 1, Lienda, Miaoli, 36003, Taiwan, ROC

    Wen-Kuei Hsu

  3. Department of Civil Engineering, Chung Yuan Christian University, No. 200, Zhongbei Rd., Taoyuan City, 32023, Taiwan, ROC

    Chi-Min Lee

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Lee, YL., Hsu, WK., Lee, CM. et al. Direct Calculation Method for the Analysis of Non-linear Behavior of Ground-Support Interaction of a Circular Tunnel Using Convergence Confinement Approach. Geotech Geol Eng 39, 973–990 (2021). https://doi.org/10.1007/s10706-020-01539-4

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