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A Simple Closed-Form Solution for Analysis of Tunnels in Mohr–Coulomb Grounds Considering Gravity Loading

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Abstract

In tunnels, convergences of sidewalls, roof and floor may be different, because of weight of the failed or plastic rock mass. In this paper, a closed-form analytical solution for a deep tunnel excavated in an elastoplastic rock mass is proposed. Using the proposed solution, the effects of the weight of the plastic or failed region developed around the tunnel are investigated. In the proposed method, brittle–plastic or elastic–perfectly plastic behavior and Mohr–Coulomb yield criterion and plastic potential function are used for the ground medium. The gravitational loading is considered as a radial body force being applied to the ground medium. Illustrative examples are given to demonstrate the performance of the proposed method and to examine the effects of the gravity loads. The results obtained by the proposed method show that the gravity loading may affect the tunnel convergence, considerably.

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Abbreviations

\(F_{r}\) :

Radial body force

\(F_{\theta }\) :

Circumferential body force

P :

Plastic potential function

\(p_{i}\) :

Internal pressure

\(R_{p}\) :

Radius of plastic region

\(r_{i}\) :

Tunnel radius

\(r\) :

Radial distance from the center of the tunnel

\(u_{r}\) :

Radial displacement

\(\varepsilon_{1}\) :

Major principal strain of rock mass

\(\varepsilon_{3}\) :

Minor principal strain of rock mass

\(\varepsilon_{\theta }\) :

Circumferential strain

\(\varepsilon_{r}\) :

Radial strain

\(\varepsilon_{\theta }^{e}\) :

Initial elastic circumferential strain

\(\varepsilon_{r}^{e}\) :

Initial elastic radial strain

\(\varepsilon_{\theta }^{p}\) :

Initial plastic circumferential strain

\(\varepsilon_{r}^{p}\) :

Initial plastic radial strain

\(\varphi_{i} ,C_{i}\) :

Material constants for original rock mass

\(\varphi_{r} ,C_{r}\) :

Material constants for residual rock mass

\(\gamma\) :

Unit weight of ground medium

\(\nu\) :

Poisson’s ratio of rock mass

\(\theta\) :

Angle measured clockwise from tunnel spring line

\(\rho\) :

Normalized radius

\(\sigma_{0}\) :

Hydrostatic field stress

\(\sigma_{1}\) :

Major principal stress

\(\sigma_{3}\) :

Minor principal stress

\(\sigma_{\theta }\) :

Circumferential stress

\(\sigma_{r}\) :

Radial stress

\(\varPsi\) :

Dilation angle

e :

Refers to elastic part

p :

Refers to plastic part

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  1. Department of civil Engineering, Estahban Higher Education Center, Estahban, Iran

    Mohammad Reza Zareifard

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Zareifard, M.R. A Simple Closed-Form Solution for Analysis of Tunnels in Mohr–Coulomb Grounds Considering Gravity Loading. Geotech Geol Eng 38, 3751–3760 (2020). https://doi.org/10.1007/s10706-020-01255-z

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