Analytical Solutions for the Construction of Deeply Buried Circular Tunnels with Two Liners in Rheological Rock
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ORIGINAL PAPER
Analytical Solutions for the Construction of Deeply Buried Circular Tunnels with Two Liners in Rheological Rock H. N. Wang • Y. Li • Q. Ni • S. Utili M. J. Jiang • F. Liu
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Received: 19 October 2012 / Accepted: 27 December 2012 Springer-Verlag Wien 2013
Abstract The construction of underground tunnels is a time-dependent process. The states of stress and strain in the ground vary with time due to the construction process. Stress and strain variations are heavily dependent on the rheological behavior of the hosting rock mass. In this paper, analytical closed-form solutions are developed for the excavation of a circular tunnel supported by the construction of two elastic liners in a viscoelastic surrounding rock under a hydrostatic stress field. In the solutions, the stiffness and installation times of the liners are accounted for. To simulate realistically the process of tunnel excavation, a time-dependent excavation process is considered in the development of the solutions, assuming that the radius of the tunnel grows from zero until its final value according to a time-dependent function to be specified by the designers. The integral equations for the supporting pressures between rock and first liner are derived according to the boundary conditions for linear viscoelastic rocks (unified model). Then, explicit analytical expressions are obtained by considering either the Maxwell or the Boltzmann viscoelastic model for the rheology of the rock mass. Applications of the obtained solutions are illustrated using two examples, where the response in terms of H. N. Wang (&) Y. Li School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, People’s Republic of China e-mail: [email protected] Q. Ni S. Utili School of Engineering, University of Warwick, Coventry CV4 7AL, UK M. J. Jiang F. Liu Department of Geotechnical Engineering, College of Civil Engineering, Tongji University, Shanghai 200092, People’s Republic of China
displacements and stresses caused by various combinations of excavation rate, first and second liner installation times, and the rheological properties of the rock is illustrated. Keywords Tunnels Rheological rock Liner Analytical research List of Symbols A, B a D1, D2 … D12 EL1 (EL2 ) Ei ðGi Þ fB1 ðfB2 Þ fM1 ðfM2 Þ GB1
GB
GL1 ðGL2 Þ GM
GK
G H
Laplace transform functions of s Function of the excavation process Constant coefficients Young’s modulus of the first (second) liner The ith Young’s (shear) modulus in the general viscoelastic model Free term of integral equation Free term of integral equation Shear elastic modulus of the Hookean element in the Boltzmann model Shear elastic modulus of the Kelvin element in the Boltzmann model Shear elastic modulus of the first (second) liner Shear elastic modulus of the Hookean element in the Maxwell model Shear elastic modulus of the Kelvin element in the Burgers model Relaxation shear modulus in the rock viscoelastic model Function defined in Eq. (13)
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H. N. Wang et al.
I kM ðkM Þ k
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