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An analytical design method for ductile support structures in squeezing tunnels

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Abstract

Ductile linings have been proved to be highly effective for tunnelling in heavy squeezing grounds. But there still has not been a well-established design method for them. In this paper, an investigation on an analytical design method for ductile tunnel linings is performed. Firstly, a solution in closed form for ground response of a circular tunnel within Burgers viscoelastic rocks is derived, accounting for the displacement release effect. Then based on the principle of equivalent deformation, the mechanical model of segmental shotcrete linings with yielding elements is established using the homogenization approach. Analytical prediction for behaviour of ductile tunnel linings is provided. Furthermore, the proposed design method for ductile tunnel linings is applied in Saint Martin La Porte access tunnel and the analytical prediction is in good agreement with field monitoring data. Finally, a parametric investigation on the influence of yielding elements on performance of ductile tunnel linings is conducted. Results show that the length of yielding elements poses a great influence on linings. It is feasible and effective to increase the length of yielding elements to obtain the pressure within the bearing capacity of linings. However, yield stress of yielding elements does not significantly affect the performance of the lining. It is suggested to apply yielding elements with relatively higher yield stress in linings for higher stability.

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Abbreviations

\(\sigma ,\sigma_{r} ,\sigma_{\theta }\) :

Compression stress, radial and tangential stresses, respectively

\(\overline{{\sigma_{\theta } }}\) :

Mean tangential stress

\(\sigma_{y}\) :

Yield stress of yielding element

\(\varepsilon ,\varepsilon_{lim}\) :

Strain and limit strain of yielding element, respectively

\(s_{ij}\) :

Tensor of stress deviator

\(e_{ij}\) :

Tensor of strain deviator

\(G_{1} ,G_{2}\) :

Elastic constants of Hookean spring

\(\eta_{1} ,\eta_{2}\) :

Viscous coefficients of Newtonian dashpot

\(\lambda\) :

Lateral pressure coefficient

\(A_{k} ,B_{k}\) :

Coefficients in Burgers constitutive equation, respectively

\(A,A_{1} ,A_{2}\) :

Cross-sectional area, cross-sectional areas of segmental lining and steel pipes, respectively

\(C_{1} ,C_{2} ,D_{1} ,D_{2}\) :

Coefficients related with Poisson’s ratio of rock

\(E_{1} ,E_{2} ,E^{*}\) :

Elastic moduli of shotcrete, yielding element and homogenized lining, respectively

\(\nu_{1} ,\nu_{2} ,\nu^{*} ,\nu_{r}\) :

Poisson’s ratios of shotcrete, yielding element, homogenized lining and rock, respectively

\(F,F_{\theta }\) :

Axial force and tangential axial force in lining, respectively

\(k\) :

Ratio of cross-sectional areas of steel pipes to segmental lining

\(K_{s}^{\left( i \right)}\) :

Stiffness of lining

\(l_{1i} ,l_{2i}\) :

Lengths of each segmental lining and yielding element, respectively

\(M,N,Q\) :

Bending moment, axial force and shear force, respectively

\(P,Q\) :

Differential time operators

\(p_{i}\) :

Internal pressure at tunnel wall or lining pressure

\(p_{0}\) :

Initial ground stress

\(R_{0}\) :

Tunnel radius

\(r\) :

Distance between element and tunnel centre

\(t_{s}\) :

Thickness of lining

\(t\) :

Time

\(u^{*}\) :

Displacement release coefficient

\(u_{R}\) :

Tunnel radial displacement or lining displacement

\(u_{\max }\) :

Maximum unlined tunnel radial displacement

\(u_{0}\) :

Tunnel radial displacement with X* = 0

\(u_{1} ,u_{2}\) :

Lining displacements responding to beginning time and finishing time of plastic strain of yielding elements

\(X^{*}\) :

Ratio of distance from tunnel face to tunnel radius

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11872287), and the Fund of Shaanxi Key Research and Development Program (No. 2019ZDLGY01-10).

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Authors and Affiliations

  1. School of Civil Engineering, Xi’an University of Architecture and Technology, Xi’an, 710055, China

    Kui Wu, Zhushan Shao & Su Qin

  2. Shaanxi Key Lab of Geotechnical and Underground Space Engineering (XAUAT), Xi’an University of Architecture and Technology, Xi’an, 710055, China

    Kui Wu, Zhushan Shao & Su Qin

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  1. Kui Wu
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Correspondence to Zhushan Shao.

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Appendix

Appendix

As previously mentioned, for steel pipes acting as yielding elements, there is a reduction in their stiffness. We define that

$$k = \frac{{A_{2} }}{{A_{1} }},$$
(34)

where A1 denotes the cross-sectional area of the segmental linings and A2 represents the actual cross-sectional area of steel pipes.

There is a relationship between compression stress and elastic modulus of material that

$$E = \frac{\sigma }{\varepsilon } = \frac{F}{\varepsilon A},$$
(35)

where σ and ε are compression stress and strain, respectively. F denotes the axial force. E and A are the elastic modulus and cross-sectional area of the material.

Considering the stress state of steel pipes in the lining, Eq. (35) can be rewritten as

$$E_{2} = \frac{{F_{\theta } }}{{\varepsilon_{2i} A_{2} }},$$
(36)

where Fθ represents the tangential axial force in the lining.

Substituting Eq. (34) into (36) provides

$$kE_{2} = \frac{{F_{\theta } }}{{\varepsilon_{2i} A_{1} }}.$$
(37)

The right part of Eq. (37) is the calculation formula for elastic modulus of normal yielding elements where a reduction in stiffness is not considered. Then, replacing E2 in Eq. (21) with kE2 can provide the elastic modulus of homogenized lining where a reduction in stiffness of steel pipes is estimated.

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Wu, K., Shao, Z. & Qin, S. An analytical design method for ductile support structures in squeezing tunnels. Archiv.Civ.Mech.Eng 20, 91 (2020). https://doi.org/10.1007/s43452-020-00096-0

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