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Analytical Solution for Seismic Response of Deep Tunnels Near a Fault Zone Subjected to SH Waves

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Abstract

A novel analytical solution for stresses and displacements of deep tunnels near a fault zone subjected to SH waves is presented. The ground, fault and liner are assumed as linear isotropic elastic, with a no-slip condition at all contact interfaces. It is assumed that the tunnel is parallel to the fault and therefore the displacements perpendicular to the tunnel axis are zero. The effect of the fault on the wave diffracted by the tunnel is neglected, to be able to obtain the solution. Validation of the proposed analytical solution is accomplished by comparing the results of the solution with those obtained from ABAQUS. The relevance of key parameters, namely: the stiffness ratio between the fault and the ground, the width of the fault, and the frequency of the incident SH wave, are discussed using the numerical results and analytical formulation.

Highlights

  • Analytical solution for response of deep tunnels near a fault zone subjected SH waves is proposed.

  • Simple formulations are given to obtain the maximum effect of faults on seismic response of tunnels.

  • Results show significant impacts of the width of the fault on tunnel seismic performances.

  • The presented solution could be an effective and efficient tool for practitioners.

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Abbreviations

x :

y, z: Cartesian coordinates

\(r, \, \theta , \, z\) :

Cylindrical coordinates

\(E_{{\text{g}}}\), \(E_{{\text{f}}}\), \(E_{{\text{s}}}\) :

Young’s modulus of the ground, the fault and the liner

\(\nu_{{\text{g}}}\), \(\nu_{{\text{f}}}\), \(\nu_{{\text{s}}}\) :

Poisson’s ratio of the ground, the fault and the liner

\(\rho_{{\text{g}}}\), \(\rho_{{\text{f}}}\), \(\rho_{{\text{s}}}\) :

Mass density of the ground, the fault and the liner

\(\theta_{1}\) :

Angle between the horizontal axis and the propagation direction of incident wave

\(\theta_{2}\) :

Angle between the vertical axis and the fault

f, \(\omega\) :

Frequency and circular frequency of the incident waves

h :

Width of the fault

d :

Distance from the center of the tunnel to the fault

r 0, r 1 :

Internal radius and external radius of the liner

\(k_{{\text{g}}}\), \(k_{{\text{f}}}\), \(k_{{\text{s}}}\) :

Wave number in the ground, the fault and the liner

\(J_{n} \left( x \right)\), \(H_{n}^{(1)} (x)\), \(H_{n}^{(2)} (x)\) :

Bessel function, first kind Hankel function and second kind Hankel function

t :

Time

\(\sigma_{{}}^{{({\text{g}}1)}}\), \(u_{z}^{{({\text{g}}1)}}\) :

Stresses and displacements of g1

\(\sigma_{{}}^{{({\text{g2}})}}\), \(u_{z}^{{({\text{g2}})}}\) :

Stresses and displacements of g2

\(\sigma_{{}}^{{({\text{f}})}}\), \(u_{z}^{{({\text{f}})}}\) :

Stresses and displacements of the fault

\(\sigma_{{}}^{{({\text{s}})}}\), \(u_{z}^{{({\text{s}})}}\) :

Stresses and displacements of the liner

\(\sigma_{x}^{{({\text{s}})}}\), \(\sigma_{xy}^{{({\text{s}})}}\), \(u_{x}^{{({\text{s}})}}\) :

Stresses and displacements components of liner

\(u_{z}^{{({\text{r1}})}}\), \(u_{z}^{{({\text{t1}})}}\), \(u_{z}^{{({\text{r2}})}}\), \(u_{z}^{{({\text{t2}})}}\) :

Displacements of waves of free-field response

\(\sigma_{z}^{{({\text{r1}})}}\), \(\sigma_{z}^{{({\text{t1}})}}\), \(\sigma_{z}^{{({\text{r2}})}}\), \(u_{z}^{{({\text{t2}})}}\) :

Stresses of waves of free-field response

\(a_{{}}^{{({\text{r1}})}}\), \(a_{{}}^{{({\text{t1}})}}\), \(a_{{}}^{{({\text{r2}})}}\) :

Complex amplitude of waves of free-field response

\(\theta_{{}}^{{({\text{r1}})}}\), \(\theta_{{}}^{{({\text{t1}})}}\), \(\theta_{{}}^{{({\text{r2}})}}\), \(\theta_{{}}^{{({\text{t2}})}}\) :

Angles of waves of free-field response

\(u_{z}^{{({\text{d}})}}\) :

Displacement of the wave diffracted by the tunnel

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Acknowledgements

The research has been supported by the Intergovernmental International Science and Technology Innovation Cooperation Key Project (2022YFE0128400), the National Natural Science Foundation of China (42177134), the Top Discipline Plan of Shanghai Universities-Class I, and the Fundamental Research Funds for the Central Universities of China.

Funding

Intergovernmental International Science and Technology Innovation Cooperation Key Project,2022YFE0128400,Haitao Yu,National Natural Science Foundation of China,42177134,Haitao Yu,Top Discipline Plan of Shanghai Universities-Class I,/,Haitao Yu,Fundamental Research Funds for the Central Universities,/,Haitao Yu

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Authors and Affiliations

  1. State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai, 200092, China

    Haitao Yu

  2. Department of Geotechnical Engineering, Tongji University, Shanghai, 200092, China

    Gong Chen

  3. Lyles School of Civil Engineering, Purdue University, West Lafayette, IN, 47907, USA

    Antonio Bobet

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  1. Haitao Yu
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  2. Gong Chen
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Correspondence to Haitao Yu.

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Yu, H., Chen, G. & Bobet, A. Analytical Solution for Seismic Response of Deep Tunnels Near a Fault Zone Subjected to SH Waves. Rock Mech Rock Eng 57, 375–388 (2024). https://doi.org/10.1007/s00603-023-03530-w

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