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Evaluation of analytical solutions and two-dimensional models in estimating the internal forces of tunnel lining against seismic loading, compared with three-dimensional analysis

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When the ground is shaken by an earthquake, forces acting on tunnel lining increase and may damage concrete lining. So far, seismic design of tunnel lining has been done using simple analytical solutions based on elasticity theory and two-dimensional numerical models. Purpose of this paper is to evaluate analytical solutions and two-dimensional numerical models for seismic analysis of tunnels by comparing their results with three-dimensional numerical models. Numerical analysis is performed using finite difference method, and ground behavior is assumed to be elastoplastic, and seismic loading is applied to numerical models as a harmonic wave. According to some of the results obtained in this paper, the internal forces induced in the tunnel lining due to seismic loading depend on the behavioral model and mechanical properties of the ground materials, the type of seismic loading, and the depth and diameter of the tunnel. Given that analytical solutions are based on simple assumptions such as linear elastic behavior for materials, they do not provide accurate estimates of internal forces. Two-dimensional numerical models also ignore energy dissipation in the third dimension, which gives a higher estimate of forces than three-dimensional models; therefore, three-dimensional numerical models are a good option for seismic analysis of tunnels. According to the seismic analysis performed for the tunnel in a type of sandy soil at different depths, it was observed that the thrust, shear force, and bending moment in the dynamic analysis are 0.3, 1.4, and 1.3 times on average the static analysis, respectively.

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The authors would like to thank the anonymous reviewers for their careful reading of the manuscript and their many insightful comments and suggestions.

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Correspondence to Ali Lakirouhani.

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Responsible editor: Murat Karakus

Appendix 1. Validation for 3D models

Appendix 1. Validation for 3D models

In this appendix, the three-dimensional model is validated with analytical relations. For this purpose, a tunnel with a radius of R = 4.5m and a depth of z = 15m with a concrete lining with a thickness of tc = 40cm is considered. The dimensions of the model and meshing are shown in Fig. 22. A quarter of the model has been intentionally removed. For comparison with analytical relations, the behavior of the materials is considered linear elastic, and uniform shear stress is applied to the boundaries of the three-dimensional model.

Fig. 22
figure 22

Domain of the problem and finite difference meshing

The specifications of the soil materials are as

$${E}_s=60 MPa,{\upsilon}_s=0.3,{\gamma}_s=\frac{1700 kg}{m^3},k=0.43.$$

To calculate the shear stress, an earthquake with specifications, amax  = 0.20g, Mw = 6.5, is considered. The soil is assumed to be soft and Cs = 116.5m/sec.

According to the calculations performed in accordance with the method provided by Hashash et al. (2001) and assuming no-slip conditions at the interface between the tunnel and the ground, the diagrams of thrust and bending moment obtained from Wang and PTTO relationships and numerical solution are presented in Fig. 23 and Fig. 24, respectively. As can be seen, assuming uniform shear stress on the boundaries of the domain as well as linear elastic behavior for the materials, a good fit between numerical models and analytical relations is observed.

Fig. 23
figure 23

Thrust distribution in tunnel lining, comparison of numerical method with analytical methods

Fig. 24
figure 24

Bending moment distribution in tunnel lining, comparison of numerical method with analytical methods

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Lakirouhani, A., Saberi, M. Evaluation of analytical solutions and two-dimensional models in estimating the internal forces of tunnel lining against seismic loading, compared with three-dimensional analysis. Arab J Geosci 15, 1110 (2022).

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