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Analytical solutions are widely adopted in the preliminary seismic design of tunnels, which gives credit to their quick response and convenience to provide the internal forces and displacements of tunnels. In the first part of this paper, five analytical solutions are extensively verified by comparing them with the reference solution which obtained by the dynamic finite element method. Some suggestions are given for their application in design according to the prediction results. The analytical solutions contain some assumptions, which may not be applicable to all possible scenarios. Therefore, the application scope of analytical solution in practical engineering is limited, and this application scope has not been discussed at present. In the second part of this paper, the applicability evaluation index is established for the first time to evaluate the critical condition for the application of analytical solution. Then, the applicability function is obtained by fitting the relationships between the critical condition and the flexibility ratios, which can be convenient to calculate the critical condition for the application of analytical solutions in engineering. Finally, the applicability of analytical solution in some engineering projects is correctly evaluated based on the proposed applicability evaluation index and applicability function. • The results of dynamic finite element analysis are used as reference solutions to verify the five analytical solution. • Applicability evaluation index is proposed to evaluate the critical condition for the application of analytical solution. • Applicability function is obtained by fitting the relationships between the critical condition and the flexibility ratios. • The applicability of analytical solution in some engineering projects is correctly evaluated. [ABSTRACT FROM AUTHOR]

A simplified analytical solution is developed for estimating the thrusts and bending moments of the circular tunnel in half space under the obliquely incident SV wave. The seismic waves impinging on the tunnel is obtained according to the elastic wave motion theory in half space firstly, which is the superposition of the incident and reflected SV waves and reflected P wave. Then, by considering the tunnel lining as a thick-wall cylinder, analytical solutions for internal forces of circular tunnel due to the incident and reflected SV waves and the reflected P wave are deduced, which enables more precise forecasts than previous analytical solutions by the thin shell model. The total tunnel internal forces are obtained finally through the superposition of the above solutions under three kinds of waves. The accuracy of the proposed analytical solution is verified by comparing with the dynamic numerical results. Moreover, the discussion about the effect of critical factors on the accuracy of the proposed analytical solutions is conducted, including the tunnel structure inertia, the seismic wavelength, the lining thickness and the tunnel depth. These results demonstrate that the proposed analytical solution performs well and can be adopted to predict the internal forces of circular tunnel under obliquely incident SV wave in seismic design. • Analytical solutions for circular tunnel in full space under SV and P waves are developed. • Analytical solution for circular tunnel in half space under obliquely incident SV wave is developed. • The present solutions are verified by comparing with dynamic numerical results. • Discussion about the effect of critical factors on the accuracy of the proposed analytical solutions is conducted. [ABSTRACT FROM AUTHOR]

Evaluating the impact of the isolation layer on the tunnel's response is crucial in the practical engineering. An analytical solution for the longitudinal seismic response of a tunnel covered by an isolation layer is first proposed in this study based on an improved elastic foundation beam model. Wherein, an equivalent beam model for the tunnel-isolation layer system is built. The tangential interaction effect at the ground-isolation layer-tunnel interfaces is taken into account by solving the ground-isolation layer-tunnel model under out-of-plane shearing stress. The proposed solution is verified by comparison with dynamic numerical results. Finally, the proposed analytical solutions are employed to investigate the seismic mitigation effects of the isolation layer on the longitudinal response of the tunnel, as well as the influence of the elastic modulus, thickness and Poisson's ratio of the isolation layer. The results indicate that the isolation layer can significantly reduce the longitudinal shearing force of the tunnel but always amplifies the longitudinal bending moment in most cases. Moreover, the mitigating effect of the isolation layer on the longitudinal response of the tunnel is sensitive to the elastic modulus and thickness, except for Poisson's ratio of the isolation layer. • An equivalent beam model for the tunnel-isolation layer system is built. • Tangential interactions at the ground-isolation-tunnel interfaces are considered. • Analytical solution of response for tunnel with isolation layer is proposed. • Seismic mitigation effects of the isolation layer are investigated. [ABSTRACT FROM AUTHOR]

The fault zone plays a vital role in the internal force responses of the tunnel. An analytical solution of the tunnel under the fault zone creeping is developed by taking account of normal and tangential interaction between the ground and tunnel. First, the elastic foundation beam model is used. Wherein, the normal interaction is considered by applying the displacements of the free field on the distal end of normal foundation springs. And the tangential interaction is transformed into bending moment applied on the beam. The analytical solution of the tunnel's response is gotten based on the model. Secondly, the numerical solutions from finite element model of 3D are provided to verify the validity of the proposed analytical solutions. Finally, the effect of the fault zone width, the elastic modulus of the fault and the concrete's grade of the lining are investigated. The results show that with the width of the fault zone increasing, the internal force decrease. The concrete's grade of the lining has opposite effects on the internal force of the tunnel. And with the growth of the elasticity modulus of the fault zone, the shear force and bending moment increase and decrease, respectively. • An analytical solution considering the tangential interaction at the ground-tunnel interface is developed. • The proposed analytical solution is verified by the numerical result obtained from the finite element model of 3D. • Effects of fault zone width, elasticity modulus of fault zone and concrete's grade of the lining are discussed. [ABSTRACT FROM AUTHOR]

The incident direction plays an important role in the seismic response of long lined tunnels. A simplified analytical solution is developed for estimating the seismic response of the circular tunnel under obliquely incident shear waves, including the principle stress and the internal forces and deformations on the cross section and longitudinal of the tunnel. Based on the elastic wave theory, the oblique incident shear wave can be decomposed into an axially propagating shear wave and a shear wave propagating transversely to the tunnel axis. A modified elastic foundation beam model is used to calculate the tunnel longitudinal response, which is based on Timoshenko beams resting on Winkler foundations with considering different tangential contact conditions. Furthermore, by adopting wave theory and Fourier transform, longitudinal response of tunnels under arbitrary seismic load can be obtained. Moreover, the flattening effect and out-plane response of tunnel cross-section caused by longitudinal bending are considered. The proposed analytical solution is verified by comparing with the dynamic numerical results. The effect of the incident angle, ground condition and contact condition on the seismic response of long lined tunnel is investigated by the simplified analytical solution. It is demonstrated that the proposed solution can be adopted to predict internal forces and deformations of circular tunnels under obliquely incident shear wave in preliminary design. • A simplified analytical solution is developed for estimating the seismic response of the circular tunnel under obliquely incident shear waves. • A modified foundation beam model is used for the tunnel longitudinal response, with considering different tangential contact conditions. • By adopting wave theory and Fourier transform, longitudinal response of tunnels under arbitrary seismic load can be obtained. • Moreover, the flattening effect and out-plane response of tunnel cross-section caused by longitudinal bending are considered. • Parameter studies are conducted to investigate the effects of incident angle, ground condition and contact condition. [ABSTRACT FROM AUTHOR]

Tunnels, as critical underground infrastructures, often intersect with faults. Field investigations have underscored the susceptibility of fault-crossing tunnels to extensive damages during earthquakes. Hence, large-scale shaking table tests are conducted to investigate the deformation pattern and failure mechanism of fault-crossing tunnels under transverse excitation, particularly in non-rupture scenarios. The dynamic behavior of the tunnel, as expected, is dominated by the surrounding strata. Acceleration responses vary notably across different regions of the fault site. The disparity increases with the increments of seismic intensity and input frequency. The tunnel section at the interface between the fault and the hanging wall manifest the strongest acceleration, where the largest strains of the tunnel and most of the lining cracks also occur, highlighting its vulnerability to seismic vibrations. The damage-induced non-linearity of the tunnel is then identified by the decreasing dominant frequency thereof. The fault-crossing tunnel exhibits unique three-dimensional shearing-torsional deformations, elucidated through the test data and an analytical model. This deformation pattern provides a better understanding of the seismic responses of fault-crossing tunnels under seismic vibrations. • Dynamic responses of fault-crossing tunnels under seismic vibrations are studied by large-scale shaking table tests. • Faults have significant influence on tunnels under seismic vibrations even without rupture. • Vulnerable region of the tunnel, which is most intensely affected by the fault, is identified. • Deformation pattern and failure mechanism of fault-crossing tunnels under seismic vibrations are revealed. [ABSTRACT FROM AUTHOR]

A circular tunnel in a homogeneous infinite ground subjected to a vertically incident shear wave with an arbitrary vibration direction in the horizontal plane is considered in this study. A simplified analytical solution with explicit formulations for the deformation and stress of the tunnel lining is presented here. The superposition of two solutions under SH and SV waves are derived first. In the analytical solutions, the tunnel lining is taken as a thick-walled cylinder, which enables more precise forecasts and is much closer to practical engineering than the thin shell model adopted in other previous solutions. Moreover, the analytical solutions consider the slippage effect at the ground-lining interface by introducing a spring-type flexibility coefficient into the interaction force-displacement relationship. The results of the analytical solutions are verified by a comparison with the results of the dynamic numerical solutions via a 3-D ground-lining interaction model under earthquakes; the results show good agreement. By using the analytical solutions, the effects of the vibration direction, ground condition, lining thickness and flexibility coefficient on the seismic response of the tunnel are investigated. Additionally, the appropriate range of seismic wavelengths for the analytical solutions is discussed. The proposed solutions can be applied to the seismic analysis and design of tunnel structures. • Analytical solution is presented for circular tunnel under vertically incident shear wave with arbitrary vibration direction. • The tunnel lining is treated as a thick-wall cylinder, which enable precise forecasts. • Different slippage effects at the ground-lining interface are considered. • Effects of vibration direction, ground condition, lining thickness and flexibility coefficient are investigated. [ABSTRACT FROM AUTHOR]