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Seismic waves propagation with an oblique angle to the tunnel axis will cause asynchronous tunnel motions and have a significant effect on the axial response. A high-precision 2.5D finite element method is established in the frequency domain to simulate the 3D seismic response of the tunnel. This method avoids the disturbance caused by the truncation of the tunnel in the longitudinal direction. Meanwhile, a 2.5D zigzag-paraxial boundary is derived to further improve the calculation efficiency of the 2.5D finite element model. Moreover, by combining the 2.5D finite element method, 2.5D zigzag boundary condition and seismic motion input methods, an obliquely incident substructure method for plane seismic waves is built by converting the plane seismic wave into equivalent nodal forces. The proposed 2.5D finite element method is verified by comparing with a reference solution. Finally, the 2.5D finite element method is applied to study the seismic response of the long lined tunnel. Parameter analyses illustrate that the wave propagation effect to the tunnel axis has a non-negligible influence on the axil deformation of long tunnels. [ABSTRACT FROM AUTHOR]

When the numerical analysis of a long lined tunnel is carried out, the calculation amount of the finite element model becomes restricted large-scale parameter analysis. In this paper, an efficient and high-precision 2.5-dimensional (2.5D) frequency-domain finite element method is used to simulate the three-dimensional response of tunnels under the action of oblique incident plane seismic waves. This method can save calculations and avoid the boundary effect caused by the longitudinal truncation of the tunnel. The 2.5D zigzag-paraxial boundary is developed. The artificial boundary is attached to the structure's surface. The substructure method for oblique plane seismic waves is established. Comparing the substructure method with the analytical solution, the correctness of the site response is verified first. The accuracy of the 2.5D finite element substructure method is further verified. The parameter analysis of different incident angles and conversion angles shows that the underground tunnel does not reach the maximum of structural seismic response when the seismic wave is vertically incident. The location of the soil–rock interface on the tunnel is further discussed. The results show that when the underground tunnel crosses the location of the soil–rock interface, the seismic response of the tunnel will be amplified. [ABSTRACT FROM AUTHOR]

Whether the value of the relative deformation of the stratum is reasonable directly affects the accuracy of the application of the reaction displacement method in the seismic design and calculation of underground structures. For the energy open site, the site model is equivalent to a one-dimensional soil column model, and a viscous boundary is set at the bottom of the model to consider the downward transfer of energy. Through the artificial synthesis ground motion method, the distribution law of the relative displacement of the stratum in the energy open site is studied according to different site types and seismic fortification intensities. At the same time, the peak shear strain per unit depth is obtained by fitting the parameters on the basis of simulation results. For the site with open energy, when the seismic response of the structure is calculated by the reaction displacement method, the relative displacement difference between the top plate and the bottom plate of the structure can be obtained by multiplying the peak shear strain on the unit depth variation by the height of the structure. [ABSTRACT FROM AUTHOR]

An analytical solution for the seismic-induced thrust and moment of the circular tunnel in half-space under obliquely incident P waves is developed in this study, which is the superposition of the solution for deep tunnels under incident and reflected P waves and the reflected SV waves. To consider tangential contact stiffness at the ground-tunnel interface, a spring-type stiffness coefficient is introduced into the force-displacement relationship. Moreover, the tunnel lining is treated as the thick-wall cylinder, providing more precise forecasts than beam or shell models used in previous analytical solution, especially for tunnels with thick lining. The reliability of the proposed analytical solution is assessed by comparing with the dynamic numerical results. Based on the proposed analytical solution, parametrical studies are conducted to investigate the effect of some critical factors on the tunnel's seismic response, including the incident angles, the tangential contact stiffness at the ground-tunnel interface, and the relative stiffness between the ground and the tunnel. The results demonstrate that the proposed analytical solution performs well and can be adopted to predict the internal forces of circular tunnels under obliquely incident P waves in seismic design. [ABSTRACT FROM AUTHOR]

The failure mechanism of tunnels crossing faults is a critical issue for tunnels located in seismically active regions. This study aims to investigate the nonlinear response of rock tunnels crossing inactive faults under obliquely incident seismic P waves. Based on the equivalent nodal force method together with the viscous-spring boundary, an incident method for the site, which contains fault and is subjected to obliquely incident seismic P waves, is developed first. Then, based on the proposed incident method, the nonlinear response and the failure process of the tunnel crossing inactive fault are numerically studied. The numerical results show that the failure mechanism of the tunnel crossing inactive fault can be attributed to the combined action of the seismic waves and its associated fault slippage. Finally, parameter studies are conducted to investigate the effects of the wave impedance ratio of the fault to the surrounding rock and the incident angle of P waves. By the parameter analysis, it can be concluded that: (1) with decreasing the wave impedance ratio of the fault to the surrounding rock, the seismic response of the tunnel increases significantly; (2) the seismic response of the tunnel increases first and then decreases with the increasing of the incident angle of P waves. This study offers the insight for further research on the seismic stability of tunnels crossing inactive faults. [ABSTRACT FROM AUTHOR]

Seismic stability of slopes has been traditionally analyzed with vertically propagated earthquake waves. However, for rock slopes, the earthquake waves might approach the outcrop still with a evidently oblique direction. To investigate the impact of obliquely incident earthquake excitations, the input method for SV and P waves with arbitrary incident angles is conducted, respectively, by adopting the equivalent nodal force method together with a viscous-spring boundary. Then, the input method is introduced within the framework of ABAQUS software and verified by a numerical example. Both SV and P waves input are considered herein for a 2D jointed rock slope. For the jointed rock mass, the jointed material model in ABAQUS software is employed to simulate its behavior as a continuum. Results of the study show that the earthquake incident angles have significance on the seismic stability of jointed rock slopes. The larger the incident angle, the greater the risk of slope instability. Furthermore, the stability of the jointed rock slopes also is affected by wave types of earthquakes heavily. P waves induce weaker responses and SV waves are shown to be more critical. [ABSTRACT FROM AUTHOR]

Simplified analytical solutions are very useful tools for the seismic preliminary design of tunnels and can quickly and easily give seismic responses to structures. This study presents simplified analytical solutions for the stress and deformation of circular composite-lined tunnels subjected to in-plane S-waves. Unlike previous solutions for single-lined tunnels, the proposed analytical solutions are suitable for tunnels with arbitrary numbers of linings. Moreover, tunnel linings are treated as multilayer thick-walled cylinders in the new solutions. Compared with thin shell or beam models used in previous analytical solutions, thick-walled cylinders provide more precise forecasts. In addition, the slippage effect at the lining-lining and ground-lining interface is considered by introducing a spring-type flexibility coefficient into the interaction force–displacement relationship. To verify the proposed analytical solutions, dynamic numerical examples are studied. The proposed solutions have good agreement with the dynamic numerical results and outperform previous analytical solutions. Finally, parameter studies are conducted to investigate the effect of the flexibility coefficient, degraded zone around the outermost lining and steel plate supporting the innermost lining on the seismically induced stresses and deformations. The proposed analytical solutions are applicable for approximately evaluating the seismic responses of circular tunnels with composite linings. [ABSTRACT FROM AUTHOR]

• A 3-D obliquely incident method for P waves was developed. • Nonlinear seismic response of a tunnel under 40 artificial accelerograms with different incident angles is analyzed. • PGV has incremental effect on the seismic response of tunnels under earthquake P waves with large incident angles. • PGA intends to increase the response of tunnels subjected to earthquake P waves with small incident angles. • PGD has little effect on the seismic response of tunnels. The response mechanism of tunnels under earthquake loads is a critical issue in tunnels located in seismically active regions. This study aims to investigate the seismic response and damage sensitivity of tunnels under earthquake motions with different peak ground acceleration, velocity and displacement (PGA, PGV and PGD). Based on the design acceleration response spectrum (ARS), four groups of total 40 artificial accelerograms are synthesized with specified PGA, PGV and PGD. In addition, for considering the wave travelling effect on the tunnel axial response, an obliquely incident method for P waves is developed by using the equivalent nodal force method. Then, the nonlinear response of tunnels subjected to artificial accelerograms with different incident angles are simulated, respectively. The numerical results indicate that the PGV and PGA have accretion effect on the seismic response of the tunnel. However, the impact of PGD on the tunnel earthquake response is less evident. The PGV has obvious incremental effect on the dynamic response of tunnels subjected to earthquake P waves with large incident angles. Compared with PGV, the PGA intends to increase the response of tunnels significantly under P waves with small incident angles. Thus, it is important to take the PGA and PGV into account during seismic analysis and design for tunnels. [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 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]

This study aims to analysis the longitudinal seismic responses of circular tunnels across soil-rock stratum longitudinally ("soil-rock" means two different stratums, not just the soil and rock ground). Based on an improved elastic foundation beam model, an analytical solution is derived to investigate the seismic response of long tunnels across soil-rock stratum, subjected to sinusoidal shear motions. The improved elastic foundation beam model adopts the Timoshenko beam and Winkler foundation, with considering tangential contact conditions at the ground-tunnel interface. The free field reaction considers the reflection and transmission by the soil-rock interface and the wave passage effect. Explicit formulations are obtained for tunnel displacement, rotation angles, bending moments and shearing forces. The proposed solution is verified by comparing with the dynamic numerical results, as well as traditional analytical solutions. Finally, a parametric study is conducted to investigate the influence of traveling wave effect, earthquake frequency, shear wave velocity ratio of two stratums, lining stiffness and contact condition on tunnel seismic responses. The proposed analytical solutions can be used to predict the longitudinal seismic response of deep circular tunnels crossing soil-rock stratum in engineering design. [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]

• The effect of soil-rock interface position on the seismic responses of a subway station structure is investigated. • The mechanism of the most detrimental soil-rock interface position for structural response occurs is explained. • The conclusions drawn from the numerical examples of this paper may be useful for seismic design of subway station structure. For subway stations in a site with thin soil layers on the engineering rock, the lower part of the subway station is embedded in the hard rock mass and the upper part is buried in the soil layers. The presence of the soil-rock interface may lead to significant increase to the seismic responses of the adjacent soil masses and underground structures. In this paper, the effect of the soil-rock interface position on the seismic response of a two-story and three-span subway station structure is investigated by numerical analyses. It is found from the numerical analyses that a most detrimental soil-rock interface position exists and makes the underground structure have maximum responses under a given artificial earthquake motion. For the upper floor, the most detrimental soil-rock interface position is near the middle slab. For the lower floor, the most unfavorable soil-rock interface position is near the bottom slab. Softer site generally leads to greater structural responses under the same intensity of ground motion excitation. On the basis of the above findings, three earthquake motions included above an artificial earthquake motion and other two real earthquake motions and three burial depths of structure are further studied. Same conclusions can be obtained for the structure with different burial depths or subjected to different input earthquake motions. The conclusions drawn from the numerical examples of this paper may be useful for seismic design of subway station structure buried in shallow soil layers on rock. [ABSTRACT FROM AUTHOR]

With the rapid development of underground space, more adverse site conditions are occurring in the construction of underground structures. For example, a subway station passes through longitudinally inhomogeneous geology. When the subway station crosses the interface of soft soil and hard soil, this may lead to the destruction of underground structures. In this paper, three-dimensional (3D) nonlinear seismic response analysis of subway stations crossing longitudinally inhomogeneous geology subjected to obliquely incident P waves is presented based on actual engineering background. The earthquake input method for the longitudinally inhomogeneous site that transforms the site seismic response into an equivalent nodal load is first developed and verified. The viscous-spring artificial boundary is used to simulate the radiation damping effect of infinite domain media. Subsequently, the topographic and geological effects on seismic response of engineering site are investigated. Finally, the failure mechanism of the subway station structure crossing longitudinally inhomogeneous geology under the obliquely incident P waves is revealed. The investigation demonstrates that the plastic zone of site usually appears along the interface between the hard and soft soil media, and appears on the side of the soft soil. The sidewall and bottom slab of the structure experience obvious tensile damage at the interface location between the hard and soft soil media. The results show that topographic and geological effects should be considered in seismic analysis and design of underground structures when the topography and geology change significantly. • Seismic analysis of subway station crossing longitudinally inhomogeneous geology is presented • Earthquake input method for longitudinally inhomogeneous site is developed and verified • Effects of topography and geology on seismic response of site and subway station is studied [ABSTRACT FROM AUTHOR]

Analytical solutions based on the elastic foundation beam model for the bending response of circular tunnels under longitudinally propagating shear waves are studied considering the tangential contact conditions at the ground-tunnel interface. First, two kinds of new analytical solutions based on Timoshenko beams resting on Winkler or Pasternak foundations are developed considering different contact conditions at the ground-tunnel interface. The contact conditions are considered by introducing a spring-type flexibility coefficient in the force–displacement relationship. Then, the new analytical solutions, as well as the traditional solutions using Euler or Timoshenko beams on Winkler or Pasternak foundations, are compared with the dynamic numerical results. The new analytical solutions exhibit good agreement with the dynamic numerical results for different contact conditions. However, due to the neglect of the tangential interaction between the tunnel and surrounding ground, traditional solutions are only suitable for predicting the seismic response of tunnels in the full-slip condition. Finally, a parametric study is conducted to investigate the effects of the earthquake frequency, ground condition, lining thickness and contact condition. The proposed analytical solutions can be used to predict the seismic response of circular tunnels under longitudinally propagating shear waves in engineering design. [ABSTRACT FROM AUTHOR]

• A substructure method for seismic multi-media interaction is developed. • An on-surface condition is proposed to provide the stiffness of truncated media. • Effect of changing groundwater level on seismic response of tunnel is evaluated. This study presents a substructure method with good accuracy and excellent efficiency for seismic underground structure-dry soil-saturated soil–bedrock interaction problem under obliquely incident earthquake. A combined zigzag-paraxial on-surface condition attached on structural surface is proposed to provide the dynamic stiffness of structural surrounding media including dry soil, saturated soil and underlying half space bedrock. The on-surface boundary condition can be implemented to any desired degree of accuracy and is compatible with numerical method such as the finite element method and the finite difference method. The condition is then coupled with structural finite element scheme and seismic wave input method to establish a substructure method. The reliability and accuracy of the proposed substructure method are verified by applying it in two degenerated interaction problems and comparing with other solutions. Finally, the proposed substructure method is employed to establish a lined tunnel-dry soil-saturated soil–bedrock interaction model for evaluating the seismic response of a lined tunnel embedded in the layered half space rich in groundwater, with an emphasis on the effects of changing groundwater levels on seismic response of tunnel lining. [ABSTRACT FROM AUTHOR]

[Display omitted] • Large-scale shaking table model tests were performed on fault-crossing tunnels. • Discrepant responses along the longitudinal direction of the fault-crossing tunnels were quantified. • Aseismic effect of grouting on the fault-crossing tunnel was investigated clearly. • Optimized seismic design parameters for grouting were discussed. Tunnels crossing inactive faults may experience large differential deformations along their lengths during earthquakes, resulting in damage. Faults always have inferior material properties for historical geological reasons. Grouting is an auxiliary construction measure in fault-crossing tunnels and has the potential to be an economical and effective aseismic measure. However, the feasibility and seismic mitigation mechanisms of grouting in fault-crossing tunnels have not been sufficiently studied. In this study, large-scale shaking table tests were conducted to verify the aseismic effect of grouting in a tunnel crossing an inactive fault and to explore the seismic mitigation mechanism behind it. On this basis, verified numerical models were developed to determine the optimized design parameters. The test results indicate that grouting reduces the acceleration response of the tunnel within the fault and reduces the differences in the acceleration responses in the longitudinal direction of the tunnel induced by the fault. As grouting increases the shear strength and stiffness of the strata around the tunnel within the fault, the strain of the tunnel invert is effectively reduced by over 80%, and the potential damage of the invert uplift is reduced simultaneously. Meanwhile, grouting effectively reduces the diameter deformation of the tunnel within the fault and reduces the differences in tunnel deformation in different areas of the fault site. By reducing the resonance caused by the frequency components of the input seismic wave that is close to the natural frequency of the strata, which contributes most to the tunnel response, grouting effectively reduces the seismic responses of the fault-crossing tunnels. Increasing the stiffness of the grouting zone is an effective approach to improve its seismic mitigation performance. In the example of this study, adjusting the relative stiffness ratio to approximately 100 between the grouting zone and the fault resulted in a favorable condition for aseismic effects, which could be deemed to be a suitable parameter for grouting. The above results can provide a reference for the seismic design of mountain tunnels crossing inactive faults. [ABSTRACT FROM AUTHOR]