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Generally speaking, the subaqueous tunnels can be regarded as the shallow-buried ones. Consequently, the classical problem of an elastic half plane with a round cavity, loaded arbitrarily along the surface boundary, can be used to obtain the stress and displacement fields of the surrounding rock for this type of tunnels. The solution uses the compl...
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Citations
... The aforementioned studies were only concerned with the problems under stress free conditions along the ground surface. Besides, the problem of tunnels subjected to the surcharge loads can be solved by complex variable theory combined with the superposition principle or Fourier series expansion [31][32][33] . ...
... (3) , (4) , (7)and(24); the axial forces, shear forces and bending moment at the cross-section of the lining: Eqs. (31) ,(32) and(33) . ...
For tunnels located at shallow depths, the adjacent surcharge loadings caused by new building constructions or overloading on the ground surface significantly influence the stability of the tunnel and lining. This study presents a new rigorous analytical solution for the stress and displacement around lined circular tunnels at a shallow depth, fully considering the interaction between the ground and rigid lining, and any distributed surcharge loadings.
A new analytical approach is provided, where the original problem is divided into two problems: Problem 1 is a semi-infinite plane subjected to surcharge loads without any holes, and Problem 2 is a tunnel contained in a semi-infinite plane loaded by the prescribed displacements along the tunnel boundary. The solutions, which strictly satisfy all the boundary and compatibility conditions of the original problem, are finally addressed by the superposition of the solutions of Problems 1 and 2 respectively obtained by the modified Flamant's solution and complex variable theory. The good agreement between the analytical and numerical results under the same assumptions verifies the correctness of the proposed analytical solutions. Furthermore, the possible deviation due to the simplification of the analytical solution is also discussed.
Based on the obtained analytical solution, a parametric study is carried out to investigate the influence of the key parameters on the stress and displacement of the ground and lining.
... For shallow buried circular tunnels, the boundary conditions include the stress boundary conditions and stress-displacement combined boundary conditions. The stress boundary conditions mean that the stress condition is regarded as boundary condition for both ground surface and tunnel cross-section simultaneously (Verruijt, 1998;Verruijt and Booker, 2000;Yu et al., 2014;Fang et al., 2015;Lu et al., 2016;Wang et al., 2018). The stress-displacement combined boundary conditions mean that the stress condition is regarded as boundary condition for the ground surface, while the displacement condition is regarded as boundary condition for the tunnel cross-section. ...
... For the problem of half plane weakened by a circular hole and with a concentrated force applied to the surface boundary, Proskura et al. [13] gave the stress and displacement fields for this problem using the bipolar coordinate system. Furthermore, for an elastic half plane with a tunnel loaded uniformly on the surface boundary, Yu et al. [14,15] and Fang et al. [16] presented an analytic solution for a circular tunnel to calculate the stress field and the settlement of the ground surface by using complex variable method. Kuliyev [17] presented an analytic solution for an elliptic hole with two rectilinear cracks based on the complex variable method and the principle of superposition. ...
An analytic stress solution is presented for a circular tunnel problem in a half plane with a concentrated force acting on any position in the field under gravity. The solution uses the complex variable method and the power series method. The influence of the unbalanced force system on the tunnel boundary is considered. The relationship between two analytic functions is established by using surface stress boundary condition. The analytic functions can be determined from the tunnel stress boundary condition. Based on the principle of superposition, the stresses of the surrounding rock can be calculated by superimposing three partial solutions which are obtained separately. The examples give contour plots of the principal stresses in the surrounding rock, focus on the stress distribution on the ground surface and the tunnel boundary and analyze the effect on the stress distribution of some main parameters.
... Different from the aforementioned works with stress-free condition of ground surface, the subaqueous tunnels are subjected to uniform vertical load along ground surface. The analytical solution for a subaqueous circular tunnel was proposed by Yu, Jing, and Wang (2014), by expanding infinite uniform surface load as Fourier series. ...
Tunnels in urban areas inevitably lie near the ground surface, and will be affected significantly by the loads on the surface. This study provides a new analytical approach to efficiently predict the time-dependent ground stresses and displacements induced by shallow tunnelling under initial stresses due to both gravity and surcharge loads. The viscoelasticity of the ground, inclination of ground surface, effect of support and its delayed installation, are all taken into account.
The elastic potentials were first derived for tunnels subjected to boundary stresses induced by gravity, surcharge loads and tunnel supports, respectively. Based on the extension of correspondence principle, the time-dependent ground stresses and displacements were then provided analytically for any type of linear viscoelastic ground. The FEM analysis was performed to verify the analytical solutions, which demonstrates a well agreement between the analytical and numerical results. Additionally, by the qualitative comparison of surface movements and initial failure zone around tunnel between analytical solution and results from other numerical model of elastoplastic cases, it was proved that the analytical solution can predict the initial failure zone and can qualitatively provide the reliable ground deformations. Finally, a parametric study was carried out to find the influence of timing for support installation, the slop angle and the surcharge loads on stresses and displacements around tunnels. The proposed solutions provide a quick alternative method to predict the time-dependent ground responses of shallow tunnelling in rocks, or medium/stiff clays.
... The reasonable boundary condition of the tunnel cross-section and the ground surface are the key to obtaining an analytical solution of a single tunnel. Compared with the stress boundary condition of the tunnel cross-section [20][21][22][23] , the displacement of the tunnel cross-section are easy to monitor in the engineering field. The displacement boundary conditions conforming to the actual engineering are constantly studied [24][25][26][27][28][29][30][31][32][33][34] . ...
A unified displacement function is used as the displacement boundary condition of the cross-section of each tunnel, which can be used to capture the asymmetrical deformation behaviors about the horizontal and vertical center line. Complex variable method is adopted to obtain the analytical solution of a single tunnel. Based on the Schwartz alternating method, the analytical solution of twin tunnels is obtained and the correctness of the analytical solution of twin tunnels is verified. A series of analyses are carried out to investigate the effects of the displacement boundary condition of the tunnel cross-section and the distance of twin tunnels on the ground surface settlement trough curve and the ground displacement field. Based on the proposed method, the parameters of displacement mode of the cross-section of twin tunnels are calibrated through field monitoring data of the Bangkok Subway Tunnel project, and the displacement field of the whole ground are analyzed. The proposed method can provides guidance in the conceptual stage of the design process of twin tunnels.
... Subaqueous tunnels are subjected to infinite uniform vertical load at the ground surface, which is different from the aforementioned studies regarding the stressfree conditions of the ground surface. The analytical solution of a subaqueous circular tunnel was proposed by Yu et al. (2014) by expanding the uniform surface load as a Fourier series. ...
Tunnels in urban areas inevitably lie near the ground surface, and will be affected significantly by the loads on the surface. In this study, a new analytical solution is presented for the first time, by means of which the ground responses around circular shallow tunnels subjected to surcharge loadings can be quickly predicted. The ground is approximately assumed as an elastic or viscoelastic medium, and arbitrary distribution of surcharge loads, as well as any viscoelastic models, are accounted for. Case (A) is modelled with surcharge loads applied before the tunnel excavation. The incremental elastic displacements and stresses which occur after the tunnel excavation are provided by Verruijt's method, combined with a Fourier expansion of the tractions. Case (B) represents the model in which the surcharge loads are applied after the tunnel excavation, and the elastic ground responses due to the presence of surcharge load are proposed by the superposition of modified Flamant's solutions and the solution of Case (A). The solutions for the viscoelastic problem of the two cases are also proposed by an extended corresponding principle, taking into account the time-dependence of the rock/soil with any types of the linear viscoelastic model. All of these analytical solutions are verified by good agreements of the comparison between the analytical solutions and FEM results. Furthermore, they are consistent with measured data qualitatively. Finally, a parametric study is carried out to investigate the influence of the uniform surcharge load on the stresses and displacements of the ground in the two cases. The proposed new solutions potentially provide an alternative approach in the preliminary designs of future shallow tunnels.
... The complex variables method was also proposed for stress and displacement around a circular tunnel in an elastic half-plane [7][8][9][10][11][12]. However, solving shallow-buried tunnel of rectangular shape using the complex function method is still difficult. ...
In mechanics, the solution of soil stresses and displacements field caused by shallow rectangular jacking pipe construction can be simplified as half-plane problem. Both the boundary conditions of the surface and the cavity boundary must be taken into account. It is the essential prerequisite for mechanical analysis of the pipe jacking with the complex variable theory that the mechanical boundary must be transformed from the half-plane with a rectangle cavity to the concentric ring. According to Riemann’s existence theorem and basic complex variable theory, a conformal mapping function is established. Both sides of boundary conditions equation are developed into Laurent series, and then the coefficients of complex stress function are solved by power series method. The derived solution is applied to an example and a comparison is made using FEM method to show the accuracy of the methods. The result shows the following: (1) the method presented in this paper is applicable to a shallow-buried rectangular tunnel; (2) the complex function method proposed in this paper is characterized by clear steps, fast convergence, and simple operation.
... Wang, Li, and Lv (2009) studied the ground deformation under different noncircular prescribed soil movement on the circular tunnel boundary. Yu, Jing, and Wang (2014) derived a semi-analytical solution of an elastic half plane with a round cavity, loaded uniformly along the ground boundary in the normal direction. Fu, Yang, Yan, and Abbas (2015) researched the deforming twin-parallel tunnels in an elastic half plane and obtained an approximate explicit analytical solution of stress field and displacement field. ...
... (4) The pore water is incompressible and steady seepage condition is considered. (5) The influences of gravity and buoyancy force on the stress field around the circular tunnel are ignored (Fu et al., 2015;Verruijt, 1997Verruijt, , 1998Yu et al., 2014). ...
Because of difficulty in mathematic solution, analysis of surrounding rock stress of shallow buried tunnel considering seepage force has always been a difficult problem. Seepage body force is introduced into governing equations and boundary conditions of an elastic plane. Then traditional complex variable method is expanded for the analysis of plane strain problems in relation to an elastic plane considering seepage effect. Subsequently, explicit analytical solutions of seepage field and stress field of a shallow underwater tunnel excavated in an elastic rock mass with uniform distribution of pressure and total water head on tunnel boundary as well as stress free on ground surface are derived. The analytical solutions are validated by numerical simulations and can be used to analyse stress variations of surrounding rock mass under seepage effect. Based on an example, influence of buried depth and water head difference between ground surface and tunnel boundary on the circumferential effective stress around the tunnel are discussed. It is found that directions of major and minor principal stress will exchange with each other with increase of total water head difference between ground surface and tunnel boundary, and stress distribution around tunnel tend to be uniform with increase of buried depth.
In this study, a new approach is presented to determine the stress and displacement field of a shallow buried circular tunnel subject to gravity and arbitrary lateral stress based on the complex variable method. The soil mass and lining are conformally mapped onto two ring regions. A more effective method than the power series method is used to solve the undetermined coefficients of complex potential functions. The interaction mechanism between the lining and the soil mass is fully considered. The boundary conditions along the inner boundary of the lining and the ground surface and the stress and displacement continuity conditions along the excavation line are used in the analytical derivation. In the verification, the stress and displacement distributions obtained in the semi-analytical solution agree with the numerical results and satisfy the boundary and continuity conditions. Using examples, the stress distributions along the inner boundary of the lining and interface are presented to examine the influence of key parameters including the Young's modulus and buried depth. The effect of the lining in reducing stress concentration and restricting tunnel convergence is quantitatively analyzed.
High temperature damage of rock is a critical problem that must be well known for underground coal gasification, underground storage of nuclear waste, repairment of underground buildings after fire disaster. In order to study the influences of loading rate and high temperature on the tensile strength of sandstone, Brazilian splitting tests were conducted on sandstone disk samples treated with five different temperature levels between 25 and 800 °C at six different loading rates between 0.01 and 10 mm/min. Test results showed that tensile strength of disk samples increases gradually and reaches to the maximum at the temperature level of 400 °C, then drop sharply. The tensile strength of sandstone samples is characterized by obvious rate effect and it increases continuously with the increase of loading rate, meeting a Logarithmic Function. The lower limit tensile strength of sandstone ignoring the influence of rate effect was calculated, and it is helpful for providing some basis for the design of rock engineering.
