Non-circular Tunnel Research Articles

Analytical solutions of noncircular tunnels in transversely isotropic rheological rock masses

In the field of tunnelling applications, it is often found that the rock masses exhibit anisotropy and rheological properties. To optimize the utilization of underground space, the use of noncircular tunnels is often preferred. However, it is important to note that these noncircular tunnels can lead to high-stress concentrations and significant displacements.This article presents a thorough analytical study on the time-dependent ground responses induced by the excavation of noncircular tunnels in transversely isotropic viscoelastic rock masses. The study considers a comprehensive set of engineering factors, including the viscoelastic characteristics of the surrounding rock, any anisotropic angle, and arbitrary tunnel shapes.Using the generalized corresponding principle of anisotropic elasticity and anisotropic viscoelasticity, an analytical model is introduced. This model can accurately and swiftly address the problem of deformation and stresses around noncircular tunnels in anisotropic rheological rock masses. The analytical solutions are verified by their good agreement with the Finite Element Method (FEM) results under identical assumptions. Moreover, the qualitative agreement between the analytical solutions and field data further validates the practical application of the analytical solution.A parametric analysis is then performed to investigate the effects of anisotropy ratio, anisotropy angle, and coefficient of lateral pressure on stresses and displacements.The proposed analytical solutions can help reveal the particular mechanical mechanism of the time-dependent ground responses due to the combination of rock anisotropy and rheology. Furthermore, they can provide a more accurate prediction of the ground response, which may be useful to optimize the design of tunnel excavation in anisotropic rheological rock masses.

Formation estimation and evolution mechanism of the pressure arch for non-circular tunnels under asymmetrical stress field

Formation estimation and evolution mechanism of the pressure arch for non-circular tunnels under asymmetrical stress field

Analytical Solution of the Non-circular Tunnel with a Void Defect in the Complex Stress Field

Analytical Solution of the Non-circular Tunnel with a Void Defect in the Complex Stress Field

The influence of existing tunnel shape and pillar distance on cross tunnel interaction

ABSTRACT Underground transportation systems often involve multiple tunnels constructed closely together. Previous studies mainly focus on interaction between circular tunnels; by contrast, interaction mechanisms involving non-circular tunnels are not well understood. In this study, four physical three-dimensional centrifuge tests were performed in dry sand, simulating the response of existing circular and horseshoe-shaped tunnels to a newly excavated tunnel. Two different ratios between pillar depth and tunnel diameter (P/D) of 0.5 and 2.0 were considered. Furthermore, three-dimensional numerical back-analyses considering small-strain stiffness were undertaken. Results reveal that the ground settlement above an existing horseshoe-shaped tunnel is less sensitive to pillar depth than for circular ones. Furthermore, for P/D = 0.5, the existing horseshoe-shaped tunnel experiences both vertical and horizontal compression; more stress reduction occurs vertically than horizontally. A circular tunnel for the same pillar depth becomes compressed vertically but elongated horizontally; stress reduction around the existing circular tunnel is less vertically than horizontally. However, for P/D = 2.0, both types of tunnel become elongated vertically and compressed horizontally because of a larger reduction in vertical stresses than horizontal ones. These results demonstrate that both pillar depth and shape profoundly affect tunnel deformation mechanisms.

Multifidelity operator learning for predicting displacement fields of tunnel linings under external loads

In practical tunnel projects, the deformation of tunnel linings affects the service performance and structural reliability of tunnels. However, currently, several monitoring points must be installed to represent the specific deformation patterns of tunnel linings accurately, incurring high labor and financial costs. In recent years, the rapid development of artificial intelligence has provided potential solutions to this issue. To solve the aforementioned problem, this study proposed a multifidelity DeepONet framework, which comprised two neural networks. The first low-fidelity network was trained with data provided by a macro-level numerical model validated by experimental campaigns to learn the physical deformation patterns of tunnel linings. The second network was trained with limited high-fidelity monitoring data to learn the correlations between observations and numerical models. Even with very limited monitoring data, the proposed framework could still predict the mechanical behavior of tunnel linings under different loading scenarios. In this study, data collected from noncircular tunnel projects were used as case studies. The results demonstrated that the final output conformed to the deformation pattern obtained with the numerical simulations and was consistent with the actual measurements, achieving seamless fusion of the experimental campaigns and numerical models.

A matrix-form complex variable method for multiple non-circular tunnels in layered media

This paper presents a matrix-form complex variable method that can predict the stress and displacement field around multiple tunnels in layered media. The proposed matrix-form method avoids complicated coefficient determinations and iterative algorithm, which effectively improves the computation efficiency. Firstly, the expressions for the stress and displacement field are given under the generalized complex variable theory. The conformal mapping technique and fast Fourier transform algorithm are used to formulate the matrix-form expressions of boundary conditions. Considering the boundary conditions of the tunnels and soil interfaces, the global matrix solution for multiple tunnels in layered media are obtained by assembling the matrix solutions of all single layers. The present theory is verified by comparison with the existing literature, finite element analysis and field data in engineering practices, and several numerical examples are given to reveal the effects of elastic properties, tunnels' position, equivalent layer and the layer thickness surrounding tunnels.

Systematic failure mechanism of an FGMs polyhedral arched liner under a fire disaster environment

This paper investigated the failure performance of an arched polyhedral liner yielding a fire disaster environment. The functionally graded materials (FGMs) are used in the liner to increase the ability of fire resistance. A concentrated load may occur at the invert of the arched liner due to the debris, which is considered in the study. The polyhedral arched liner deforms in a single-lobe shape, and this deformation is expressed by a trigonometric function. The nonlinear equilibrium equations are presented by applying the thin-walled shell theory and the principle of minimum potential energy, respectively. Based on the above investigations, two innovative contributions of the present study are addressed: (1) a novel FGMs polyhedral liner is introduced to rehabilitate the cracked non-circular tunnel in the combination of the mechanical loading and fire environment; and (2) the critical buckling load, as well as the load-displacement nonlinear equilibrium curves, is derived explicitly by solving the nonlinear equilibrium equations. The effect of the fire disaster on the buckling load is quantized to describe the destabilization behavior of the arched liner in a fire environment, respectively. Then, the present study is compared efficiently with other closed-form formulations when the polyhedral FGM arched liner simplifies to a circular homogeneous liner. It is shown that a smaller number of polyhedral edges and a higher content of ceramic are beneficial to resist external loadings and fire disasters in engineering practices. Finally, the effects of geometric parameters, volume fraction exponents, and temperature variations on the nonlinear buckling behavior of the polyhedral arched liner are evaluated.

Complex Function Solution of Stratum Displacements and Stresses in Shallow Rectangular Pipe Jacking Excavation Considering the Convergence Boundary

The construction of pipe jacking has little impact on the environment and is usually used to build underground passages with shallow buried depths and short lengths. Compared with circular pipe jacking, rectangular pipe jacking has the advantages of shallow buried depth and high space utilization. Therefore, research on the excavation of rectangular pipe jacking is necessary. This paper establishes a cross-section model of shallow buried rectangular pipe jacking excavation. Taking advantage of complex functions for solving problems involving non-circular tunnels, an analytical solution is obtained using an approximate mapping function and potential functions in series forms for the stress and displacement of the stratum with a displacement condition at the excavation boundary and a stress condition at the ground surface boundary. The finite element simulation results and the engineering-measured data are used for comparisons and verifications. With the analytical solution of the complex function, the influence of selecting control points for the mapping function on the accuracy is calculated and analyzed, as well as the influence of the stratum loss rate, span, buried depth, and stratum unit weight on surface subsidence and major principal stress of the excavation boundary. The proposed analytical solution can be applied to the construction of rectangular pipe jacking tunnels.

Implementing a simple 2D constitutive model for rocks into finite element method

Our research group has developed a 2D constitutive model for rocks that incorporates a strain-dependent elastic modulus. This model efficiently represents nonlinear stress–strain curves using just three equations and four parameters. In this study, we have implemented the simple 2D model into finite element codes to tackle engineering problems. We have calculated and compared stresses and displacements around a pressurized thick-walled hollow cylinder, a lined non-circular tunnel, and a rock slope using the elastic, simple 2D, and elasto-plastic models. The results confirmed that the simple 2D model closely matches the stress distribution obtained from the elastic solution at low-stress levels. Additionally, the simple 2D model produces a plastic zone with a larger inward displacement than the elasto-plastic model at higher stress levels. Only the simple 2D model captured sidewall convergence, roof sag, and floor heave in non-circular tunnels, and the toppling failure for 90° slope and toe translational slide for 60° slope in rock slopes with smooth critical slip surface. We did not encounter any convergence difficulties while solving the simple 2D model.

Excavation-Induced Groundwater Evolution of Noncircular Tunnels in Mountainous Regions: Analytical and Numerical Investigation

Excavation-Induced Groundwater Evolution of Noncircular Tunnels in Mountainous Regions: Analytical and Numerical Investigation

Elastoplastic solutions for deep-buried twin tunnels with arbitrary shapes and various arrangements under biaxial in-situ stress field based on Mohr-Coulomb and generalized Hoek-Brown criteria

Theoretical solutions for elastoplastic analysis of deep-buried twin tunnels with arbitrary shapes and various arrangements are developed under biaxial in-situ stress field based on Mohr-coulomb (M–C) and generalized Hoek-Brown (G-H-B) criteria. The boundary stress transformation formulas adapted for arbitrary stress conditions on non-circular tunnel wall are provided in terms of conformal transformations. Regarding the plastic analysis, analytical expressions for stress characteristic relationships along slip lines under generalized Hoek-Brown criterion are derived. Subsequently, the semi-analytical solutions for plastic stresses of rock surrounding arbitrary-shaped tunnels are constructed based on Mohr-Coulomb and generalized Hoek-Brown criteria applying conformal mapping technique, slip-line method and finite difference-numerical iterative approach. On the basis of the discrete plastic stresses, interpolation functions are generated according to three-dimensional linear interpolation technique. And then, solutions for elastic stress function and plastic radius of twin tunnels with various arrangements are developed with improved mapping functions and newly defined fitness functions through differential evolution method. The accuracy of the semi-analytical solutions is verified by the numerical simulation. Meanwhile, the proposed solutions have the advantages of meshless characteristic, better convergence and higher computing efficiency comparing with the numerical simulation. From the point of engineering, the developed elastoplastic solutions can provide theoretical supports for the stress analysis, plastic radius estimation, safety clearances optimizations of deep-buried twin tunnels.

A novel complex variable solution for the stress and displacement fields around a shallow non-circular tunnel

A novel complex variable solution is derived in this paper for the stress and displacement fields around a shallow non-circular tunnel. The proposed method allows explicit computations in matrix form and has advantages of considerable efficiency and high accuracy. Firstly, the stress and displacement boundary conditions for the shallow non-circular tunnel are obtained using the generalized complex variable theory and conformal mapping technique. Then, the fast Fourier transform algorithm is employed to effectively perform the Fourier series fitting for these boundary conditions, and matrix equations for the potential functions in the frequency domain are established. Later, a matrix solution for the potential functions is formulated to determine the stress and displacement fields around the shallow non-circular tunnel. Finally, select numerical analyses are conducted to verify the proposed method, discuss the key parameters in the algorithm, and reveal the effect of the acceleration strategy using FFT as well as the effect of non-circular geometries on the stress and displacement fields.

Stability assessment of a non-circular tunnel face with tensile strength cut-off subject to seepage flows: A comparison analysis

This work aims to develop a numerical-analytical framework within the kinematic approach of limit analysis to assess the stability of non-circular tunnel faces under seepage conditions with tensile strength cut-off from a discretization-based perspective. Considering the overestimation of the tensile strength of the soil by the Mohr-Coulomb criterion, a modification based on the concept of tensile strength cut-off is recommended to reduce or eliminate the tensile strength. A modified 3D rotational discretization-based failure mechanism incorporating the tension cut-off is generated point-to-point starting from a real non-circular tunnel face to model the face failure, where the failure surface on the top intersects more rapidly due to the increasing rupture angle in the tensile regime. The seepage towards the non-circular tunnel face is numerically simulated, and then the numerical seepage field is extracted into the 3D discretization-based failure mechanism. Within the kinematic approach, the seepage effect is introduced from two aspects: pore pressure and seepage force. A hybrid optimization routine is employed to search for the maximum required face pressure. The developed framework is validated by comparisons with numerical fluid-mechanical simulations and previous studies. Afterward, parametric analyses are carried out on different seepage conditions, different face pressure distributions, different soil strengths, and different tension cut-off conditions. The developed framework can give insight into the combined effect of the tension cut-off and seepage on the stability of non-circular tunnel faces under various conditions.

A quantitative analysis procedure for solving safety factor of tunnel preliminary support considering the equivalence between Hoek–Brown and Mohr–Coulomb criteria

There is a development trend from qualitative towards quantitative analysis for tunnel stability. The widely accepted convergence-confinement method (CCM) provides a quantitative framework for tunnel stability analysis, while the conventional CCM based on closed-form functions is essentially an ideal model without considering the tunnel section shape and rock weight in plastic zone. Herein, based on the CCM framework, a quantitative analysis procedure formulated through numerical simulations for more practical situations was systematically proposed to calculate the safety factor of spray-anchor support system in non-circular tunnel. Besides, similarities and differences between the most widely used Hoek-Brown (HB) and equivalent Mohr-Coulomb (EMC) criteria were considered for practical reference. The proposed method was utilized to conduct quantitative analysis of LDPs, mechanical responses of surrounding rock and safety factors for four typical tunnels, and the consistency between tunnels based on HB criterion (TBHBC) and EMC criterion (TBEMCC) was analyzed. The results indicated that the maximum radial convergence of LDP, plastic zone area, and safety factor of TBEMCC were 34.54%, 37.29%, and 47.71% lower than those of TBHBC respectively. The reasons for differences between results were revealed. The spatial effect induced by the face excavation of TBEMCC was weakened as compared to TBHBC, leading to a drop of LDP. Consequently, the safety factor was underestimated. The method systematically provided an effective tool for quantitative stability analysis and design optimization of support system for an underground excavation.

Visco-Elastic Interface Effect on the Dynamic Stress of Symmetrical Tunnel Embedded in a Half-Plane Subjected to P Waves

A semi-analytical method is developed to study the visco-elastic interface effect on the dynamic stress of a non-circular symmetrical tunnel in a half-plane subjected to P waves, and the interaction between the visco-elastic interface around the tunnel and the traction free boundary at the half-plane is analyzed. The complex variable function method combined with the wave function expansion method is introduced to obtain the theoretical expressions of waves around the non-circular tunnel. To satisfy the traction-free boundary condition at the half-plane, the large circle assumption is applied. The expanded coefficients are determined by using appropriate boundary conditions with stiffness parameters and viscosity coefficients of the interface. Selected numerical results are presented, and the interface effect on the dynamic stress under different embedded depths is discussed. It is found that the interacting effect of the visco-elastic interface and the traction-free boundary is quite related with the wave frequency. Comparison with existing results is also given to validate the semi-analytical method.

Analytical stress and displacement of twin noncircular tunnels in elastic semi-infinite ground

In general, twin noncircular tunnels may cause high stress concentrations and large settlements because of the strong interaction between tunnels and their noncircular shape. In this study, new analytical solutions are presented to efficiently predict the elastic stresses and displacements around shallow twin noncircular tunnels in rocks. The key factors, i.e., the real interactions between tunnels, arbitrary tunnel shapes, arrangements and pressures exerted at the internal tunnel boundaries due to liners or water pressures, are fully taken into account.The Schwartz alternating method is used to reduce the twin noncircular tunnels problem to a series of problems where only one noncircular tunnel is contained in the half-plane. Then, the solutions to the problem of a noncircular tunnel in the half-plane are obtained by complex variable theory combined with conformal mapping. Finally, convergent and highly accurate analytical solutions of twin noncircular tunnels are achieved by superposing the solutions of the reduced single tunnel problems. The analytical solutions are verified by the good agreement between the analytical and FEM results under the same assumptions. The proposed analytical solution can help to reveal the mechanical mechanisms of ground responses due to twin tunnelling in an elastic semi-infinite ground.

Plastic Displacement Solutions for Two-Way Unequal-Pressure Non-Circular Tunnels Based on Mogi–Coulomb Criterion and Applications

Plastic Displacement Solutions for Two-Way Unequal-Pressure Non-Circular Tunnels Based on Mogi–Coulomb Criterion and Applications

Three-dimensional stability assessments of a non-circular tunnel face reinforced by bolts under seepage flow conditions

This paper aims to develop a three-dimensional (3D) analytical framework to assess the safety factors of realistic non-circular tunnel faces reinforced by longitudinal bolts under seepage conditions from a discretization-based perspective. The axial failure mode of bolts (including the tensile failure of the bolt and the pull-out failure at the bolt-soil interface) paired with the introduction of the interaction zone is employed to describe the reinforcement effect of bolts. A numerical fluid calculation is performed to simulate the 3D seepage towards the non-circular tunnel face, and the resulting pore pressure distribution is extracted and further interpolated to determine the pore pressure of each discrete point on the 3D discretization-based failure mechanism. Within the kinematical approach of limit analysis, the work rate balance equation incorporating the role of the axial force along the interaction zone and the pore pressure due to the 3D seepage is established based on the discretization-based failure mechanism. Afterward, combining the strength reduction method with the dichotomy, the upper-bound estimations of safety factors can be obtained by an optimization process. The developed framework is compared with numerical fluid-mechanical calculations and other analytical methods with respect to an actual horseshoe tunnel, which proves that the developed framework is effective in providing fast and reasonable estimates of reinforced tunnel face under seepage conditions. Parametric analyses are performed to investigate the effect of bolt layouts (including the length and density of bolts) and seepage conditions (including the groundwater level and permeability anisotropy). Finally, a series of design nomograms for various water levels and soil strength parameters are provided for reference.

Initial support distance of a non-circular tunnel based on convergence constraint method and integral failure criteria of rock

Initial support distance of a non-circular tunnel based on convergence constraint method and integral failure criteria of rock

Analytical Solution to the Seepage Field of Two Parallel Noncircular Tunnels in Permeable Anisotropic Ground

This paper provides an analytical solution and solving procedure for the seepage field around two noncircular tunnels in anisotropic permeable ground. The anisotropic problem is first made equivalent to an isotropic problem by coordinate transformation. Then, by using Schwartz alternate method and conformal mapping, a precise analytical solution is obtained based on the anisotropic problem of single tunnel with redundant hydraulic heads at tunnel boundary. The noncircular tunnel interaction and the permeable anisotropy are considered accurately. The iterative procedure is simple and efficient in its calculations, and achieves good convergence. The results of the analytical model are compared with the finite element results, and show good agreement. Finally, some parametric studies are present to research the influences of tunnel shapes and the anisotropic permeability ratio on the seepage field, and the effect of the location of the pilot tunnel on the seepage field of the primary tunnel is also investigated.