Analytical stress and displacement due to twin tunneling in an elastic semi‐infinite ground subjected to surcharge loads

The construction of twin tunnels at shallow depth has become increasingly common in urban areas. In general, twin tunnels are usually near each other, in which the interaction between tunnels is too significant to be ignored on their stability. The equivalent arbitrarily distributed loads imposed on ground surface were considered in this study, and a new analytical approach was provided to efficiently predict the elastic stresses and displacements around the twin tunnels. The interaction between 2 tunnels of different radii with various arrangements was taken into account in the analysis. We used the Schwartz alternating method in this study to reduce the twin‐tunnel problem to a series of problems where only 1 tunnel was contained in half‐plane. The convergent and highly accurate analytical solutions were achieved by superposing the solutions of the reduced single‐tunnel problems. The analytical solutions were then verified by the good agreement between analytical and numerical results. Furthermore, by the comparison on initial plastic zone and surface settlement between analytical solution and numerical/measured results of elastoplastic cases, it was proven that the analytical solution can accurately predict the initial plastic zone and its propagation direction and can qualitatively provide the reliable ground settlements. A parametric study was finally performed to investigate the influence of locations of surcharge load and the tunnel arrangement on the ground stresses and displacements. The new solution proposed in this study provides an insight into the interaction of shallow twin tunnels under surcharge loads, and it can be used as an alternative approach for the preliminary design of future shallow tunnels excavated in rock or medium/stiff clay.     

An analytical solution for deforming twin‐parallel tunnels in an elastic half plane

The interaction between twin‐parallel tunnels affects the tunnelling‐induced ground deformation, which may endanger the nearby structures. In this paper, an analytical solution is presented for problems in determining displacements and stresses around deforming twin‐parallel tunnels in an elastic half plane, on the basis of complex variable theory. As an example, a uniform radial displacement was assumed as the boundary condition for each of the two tunnels. Special attention was paid to the effects of tunnel depth and spacing between the two tunnels on the surface movement to gain deep insight into the effect of the interaction between twin‐parallel tunnels using the proposed analytical approach. It is revealed that the influence of twin tunnel interaction on surface movements diminishes with both the increase of the tunnel depth and the spacing between the two tunnels. The presented analytical solution manifests that, similar to most of the existing numerical results, the principle of superposition can be applied to determine ground deformation of twin‐parallel tunnels with a certain large depth and spacing; otherwise, the interaction effect between the two tunnels should be taken into account for predicting reliable ground movement. Copyright © 2014 John Wiley & Sons, Ltd.     

Analytical Solutions for the Construction of Deeply Buried Circular Tunnels with Two Liners in Rheological Rock

The construction of underground tunnels is a time-dependent process. The states of stress and strain in the ground vary with time due to the construction process. Stress and strain variations are heavily dependent on the rheological behavior of the hosting rock mass. In this paper, analytical closed-form solutions are developed for the excavation of a circular tunnel supported by the construction of two elastic liners in a viscoelastic surrounding rock under a hydrostatic stress field. In the solutions, the stiffness and installation times of the liners are accounted for. To simulate realistically the process of tunnel excavation, a time-dependent excavation process is considered in the development of the solutions, assuming that the radius of the tunnel grows from zero until its final value according to a time-dependent function to be specified by the designers. The integral equations for the supporting pressures between rock and first liner are derived according to the boundary conditions for linear viscoelastic rocks (unified model). Then, explicit analytical expressions are obtained by considering either the Maxwell or the Boltzmann viscoelastic model for the rheology of the rock mass. Applications of the obtained solutions are illustrated using two examples, where the response in terms of displacements and stresses caused by various combinations of excavation rate, first and second liner installation times, and the rheological properties of the rock is illustrated.     

Analytical and Numerical Study of the Mechanics of Rockbolt Reinforcement around Tunnels in Rock Masses

Summary  This paper addresses the problem of quantifying the mechanical contribution of rockbolts installed systematically around tunnels excavated in rock masses. The mechanical contribution referred to here is that of increased stress confinement and decreased tunnel convergences as compared with corresponding stresses and displacements obtained for non-reinforced tunnels. The problem is treated analytically first by presenting a closed-form solution for stress and displacement distributions around a circular tunnel excavated in elastic material and reinforced by grouted or anchored rockbolts. The analytical solution assumes that rockbolts are regularly spaced around the tunnel and that axi-symmetry conditions of geometry and loading apply. The results obtained with the closed-form solution are shown to be equivalent to the results of the same problem solved with traditional numerical methods. Based on the analytical and numerical results and by introducing dimensionless ratios that allow to quantify the increase of radial stresses and the decrease of radial displacements in the reinforced region of the tunnel, the paper shows that reinforcement can have a significant mechanical effect (i.e., increasing the confinement and decreasing the convergences) in tunnels excavated in rock masses of poor to very poor quality. The paper analyzes then the mechanical contribution of rockbolt reinforcement when the rock mass is assumed to behave elasto-plastically. For this case, it is shown that rockbolt reinforcement can also have a critical effect in controlling the extent of the plastic failure zone and the convergences of the tunnel. Correspondence: C. Carranza-Torres, Department of Civil Engineering, University of Minnesota, Duluth Campus, 1305 Ordean Court, Duluth, USA     

基于量测位移的隧洞围岩弹性抗力系数反演方法

在水工隧洞工程中,岩石弹性抗力系数K反映了岩石弹性抗力的大小,是设计环节中需要确定的一个极为重要的基础计算参数。水工隧洞工程中需用水压法试验测定弹性抗力系数,由于水压法测定抗力系数不仅耗时、费工、且试验工艺复杂。本文通过对比分析弹性状态下深埋圆形隧洞开挖支护与水压法试验力学效应关系,提出一种基于量测隧洞位移的围岩抗力系数的反演方法,该法可根据量测隧洞位移直接反演得到弹性抗力系数。结合北京市南水北调配套工程南干渠工程试验段工程实践,对浅埋水工隧洞位移进行监测,利用提出的反演方法,得到了隧洞围岩的弹性抗力系数。提出的反演方法为复杂水工隧洞不同围岩应力状态问题的弹性抗力系数反演提供参考和借鉴。     

Vertical vibration of a rigid strip footing on viscoelastic half-space

This paper presents an analytical method for modeling the dynamic response of a rigid strip footing subjected to vertical-only loads. The footing is assumed to rest on the surface of a viscoelastic half-space; therefore, effects of hysteretic soil damping on the impedance of the foundation and the generated ground vibrations are considered in the solution. To solve the mixed boundary value problem, we use the Fourier transform to cast a pair of dual integral equations providing contact stresses, which are solved by means of Jacobi orthogonal polynomials. The resulting soil and footing displacements and stresses are obtained by means of the Fourier inverse transform. The solution provides more realistic estimates of footing impedance, compared to existing solutions for elastic soil, as well as of the attenuation of ground vibrations with distance of the footing. The latter is important for the estimation of machine vibration effects on nearby structures and installations.     

A new closed‐form solution for analysis of unlined pressure tunnels under seepage forces

In this study, a simplified analytical closed‐form solution, considering plane strain and axial symmetry conditions, for analysis of a circular pressure tunnel excavated underwater table, is developed. The method accounts for the seepage forces with the steady‐state flow and is based on the generalized effective stress law. To examine the effect of pore pressure variations and also the boundary conditions at the ground surface, the formulations are derived for different directions around the tunnel. The proposed method can be applied for analysis and design of pressure tunnels. Illustrative examples are given to demonstrate the performance of the proposed solution and also to examine the effect of seepage forces on the stability of tunnels. The simplified analytical solution derived in this study is compared with numerical analyses. It is concluded that the classic solutions (Lame's thick‐walled solution), considering the internal pressure as a mechanical load applied to the tunnel surface, are not applicable to pervious media and can result in an unsafe design. Copyright © 2012 John Wiley & Sons, Ltd.     

Analytical solutions for stresses and displacements around tunnels driven in cross-anisotropic rocks

In this paper, closed-form solutions for the stresses and displacements around unlined circular tunnels excavated in an elastic, orthotropic (cross-anisotropic) medium are developed. The effect of both the elastic parameters characterizing the behaviour of the medium and the anisotropy of the initial stress system on the stresses and displacements induced are evaluated. An example of utilizing the theoretical solution for design analysis is given. For convenience of application, design charts are prepared for the determination of stresses and displacements for given values of initial stresses and the elastic parameters. Copyright © 1999 John Wiley & Sons, Ltd.     

Radial Deformations Induced by Groundwater Flow on Deep Circular Tunnels

Summary. The magnitude and distribution of ground deformations around a tunnel are often monitored during construction and provide key information about ground-support interaction and ground behavior. Thus it is important to determine the effects of different parameters on ground deformations to accurately and effectively evaluate what contributes to ground and support behavior observed during excavation. This paper investigates one such relation: the effects of seepage on radial deformations. A number of numerical analyses have been conducted with the following assumptions: deep circular unsupported tunnel, elastic ground, isotropic far field stresses, dry ground or saturated ground with steady-state water seepage. The analyses cover a wide range of tunnel sizes, effective stresses, and pore pressures. Results from the numerical simulations confirm previous analytical solutions for normalized radial deformations behind the face (i.e. on the tunnel side of the face) of a tunnel excavated in dry ground, and have been used to propose a new analytical formulation for normalized radial displacements ahead and behind the tunnel face for both dry and saturated ground with water flow. Water seepage substantially increases the magnitude and distribution of the normalized radial deformations ahead of the face and at the tunnel face, but does not change much the displacement distribution behind the tunnel face.     

爆炸荷载作用下深埋圆形隧洞的饱和黏弹性土-衬砌系统动力响应

假定土骨架服从标准线性固体黏弹性本构关系,研究了深埋圆形隧洞的饱和黏弹性土-弹性衬砌耦合系统在轴对称爆炸作用下的瞬态动力响应。首先,基于饱和土的Biot模型和衬砌的弹性理论,通过引入势函数和Laplace变换,利用弹性衬砌和饱和黏弹性土界面处的连续性条件以及边界条件,得到饱和黏弹性土体和弹性衬砌位移、应力和孔隙水压力等在Laplace变换域中的解析解。其次,利用Laplace数值Crump逆变换得到耦合系统在时间域的动力响应,数值分析了不同土体模型下土体-衬砌耦合系统的径向位移和环向应力以及土体孔隙水压力等。结果表明：对不同土体模型的土体-衬砌耦合系统,其在爆炸载荷作用下的动力响应性态基本一致,但动力响应的振动周期和幅值等具有明显的差异。同时,对于饱和黏弹性土-弹性衬砌系统,土体黏性参数对土体径向位移和孔隙水压力有明显的影响,但对土体环向应力影响较小。     

深埋圆形隧道弹塑性位移统一解及其比较分析

基于统一强度理论和非关联线性流动法则,考虑塑性区的真实弹性应变,推导了深埋圆形隧道弹塑性位移统一解。位移统一解合理地考虑了中间主应力效应和围岩剪胀特性的影响,具有广泛的理论意义。经与文献简化位移解析解比较分析知,围岩塑性区的弹性应变对隧道位移影响显著,简化解析解所得结果偏小,简化解析解和文中统一解之间的差异与中间主应力效应和围岩剪胀特性密切相关;考虑中间主应力的影响,能更加充分发挥围岩的强度潜能;隧道支护设计应考虑围岩剪胀特性的重要影响     

活动性断裂带错动下隧道纵向响应的解析解

震害调研表明，活动性断裂带区域的隧道灾害最为严重。针对错动作用下穿越活动性断裂带隧道的纵向响应进行了研究，推导了隧道纵向力学响应的解析解并进行了验证。考虑断裂破碎带围岩力学性质较差且处于错动变形的主要影响区，将隧道沿纵向进行分区，包括错动影响区、过渡影响区和非影响区。采用Pasternak双参数弹性地基梁，假定不同分区的地基参数和计算模式不同，建立了满足变形和内力连续的隧道纵向力学解析模型并进行了求解。解析计算结果与数值模拟结果、室内试验数据基本一致，验证了解析解的正确性。结果表明：错动作用下，活动性断裂带区域的隧道内力和变形发生了显著变化；隧道纵向挠曲变形与错动方向一致，但在断裂带与上下盘交界区域发生了反向的挠曲；在正断层错动下，纵向弯矩在断裂带与上下盘交界区域达到最大值，且上、下盘区域的隧道拱顶分别出现受拉和受压区域；断裂带区域内的剪力远大于其他区域，且受到较大弯矩，隧道结构易发生破坏。上述计算结果与实际调研结果相一致，表明了提出的解析计算方法可用于活动性断裂带错动下的隧道纵向响应分析。最后，针对地基系数和断裂带宽度两个关键参数进行了敏感性分析，得到了有益规律，可为该类区域的隧道设计和施工提供技术支撑。     

Viscoelastic solutions for stresses and displacements around non-circular tunnels sequentially excavated at great depths

Acta Geotechnica - This research study presents analytical solutions for the stresses and displacements around deeply buried non-circular tunnels, taking into account the viscoelasticity of the...     

Progressive degradation of rock properties and time-dependent behavior of deep tunnels

Time-dependent response of deep tunnels is studied considering the progressive degradation of the mechanical properties of the rock mass. The constitutive model is based on a rock-aging law for the uniaxial strength of the rock and for the Young’s modulus. A semi-analytical solution is developed for the stresses and displacements around a deep circular tunnel taking into account the face advance. The evolution of the plastic and damage zones over time is determined. Numerical examples are presented for the case of Saint-Martin-La-Porte access adit in France of the Lyon–Turin Base Tunnel. The computed results which are compared with the field data in terms of the convergence of tunnel wall and of the displacements inside the rock mass monitored by multi-point extensometers show the efficiency of the approach to simulate the time-dependent deformation of a tunnel excavated in squeezing ground. Simple relationships are proposed to evaluate the parameters of the constitutive model directly from those of the empirical convergence law presented in previous work.     

二郎山公路隧道轴线地应力测试研究

介绍了二郎山公路隧道轴线上3个钻孔的地应力测试结果,应用水压致裂法在钻孔中测试,所测的两个不同大小的水平主应力都大于垂直主应力（除了地表以下很浅处外）,这个地应力与区域大地构造方向一致,在此基础上评价了隧道碉室围岩的稳定性。     

软土地层浅埋圆形隧道非达西渗流场解析研究

软土在低水力坡降下的渗流会偏离达西定律,即为非达西渗流模式。假设孔隙水渗透服从指数渗流模式,采用镜像法原理推导了浅埋单孔和双孔圆形隧道非达西渗流场的解析解。结合算例,对浅埋圆形隧道非达西渗流解析解与达西渗流解析解进行了对比分析与验证,并对非达西渗流指数、隧道周围土体与衬砌渗流系数比值对隧道渗流场的影响进行了讨论。结果表明：非达西渗流指数、渗流系数比值对隧道渗流量和周围土体孔压均有较大的影响;随着渗流指数逐渐增大,土体内水头损失加快,隧道周围土体孔压及渗流量逐渐减小;随着土体与衬砌渗流系数比值逐渐增大,衬砌排水能力增强,隧道渗流量逐渐增大,隧道周围土体孔压减小更大。     

弹性矩形板下横观各向同性多层地基分析

利用弹性矩形板与多层地基表面的竖向位移协调条件与光滑接触条件,由横观各向同性多层地基应力与位移非耦合的传递矩阵解,推导出弹性矩形板下竖向应力和位移的解析解。在此基础上,编制了相应的程序,并进行了数值计算。计算结果表明：矩形板刚度对板底竖向位移及板中心下的竖向应力有着较为显著的影响;板底竖向位移及板中心下的竖向应力随着板刚度的增加而减小,相同荷载作用下横观各向同性地基与均匀各向同性地基模型的计算结果差异较大,实际工程中很有必要采用更符合土体性质的横观各向同性地基模型     

考虑椭圆化地层变形影响的浅埋隧道弹性解

研究了浅埋隧道周围地层最终变形的计算问题,提出了考虑椭圆化变形影响的计算模型和浅埋隧道位移边界条件,并利用Airy应力函数得到了该问题弹性解析解。在此基础上,通过算例分析了椭圆化变形对竖向位移和水平位移的影响。结果表明,椭圆化变形对水平位移的影响远大于竖向位移。     

Three‐dimensional finite element analysis of lined tunnels

This paper describes finite element procedures that have been developed to model the ground movements that occur when a shallow tunnel is installed in a clay soil. This study is part of a wider project concerned with the development of new methods to predict the likely extent of damage to surface structures caused by nearby shallow tunnelling. This particular paper, however, is concerned only with the numerical model of tunnel installation. The structural liner is an important component of this tunnel installation model; two different ways of modelling the liner (based on continuum elements and shell elements) are discussed in the paper. A test problem consisting of the installation of a lined tunnel in an elastic continuum is used to investigate the merits of these different approaches. When continuum elements are used to model the liner, the numerical results agree well with an analytical solution to the problem. When shell elements are used to model the liner, however, the results were found to be significantly influenced by the particular formulation adopted for the shell elements. Example analyses, involving incremental tunnel construction in a clay soil where the soil is modelled using a kinematic hardening plasticity model, are described. These analyses confirm that a thin layer of continuum elements may be used, satisfactorily, to model tunnel linings in a soil–structure interaction analysis of this sort. Copyright © 2001 John Wiley & Sons, Ltd.     

三向地震作用下叠交隧道地震响应振动台试验研究

叠交隧道是涉及隧道之间、隧道与土体相互作用的复杂体系,其安全性将严重影响城市轨道交通建设。目前,对于叠交隧道振动台试验的研究集中在水平平行和交叉叠交隧道、单向和双向地震动输入。鉴于此,利用自行设计的层状三向剪切模型箱,对竖直平行叠交隧道开展三向地震作用下的振动台模型试验,研究地基土−叠交隧道模型体系动力特性、地基土加速度、叠交隧道加速度、地表沉降、地基土孔压、叠交隧道动土压力及叠交隧道应变等地震响应。结果表明：随着震波峰值加速度（peak ground acceleration,简称PGA）依次增加,地基土−叠交隧道模型体系的自振频率随之减小,而阻尼比随之增大;叠交隧道周围地基土加速度和孔压的梯度差随着地震波PGA的增大而增大,且上隧道周围梯度差比下隧道更大;地基土对加速度的放大效应随着地震波PGA的增大而减弱;相同地震波作用下,相同位置处的叠交隧道加速度傅里叶谱形状相似,但幅值随着地震波PGA的增大而增大。此外,与顶部和底部位置相比,腰部位置加速度傅里叶谱频段范围变宽,幅值峰值有所降低;地表沉降峰值随着地震波PGA的增大而减小,相比地基土两侧位置,中心位置的沉降峰值明显较小;地震波的类型对叠交隧道动土压力峰值和应变峰值影响较小;对于动土压力峰值,两隧道的最大值均为腰部,而上、下隧道的最小值分别为底部、顶部;对于应变峰值,上隧道在腰部明显大于顶部和底部,而下隧道在4个位置相差不大。