深埋马蹄形隧道开挖围岩应力与位移的复变函数解

利用复变函数解法中的柯西积分法,求解工程中常用的单心圆仰拱马蹄形隧道在弹性半空间内任意一点处的应力值和位移值解析解表达式。由于是深埋隧道,且埋深与孔径之比较大,故不考虑重力梯度影响,直接把重力作用化为无限远处作用有P1、P2的外载;求解出马蹄形隧道孔洞在弹性半空间内任意一点处的应力值和位移值解析解表达式。结合典型断面,利用三维有限元分析软件MADIS/GTS建立二维平面应变模型,对理论推导单心圆马蹄形隧道在弹性平面内的解析解公式进行验证。分析表明,有限元结果和解析解结果有较好的吻合性,证明了新方法的准确性,针对深埋马蹄形隧道开挖工程,可以快捷地评估围岩应力状态及位移变形。     

流变岩体中既有隧道与新建平行隧道相互影响的理论解

在流变岩体中进行隧道施工时,既有隧道与新建隧道的相互作用将使围岩应力、位移分布与单隧道问题有明显不同,且与时间相关。针对黏弹性岩体中深埋双圆形隧洞考虑施工顺序问题,用复变函数方法、Laplace变换、黏弹性叠加关系导出两隧道周边开挖增量位移和应力场的求解方法和理论解答,并与有限元解进行了比对。根据解答分析了既有和新建隧道孔边增量位移与全应力分布特点;隧道周边增量位移随时间的变化;周边位移随隧道间距的变化规律。可用于黏弹性岩体中双圆形隧道顺序开挖的施工分析。相比数值方法,理论解可更方便地进行参数分析和初步设计。     

含多孔洞无限平面体弹性应力场解析逼近方法

 将含单孔洞无限平面体弹性应力场解析逼近方法进行推广,求解无限平面体内含多个任意排列、任意形状凸多边形孔洞在内边界上作用任意荷载下的弹性应力场。基于应力叠加原理,将含m个孔洞的无限平面体应力场视为m个仅含单个孔洞(称隔离孔洞)的无限平面体在洞口内边界上作用虚拟面力产生的应力场的叠加。对于每个隔离孔洞,其内边界上虚拟面力会在其他隔离孔洞内边界位置产生附加面力;一个孔洞内边界上由其他隔离孔洞产生的附加面力与其上虚拟面力之和最终等于实际作用外力。提出一种迭代方法求解所有隔离孔洞内边界上虚拟面力直至收敛,进而得到多孔洞外域弹性应力解。算例分析表明该方法获得的工程尺度范围的孔洞外域应力场与有限元法计算结果吻合良好;同时可计算孔洞边角处极近场应力,据此拟合得到应力奇异性次数与广义应力强度因子。该方法原理与计算过程简单,由于基于弹性力学解析解和高精度数值积分,其最终结果属解析逼近解。     

Two- and three-dimensional analysis of closely spaced double-tube tunnels

The two tubes of underground railways in cities are constructed rather close together because of the station configurations. The two tunnel tubes are driven one after the other. The stresses in the linings and in the ground are affected by the driving sequences, especially in soft shaund of shallow tunnels. Hence, a consistent stres--deformation analysis for the tunnels has to take into account the mutual influence of the consecutive tube driving. The paper reports on the results of a finite element approach to these problems for shield-driven tunnels as well as for tunnels driven by excavation and shortcreting. The results are presented in diagrams that show the relative changes in stresses and deformations, in comparison with a single-tunnel analysis. These results show that single-tunnel solutions can be used to find double-tube solutions.     

Analytical solutions for steady seepage into an underwater circular tunnel

Based on the conformal mapping of the complex variable methods, analytical solutions are derived for two-dimensional, steady ground water flow into a horizontal tunnel in a fully saturated, homogeneous, isotropic, and semi-infinite aquifer. Main application of the equations is to calculate the distribution of the hydraulic head and the pore pressure in the rock mass surrounding the tunnel, and it takes constant hydraulic head and constant water pressure boundary conditions at the tunnel perimeter into consideration. Moreover, a comparison of the analytical solution and FLAC^3d shows that the solutions match well for points around the tunnel. When Harr’s solution is compared to the solution presented here, it appears that the surface has a large effect on the water pressure around the tunnel, even for relatively deep tunnels.     

深埋双隧洞开挖的解析延拓法求解

 将隧洞所处的地下空间转换为平面应变情况下的弹性半空间,考虑两隧洞开挖次序的影响,利用解析延拓法及Schwarz交替法求解出弹性半空间内2个任意形状任意尺寸的双隧洞在任意相对位置下其半空间内任意一点的应力解和位移解的隐式表达式。将双圆形隧洞作为特例,考虑到隧洞埋深与孔径相比比较大,从而不考虑重力梯度的影响,将重力化为作用在边界上的均布外荷载,考虑2个隧洞衬砌法向反力的影响,求解出弹性半空间内圆形双隧洞任意一点处的应力值和位移值解析解的具体显式表达式。最后利用二维有限元分析对理论推导的两圆形孔洞解析解进行验证,表明解的可靠性。     

A closed-form elastic solution for stresses and displacements around tunnels

A closed-form plane strain solution is presented for stresses and displacements around tunnels based on the complex potential functions and conformal mapping representation. The tunnel is assumed to be driven in a homogeneous, isotropic, linear elastic and pre-stressed geomaterial. Further, the tunnel is considered to be deep enough such that the stress distribution before the excavation is homogeneous. Needless to say that tunnels of semi-circular or “D” cross-section, double-arch cross-section, or tunnels with arched roof and parabolic floor, have a great number of applications in soil/rock underground engineering practice. For the specific type of semi-circular tunnel the distribution of stresses and displacements around the tunnel periphery predicted by the analytical model are compared with those of the FLAC^2D numerical model, as well as, with Kirsch's “circular” solution. Finally, a methodology is proposed for the estimation of conformal mapping coefficients for a given cross-sectional shape of the tunnel.     

端锚式锚杆-围岩耦合流变模型研究

针对端锚式锚杆-围岩结构体在长时条件下支护作用的演化机制,建立了端锚式锚杆-隧洞围岩耦合作用的结构模型。进行了结构模型的基本假设：①圆形隧洞;②深埋;③各向等压原岩应力;④均质且各向同性黏弹性围岩模型;⑤一维黏弹性锚杆模型;⑥锚杆对围岩作用力整体为面力。基于基本假设建立了端锚式锚杆-围岩耦合流变理论模型。假设围岩和锚杆均为Maxwell模型时,求解了圆形隧洞围岩应力和位移的径向分布随时间变化的解析解,获得了锚杆轴力随时间演变的理论公式。基于锚杆（索）流变模型,进行了FLAC^3D数值模拟软件的二次开发;并通过数值模拟与理论计算的对比分析验证了理论模型的合理性,分析了端锚式锚杆-围岩耦合流变规律及其影响因素。该模型对于研究地下隧洞的流变力学行为,分析锚固支护结构的长期稳定性,指导工程支护设计具有重要的基础理论价值。     

Effect of pore water pressure on tunnel support during static and seismic loading

The support of underground structures must be designed to withstand static overburden loads as well as seismic loads. New analytical solutions for a deep tunnel in a saturated poroelastic ground have been obtained for static and seismic loading. The static solution accounts for drainage and no-drainage conditions at the ground–liner interface. Linear elasticity of the liner and ground, and plane strain conditions at any cross-section of the tunnel are assumed. For tunnels in which ground stresses and pore pressures are applied far from the tunnel center, the drainage conditions at the ground–liner interface do not affect the stresses in the liner. The analytical solution shows that the stresses in the liner are exactly the same whether there is drainage or not at the ground–liner interface. Hence, if the drainage conditions in the tunnel are changed from full drainage to no-drainage or vice versa the stresses in the liner are not affected. However, the stresses and displacements in the ground change significantly from drainage to no-drainage conditions. For seismic loading a new analytical formulation is presented which provides the complete solution for the ground and the liner system for both dry and saturated ground conditions. The formulation is based on quasi-static seismic loading and elastic ground response; for a saturated ground, undrained conditions are assumed which indicate that the excess pore pressures generated during the seismic event do not dissipate. The results show that the racking deformations of a liner in dry or saturated ground are highly dependent on the flexibility of the liner.     

Analytical solution for tunnelling-induced ground movement in clays

Elastic solutions are presented to predict the tunnelling-induced undrained ground movements for shallow and deep circular tunnels in soft ground, by imposing the oval-shaped ground deformation pattern as the boundary condition of the displacement around the tunnel opening. The gap parameter is used to describe the displacement at the opening. The difference between uniform radial and oval-shaped ground deformation patterns on ground deformations is investigated and different definitions for ground loss are discussed. The applicability of the proposed analytical solutions is checked with five case studies. Generally good agreement of the predicted ground deformations can be seen with field observations for tunnels in uniform clay.     

A new elasto-plastic solution for analysis of underwater tunnels considering strain-dependent permeability

An analytical–numerical solution is proposed in this paper for calculating the mechanical and hydraulic characteristics of underwater tunnels, excavated in an elasto-plastic strain-softening and Hoek–Brown rock material considering strain-dependent permeability. The problem is considered with axial-symmetry condition and thus, the initial stress state is assumed to be hydrostatic. Hydraulic and mechanical differential equations governing the plastic and the elastic rock mass around the tunnel are derived. As the derived equations do not have closed-form solutions, a computer program has been prepared, in order to solve the corresponding equations numerically and to examine the analysis. The results obtained are compared with the results obtained using a commercial finite difference code for investigating the effects of utilised simplifying assumptions. The proposed approach is also used to analyse the response of tunnels under different hydro-mechanical conditions. The results show that the proposed solution can be used for the actual design of underwater tunnels, hydraulically and mechanically.     

Effect of drainage conditions on porewater pressure distributions and lining stresses in drained tunnels

This study deals with the effect of drainage conditions on porewater pressure distributions and lining stresses in drained tunnels. Firstly, simple closed-form analytical solutions for the steady-state porewater pressure are re-derived within a common theoretical framework for two different boundary conditions (one for zero water pressure and the other for a constant total head) along the tunnel circumference by using the conformal mapping technique. The difference in porewater pressure distributions among the analytical solutions is investigated. The numerical simulation of a drained circular tunnel under the steady-state groundwater flow condition is made to investigate the effect of porewater pressure distributions on lining stresses. Secondly, the case study of the planned South Blue Line Extension subway tunnel under the steady-state groundwater flow condition is performed for four different drainage conditions (sealed, fully-drained with zero water pressure or a constant total head, and invert only-drained) along the tunnel circumference. The effects of different drainage conditions on porewater pressure distributions, flow nets, and lining stresses are investigated.     

圆形巷道围岩变形压力新解法

将巷道围岩塑性区内的岩体视为塑性软化介质，以岩体的全应力─应变曲线后破坏段的应力值作为岩体的塑性软化强度，得到了关于巷道围岩塑性区半径和应力的一般解，适用于具有理想弹塑性、应变软化和应变硬化特征岩体中巷道变形压力的分析计算。由于本文力学模型中引入了岩体的全应力─应变曲线，很好地反映了岩体出现塑性变形后强度降低的特点，所以计算结果对工程实践具有指导价值。著名的Ｈ．Ｋａｓｔｎｅｒ公式可以看成是本文公式在岩体出现塑性变形后强度不变的条件下的一个特解。     

流变岩体中任意形状隧道分部开挖响应的理论解

针对流变岩体中任意断面形状隧道的任意复杂分部开挖问题,根据复变函数理论结合Laplace变换建立黏弹性位移、应力与复位势之间的关系,给出了各施工阶段力学响应(位移与应力)理论解的一般求解方法和解答。从解答可以看出,各步增量位移是新开挖边界处释放面力作用下的流变位移与之前所有各施工步边界释放面力在本步时段中的增量流变位移的和,与施工路径相关。以圆形断面隧道分两阶段(从半圆形到圆形)任意分部开挖为例,导出了各步开挖过程围岩位移与应力的解析解。为验证解答的正确性,针对圆形断面上下开挖方式,将理论结果与有限元结果进行了对比,两者一致。利用解答对上下开挖、右左开挖两种方案下圆形隧道的位移、应力分布特点进行了分析。根据本文方法,可以得到任意黏弹性模型岩体中,复杂孔型任意分部开挖过程的解答。依据解答可建立快速预测系统,给出更方便、快捷地进行相似工程条件下初步设计的可选方法。     

Excavation-induced hydraulic conductivity reduction around a tunnel – Part 1: Guideline for estimate of ground water inflow rate

Field observations indicate that current engineering practice does not consistently estimate ground water flows into unlined rock excavated tunnels due to various factors that analytical solutions do not take into account. These factors include significant geological features, groundwater drawdown, inadequate estimates of hydraulic conductivity from packer tests, and stress-induced rock-mass permeability reduction in the vicinity of tunnel (lining-like zone). A key variable that is not properly accommodated in current practice, is the hydro-mechanical interaction within the joints in the surrounding rock mass. The significance of this variable is discussed in the 1st part of the paper which presents an analytical solution assessing ground water inflow rate into a tunnel using a mathematical derivation that takes into account the excavation-induced rock permeability reduction in the vicinity of a tunnel based on hydro-mechanical coupling effect. In the 2nd part of the paper, results from numerical analysis are presented to verify the proposed analytical solution for estimating ground water inflow rate into a tunnel. Further studies are currently underway to identify other key variables and their impact on the behavior of unlined tunnels and hydrological flow regime in the surrounding fractured rock mass using a distinct element method program which can fully consider hydro-mechanical coupled behavior of joints.     

浅埋隧道围岩应力及位移的显式解析解

浅埋隧道施工引起的地层位移将传递至地表进而形成地表沉降,施工前准确预测地层位移是有效控制地表沉降的前提。基于复变函数法,利用逆映射函数求解z平面复势函数的级数形式,并结合柯西-黎曼方程（C-R条件）对Verruijt提出的浅埋隧道围岩应力及位移隐式解析解中的解析函数求导,得出了浅埋隧道应力及位移函数的级数显式表达式;与Verruijt隐式解析解相比,该显式解析解直观,便于被工程人员使用,其编程计算量也较小;为确定显式解析解中的未知系数,基于浅埋隧道洞室变形产生的原因,提出了3种隧道洞室边界的变形模式和2种变形比率,合理调整变形比率可以实现多种边界条件组合。工程实测地层位移与显示解析解的计算结果吻合度高,验证了所提显式解析解的正确性和实用性。     

Ovaling deformations of circular tunnels under seismic loading, an update on seismic design and analysis of underground structures

Two analytical solutions for estimating the ovaling deformation and forces in circular tunnels due to soil–structure interaction under seismic loading are widely used in engineering practice. This paper addresses an unresolved issue related to discrepancy between the two solutions. A comparison of the two solutions shows that the calculated forces and displacements are identical for the condition of full-slip between the tunnel lining and ground. However, the calculated lining thrusts differ by an order of magnitude when assuming no-slip between the tunnel lining and the ground. The analytical solutions are compared to numerical analyses of the no-slip condition using the finite element method to validate which of the two solutions provide the correct solution. Numerical analysis results agree with one of the analytical solution that provides a higher estimate of the thrust on the tunnel lining, thus highlighting the limitation of the other analytical solution.     

Analytical algorithm for longitudinal deformation profile of a deep tunnel

To investigate the longitudinal deformation profile(LDP) of a deep tunnel in non-hydrostatic condition,an analytical model is proposed in our study.In this model,the problem is considered as a superposition of two partial models,and the displacement field of the second partial model is the same as that of the concerned problem.Therefore,the problem can be solved by a model with simple boundary conditions.We obtain the solutions for the stress and displacement fields of an infinite body caused by arbitrary surface tractions on the boundary of the coming tunnel(zone inside the tunnel before excavation) by integrating the extended Kelvin solution over the boundary.The obtained stress solution is used to solve the specific surface tractions,which can satisfy the boundary conditions of the second partial model,on the boundary of the coming tunnel in an infinite body.Then,the specific surface tractions are substituted into the obtained displacement solution to solve the displacement on the wall and face of the tunnel.Therefore,the LDP can also be calculated.The proposed solution is verified by both numerical simulation and the LDP functions recommended by other researchers.The major advantage of our analytical model is that it can consider the effects of the axial and horizontal lateral pressure coefficients.It is revealed that the horizontal lateral pressure coefficient majorly affects the LDP behind the tunnel face,while the axial lateral pressure coefficient dominates the LDP ahead of the tunnel face.Furthermore,the deformation characteristics of the LDPs ahead of the face and the unexcavated core are investigated.The axial displacements of the excavation face can be used to predict the crown displacements ahead of the face.     

重叠隧道的施工力学研究

由于地下空间的不断利用，出现了重叠隧道。这种形式布局的隧道在第1孔隧道完成之后，引起了地层内初始应力的变化，而且由于地层中已有一个相邻的已衬砌隧道，地层的刚度条件不再与单孔隧道的垂直轴对称，第2孔隧道的修建又要对已建的第1孔隧道发生影响。应用ANSYS程序对深圳地铁1期工程中的3种典型的重叠断面的计算表明：重叠隧道地表和拱顶下沉及两隧道间土体的稳定难于控制，上面第1孔隧道易发生结构整体下沉。     

Displacement characteristics of high-speed railway tunnel construction in loess ground by using multi-step excavation method

The sequential excavation method (SEM), in which the tunnel face is partitioned into several temporary drifts, can promote face stability and reduce ground displacement. Three-bench seven-step excavation method (TSEM) as a kind of the SEM, which can stabilize the tunnel face but does not require any additional support, has been widely used in loess tunnels with large cross-sections for high-speed railway (HSR) in China. This study focuses on the effects of TSEM on the displacement characteristics for HSR tunnels in loess ground. A series of in-situ tests and numerical simulations were conducted to reveal displacement characteristics and obtain optimal construction approaches for large-span loess tunnels in China. The strata around the loess tunnels have undergone significant settlement either at the arch part or at the ground surface due to loess’s metastable structure especially in the new loess ground. Upper strata above loess tunnels subsided as a whole, and the subsidence developed suddenly and devastatingly. Large residual ground surface settlements still occurred after the ring closure of the primary support even after the application of the upper lining. The best construction approach of TSEM was proposed for its potential to limit surface and subsurface ground displacements, as revealed both by numerical simulation and field monitoring.