Analytical solutions for the stresses and deformations of deep tunnels in an elastic-brittle-plastic rock mass considering the damaged zone

The excavation impact (e.g. due to blasting, TBM drilling, etc.) induces an excavation damaged or disturbed zone around a tunnel. In this regard, in drill and blast method, the damage to the rock mass is more significant. In this zone, the stiffness and strength parameters of the surrounding rock mass are different. The real effect of a damage zone developed by an excavation impact around a tunnel, and its influence on the overall response of the tunnel is of interest to be quantified. In this paper, a fully analytical solution is proposed, for stresses and displacements around a tunnel, excavated in an elastic–brittle–plastic rock material compatible with linear Mohr–Coulomb criterion or a nonlinear Hoek–Brown failure criterion considering the effect of the damaged zone induced by the excavation impact. The initial stress state is assumed to be hydrostatic, and the damaged zone is assumed to have a cylindrical shape with varied parameters; thus, the problem is considered axial-symmetric. The proposed solution is used to explain the behavior of tunnels under different damage conditions. Illustrative examples are given to demonstrate the performance of the proposed method, and also to examine the effect of the damaged zone induced by the excavation impact. The results obtained by the proposed solution indicate that, the effects of the alteration of rock mass properties in the damaged zone may be considerable.     

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 long lined tunnels subjected to travelling loads

An analytical solution is derived for dynamic response of long lined tunnels subjected to travelling loads. For the derivation, the long lined tunnel is assumed to be infinitely long with a uniform cross-section resting on a viscoelastic foundation. Fourier and Laplace transforms are utilized to simplify the governing equation of the tunnel to an algebraic equation, so that the solution can be conveniently obtained in the frequency domain. The convolution theorem is employed to convert the solution into the time domain. Final solutions of tunnel responses investigated are deflection, velocity, acceleration, bending moment, and shear force. The proposed solution is verified by providing comparisons between its results and those from the Finite Element program ABAQUS. Further parametric analysis, such as the influence of wave velocity and frequency on dynamic responses of the tunnel is presented with the analytical solution. These relationships can be an effective tool for practitioners.     

基于应变非线性软化的衬砌压力隧洞弹塑性解析解

基于岩体应变非线性软化本构模型,考虑中间主应力的影响,对等压荷载作用下的衬砌压力隧洞进行弹塑性分析。将围岩划分为裂缝区、塑性区、弹性区,推导出了围岩裂缝区、塑性区、弹性区及衬砌内任一点的应力和位移解析计算公式,并得到了使衬砌边缘岩体开始产生塑性变形的临界内压力。所得到的解与基于理想弹塑性体的Kaster’s解相比,更接近实际隧洞围岩受力变形情况。     

膨胀流变性围岩隧洞预制钢筋混凝土衬砌稳定性分析

 针对一实际流变、膨胀性围岩的室内实验结果及初步设计与施工方案, 应用引进的岩土工程仿真分析软件F INAL 对隧洞施工期、运行期的围岩变形与薄拱片钢筋混凝土衬砌结构的内力、强度特性进行了较系统的数值仿真分析。分析结果说明: 膨胀、流变性围岩一次开挖后按距掌子面2～ 3 倍洞径时施作薄拱片钢筋混凝土衬砌的开挖与支护方案是可行的; 在个别岩性岩层复杂的洞段, 由于衬砌内力受力不均匀可能导致弯矩较大而使衬砌在个别部位出现拉裂缝; 在所讨论的情况下, 衬砌内力受围岩膨胀性影响最大, 外水压力荷载次之, 流变影响最小。     

大断面黄土隧道中型钢与格栅适应性的对比试验

正在修建的郑州-西安铁路客运专线大断面黄土隧道,开挖面积达160m2,因此,开展型钢拱架与格栅拱架研究,明确型钢拱架和格栅拱架的适用条件对设计和施工都具有重要的指导意义。研究方法采用现场对比试验,为了使试验结果有可比性,选取试验条件基本相同的贺家庄隧道洞身段作为试验段,分别设置型钢拱架段40m和格栅拱架段40m。测试内容有:拱顶下沉、拱脚下沉、水平收敛、围岩压力、初支钢架应力等。试验结果表明:格栅拱架比型钢拱架的沉降略大,水平收敛基本相等;格栅拱架比型钢拱架应力小,且应力分布相对均匀;土压力普遍很小,但格栅拱架土压力更小些;隧道掘进四榀钢架(3.2m)后,钢架沉降达总沉降的26%左右,最大应力达总应力的33%左右。经综合分析认为型钢拱架及格栅拱架均能适用于IV级黏质老黄土隧道,但格栅拱架更具优越性。     

Thin spray-on liner adhesive strength test method and effect of liner thickness on adhesion

The adhesive strength between a thin spray-on liner (TSL) and rock is an important property controlling the design and performance of TSL rock support systems. The advantages and disadvantages of various adhesion test methods used by the coatings, microelectronics and concrete industries with respect to TSLs are discussed. A suggested test method for TSL materials is presented, which is based on direct pull-off of an elevator bolt that is attached to a liner with strong epoxy. To interpret the test results correctly, the effect of the liner thickness on adhesive strength must be assessed. This was shown experimentally and was supported by an analytical solution. The adhesive strength is inversely proportional to the square root of the liner thickness. Tests were performed using Tekflex as a liner material, which was applied to concrete or granite substrates.     

An elastic stress–displacement solution for a lined tunnel at great depth

By using the complex potential theory proposed by Muskhelishvili, an elastic plane strain solution for stresses and displacements around a lined tunnel under in situ stresses is presented. The tunnel is assumed to be driven in a homogeneous and isotropic geomaterial, and the tunnel construction sequence is properly taken into consideration. Numerical analyses indicate that, in order to simultaneously meet the needs of structural strength and rigidity, the relative rigidity and thickness of liner should be in an appropriate range. Either too high or low values for these parameters are unfavorable for the structural stability. The variations of stresses in the geomaterial intensively rely on the relative rigidity and thickness of liner when the ratio between the distance of the point under investigation to the tunnel axis and the outer radius of the liner is in the range from 1 to 2. In addition, the present solutions contain previously known results as the special cases.     

考虑岩体时效深埋隧洞施工过程的理论解析——开挖、锚喷与衬砌支护的全过程模拟与解答

对具有时效特性的流变岩体,施工过程对围岩力学响应有显著影响。针对黏弹性时效岩体中深埋圆形隧洞的断面及纵向开挖、锚喷和衬砌施工,用圆柱型正交各向异性弹性体模拟锚固后岩体,建立材料特性变化后锚固区力学响应的计算方法,进而导出Burgers岩体全施工过程的封闭解析解。根据解答分析了锚固区各向异性程度、锚固及衬砌支护时刻、锚固厚度对锚固区应力、位移的影响。分析表明：锚固区环向刚度的改变可显著影响岩体力学响应,而改变径向刚度则变化极小;衬砌受力与其施加时刻相关,而受锚固时刻影响不大;增加锚固区厚度可减小位移,优化锚固区受力,但对衬砌的受力影响很小。本解答可进行相似条件下隧洞初步设计。     

隧道开挖问题的水力耦合计算分析

隧道工程中的地下水问题是富水地层中普遍存在的重要问题,地下水流动对隧道围岩稳定性有重要影响。本文利用解析法给出了饱和孔隙介质地层中考虑地下水渗流力作用下的应力。位移表达式;提出了隧道水力耦合数值分析中的耦合计算模型的建立方法并进行数值模拟分析。数值计算结果表明在考虑渗流影响条件下,隧道周边位移最大增加17．0％,最大剪应力最大增加10．3％。在不考虑衬砌条件下,由于渗流引起的渗流力增加了围岩的应力、位移,从围岩-支护结构共同作用原理考虑,进行隧道支护结构设计时是应该考虑渗流效应的。     

Analytical prediction for ground movements and liner internal forces induced by shallow tunnels considering non-uniform convergence pattern and ground-liner interaction mechanism

Current studies for tunneling-induced ground movements provide little information on the complex interaction between the tunnel liner and the surrounding soils. In particular, little attention is paid to the internal force analyses for segment liners based on the non-uniform convergence deformation pattern. This paper presents a closed-form analytical solution for the prediction of ground deformation and liner internal forces induced by shallow tunnels in clays. The non-uniform convergence deformation pattern is incorporated as the boundary condition of displacements between the ground and the liner. We investigated the difference between uniform radial and non-uniform convergence deformation patterns on the surface settlements and lateral deformation of soils. In general, good alignment was obtained between the predicted ground deformation caused by the non-uniform convergence model and field observations for tunnels in clays. In addition, the influences of sensitive parameters on the ground movements induced by tunneling were evaluated based on the non-uniform convergence pattern, including the soil and liner properties, the geometric properties of tunnel, the tunnel depth, the excavation gap and other main parameters. Furthermore, these solutions offer a more comprehensive framework for understanding the ground-liner interaction mechanism and the circumferential distribution of internal forces for segment liners. Parametric analyses were used to measure the influences of the lateral earth pressure coefficient on the axial forces and bending moments of tunnel liners. Results are provided on a theoretical basis to estimate the interaction behavior between the tunnel liner and the surrounding soils correctly.     

Analytical and numerical study of the effect of water pressure on the mechanical response of cylindrical lined tunnels in elastic and elasto-plastic porous media

This paper revisits the classical problem of excavating a deep cylindrical tunnel in a saturated elastic or elasto-plastic porous ground that obeys Terzaghi's effective stress principle. A generalized form of the classical two-dimensional analytical solution by Lamé for the problem of excavating a cylindrical tunnel in elastic medium subject to axi-symmetric loading is presented. The solution considers the long-term (i.e., post-transient) mechanical effect of changes in pore pressure due to drainage of the ground around the opening on the mechanical response of the tunnel. The proposed solution together with the analytical solution for axi-symmetric loading of an annular ring representing a liner are used to explain the fundamental differences in support loading obtained for various hydro-mechanical conditions. These conditions involve excavating the tunnel with a permeable liner, with and without removal of water from the tunnel, excavating the tunnel with an impermeable liner, with subsequent drainage of water in the ground around the tunnel, etc. The results obtained with the analytical solution are compared with results obtained with the hydro-mechanical option in a commercial finite difference code. The finite difference approach is also used to obtain results and explain the behavior of the tunnel under different hydro-mechanical conditions when the ground is assumed to behave elasto-plastically.     

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。     

Behaviour of tunnel lining strengthened by textile-reinforced concrete

In this paper, the effectiveness of textile-reinforced concrete (TRC), as a means of increasing the bending resistance of tunnel lining, was experimentally and analytically investigated. The short RC column strengthened by TRC on the side farthest from the axial load was designed to investigate the behaviour of tunnel lining strengthened by TRC. The parameters under study included the loading eccentricities, the number of textile layers and the preload ratio of the short RC columns. The experimental programme was composed of two control RC columns with different eccentricities. Five RC columns which were externally upgraded by TRC sheets were also analysed for their enhanced flexural capacity. Experimental responses of the reinforced concrete members strengthened in bending indicated that TRC substantially enhanced the bending resistance. Specifically, the gain increased with a concurrent increase in both the number of layers and eccentricity. The gain greatly worsened, however, with an increase in the preload ratio. An analytical method considering the non-linear behaviour of the materials was also proposed and validated through the test results. The study was extended analytically to include additional cases of TRC-strengthened specimens with some more parameters, including different concrete and steel types.     

高速公路泥岩隧道围岩压力试验研究

以宝天高速公路天水过境段梁家山泥岩隧道为工程背景,通过现场测试,研究了围岩拱顶下沉、围岩与初期支护接触压力、钢拱架应力和初衬与二衬接触压力随时间变化规律及分布特征,并应用现有的理论和方法对实际工程安全状态进行评估。研究结果表明:以隧道开挖空间效应和Chern经验公式两种理论方法对拱顶沉降数据进行分析处理,所得到的隧道极限位移相差较小;围岩与初期支护压力的分布具有不均匀性特点;钢拱架外缘应力平均值大于内缘,拱顶处应力大于其他部位,按其推算得到的拱顶沉降极限位移更接近于实际情况;初衬与二衬间接触压力先迅速增加后逐渐较小最后趋于稳定,其稳定值整体偏小。     

公路隧道衬砌混凝土开裂的外水压力原因分析

在公路隧道检修时,发现衬砌混凝土产生多处纵向裂缝,开裂较为严重。由于该隧道处在地下水丰富的地层中,文章采用三维有限元法、应用渗流应力耦合理论,对可能导致衬砌混凝土开裂的水荷载原因进行分析研究,并从降低渗流荷载引起衬砌拉应力的角度,提出了修补建议。     

喷射钢纤维混凝土加固机理及其在隧道溶洞群处治中的应用

利用喷射混凝土中掺入钢纤维能极大地提高混凝土的抗拉弯、抗剪强度和韧性的机理特性,可用来处治隧道溶洞的病害,达到增强围岩结构稳定性的目的。结合具体工程实例,阐述了喷射钢纤维混凝土处理大断面公路隧道小溶洞群病害的施工关键技术,并从技术、安全、经济效果等方面与常规的挂网喷锚加固技术进行对比分析,得出其综合经济效益的显著性,其成功的工程实践应用也证实了该方法的可行性,可应用于其它类似溶洞处理中。     

Hydro-mechanical interaction analysis of high pressure hydraulic tunnel

The reinforced concrete lining of hydraulic tunnel cracks under high internal water pressure and the high pressure water will leak off from the crack surface, which is a typical hydro-mechanical interaction problem. Unfortunately, the analytical and numerical methods used in current engineering practice for simulating the hydro-mechanical interaction of the lining and surrounding rock do not adequately account for the effects of the cracking of lining as well as the conditional cooperation between them. When lining cracks under high internal water pressure, the tensile strain of concrete and reinforcement presents to be nonuniformity and the lining would detach from the surrounding rock in a certain condition. In this paper, the nonuniformity coefficients of reinforcement strain and concrete strain are introduced to compute the secondary permeability of cracked lining and a water-filled joint (WFJ) element is developed to simulate the conditional cooperation between the reinforced concrete lining and surrounding rock. Thus an equivalent coupling method is adopted to calculate the hydro-mechanical interaction in hydraulic tunnel. The results show that the reinforcement stress obtained by the new method and the observed data match well. The phenomenon that the reinforcement stress stays at a lower value in the reinforced concrete tunnel with high internal water pressure is verified.     

早强锚杆在高地应力层状软岩隧道中的应用研究

以在建的成都-兰州铁路杨家坪隧道为工程依托,选取条件基本相同的30m典型围岩区段为试验段,对普通锚杆、早强锚杆支护时的洞周位移、围岩与初支接触压力、型钢拱架应力及其锚杆轴力进行实测对比分析,探讨了早强锚杆在高地应力陡倾层状软岩隧道中的作用机制。结果表明：高应力软岩隧道中锚杆轴力为拉力,早强锚杆比普通锚杆轴力更大,可以使隧道洞周位移减小40%|早强锚杆使隧道边墙围岩压力和钢架拱顶应力减小,围岩压力分布和钢架受力趋于均匀|早强锚杆通过注浆材料深入围岩,可以提高围岩层面强度|及时发挥锚固作用,抑制了围岩渐进破坏过程,从而减小围岩塑性区|加长了锚杆的拉拔长度,减小围岩与初支接触压力,改善隧道支护的受力状况,有效地控制隧道变形。     

Residual stresses in press-braked stainless steel sections,I: Coiling and uncoiling of sheets

The manufacturing process of cold-formed thin-walled steel members induces cold work which can be characterized by the co-existent residual stresses and equivalent plastic strains and has a significant effect on their structural behaviour and strength. The present paper and the companion paper are concerned with the prediction of residual stresses and co-existent equivalent plastic strains in stainless steel sections formed by the press-braking method. This manufacturing process consists of the following two distinct stages: (i) coiling and uncoiling of the sheets, and (ii) press-braking operations. This paper presents an analytical solution for the residual stresses and co-existent equivalent plastic strains that arise from the first stage. In the analytical solution, the coiling–uncoiling stage is modelled as an inelastic plane strain pure bending problem; the stainless steel sheets are assumed to obey Hill’s anisotropic yield criterion with isotropic hardening to account for the effects of material anisotropy and nonlinear stress–strain behaviour. The accuracy of the solution is demonstrated by comparing its predictions with those obtained from a finite element analysis. The present analytical solution and the corresponding analytical solution for press-braking operations presented in the companion paper form an integrated analytical model for predicting residual stresses and equivalent plastic strains in press-braked stainless steel sections.