有压隧洞结构稳定性力学模型研究
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摘要
隧洞作为一种主要的地下结构,广泛的应用于交通、水利、采矿、石油以及水电和通信等工程中。随着对于基础建设的发展和升级换代的要求,隧洞建设在世界各地蓬勃发展。而所建隧洞的安全性和经济性也越来越受到关注。一般的,隧洞设计应满足如下三个要求:隧洞在建过程中要满足稳定、安全性的要求;尽量减少对附近结构的不利影响;达到隧洞的既有设计功能。其中,第一条主要是针对隧洞支护体系的合理设计。
众所周知,隧洞开挖后地层中初始应力释放,应力重新分布形成二次应力场,从而引起隧洞的变形。为了确保隧洞开挖后的稳定性,工程中常采用衬砌材料进行支护。然而,大多数力学模型可以退化为这样的情形:隧洞开挖,衬砌安装完,而后结构再受到地应力的影响。而事实上,岩(土)体是赋存了一定的初始应力,开挖导致隧洞发生部分变形,而后再安装衬砌,因而,必须提出一种合理的力学模型。
另外,当作用于岩体的初始应力较大或岩体自身的强度较低,洞室开挖后,洞周的部分岩体应力超出了岩体的屈服应力,使岩体进入了塑性状态。随着与洞壁距离的增大,最小主应力也随之增大,进而提高了岩体的强度,并使岩体的应力转化为弹性状态,岩体的二次应力状态呈现弹塑性并存的状态。虽然,各种基于弹性理论的围岩-衬砌干涉作用模型很多,而经典的弹塑性解答中,一般都未考虑衬砌本身对于围岩的制约作用,实际上,围岩压力是通过衬砌和围岩的共同作用来承担的;再者,屈服条件中的第一主应力是以径向应力的情况下导出的,这样做还不够全面,必须根据不同的工况和不同的地应力条件,正确选择屈服条件中的第一主应力,导出衬砌和围岩的屈服范围和应力计算公式,给出屈服区和应力计算公式的适用范围。
许多隧洞,包括新奥法施工的隧洞,需要的支护形式为:初期支护和二期支护。初期支护主要来承受开挖荷载,而二次衬砌支护需要承受隧洞运行期间的附加荷载以及围岩、初期支护强度随时间的降低,或者来自围岩中的其它附加荷载作用。地下水流的变化则是其中的主要附加荷载之一。隧洞突水无论是在隧洞的施工期还是在运行期均可发生。隧洞突水会造成停工、环境破坏以及结构的沉陷等严重问题。另外,为了评估相关的问题,隧洞突水也必须尽量做到提前预警。虽然孔隙水压力对隧洞支护的影响有许多的研究成果,但是,各种水力条件变化下的隧洞支护设计标准仍然急需在这方面进行多方面的深入研究。
弹性问题的复势方法是迄今唯一能从已知到未知顺次求解的分析方法;它统一了弹性力学中的三种基本方法(位移法、应力法、应力函数法)和三类边界(力边界、位移边界、混合边界)问题;而弹性场本身的应力变形特征与复势函数之间必定存在一种确定的关系,找到这种关系无疑对于具体问题中复势函数的求解是重要的。本文的主要研究成果如下:
1.利用基本的平面弹性方程,结合解析函数的性质,首次得到了平面弹性复势中的几个重要性质,其内容为:对于平面、反平面弹性场,当应力状态关于原点对称时,复势为偶函数;当应力状态关于x轴对称时,平面弹性场中的复势具有实系数;当应力状态关于x轴反对称时,反平面弹性场的复势具有虚系数。利用所得基本性质可以简化复势函数的求解过程。
2.根据围岩和衬砌联合作用的机理,研究了深埋圆形衬砌隧洞在多种荷载作用下的弹性应力、位移场的解析解答。利用以上复势函数性质,构造合适的复势函数,然后,根据边界上力、位移的连续性条件,获得了全场复势的解析表达。详细讨论了非静水压力条件下的深埋衬砌压力隧洞的这一情形。结果表明:为了减少隧洞开挖对原有应力、位移场的扰动,施作衬砌是一种有效措施,但是,衬砌的厚度和刚度应该针对具体情况保持在一定范围,过大过小的值都是不合适的。另外发现,隧洞开挖对位移场的影响要远远大于对应力场的影响。就应力场而言,当隧洞埋深和隧洞内径的比值大于5时,就完全可以应用本文的结论;然而,就位移场而言,只有当隧洞埋深和隧洞内径的比值大于20时,本文的结论才可以应用。
3.研究了隧洞处于弹塑性条件的围岩稳定性问题。考虑地应力的释放和衬砌安装的时序,提出合理模拟隧洞修筑过程的力学模型。利用边界上位移和力的连续性条件,得到围岩和衬砌内的复应力函数表达式。根据不同的工况和地应力条件,基于Mohr-Coulomb屈服准则,正确选择屈服条件中的第一主应力,导出衬砌和围岩的屈服范围和应力计算公式,并给出屈服范围和应力计算公式的适用条件。重点抓住影响地下结构稳定性的几个主要参数,即围岩、衬砌的弹性(或变形)模量,地应力侧压系数,衬砌的厚度,以及到洞轴线距离等,来分析其对结构的强度、变形影响。经典的Fenner公式和Lamé解答等许多已往解答,可以作为其特殊情况退化得到。
4.将深埋圆形隧洞各影响因素简化为轴对称。首先,求解得到渗流场;然后,以渗透体积力方式作用在围岩应力场,求解得到衬砌作为渗透材料和不渗透材料情形下的全场位移和应力解析表达式。根据所得解答,可以用来解释各种围岩水力条件以及隧洞排水条件下的隧洞支护压力的不同。综合考虑了衬砌具有渗透性、不具有渗透性两种情形,且对得到的结果进行了对比。
Tunnel is a main underground structure and widely used for transportation, water passage, mining, petroleum, geophysical engineering practice and other purposes such as electricity or communication cable installation. With the development and upgrade of infrastructures, the demand for tunnel construction is increasing all over the world and the importance of the safety and economics of tunnel construction are recognized in tunnel engineering. In general, in relation to tunnel construction, it refers to three issues, which are: (1) maintaining stability and safety during construction; (2) minimizing unfavorable impact on adjacent structures; (3) performing the intended function over the life of a project. Among the issues, the first one is directly related to the appropriate design of tunnel support system.
The excavation process reduces the confining stress to zero on the boundary of the opening. Thus, tunnel excavation always causes stress relief and deformation at the tunnel boundary before the liner is installed, which is well-known. The support system of underground facilities must be designed to withstand static overburden loads as well as to accommodate the additional deformations imposed by the earthquake induced motions. It is essential to understand the interaction between the liner and the surrounding geomaterial with the construction sequence taken into consideration. The real problem is that most of the analyses in the past completely misunderstand the history of stresses on the tunnel, which are essentially a "mechanical engineering" calculation in which one imagines an initially unstressed body (tunnel and liner) and then the stresses are applied at the outer boundaries. In fact of course certain pre-existing stresses exist in the ground. The tunnel is then opened, causing stress relief at the tunnel boundary (and consequent displacements), the liner is then installed, and finally the concrete of the liner cures. The effect of this construction sequence on the stresses in and around the liner was taken no account in the past.
It is normal practice in the design of tunnel liners using surrounding geomaterial-support interaction models to consider that the geo-material behaves elasto-plastically. This means that under loading the material will first behave elastically, but when (and if) the elastic limit or plastic strength of the material (liner and surrounding geomaterial) is reached, then the material will start behaving plastically. In most of the existing elasto-plastic solutions, the first principal stress in the yield range was generally assumed to be the radial stress. In addition, the interaction between the liner and the surrounding geomaterial was not appropriately considered. Although the surrounding geomaterial-liner interaction based on elastic theory has been studied by numerous researchers. Furthermore, the variations of yield radius with the internal pressure acting on the inner boundary of the liner, the liner thickness and rigidity were not adequately discussed in the past literatures. In fact, certain pre-existing stresses exist in the geomaterail before the tunnel is excavated, which causes stress relief at the tunnel boundary (and consequent displacements). The liner is then installed, and finally the concrete of the liner cures. It is essentially important to understand how the surrounding geomaterial around a tunnel deform due to changes in stresses with the construction sequence taken into account properly. Moreover, the variations of the internal pressure acting on the inner boundary of the liner, the liner thickness and rigidity may result in the variations of the yield radius, the first principal stress and the boundary condition. Such factors should be properly considered in order to have a full picture of the elasto-plastic behavior for underground engineering problems.
Many tunnels, including NATM tunnels, require two types of support: primary and secondary support. The role of the primary support is to withstand the loads that may arise during excavation. The secondary support has to withstand those loads that arise from tunnel operation, deterioration of ground strength and primary support with time, or from other changes of stresses in the ground around the tunnel. An important source for such additional loads is the change of groundwater flow around the tunnel. Water ingress into tunnels may be encountered as well in the construction phase of any tunnel as in the operation phase of hydraulic tunnels. Water ingress in the construction and in the operational phases may cause severe difficulties, downtimes and environmental impact by altering the groundwater regime and causing settlements of structures on the surface. To assess the related problems, the water ingress must be somehow predicted in advance. Even though some of the effects of pore water pressure on tunnel support have been investigated, there are many aspects that require further scrutiny, in particular a reasonable design criterion for tunnel support is needed where groundwater flow conditions are included.
As an important analytical method, the complex variable method has been widely used to analyze problems associated with underground construction. The advantage of the complex variable method in solving elastic tunnelling problems is that both the stresses and displacements are solved simultaneously, enabling the solution of problems for various types of boundary conditions, and that the boundary conditions for the displacements and stresses are similar. An important issue is to find the deterministic relations between the stress-deformation conditions and the complex potentials. The work in the present paper can be summarized in the following:
1. Four important properties for the complex potentials expressed in series expansion in the plane and anti-plane elastic fields is obtained by applying the basic equations of plane and anti-plane problems, combining the theory of analytical function, and recalling the feature of stress and deformation. Firstly, the complex potentials in the plane or anti-plane elastic fields are real functions when the stress states are symmetrical with regard to the origin. In addition, the complex potentials in the plane elastic field have real coefficient when the stress states are symmetrical with regard to the x-axis. Finally, the complex potentials in the anti-plane elastic field have imaginary coefficient when the stress states are anti-symmetric with regard to the x-axis. Then, based on the conclusions of the paper, some classic solutions are rearrived at, which indicates the results derived in this paper are right and simplify the process of constructing and solving the complex potential functions. The present conclusions provide an efficient tool to discovery the solutions in the sophisticated state.
2. By applying the series expansion technique in the complex variable method established by Muskhelishvili, the plane elasticity problem for the stress and displacement field around a lined circular tunnel in conjunction with the consideration of misfit and interaction between the liner and the surrounding geomaterial is dealt with. The tunnel is assumed to be driven in a homogeneous and isotropic geomaterial. The coefficients in the Laurent series expansion of the stress functions are determined. The complex potentials in the liner and the surrounding geomaterial are explicitly derived, respectively. An elastic solution is obtained under a wide class of loading conditions, the main limitation being that internal stress sources (if any) are located in the geomaterial. Loading by misfit alone, by in situ stresses, by far-field shear stresses, and by a concentrated force, are explicitly treated as special cases. As an example, the case of a lined circular tunnel located in an isotropic initial stress field but subjected to uniform internal pressure is numerically considered. Numerical results indicate that the installation of tunnel liner can reduce the influences of the tunnel excavation on the in situ displacement and stress fields. However, the relative thickness and rigidity of the liner should be in an appropriate range. In addition, the effect of the tunnel excavation upon the displacement field is more significant than that upon the stress field. As far as the stress field in the surrounding geomaterial is concerned, when the ratio between the cover depth of tunnel and the tunnel radius is larger than 5, the results for the stress field in the paper are applicable. When the ratio between the tunnel depth and the tunnel radius is larger than 20, the results are applicable for the displacement field.
3. The elasto-plastic problem for the stress fields around a lined circular tunnel is investigated in this paper. An appropriate mechanical model considering the stress relief and the installation of the liner is proposed to model the tunnel construction sequence. By considering the continuity conditions for the stresses and displacements along the boundaries, the complex potentials in the liner and the surrounding geomaterial are explicitly derived, respectively. Based on the linear Mohr-Coulomb (M-C) yield criterion, for two cases that the first principal stress is the radial stress or the tangential stress, respectively, the stress solutions are given when plastic deformation occurs in the liner and/or the surrounding geomaterial. Numerical results indicate that the influences of the internal pressure, liner thickness and rigidity on the yield range are significant. The first principal stress may vary with the variations of the above parameters and is not always the radial stress. In addition, it is shown that the variations of stresses in the surrounding geomaterial intensively rely on the relative liner rigidity, thickness and distance to the tunnel axis when the ratio between the distance of these points under investigation to the tunnel axis and the outer radius of the liner ranges from 1 to 2. Classical Fenner's solution and Lame's solution can be considered as the special cases of the present solutions.
4. It is assumed that all the influence factors surrounding deep-buried circle tunnel are axially symmetrical. Firstly, the solution of fluid flow field is resolved. Secondly, in conjunction with two cases of the material character of the liner, i.e., a permeable liner or an impermeable liner, analytical solutions of elastic stress and displacement are obtained by considering the seepage force in the surrounding geomaterial as a body force. 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. In addition, the results of the liner as a permeable material are compared with those of the liner as an impermeable material.
众所周知,隧洞开挖后地层中初始应力释放,应力重新分布形成二次应力场,从而引起隧洞的变形。为了确保隧洞开挖后的稳定性,工程中常采用衬砌材料进行支护。然而,大多数力学模型可以退化为这样的情形:隧洞开挖,衬砌安装完,而后结构再受到地应力的影响。而事实上,岩(土)体是赋存了一定的初始应力,开挖导致隧洞发生部分变形,而后再安装衬砌,因而,必须提出一种合理的力学模型。
另外,当作用于岩体的初始应力较大或岩体自身的强度较低,洞室开挖后,洞周的部分岩体应力超出了岩体的屈服应力,使岩体进入了塑性状态。随着与洞壁距离的增大,最小主应力也随之增大,进而提高了岩体的强度,并使岩体的应力转化为弹性状态,岩体的二次应力状态呈现弹塑性并存的状态。虽然,各种基于弹性理论的围岩-衬砌干涉作用模型很多,而经典的弹塑性解答中,一般都未考虑衬砌本身对于围岩的制约作用,实际上,围岩压力是通过衬砌和围岩的共同作用来承担的;再者,屈服条件中的第一主应力是以径向应力的情况下导出的,这样做还不够全面,必须根据不同的工况和不同的地应力条件,正确选择屈服条件中的第一主应力,导出衬砌和围岩的屈服范围和应力计算公式,给出屈服区和应力计算公式的适用范围。
许多隧洞,包括新奥法施工的隧洞,需要的支护形式为:初期支护和二期支护。初期支护主要来承受开挖荷载,而二次衬砌支护需要承受隧洞运行期间的附加荷载以及围岩、初期支护强度随时间的降低,或者来自围岩中的其它附加荷载作用。地下水流的变化则是其中的主要附加荷载之一。隧洞突水无论是在隧洞的施工期还是在运行期均可发生。隧洞突水会造成停工、环境破坏以及结构的沉陷等严重问题。另外,为了评估相关的问题,隧洞突水也必须尽量做到提前预警。虽然孔隙水压力对隧洞支护的影响有许多的研究成果,但是,各种水力条件变化下的隧洞支护设计标准仍然急需在这方面进行多方面的深入研究。
弹性问题的复势方法是迄今唯一能从已知到未知顺次求解的分析方法;它统一了弹性力学中的三种基本方法(位移法、应力法、应力函数法)和三类边界(力边界、位移边界、混合边界)问题;而弹性场本身的应力变形特征与复势函数之间必定存在一种确定的关系,找到这种关系无疑对于具体问题中复势函数的求解是重要的。本文的主要研究成果如下:
1.利用基本的平面弹性方程,结合解析函数的性质,首次得到了平面弹性复势中的几个重要性质,其内容为:对于平面、反平面弹性场,当应力状态关于原点对称时,复势为偶函数;当应力状态关于x轴对称时,平面弹性场中的复势具有实系数;当应力状态关于x轴反对称时,反平面弹性场的复势具有虚系数。利用所得基本性质可以简化复势函数的求解过程。
2.根据围岩和衬砌联合作用的机理,研究了深埋圆形衬砌隧洞在多种荷载作用下的弹性应力、位移场的解析解答。利用以上复势函数性质,构造合适的复势函数,然后,根据边界上力、位移的连续性条件,获得了全场复势的解析表达。详细讨论了非静水压力条件下的深埋衬砌压力隧洞的这一情形。结果表明:为了减少隧洞开挖对原有应力、位移场的扰动,施作衬砌是一种有效措施,但是,衬砌的厚度和刚度应该针对具体情况保持在一定范围,过大过小的值都是不合适的。另外发现,隧洞开挖对位移场的影响要远远大于对应力场的影响。就应力场而言,当隧洞埋深和隧洞内径的比值大于5时,就完全可以应用本文的结论;然而,就位移场而言,只有当隧洞埋深和隧洞内径的比值大于20时,本文的结论才可以应用。
3.研究了隧洞处于弹塑性条件的围岩稳定性问题。考虑地应力的释放和衬砌安装的时序,提出合理模拟隧洞修筑过程的力学模型。利用边界上位移和力的连续性条件,得到围岩和衬砌内的复应力函数表达式。根据不同的工况和地应力条件,基于Mohr-Coulomb屈服准则,正确选择屈服条件中的第一主应力,导出衬砌和围岩的屈服范围和应力计算公式,并给出屈服范围和应力计算公式的适用条件。重点抓住影响地下结构稳定性的几个主要参数,即围岩、衬砌的弹性(或变形)模量,地应力侧压系数,衬砌的厚度,以及到洞轴线距离等,来分析其对结构的强度、变形影响。经典的Fenner公式和Lamé解答等许多已往解答,可以作为其特殊情况退化得到。
4.将深埋圆形隧洞各影响因素简化为轴对称。首先,求解得到渗流场;然后,以渗透体积力方式作用在围岩应力场,求解得到衬砌作为渗透材料和不渗透材料情形下的全场位移和应力解析表达式。根据所得解答,可以用来解释各种围岩水力条件以及隧洞排水条件下的隧洞支护压力的不同。综合考虑了衬砌具有渗透性、不具有渗透性两种情形,且对得到的结果进行了对比。
Tunnel is a main underground structure and widely used for transportation, water passage, mining, petroleum, geophysical engineering practice and other purposes such as electricity or communication cable installation. With the development and upgrade of infrastructures, the demand for tunnel construction is increasing all over the world and the importance of the safety and economics of tunnel construction are recognized in tunnel engineering. In general, in relation to tunnel construction, it refers to three issues, which are: (1) maintaining stability and safety during construction; (2) minimizing unfavorable impact on adjacent structures; (3) performing the intended function over the life of a project. Among the issues, the first one is directly related to the appropriate design of tunnel support system.
The excavation process reduces the confining stress to zero on the boundary of the opening. Thus, tunnel excavation always causes stress relief and deformation at the tunnel boundary before the liner is installed, which is well-known. The support system of underground facilities must be designed to withstand static overburden loads as well as to accommodate the additional deformations imposed by the earthquake induced motions. It is essential to understand the interaction between the liner and the surrounding geomaterial with the construction sequence taken into consideration. The real problem is that most of the analyses in the past completely misunderstand the history of stresses on the tunnel, which are essentially a "mechanical engineering" calculation in which one imagines an initially unstressed body (tunnel and liner) and then the stresses are applied at the outer boundaries. In fact of course certain pre-existing stresses exist in the ground. The tunnel is then opened, causing stress relief at the tunnel boundary (and consequent displacements), the liner is then installed, and finally the concrete of the liner cures. The effect of this construction sequence on the stresses in and around the liner was taken no account in the past.
It is normal practice in the design of tunnel liners using surrounding geomaterial-support interaction models to consider that the geo-material behaves elasto-plastically. This means that under loading the material will first behave elastically, but when (and if) the elastic limit or plastic strength of the material (liner and surrounding geomaterial) is reached, then the material will start behaving plastically. In most of the existing elasto-plastic solutions, the first principal stress in the yield range was generally assumed to be the radial stress. In addition, the interaction between the liner and the surrounding geomaterial was not appropriately considered. Although the surrounding geomaterial-liner interaction based on elastic theory has been studied by numerous researchers. Furthermore, the variations of yield radius with the internal pressure acting on the inner boundary of the liner, the liner thickness and rigidity were not adequately discussed in the past literatures. In fact, certain pre-existing stresses exist in the geomaterail before the tunnel is excavated, which causes stress relief at the tunnel boundary (and consequent displacements). The liner is then installed, and finally the concrete of the liner cures. It is essentially important to understand how the surrounding geomaterial around a tunnel deform due to changes in stresses with the construction sequence taken into account properly. Moreover, the variations of the internal pressure acting on the inner boundary of the liner, the liner thickness and rigidity may result in the variations of the yield radius, the first principal stress and the boundary condition. Such factors should be properly considered in order to have a full picture of the elasto-plastic behavior for underground engineering problems.
Many tunnels, including NATM tunnels, require two types of support: primary and secondary support. The role of the primary support is to withstand the loads that may arise during excavation. The secondary support has to withstand those loads that arise from tunnel operation, deterioration of ground strength and primary support with time, or from other changes of stresses in the ground around the tunnel. An important source for such additional loads is the change of groundwater flow around the tunnel. Water ingress into tunnels may be encountered as well in the construction phase of any tunnel as in the operation phase of hydraulic tunnels. Water ingress in the construction and in the operational phases may cause severe difficulties, downtimes and environmental impact by altering the groundwater regime and causing settlements of structures on the surface. To assess the related problems, the water ingress must be somehow predicted in advance. Even though some of the effects of pore water pressure on tunnel support have been investigated, there are many aspects that require further scrutiny, in particular a reasonable design criterion for tunnel support is needed where groundwater flow conditions are included.
As an important analytical method, the complex variable method has been widely used to analyze problems associated with underground construction. The advantage of the complex variable method in solving elastic tunnelling problems is that both the stresses and displacements are solved simultaneously, enabling the solution of problems for various types of boundary conditions, and that the boundary conditions for the displacements and stresses are similar. An important issue is to find the deterministic relations between the stress-deformation conditions and the complex potentials. The work in the present paper can be summarized in the following:
1. Four important properties for the complex potentials expressed in series expansion in the plane and anti-plane elastic fields is obtained by applying the basic equations of plane and anti-plane problems, combining the theory of analytical function, and recalling the feature of stress and deformation. Firstly, the complex potentials in the plane or anti-plane elastic fields are real functions when the stress states are symmetrical with regard to the origin. In addition, the complex potentials in the plane elastic field have real coefficient when the stress states are symmetrical with regard to the x-axis. Finally, the complex potentials in the anti-plane elastic field have imaginary coefficient when the stress states are anti-symmetric with regard to the x-axis. Then, based on the conclusions of the paper, some classic solutions are rearrived at, which indicates the results derived in this paper are right and simplify the process of constructing and solving the complex potential functions. The present conclusions provide an efficient tool to discovery the solutions in the sophisticated state.
2. By applying the series expansion technique in the complex variable method established by Muskhelishvili, the plane elasticity problem for the stress and displacement field around a lined circular tunnel in conjunction with the consideration of misfit and interaction between the liner and the surrounding geomaterial is dealt with. The tunnel is assumed to be driven in a homogeneous and isotropic geomaterial. The coefficients in the Laurent series expansion of the stress functions are determined. The complex potentials in the liner and the surrounding geomaterial are explicitly derived, respectively. An elastic solution is obtained under a wide class of loading conditions, the main limitation being that internal stress sources (if any) are located in the geomaterial. Loading by misfit alone, by in situ stresses, by far-field shear stresses, and by a concentrated force, are explicitly treated as special cases. As an example, the case of a lined circular tunnel located in an isotropic initial stress field but subjected to uniform internal pressure is numerically considered. Numerical results indicate that the installation of tunnel liner can reduce the influences of the tunnel excavation on the in situ displacement and stress fields. However, the relative thickness and rigidity of the liner should be in an appropriate range. In addition, the effect of the tunnel excavation upon the displacement field is more significant than that upon the stress field. As far as the stress field in the surrounding geomaterial is concerned, when the ratio between the cover depth of tunnel and the tunnel radius is larger than 5, the results for the stress field in the paper are applicable. When the ratio between the tunnel depth and the tunnel radius is larger than 20, the results are applicable for the displacement field.
3. The elasto-plastic problem for the stress fields around a lined circular tunnel is investigated in this paper. An appropriate mechanical model considering the stress relief and the installation of the liner is proposed to model the tunnel construction sequence. By considering the continuity conditions for the stresses and displacements along the boundaries, the complex potentials in the liner and the surrounding geomaterial are explicitly derived, respectively. Based on the linear Mohr-Coulomb (M-C) yield criterion, for two cases that the first principal stress is the radial stress or the tangential stress, respectively, the stress solutions are given when plastic deformation occurs in the liner and/or the surrounding geomaterial. Numerical results indicate that the influences of the internal pressure, liner thickness and rigidity on the yield range are significant. The first principal stress may vary with the variations of the above parameters and is not always the radial stress. In addition, it is shown that the variations of stresses in the surrounding geomaterial intensively rely on the relative liner rigidity, thickness and distance to the tunnel axis when the ratio between the distance of these points under investigation to the tunnel axis and the outer radius of the liner ranges from 1 to 2. Classical Fenner's solution and Lame's solution can be considered as the special cases of the present solutions.
4. It is assumed that all the influence factors surrounding deep-buried circle tunnel are axially symmetrical. Firstly, the solution of fluid flow field is resolved. Secondly, in conjunction with two cases of the material character of the liner, i.e., a permeable liner or an impermeable liner, analytical solutions of elastic stress and displacement are obtained by considering the seepage force in the surrounding geomaterial as a body force. 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. In addition, the results of the liner as a permeable material are compared with those of the liner as an impermeable material.
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