双孔平行地铁隧道开挖的复变函数解析解与数值分析
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摘要
随着我国城市基础设施建设的迅速发展，地铁正以其环保、高效带来了良好的社会、经济效益，逐渐成为现代城市交通的主体。受客观条件的限制，地铁隧道往往采用双孔平行隧道形式，且相邻隧道间距较小，相互影响明显，从而产生地表沉降较大、地面房屋开裂、隧道变形过大等一系列复杂问题。由于双孔平行隧道开挖问题的普遍性和复杂性，以及开展的研究尚不够深入，因此，开展城市地铁双孔平行隧道相互影响及对地表沉降影响的研究，具有重要的理论意义和工程应用价值。论文利用数学、力学、数值分析、现场实测分析等多种手段进行研究，主要包括以下七个方面：
(1)利用复变函数中的解析延拓法，并考虑隧道支撑法向力，推导求解地下半空间任意形状、任意布置下的单孔隧洞在弹性半空间内任意一点处的应力值和位移值解析解；以圆形隧道作为简化条件下的特例，求出在弹性半空间内任意一点处的应力值和位移值解析解具体显式表达式。
(2)利用复变函数柯西积分法结合最优化理论，推导地下半空间典型断面的单心圆仰拱马蹄形隧道，在其所在的弹性半空间内任意一点的应力解和位移解的解析解表达式，通过有限元方法验证了公式的正确性。
(3)利用复变函数中的解析延拓法与Schwarz交替法，求解了考虑开挖顺序、原岩应力、隧道管片法向支撑力等因素的影响下，半无限平面内任意形状双孔隧道开挖后围岩的应力和位移解；其后简化为半无限弹性平面内的双孔平行圆形隧道，给出隧道开挖后围岩的应力和位移解的解析解具体表达式，并且通过有限元分析进行了验证。
(4)利用上述得到的双隧道的解析解公式进行参数敏感性分析，研究隧道半径、隧道间距、原岩应力、隧道管片法向支撑力等因素对双隧道周边的应力和位移的影响。分析结果表明原岩应力的变化对隧道周边围岩的应力值和位移值的影响非常大；隧道之间圆心距的影响次之；隧道半径、管片法向支撑力的影响较小。
(5)建立了三维数值模型，研究异步同向开挖、同步同向开挖时隧道之间的相互影响，并探讨其断面收敛曲线特征。结果表明：双孔隧道的分布方式不同，影响距离也不同、收敛曲线特征也不同。
(6)建立了三维数值模型，分析双孔隧道同步开挖在双孔并行、垂直重叠、倾斜重叠三种不同分布情况下，引起地表沉降及隧道周边土体变形的规律；分析不同的隧道分布、间距、埋深对双孔隧道开挖性状的影响。结果表明：垂直分布对隧道施工的影响最大，倾斜分布次之，水平分布的影响最小；随着隧道间距的不断增大，两隧道之间的相互影响就越来越小；随着隧道埋深的不断增大，对隧道施工的影响就越来越大。
(7)以深圳地铁3号线为工程背景，建立数值模型，分析双孔垂直重叠（不考虑建筑物荷载）、倾斜重叠（不考虑建筑物荷载）、垂直重叠（考虑建筑物荷载）三种情况下，隧道开挖顺序对地面沉降规律的影响，提出双孔地铁隧道地面沉降的刚度修正方法，与计算值、实测值进行对比，分析城市地铁双孔隧道地表沉降规律。结果表明：对倾斜重叠或垂直重叠分布的双孔隧道，宜采用先下洞后上洞的开挖顺序，有利于减少地表沉降。
本文的解析解拓展了复变函数解析延拓法的应用，推动了隧道力学理论的发展，为成功解决复杂形状的单孔、双孔隧道开挖问题迈出了重要一步；数值模拟分析全面研究双孔隧道的分布形式，埋深、间距等对隧道施工的影响，具有一定的工程指导意义；提出的计算地表沉降的刚度修正法，综合考虑影响地表沉降的各种因素，计算简单便捷，具有工程适用性和实用性。
With rapid development of China’s urban infrastructure construction,subway brings good social and economic benefits with its environmentallyfriendliness and high efficiency, and is gradually becoming main body formodern urban transportation. Restricted by objective conditions, subwaytunnels commonly adopt two holes parallel tunnel form, with small distancebetween nearby tunnels, and evident inter influences. Hence, a series ofcomplex problems such as surface subsidence, ground housing cracking, andtunnel over-large deformation occur. Given the universality and complexity oftwo holes parallel tunnel digging problem, and superficial research upon it,developing the research on urban subway two holes parallel tunnel interinfluences and surface subsidence impact, is of theoretical significance andengineering application value. The paper adopts various methods ofmathematics, mechanics, numerical analysis, field measurement analysis toconduct research, including the following seven aspects:
(1)Utilize analytical continuation of complex variables function, considertunnel support force, deduce the stress value and displacement valueanalytical solution of the underground half space arbitrary shape and arbitrarysetting single hold tunnel in elastic half space at an arbitrary point; take roundtunnel as special case under simplified conditions, calculate the stress valueand displacement value analytical solution concrete explicit formula of anarbitrary point in elastic half space.
(2)Utilize complex variables functions Cauchy integral test, combinewith optimized theories, deduce the stress value and displacement valueanalytical solution formula of an arbitrary point in elastic half space of thesingle heart round inverted arch U-shape tunnel at underground half spacesection, and verify the correctness of the formula via finite element method.
(3)Utilize analytical continuation of complex variables functions andSchwarz alternating method, solve surrounding rock stress and displacement after half infinite plane arbitrary form two holes tunnel digging, under theinfluence of digging sequence, virgin rock stress, tunnel duct piece normalforce. Simplify to two holes parallel round tunnel in half infinite plane,present surrounding rock stress and displacement concrete formula, and verifyvia finite element method.
(4)Utilize double tunnel analytical solution formula obtained from above,conduct parameter sensitivity analysis, and study the influence of tunnelradius, tunnel spacing, virgin rock stress, tunnel duct piece normal support ondouble tunnel surrounding stress and displacement. Analysis shows that thechanges of virgin rock stress have huge influence on tunnel surrounding rockstress value and displacement; next comes the influence of center distancebetween tunnels; and last are the influence of tunnel radius, duct piece normalsupport.
(5)Establish three-dimensional numerical model, study the interinfluences of asynchronous syntropy digging and synchronized syntropydigging, and discuss the features of section convergence curve. Results show,different two holes tunnels distributions have different influence distance anddifferent convergence curve features.
(6)Establish three-dimensional numerical model, analyze the rulessurface subsidence and soil deformation around the tunnels caused by twoholes tunnel synchronized digging in the three distributions of two holesparallel, vertical overlap, tilted overlap. Analyze the influence of differenttunnels distributions, spacing, depth on two holes tunnel digging. Resultsshow, vertical distribution has the biggest impact on tunnel construction,tilted distribution has less impact, and horizontal distribution has the leastimpact. With the increase in tunnel spacing, the inter influences of the twotunnels become smaller. With the increase in tunnel depth, the impact ontunnel construction grows.
(7)Take Shenzhen Line3Subway as project background, establish valuemodel, analyze the impact of tunnel digging sequence on surface subsidence,under the three conditions of two holes vertical overlap (regardless of building load), tilted overlap (regardless of building load), vertical overlap(taking building load into consideration), proposes rigidity correction methodfor two holes subway tunnel surface subsidence, compare calculation valuewith actual measurement value, and analyze rules of urban subway two holestunnel surface subsidence. Results show, for tilted overlap or verticaloverlap distribution two holes tunnels, it is better to adopt digging sequenceof lower cave first, upper cave later so as to reduce surface subsidence.
The analytical solution of the paper expands the application of complexvariables function analytic continuation method, promote the development oftunnel mechanics theory, and takes an important step for the successfulresolution to the complex form single hole and two holes tunnel diggingproblem. Numerical simulation analysis compressively studies the influenceof distribution forms, depth, spacing of two holes tunnel on tunnelconstruction, which possesses some engineering guidance significance. Thepaper also proposes rigidity correction method for calculating surfacesubsidence, taking consideration of various factors affecting surfacesubsidence. The calculation is simple and convenient, with engineeringapplicability and practicality.
(1)利用复变函数中的解析延拓法，并考虑隧道支撑法向力，推导求解地下半空间任意形状、任意布置下的单孔隧洞在弹性半空间内任意一点处的应力值和位移值解析解；以圆形隧道作为简化条件下的特例，求出在弹性半空间内任意一点处的应力值和位移值解析解具体显式表达式。
(2)利用复变函数柯西积分法结合最优化理论，推导地下半空间典型断面的单心圆仰拱马蹄形隧道，在其所在的弹性半空间内任意一点的应力解和位移解的解析解表达式，通过有限元方法验证了公式的正确性。
(3)利用复变函数中的解析延拓法与Schwarz交替法，求解了考虑开挖顺序、原岩应力、隧道管片法向支撑力等因素的影响下，半无限平面内任意形状双孔隧道开挖后围岩的应力和位移解；其后简化为半无限弹性平面内的双孔平行圆形隧道，给出隧道开挖后围岩的应力和位移解的解析解具体表达式，并且通过有限元分析进行了验证。
(4)利用上述得到的双隧道的解析解公式进行参数敏感性分析，研究隧道半径、隧道间距、原岩应力、隧道管片法向支撑力等因素对双隧道周边的应力和位移的影响。分析结果表明原岩应力的变化对隧道周边围岩的应力值和位移值的影响非常大；隧道之间圆心距的影响次之；隧道半径、管片法向支撑力的影响较小。
(5)建立了三维数值模型，研究异步同向开挖、同步同向开挖时隧道之间的相互影响，并探讨其断面收敛曲线特征。结果表明：双孔隧道的分布方式不同，影响距离也不同、收敛曲线特征也不同。
(6)建立了三维数值模型，分析双孔隧道同步开挖在双孔并行、垂直重叠、倾斜重叠三种不同分布情况下，引起地表沉降及隧道周边土体变形的规律；分析不同的隧道分布、间距、埋深对双孔隧道开挖性状的影响。结果表明：垂直分布对隧道施工的影响最大，倾斜分布次之，水平分布的影响最小；随着隧道间距的不断增大，两隧道之间的相互影响就越来越小；随着隧道埋深的不断增大，对隧道施工的影响就越来越大。
(7)以深圳地铁3号线为工程背景，建立数值模型，分析双孔垂直重叠（不考虑建筑物荷载）、倾斜重叠（不考虑建筑物荷载）、垂直重叠（考虑建筑物荷载）三种情况下，隧道开挖顺序对地面沉降规律的影响，提出双孔地铁隧道地面沉降的刚度修正方法，与计算值、实测值进行对比，分析城市地铁双孔隧道地表沉降规律。结果表明：对倾斜重叠或垂直重叠分布的双孔隧道，宜采用先下洞后上洞的开挖顺序，有利于减少地表沉降。
本文的解析解拓展了复变函数解析延拓法的应用，推动了隧道力学理论的发展，为成功解决复杂形状的单孔、双孔隧道开挖问题迈出了重要一步；数值模拟分析全面研究双孔隧道的分布形式，埋深、间距等对隧道施工的影响，具有一定的工程指导意义；提出的计算地表沉降的刚度修正法，综合考虑影响地表沉降的各种因素，计算简单便捷，具有工程适用性和实用性。
With rapid development of China’s urban infrastructure construction,subway brings good social and economic benefits with its environmentallyfriendliness and high efficiency, and is gradually becoming main body formodern urban transportation. Restricted by objective conditions, subwaytunnels commonly adopt two holes parallel tunnel form, with small distancebetween nearby tunnels, and evident inter influences. Hence, a series ofcomplex problems such as surface subsidence, ground housing cracking, andtunnel over-large deformation occur. Given the universality and complexity oftwo holes parallel tunnel digging problem, and superficial research upon it,developing the research on urban subway two holes parallel tunnel interinfluences and surface subsidence impact, is of theoretical significance andengineering application value. The paper adopts various methods ofmathematics, mechanics, numerical analysis, field measurement analysis toconduct research, including the following seven aspects:
(1)Utilize analytical continuation of complex variables function, considertunnel support force, deduce the stress value and displacement valueanalytical solution of the underground half space arbitrary shape and arbitrarysetting single hold tunnel in elastic half space at an arbitrary point; take roundtunnel as special case under simplified conditions, calculate the stress valueand displacement value analytical solution concrete explicit formula of anarbitrary point in elastic half space.
(2)Utilize complex variables functions Cauchy integral test, combinewith optimized theories, deduce the stress value and displacement valueanalytical solution formula of an arbitrary point in elastic half space of thesingle heart round inverted arch U-shape tunnel at underground half spacesection, and verify the correctness of the formula via finite element method.
(3)Utilize analytical continuation of complex variables functions andSchwarz alternating method, solve surrounding rock stress and displacement after half infinite plane arbitrary form two holes tunnel digging, under theinfluence of digging sequence, virgin rock stress, tunnel duct piece normalforce. Simplify to two holes parallel round tunnel in half infinite plane,present surrounding rock stress and displacement concrete formula, and verifyvia finite element method.
(4)Utilize double tunnel analytical solution formula obtained from above,conduct parameter sensitivity analysis, and study the influence of tunnelradius, tunnel spacing, virgin rock stress, tunnel duct piece normal support ondouble tunnel surrounding stress and displacement. Analysis shows that thechanges of virgin rock stress have huge influence on tunnel surrounding rockstress value and displacement; next comes the influence of center distancebetween tunnels; and last are the influence of tunnel radius, duct piece normalsupport.
(5)Establish three-dimensional numerical model, study the interinfluences of asynchronous syntropy digging and synchronized syntropydigging, and discuss the features of section convergence curve. Results show,different two holes tunnels distributions have different influence distance anddifferent convergence curve features.
(6)Establish three-dimensional numerical model, analyze the rulessurface subsidence and soil deformation around the tunnels caused by twoholes tunnel synchronized digging in the three distributions of two holesparallel, vertical overlap, tilted overlap. Analyze the influence of differenttunnels distributions, spacing, depth on two holes tunnel digging. Resultsshow, vertical distribution has the biggest impact on tunnel construction,tilted distribution has less impact, and horizontal distribution has the leastimpact. With the increase in tunnel spacing, the inter influences of the twotunnels become smaller. With the increase in tunnel depth, the impact ontunnel construction grows.
(7)Take Shenzhen Line3Subway as project background, establish valuemodel, analyze the impact of tunnel digging sequence on surface subsidence,under the three conditions of two holes vertical overlap (regardless of building load), tilted overlap (regardless of building load), vertical overlap(taking building load into consideration), proposes rigidity correction methodfor two holes subway tunnel surface subsidence, compare calculation valuewith actual measurement value, and analyze rules of urban subway two holestunnel surface subsidence. Results show, for tilted overlap or verticaloverlap distribution two holes tunnels, it is better to adopt digging sequenceof lower cave first, upper cave later so as to reduce surface subsidence.
The analytical solution of the paper expands the application of complexvariables function analytic continuation method, promote the development oftunnel mechanics theory, and takes an important step for the successfulresolution to the complex form single hole and two holes tunnel diggingproblem. Numerical simulation analysis compressively studies the influenceof distribution forms, depth, spacing of two holes tunnel on tunnelconstruction, which possesses some engineering guidance significance. Thepaper also proposes rigidity correction method for calculating surfacesubsidence, taking consideration of various factors affecting surfacesubsidence. The calculation is simple and convenient, with engineeringapplicability and practicality.
引文
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