基于复变函数方法的水下隧道围岩弹性分析
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摘要
水下隧道属于浅埋隧道,故其围岩的应力位移分析从力学角度可简化为含圆孔半平面弹性体在水平边界上受任意分布荷载的问题。与深埋隧道不同,浅埋隧道围岩分析在数学处理上历来存在较大的困难。借助Verruijt提供的共形映射函数,把含圆孔半无限平面映射为圆环域,然后将像平面上的解析函数展成Laurent级数,利用Muskhelishvili的复变函数解法,求得问题的应力场和位移场,最后用Fortran语言编写了计算程序。利用该程序给出一个水下隧道算例的围岩应力、位移结果,并分析了其受力变形特点。
The underwater tunnels can be regarded as shallow tunnels.So the classical problem of an elastic half plane with a circular cavity,loaded arbitrarily on the horizontal boundary can be used to analyse the stress and displacement fields of the surrounding rock for underwater tunnels.As is known to all,it is much more difficult to deal with shallow tunnels than deep tunnels via the analytical way.Using Verruijt's conformal mapping function,the region of exclusion of a hole in a half plane is tansfered into a cirque.Then the analytic functions can be expanded as Laurent series in this region.The stress and displacement can be solved by the function of complex varibles method founded by Muskhelishvili.The corresponding calculation program is coded by Fortran.As an example,a specific underwater tunnel is analysed by the Fortran calculation program eventually.
The underwater tunnels can be regarded as shallow tunnels.So the classical problem of an elastic half plane with a circular cavity,loaded arbitrarily on the horizontal boundary can be used to analyse the stress and displacement fields of the surrounding rock for underwater tunnels.As is known to all,it is much more difficult to deal with shallow tunnels than deep tunnels via the analytical way.Using Verruijt's conformal mapping function,the region of exclusion of a hole in a half plane is tansfered into a cirque.Then the analytic functions can be expanded as Laurent series in this region.The stress and displacement can be solved by the function of complex varibles method founded by Muskhelishvili.The corresponding calculation program is coded by Fortran.As an example,a specific underwater tunnel is analysed by the Fortran calculation program eventually.
引文
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[18]MUSKHELISHVILI N I.Mathematical theroy of elasticity[M].Leyden:International Publishing,1954.
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[2]王梦恕.台湾海峡海底铁路隧道建设方案[J].隧道建设,2008,28(5):517-526.WANG Meng-shu.Construction scheme of Taiwan Strait subsea railway tunnel[J].Tunnel Construction,2008,28(5):517-526.
[3]薛守义,刘汉东.岩体工程学科性质透视[M].郑州:黄河水利出版社,2002.
[4]李术才,李树忱,徐帮树,等.海底隧道最小岩石覆盖厚度确定方法研究[J].岩石力学与工程学报,2007,26(11):2289-2295.LI Shu-cai,LI Shu-chen,XU Bang-shu,et al.Study on determination method for minimum rock cover of subsea tunnel[J].Chinese Journal of Rock Mechanics and Engineering,2007,26(11):2289-2295.
[5]张顶立,李兵,房倩,等.基于风险系数的海底隧道纵断面确定方法[J].岩石力学与工程学报,2009,28(1):9-19.ZHANG Ding-li,LI Bing,FANG Qian,et al.Method for determining longitudinal section of subsea tunnel based on risk coefficient[J].Chinese Journal of Rock Mechanics and Engineering,2009,28(1):9-19.
[6]蔚立元,徐帮树,李术才,等.确定水下隧道覆盖层厚度的经验公式及其应用[J].地下空间与工程学报,2011,7(3):497-503.YU Li-yuan,XU Bang-shu,LI Shu-cai,et al.Empirical formulas for deciding overburden thickness of underwater tunnel and relevant applications[J].Chinese Journal of Underground Space and Engineering,2011,7(3):497-503.
[7]SAGASETA C.Analysis of undrained soil deformation due to ground loss[J].Geotechnique 1987,37(4):301-326.
[8]JEFFERY G B.Plane stress and plane strain in bipolar coordinates[J].Transactions of the Royal Society of London,(Series A),1920,221:265-293.
[9]MINDLIN R D.Stress distribution around a tunnel[J].Transactions of the ASCE,1940,1117-1153.
[10]MINDLIN R D.Stress distribution around a hole near the edge of a plate under tension[C]//Proceedings of the Society of Experimental Stress Analysis,[S.l.]:[s.n.],1948,5:56-57.
[11]BOBET A.Analytical solutions for shallow tunnels in saturated ground[J].ASCE Journal Engineering Mechanics,2001,127:1258-1266.
[12]CHOU W I,BOBET A.Predictions of ground de-formations in shallow tunnels in clay[J].Tunnelling and Underground Space Technology 2002;17:3-19.
[13]VERRUIJT A,BOOKER J R.Surface settlements due to deformation of a tunnel in an elastic half plane[J].Geotechnique,1996,46:753-756.
[14]VERRUIJT A.Deformations of an elastic half plane with a circular cavity[J].International Journal of Solids and Structures 1998;35(21):2795-2804.
[15]VERRUIJT A.A complex variable solution for a deforming circular tunnel in an elastic half-plane[J].International Journal for Numerical and Analytical Methods in Geomechanics 1997,21:77-89.
[16]STRACK O E,VERRUIJT A.A complex variable solution for a deforming buoyant tunnel in a heavy elastic half-plane[J].International Journal for Numerical and Analytical Methods in Geomechanics,2002;26:1235-1252.
[17]王立忠,吕学金.复变函数分析盾构隧道施工引起的地基变形[J].岩土工程学报,2007,29(3):319-327.WANG Li-zhong,LüXue-jin.A complex variable solution for different kinds of oval deformation around circular runnel in an elastic half plane[J].Chinese Journal of Geotechnical Engineering,2007,29(3):319-327.
[18]MUSKHELISHVILI N I.Mathematical theroy of elasticity[M].Leyden:International Publishing,1954.
[19]刘建亚.复变函数与积分变换[M].北京:高等教育出版社,2005.
[20]朱镜清,周建.海底地震动估计的一个流体力学基础[J]地震工程与工程振动,1991,11(3):87-93.ZHU Jing-qing,ZHOU Jian.A fluid mechanics basis for estimating undersea ground motion[J]Earthquake Engineering and Engineering Vibration,1991,11(3):87-93.
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