圆形偏压隧道应力场的弹性解析
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摘要
中国地域辽阔,地形复杂多样,高等级公路建设中遇到大量的隧道工程问题。受地形的限制和影响,越来越多的隧道出现偏压问题,其理论解析还存在很多难点。本文基于平面弹性复变理论,对地形引起的圆形偏压隧道二次应力场进行了理论解析。主要研究内容和成果包括：
(1)将偏压隧道问题应力场的求解分解为三部分。第一部分为偏压隧道的初始应力场；第二部分为体力为零的半平面内作用一个集中力时的应力场,该集中力大小等于开挖洞室部分材料的重力；第三部分为隧道洞周作用一应力张量时的应力场,该应力张量是为了平衡前两部分求解结果以满足应力边界条件。
(2)采用平面弹性复变理论和Fourier变换,首次得到了圆形偏压隧道的应力场解析解。引入Mobius保角映射,将所求域映射到一圆环内,分别求出三部分情况下的应力场。最后将三部分结果叠加,得到该问题的闭合形式解。
(3)采用本文介绍的方法,对圆形偏压隧道应力场进行参数分析。编写计算程序,实现对问题求解的程序化。对势函数的收敛性进行了探讨。系数的可根据边界条件求得,根据洞周和上边界的应力边界条件,结合无穷远处的位移边界条件可知,在变量k值趋近于无穷大时,系数的ak值应为零,根据这个条件可对系数进行迭代求解出ak,然后根据边界条件求得其他系数,即求出了势函数。由于应力场必须收敛,则要求势函数必须收敛,势函数的系数必须收敛。通过系数的求解,发现势函数的系数随着变量k值的增大急剧收敛,在k=6时,系数的实部和虚部都已经收敛,且大小趋近于零。
(4)采用本文介绍的方法,对圆形偏压隧道应力场进行参数分析。探讨了材料和几何参数对计算结果的影响,其中重点分析了泊松比、自由半平面的倾角及隧道埋深等因素的影响。计算结果表明,随着倾角的增大,隧道洞周应力越大,但是最大值一般都出现在洞顶的两侧附近区域；随着埋深的减小,应力在地面处受影响的范围越小；泊松比对偏压隧道应力场的影响较小。
China has a vast territory, and a complex and diverse topography, high-grade roads construction engineering encountered in a large number of problems in tunnels. Subject to the restrictions and effects of topography, more and more tunnels appears the problems of unsymmetrical pressure problem, the theoretical analysis of this problem still has a lot of difficulties. Based on plane elastic complex variable theory,this Article theoretically analysised the secondary stress distribution caused by topography in circular unsymmetrical tunnel. The main contents and results include:
(1) In this study, the integral stress distribution is divided into three sub-distributions. The first one is the initial stress distribution in the unsymmetrical tunnels; and the second is that of a concentrates stress, which is equal to the weight of materials dug out, in a half plane excluding the field of body forces. The third is that caused by stresses weighting on the tunnel's periphery. This stress tension is used to balance the previous two segments and to adaptive the whole distribution to satisfy the initial boundary conditions.
(2) Using plane elastic complex variable method and of Fourior transform, the stress distribution in round unsymmetrical tunnels is obtained at the first time. And the Mobius conformal mapping is adopted. The solution domain is represented by a circle and the three sub-distributions are obtained. The closed form is gained by summing them.
(3) Using the methods this article describes, the parameters of the circular bias tunnel's stress field can be got. Writing programs and procedures to achieve the solution of the problem. The convergence of the potential function is discussed. Coefficients can be obtained under the boundary conditions, according to tunnel on the border weeks and the stress boundary conditions, combined infinite and known boundary conditions, the value of the variable k approaches infinity, the coefficient of ak value should be zero, according to the conditions of the coefficients can be solved out of iterative ak, and then obtained the boundary conditions of other factors, to derive the potential function. As the stress field must be convergence, requires the potential function must converge, the coefficient of the potential function must converge. Through coefficient is found that the potential function of the coefficient of variable k with the rapid increase of the value of convergence, in the k= 6, the coefficient of the real and imaginary parts have convergence, and the size of the near zero.
(4)Last, apply the methods this article describes to analysis stress field parameters of the circular unsymmetrical tunnel. Investigate the influence of material and geometric parameters on the calculation results, which focus on analysis of Poisson's ratio, free half-plane angle and tunnel depth and some other factors. The results show that with increasing inclination, the greater the tunnel on stress, but the maximum generally appears in the top of the cave near the region on both sides; as the depth decreases, the stress get smaller infuence area in the surface; Poisson's ratio has little effect on the stress distribution of unsymmetrical tunnel.
(1)将偏压隧道问题应力场的求解分解为三部分。第一部分为偏压隧道的初始应力场；第二部分为体力为零的半平面内作用一个集中力时的应力场,该集中力大小等于开挖洞室部分材料的重力；第三部分为隧道洞周作用一应力张量时的应力场,该应力张量是为了平衡前两部分求解结果以满足应力边界条件。
(2)采用平面弹性复变理论和Fourier变换,首次得到了圆形偏压隧道的应力场解析解。引入Mobius保角映射,将所求域映射到一圆环内,分别求出三部分情况下的应力场。最后将三部分结果叠加,得到该问题的闭合形式解。
(3)采用本文介绍的方法,对圆形偏压隧道应力场进行参数分析。编写计算程序,实现对问题求解的程序化。对势函数的收敛性进行了探讨。系数的可根据边界条件求得,根据洞周和上边界的应力边界条件,结合无穷远处的位移边界条件可知,在变量k值趋近于无穷大时,系数的ak值应为零,根据这个条件可对系数进行迭代求解出ak,然后根据边界条件求得其他系数,即求出了势函数。由于应力场必须收敛,则要求势函数必须收敛,势函数的系数必须收敛。通过系数的求解,发现势函数的系数随着变量k值的增大急剧收敛,在k=6时,系数的实部和虚部都已经收敛,且大小趋近于零。
(4)采用本文介绍的方法,对圆形偏压隧道应力场进行参数分析。探讨了材料和几何参数对计算结果的影响,其中重点分析了泊松比、自由半平面的倾角及隧道埋深等因素的影响。计算结果表明,随着倾角的增大,隧道洞周应力越大,但是最大值一般都出现在洞顶的两侧附近区域；随着埋深的减小,应力在地面处受影响的范围越小；泊松比对偏压隧道应力场的影响较小。
China has a vast territory, and a complex and diverse topography, high-grade roads construction engineering encountered in a large number of problems in tunnels. Subject to the restrictions and effects of topography, more and more tunnels appears the problems of unsymmetrical pressure problem, the theoretical analysis of this problem still has a lot of difficulties. Based on plane elastic complex variable theory,this Article theoretically analysised the secondary stress distribution caused by topography in circular unsymmetrical tunnel. The main contents and results include:
(1) In this study, the integral stress distribution is divided into three sub-distributions. The first one is the initial stress distribution in the unsymmetrical tunnels; and the second is that of a concentrates stress, which is equal to the weight of materials dug out, in a half plane excluding the field of body forces. The third is that caused by stresses weighting on the tunnel's periphery. This stress tension is used to balance the previous two segments and to adaptive the whole distribution to satisfy the initial boundary conditions.
(2) Using plane elastic complex variable method and of Fourior transform, the stress distribution in round unsymmetrical tunnels is obtained at the first time. And the Mobius conformal mapping is adopted. The solution domain is represented by a circle and the three sub-distributions are obtained. The closed form is gained by summing them.
(3) Using the methods this article describes, the parameters of the circular bias tunnel's stress field can be got. Writing programs and procedures to achieve the solution of the problem. The convergence of the potential function is discussed. Coefficients can be obtained under the boundary conditions, according to tunnel on the border weeks and the stress boundary conditions, combined infinite and known boundary conditions, the value of the variable k approaches infinity, the coefficient of ak value should be zero, according to the conditions of the coefficients can be solved out of iterative ak, and then obtained the boundary conditions of other factors, to derive the potential function. As the stress field must be convergence, requires the potential function must converge, the coefficient of the potential function must converge. Through coefficient is found that the potential function of the coefficient of variable k with the rapid increase of the value of convergence, in the k= 6, the coefficient of the real and imaginary parts have convergence, and the size of the near zero.
(4)Last, apply the methods this article describes to analysis stress field parameters of the circular unsymmetrical tunnel. Investigate the influence of material and geometric parameters on the calculation results, which focus on analysis of Poisson's ratio, free half-plane angle and tunnel depth and some other factors. The results show that with increasing inclination, the greater the tunnel on stress, but the maximum generally appears in the top of the cave near the region on both sides; as the depth decreases, the stress get smaller infuence area in the surface; Poisson's ratio has little effect on the stress distribution of unsymmetrical tunnel.
引文
[1]刘佑荣,唐辉明.岩体力学.武汉：中国地质大学大学出版社,1998,125-130
[2]王磊.地质偏压隧道围岩压力分布及衬砌安全性分析：[西南交通大学硕士学位论文].成都：西南交通大学桥梁与隧道工程系,2008,1-7
[3]黄强.小净距公路隧道施工中围岩地应力分析研究：[武汉理工大学硕士学位论文].武汉：武汉理工大学,2009,1-6
[4]于跃勋.地质顺层偏压隧道施工力学研究：[西南交通大学硕士学位论文]成都：西南交通大学桥梁与隧道工程系,2004,1-8
[5]李志业,曾艳华.地下结构设计原理与方法.成都：西南交通大学大学出版社,2003,21-23
[6]房营光,孙钧.地面荷载下浅埋隧道围岩的粘弹性应力和变形分析.岩石力学与工程学报,1998,17(3)：239-247
[7]徐芝纶.弹性力学(上册).北京：人民教育出版社,1982,132-156
[8]黄文彬,曾国平.论弹性理论中半无限平面问题的一些解答.力学与实践,1992,14(2)：40-41
[9]薛福林.也论弹性力学中半无限平面问题的一些解答.力学与实践,1996,18(3)：59-61
[10]禹奇才,倪国荣.弹性力学中某些问题的探讨.长沙铁道学院学报,1991,9(1)：106-110
[11]王兵,谢锦昌.偏压隧道模型试验及可靠分析.工程力学,1998,15(1)：85-92
[12]王磊,徐丽芬.偏压隧道围岩压力与初期支护受力数值分析.路基工程,2009,(6)：156-158
[13]王省哲,怡晓玲.弹性力学问题复变函数的应用与发展.力学与实践,2008,30：110-113
[14]路可见.平面弹性复变方法.武汉：武汉大学出版社,1986,78-86
[15]Muskhelishvili N. Some Basis Problems of the Mathmatical Theory of Elasticity. Russian,1953,232-241
[16]Mindlin R. Stress distribution around a tunnel. Trans,ASCE.1940:1117-1153
[17]Verruijt A., Booker. Surface settlements due to deformation of a tunnel in an elastic half plane. Geotechnique,1996,46(4):753-756
[18]Verruijt A. A complex variable solution for a deforming circular tunnel in an elastic half plane.Int. J.Number. Methods Geomechanics,1997,21:77-89
[19]Verruijt A. Deformation of an elastic half plane with acircular cavity. International Journal of Solids and Structures,1998,35(21):2795-2804
[20]Verruijt A. Buoyancy of tunnel in sofe soils. Geotechnique,1996,58(6): 513-515
[21]王立忠,郭东杰.偏压隧道二次应力场分析及应用力学与实践.力学与实践,2000,22,25-27
[22]王立忠,郭东杰.倾斜地表下深埋隧道受力弹性分析.地下空间,1997,17(4)：211-215,266.
[23]王立忠,冯永兵.倾斜坡体中水工高压隧道围岩应力场特性分析.岩石力学与工程学报,2004,23(23)：4038-4046
[24]Li Shu-cai, Wang Ming-bin. An elastic stress-displacement solution for a liner tunnel at great depth. International Journal of Rock Mechanics&Mining Sciences,2007,7:1-10
[25]Li Shu-cai, Wang Ming-bin. Elastic analysis of stress-displacement field for a lined circular tunnel at great depth due to loads and internal pressure.Tunnelling and Underground Space Technology,2008,23:609-610.
[26]Li Shu-cai, Wang Ming-bin. An elastic stress-displacement solution for a lined circular tunnel at great depth. International Journal of Rock Mechanics&Mining Sciences,2008,45:486-494
[27]童磊,谢康和,程永峰,等.考虑椭圆化地层变形影响的浅埋隧道弹性解.岩土力学,2009,30(2)：393-398
[28]陆文超.地面荷载下浅埋隧道围岩应力的复变函数解法.江南大学学报(自然科学版),2002,1(4)：409-413
[29]姜勇,朱合华.岩石偏压隧道的动态分析及相关研究.地下空间,2004,24(3)：312-331
[30]刘铭.高速公路浅埋偏压隧道力学计算分析.公路工程,2007,32(5)：1 35-138
[31]杨小礼,李亮,刘宝琛.偏压隧道结构稳定性评价的信息优化分析.岩土力学与工程学报,2002,21(4)：484-488
[32]王兵,谢锦昌.偏压隧道模型试验及可靠分析.工程力学,1998,15(1)：85-92
[33]潘洪科,杨林德,黄慷.公路隧道偏压效应与衬砌裂缝的研究.岩石力学与工程学报,2005,24(18)：3311-3315
[34]杨峰,阳军生.浅埋隧道围岩压力确定的极限分析方法.工程力学,2008,25(7)：179-184
[35]沈习文,邓荣贵,邓林.小净距偏压公路隧道围岩应力分析.路基工程,2007,4：90-91
[36]沈习文.小净距偏压公路隧道围岩压力分析：[西南交通大学硕士学位论文]成都：西南交通大学桥梁与隧道工程系,2007,1-15
[37]李毕华,李永盛,张亮,等.错台偏压小间距隧道施工现场监测研究.岩土工程界,2007,(3)：68-69
[38]梁波,王志勇.浅埋偏压隧道地震动力特性及稳定性分析.公路交通技术,2008,(6)：96-99
[39]汪宏,蒋超.浅埋偏压隧道洞口坍方数值分析与处治.岩土力学,2009,30(11)：3481-3485
[40]高飞,李云鹏.长哨浅埋偏压隧道施工顺序与支护力学行为分析.隧道建设,2009,29(11)：19-23
[41]胥润东,徐林生.隧道围岩应力的线弹性分析.岩土工程技术,2005,19(3)：127-129
[42]樊大钧等.空间弹性力学—复变函数论的应用.成都：四川科学技术出版社,1985,86-93
[43]钟阳,郭大智,张肖宁.对称弹性半空间问题一般解的新方法.哈尔滨建筑大学学报,1995,28(2)：23-27
[44]贾普荣.弹性力学空间问题的复变函数解.力学与实践,1995,17(3)：73-75
[45]卜万奎,董正筑,卢爱红.弹性理论中半无限平面问题的应力解答.黑龙江大学自然科学学报,2008,25(4)：452-454,458
[46]艾志勇,杨敏.多层地基内部作用一水平力时的扩展Mindlin解.同济大学学报,2000,28(3)：272-276
[47]胡晨,王建华.水平集中荷载作用下两层弹性半空间的基本解.上海交通大学学报,2001,35(4)：618-620
[48]赵凯,刘长武,张国良.用弹性力学的复变函数法求解矩形硐室周边应力.采矿与安全工程学报,2007,24(3)：361-365
[49]王志良,杨影丹,张立翔.用计算复变函数法求解含圆孔板的应力场.昆明理工大学学报(理工版),2007,32(2)：73-77
[50]杨丽红,何蕴增.无限平面矩形开孔的应力场分析.哈尔滨工程大学学报,2002,23(2)：106-110
[51]龙述尧,许敬晓.弹性力学问题的局部边界积分方程方法.力学学报,2000,32(5)：566-577
[52]文国庆,陈华荣.弹性半空间地基上的板载轴对称荷载作用下的一般解.岩土工程学报,1989,11(5)：109-119
[53]杨小礼.浅埋小净距偏压隧道施工工序的数值分析.中南大学学报(自然科学版),2007,38(4)：764-770
[54]王祥秋,杨林德,高文华.高速公路偏压隧道施工动态监测与有限元仿真模拟.岩石力学与工程学报,2005,24(2)：284-289
[55]毕继红,钟建辉,丛蓉.浅埋洞室围岩压力有限元分析.铁道工程学报,2004,(4)：77-81,106
[56]马步方,王宝先.公路偏压隧道施工监控量测与模拟分析.企业技术开发,2009,28(1)：44-46,56
[57]张敏,黄润秋.浅埋偏压隧道出口变形机理及稳定性分析.工程地质学报,2008,16(4)：482-488
[58]胡光进,刘登聪.公路双连拱隧道常见病害防治.探矿工程.2008,4：69-73
[59]段方情,张可能,胡新红等.某偏压隧道洞口段施工出现的问题及处理.西部探矿工程,2008,5：150-152
[60]孙钧,侯学渊.地下结构(上册).北京：科学出版社,1987
[61]Sneddon D. Fourier Transforms. Mcgraw-hill Book Company,INC. New York, 1951,35-48
[62]Fryba L. Vibration of solids and structures under moving loads. Publishing House of the Czechoslovak Academy of Sciences Prague,1963,89-96
[63]Poulous H.G., Davis E.H. Elastic solution for soil and rock mechanics. John Wiley&Sons Incorporatde,1974,173-179
[64]弗洛林.土力学原理.北京：中国工业出版社,1965,125-132
[65]张雯雯,刘黎平.一种简化的分数阶傅里叶变换及其应用.数据采集与处理,2009,24(6)：814-818
[66]刘齐建.软土地铁建筑结构抗震设计计算理论的研究：[同济大学博士学位论文].上海：同济大学,2005,85-99
[67]Verruijt A, Booker R. Complex variable solution of Mindlin's problem of an excavated tunnel. http://geo.verruijt.net,2007-8
[68]Verruijt A, Booker R. Complex variable analysis of Mindlin's tunnel problem. http://geo.verruijt.net,2007-8
[69]Verruijt A, Booker R. Complex variable solutions of elastic tunneling problems. http://geo.verruijt.net,2007-8
[2]王磊.地质偏压隧道围岩压力分布及衬砌安全性分析：[西南交通大学硕士学位论文].成都：西南交通大学桥梁与隧道工程系,2008,1-7
[3]黄强.小净距公路隧道施工中围岩地应力分析研究：[武汉理工大学硕士学位论文].武汉：武汉理工大学,2009,1-6
[4]于跃勋.地质顺层偏压隧道施工力学研究：[西南交通大学硕士学位论文]成都：西南交通大学桥梁与隧道工程系,2004,1-8
[5]李志业,曾艳华.地下结构设计原理与方法.成都：西南交通大学大学出版社,2003,21-23
[6]房营光,孙钧.地面荷载下浅埋隧道围岩的粘弹性应力和变形分析.岩石力学与工程学报,1998,17(3)：239-247
[7]徐芝纶.弹性力学(上册).北京：人民教育出版社,1982,132-156
[8]黄文彬,曾国平.论弹性理论中半无限平面问题的一些解答.力学与实践,1992,14(2)：40-41
[9]薛福林.也论弹性力学中半无限平面问题的一些解答.力学与实践,1996,18(3)：59-61
[10]禹奇才,倪国荣.弹性力学中某些问题的探讨.长沙铁道学院学报,1991,9(1)：106-110
[11]王兵,谢锦昌.偏压隧道模型试验及可靠分析.工程力学,1998,15(1)：85-92
[12]王磊,徐丽芬.偏压隧道围岩压力与初期支护受力数值分析.路基工程,2009,(6)：156-158
[13]王省哲,怡晓玲.弹性力学问题复变函数的应用与发展.力学与实践,2008,30：110-113
[14]路可见.平面弹性复变方法.武汉：武汉大学出版社,1986,78-86
[15]Muskhelishvili N. Some Basis Problems of the Mathmatical Theory of Elasticity. Russian,1953,232-241
[16]Mindlin R. Stress distribution around a tunnel. Trans,ASCE.1940:1117-1153
[17]Verruijt A., Booker. Surface settlements due to deformation of a tunnel in an elastic half plane. Geotechnique,1996,46(4):753-756
[18]Verruijt A. A complex variable solution for a deforming circular tunnel in an elastic half plane.Int. J.Number. Methods Geomechanics,1997,21:77-89
[19]Verruijt A. Deformation of an elastic half plane with acircular cavity. International Journal of Solids and Structures,1998,35(21):2795-2804
[20]Verruijt A. Buoyancy of tunnel in sofe soils. Geotechnique,1996,58(6): 513-515
[21]王立忠,郭东杰.偏压隧道二次应力场分析及应用力学与实践.力学与实践,2000,22,25-27
[22]王立忠,郭东杰.倾斜地表下深埋隧道受力弹性分析.地下空间,1997,17(4)：211-215,266.
[23]王立忠,冯永兵.倾斜坡体中水工高压隧道围岩应力场特性分析.岩石力学与工程学报,2004,23(23)：4038-4046
[24]Li Shu-cai, Wang Ming-bin. An elastic stress-displacement solution for a liner tunnel at great depth. International Journal of Rock Mechanics&Mining Sciences,2007,7:1-10
[25]Li Shu-cai, Wang Ming-bin. Elastic analysis of stress-displacement field for a lined circular tunnel at great depth due to loads and internal pressure.Tunnelling and Underground Space Technology,2008,23:609-610.
[26]Li Shu-cai, Wang Ming-bin. An elastic stress-displacement solution for a lined circular tunnel at great depth. International Journal of Rock Mechanics&Mining Sciences,2008,45:486-494
[27]童磊,谢康和,程永峰,等.考虑椭圆化地层变形影响的浅埋隧道弹性解.岩土力学,2009,30(2)：393-398
[28]陆文超.地面荷载下浅埋隧道围岩应力的复变函数解法.江南大学学报(自然科学版),2002,1(4)：409-413
[29]姜勇,朱合华.岩石偏压隧道的动态分析及相关研究.地下空间,2004,24(3)：312-331
[30]刘铭.高速公路浅埋偏压隧道力学计算分析.公路工程,2007,32(5)：1 35-138
[31]杨小礼,李亮,刘宝琛.偏压隧道结构稳定性评价的信息优化分析.岩土力学与工程学报,2002,21(4)：484-488
[32]王兵,谢锦昌.偏压隧道模型试验及可靠分析.工程力学,1998,15(1)：85-92
[33]潘洪科,杨林德,黄慷.公路隧道偏压效应与衬砌裂缝的研究.岩石力学与工程学报,2005,24(18)：3311-3315
[34]杨峰,阳军生.浅埋隧道围岩压力确定的极限分析方法.工程力学,2008,25(7)：179-184
[35]沈习文,邓荣贵,邓林.小净距偏压公路隧道围岩应力分析.路基工程,2007,4：90-91
[36]沈习文.小净距偏压公路隧道围岩压力分析：[西南交通大学硕士学位论文]成都：西南交通大学桥梁与隧道工程系,2007,1-15
[37]李毕华,李永盛,张亮,等.错台偏压小间距隧道施工现场监测研究.岩土工程界,2007,(3)：68-69
[38]梁波,王志勇.浅埋偏压隧道地震动力特性及稳定性分析.公路交通技术,2008,(6)：96-99
[39]汪宏,蒋超.浅埋偏压隧道洞口坍方数值分析与处治.岩土力学,2009,30(11)：3481-3485
[40]高飞,李云鹏.长哨浅埋偏压隧道施工顺序与支护力学行为分析.隧道建设,2009,29(11)：19-23
[41]胥润东,徐林生.隧道围岩应力的线弹性分析.岩土工程技术,2005,19(3)：127-129
[42]樊大钧等.空间弹性力学—复变函数论的应用.成都：四川科学技术出版社,1985,86-93
[43]钟阳,郭大智,张肖宁.对称弹性半空间问题一般解的新方法.哈尔滨建筑大学学报,1995,28(2)：23-27
[44]贾普荣.弹性力学空间问题的复变函数解.力学与实践,1995,17(3)：73-75
[45]卜万奎,董正筑,卢爱红.弹性理论中半无限平面问题的应力解答.黑龙江大学自然科学学报,2008,25(4)：452-454,458
[46]艾志勇,杨敏.多层地基内部作用一水平力时的扩展Mindlin解.同济大学学报,2000,28(3)：272-276
[47]胡晨,王建华.水平集中荷载作用下两层弹性半空间的基本解.上海交通大学学报,2001,35(4)：618-620
[48]赵凯,刘长武,张国良.用弹性力学的复变函数法求解矩形硐室周边应力.采矿与安全工程学报,2007,24(3)：361-365
[49]王志良,杨影丹,张立翔.用计算复变函数法求解含圆孔板的应力场.昆明理工大学学报(理工版),2007,32(2)：73-77
[50]杨丽红,何蕴增.无限平面矩形开孔的应力场分析.哈尔滨工程大学学报,2002,23(2)：106-110
[51]龙述尧,许敬晓.弹性力学问题的局部边界积分方程方法.力学学报,2000,32(5)：566-577
[52]文国庆,陈华荣.弹性半空间地基上的板载轴对称荷载作用下的一般解.岩土工程学报,1989,11(5)：109-119
[53]杨小礼.浅埋小净距偏压隧道施工工序的数值分析.中南大学学报(自然科学版),2007,38(4)：764-770
[54]王祥秋,杨林德,高文华.高速公路偏压隧道施工动态监测与有限元仿真模拟.岩石力学与工程学报,2005,24(2)：284-289
[55]毕继红,钟建辉,丛蓉.浅埋洞室围岩压力有限元分析.铁道工程学报,2004,(4)：77-81,106
[56]马步方,王宝先.公路偏压隧道施工监控量测与模拟分析.企业技术开发,2009,28(1)：44-46,56
[57]张敏,黄润秋.浅埋偏压隧道出口变形机理及稳定性分析.工程地质学报,2008,16(4)：482-488
[58]胡光进,刘登聪.公路双连拱隧道常见病害防治.探矿工程.2008,4：69-73
[59]段方情,张可能,胡新红等.某偏压隧道洞口段施工出现的问题及处理.西部探矿工程,2008,5：150-152
[60]孙钧,侯学渊.地下结构(上册).北京：科学出版社,1987
[61]Sneddon D. Fourier Transforms. Mcgraw-hill Book Company,INC. New York, 1951,35-48
[62]Fryba L. Vibration of solids and structures under moving loads. Publishing House of the Czechoslovak Academy of Sciences Prague,1963,89-96
[63]Poulous H.G., Davis E.H. Elastic solution for soil and rock mechanics. John Wiley&Sons Incorporatde,1974,173-179
[64]弗洛林.土力学原理.北京：中国工业出版社,1965,125-132
[65]张雯雯,刘黎平.一种简化的分数阶傅里叶变换及其应用.数据采集与处理,2009,24(6)：814-818
[66]刘齐建.软土地铁建筑结构抗震设计计算理论的研究：[同济大学博士学位论文].上海：同济大学,2005,85-99
[67]Verruijt A, Booker R. Complex variable solution of Mindlin's problem of an excavated tunnel. http://geo.verruijt.net,2007-8
[68]Verruijt A, Booker R. Complex variable analysis of Mindlin's tunnel problem. http://geo.verruijt.net,2007-8
[69]Verruijt A, Booker R. Complex variable solutions of elastic tunneling problems. http://geo.verruijt.net,2007-8