|
|
|
| A non-iterative analytical method for mechanical analysis of surrounding rock with arbitrary shape holes |
| FANG Huangcheng，ZHANG Dingli，WEN Ming，HU Xinyu |
| (Key Laboratory for Urban Underground Engineering of Education Ministry，Beijing Jiaotong University，Beijing 100044，China) |
|
|
|
|
Abstract The mechanical analysis of surrounding rock with arbitrary shape holes is of great significance for the safety of tunnel construction. In traditional analytical methods，the Schwarz alternating method is usually used to solve the multiple-hole problem，but its accuracy and efficiency are seriously affected by the number of iterations. Therefore，the Schwarz alternating method is not suitable for engineering applications. To solve this problem，in this paper，a non-iterative analytical method for the mechanical analysis of surrounding rock with arbitrary shape holes was proposed based on the theory of multiple Laurent series expansions. This method introduces several groups of analytic functions to accurately reformulate the stress and displacement fields of the multiply-connected domain，and establishes the stress and displacement boundary conditions by performing the conformal mapping on each hole individually. The conformal transformations of different holes are independent of each other，which makes it possible to handle the complex boundaries. Finally，the Fourier transform technique is used to transform the boundary conditions into the frequency equations，and then all unknowns in the analytic function can be uniquely determined. The proposed method can directly obtain the accurate solution of stress and displacement without any iteration，and hence is simple，efficient and suitable for engineering applications.
|
|
|
|
|
|
[1] 张顶立，孙振宇，宋浩然，等. 海底隧道突水演化机制与过程控制方法[J]. 岩石力学与工程学报，2020，39(4)：649–667.(ZHANG Dingli，SUN Zhenyu，SONG Haoran，et al. Water inrush evolutionary mechanisms of subsea tunnels and process control method[J]. Chinese Journal of Rock Mechanics and Engineering，2020，39(4)：649–667.(in Chinese))
[2] 李 然，张顶立，房 倩，等. 深埋3孔小净距隧道围岩压力计算方法及其工程应用[J]. 中国铁道科学，2020，41(2)：81–90.(LI Ran，ZHANG Dingli，FANG Qian，et al. Calculation method for surrounding rock pressure of deeply buried closely spaced triple tunnels and its engineering application[J]. China Railway Science，2020，41(2)：81–90.(in Chinese))
[3] 张成平，张 旭，李 贺，等. 浅埋隧道施工扰动下含空洞地层破坏演化规律试验研究[J]. 岩土工程学报，2016，38(2)：263–270. (ZHANG Chengping，ZHANG Xu，LI He，et al. Model tests on failure laws of ground with voids induced by shallow tunneling[J]. Chinese Journal of Geotechnical Engineering，2016，38(2)：263–270.(in Chinese))
[4] MUSKHELISHVILI N I. Some basic problems of the mathematical theory of elasticity[M]. Noordhoff，Groningen：Springer，1963：5–20.
[5] VERRUIJT A，BOOKER J R. Surface settlements due to deformation of a tunnel in an elastic half plane[J]. Geotechnique，1996，46(4)：753–756.
[6] 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(2)：77–89.
[7] STRACK O E，VERRUIJT A. A complex variable solution for a deforming buoyant tunnel in a heavy elastic half-plane，International Journal for Numerical and Analytical Methods in Geomechanics，2002，26(12)：1 235–1 252.
[8] LU A Z，ZHANG L Q，ZHANG N. Analytic stress solutions for a circular pressure tunnel at pressure and great depth including support delay[J]. International Journal of Rock Mechanics and Mining Sciences，2011，48(3)：514–519.
[9] LU A Z，ZENG X T，XU Z，Solution for a circular cavity in an elastic half plane under gravity and arbitrary lateral stress[J]. International Journal of Rock Mechanics and Mining Sciences，2016，89(11)：34–42.
[10] HEFNY A M，LO K Y. Analytical solutions for stresses and displacements around tunnels driven in cross-anisotropic rocks[J]. International Journal for Numerical and Analytical Methods in Geomechanics，1999，23(3)：161–177.
[11] FANG Q，SONG H R，ZHANG D L. Complex variable analysis for stress distribution of an underwater tunnel in an elastic half plane[J]. International Journal for Numerical and Analytical Methods in Geomechanics，2015，39(16)：1 821–1 835.
[12] 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 Lizhong，LU Xuejin. A complex variable solution for different kinds of oval deformation around circular tunnel in an elastic half plane[J]. Chinese Journal of Geotechnical Engineering，2007，29(3)：319–327.(in Chinese))
[13] BOBET A. Analytical solutions for shallow tunnels in saturated ground[J]. Journal of Engineering Mechanics-ASCE，2001，127(12)：1 258–1 266.
[14] 陈子荫. 围岩力学分析中的解析方法[M]. 北京：煤炭工业出版社，1994：21–23.(CHEN Ziyin. Analytical methods in analysis of rock mechanics[M]. Beijing：China Coal Industry Publishing House，1994：21–23.(in Chinese))
[15] 苏 锋，陈福全，施有志. 深埋双隧洞开挖的解析延拓法求解[J]. 岩石力学与工程学报，2012，31(2)：365–374.(SU Feng，CHEN Fuquan，SHI Youzhi. Analytical continuation solution of deep twin- tunnels[J]. Chinese Journal of Rock Mechanics and Engineering，2012，31(2)：365–374.(in Chinese))
[16] 张治国，张成平，奚晓广. 双线隧道不同布置方式下相互作用影响的地层位移解析[J]. 岩土工程学报，2019，41(2)：262–271. (ZHANG Zhiguo，ZHANG Chengping，XI Xiaoguang. Closed solutions to soil displacements induced by twin-tunnel excavation under different layout patterns[J]. Chinese Journal of Geotechnical Engineering，2019，41(2)：262–271.(in Chinese))
[17] 吕爱钟，张路青. 地下隧洞力学分析的复变函数方法[M]. 北京：科学出版社，2007：127–153.(LU Aizhong，ZHANG Luqing. Complex function method on mechanical analysis of underground tunnel[M]. Beijing：Science Press，2007：127–153.(in Chinese))
[18] 晏 莉，阳军生，刘宝琛. 浅埋双孔平行隧道开挖围岩应力和位移分析[J]. 岩土工程学报，2011，33(3)：413–419.(YAN Li，YANG Junsheng，LIU Baochen. Stress and displacement of surrounding rock with shallow twin-parallel tunnels[J]. Chinese Journal of Geotechnical Engineering，2011，33(3)：413–419.(in Chinese))
[19] 杨公标，张成平，闵 博，等. 浅埋含空洞地层圆形隧道开挖引起的位移复变函数弹性解[J]. 岩土力学，2018，39(增2)：25–36. (YANG Gongbiao，ZHANG Chengping，MIN Bo，et al. Elastic solution of soil displacement induced by shallow circular tunnel with a cavern in a stratum using function of complex variable method[J]. Rock and Soil Mechanics，2018，39(Supp.2)：25–36.(in Chinese))
[20] CHEN F Q，LIN L B，LI D Y. Analytic solutions for twin tunneling at great depth considering liner installation and mutual interaction between geomaterial and liners[J]. Applied Mathematical Modelling，2019，73(19)：412–441.
[21] LING C B. On the stresses in a plate containing two circular holes[J]. Journal of Applied Physics，1948，19(4)：77–82. |
|
|
|
2020, Vol. 39 
