Abstract：Under the assumption of pure slip contact along the lining-rock interface and the requirements of the net size and the lining thickness，derivation of shape optimization of a tunnel support was carried out with the objective of the greatest tangential stress along the inner edge of the support being minimized. The optimizing process is to solve a class of inverse geometry problems，and the corresponding forward problem can be solved with the method of conformal mapping of complex variable function in plane elasticity. During the process，the tangential stress along the inner edge of the support was selected as the objective function and coefficients of the mapping function were taken as the design variables. The method of mixed penalty function was used to minimize the objective function. As the objective function reached the minimum value，the optimal shape of support can be obtained. With the optimal shape，stress state in tunnel support was improved most effectively and the tangential stress concentration along the inner edge of support was minimized.
吕爱钟，陈虹宇，覃 媛，张 宁. 光滑接触条件下隧洞支护断面形状优化[J]. 岩石力学与工程学报, 2014, 33(8): 1563-1571.
LU Aizhong，CHEN Hongyu，QIN Yuan，ZHANG Ning. SHAPE OPTIMIZATION OF TUNNEL SUPPORT SECTION UNDER CONTACT CONDITION OF PURE SLIP. , 2014, 33(8): 1563-1571.
[1] CRISTESCU N D，PARASCHIV I. The optimal shape of rectangular-like caverns[J]. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts，1995，32(4)：285–300.
[2] 吕爱钟. 高地应力区地下硐室断面形状的选择方法[J]. 煤炭学报，1997，22(5)：495–498.(LU Aizhong. Shape selection method of the underground caverns under high ground stresses[J]. Journal of China Coal Society，1997，22(5)：495–498.(in Chinese))
[3] 吕爱钟. 地下洞室最优开挖形状的确定方法[J]. 岩石力学与工程学报，1996，15(3)：275–281.(LU Aizhong. The method for optimum shapes of tunnels and cavities[J]. Chinese Journal of Rock Mechanics and Engineering，1996，15(3)：275–281.(in Chinese))
[4] 张志强，师晓权，何 川. 基于流固耦合的水底隧道仰拱受力分析与优化[J]. 铁道学报，2011，33(1)：108–133.(ZHANG Zhiqiang，SHI Xiaoquan，HE Chuan. Analysis of optimizing cross-section shape of subsea tunnel[J]. Journal of the China Railway Society，2011，33(1)：108–133.(in Chinese))
[5] 房 倩，张顶立，李鹏飞，等. 基于施工安全性的海底隧道断面优化研究[J]. 北京工业大学学报，2010，36(3)：321–327.(FANG Qian，ZHANG Dingli，LI Pengfei，et al. Study on the optimization of the shape design for subsea tunnel based on construction safety[J]. Journal of Beijing University of Technology，2010，36(3)：321–327.(in Chinese))
[6] 刘元雪，谢 锋，蒋树屏，等. 棚洞结构洞形优化计算分析[J]. 岩土力学，2007，28(增1)：490–492.(LIU Yuanxue，XIE Feng，JIANG Shuping，et al. Numerical analysis of the shed-tunnel structrue on the tunnel shape optimizing[J]. Rock and Soil Mechanics，2007，28(Supp.1)：490–492.(in Chinese))
[7] 石广斌，李 宁. 用灰色局势决策法优选高地应力下大型地下洞室拱形[J]. 岩石力学与工程学报，2005，24(12)：2 119–2 123.(SHI Guangbin，LI Ning. Selecting arch shape of large caverns under high in-situ stress by method of gray condition critertion[J]. Chinese Journal of Rock Mechanics and Engineering，2005，24(12)：2 119–2 123.(in Chinese))
[8] REN G，SMITH J V，TANG J W，et al. Underground excavation shape optimization using an evolutionary procedure[J]. Computers and Geotechnics，2005，32(2)：122–132.
[9] SOBÓTKA M，?YD?BA D，RÓ?A?SKI A. Shape optimization of underground excavation by simulated annealing[J]. Studia Geotechnica et Mechanica，2013，35(1)：209–218.
[10] RÓ?A?SKI A，SOBÓTKA M. Warunki optymalizacji kszta?tu wyrobisk podziemnych[J]. Górnictwo i Geoin?ynieria，2009，33(1)：519–529.(in Polish)
[11] BJORKMAN G S，RICHARDS R. Harmonic holes—an inverse problem in elasticity[J]. ASME Journal of Applied Mechanics，1976，43(3)：414–418.
[12] BJORKMAN G S，RICHARDS R. Harmonic holes for nonconstant fields[J]. ASME Journal of Applied Mechanics，1979，46(3)：573–576.
[13] DHIR S K. Optimization in a class of hole shapes in plate structures[J]. ASME Journal of Applied Mechanics，1981，48(4)：905–908.
[14] 孙焕纯，张巨勇，严国梅，等. 弹性平面孔洞形状优化复变函数方法[J]. 应用数学和力学，1987，8(2)：133–141.(SUN Huanchun，ZHANG Juyong，YAN Guomei，et al. Optimization of the shape of a elastic plane hole based on complex variable function method[J]. Applied Mathematics and Mechanics，1987，8(2)：133–141.(in Chinese))
[15] 吕爱钟. 以孔边绝对值最大的切向应力最小为优化准则的孔洞形状优化[J]. 固体力学学报，1996，17(1)：73–76.(LU Aizhong. Shape optimization based on the criterion of making the absolute maximum tangential stress of the hole be the minimum[J]. Chinese Journal of Solid Mechanics，1996，17(1)：73–76.(in Chinese))
[16] 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.
[17] ATKINSON C，EFTAXIOPOULOS D A. Numerical and analytical solutions for the problem of hydraulic fracturing from a cased and cemented wellbore[J]. International Journal of Solids and Structures，2002，39(6)：1 621–1 650.
[18] 万耀青，梁庚荣. 最优化计算方法常用程序汇编[M]. 北京：工人出版社，1983：282–344.(WAN Yaoqing，LIANG Gengrong. Common programs for optimization calculation[M]. Beijing：Workers? Press，1983：282–344.(in Chinese))
[19] FLETCHER R. Practical methods of optimization：constrained optimization[M]. [S. l.]：Wiley-Interscience，1981：46–78.
[20] 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，2007：67–73.(LU Aizhong，ZHANG Luqing. Complex variable function method on mechanical analysis of underground tunnel[M]. Beijing：Science Press，2007：67–73.(in Chinese))
[21] 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 the mechanical analysis of surrounding rock[M]. Beijing：Coal Industry Publishing House，1994：21–23.(in Chinese))
[22] ENGLAND A H. Complex variable methods in elasticity[M]. [S. l.]：Wiley-Interscience，2012：161–168.
2014, Vol. 33 
