考虑围岩和衬砌保角映射差异的深埋非圆形隧道力学解析  

Mechanical analysis of deep buried non-circular tunnel considering conformal mapping difference between rock and lining

作　　者：陈文博[1] 房倩[1] 张顶立[1] 陈炫皓 CHEN Wenbo;FANG Qian;ZHANG Dingli;CHEN Xuanhao(Key Laboratory for Urban Underground Engineering of Ministry of Education,Beijing Jiaotong University,Beijing 100044,China)

机构地区：[1]北京交通大学城市地下工程教育部重点实验室,北京100044

出　　处：《中南大学学报（自然科学版）》2023年第6期2349-2359,共11页Journal of Central South University:Science and Technology

基　　金：中央高校基本科研业务费专项资金资助项目(2021YJS110)。

摘　　要：非圆形隧道围岩和衬砌区域的保角映射函数不同,接触面边界上不同区域的同一点位在映射后的位置差异较大。因此,在求解非圆形隧道的力学解析表达式时,有必要考虑该映射差异。首先,引入基于边界积分的保角映射理论,编制了以多段圆弧为边界的有限双连通域和无限双连通域的保角映射程序;其次,考虑到围岩和衬砌保角映射差异,提出多点逼近的应力函数求解方法,并采用LSQR算法计算应力函数系数的最小二乘解,以满足隧道临空面边界和接触面边界上位移和应力条件;最后,通过多心圆隧道的算例,开展验证和分析。研究结果表明:所求围岩和衬砌保角映射函数具有良好的保角性和保圆性;随着应力函数项数增加,在边界上的应力误差和位移误差显著降低;当侧压力系数为0.50时,多心圆隧道洞周收敛变形的差异显著,拱顶和拱底变形远高于拱脚位置的变形,在衬砌拱脚位置所受围岩压力和切向力最大,并出现显著的应力集中现象;衬砌内外边界上应力和位移的解析解与数值解十分接近,验证了本文所提力学解析方法的正确性。The conformal mapping functions of the surrounding rock and lining domains of non-circular tunnels are different,and the points of two domains at the same positions on the contact boundary vary far away after mapping.Therefore,this difference should be considered when solving the analytical solutions of non-circular tunnels.Firstly,the conformal mapping theory was introduced based on boundary integral,and the conformal mapping program of finite biconnected domains and infinite biconnected domains bounded by multiple arcs were compiled.Secondly,a multi-point approximation method was proposed to solve the complex variable stress functions by taking into account the conformal mapping difference of two domains.The least squares solution of the stress function coefficients was calculated using the LSQR algorithm to meet the displacement and stress conditions on the free surface boundary and the contact boundary.Finally,the verification and analysis were carried out through the calculation example of one multi-center tunnel.The results show that the conformal mapping function calculated by the conformal mapping program has good conformal and circularity.The stress errors and displacement errors of stress functions on boundaries significantly decrease as the number of stress function items increases.When the lateral pressure coefficient is 0.50,the convergence deformation around the tunnel is significantly different.The displacements at the vault and arch bottom are much higher than those at arch foot.The surrounding rock pressure and tangential force near the arch foot of lining are the largest,and there is a significant stress concentration phenomenon.The analytical solutions with numerical solutions of the stress and displacement on the inner and outer boundaries of the lining are very close,which verifies the correctness of the mechanical analysis method.

关 键 词：非圆形隧道 力学解析 边界积分方程 保角映射 应力函数 

分 类 号：U452[建筑科学—桥梁与隧道工程]