深埋公路隧洞围岩应力和位移分布的复变函数解  

Complex Variable Function Solutions for Stress and Displacement Distributions in Surrounding Rock of Deep Buried Highway Tunnels

作　　者：刘淑红 王自豪 甄卫刚 LIU Shuhong;WANG Zihao;ZHEN Weigang(Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, P.R. China;The Key Laboratory for Health Monitoring and Control of Large Structures of Hebei Province, Shijiazhuang 050043, P.R. China)

机构地区：[1]石家庄铁道大学工程力学系,石家庄050043 [2]河北省大型结构健康诊断与控制重点实验室,石家庄050043

出　　处：《应用数学和力学》2018年第5期548-557,共10页Applied Mathematics and Mechanics

基　　金：河北省自然科学基金(A2015210089);河北省高等学校科学技术研究重点项目(ZD2015101)

摘　　要：为了优化公路隧道的设计和确保施工安全,必须明确公路隧洞开挖时围岩的力学行为.利用复变函数方法,通过保角映射函数把隧洞外域变换为单位圆外域.利用Cauchy(柯西)积分和留数定理求出两个应力函数,从而得到围岩的应力与位移的平面应变问题的解析解.结合曲墙马蹄形断面,通过数学软件MATLAB编程计算,分别给出了应力和仅考虑开挖引起的位移沿隧洞边和坐标轴方向的分布.利用有限元软件ANSYS建立二维平面应变模型,对理论推导得到的应力和位移的分布进行验证,数值解结果与近似解析解结果吻合性很好.研究结果表明:最大的环向应力发生在隧洞拱脚处,最大水平位移发生在拱腰处,最大的沉降和隆起分别发生在拱顶和仰拱中心处.沿坐标轴的正应力在隧洞附近变化较大,不一定在洞边取得最大值,离洞边不到10 m的距离,便分别趋于所加外荷载.位移值在洞边最大,随着离洞边距离的增大,逐渐单调趋于0.In order to optimize the design of highway tunnels and ensure the safety of construc- tion, mechanical behavior of rock shall be made clear during highway tunnel excavation. With the complex variable method, fmstly, the exterior domain of the tunnel was transformed into a unit-circle domain through the conformal mapping function; then 2 stress functions were de- rived with the Cauchy integral formula and the residue theorem; thereby, closed-form plane strain solutions of the surrounding rock were obtained for stresses and displacements. The curved-wall horseshoe-shaped cross section was adopted, and the distributions of stresses and excavation displacements along the tunnel boundary and the coordinate axes were calculated re- spectively with mathematical software MATLAB. To verify the accuracy of the stress and dis- placement distributions derived with analytical solutions, a 2D plane strain model was estab- fished with finite element software ANSYS. Comparison between the numerical results and the approximate analytical results shows good agreement. The results show that, the maximum hoop stress occurs at the arch foot, the maximum horizontal displacement occurs at the hance, the largest settlement and uplift are at the centers of the vault and the invert, respectively. Nor- real stresses along coordinate axes change obviously near the tunnel, and the maximum stress doesn＇ t always occur at the tunnel boundary. The maximum stress approaches the applied load at a distance less than 10 m from the tunel boundary. The largest displacement occurs at the tunnel boundary, and gradually reaches zero with an increasing distance from the tunnel bound- ary.

关 键 词：公路隧洞 近似解析解 复变函数 应力 位移 

分 类 号：U451[建筑科学—桥梁与隧道工程]