基于复变函数理论的非圆形隧道解析解

李岩松; 陈寿根

doi:10.3969/j.issn.0258-2724.20180243

基于复变函数理论的非圆形隧道解析解

doi: 10.3969/j.issn.0258-2724.20180243

详细信息

作者简介:
李岩松（1989—），男，博士研究生，研究方向为隧道及岩石力学，E-mail：liyansong@my.swjtu.edu.cn

中图分类号: TU452
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- 被引次数: 0
出版历程
- 收稿日期: 2018-03-30
- 修回日期: 2018-07-23
- 网络出版日期: 2019-01-11
- 刊出日期: 2020-04-01

Lining and Surrounding Rock in Non-circular Tunnel Based on Complex Variable Method

摘要

摘要: 为了得到考虑衬砌支护的非圆形隧道衬砌和围岩应力及变形的解析解，基于复变函数理论提出了一种求解考虑衬砌的非圆形隧道衬砌和围岩应力及变形的方法. 首先，为了克服非圆形隧道断面几何形状和考虑衬砌支护造成的计算困难问题，引入了保角变换，通过采取最优化解法确定映射函数中的各项系数，得到映射函数；其次，采用幂级数复变函数法克服隧道衬砌带来的多连通域问题，确定应力函数中的各项系数，建立方程求解；最后，通过Flac有限差分软件进行数值模拟证明解析解的正确性. 研究发现：弹性范围内解析解与Flac有限差分软件计算得到的应力、位移解有较好的吻合性，表明弹性解析法的结果是可靠、合理的；深埋条件下，弹性解析法无需根据埋深、工况建立计算模型，只需明确边界条件和映射函数就可计算非圆形隧道应力、位移，弹性解析法克服了计算软件在计算中由于网格划分尺寸等问题造成计算结果不精确、计算慢等问题，为非圆形隧道开挖问题提出了一种快速、准确的弹性计算方法.
- 非圆形隧道 /
- 衬砌 /
- 复变函数 /
- 弹性力学 /
- 数值模拟 /
- 解析解
Abstract: A complex variable method is presented of stress and displacement problems for a non-circular deep tunnel with a certain given boundary conditions at infinity. Firstly, in order to overcome the complex problems caused by non-circular geometric configurations and the lining supports, optimal design method are used to determine coefficients of the conformal mapping function. Secondly, The problem of the multiply connected region is overcome by Power series complex function method, which determine stress and displacement within tunnel lining and within surrounding rock. The coefficients in the stress functions are determined by complex variable method. Finally, the complex variable method is validated by FLAC finite difference software through an example. Both the complex variable method and the numerical simulation obtain the similar results of the stress concentration and the minimum radial displacement occurs at a similar place of tunnel. It is demonstrated that the complex variable complex variable method is reliable and reasonable. Under deep buried conditions, the complex variable method does not need to establish models and mesh according to different burial depth and working conditions. Only mapping function and boundary conditions are required for the calculation. The complex variable method overcomes the problems of inaccuracy and slow calculation caused by meshing dimension and other problems in software calculation. And it also provides another way for solving non-circular tunnel excavation problems in the range of elasticity in a fast and accurate way.
- non-circular tunnel /
- lining /
- complex variables /
- elasticity /
- numerical simulation /
- analytical solution

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图 1 z平面保角变换为 ζ 平面示意

Figure 1. Conformal mapping of tunnel in z-plane in to two concentric circles in ζ-plane

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图 2 隧道计算简图（单位：m）

Figure 2. Tunnel calculation diagram (unit: m)

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图 3 衬砌与围岩接触面上环向应力

Figure 3. Circumferential stress along rock-lining interface

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图 4 衬砌与围岩接触面上径向应力

Figure 4. Normal stress along rock-lining interface

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图 5 衬砌与大气接触面上衬砌一侧应力

Figure 5. Stress along the inner lining periphery

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图 6 衬砌与围岩接触面上径向位移解析解

Figure 6. Radial displacement along rock-lining interface predicted by analytical solution

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图 7 衬砌与大气接触面上径向位移解

Figure 7. Radial displacement along inner lining periphery

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表 1 地层及材料参数

Table 1. Main physical parameter for tunnel calculation

名称弹性模
量/MPa 泊松比重度/
(kN•m⁻³) 侧压力
系数

板岩 25 000 0.3 26 0.5
衬砌 30 000 0.2 25 0.5

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