Analytical Solution of Mechanical Response inShallow Non-circular Tunnels
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摘要:
浅埋非圆形隧道的力学分析对城市地铁隧道的施工安全具有重要意义，其求解难点在于如何考虑重力场的影响以及圆环域保角映射函数的确定.为此，在复变函数理论框架下通过重构解析函数的表达，提出了重力条件下浅埋非圆形隧道力学分析的解耦保角映射方法，该方法将解析函数拆分成两组子函数，并采用不同的局部坐标系进行表达，两组子函数可以分别表达地表边界内与隧道边界外的应力与位移场，同时地表边界与隧道边界也可以单独进行保角映射；利用快速Fourier变换将用于确定解析函数的边界条件方程转化为频域方程进行求解；将本文方法应用于浅埋隧道开挖引起的地表沉降分析. 研究结果表明：隧道形状会对地表沉降的大小造成显著影响，其主因素来源于隧道的高跨比；隧道埋深会同时对地表沉降量以及沉降槽宽度产生影响，埋深越小其敏感程度越大；侧压力系数的改变对地表沉降槽的宽度影响较小；与有限元方法相比，本文方法的程序体量极小、计算速度更快且精度更高.
Abstract:The mechanical analysis of shallow noncircular tunnels is of great significance to the construction safety of urban subway tunnels. The difficulty in solving this problem originates from the influence of gravity and the determination of conformal mapping. To this end, a decoupling conformal mapping method with the framework of complex variable theory was proposed. In this method, the analytic function was decomposed into two groups of sub-functions, which were expressed in different local coordinate systems. These two sub-functions can exactly express the mechanical field of the computational domains inside the ground surface and outside the tunnel boundary. Then the ground and tunnel boundaries can be mapped independently by conformal transformation. Furthermore, the boundary condition equation used to determine the analytic function was transformed into the frequency equation using the fast Fourier transform method. Finally, the proposed method is applied to the analysis of ground settlement caused by the shallow tunnel excavation. From the results, the following conclusions can be drawn: 1) The tunnel shape has a significant effect on the ground settlement, and its main factor is the height-to-span ratio. 2) The buried depth affects both the ground settlement and the width of the settlement trough. Its sensitivity increases with the decrease of buried depth. 3) The lateral pressure coefficient has little effect on the width of the ground settlement trough. 4) Compared with the finite element method, the proposed method can obtain high-precision results at a small computational cost.
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Key words:
- tunnel engineering /
- decoupling conformal mapping method /
- complex variable theory /
- shallow /
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图 4 浅埋非圆形隧道力学分析（单位：m）
Figure 4. Mechanical analysis of shallow non-circular tunnel (unit：m)
图 6 隧道开挖引起地表沉降的影响因素分析（单位：m）
Figure 6. Influencing factors analysis of the ground settlement due to tunnel excavation (unit：m)
图 9 侧压力系数对地表沉降的影响
Figure 9. Influence of lateral pressure coefficient on ground settlement
表 1 矩形隧道应力计算结果对比
Table 1. Comparison of stress for rectangular tunnel
有限元法 本文方法 网格尺寸/m 应力/MPa 项数/项 应力/MPa 0.12 0.13 30 1.80 0.06 1.80 60 2.70 0.03 2.40 120 2.70 
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