Analytical solution for seismic response of deep tunnels with arbitrary cross-section shapes
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摘要: 考虑地层-结构相互作用的拟静力法广泛应用于隧道抗震设计和分析,而以拟静力法为基础推导出的解析解则更是备受工程设计人员所青睐,但现有解析方法仅适用于圆形或矩形隧道。以均质地层中的任意形状深埋隧道为对象,将衬砌简化为均质地层中的壳体,并考虑隧道内部中隔墙的影响,建立了远场剪切波作用下隧道结构和地层各自的控制方程,结合地层与结构之间不滑移和全滑移两种典型接触边界,运用复变函数及保角映射方法,推导出均质地层中任意形状深埋隧道地震响应的解析解。通过与有限元基准模型在相同计算条件下的结果对比,验证了本文解析方法的正确性。最后应用该解析表达式开展多工况参数化分析,揭示了地层–结构相对刚度比和中隔墙厚度等因素对不规则断面隧道结构的地震响应的影响规律。结果表明：隧道结构内力响应随地层-结构相对刚度的增大而增大,而位移响应则相反；不滑移条件下隧道的内力响应要大于全滑移条件的内力响应,而位移响应相比全滑移条件则较小；与不考虑中隔墙的隧道相比,当中隔墙厚度大于隧道衬砌厚度1.5倍时,中隔墙的存在将显著增大衬砌的最大内力响应。Abstract: The pseudo-static method considering ground-structure interaction is widely used in seismic design and analysis of tunnels, and especially, the analytical solution based on this method can serve as an effective tool for practitioners. However, the existing methods in literatures are only applicable for circular or rectangular tunnels. An analytical solution for seismic response of deep tunnels with arbitrary cross-section shapes in homogeneous ground is presented. The lining is assumed as a shell, and the effects of the middle wall as well as two typical soil-structure contact conditions (i.e., no-slip and full-slip) are also considered. First, the governing equations for the tunnel structure and the ground are introduced. Then, the complex variable function and the conformal mapping method are employed to obtain the analytical solution. The validation of the proposed solution is verified by comparison with the results from FEM. Finally, parametric analyses are carried out using the solution, and the influences of the relative stiffness ratio of the ground to the structure as well as the thickness of middle wall are analyzed. The results show that the internal force of the tunnel increases with the increase of the ground-structure relative stiffness ratio, while the deformation of the tunnel decreases. The internal force of the tunnel under no-slip condition is larger than that under full-slip condition, while, oppositely, the structural deformation under no-slip condition is less than that under full-slip condition. Compared to that of the case neglecting the middle wall in the tunnel, the maximum internal force of the tunnel significantly increases when the thickness of the middle wall is 1.5 times larger than the thickness of the tunnel lining.
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图 1 远场剪应力作用下的任意形状深埋隧道
Figure 1. Deep tunnel with arbitrary cross-section shapes subjected to far-field shear loading
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图 4 直墙拱形隧道地震响应对比验证
Figure 4. Comparison of seismic responses of straight-wall-arch shaped tunnel by FEM model and proposed analytical solution
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图 5 类矩形隧道地震响应对比验证
Figure 5. Comparison of seismic responses of quasi-rectangular tunnel by FEM model and proposed analytical solution
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图 6 柔度比对隧道最大内力响应和位移响应的影响
Figure 6. Effects of flexibility ratio on maximum stress and displacement responses of tunnel
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