Research on the dynamic response of shallow-buried circular non-complete bonded tunnels under anti-plane line source loading
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摘要: 为加深理解波源距离和非完全粘结对地震波散射的影响规律，结合位移不连续模型、波函数展开法、Graf公式和镜像方法推导了反平面线源荷载下浅埋圆形非完全粘结隧道动力响应的级数解，并通过衬砌内外边界条件残余量与级数解截断项数的关系校验了该解的精度。通过对该级数解进行参数分析，系统地探讨了衬砌与围岩的接触刚度、衬砌模量、衬砌厚度、隧道埋深和波源距离等因素对衬砌内表面位移和周向剪应力的影响。结果表明：衬砌与围岩的接触刚度对隧道的动力响应具有显著的影响，尤其在某些较小接触刚度情况下隧道动力响应幅值可能非常大；增大衬砌模量会减小位移，但同时会导致周向剪应力增加；增大衬砌厚度能同时减小位移和周向剪应力；增大隧道埋深会使最大位移和周向剪应力向隧道拱顶附近移动；增大线源与隧道的水平距离会使隧道背波侧相对幅值增大。Abstract: The scattering of seismic waves by shallow-buried underground structures has significant theoretical value in the engineering field. However, previous studies have mainly focused on the case of plane waves or the case of complete bonding between lining and surrounding rock, with little consideration of the effects of source distance and non-complete bonding between lining and surrounding rock. In order to deepen the understanding of the influence of source distance and non-complete bonding on seismic wave scattering, the series solution of the dynamic response of shallow-buried circular non-complete bonded tunnels under the loading of anti-plane line source was derived based on the displacement discontinuity model, wave function expansion method, Graf formula and mirror method. The accuracy of the obtained solution was verified by the relationship between the residuals of the inner and outer boundary conditions of the lining and the number of truncated terms in the series solution. By systematically analyzing the parameters of this series solution, the influence of factors such as the contact stiffness between lining and surrounding rock, lining modulus, lining thickness, tunnel depth and source distance on the displacement and circumferential shear stress on the inner surface of the lining was discussed. The results show that the contact stiffness between lining and surrounding rock has a significant influence on the dynamic response of the tunnel, especially in cases with relatively low contact stiffness, where the amplitude of the dynamic response of the tunnel can be very large. Increasing the lining modulus reduces the displacement but increases the circumferential shear stress. Increasing the lining thickness can simultaneously reduce the displacement and circumferential shear stress. As the tunnel depth increases, the maximum displacement and circumferential shear stress on the inner surface of the lining shifts towards the apex of the tunnel. Increasing the horizontal distance between the line source and the tunnel increases the relative amplitude of the tunnel's back wave side.
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图 2 当η = 1.0和Ks/μ1 = 1.0 m−1时衬砌内外边界的位移和应力残差与截断项数N之间的关系
Figure 2. The relationship between the displacement and stress residuals of the inner and outer boundaries of the lining and the number of truncated terms N when η=1.0 and Ks/μ1=1.0 m−1
图 3 当η = 3.0和Ks/μ1 = 0.1 m−1时衬砌内外边界的位移和应力残差与截断项数N之间的关系
Figure 3. The relationship between the displacement and stress residuals of the inner and outer boundaries of the lining and the number of truncated terms N when η=3.0 and Ks/μ1=0.1 m−1
图 4 衬砌内表面归一化位移和周向剪应力与衬砌和围岩的接触刚度的关系
Figure 4. The relationship between normalized displacement and circumferential shear stress on the inner surface of lining and the contact stiffness between lining and surrounding rock
图 5 衬砌内表面归一化位移和周向剪应力与衬砌模量的关系
Figure 5. The relationship between normalized displacement and circumferential shear stress on the inner surface of lining and lining modulus
图 6 衬砌内表面归一化位移和周向剪应力与衬砌厚度的关系
Figure 6. The relationship between normalized displacement and circumferential shear stress on the inner surface of lining and lining thickness
图 7 衬砌内表面归一化位移和周向剪应力与隧道埋深的关系
Figure 7. The relationship between normalized displacement and circumferential shear stress on the inner surface of lining and tunnel burial depth
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