Tunnels built in areas subject to earthquake activity must withstand seismic loadings. Covering tunnel liners with a coating layer will be a possible way to mitigate seismic damage to tunnels. In this paper, an analytical solution is developed for the seismic response of deep circular tunnels covered by an isolation layer. Since the cross-section dimension of tunnels is normally much smaller than the wavelength of ground peak velocities, the inertial forces can be neglected, and the structure can be designed using the pseudo-static approach, where the seismic-induced loads or deformations can be approximated by far-field shear stresses. The ground and the isolation layer are assumed to be elastic, homogeneous, and isotropic in plane strain conditions, and the tunnel lining is represented as an elastic shell. Both the full-slip and no-slip conditions are considered for the contact between the tunnel and the isolation layer, while the interface between the isolation layer and the ground is assumed to be continuous. The relative stiffness method, proposed by Einstein and Schwartz (1979), is employed to obtain the closed-form solutions for tunnel distortion and internal forces, including axial force, bending moment, and shear force. The proposed solution is verified by providing comparisons between its results and those from the known results in literature and the Finite Element program. Parametric analyses are presented where the seismic mitigation effects of the isolation layer with different properties such as thickness, elastic modulus, and Poisson’s ratio. Results show that the elastic modulus and thickness of the isolation layer, as well as the tunnel-isolation layer interface conditions (i.e., full-slip and no-slip), have significant influences on the seismic mitigation effect, except for the Poisson’s ratio of the isolation layer. The proposed solution can be used as an effective tool for the design optimization of tunnel structures with an isolation layer.

在易受地震活动影响的地区建造的隧道必须承受地震载荷。用涂层覆盖隧道衬砌将是减轻隧道地震破坏的一种可能方法。在本文中，为被隔离层覆盖的深圆形隧道的地震响应开发了一种解析解。由于隧道的横截面尺寸通常远小于地面峰值速度的波长，惯性力可以忽略不计，可以使用拟静力方法设计结构，其中地震引起的载荷或变形可以由远场剪应力近似。假定地面和隔离层在平面应变条件下是弹性的、均匀的和各向同性的，隧道衬砌表示为弹性壳。隧道与隔离层的接触考虑全滑移和无滑移两种情况，而隔离层与地面的界面假设为连续的。Einstein 和 Schwartz (1979) 提出的相对刚度法用于获得隧道变形和内力（包括轴向力、弯矩和剪切力）的封闭形式解。通过提供其结果与文献中已知结果和有限元程序的结果之间的比较来验证所提出的解决方案。参数分析显示了具有不同属性（例如厚度、弹性模量和泊松比）的隔离层的减震效果。结果表明，隔离层的弹性模量和厚度，除了隔离层的泊松比外，隧道-隔震层界面条件（即全滑移和无滑移）对减震效果有显着影响。所提出的解决方案可用作具有隔离层的隧道结构设计优化的有效工具。