Vibration Characteristic Analysis of Tunnel Structure Buried in Elastic Foundation Based on Dynamic Stiffness Matrix

In this study, the free vibration characteristics of a non-circular tunnel buried in a Winkler foundation are first investigated based on a dynamic stiffness approach and the Wittrick–Williams algorithm. The non-circular tunnel was modeled as a shell composed of multipartial circular cylinders based on the first shear deformation Flügge shell theory, by which the effects of shear deformation and moment of inertia were considered. The transfer function of the state variable was derived, and the elementary dynamic stiffness matrices were formulated based on this transfer function. Similar to the finite element method, the global dynamic stiffness matrix was established by assembling each elementary dynamic stiffness matrix. By solving the global dynamic stiffness matrix using the Wittrick–Williams algorithm, the natural frequencies and the corresponding mode shapes were determined. Subsequently, the calculated results were compared with literature and finite element analysis results, thus validating the accuracy and reliability of the proposed method. A parametric analysis was conducted to investigate the effects of the cross-section, stiffness coefficient, thickness, and length of the tunnel composed of multipartial circular cylinders. It was concluded that among the four types of cross-sections, the free vibration characteristics of the shells with close isoperimetric radii are similar. The natural frequencies of the shells buried in a moderately stiff Winkler foundation increase rapidly with its stiffness coefficient, whereas those of the shells buried in softer and stiffer foundations remain practically constant. The increase in shell thickness progressively increases the natural frequencies. In contrast, the natural frequencies decrease sharply with the increase in shell length; they first decrease to the lowest limits and subsequently remain unaltered. Thus, the free vibration characteristics of an actual tunnel structure resting on a Winkler foundation can be obtained by conducting the free vibration analysis on a sufficiently long shell segment, which is of major significance in the dynamic design of tunnel structures.