In this study, the segment joint and ring joint of the lining of the straight-jointed tunnel are simplified as an equivalent open cylindrical shell with a small central angle and an equivalent closed cylindrical shell with a small width, and the lining of the straight-jointed tunnel is thus simplified as an equivalent continuous periodic shell (ECPS) described by the cylindrical shell theory. Since the ECPS lining is periodic along the longitudinal direction, the tunnel-soil system can thus be treated as a periodic system, which is referred to as the periodic tunnel-soil system (PTSS) in this study. Based on the proposed ECPS lining model, an analytical method for the tunnel with the ECPS lining (ECPS tunnel) under seismic waves is established in this study. By employing the periodicity condition for the PTSS as well as the wave function expansion method, the representation for the wave field in the soil is established. With the aforementioned cylindrical shell theory and Fourier series expansion method, the convolution type constitutive relation for the ECPS lining and the Fourier space equations of motion are derived. By using the soil-lining continuity condition and aforementioned formulations for the ECPS lining and soil, the coupled Fourier space equations of motion for the PTSS are established, with which the response of the ECPS lining and scattered waves in the soil can be determined.

This paper aims to elucidate the clear visibility of attenuating seismic waves (SWs) with forest trees as natural metamaterials known as forest metamaterials (FMs) arranged in a periodic pattern around the protected area. In analyzing the changeability of the FM models, five distinct cases of metawall configurations were considered. Numerical simulations were conducted to study the characteristics of bandgaps (BGs) and vibration modes for each model. The finite element method (FEM) was used to illustrate the generation of BGs in low frequency ranges. The commercial finite element code COMSOL Multiphysics 5.4a was adopted to carry out the numerical analysis, utilizing the sound cone method and the strain energy method. Wide BGs were generated for the Bragg scattering BGs and local resonance BGs owing to the gradual variations in tree height and the addition of a vertical load in the form of mass to simulate the tree foliage. The results were promising and confirmed the applicability of FEM based on the parametric design language ANSYS 17.2 software to apply the boundary conditions of the proposed models at frequencies below 100 Hz. The effects of the mechanical properties of the six layers of soil and the geometric parameters of FMs were studied intensively. Unit cell layouts and an engineered configuration for arranging FMs based on periodic theory to achieve significant results in controlling ground vibrations, which are valuable for protecting a large number of structures or an entire city, are recommended. Prior to construction, protecting a region and exerting control over FM characteristics are advantageous. The results exhibited the effect of the 'trees' upper portion (e.g., leaves, crown, and lateral bulky branches) and the gradual change in tree height on the width and position of BGs, which refers to the attenuation mechanism. Low frequency ranges of less than 100 Hz were particularly well suited for attenuating SWs with FMs. However, an engineering method for a safe city construction should be proposed on the basis of the arrangement of urban trees to allow for the shielding of SWs in specific frequency ranges.