Antiplane SH-Waves Scattering by a Circular Lined Cavity

According to the extensive development of urban texture and the vital necessity of lifelines, infrastructure and underground openings have found an important role in human societies. A full understanding of the behaviors of underground tunnels including tunnels for transportation, water, and facilities, can assist in presenting an optimum layout. The importance of this issue has increased because of the complex performance of the tunnels against seismic loads. The seismic analysis of underground tunnels has been used by the researchers for almost halfa century. Technically speaking, there are some analytical as well as semi-analytical methods for analyzing the ground response subjected to subsurface structures such as lined/unlined tunnels. Although these methods have high accuracy and used in the benchmark purposes, arbitrary shaped models with different boundary conditions cannot be easily applied in establishing the real problems. On the other hand, by developing the computers and software knowledge, the numerical approaches are frequently used for analyzing the continuum media in recent two decades, especially in soil mechanics. Based on the formulation, numerical approaches used in dynamic problems analysis can be usually divided into domain and boundary methods. In domain methods, such as finite element method (FEM) and finite difference method (FDM), it is required to discretize the whole body including internal parts and its boundaries. Although the simplicity of domain methods makes them favorite for analyzing seismic finite mediums, the models are complicated by discretizing the whole body and closing the boundaries in a distance far away from the interested zone for dynamic analysis of soil medium. In the response to the mentioned issues, the boundary element method (BEM) can be practically used for analyzing infinite/semi-infinite soil mediums in a better manner, due to discretizing only the boundaries of the problem. The BEM approaches can be formulated in the classes of full-plane (FBEM) and half-plane (HBEM) boundary element method.In this study, a direct half-plane time-domain boundary element method (HBEM) was developed and successfully applied to analyze the transient response of ground surface in the presence of circular lined tunnels, embedded in a linear elastic half-plane, subjected to propagating incident plane SH-waves. The Fundamental solutions for the case of propagating SH-waves in single-medium environments in the presence of unlined tunnels and ground surface topographies have been developed by Panji et al. [30]. For Multi-medium problems such as lined tunnels embedded in deep/shallow soil, the use of single-medium fundamental solutions was expanded by considering substructure procedure such that the problem was decomposed into a pitted half-plane and a closed ring-shaped domain.To solve the model, only the interface and inner boundary of the lining need to be discretized. After computing the matrices and satisfying the compatibility as well as boundary conditions, the coupled equations were solved to obtain the boundary values. There are two types of boundary conditions on interface boundaries, the first of which is the equality of displacement between mediums, and the second one is the equality of normal and shear stresses along the interface. These two conditions must be considered when the boundary equation is formed. In addition, to boundary conditions it is required to apply free-field displacement effects on the displacement of boundary and internal points. Free-field displacement provides the effects of wave incident and reflectance when the wave arrives to boundary or internal nodes. For Multi-medium problems such as lined tunnels embedded in deep/shallow soil, the free-field displacement accomplishes in the same way that it was done for single-medium problems.
 The mentioned method was successfully implemented in a developed computer code called DASBEM [30]. To validate the responses, a practical example was analyzed and compared with those of the published works. Manoogian [13] has obtained ground surface frequency domain responses for the case of embedded lined circular tunnel in shallow soil medium. By implementing the problem into the DASBEM, the results were obtained and compared with those of Manoogian [13]. The results showed that the analytical and numerical responses are in good agreement. Finally, in a parametric study, a circular lined tunnel was evaluated and the effect of depth and incident wave angle on ground surface response was analyzed. The results mainly showed that the ground surface frequency response due to tunnels without lining is more than that of lined tunnels. The method used in this paper is recommended to obtain the transient response of underground structures in combination with other numerical methods.