Seismic Response of a Shallow Asymmetrical V-Shaped Canyon with Cliffs under SH Waves Propagation

Irregular topographies are widely distributed in nature, particularly in mountainous areas. The cliff on the canyon top forming over long periods of geological evolution significantly influences the amplification or reduction of ground motion. In this study adopting the wave function expansion method, a comprehensive analytical solution is formulated to address the scattering issue under shear horizontal (SH) waves induced by a shallow asymmetrical V-shaped canyon with cliffs. In general, employing the appropriate Graf’s addition formula, wave function expressions in different polar coordinate systems can be unified. According to continuity conditions of stress and displacement on the auxiliary boundary, the region-matching technique (RMT) has also been adopted to determine the unknown coefficients in the algebraic equation. Further, by comparing the results obtained in this study with those from previous studies, the correctness and applicability of the proposed theoretical model and analytical solution have been verified. A significant finding indicates that the maximum difference in peak displacement under a canyon with cliffs (d1/a = 0.4) can reach 1.81 times that without cliffs (d1/a = 0). Meanwhile, the asymmetric effect in this study is mainly concentrated on the ground motion of the canyon inner surface (−1 ≤ x/a ≤ 1), as opposed to a symmetrical V-shaped canyon with cliffs and an asymmetrical V-shaped canyon without cliffs. The proposed analytical solution not only enriches the category of well-known canyon problems related to SH-waves scattering, but also can serve as a reference for numerical verification and engineering practice.