Analytical models and amplification effects of seismic wave propagation in canyon sites

Morphologically, there are three types of topographic and site conditions in nature: flat, convex and concave sites. The canyon (or valley) is a common concave site, and a large number of structures such as earth-rock dams and bridges have been built in such a site. Investigation of actual earthquake damage shows that the topographic and site conditions have great influences on earthquake disasters. Aiming at the analytical models and amplification effects of seismic wave propagation in canyon sites, the long-term research achievements of the author and his research group are summarized comprehensively. They include four aspects: (1) The concept of near-source topographic and site effects is proposed by simulating the incident seismic waves with a line source of cylindrical SH waves. The plane wave is a special case of its far field incidence. The freewave field under the line source of cylindrical SH waves is constructed to realize the curvature of the incident wave front. The amplification factor is defined to describe the topographic effects under near-source excitation, which opens new possibilities for studying the near-source amplification effects of other topographies and sites. (2) An analytical model for seismic wave propagation in non-symmetrical V-shaped canyon is constructed, including the Helmholtz equation, traction-free boundary conditions on canyon surface, and continuity conditions of traction and displacement on the auxiliary boundary. A two-step strategy for region decomposition and region matching is proposed. Firstly, the whole region is decomposed into three sub-regions in accordance with the corresponding polar coordinate systems. The corresponding wave fields (including unknown coefficients) are obtained by solving the equation of motion in the sub-regions. Then, the wave fields of each sub-region are matched at the boundary, and the unknown coefficients are solved by using the boundary conditions. The wave-field solutions of the whole region and the two-dimensional scattering patterns of cylindrical SH waves are obtained. The differential amplification effects of the non-symmetrical V-shaped canyon are revealed, which will have an unignorable influence on the large-span projects built in it. (3) U-shaped canyons are ubiquitous on the earth's surface. Due to the lack of actual seismic records and theoretical researches, the topographic amplification effects of the U-shaped canyons are still unknown. The analytical model for a U-shaped canyon is constructed, which is essentially the boundary value problem of Helmholtz equation. The wave function series solution to the problem is obtained. The anomalous amplification of seismic waves at the bottom of U-shaped canyon has been found. It has changed the incomplete understanding that the ground motion at the bottom of a concave topography is bound to attenuate, and has been used to explain the large number of rockfalls and landslides in Arizona during the warm period of the Middle Ages. (4) Sediments (overburden layers) often occur in canyons, which may further aggravate the amplification effects of earthquakes. An analytical model for a partially filled semi-circular alluvial valley under a line source of cylindrical SH waves is constructed, and its analytical series solution is given. It is found that the overburden layers have obvious amplification effects on the seismic waves, especially for those with a small damping ratio, which will aggravate the damage of engineering structures. Finally, the seismic stability analysis of the canyon or valley slopes, the seismic response analysis of earth and rockfill dams as well as the seismic stability analysis of the dam slopes are carried out considering the seismic amplification effects of the canyon or valley sites. It is believed that the seismic amplification effects of canyons or valleys have important influences on the seismic analysis of slope and dam engineering.