The anti-plane surface motion of valleys in coastal cities induced by earthquakes

Introduction: Valleys are a common topographical feature of coastal cities, and some coastal cities are located within the Circum-Pacific and Eurasian earthquake belts. In addition, anti-plane surface motion can damage engineering structures and threaten human safety during an earthquake. A novel and significant framework is therefore proposed for computing anti-plane surface motion of valleys induced by earthquakes in order to enhance the resilience of coastal cities against earthquakes.

Methods: 1. A valley under earthquake excitation is simplified as an arc-shaped mathematical model for incident SH waves. 2. The elastic wave motion equation is generated without factors of time. 3. Based on the variable-separation method, wave functions satisfying the equation are derived. 4. Wave functions meeting boundary conditions can be obtained using zero-stress boundary conditions of the mathematical model. 5. Complex displacement of surface motion is obtained through wave functions' linear calculations. 6. Amplitudes and phases of surface motion are determined using complex-number calculative formulae.

Results and Discussion: As a result, anti-plane surface motion of valleys can be computed by the newly presented framework. In the next study, angles ranging from 30 to 60 degrees and frequencies ranging from 0 to 10 HZ will be evaluated. Amplitudes and phases of surface motion are given as functions of various angles and frequencies of incident SH waves. It is found that amplitudes and phases of surface motion of valleys usually vary with angle and frequency.

Conclusions: For oblique incidence, displacement amplitudes on a surface facing waves are often higher than those on a shadow surface. In addition, the number of oscillations of amplitudes on a surface facing waves is often greater than that on a shadow surface. However, regular patterns of phases are nearly consistent. These findings suggest that important buildings, such as cable cars, bridges and railways, can be built better on shadow surfaces of valleys in coastal cities.