Semi-analytical solution for ground vibrations of a finite soil layer induced by symmetrical vertical distributed loads

This study presents a semi-analytical solution for ground vibrations of a three-dimensional elastic soil layer induced by vertically distributed loads based on the wave function expansion method and the Fourier–Bessel expansion method. The generating wave fields are constructed using the method of the separation of variables. General expressions for the displacement and stress components are derived based on Hooke’s law and the theory of small-strain elasticity. Wave potentials and the vertically distributed loads are transformed into Fourier–Bessel series with the given calculation range. A boundary value problem involves displacement-fixed boundary conditions at the rigid base and stress-given boundary conditions along the ground surface, resulting in a series of algebraic equations. Unknown coefficients are numerically solved by truncating the series numbers. Contour integrals, including branch points and poles in Lamb-type problems, are circumvented using Fourier–Bessel series expansion methods. A parametric study is conducted to investigate the ground vibrations of the soil layer under time-harmonic vertically distributed loads. Numerical results reveal that the soil damping ratio, slenderness ratio, and non-dimensional frequency of loading have a significant influence on the ground motions. Resonance occurs when the load frequency matches the first natural frequency of the soil layer. Higher load frequencies result in more concentrated dynamic effects but narrower vibration ranges.