Temporal Analysis of Vertical Vibration for a Rigid Strip Footing on a Half-Space

This study provides a novel solution to dynamic response of a rigid strip footing on a half-space under vertical loading in time domain. The wave-like equations are established based on the fractional-order constitutive model of the soil. To address the mixed boundary value problem, Laplace and Fourier transforms are applied to convert the wave-like equations into a pair of dual integral equations. These equations are then solved using Jacobi orthogonal polynomials, yielding the dynamic stiffness and contact stress in the Laplace domain. The dynamic stiffness and contact stress in the Laplace domain are converted to the time domain through numerical inverse Laplace transformation. Numerical results show that the dynamic stiffness and contact stress in time domain undergo initial oscillations, followed by a gradual rise and eventual stabilization. The proposed dynamic stiffness formulation is applicable to both elastic and viscoelastic soils, offering practical utility for assessing structural impacts due to machine vibrations.