Dynamic Stress Bulbs for Harmonic Vertical Point Loads in Homogeneous Soil

The assessment of earthquake-induced demand in the soil supporting a structure is commonly carried out in terms of stresses and should consider three components: (1) geostatic stresses; (2) stresses due to wave propagation in the absence of a structure; and (3) stresses due to dynamic soil-structure interaction (SSI). Engineers presently lack a consolidated procedure for assessing SSI-related stresses, mainly due to the absence of simple solutions for dynamic stress bulbs, analogous to those available for static loads. In this work, we revisit the fundamental problem of a harmonic vertical point load in the interior of a homogeneous viscoelastic full space (Stokes problem). Due to antisymmetry, the full-space can be conceptually divided into two vertically loaded, bonded half-spaces separated by a fictitious planar interface that is free of normal stresses. While shear stresses may still be present on the interface (and this differentiates the full-space from the half-space problem), in the important case of an incompressible medium, the interface shear stresses vanish and the two problems become equivalent. For other values of Poisson's ratio the equivalence is not perfect, yet the two problems remain sufficiently similar so the main observations from the analysis of the full-space problem can be readily extended to the case of surface loads. We present an extensive set of dimensionless graphs for displacements, strains, and stresses as functions of dimensionless frequency-distance (ωR/Vs) and aperture angle (φ) from the axis of the load, and we demonstrate that the solution is best expressed in spherical coordinates. The limit behavior at low and high frequencies and the effect of damping and Poisson's ratio are discussed. It is shown that for small values of ωR/Vs (0 to 1), the stresses from dynamic loads are similar to static values, whereas higher values of ωR/Vs produce larger dynamic stresses due to wave interference. Application examples are presented.