###### Abstract  
This paper focuses on improvement effects on soil foundations under dynamic compaction (DC). Firstly, a confocal ellipsoidal densification model (CEDM) composed of a heavy compacted zone (HCZ) and a weak compacted zone (WCZ) was proposed to describe the subarea characteristic of an improvement range. Next, based on a confocal assumption of HCZ and WCZ ellipses, a mass balance equation considering changes in soil dry density in different compacted zones was established for solving the ellipsoidal parameters. Then, a designed laboratory test was conducted and a two-dimensional (2D) finite element model (FEM) established. The simulated crater depth and dynamic stress agreed well with testing results, confirming that the established FEM could be used for investigating the DC process. Finally, the applicability of the solution procedure for the proposed CEDM was verified. The predicted HCZ and WCZ were in close agreement with the simulated results, indicating that the proposed CEDM could be used for estimating the soil improvement range. With increases in tamping times, the HCZ ellipse moved down in the vertical direction without volumetric expansion, while the WCZ ellipse expanded along the depth and lateral directions. These findings may offer some guidelines for research into improvement effects on soil foundation under DC. dynamic compaction (DC); confocal ellipsoidal densification model (CEDM); dry density; improvement range; laboratory test; finite element model (FEM) Article  
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[PERSON] et al. [3] collected 120 site measurements and obtained general trends involving variations in crater size and depth of influence with damping energy, which was also selected as a key index by [PERSON] [4] and by [PERSON] et al. [5] for calculating crater depth. [PERSON] et al. [6; 7] reported the effectiveness of high-energy DC and suggested different empirical factors for coarse-grained soil and desert sand. [PERSON] et al. [8; 9] analyzed the effect of normalized impact energy on crater depth for dry sand through model tests and numerical simulations. [PERSON] et al. [10; 11] used ram momentum to calculate crater depths under heavy tamping based on centrifugal model tests. [PERSON] et al. [12] studied how the evolution of crater depth varied with the number of blows. In addition, numerical analyses have been conducted to estimate crater depth under the influence of DC operating parameters [13], as well as soil constitutive models [14; 15]. Although there are some differences among pre-estimations of DC design and construction parameters, these extended empirical formulas can provide easy and convenient guidelines for determining relevant DC parameters in terms of single-point tamping. However, the dynamic stress wave generated by DC does not only densify soil beneath the tamping location, but also in the area around this location [16; 17]. Thus, the estimation of improvement depth and lateral extension also plays an important role in DC design and construction, especially for multiple-point tamping. It can help in ascertaining tamping spacing, tamping order, and the grid pattern. Accordingly, various empirical relationships have been developed to predict improvement depth and lateral extension based on simplified theoretical models [18; 19] and field measurements [20]. Moreover, numerous numerical methods have been adopted to evaluate improvement ranges. [PERSON] et al. [9] obtained the dimensions of volumetric strain contours for dry sand using a finite element model (FEM). [PERSON] et al. [14] embedded a cap model into the ABAQUS program for simulating successive DCs. [PERSON] et al. [21] developed a fully coupled hydromechanical FEM to evaluate the DC effect on saturated granular soils. Since then, FEM simulations have been used to describe improvement ranges with considerations of energy efficiency [22], multi-point tamping [23; 24], and the groundwater table [25].