# Evaluation on Improvement Zone of Foundation after Dynamic Compaction
## 1 Introduction  
Dynamic compaction (DC) is a commonly used method to improve the bearing capacity of various kinds of soils [1, 2, 3, 4, 5]. At present, the most commonly used design approach is [PERSON]'s [6] framework, which is entirely empirical with low accuracy for predicting the improvement depth. Moreover, site-dependent parameters such as soil properties as well as equipment factors [7, 8] like tamping distance, tamper radius, tamper shape, and drop number may also influence the improvement depth of the compacted soil. Up to now, a significant amount of numerical studies [9, 10, 11, 12], physical model tests [13, 14], and field tests [1, 15, 16, 17, 18] have been reported on the improvement depth of DC; however, the majority of these works mainly focused on the improvement range below the tamper and only dealt with the soil response by single-point tamping. In reality, the layout of DC improvement is typically in a rectangular or triangular grid pattern to conduct multipoint tamping. In terms of multipoint tamping, [PERSON] et al. [19] proposed an empirical method for assessing the compaction effect between impact points, which is based on the relationship between normalized friction angle and dimensionless distance by single-point tamping. Two design approaches to choose the tamping distance of DC were presented. However, the assumption of a constant frictional angle along with depth may lead to an inaccurate prediction of the tamping distance. [PERSON] [20] proposed a novel method to determine the corresponding tamping distance. Although the forecasting methods have made great progress, compared with the method of [PERSON] et al. [19], the assumption of asymmetry in a compacted zone around the centerline of the impact point causes limitationson the feasibility of damping distance. [PERSON] et al. [21] used the LS-DYNA software to study multipoint tamping in granular soils. However, they only modeled two blows at each impact point, which is far from enough to ensure the stabilization of the DC. [PERSON] [22] also proposed a method to estimate the relative density with the depth between impact points. However, the assessment curve must be benchmarked by five special points and the relevant crater depth has to be estimated first. Several sandy field tests have shown that the triangular grid patterns are more efficient than the rectangular grid patterns in the case of small areas of DC treatment [23; 24; 25; 26]. However, those field investigations aim to come up with the DC construction parameters based on some specific project cases, which may be not suitable for other projects. In general, a reasonably comprehensive design framework that can account for different sandy soil properties and construction parameters to guide the practical DC design for multipoint tamping remain unavailable.  
In this paper, a three-dimensional finite element analysis with an improved cap soil model was conducted to simulate the single-point and the multipoint tamping on dry sandy ground. The numerical analysis was first validated by using centrifuge model results from the published literature, then the influences of drop energy, drop number, tamper radius, tamping distance, as well as drop momentum on the relative degree of improvement, were discussed in detail. After that, a prediction formula considering the factors affecting relative degree of sandy soil improvement was proposed based on the simulation results. Finally, the accuracy and reliability of the proposed formula were validated by using the sandy field test results. It should be noted that the soils studied in this paper are moist and dry granular soils with high permeability, which can be assumed to be dry soil; saturated or unsaturated soils are not involved.