# Numerical Analysis of the Settlement Behavior of Soft Soil Improved with Stone Columns
## 1 Introduction  
The design of foundations and embankments over soft soils is a challenging problem for geotechnical engineers due to the undesirable settlement and bearing capacity performance. Several ground improvement techniques are used to overcome the difficulties associated with the high compressibility and low-shear strength of soft soils [1]. Piled embankments and vibro-stone columns are considered the most popular solutions for the construction of roads, railways, and foundations over soft soils. These ground stabilization methods surpass other conventional ground improvement methods (e.g., preloading) as they are relatively rapid to construct, and subsequent structural works can follow very quickly [2]. Stone columns can be installed to depths up to 15 m and offer an economical and environmentally friendly alternative to piled foundations when weak soil has to be improved [3]. Experimental studies have shown that stone columns have different modes of failure, such as bulging, punching, shear, and bending [4]. The hypothesis of the upper part of a single stone column failing by bulging, with a bulging zone of four column diameters, was confirmed by experimental [5] and field tests [6]. Empirical and semi-empirical design methods are still being used in practice with most of them implementing the unit cell idealization [7; 8; 9; 10; 11; 12]. However, these theoretical methods tend to be conservative and do not take into account the presence of a stiff crust at shallow depths that is common in weak soils and beneficial to the deformational mode of stone columns [13]. The performance of an ordinary stone column (OSC) can be improved by adding a cylindrical encasement of geosynthetic material. Experimental studies [14; 15; 16; 17] and numerical simulations [18; 19; 20; 21; 22] have investigated the effectiveness of encased stone columns (ESC) under vertical loads. It has been shown that the load-carrying capacity is increased and settlement is reduced when the stone column is surrounded by a cylindrical encasement. The circumferential or hoop forces developed in the geosynthetic material prevent excessive bulging at shallow depths, and the stiffness of the geogrid material can have a positive impact on the performance of the ESC. The progress of computer power has made numerical analyses of complex problems in geotechnical engineering a popular approach. Researchers have adopted different simplified numerical models to predict the performance of stone columns, the unit cell model [23], homogenization method [24], plane strain technique [10; 25], axial symmetry technique [26; 27], and full 3D model [28; 29]. Most of these studies have implemented the finite element (FE) method to simulate a large array of stone columns supporting a widespread loaded area (e.g., embankments and slabs). The length and diameter of the stone column were found to have a significant impact on the settlement performance with improved behavior identified for a length over diameter ratio beyond ten [29]. The installation of stone columns alters the surrounding soil, leading to an increase in horizontal stress which needs to be introduced in the numerical modeling. Increasing the lateral earth pressure coefficient at rest, K\({}_{o}\), from 0.75 to 1.5 is commonly used to change the initial stress state due to installation [29; 30]. The application of a cylindrical cavity expansion controlled by displacement or stress is considered as another method to simulate column installation effects [31]; however, it was found that excess pore water pressures cannot always be captured properly [32]. The bearing capacity and settlement performance are the two main design criteria that control the design of stone columns. It has been shown that a single stone column in soft clay fails by bulging at an upper zone with a critical length of about four times the column diameter [5; 33]. Researchers have proposed design methods for large groups of stone columns installed in soft clay [13; 34; 35].