## 1 Introduction  
Soil improvement is becoming increasingly important due to the lack of quality land for the development of structures and transport infrastructure [1; 2; 3; 4], and at the same time, the need for rapid construction of embankments for highways and railways over soft ground [5; 6; 7; 8]. There are various methods of soil improvement, and among these methods, columnar inclusion is one of the most effective techniques [8; 9]. The columnar inclusions such as stone columns, sand compaction piles, jet grout columns, etc., are common and efficient techniques to improve the soft and weak ground with low bearing capacity, high compressibility, and large settlement [10; 11]. Previous studies have shown that columnar inclusions can significantly enhance the bearing capacity [12; 13; 14; 15; 16], reduce the amount of settlement [17; 18], accelerate consolidation [19; 20], improve slope stability [21], and avoid liquefaction of soil [22; 23; 24]. The role of columnar inclusion in limiting settlement is essential in some cases, for example, road embankments and land reclamations. Many theoretical or semi-empirical approaches have been proposed to study the behavior of column-reinforced foundationsand to evaluate ground settlement [25]. The simplest analytical approach to columnar inclusion is known as the \"equilibrium method\". This approach is based on elasticity theory and has been depicted by [PERSON] et al. [26]; it assumes that vertical equilibrium between the columns and the soils satisfied vertical equilibrium with oedometric conditions in the soil and ignores the axial friction was ignored [27]. Furthermore, [PERSON] and [PERSON] [28] proposed a method that can be used to get a closed-form analytical solution. Subsequently, [PERSON] and [PERSON] [29] proposed an iterative approach that requires a numerical implementation to obtain a solution. Based on a semi-experiential approach, [PERSON] [30] developed one of the most popular design methods in European practice to estimate the settlement of subsoil reinforced with an end-bearing stone column. In addition, [PERSON] and [PERSON] [31] proposed an analytical method to analyze the behavior of rigid foundations resting on soft soil stabilized by a large number of end-bearing stone columns. [PERSON] et al. [32] proposed an analytical solution to calculate the deformation of the composite foundations reinforced with stone columns by considering the radial expansion of the stone column among the surrounding soil near the top portion of the column. However, these methods can only provide results with equal vertical strain based on the \"equal strain\" assumption. For the case with different vertical strains, the approaches with the \"equal strain\" assumption cannot provide the actual behavior of the column-reinforced foundations. [PERSON] et al. [33] indicated that, for the column-reinforced embankment, the deformations of the stone column and surrounding soil are different, and cannot be described by the \"equal strain\" assumption. [PERSON] et al. [34] first paid attention to this problem and provided the deformation analysis on the reinforced foundation with columnar inclusion subjected to a uniform surcharge. The deformation field proposed by [PERSON] et al. was used by many following researchers to study the behavior of column-reinforced foundations. In particular, [PERSON] and [PERSON] [35] and [PERSON] et al. [33; 36] adopted [PERSON] et al. 's solution for the deformation of the reinforced foundation to analyze the column-supported embankment, taking into account the effect of soil arching from the embankment. The analytical method of [PERSON] et al. provided a technique to consider the effects of the column modulus on the settlement process and the stress distribution on the ground surface, which provide useful information for further analysis of soil arching in the embankment. However, the deformation pattern of the reinforced foundation in [PERSON] et al. was assumed to follow a certain shape function, from which strain and stress were derived.