## 1 Introduction  
Pile inclusion is considered as one of the most versatile and cost-effective soft soil improvement techniques for its distinct advantages of high construction speed, small total/differential settlement, and low construction cost [1; 2; 3; 4; 5; 6; 7]. This technique consists of a grid of piles driven through the soft layer and embedded in an appropriate substratum or the bedrock with an embankment placement above the pile, as shown in Figure 1.  
The embankment load is transferred to substratum via the pile. To strengthen the effect of the pile, the pile head area should be increased while the pile shaft area remains unchanged. A common technique is to set square or circular concrete slabs on the pile top [8; 9]. However, the major limitation of this technique is that it necessitates two separate processes (completion of the pile and installation of the precast slab) and is not easy to carry out [10]. Therefore, a circular pile with a conical head, a type of the variable section pile, can be designed under the concept of the variable section pile. The pile can be constructed at one time, which sec  
Figure 1: Types of pile-supported embankments: (**a**) End-bearing pile (**b**) Floating pile. Compared with the equal section pile, the pile with a cone cap (a type of variable section pile) can convert partial vertical load into horizontal load due to its unique geometry. This characteristic can effectively reduce the negative friction and improve the bearing capacity of the composite foundation. [PERSON] [11], [PERSON] [12] have conducted a series of tests on the tapered pile (a type of variable section pile). According to their experiment results, it indicates that the bearing capacity of the tapered pile is far superior to that of the equal section pile. [PERSON] [13] carried out load tests on several types of piles in permanently frozen soil, such as the tapered pile and the equal section pile. Based on the experiment results, we can see that the bearing capacity of the tapered pile is far superior to that of the other types of pile. [PERSON] [14] also has conducted a series of model tests on the pile with a tapered cap in sandy soil, with the conclusion that little negative skin friction is generated within the tapered cap zone. The circular pile with a conical head, a type of variable section pile, has been successfully applied in a pile-supported embankment near Bourgoin-Jallieu (France). As for this, [PERSON] [15] made an introduction about the construction of a circular pile with a conical head and conducted a numerical study. In summary, the researches on the variable section piles are carried out on the bases of experiments and numerical analysis, and rarely on theoretical analysis. Generally, the theoretical analysis of the pile-supported embankment includes two calculation steps: one is the soil arching effect; the other is the pile-soil interaction. Theoretically, the existing soil arching models can be roughly classified into three groups [4; 8; 16]. The first is called the rigid arch model, such as the enhanced arching models [17] and Scandinavian models [18; 19; 20; 21]. It is assumed in the model that an arch of a fixed shape can be formed. The load above the soil arch is delivered directly to the pile and the weight below the soil arch is carried by the soft soil. However, in this model, it fails to consider the influence of some physical properties of the fill, such as the friction angle and the compactness. The second is called the limit equilibrium model, such as the Hewlett and Randolph model [22] and the concentric arch model [8]. In the model, it is assumed that the height of the soil arch is 0.5 times the net spacing of pile and the dome or arch feet is in an ultimate stress state. Another is the frictional model [9; 23; 24; 25], which assumes that the friction exists in the vertical plane along the edge of the pile cap and the equilibrium of the soil is considered based on the [PERSON] [26] theory.