## 1 Introduction  
Loess is a kind of loose deposit formed by wind transport in the quaternary [1]. The distribution range of loess on the earth's surface is extensive, covering more than one-tenth of the world's land area, and most of the loess is distributed in the mid-latitude region with more arid climate [2, 3]. The thickness of the Loess Plateau in China is between 0 and 350 m, and its mean thickness is 92.2 m [4]. Silt and clay particles are the main granularity compositions of the loess. The mineral compositions of the loess consist of detrital minerals and clay minerals. The content of detrital minerals exceeds 70%, mainly including quartz (SiO\({}_{2}\)), feldspar (mainly consists of alkali metal oxide), and carbonate minerals [5, 6]. Loess has high shear strength in its natural state. However, the shear strength of the loess will decrease rapidly after encountering water, which means that loess shows collapsibility [7]. According to [PERSON] et al. [8], about 60% of loess is collapsible, which means that under the influence of water and overlying pressure, the cementation between loess particles is destroyed and pores collapse, eventually leading to quick settlement [9]. The collapsibility does extreme harm to engineering constructions. For pile bases, the collapsibility of loess will significantly reduce the bearing capacity of pile bases, produce uneven settlement of building bases, and cause the collapse of superstructure in severe cases, leading to a series of engineering accidents [3; 9; 10]. Therefore, it is of great significance to study the mechanical properties of adjacent pile bases in collapsible loess [11]. Four main methods exist to research the pile-soil interaction: model tests, field tests, theoretical analysis, and numerical simulation. Lots of studies about pile-loess interaction have been carried out through model tests and field tests. [PERSON] et al. [7] researched the interaction between concrete and steel piles with and without drive pipe and collapsible loess with and without immersion through model tests. [PERSON] et al. [12] compared the interaction results between a single pile and group piles with collapsible loess through centrifugal model tests. [PERSON] et al. [13] explored the influence of cumulative relative collapse amount, pile type, and change of loess collapsibility on the results of the interaction between group pile and collapsible loess through field testing. [PERSON] et al. [14] analyzed the effect of ground flooding on the bearing capacity of bridge pile bases in the loess area by conducting field immersion tests in the arid loess area. In addition, more and more researchers have found that the collapsibility of loess has a detrimental impact on the bearing capacity and settlement of pile bases through laboratory and field tests. According to these studies, friction pile is the most common bearing type in collapsible loess areas. The collapsibility of loess will make the value of shaft friction transform from positive to negative, which can lead to a remarkable decrease in load borne by side friction resistance [15; 16; 17; 18; 19]. Based on many model and field tests, researchers have begun to select the approach of theoretical analysis to study and forecast the regulation of bearing capacity of the piles in collapsible loess stratum. [PERSON] et al. [20] proposed a prediction method to evaluate the collapsibility of loess around piles by considering the collapsibility deformation characteristics of the loess-pile interface and side friction and verified it in three engineering examples. [PERSON] et al. [21] presented formulas for calculating the interaction between the pile and surrounding soil when loess is immersed in water on the basis of the results of laboratory and field tests. These theoretical formulas, founded on test results, have relatively high guiding significance for engineering practice. Loess covers an area of 640,000 km\({}^{2}\) in China; it is mainly concentrated in the western region of China, and most loess in China is collapsible [11].