## 1 Introduction  
Pile foundations are commonly used for large bridges, buildings, and harbors due to their reliability. Variable-section piles offer advantages such as a high bearing capacity, minimal settlement and deformation, rapid construction, and strong horizontal resistance, making them widely adopted [1]. In engineering, vertical piles can also become inclined due to lateral loads, leading to different mechanical behavior compared to strictly vertical piles. The factors influencing inclined piles are complex. However, due to the intricate nature of variable-section piles, traditional pile foundation analysis methods are inadequate for understanding their mechanical behavior. In their study on variable-section piles under horizontal loading, [PERSON] et al. (2010) [2] discovered that increasing the size of the pile body's variable section significantly enhances the pile's lateral bearing capacity and reduces displacement at the pile top. They attributed this improvement to the increased bending strength resulting from the larger cross-sectional area of the pile. Field tests on the horizontal bearing characteristics of variable-section piles reinforced the need for a comprehensive analysis of the pile foundation mechanisms, emphasizing the interplay between the pile bearing capacity and co-ordinated pile-soil deformation [3, 4]. The horizontal loading and vertical loading performance of inclined piles in sandy soil were investigated by a model test, which is of great significance for recognizing the bearing and deformation characteristics of inclined piles in sandy soil [5]. Some scholars have combined variable-section piles with engineering to replace steel pipe pile foundations with variable-section piles to provide the required bearing capacity for offshore oil platforms [6]. [PERSON] et al. (2016) [7] carried out a model test of a retaining wall combined with an inclined pile support pit in marine clay, and the results showed that the inclined pile reduced the lateral displacement of the support structure by about 40%. [PERSON] et al. (2001) [PERSON] et al. (2001) gave a numerical analytical solution for displacement by theoretical calculations for the force characteristics of inclined piles, developed a finite element model for a static and cyclic load analysis using an incremental iteration method, and gave a numerical solution for single-pile load displacement. [PERSON] et al. (2016) [PERSON] et al. (2016) studied the properties of horizontally loaded inclined piles, pull-out inclined piles, and vertically loaded inclined piles in sandy soils through model tests, which is of great significance in recognizing the bearing and deformation characteristics of inclined piles in sandy soils. The current studies have ignored the soil crowding effect of the immersed piles, and are unable to accurately analyze how the disturbance of the surrounding soil during pile filling affects the bearing capacity of the foundation piles. The real deformation of the soil body cannot be observed using conventional model tests and field tests. By laying earth pressure transducers inside the soil body, not only is it easy to be damaged, but it also has a large error, which makes it impossible to obtain accurate test results, and the number of data points that can be observed is also limited ([PERSON] et al., 2016). Transparent materials mimicking natural soil were chosen to simulate its properties. Various experiments were conducted to determine the physical and mechanical characteristics of these materials and apply them in model experiments studying soil deformation and displacement ([PERSON] et al., 2016; [PERSON] et al., 2017; [PERSON] et al., 2018; [PERSON] et al., 2018; [PERSON] et al., 2018). In the fields of agriculture and ecological engineering, the motion of granular media can be better understood by measuring the displacement field of the granular media and trends associated with the displacement can be better determined ([PERSON] et al., 2018; [PERSON] et al., 2018; [PERSON] et al., 2018).