## 1 Introduction  
The variations in pipe diameter are one of the most important performance indicators of buried pipe systems [1, 2]. It is therefore, a design criterion of 5% or 7.5% of the pipe diameter is recommended for flexible pipes [3, 4, 5]. Notably, [PERSON] and [PERSON] [6] conducted a comprehensive study using laser scanners and video recordings to assess the deformation characteristics of buried HDPE pipelines spanning a total length of 4572 m across 96 pipelines. Their findings revealed that a substantial majority, 63%, exceeded the deformation limit proposed by AASHTO [5]. In buried pipe systems, the arching phenomenon is employed to reduce stresses acting on the pipe and pipe deflection. Pipe deflection can be defined as either vertical expansion and horizontal shortening or vertical shortening and horizontal expansion of the pipe diameter. To achieve this, researchers recommend the use of different low-stiffness materials (such as EPS) in different buried structures and burial depths, under different loading conditions and in various geometries [7, 8, 9, 10, 11, 12]. While these materials significantly reduce stresses and pipe deflection, the resulting surface displacements are larger than of the original case [13]. [PERSON] et al. [14] explored the use of geocell to minimise surface displacements resulting from the use of EPS in shallow buried pipes with successful outcomes. However, they also pointed out that the combined use of EPS and geocell was not economical. One design option involves predicting new stress distributions at key points of the pipe section due to pipe deflection and incorporate this into the design. In this context,an understanding the parameters influencing pipe deflection behaviour is essential. In shallow buried flexible pipes, several parameters impact pipe deflection. These include the depth and stiffness of pipe burial, the properties of the backfill surrounding the pipe, the compaction technique and degree, the soil support supplied to the pipe, the location and frequency of load application, the stiffness of the pavement, and the distribution of stresses around the pipe [15; 16; 17; 18; 19; 20; 21; 22]. Flexible pipes exhibit either deflection or vertical extension behaviour when loaded. If horizontal stresses (\(\sigma_{\text{h}}\)) and vertical stresses (\(\sigma_{\text{v}}\)) around the pipe are equal, variations in the pipe section is unremarkable (Figure 1a). However, if \(\sigma_{\text{h}}\) is more dominant than \(\sigma_{\text{v}}\), the pipe will extend vertically (Figure 1b). Conversely, when \(\sigma_{\text{v}}\) is more dominant than \(\sigma_{\text{h}}\), the pipe deforms and expands laterally (Figure 1c). With the deflection of the pipe, some of the load it carries is transferred to the soil prism on the sides of the pipe. This load transfer mechanism involves the pipe displacing more downward than the surrounding soil, leading the central soil prism to move along with the pipe. As the central soil prism tends to move downwards, it interacts with the surrounding soil prisms in this case, facilitating the transfer of some of the load to the adjacent soil through friction forces (Fv) (Figure 1d). This process is a phenomenon known as \"positive arching\" [23; 24; 25; 26; 27; 28]. [PERSON] [25; 26] developed a test setup (trapdoor) with a moving door underneath and used it to investigate the effects of soil arching. During the test, the trapdoor was gradually moved downwards. At each stage, both the downward displacement of the trapdoor and the soil loads acting on the trapdoor were measured. Observations showed that the soil load decreased rapidly as the trapdoor moved downwards. [PERSON] [25; 26] noted that the soil loads acting on the trapdoor were equal to the difference between the weight of the soil between the shear planes and the \(\text{Fv}\) generated along these planes. The \(\text{Fv}\) occurring at a specific height is the basic parameter creating the arching (Figure 1d).