## 1 Introduction  
As a particular soil, loess has unique engineering geological characteristics [1, 2]. During the construction of loess tunnels, problems such as surface cracking, primary lining misalignment, lining deformation, and even landslides are common and need to be urgently addressed. In recent years, with the construction of railway line projects such as Zhengxi Passenger College in China, experts have conducted a series of technical research and achieved certain results. [PERSON] et al. systematically summarized the technical characteristics and main problems of the large-section loess tunnels on China's high-speed railway [3]. [PERSON] et al. studied the supporting theory, design method, and construction technology of highway loess tunnels [4]. They proposed the theory that the systematic bolt cannot restrain the deformation of the loess tunnels and established the method of combining steel frame and foot steel pipe (FSP) in the loess tunnels to control deformation. However, the supporting means are still not perfect, and the significant deformation problem of loess tunnels still occurs from time to time. The FSP is essential to the soft foundation tunnel support. It has the characteristics of convenient and fast construction, high rigidity, and remarkable effect and is often used for vault settlement control [5]. Many studies have been carried out on FSP at present. Regarding the stress and deformation characteristics of the FSP, [PERSON] et al. believed that when the FSP is subjected to external loads, the deformation characteristics are similar to those of the bearing piles subjected to horizontal loads [5]. The supporting force mainly comes from three aspects: the normal pressure generated by the relative extrusion with the side wall of the FSP and the surrounding rock, the frictional resistance caused by the relative sliding of the surrounding rock and the side wall of the steel conduit, and the supporting force generated by the surrounding rock at the bottom of the FSP. [PERSON] et al. used the elastic foundation beam to establish the mechanical calculation model of the FSP [6]. They used the series solution method to solve the beam's differential equation to calculate the FSP's stress state. Based on the ultimate bond strength between the surrounding rock and the cement mortar, it is judged whether there is penetration failure between the FSP and the surrounding rock when the anchor bolt is subjected to the axial force. [PERSON] used the variable foundation coefficient method to derive the deformation characteristics of the FSP [7]. The research of [PERSON] et al., [PERSON] et al., and [PERSON] et al. on reinforced concrete single piles in soft clay strata also have reference value for the study of FSP [8; 9; 10]. Mastering the force and deformation characteristics of the FSP helps evaluate and analyze its application effect in practical engineering. Regarding the field application of FSP in loess tunnels, [PERSON] et al. believed that FSP was not loaded independently but was welded together with the steel arch to carry the load as a whole [11]. [PERSON] et al. also used structural mechanics and elastic foundation beam theory to establish a theoretical model of the overall bearing capacity of the tunnel support system under the shallow and weak surrounding rock [12]. [PERSON] et al. used numerical analysis software to establish a three-dimensional calculation model of a loess tunnel, selected beam elements to simulate the FSP, and analyzed the influence of the length and insertion angle of the FSP on the deformation of the surrounding rock [13]. [PERSON] found through simulation that increasing the downward slope angle of the FSP has a certain effect on controlling the vertical displacement of the surrounding rock [14]. However, a smaller downward slope angle of the FSP is beneficial for reducing the horizontal convergence of the tunnel. The above analysis shows that different design parameters of the FSP lead to different supporting effects of surrounding rock.