## 1 Introduction  
The large settlement of tunnel vaults often occurs in soft-rock tunnels, mainly due to primary lining arch foot subsidence. The stability of the tunnel is a critical issue in the construction process [1, 2]. If the settlement of the vault is not controlled, it may cause a large-scale collapse. At present, the most commonly adopted methods to control the subsidence of the arch foot include enlarging the arch foot, connecting the longitudinal beams, and installing feet-lock steel pipes (FLSPs) [3]. Among these methods, FLSP has the advantages of simple construction and a remarkable ability to control the settlement [4] and has been extensively utilized to control the settlement of tunnel vault in soft-rock areas [5, 6, 7]. The TFP is an inclined downward steel pipe with one end inserted into the surrounding stratum, and the other end welded to the steel arch [8], as shown in Figure 1. To explore and fully exploit the function of FLSP, it is necessary to carry out an in-depth study on the mechanical characteristics of FLSP and the selection of the relevant parameters.  
At present, domestic and foreign scholars have carried out a series of studies on the mechanical characteristics of FLSP, utilizing theoretical analysis [9, 10, 11], numerical simulation [12, 13, 14], and field-test methods [15, 16]. [PERSON] et al. [11] proposed an analytical method to predict the vertical load and settlement at the feet of steel ribs with the support of the feet-lock pipe and validated this method with the results of field measurements. [PERSON] et al. [14] explored the mechanism of action of foot-side reinforcement piles (FRSP)through model tests and numerical simulations and determined reasonable parameters. The results show that when the FRSP is long enough to cross the shear line, it can play a role in shear reinforcement, load redistribution, and internal pressure. FRSP effectively prevents the settlement of the tunnel and its surrounding foundations. [PERSON] et al. [17] compared the effect of the feet-lock bolt and the systematic bolt under the condition of a cracked surrounding rock tunnel through a field test and a numerical simulation. The results show that applying feet-lock bolts is a feasible option to replace systematic bolts in cracked surrounding rock tunnels. [PERSON] et al. [15] obtained the distribution law of the axial strain and the bending moment of the FLSP through electrical measurement and verified the accuracy of their test method by comparing the deformation of the pipe body before and after loading.  
In summary, it can be seen that the theoretical research on FLSPs has been relatively comprehensive. However, there is a lack of research on the parameter selection and installation quantity of FLSPs in actual projects. FLSPs need to undergo a certain level of deformation before providing the corresponding bearing capacity. Based on the concept of prestressed steel bars, the active load-bearing of FLSP is worth exploring. In order to guide the construction and design of FLSPs, a design optimization method is proposed in this paper. Firstly, the problem of the load-deformation inconsistency of FLSPs in controlling the vault settlement is raised. Next, the process of the design optimization method is explained. Finally, the design method is applied to the Yulinzi Tunnel. The change in the vault settlement before and after the design optimization was analyzed by monitoring and measurement and is presented here.