## 1 Introduction  
The construction of urban subway stations frequently occurs in densely populated areas. The excavation of foundation pits can lead to significant risks, including damage to existing subway tunnels, ground subsidence, structural damage to buildings, the uplift of diaphragm walls, and the failure of support systems [1, 2, 3, 4, 5, 6, 7, 8]. The iterative update of subway construction methods has greatly promoted the rapid development of infrastructure. [PERSON] et al. [9] compared conventional and widely used methods; an appropriate and safe support system was proposed for excavation in urban areas depending on construction, structural, and economic conditions. [PERSON] et al. [10] established a three-dimensional finite element model by considering a top-down construction method, a supporting system of concrete slabs, and the influence of the soil modulus. [PERSON] et al. [11] studied the influence of lime-cement stability on foundation pit support by using the finite element method. The 2D finite element analyses were performed with advanced constitutive models for both natural clay and stabilized soil, performing a laboratory test made on both field-mixed and laboratory-mixed stabilized clay samples. [PERSON] et al. [12] carried out a 3D numerical simulation study of the Chiaia station of the Naples Metro Line 6 to predict and control the deformation of a deep foundation pit in order to safeguard the safety of the adjacent Basilica Santa Maria degli Angeli. [PERSON] et al. [13] analyzed the similarities and differences in force and deformation between foundation pits excavated by the zonal method and those excavated by the conventional method. Nevertheless, the considerable excavation depth associated with foundation pits continues to pose challenges, particularly concerning excessive deformation. Previous studies have indicated that the deformation resulting from the cover-and-cut method is generally smaller than that associated with the cut-and-cover method; however, the mechanism of the deformation of the surrounding environment caused by different excavation methods is still not yet fully understood [14,15]. For instance, [PERSON] et al. [16] examined a foundation pit in Bangkok using the top-down method and analyzed deformation and stress patterns when employing a deep mixing cement-soil wall as a retaining structure. Their findings revealed that the construction of the structural slab effectively mitigates the horizontal displacement of the retaining wall. [PERSON] et al. [17] proposed the channel excavation method. [PERSON] et al. [18] compared a foundation pit project constructed using the top-down method with relevant international case studies to analyze the mechanical behaviors observed during excavation. [PERSON] et al. [19] explored deformation and stress characteristics throughout the top-down construction process by measuring the stress of the middle columns, the displacement and bending moment of the diaphragm wall, and the ground surface settlement. Their observational results emphasize the necessity of considering creep factors during excavation analysis in soft clay conditions. [PERSON] et al. [20] proposed a deformation algorithm for retaining structures in a foundation pit constructed via the nonlinear top-down method and evaluated its predictive accuracy through comparison with the measured data. [PERSON] et al. [21] present rational design strategies based on field and numerical analyses to optimize stability while reducing the length of soil nails for a comprehensive understanding of deformation control in deep excavations, particularly in the context of alternative retaining systems such as soil nail walls. They demonstrated that numerical simulations can effectively predict soil deformation. The intermediate columns represent a unique structural system in the top-down method. During the construction process, these columns endure substantial vertical loads, which prompts extensive research on their displacement characteristics and differential settlement patterns.