## 1 Introduction  
The construction of subway tunnel shielding in urban settings disturbs the original stress state of the surrounding soil [1; 2], which may lead to the cracking, tilting, and even collapse of preexisting buildings [3; 4; 5; 6; 7; 8]. Therefore, it is essential to protect buildings from damage during shielding excavation. Engineers need to possess adequate specialized knowledge in order to estimate the potential ground movements so that they can assess whether the damage of neighboring buildings would be serious. Early studies focus on using elastic mechanics and soil mechanics to solve the analytical solutions of ground deformation considering simplified tunnel excavation boundaries [9]. For example, [PERSON] [10] used the Galerkin method to derive the formation of stress fields and displacement fields under the action of concentrated forces in elastic half spaces. Based on this form, [PERSON] [11] proposed a general solution in the form of Airy's function to describe the ground deformation caused by tunnel excavation. [PERSON] [12] assumed that the soil was an incompressible, isotropic, elastic semi-infinite body and adopted the mirror image method to eliminate the influence of the top free boundary. The soil strain and stress field caused by formation loss at a depth below the surface was then analyzed. [PERSON] [13] adopted a complex function method to derive the analytical deformation solution of the ground caused by circular tunnel excavations in elastic half spaces. Empirical solutions can extend application conditions of the proposed analytical solution. The empirical formula for predicting ground settlement caused by tunnel excavations was first proposed by [PERSON] [14], and the Gaussian normal distribution curve was used to describe the ground settlement profile. Later, many scholars further improved [PERSON]'s [14] equation according to different geological conditions, such as [PERSON] et al. [15], [PERSON] et al. [16], and [PERSON] et al. [17]. [PERSON] et al. [18] pointed out that [PERSON]'s [14] equation and its modified forms have unavoidable calculation errors due to not considering the soil's three-dimensional (3D) space effect. A 3D numerical simulation method was gradually developed to analyze ground deformation [19; 20; 21; 22]. [PERSON] et al. [23] presented a case study of ground settlements induced by twin shield tunneling in Copenhagen using analytical and numerical methods. Although the numerical method has advantages in simulating complex boundary conditions and capturing the 3D space effect, it has shortcomings that make it highly time consuming challenging to model ([24; 25]). Therefore, it is desirable to develop a simple and practical method for calculating the ground movements. In addition, there are imbalances in many large soil-tunnel structures caused by shield tunnels passing under existing tunnels, which is another common and important problem that has been studied by many scholars [26; 27; 28; 29; 30]. [PERSON] et al. [31] studied the effect of shield excavation on existing pipelines by using the boundary integral method, and gave the normalized solution of calculating the maximum bending moment and angle. [PERSON] et al. [32] studied the soil disturbance caused by multi-line tunnels and the complex overlapping interaction mechanisms of adjacent tunnels. Combined with the field monitoring data of tunnels in Shanghai, the theoretical solution of the existing tunnel is obtained. [PERSON] et al. [33] pointed out that shield tunneling affected the stability of existing tunnels, and it was crucial to make a reasonable supporting scheme. At present, the protection measures for existing tunnels are building enclosure structures to cut off building and existing tunnels [34; 35] and strengthening the surrounding soil with grout [33; 36]. Grout fluid is difficult to control and easy to lose, so enclosure structures can better control the soil and the existing tunnel's deformation. Shield tunneling would produce additional thrust acting on the enclosure structure.