## 1 Introduction  
Since the 21 st century, tunnel and underground engineering in China have made great progress [1; 2; 3; 4; 5; 6; 7]. With the continuous development of urbanization in China, the urban population is growing, and urban traffic pressure is also increasing. In order to alleviate the urban traffic pressure, city managers have taken a lot of measures, such as limited number driving, expansion, and transformation of existing roads, but the traffic demand is still far greater than the current urban ground carrying capacity. Metro has the advantages of large passenger capacity, fast speed, and full use of underground space, which is more and more popular in modern society. However, the urban subway has strict requirements for its construction methods. Because of the high density of buildings in cities, the construction of the metro will cause stratum movement and bring adverse effects on ground buildings. Especially when passing through high-rise buildings, the disturbance caused by construction on stratum must be more strictly controlled. The construction period of Metro is long, and the interruption of traffic should be avoided as far as possible when crossing the road. Usually, subway construction methods include the open-cut method, shallow-buried excavation method, and shield tunneling method. Among them, the shield tunneling method has become the preferred construction scheme for metro construction because of its great advantages of fast construction speed, a high degree of automation, and small disturbance to stratum [8; 9]. Although shield tunneling has many advantages mentioned above, and the construction technology has made great progress after many years of development, due to the defects of geological conditions and construction technology, the advance of shield tunneling will inevitably make a disturbance, change the stress state of soil, and cause stratum displacement, as shown in Figure 1. If the disturbance of stratum cannot be clearly understood, it will affect the safety of nearby underground structures and aboveground buildings. For example, the adjacent metro tunnels may cause deformation restrictions beyond the normal operation of the metro, or even lead to train derailment; it may also lead to the breakage of underground pipelines, resulting in a series of problems, such as gas leakage, interruption of urban water use, interruption of communication power system, and so on, affecting the daily life of the city [10; 11; 12]. Therefore, a reliable prediction of stratum movements is important. Over the years, previous researchers have done numerous studies on the effects of shield tunnel construction on stratum displacement [13; 14; 15; 16; 17; 18]. In the direction of theoretical research, [PERSON] assumed that the curve of land settlement trough satisfies the conditions of Gauss distribution and invariant stratum volume, and deduced [PERSON]'s empirical formula based on many field monitoring data [19]. [PERSON] et al. used [PERSON]'s empirical formula to simulate the free displacement of soil at the location of existing tunnels [20; 21; 22; 23; 24], and then deduced the displacement of existing tunnels with elastic foundation beams and other models. However, [PERSON] and [PERSON] [25; 26] gradually found that the Peck curve could not accurately describe soil settlement trough in many cases. [PERSON], [PERSON] et al. [27; 28; 29; 30] have proposed fitting curves from different research angles to fit the free displacement field of soils. [PERSON] [31] put forward the theory of random medium through a sand box test. [PERSON], [PERSON], et al. [32; 33; 34] introduced this theory into the prediction of stratum displacement caused by the excavation of geotechnical tunnels. Although these studies have the advantages of concise calculation formula and convenient use, they lack a clear theoretical basis of mechanics. They can only be called a mathematical empirical method, not an analytical solution of soil settlement.