## 1 Introduction  
In recent years, the increasing urban population density and residents' travel demand have brought huge traffic load to the limited ground space. The underground railway constructed by shield tunnelling has advantages of large unit transport, fast transport speed, high punctuality, and almost no influence from external weather, which can effectively mitigate the ground traffic congestion and gradually become the first choice for residents to travel in the urban area. However, shield tunnelling often needs to pass through a huge number of existing pile foundations at a close distance due to the large quantity of infrastructure in metropolitan areas [1; 2; 3; 4; 5]. Tunnelling-induced ground surface settlement, inclination, and discontinuous displacement could affect or even damage adjacent superstructures. The isolation pile between the existing pile and the tunnel is widely proven as an efficient protective measure for controlling the displacement of building's pile foundation induced by shield tunnelling [6; 7; 8; 9]. In general, the surface settlement trough induced by shield tunnelling is approximately a Gaussian distribution in transverse direction [10]. However, when the isolation pile is pre-installed, the stiffness of the isolation pile is substantially larger than that of the surrounding soil, resulting in an asymmetrical distribution of surface settlement trough. Currently, various scholars have focused on this issue and conducted several studies. [PERSON] et al. [11] first presented a plane strain finite element analysis, in which a continuous diaphragm made by adjacent panels was established to evaluate the effects of various geometrical and mechanical parameters. [PERSON] et al. [12] held the opinion that surface settlement trough based on Peck formula was not reasonable. Surface settlement trough will present cork distribution, skewed distribution, and normal distribution according to the scope of the disturbance when shield tunnelling passes through adjacent buildings. [PERSON] et al. [13] highlighted that the surface settlement trough curve presented a 'hanging phenomenon' near the adjacent pile foundation. The existence of pile foundation will restraint the displacement of soil around pile induced by shield tunnelling. [PERSON] et al. [14] presented a numerical study to evaluate the efficiency of different schemes of embedded barriers. The results illustrated that the diaphragm wall could reduce both surface settlements and curvature of surface settlement trough beyond its location. [PERSON] et al. [15] conducted a series of soil-structure interaction finite element analyses to evaluate the efficiency of a barrier made of a line of bored piles, comparing with field monitoring data from a test site in Rome and simple empirical equations. [PERSON] et al. [16] proposed a three-dimensional linear elastic prediction method to evaluate the protective action of pile walls against surface and subsurface ground movements due to tunnel excavation. The barrier efficiency in reducing settlements is explored by comparing pile walls and diaphragm walls. [PERSON] et al. [17] established three-dimensional finite element models to analyze the effects of different pile diameters and different distance between pile and tunnel on the surface settlement trough. An asymmetric skewness distribution curve was proposed by fitting the simulated data. [PERSON] et al. [18] proposed an elastic solution for evaluating tunneling-induced vertical ground displacements due to the restraint of embedded isolation piles. Parametric analysis indicates that parameters associated with the isolation pile and the soil have significant influence on the surface settlement troughs and the barrier efficiency. To sum up, the above theoretical studies, numerical simulations, centrifugal tests, and field measured data adequately describe the restraint effect of isolation piles.