## 1 Introduction  
With the further development of rail transit, the excavation projects of foundations adjacent to shield tunnels are becoming more common. For example, the nearest distance between the section tunnel of Shanghai Rail Transit Line 1 and the building foundation pit is only 7 m [1]. The closest distance was about 11 m between a construction foundation pit and Hangzhou Metro Line 1 [2]. The unloading effect on the sidewall due to the excavation of foundation pit leads to the deformation of the enclosure structure, which will generate additional loads on the adjacent tunnel. The excessive loads will endanger the safety of the tunnel. It has been reported an engineering case in which the tunnel section of the Panchiao line was damaged due to adjacent foundation pit excavation during the construction of the Taipei MRT system [3]. As a result, it is significant to predict the deformation of ad-jacent shield tunnel due to the foundation pit excavation. Such problems have been concerned domestically and internationally. The main research methods include theoretical calculation [4; 5; 6], numerical simulation [7; 8], centrifuge test [9] and measured data analysis [10; 11]. However, the theoretical calculation methods are almost based on the stress release method. The construction of foundation pit is regarded as the process of soil stress release. The additional stress acting on the side shield tunnel caused by sidewall unloading is calculated to establish the synergistic deformation equation of tunnel and soil for obtaining the deformation of the side shield tunnel. [PERSON] et al. [4] calculated the extra stress on the shield tunnel adjacent to a foundation pit engineering based on the Mindlin solution, the Winkler foundation model was introduced to obtain the calculation equation of tunnel displacement and internal force. However, the study considered that the stresses on the four sidewalls of the foundation pit were all released, which was inconsistent with the engineering practice. [PERSON] and [PERSON] [12] further considered the partial release of foundation pit sidewall stress under the action of retaining structure. The stress loss rate of enclosure structure \(\beta\) is introduced in the research. However, the deformation of the retaining structure was not considered in the study. [PERSON] et al. [13] predicted the deformation form of the enclosure structure as \"bow-shaped\" to get the sidewall unloaded. However, the sidewall unloading stress could not be verified. The virtual image technique [14] has been used to calculate soil displacement caused by tunnel construction or foundation pit excavation, because it can make full use of the field monitoring data and precisely calculate the soil displacement at any point caused by soil loss. Based on the virtual image technique, [PERSON] et al. [15] obtained the soil deformation due to tunnel excavation considering the compressibility of soil. [PERSON] and [PERSON] [16] considered the deformation of the enclosure structure as a quadratic parabola to calculate the soil displacement at any point outside the foundation pit due to the excavation based on the virtual image technique. Some scholars further considered the impact of soil deformation caused by construction activities on adjacent structures. [PERSON] et al. [17] obtained the displacement of soil due to foundation pit construction and then get the displacement of the adjacent pile foundation on the Winkler foundation model. [PERSON] et al. [18] further considered the Kerr foundation model to assess the impact on the pile adjacent to the foundation pit more accurately. Yet, the virtual image technique has not been applied to analyze the effect on shield tunnels beside the foundation pit excavation. To correctly evaluate the impact on the shield tunnel because of the foundation pit construction, a simplified analytical approach that can consider the deformation of the enclosure structure is presented in this study, which has the following innovations:  
1.