## 4 Conclusions  
1. A new method for predicting the stratum movements caused by EPB shield tunnel construction is developed that combines an elastic half-space model of mirror source-sink method with the use of modified analytical method. Compared with the effect of a physical gap, the stratum movements caused by the additional force on the shield working face and the frictional force between the outer surface of the shield and the surrounding soil during tunnel construction is very small. Therefore, this paper neglects the effects of the latter two factors when analyzing, and mainly studies the effects of the physical gap. The method also takes into account non-equivalent radial ground loss.
2. [PERSON]'s method assuming equivalent radial movement pattern can only obtain the surface movements from the tunnel excavation. Considering non-equivalent radial displacement model, [PERSON]'s solution modifies [PERSON]'s analytical formula by exponential function, but it is only applicable to plane strain cases. The two methods have great limitations. For example, they cannot show the soil displacement of the x = y plane. If there is a new shield tunnel skewed with an old tunnel or pipeline and the displacement effect at the old tunnel or pipeline needs to be calculated, neither of the above two methods will apply. This paper combines the advantages of the two methods to derive a formula for predicting the three-dimensional stratum displacement caused by shield tunneling.
3. Vertical displacement contours in the central part show good consistency and regularity, with a larger difference at the edge. Centrifugal contours are nearly straight-line contours, while the improved solution contours are arc. Vertical displacement has a relatively small numerical value and its regularity is not obvious. The shape and distribution position of centrifugal contours and improved contours are similar. The difference also shows an increasing trend from the center to the edge, and the improved solution contours shift to the right a little and has a wider range. One of the important reasons is that sand soil is chosen in the centrifugal test but part of [PERSON]'s solution is fitted with clay data and is only applicable to clay soil. The current centrifugal tests are mostly based on sand. However, suitable clay centrifugal tests are hard to find. Considering the inevitable errors in the centrifugal test and theoretical calculations, the comparison results are acceptable. The values of [PERSON]'s contours are larger than the other two, because [PERSON]'s solution is the plane strain method; namely, it is a final stratum displacement from the excavation of an infinite tunnel. However, in the test and the improved method, the tunnel length is finite, the distance between the measuring plane and the tunnel face is much shorter, so their displacements are smaller than the final displacement.  
**Author Contributions:** [PERSON] wrote the article; [PERSON] carried out the calculations and analyzed the calculation results; [PERSON] and [PERSON] processed the data; [PERSON] offered some useful suggestions for the preparation.  
**Funding:** The study was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 51508037.  
**Acknowledgments:** We also highly appreciate reviewers for useful comments and editors for improving the manuscript.  
**Conflicts of Interest:** The authors declare no conflict of interest.