## 5 Conclusions  
In this study, a simplified analytical solution for a tunnel passing through soft-hard stratum equipped with a transition tunnel and flexible joints was proposed, and the seismic responses of the tunnel under earthquake motion were studied by the proposed analytical solution and a series of shaking table tests. Comparison of the analytical solutions with the model test results were carried out, and their results are in agreement, which indicates that the analytical solutions are workable and effective. The following conclusions can be drawn:  
(1) The influence of the soft-hard stratum junction on the tunnel responses is remarkable. The largest bending moment is located at the side of soft rock near the sharp contact area and the maximum shear force appears at contact. The maximum amplified factor is in relatively hard rock for the bending moment, and is in soft rock for the shear force.  
(2) The joints efficiently reduce the bending moment and shear force of the tunnel through soft-hard stratum. The region affected by the joint is approximately 4.5 times the tunnel diameter on both sides of the stratum interface. The transition tunnel could mitigate the potential adverse effects of the sharp change of ground properties. The length of transition Region 2 with 24 m or 36 m can be taken as the optimal length.  
(3) The soft-hard interface has a significant impact on the tunnel strain responses. The influence of sharp change of ground layers is more remarkable with a larger excitation amplitude. The deformation effect in soft rock is greater, while the deformation effect is relatively small in relatively hard rock.  
(4) The peck acceleration of the soft rock is larger than that of the hard one. The sharp change of ground properties affects the frequency components. The peak acceleration and spectrum amplitude increase monotonically as the measuring point proceeds from the bottom to the surface. The frequency characteristics of the model soil depend on the frequency characteristics of both input earthquake motions and the model soil itself.  
It should be noted that the results may change for geometries and parameters other than those analyzed; however, the proposed method, the design of tunnel through soft-hard stratum, and hopefully many of the conclusions of this study are generally applicable to other similar tunneling projects. Although the analytical solutions could provide approximate results quickly and easily, it still needs to be further improved by a numerical method or experimental test because of limitations of the test monitoring data in this paper.  
Figure 15: Acceleration records and Fourier spectra of monitoring positions in the shaking table tests. (**a**) Acceleration time history and Fourier spectrum of monitoring point A1; (**b**) acceleration time history and Fourier spectrum of monitoring point A3; (**c**) acceleration time history and Fourier spectrum of monitoring point A4; (**d**) acceleration time history and Fourier spectrum of monitoring point A5.  
**Author Contributions:** Conceptualization, G.Y.; methodology, G.Y.; software, G.Y.; validation, B.Z.; writing--original draft preparation, G.Y.; writing--review and editing, G.Y. and B.Z.; supervision, B.Z.; funding acquisition, G.Y. and B.Z. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the Fundamental Research Funds for the Central Universities (No. 2021 JBM038) and the National Natural Science Foundation of China (No. 51778046 and 51808035).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest.