## 4 Conclusions  
The use of advanced constitutive models is limited in practice due to the difficulty in calibrating several constitutive parameters. Here, a linear elastic model is used for tunnels, and a linear elastic-perfectly plastic Mohr-Coulomb model and a nonlinear kinematic hardening model are used as advanced constitutive models for clay soil. Two different methods--one using data from a cyclic shear test simulation and the experimental curve of Ishibashi and Zhang [32] and the other using experimental data and some data from a simulation of an undrained cyclic triaxial test (which is a new method)--are used to calibrate the parameters of a kinematic hardening constitutive model. The results of this research can be summarized as follows.  
1. Choosing the appropriate constitutive model to achieve logical results is key to evaluating the seismic behavior of the tunnel in numerical methods. The use of a nonlinear kinematic hardening model that considers the effect of soil stiffness degradation is appropriate for structures under cyclic loads.
2. Calibrating the parameters of the kinematic hardening constitutive model using the data from undrained cyclic triaxial tests afford results that are almost identical to those from using the Ishibashi and Zhang equations and data from simulation of the cyclic shear tests.
3. The nonlinear kinematic hardening constitutive model can better demonstrate the cyclic deformation behavior of soils under cyclic loading than conventional and simple models such as the linear elastic-perfectly plastic Mohr-Coulomb model.  
Figure 18: Bending moment during the earthquake loading on the top of the circular tunnel lining.  
4. The Mohr-Coulomb model overestimates the design and it is not economically acceptable. For example, the bending moments created in the tunnel lining are much larger in the Mohr-Coulomb model than in the kinematic model.
5. The plastic strain of soil increases in both the kinematic hardening and Mohr-Coulomb models as the intensity of the earthquake increases from 0.4 to 0.83 g, but this increase is greater in the Mohr-Coulomb model due to its inability to create of hysteresis loops. Therefore, the Mohr-Coulomb model can't be used to accurately predict the behavior of soil under earthquake loading.
6. Changing the depth of a tunnel has no effect on the maximum acceleration response on the soil surface, but as the depth increases, greater forces are applied to the lining, and the bending moment created on the tunnel lining also increases. The lining displacement also decreases with the increasing tunnel depth.
7. Another parameter that affects the dynamic behavior of tunnels is the tunnel-lining thickness. The bending moment in the tunnel lining increases with its thickness so that as the dimensions of the tunnel increase, the EI increases and the structure becomes more stiff and able to withstand more forces. The resistance to deformation and the flexibility also increase under dynamic loads.
8. The shape of the tunnel dictates the overall response of the soil-tunnel system under dynamic loads. Circular tunnels show better performance than square tunnels against seismic loads. Generally, rounding the corners in square tunnels causes the tunnels to perform better under the imposed loads.  
[PERSON] and [PERSON] equally contributed to the conceptualization, analysis, and methodology [PERSON] wrote the original draft; [PERSON] contributed mainly to the conceptualization, review, and editing. All authors have read and agreed to the published version of the manuscript.  
The first author of this article is financially supported by Chines Government Scholarship (CSC).  
The authors declare no conflicts of interest.