## 4 Conclusions  
A 2D numerical study allowed an investigation of the behavior of sub-rectangular tunnel linings under seismic loadings. The influences of parameters, such as soil deformation, maximum horizontal acceleration, lining thickness, and soil-lining interface conditions, on the circular- and sub-rectangular-shaped tunnel behavior under seismic loading were investigated. Considerable differences in the response of these tunnels were observed. Based on the research results, the following conclusions can be drawn:  
The horizontal acceleration \(a_{H}\), the soil Young's modulus \(E\), and lining thickness \(t\) have a great effect on the incremental internal forces induced in both sub-rectangular and circular tunnels for both no-slip and full-slip conditions.  
In general, a higher seismic loading induced by a higher horizontal acceleration \(a_{H}\) will induce higher incremental bending moments and normal forces in both circular and sub-rectangular tunnels. The relationship is quite linear.  
The results proved that soil-lining interface conditions have a great influence on the behavior of sub-rectangular tunnels. This is completely different when comparing the behavior of circular-shaped tunnels. Indeed, while the absolute extreme incremental bending moments of a circular tunnel for the no-slip condition are smaller than the corresponding full-slip ones, the absolute extreme incremental bending moments of sub-rectangular tunnels for the no-slip condition are greater than the corresponding full-slip ones. This represents an opposite trend to what can be observed in circular tunnel linings.  
For all investigated case studies, absolute incremental normal forces for the no-slip conditions are always larger than the full-slip ones, for both the circular and sub-rectangular tunnels cases. Absolute extreme incremental normal forces in sub-rectangular tunnels are approximately 10% smaller than the circular ones.  
The dependency of the absolute extreme incremental bending moments induced on the sub-rectangular tunnels on the soil Young's modulus (\(E\)) is insignificant compared with that on the circular ones. The soil Young's modulus of 50 MPa could be considered as a critical value for both tunnel shape cases. Beyond this value, the E increase induces a decrease in the absolute extreme incremental bending moments. However, below this  
Figure 12: Effect of the lining thickness on the incremental internal forces of the circular and sub-rectangular tunnel linings. (**a**) Incremental bending moments; (**b**) incremental normal forces.  
value, an increase in \(E\) value induces an increase in the absolute extreme incremental bending moments.  
An increase in the soil Young's modulus (\(E\)) causes a significant corresponding increase in the absolute extreme incremental normal forces for both sub-rectangular and circular tunnels (no-slip condition). An insignificant change in the absolute extreme incremental normal forces is observed for the full-slip conditions.  
The numerical results obtained in the present study are useful for the preliminary design of circular- and sub-rectangular-shaped tunnel linings under seismic loadings. The joint distribution influence in the segmental lining on the tunnel behavior will be considered in future research.  
V.V.P.: Methodology, software and validation, investigation, data curation, writing--original draft preparation. N.A.D.: Conceptualization, methodology, software and validation, writing--review and editing. D.D.: Conceptualization, methodology, writing--review and editing, supervision. All authors have read and agreed to the published version of the manuscript.  
This research is supported by the Vietnam Ministry of Education and Training.  
Not applicable.  
The authors declare no conflict of interest.