_Applied_  
_sciences_  
_Sub-Rectangular Tunnel Behavior under Seismic Loading_  
**[PERSON] 1\({}^{\copyright}\)**, [PERSON] 1,2\({}^{\copyright}\) and [PERSON] 3,4,*(\({}^{\copyright}\)  
\({}^{1}\) Department of Underground and Mining Construction, Faculty of Civil Engineering, Hanoi University of Mining and Geology, Hanoi 100000, Vietnam; [EMAIL_ADDRESS] (V.V.P. ); [PERSON] [EMAIL_ADDRESS] (N.A.D.) \({}^{2}\) Sustainable Development in Underground Engineering Research Team, Hanoi University of Mining and Geology, Hanoi 100000, Vietnam  
\({}^{3}\) Laboratory 38S, Grenoble Alpes University, 38000 Grenoble, France  
\({}^{4}\) Geotechnical Expert, Antea Group, 92160 Antony, France  
\({}^{*}\) Correspondence: daniel.dias@3 sr-grenoble.fr  
**Abstract** Circular and rectangular tunnel shapes are usually chosen when excavating at shallow depths in urban areas. However, special-shaped tunnels such as sub-rectangular tunnels have recently been used to overcome some drawbacks of circular and rectangular tunnels in terms of low space utilization efficiency and stress concentration, respectively. In the literature, experimental studies as well as analytical and numerical models have been developed for the seismic analysis and vulnerability assessment of circular and rectangular tunnels since the early 1990s. However, knowledge gaps regarding the behavior of sub-rectangular tunnels under seismic loading remain and still need to be bridged. The present paper focuses on introducing a numerical analysis of sub-rectangular tunnels under seismic loading. The numerical model of sub-rectangular tunnels is developed based on the numerical analyses of circular tunnels validated by comparing well-known, analytical solutions. This paper aims to highlight the differences between the behavior of sub-rectangular tunnels compared with circular tunnels when subjected to seismic loadings. Special attention is paid to the soil-lining interface conditions. The influence of parameters, such as soil deformations, maximum horizontal acceleration, and lining thickness, on sub-rectangular tunnel behavior under seismic loading is also investigated. The results indicate a significant behavior difference between sub-rectangular and circular tunnels. The absolute extreme incremental bending moments for a circular tunnel (no-slip condition) are smaller than that for the corresponding full-slip condition. The absolute extreme incremental bending moments of sub-rectangular tunnels (no-slip condition) are, however, greater than the corresponding full-slip conditions. sub-rectangular tunnel; seismic load; tunnel lining; numerical analysis  
**Highlights**  
* Two-dimensional numerical analyses of circular and sub-rectangular tunnels under seismic loading. * The behavior of a sub-rectangular tunnel under seismic loading is significantly different from that of a circular tunnel. * Effects of horizontal acceleration, the soil Young's modulus, lining thickness, and soil-lining interface are presented. **1. Introduction**  
Circular- and rectangular-shaped tunnels are frequent choices when such structures are excavated at shallow depths [1, 2, 3, 4], particularly in urban areas. The main disadvantage of a circular tunnel is its small space utilization ratio. Compared to circular tunnels, rectangular tunnels provide a major part of their cross-section area for utilization purposes. However, their shape can induce stress concentrations at the tunnel corners, which can be followed by failures of the supporting structure. Special-shaped tunnels based on a combination of circular and rectangular shapes can be a good solution, as they have a higher space utilization efficiency than that of circular tunnels. They also help to eliminate stress concentrations at the corners [5] and can have a greater bearing capacity. A special-shaped tunnel, namely, the sub-rectangular tunnel, was therefore recently used to improve the efficiency of underground space use [6; 7; 8].