Sub-Rectangular Tunnel Behavior under Seismic Loading

: 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.


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 Figure 1. Sub-rectangular express tunnel in Shanghai [22], distances in millimeters.  Sub-rectangular express tunnel in Shanghai [22], distances in millimeters.

Figure 2.
Circular tunnel with the same utilization space area, distances in millimeters.

Numerical Model for the Circular Tunnel
Firstly, a numerical model for circular tunnels was developed using a finite difference program (FLAC 3D ) [24]. The purpose was to investigate the behavior of circular tunnel linings under quasi-static loading and make a comparison with those obtained by an analytical solution.
A 2D plane strain model was created (Figure 3). The soil mass is discretized into hexahedral zones. The tunnel lining is modeled using embedded liner elements, which are attached to the zone faces along the tunnel boundary with interfaces. Two interface conditions were considered: no slip between the soil and lining and full slip. The liner-zone interface stiffness (normal stiffness kn and tangential stiffness ks) is determined based on a given rule of thumb [24]. kn and ks are set equal to 100 times the equivalent stiffness of the stiffest neighboring zone [24] for the no-slip condition case. When considering the fullslip condition, ks is assigned to be equal to zero.

Numerical Model for the Circular Tunnel
Firstly, a numerical model for circular tunnels was developed using a finite difference program (FLAC 3D ) [24]. The purpose was to investigate the behavior of circular tunnel linings under quasi-static loading and make a comparison with those obtained by an analytical solution.
A 2D plane strain model was created (Figure 3). The soil mass is discretized into hexahedral zones. The tunnel lining is modeled using embedded liner elements, which are attached to the zone faces along the tunnel boundary with interfaces. Two interface conditions were considered: no slip between the soil and lining and full slip. The liner-zone interface stiffness (normal stiffness k n and tangential stiffness k s ) is determined based on a given rule of thumb [24]. k n and k s are set equal to 100 times the equivalent stiffness of the stiffest neighboring zone [24] for the no-slip condition case. When considering the full-slip condition, k s is assigned to be equal to zero.
The mesh is composed of a single layer of zones in the y-direction, and the elem dimension increases as one moves away from the tunnel. It reaches the maximum siz 2.6 x 1 m at the model boundary ( Figure 3). The boundary conditions of the nume model are 120 m in the x-direction and 40 m in the z-direction. It consists of approxima 4800 zones and 9802 nodes. The bottom of the model was blocked in all directions, the vertical sides were fixed in the horizontal one.
In this study, similar to the research work of Sederat et al. [15], Do et al. [25], Naggar and Hinchberger [26], ovaling deformations due to the seismic loading are posed as inverted triangular displacements, along with the model lateral boundaries. The mesh is composed of a single layer of zones in the y-direction, and the element's dimension increases as one moves away from the tunnel. It reaches the maximum size of 2.6 × 1 m at the model boundary ( In this study, similar to the research work of Sederat et al. [15], Do et al. [25], and Naggar and Hinchberger [26], ovaling deformations due to the seismic loading are imposed as inverted triangular displacements, along with the model lateral boundaries. Uniform horizontal displacements are applied along the top boundary ( Figure 3). The magnitude of the prescribed displacements assigned at the top of the model is dependent on the maximum shear strain γ max , estimated based on the horizontal acceleration a H . The bottom of the model is restrained in all directions.
Before applying the ovaling deformation due to seismic loading, a steady state of the excavated tunnel under static conditions was established. When the tunneling process is performed in a 2D plane strain model, it is assumed that ground displacements surrounding the tunnel boundary prior to the lining installation are simulated by using the convergence confinement method with a relaxation factor, λ d , equal to 0.3 [25,27]. The numerical simulation of the tunnel ovaling was therefore performed through the following steps:

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Step 1: In situ state of stresses before tunnel construction.

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Step 2: Excavation of the tunnel and use of the convergence confinement process with a relaxation factor, γ d , of 0.3. The concrete lining is then activated on the tunnel's periphery.

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Step 3: Assigning ovaling deformations caused by the seismic loading on the model boundaries using the prescribed displacements previously mentioned.
It should be noted that only incremental internal forces are presented in this study. They are determined by subtracting the total lining forces computed at the end of the static loading (step 2) from those determined at the end of the ovaling step (step 3).

Comparison of the Numerical and Analytical Model for the Circular Tunnel Case Study
For validation purposes of the numerical model subjected to quasi-static loading, the well-known analytical solution proposed by Wang [9], and thereafter improved by Kouretzis [16], was used for comparison with the results obtained from the numerical model. Several authors showed that this solution is efficient and can be used for the seismic design of circular tunnels [12,25]. It should be noted that Kouretzis [16] proposed an expression of the maximum incremental bending moment under the no-slip condition, which was not mentioned by Wang [9]. Parameters shown in Table 1 were adopted in this section as the reference case. The soil and tunnel lining material properties in numerical models were assumed to be linear elastic and massless. These assumptions were also adopted in the analytical solution. An anisotropic stress field was assigned in the numerical model with a lateral earth pressure coefficient at rest, K 0 , of 0.5. Figure 4 illustrates the distribution of the incremental internal forces induced in the tunnel lining under seismic loading. The conditions of lining and soil interaction, when using the Wang solution and FLAC 3D , were considered for both cases of no slip and full slip. The soil and tunnel lining parameters fed into the model are presented in Table 1. The results obtained by numerical and analytical models are in good agreement. Figure 4a,c show that the maximum incremental bending moment in the full-slip case is 14% larger than that obtained in the no-slip case. In contrast, the maximum incremental normal forces in the full-slip case are smaller than those in the no-slip case (Figure 4b,d). In the section below, a parametric study is conducted to highlight the behavior of circular tunnel lining subjected to quasi-static loadings considering the effect of Young's modulus E, horizontal seismic acceleration a H , and tunnel lining thickness t variations. Parameters of the soil and tunnel lining presented in Table 1 are adopted for the reference case study.
Both maximum and minimum incremental bending moments are presented. They are both labelled "extreme incremental bending moment". Similarly, extreme incremental normal forces representing both the maximum and minimum incremental normal forces induced in the tunnel lining are presented. the analytical solution. An anisotropic stress field was assigned in the numerical model with a lateral earth pressure coefficient at rest, K0, of 0.5.   Table 1. The results obtained by numerical and analytical models are in good agreement. Figure  4a,c show that the maximum incremental bending moment in the full-slip case is 14% larger than that obtained in the no-slip case. In contrast, the maximum incremental normal forces in the full-slip case are smaller than those in the no-slip case (Figure 4b   In the section below, a parametric study is conducted to highlight the behavior of circular tunnel lining subjected to quasi-static loadings considering the effect of Young's modulus E, horizontal seismic acceleration aH, and tunnel lining thickness t variations. Parameters of the soil and tunnel lining presented in Table 1 are adopted for the reference case study.
Both maximum and minimum incremental bending moments are presented. They are both labelled "extreme incremental bending moment". Similarly, extreme incremental normal forces representing both the maximum and minimum incremental normal forces induced in the tunnel lining are presented.

Effect of the Peak Horizontal Seismic Acceleration (aH)
A parametric study was conducted to investigate the seismic loading magnitude effects represented here by the maximum horizontal acceleration, aH. The maximum horizontal acceleration is varied in the range between 0.05 and 0.75 g, corresponding to the respectively maximum shear strains, γmax, of 0.038 and 0.58%. The other parameters of the reference case in Table 1 are used. The following conclusions can be deduced from Figure  5: For both the no-slip and full-slip conditions, numerical results show good agreement with the analytical solution. A discrepancy of approximately 1% for both the extreme incremental bending moments and normal forces is obtained.
The absolute values of the extreme incremental bending moments and normal forces increased gradually with the aH increase from 0.05 to 0.75 g. Incremental bending moments for both no-slip and full-slip conditions are strongly dependent on the aH value ( Figure  5a). However, while incremental normal forces in the tunnel lining for the no-slip condition are strongly affected by the aH value, insignificant incremental normal forces variations due to aH for the full-slip condition are observed ( Figure 5b).

Effect of the Peak Horizontal Seismic Acceleration (a H )
A parametric study was conducted to investigate the seismic loading magnitude effects represented here by the maximum horizontal acceleration, a H . The maximum horizontal acceleration is varied in the range between 0.05 and 0.75 g, corresponding to the respectively maximum shear strains, γ max, of 0.038 and 0.58%. The other parameters of the reference case in Table 1

Effect of the Soil Young's Modulus E
The soil Young's modulus is assumed to fall in a range from 10 to 350 MPa. The other parameters presented in Table 1, based on the reference case study, were used as the input data. The numerical results obtained by using the FLAC 3D comparison with Wang's analytical method for the full-slip and no-slip conditions are presented in Figure 6. The observations discussed below were made. For both the no-slip and full-slip conditions, numerical results show good agreement with the analytical solution. A discrepancy of approximately 1% for both the extreme incremental bending moments and normal forces is obtained.
The absolute values of the extreme incremental bending moments and normal forces increased gradually with the a H increase from 0.05 to 0.75 g. Incremental bending moments for both no-slip and full-slip conditions are strongly dependent on the a H value (Figure 5a). However, while incremental normal forces in the tunnel lining for the no-slip condition are strongly affected by the a H value, insignificant incremental normal forces variations due to a H for the full-slip condition are observed (Figure 5b).

Effect of the Soil Young's Modulus E
The soil Young's modulus is assumed to fall in a range from 10 to 350 MPa. The other parameters presented in Table 1, based on the reference case study, were used as the input data. The numerical results obtained by using the FLAC 3D comparison with Wang's analytical method for the full-slip and no-slip conditions are presented in Figure 6. The observations discussed below were made. The soil Young's modulus is assumed to fall in a range from 10 to 350 MPa. The other parameters presented in Table 1, based on the reference case study, were used as the input data. The numerical results obtained by using the FLAC 3D comparison with Wang's analytical method for the full-slip and no-slip conditions are presented in Figure 6. The observations discussed below were made.   Figure 6 shows good agreement of the incremental bending moments and normal forces induced in the tunnel lining under seismic loading, obtained by the numerical model and the analytical solution for both no-slip and full-slip conditions when considering the E variation. The maximum difference is smaller than 2%.
The extreme incremental bending moments are strongly dependent on the E value as seen in Figure 6a. The absolute values of the extreme incremental bending moments are reached for E values close to 50 MPa. There is a rapid decrease in the absolute extreme incremental bending moments when the E value reduces from 25 to 10 MPa. When the E values are larger than 50 MPa, an increase in E induces a decrease in the absolute extreme incremental bending moments. This correlation of the extreme incremental bending moments is observed in both full-slip and no-slip conditions. It should be noted that for the same E value, the absolute extreme incremental bending moments induced in the tunnel lining for the no-slip condition are always 10 to 15% smaller than the full-slip ones.
While the extreme incremental normal forces in the full-slip conditions are insignificantly dependent on the E value (Figure 6b), an increase in E can cause a rapid increase in the maximum and minimum incremental normal forces in the tunnel lining for the no-slip condition. As predicted, incremental normal forces for the no-slip condition are always larger than the full-slip ones.

Effect of the Lining Thickness
The tunnel lining thickness was assumed to vary between 0.2 to 0.8 m, corresponding to a lining thickness vs. tunnel dimension ratio of 3 to 8.5% [28], while the other parameters are based on the reference case assumed in Table 1. Similar to what happens when considering the soil Young's modulus E and horizontal seismic acceleration a H effects, the results presented in Figure 7 show good agreement between analytical and numerical models for both no-slip and full-slip conditions. The discrepancy is under 1% for both the incremental bending moments and normal forces.
considering the soil Young's modulus E and horizontal seismic acceleration aH effects, the results presented in Figure 7 show good agreement between analytical and numerical models for both no-slip and full-slip conditions. The discrepancy is under 1% for both the incremental bending moments and normal forces.
In general, the absolute extreme incremental bending moment and normal force values gradually increase when the lining thickness t increases from 0.2 to 0.8 m. This concerns both the full-slip and no-slip conditions. The incremental bending moments for the no-slip condition are always smaller than the full-slip ones (Figure 7a). The biggest difference of 14% was obtained for a lining thickness of 0.8 m. It should be noted that the incremental normal forces variations caused by the lining thickness increase are less significant when compared to the incremental bending moment ones (Figure 7a,b). Based on the above comparison between the analytical solution and numerical model when considering Young's modulus E, horizontal seismic acceleration aH, and tunnel lining thickness t, which show good agreement between the analytical solution and numerical model, it is reasonable to conclude that the circular tunnel numerical model developed can be used to investigate the behavior of circular tunnels subjected to seismic loadings. In general, the absolute extreme incremental bending moment and normal force values gradually increase when the lining thickness t increases from 0.2 to 0.8 m. This concerns both the full-slip and no-slip conditions. The incremental bending moments for the no-slip condition are always smaller than the full-slip ones (Figure 7a). The biggest difference of 14% was obtained for a lining thickness of 0.8 m. It should be noted that the incremental normal forces variations caused by the lining thickness increase are less significant when compared to the incremental bending moment ones (Figure 7a,b).

Development of a Numerical Model for Sub-Rectangular Tunnels
Based on the above comparison between the analytical solution and numerical model when considering Young's modulus E, horizontal seismic acceleration a H , and tunnel lining thickness t, which show good agreement between the analytical solution and numerical model, it is reasonable to conclude that the circular tunnel numerical model developed can be used to investigate the behavior of circular tunnels subjected to seismic loadings.

Development of a Numerical Model for Sub-Rectangular Tunnels
In this section, a numerical model was developed for the sub-rectangular tunnel case using similar soil parameters, lining material, and modeling processes to consider the static and seismic loadings introduced above. The tunnel shape is modified into a sub-rectangular geometry, and the gravity effect is taken into consideration. The mesh consists of a single layer of zones in the y-direction, and the dimension of the elements increases as one moves away from the tunnel (Figure 8). The geometry parameters of sub-rectangular tunnels are presented in Figure 1, and other parameters presented in Table 1  of a single layer of zones in the y-direction, and the dimension of the elements increases as one moves away from the tunnel (Figure 8). The geometry parameters of sub-rectangular tunnels are presented in Figure 1, and other parameters presented in Table 1 are adopted. The numerical model is 120 m wide in the x-direction, 40 m high in the z-direction, and consists of approximately 5816 elements and 11,870 nodes. The bottom of the model was blocked in all directions, and the vertical sides were fixed in the horizontal one.  Figure 9 introduces the incremental bending moments and normal forces induced in the sub-rectangular tunnel linings subjected to seismic loadings and considers both noslip and full-slip conditions. Parameters of the reference case presented in Table 1 are adopted. Figures 4 and 9 provide a clear understanding of the behavior of circular and subrectangular tunnel linings under seismic loadings. The positions at the tunnel periphery, where the extreme incremental internal forces are reached, are positioned. It can be seen from Figure 9 that extreme incremental bending moments and normal forces observed in the sub-rectangular tunnel appear at the tunnel lining corners where the smaller lining radii are located. In the following sections, a numerical investigation was conducted to highlight the behavior of a sub-rectangular tunnel compared with a circular shape. These two tunnels have the same utilization space area and are twice subjected to seismic loadings while considering the effect of parameters, such as the horizontal seismic acceleration, soil deformation modulus, and lining thickness. The effects of the soil-lining interface condition are also investigated.  Figure 9 introduces the incremental bending moments and normal forces induced in the sub-rectangular tunnel linings subjected to seismic loadings and considers both no-slip and full-slip conditions. Parameters of the reference case presented in Table 1 are adopted.

Effect of the Peak Horizontal Seismic Acceleration (aH)
Shear strain values corresponding to a range of a maximum horizontal acceleration varying from 0.05 and 0.75 g were adopted in this study. In general, high seismic loadings are implied by a high horizontal acceleration aH, and therefore shear strain values of γmax, and result in high absolute extreme incremental bending moments and normal forces. The relationship is quite linear (Figure 10).
The results presented in Figure 10a show that, for the no-slip condition, absolute extreme incremental bending moments in the sub-rectangular lining are 20% larger than the circular ones. Nevertheless, for the full-slip condition, absolute extreme incremental bending moments in the circular lining are approximately 4% greater than the sub-rectangular ones. In the case of sub-rectangular linings, absolute extreme incremental bending moments for the full-slip condition are always lower by about 10% than the no-slip ones. This relationship is opposite to that observed in the cases of the circular-shaped tunnel ( Figure  10a).
It can be seen in Figure 10b that for both shapes of tunnels, the absolute extreme incremental normal forces for the no-slip condition are approximately 80% larger than the full-slip ones. Unlike the incremental bending moments mentioned above, the absolute extreme incremental normal forces of the sub-rectangular lining are approximately 9% lower than the circular lining ones, for both the no-slip and full-slip conditions and when changing the horizontal acceleration (Figure 10b).  Figure 9 that extreme incremental bending moments and normal forces observed in the sub-rectangular tunnel appear at the tunnel lining corners where the smaller lining radii are located. In the following sections, a numerical investigation was conducted to highlight the behavior of a sub-rectangular tunnel compared with a circular shape. These two tunnels have the same utilization space area and are twice subjected to seismic loadings while considering the effect of parameters, such as the horizontal seismic acceleration, soil deformation modulus, and lining thickness. The effects of the soil-lining interface condition are also investigated.

Effect of the Peak Horizontal Seismic Acceleration (a H )
Shear strain values corresponding to a range of a maximum horizontal acceleration varying from 0.05 and 0.75 g were adopted in this study. In general, high seismic loadings are implied by a high horizontal acceleration a H , and therefore shear strain values of γ max , and result in high absolute extreme incremental bending moments and normal forces. The relationship is quite linear (Figure 10).

Effect of the Soil Young's Modulus (E)
The soil Young's modulus values are assumed to vary in the range from 10 to 350 MPa while keeping K0 equal to 0.5 and aH of 0.5 g. The other parameters based on the reference case are assumed ( Table 1). The following can be seen from Figure 11.
For low E values smaller than 50 MPa, an increase in E induces an increase in the absolute extreme incremental bending moments. When the E value is greater than 50 MPa, the increase in E causes a decrease in the absolute extreme incremental bending moments (Figure 11a). It should be noted that the dependency of the absolute extreme incremental bending moments in the sub-rectangular tunnels on the E value is insignificant compared to the circular-shaped tunnels (Figure 11a). It is also worth highlighting that while the absolute extreme incremental bending moments of the circular tunnel for the no-slip condition are smaller than the corresponding full-slip ones [9,25], the absolute extreme incremental bending moments of the sub-rectangular tunnel for the no-slip condition are greater than the corresponding full-slip ones. The behavior of sub-rectangular tunnels is different from the circular-shaped tunnels. The same conclusion was also obtained when considering the horizontal seismic acceleration aH. Figure 11a also shows greater absolute extreme incremental bending moments induced in sub-rectangular tunnels for the no-slip condition compared with circular tunnels with the same utilization space area. However, in the full-slip condition, absolute extreme incremental bending moments in the circular tunnel are greater than the sub-rectangular ones for E values smaller than approximately 150 MPa. When E values are larger than 150 MPa, absolute extreme incremental bending moments developed in circular tunnels are smaller than in sub-rectangular tunnels. Figure 11b indicates that an increase in E value causes a significant corresponding increase in the absolute extreme normal forces in both sub-rectangular and circular tunnels for the no-slip condition. However, it induces an insignificant change in absolute extreme incremental normal forces for the full-slip condition. Absolute extreme incremental normal forces in the sub-rectangular tunnels are generally 10% smaller than for the circular tunnels. The results presented in Figure 10a show that, for the no-slip condition, absolute extreme incremental bending moments in the sub-rectangular lining are 20% larger than the circular ones. Nevertheless, for the full-slip condition, absolute extreme incremental bending moments in the circular lining are approximately 4% greater than the sub-rectangular ones. In the case of sub-rectangular linings, absolute extreme incremental bending moments for the full-slip condition are always lower by about 10% than the no-slip ones. This relationship is opposite to that observed in the cases of the circular-shaped tunnel (Figure 10a).
It can be seen in Figure 10b that for both shapes of tunnels, the absolute extreme incremental normal forces for the no-slip condition are approximately 80% larger than the full-slip ones. Unlike the incremental bending moments mentioned above, the absolute extreme incremental normal forces of the sub-rectangular lining are approximately 9% lower than the circular lining ones, for both the no-slip and full-slip conditions and when changing the horizontal acceleration (Figure 10b).

Effect of the Soil Young's Modulus (E)
The soil Young's modulus values are assumed to vary in the range from 10 to 350 MPa while keeping K 0 equal to 0.5 and a H of 0.5 g. The other parameters based on the reference case are assumed ( Table 1). The following can be seen from Figure 11.
For low E values smaller than 50 MPa, an increase in E induces an increase in the absolute extreme incremental bending moments. When the E value is greater than 50 MPa, the increase in E causes a decrease in the absolute extreme incremental bending moments (Figure 11a). It should be noted that the dependency of the absolute extreme incremental bending moments in the sub-rectangular tunnels on the E value is insignificant compared to the circular-shaped tunnels (Figure 11a). It is also worth highlighting that while the absolute extreme incremental bending moments of the circular tunnel for the no-slip condition are smaller than the corresponding full-slip ones [9,25], the absolute extreme incremental bending moments of the sub-rectangular tunnel for the no-slip condition are greater than the corresponding full-slip ones. The behavior of sub-rectangular tunnels is different from the circular-shaped tunnels. The same conclusion was also obtained when considering the horizontal seismic acceleration a H . Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 15 (a) (b) Figure 11. Effect of the E value on the internal forces for the circular and sub-rectangular tunnel linings. (a) Incremental bending moments; (b) incremental normal forces.

Effect of the Lining Thickness (t)
The lining thickness t is assumed to vary in the range between 0.2 and 0.8 m while keeping K0 value of 0.5, an aH value of 0.5 g, and an E value of 100 MPa. Other parameters introduced in Table 1 were adopted. The results presented in Figure 12 indicate that the lining thickness has a great effect on the incremental internal forces for both sub-rectangular and circular tunnels and in both no-slip and full-slip conditions. The relationship between the lining thickness and the incremental internal forces for the considered cases is quite linear. For the no-slip condition, absolute extreme incremental bending moments of the subrectangular linings are always larger than the circular ones (Figure 12a). The discrepancy declined gradually from 124 to 6%, corresponding to the lining thickness increase from 0.2 to 0.8 m. In the full-slip conditions, considering a lining thickness smaller than approximately 0.5 m, the absolute extreme incremental bending moments of the sub-rectangular Figure 11a also shows greater absolute extreme incremental bending moments induced in sub-rectangular tunnels for the no-slip condition compared with circular tunnels with the same utilization space area. However, in the full-slip condition, absolute extreme incremental bending moments in the circular tunnel are greater than the sub-rectangular ones for E values smaller than approximately 150 MPa. When E values are larger than 150 MPa, absolute extreme incremental bending moments developed in circular tunnels are smaller than in sub-rectangular tunnels. Figure 11b indicates that an increase in E value causes a significant corresponding increase in the absolute extreme normal forces in both sub-rectangular and circular tunnels for the no-slip condition. However, it induces an insignificant change in absolute extreme incremental normal forces for the full-slip condition. Absolute extreme incremental normal forces in the sub-rectangular tunnels are generally 10% smaller than for the circular tunnels.

Effect of the Lining Thickness (t)
The lining thickness t is assumed to vary in the range between 0.2 and 0.8 m while keeping K 0 value of 0.5, an a H value of 0.5 g, and an E value of 100 MPa. Other parameters introduced in Table 1 were adopted. The results presented in Figure 12 indicate that the lining thickness has a great effect on the incremental internal forces for both sub-rectangular and circular tunnels and in both no-slip and full-slip conditions. The relationship between the lining thickness and the incremental internal forces for the considered cases is quite linear.
For the no-slip condition, absolute extreme incremental bending moments of the subrectangular linings are always larger than the circular ones (Figure 12a). The discrepancy declined gradually from 124 to 6%, corresponding to the lining thickness increase from 0.2 to 0.8 m. In the full-slip conditions, considering a lining thickness smaller than approximately 0.5 m, the absolute extreme incremental bending moments of the sub-rectangular linings are still larger than the circular ones, similar to the no-slip condition presented earlier. However, when the lining thickness is larger than 0.5 m, the opposite result is obtained, as shown in Figure 12a. Thus, larger absolute extreme incremental bending moments on the circular tunnels are observed.
keeping K0 value of 0.5, an aH value of 0.5 g, and an E value of 100 MPa. Other parameters introduced in Table 1 were adopted. The results presented in Figure 12 indicate that the lining thickness has a great effect on the incremental internal forces for both sub-rectangular and circular tunnels and in both no-slip and full-slip conditions. The relationship between the lining thickness and the incremental internal forces for the considered cases is quite linear. For the no-slip condition, absolute extreme incremental bending moments of the subrectangular linings are always larger than the circular ones (Figure 12a). The discrepancy declined gradually from 124 to 6%, corresponding to the lining thickness increase from 0.2 to 0.8 m. In the full-slip conditions, considering a lining thickness smaller than approximately 0.5 m, the absolute extreme incremental bending moments of the sub-rectangular It can be seen in Figure 12b that the incremental normal forces in the no-slip condition are always larger than the full-slip ones. In comparison with the incremental normal forces of the circular lining, the incremental normal forces in the sub-rectangular lining are lower by about 9 and 25% for no-slip and full-slip conditions, respectively (Figure 12b).

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 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.