Keywords

1 Introduction

Improvement of transportation infrastructure in urban areas for the transport of goods and mass rapid transit has increased the research interest in the study of tunnels [1,2,3]. Amorosi and Boldini carried out the study of the dynamic behaviour of the circular tunnel in clayey soil using the finite element method [4]. The authors further concluded that a seismic event can produce a substantial modification of loads acting in the lining. Yu et al. studied the longitudinal seismic response of tunnel liner using an analytical approach [5].

The shape of the tunnel plays a pivotal role when deformation study is carried out. Several researchers have studied the effect of tunnel shape on the stability of the tunnel [6,7,8,9,10,11]. Augarde and Burd described the finite element for the deformations in case of the shallow tunnel and concluded that a thin layer of the continuum may be used for modelling a tunnel lining for soil–structure interaction problems [12]. Chungsik Yoo studied the longitudinal reinforced face of the tunnel using finite element analysis and concluded that the face deformations can be reduced by the application of longitudinal pipes as face reinforcement [13].

Nevertheless, authors have not found any open literature dealing with the effect of weathering of a rock mass on the behaviour of the tunnel. In the present study, weathering effect of rock, the mass has been incorporated with different opening shapes, different magnitude seismic loadings and varying depth of tunnels. The present paper provides a comparative study by considering four different stages of weathering for three different shapes of tunnels, i.e. circular-shaped tunnel, arch-shaped tunnel and horseshoe-shaped tunnel. These cases have been analysed under four different magnitude earthquake loadings. The effect of depth of overburden has also been considered for each case by varying the overburden depth in three intervals, i.e. 5, 10 and 17.5 m.

2 Numerical Model

The dynamic analysis has been carried out for the 2D plane strain model of 42 m × 42 m size using finite element software Abaqus/Dynamic. Arch-shaped, circular-shaped and horseshoe-shaped tunnels are drawn having a 7 m diameter, and thickness of tunnel lining has been taken as 120 mm. The three depths of overburden, 5, 10, and 17.5 m, are considered to compare the effect of overburden depth under different (4.6, 5.6, 6.5 and 7.4 M) magnitudes of earthquakes for different stages of rock mass weathering (fresh basalt, slightly weathered, medium weathered and highly weathered basalt). The CPE3R-three-noded plane strain linear triangular element with reduced integration and hourglass control element type has been used for meshing the model. 500 m of overburden due to the already present rock mass has been taken into account in the form of pressure at the top line of the model. The infinite boundary condition has been adopted at the side faces of the tunnel model [14]. The CINPE4, a four-noded linear infinite quadrilateral element, was used for the infinite boundary condition meshing. 1.5 m mesh size has been used for meshing the rock mass and 0.1 m for meshing the tunnel lining (Fig. 1).

Fig. 1
figure 1

The geometry of tunnel and meshed model

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3 Input Properties

The Mohr–Coulomb model of elasto-plastic material has been considered for the behaviour of the rockmass. Various properties of different rockmass taken in the present study are shown in Table 1 [15,16,17,18,19,20,21]. The different weathering stages considered in the present study are fresh basalt (W0), slightly weathered basalt (W1), medium weathered basalt (W2) and highly weathered basalt (W3).

Table 1 Properties of different weathering stages [22]
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3.1 Analysis

The model was prepared using Abaqus/CAE and boundary conditions were applied. The vertical sides of the model were restrained in position by creating infinite boundary conditions. The earthquake loading has been applied at the base of the model in the form of the acceleration-time history of different earthquakes. The acceleration versus time history plot is shown in Fig. 2.

Fig. 2
figure 2

Acceleration-time history of a 4.6 M, b 5.6 M, c 6.5 M and d 7.4 M magnitude of earthquake [23]

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4 Results and Discussion

In the present paper, different shapes of the tunnel are analysed under different earthquake conditions. The weathering of the basalt rock has also been considered. The four weathering stages of basalt rock adopted in the present paper are; fresh basalt (W0), slightly weathered basalt (W1), medium weathered basalt (W2) and highly weathered basalt (W3). The depth variation has also been taken into account. There are three depths of overburden 5, 10 and 17.5 m adopted in the present analysis. The finite element software Abaqus/Explicit has been adopted for the present study. The paper focuses on the comparison of the stability of different shapes in different magnitudes of earthquakes.

The comparison of the depth of overburden has been shown in Table 2. It can be seen from the table that with the increase in the depth of overburden the deformation decreases. For the 4.6 M magnitude of the earthquake, a 5% decrease in deformation has been observed for the increase in depth of overburden from 5 m to 10 m.

Table 2 Deformation for arch-shaped tunnel for medium weathered (W2) basalt rock at right springer of tunnel
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For an increase in depth of overburden from 10 to 17.5 m, a 3.5% decrease in deformation has been observed. Similarly, for the 5.6 M magnitude of the earthquake, an increase in depth of overburden from 5 to 10 m and 10 to 17.5 m shows 12 and 1% decrease in the values of deformations, respectively. And for the 6.5 M magnitude of the earthquake, 16 and 3% decrease in the value of deformation has been observed for the similar increase in depth of overburden. Lastly, for 7.4 M magnitude, 28 and 8% decrease in the value of deformation has been observed for the increase in the depth of overburden from 5 m to 10 m and 10 m to 17.5 m, respectively.

The deformation has been plotted against each weathering stage for comparing the depth of overburden in Fig. 3 for the arch-shaped tunnel. This shows that weathering has a significant effect on deformation. Also, as the depth of overburden increases the deformation of the tunnel decreases. Figure 3 shows the results for 4.6 M of earthquake for different shapes of the tunnel.

Fig. 3
figure 3

Comparison of depths for different stages of weathering for a arch-shaped tunnel, b circular-shaped tunnel and c horseshoe tunnel under 4.6 M earthquake loading

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The graph shows that as the weathering stage of a basalt rock increases, it results in an increase in deformation at each depth of overburden. Also, as the depth of overburden increases, the deformation reduces as the situation results to lead towards the lithostatic condition.

Table 3 shows the comparison of different shapes of tunnels considered in the present study. It is noted that in the case of fresh basalt (W0), an arch-shaped tunnel is the safest and the horseshoe-shaped tunnel is most unstable. Similar is the case with slightly weathered basalt (W1). But, as the weathering of rock increases, a circular-shaped tunnel becomes the safest and the arch-shaped tunnel is the most unstable tunnel.

Table 3 Comparison of tunnel shape for the 5 m depth of overburden under 7.4 M earthquake loading for different stages of weathering
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The magnitude of an earthquake similar to or equal to the 7.4 M makes the comparison clearer, as shown in Fig. 4. In this case, the arch-shaped tunnel is most stable at every depth of overburden. The horseshoe-shaped tunnel has maximum deformation, and thus it is the most unstable shape of a tunnel.

Fig. 4
figure 4

Comparison of shapes of the tunnel in different stages of weathering for a 5 m, b 10 m and c 17.5 m under 7.4 M earthquake loading

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Figure 5 shows a graph for the comparison of the magnitude of an earthquake for the horseshoe shape of the earthquake at 5, 10 and 17.5 m of overburden depth. The magnitude of the 4.6 M earthquake has the least deformation and the 7.4 M magnitude of earthquake has maximum deformation. Figure 5 has been plotted for horseshoe-shaped tunnel, irrespective of shape; the deformation pattern is similar for all shapes of tunnels.

Fig. 5
figure 5

Comparison of deformation under the different magnitudes of the earthquake in different stages of weathering for a 5 m, b 10 m and c 17.5 m of depth for the horseshoe-shaped tunnel

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Table 4 shows the comparison between different magnitudes of earthquakes for the deformation readings taken at the right springer of the tunnel. For fresh basalt rock (W0), it can be noted that an increase of the magnitude of the earthquake from 4.6 to 5.6 M increases the deformation by 27, 33% for an increase in magnitude from 5.6 to 6.5 M and 7.4% increase for 6.5 to 7.4 M increase in the magnitude of the earthquake.

Table 4 Deformation comparison under the different magnitudes of earthquakes for circular-shaped tunnel at a 5 m depth of overburden (in metres)
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5 Conclusion

A two-dimensional finite element analysis has been performed to study the behaviour of rock tunnels considering weathering effects. Few important observations may be concluded as follows:

  • As the weathering stage of a basalt rock increases, due to a loss in structural strength, it results in an increase of deformation in the rock tunnel at each depth of overburden.

  • As the depth of overburden increases, the deformation in the rock tunnel reduces. It is may be due to the development of a lithostatic condition.

  • In the case of fresh basalt (W0), arch-shaped tunnel is the safest and the horseshoe-shaped tunnel is most unstable, similarly for slightly weathered basalt (W1).

  • As the weathering of rock increases, circular-shaped tunnel becomes the safest and the arch-shaped tunnel is the most unstable tunnel.

  • In case of an earthquake of a magnitude of 7.4 M, the arch-shaped tunnels are most stable at every depth of overburden and the horseshoe-shaped tunnels have maximum deformation, and thus it is the most unstable shape of a tunnel.

  • Deformation increases with an increase in the magnitude of the earthquake.