1 Introduction

In tunnel engineering, asymmetric phenomena are common. Especially at the entrance and exit of a tunnel, asymmetrical loads are particularly prevalent (Yang et al. 2020; Hu et al. 2021; Liu et al. 2017). In contrast to symmetric tunnels, asymmetric tunnels are more complex in tunnel design and support structures (Xiao et al. 2014, 2016; Papanikolaou and Kappos 2014). Some scholars have studied the surrounding rock stress and the failure mechanism of asymmetrical loading tunnels, and many achievements have been obtained (Lei et al. 2015a, b; Lu et al. 2021; Zhao et al. 2021; Zhang et al. 2014; Qiu et al. 2021).

The first aspect is about the study of surrounding rock pressure calculation. Based on the calculation method of the code, modified computational equations of vertical load and horizontal load were proposed, and a simplified analytical method for surrounding rock pressure was given based on the new assumption condition and load simplified model (Zuo et al. 2011; Zhang et al. 2016). According to the bipolar coordinate system, Zou et al. (2018) proposed the Mohr–Coulomb failure criterion, the gravitational effect and elastic–plastic solutions for the rock surrounding a shallow tunnel in an elastic–plastic semi-infinite space. Using the complex variable method and discrete Fourier transform, Lin et al. (2019) presented a solution for the sectional estimation of the stress and displacement fields of a liner and a geomaterial near a shallow buried tunnel. In addition, Nomikos et al. (2006) presented an analytical method to calculate the load-carrying capacity of two-dimensional asymmetric rock wedges when the load on joint faces was not symmetric, such as the case of an asymmetric wedge formed in the roof of a circular tunnel in an inclined stress field. Furthermore, under the action of asymmetric loads, some scholars have proposed upper limit solutions of surrounding rock pressure, elastic solutions of plane strain and analytical solutions for the interaction of shallow buried tunnels (Lei et al. 2015a, b; Cao et al. 2019; Yang et al. 2013). By considering the mutation theory, strain hardening and softening and gravity of the cavity, some researchers have proposed the mutation model, rheological constitutive model and analytical model (Yang et al. 2019; Zuo et al. 2015).

The second aspect is about the study of failure mechanism and failure characteristics. Many scholars have obtained the stress distributions and variation trends of the surrounding rock by mechanical tests and numerical simulations (Li et al. 2020; Liu et al. 2017). Using numerical code realistic failure process analysis (RFPA), Wang et al. (2012) studied the failure mechanism for a circular tunnel in transversely isotropic rock. Based on field monitoring data, Xiao et al. (2014) conducted three-dimensional numerical modeling to simulate the process of tunnel construction and to analyze the mechanism of cracking in the secondary lining. The existence of an asymmetric load transforms the structural stress state, and the failure form differs depending on location. Using a tunnel excavation simulation testing system, the change laws and distribution forms of structural stress and surrounding rock pressure and the failure mechanism of the lining and surrounding rock in shallow buried tunnels under asymmetrical loading were studied systematically (Lei et al. 2016; Lei et al. 2015a, b). Based on similarity theory, a series of physical testing models and shaking table tests were designed and manufactured, and the dynamic response characteristics of shallow-buried tunnels with asymmetrical pressure distributions were obtained (Jiang et al. 2018a, b; Jiang et al. 2018a, b). Moreover, a unified displacement function is used as the displacement boundary condition of the cross section of each tunnel and can be used to capture the asymmetric deformation behaviors about the horizontal and vertical centerlines (Kong et al. 2019). Based on the Lianchengshan tunnel, Chen et al. (2020) analyzed the change in rock mass pressure with deformation, time and distance from the excavation face by field monitoring and numerical simulation. Furthermore, with the DDA (discontinuous deformation analysis) method, Do et al. (2020) studied the mechanical response caused by horizontal excavation under different conditions (such as different inclinations and positions of the tunnel face). Using the direct boundary element method (BEM), Panji et al. (2016) studied the stress behavior of shallow buried tunnels under simultaneous nonuniform surface traction and asymmetric gravity loading, and the study showed that the shallow eccentricity of the load had a significant impact on the displacement and stress around the tunnel. By field observations, monitoring and three-dimensional (3D) numerical simulation, Yang et al. (2020) concluded that the retaining wall at the shallow side of a tunnel presents counterclockwise rotation deformation characteristics along the toe of the wall, and the covering soil at the wall surface can apply counterpressure to the retaining wall. Considering the most important parameters on overbreak, many ABC-ANN models have been constructed based on their effective parameters, and a new hybrid model, namely an artificial bee colony (ABC)–artificial neural network (ANN), has been developed to predict overbreak (Koopialipoor et al. 2019).

The third aspect is about the study of stress ratio quantification, as shown in Table 1. By examining an example of a seismic cluster in a deep mine in Canada, a relative apparent stress ratio (ASR) was proposed to quantify the apparent stress of a seismic group, and the overall apparent stress of earthquakes was quantitatively calculated (Brown et al. 2017). To identify the deformation stage and failure process of a rock containing a single fissure, Lv et al. (2017) introduced the ratio of the concentrated stress to the peak stress of samples during experiments. In addition, Abdollahipour et al. (2012) defined the ratio of lateral stress to vertical stress as the stress ratio, and by the change in the ratio and the displacement of key points, the stability of the cavern was evaluated. Karatela et al. (2016) investigated the effect of fracture orientation and the in situ horizontal stress ratio on the stability of the rock mass around the borehole, and the results showed that when the in situ stress ratio increases, the rock blocks at the borehole wall tend to move toward the center of the borehole. Yu et al. (2020) defined the ratio of vertical stress as the stress ratio, and based on the Chinese Railway Tunnel Design Code, they quantitatively analyzed the stress ratio of two sides of asymmetrical loading tunnels. Furthermore, the influence of the dip angle and the ratio of horizontal and vertical stress on the bearing capacity of prestressed concrete-lined pressure tunnels was studied using a two-dimensional finite element model, and the study revealed that the in situ stress ratio significantly affects the load sharing between the rock mass and the lining (Simanjuntak et al. 2016). In addition, studies have shown that when the tunnel diameter is fixed, the deeper the ground covering depth is, the greater the required limit support pressure. However, the ground covering depth has almost no effect on the ultimate support pressure, even if the tunnel is deep (Zhang et al. 2021). According to the Chinese Code for the Design of Railway Tunnels, determining whether a tunnel is subject to asymmetric loads is based on empirical values that are determined by maximum burial depths and surface slope angles (TB 1003–2019).

Table 1 Summary of stress ratio
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These studies have increased our understanding of the surrounding rock pressure calculation, failure mechanism, failure characteristics and quantification of stress ratio. However, very few studies have quantitatively analyzed the asymmetric loads of tunnels using the asymmetric stress ratio index. Furthermore, in the Chinese Code, asymmetrical loading tunnels are given by experience from the maximum burial depths and surface slope angles of the tunnel, and the continuous discrimination of asymmetric tunnels is not realized. In this study, numerical simulations were used to quantitatively analyze the stress ratio of the surrounding rock, and a new approach for achieving the continuous discrimination of asymmetric tunnels was proposed. Compared to the Chinese Code, the new approach is relatively cumbersome. However, this method breaks through the constraints of a specific slope angle and buried depth, overcomes human subjective factors and realizes the continuous discrimination of asymmetrical loading tunnels. The methodology overview of this study is shown in Fig. 1. The results of this study provide significant insights for the identification of asymmetrical loading tunnels, design of support structures and safety risk assessment of tunnels.

Fig. 1
figure 1

Methodology overview

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2 Asymmetric Tunnel Stress Ratio Model

To eliminate the limitation and subjectivity of relying only on the empirical values listed in the Chinese Code (TB 1003–2019) to judge asymmetrical loading tunnels, finite element software was used to establish the 269 kinds of models with different burial depths and different ground slopes under different surrounding rock grades, and the asymmetric stress ratio of the arch shoulder under different conditions was quantitatively analyzed.

2.1 Elastic–plastic Mechanics Solution Model

In this study, the surrounding rock was considered to be an elastic–plastic material. When the stress state at a certain point in space enters the yield stage, the stress–strain relationship is nonlinear, and the increment of the total strain is made up of elastic strain and plastic strain on the basis of elastic–plastic theory (Wang et al. 2007):

$$d\left\{ \varepsilon \right\} = d\left\{ \varepsilon \right\}^{e} + d\left\{ \varepsilon \right\}^{p}$$
(1)

The elastic–plastic constitutive relation of the surrounding rock is as follows:

$$\left\{ {d\sigma } \right\} = \left[ {\left[ {D_{{\text{e}}} } \right] - \frac{{\left[ {D_{{\text{e}}} } \right]\left\{ {\frac{\partial F}{{\partial \sigma }}} \right\}\left\{ {\frac{\partial F}{{\partial \sigma }}} \right\}^{T} \left[ {D_{\varepsilon } } \right]}}{{\left\{ {\frac{\partial F}{{\partial \sigma }}} \right\}^{T} \left[ {D_{{\text{e}}} } \right]\left\{ {\frac{\partial F}{{\partial \sigma }}} \right\} + A}}} \right] \,d\left\{ \varepsilon \right\}$$
(2)

Or abbreviated as:

$$\left\{ {d\sigma } \right\} = \left( {\left[ {D_{{\text{e}}} } \right] - \left[ {D_{{\text{p}}} } \right]} \right)d\left\{ \varepsilon \right\} = \left[ {D_{{{\text{ep}}}} } \right]d\left\{ \varepsilon \right\}$$
(3)

where \(\left[ {D_{{\text{e}}} } \right]\) is the elastic matrix; \(\left[ {D_{{\text{p}}} } \right]\) is the plastic matrix; \(\left[ {D_{{{\text{ep}}}} } \right]\) is the elastic–plastic matrix; and A = -\(\left\{ {\frac{\partial F}{{\partial \sigma }}} \right\}\left\{ H \right\}\) is the characteristic parameter that can reflect the soft and hard characteristics of rock and soil materials, where A = 0 corresponds to an ideal elastic–plastic material.

The expression of the elastic matrix \(\left[ {D_{{\text{e}}} } \right]\) of a material is as follows:

$$\left[ {D_{{\text{e}}} } \right] = \left[ {\begin{array}{*{20}c} {\lambda + 2G} & \lambda & \lambda & 0 & 0 & 0 \\ \lambda & {\lambda + 2G} & \lambda & 0 & 0 & 0 \\ \lambda & \lambda & {\lambda + 2G} & 0 & 0 & 0 \\ 0 & 0 & 0 & G & 0 & 0 \\ 0 & 0 & 0 & 0 & G & 0 \\ 0 & 0 & 0 & 0 & 0 & G \\ \end{array} } \right]$$
(4)

In contrast, the expression of the plastic matrix \(\left[ {D_{{\text{p}}} } \right]\) of a material is as follows:

$$\left[ {D_{{\text{p}}} } \right] = \frac{{\left[ {D_{{\text{e}}} } \right]\left\{ {\frac{\partial F}{{\partial \sigma }}} \right\}\left\{ {\frac{\partial F}{{\partial \sigma }}} \right\}^{T} \left[ {D_{{\text{e}}} } \right]}}{{\left\{ {\frac{\partial F}{{\partial \sigma }}} \right\}^{T} \left[ {D_{{\text{e}}} } \right]\left\{ {\frac{\partial F}{{\partial \sigma }}} \right\} + A}}$$
(5)

where λ is the lame constant and G is the shear modulus (kPa).

The plastic yield condition of rock and soil can be described according to the Mohr–Coulomb yield criterion as:

$$\tau_{{\text{n}}} = - \sigma_{{\text{n}}} \tan \varphi + c$$
(6)

With the yield function (f) of:

$$f = \frac{1}{2}\left( {\sigma_{1} - \sigma_{3} } \right) + \frac{1}{2}\left( {\sigma_{1} + \sigma_{3} } \right)\sin \varphi - c\cos \varphi$$
(7)

where \(\tau_{{\text{n}}}\) is the shear stress (kPa), \(\sigma_{{\text{n}}}\) is the normal stress (kPa), \(\varphi\) is the internal friction angle, c is the cohesion coefficient (\({\text{kN}}/{\text{m}}^{2}\)), and \(\sigma_{1}\) and \(\sigma_{3}\) are the first and third principal stresses, respectively.

2.2 Boundary Conditions and Model Parameters

According to the geological conditions and the design parameters of the Shidaoyangcha tunnel, the span of the tunnel was taken to be 12.6 m and the height to be 10.0 m. Based on the principles of numerical model establishment, the width of the model in the X direction is 100 m, and the height of the model in the Z direction is 40 m from the bottom of the model to the center of the tunnel. The boundary is the displacement constraint. The two side boundaries are horizontal displacement constraints, the bottom boundary is x and y constraints, and the top boundary condition is an unconstrained boundary.

According to the support parameters of the Shidaoyangcha tunnel, the support structure is simulated by a linear elastic beam element and supported by C25 concrete with a thickness of 30 cm. The bolts are modeled as embedded truss elements with a length of 3.5 m and a spacing of 1.0 m. Depending on the characteristics of the surrounding rock, the initial geostress field considers only the overburden pressure of the surrounding rock, and other mechanical influences are ignored. The model assumes that the surrounding rock is an isotropic and homogeneous material. Tunnel excavation is simulated by the two-step method. A schematic of the calculation model is shown in Fig. 2a, and the finite element method (FEM) model is shown in Fig. 2b.

Fig. 2
figure 2

Diagrams of the calculation model and FEM model for a tunnel under asymmetric loads: a geometric dimensional drawing of the tunnel model; b sectional drawing of the tunnel model

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According to the Chinese Code for the Design of Highway Tunnels (JTG 3370.1–2018), the physical and mechanical parameters of the surrounding rocks are shown in Table 2.

Table 2 Physical and mechanical parameters of surrounding rocks
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2.3 Verifying the Model for Numerical Simulation

To verify the correctness of the numerical model, the stresses between the surrounding rock and initial lining of the shallow buried tunnel under asymmetric loads are monitored. Measuring points should be laid on the key points of representative sections, and the monitoring data of the arch shoulder were selected for analysis, as shown in Fig. 3a.

Fig. 3
figure 3

Layout of stress monitoring points of the tunnel: a position of pressure sensors; b arrangement of pressure sensors between the surrounding rock and initial lining

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The compressive face of the pressure sensor should be oriented toward the surrounding rock, and the pressure sensor should be fixed on the steel arch frame before the initial lining is completed. Then, the shotcrete layer should be carefully applied to prevent the formation of a gap between the shotcrete and the pressure sensor, as shown in Fig. 3b.

According to the numerical simulation, the corresponding simulated values of the asymmetric stress were obtained. A comparison of the stress monitoring results and the numerical simulation results is shown in Table 3. The results show that the numerical analysis agrees with the field measurements and that the numerical model is reliable.

Table 3 Comparison of the stress monitoring results and numerical simulation results
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3 Stress Ratio and Numerical Models of Asymmetric Tunnels

3.1 Stress Ratio of Asymmetric Tunnels

At present, determining whether a tunnel is subject to asymmetric loads is based on the empirical values of the maximum burial depths and surface slope angles listed in the Chinese Code for the Design of Railway Tunnel (TB 1003–2019), which has great limitations and subjectivity. Therefore, to analyze the degree of asymmetric loads applied on the tunnel and realize the continuous discrimination of asymmetric tunnels, a quantitative approach to assess asymmetrical loading tunnels was presented. In particular, a new discriminant indicator of asymmetric tunnels, named the asymmetric stress ratio, is introduced.

Based on the group's previous research, regardless of whether the surrounding rock is of grade III, IVRock, IVSoil or V, the stress ratio of the arch wall is approximately 1.0, and the stress ratio of the arch foot has no obvious variation with increasing burial depth and decreasing surface slope angle. However, the stress ratio of the arch shoulder changes obviously. Due to space limitations, this article only introduces some of the main overviews of the stress ratio of the arch shoulder.

The asymmetric stress ratio is the ratio of the von Mises stress on symmetric left- and right-handed positions in the tunnel. Figure 2a shows points 1 and 2 that selected around the tunnel structure to analyze the asymmetric stress ratio. These points correspond to the left and right sides of the arch shoulder, respectively.

The stress ratio of the surrounding rock between points 1 and 2 of the arch shoulder is defined as \(\Delta_{1}\):

$$\Delta_{1} = \frac{{\sigma_{2} }}{{\sigma_{1} }}$$
(8)

where \(\sigma_{1}\) and \(\sigma_{2}\) are the von Mises stresses at key points 1 and 2, respectively.

3.2 Numerical Models of the Calculation

To give the quantitative criterion for discriminating asymmetric loading tunnels, the 269 kinds of models with different burial depths and different ground slopes under different surrounding rock grades were established, which consist of two types.

First, the 256 kinds of numerical models were established: the surrounding rock grades are III, IVRock, IVSoil and V; the burial depths are 5 m, 10 m, 15 m, 20 m, 25 m, 30 m, 35 m and 40 m; and the surface slope angles are 5°, 10°, 15°, 20°, 25°, 30°, 35° and 40°. Second, the13 kinds of numerical models based on the critical conditions of asymmetrical loading tunnels listed in the Chinese Code for the Design of Railway Tunnels (TB 1003–2019) were established. Through the model calculations, the arch shoulder stresses of the surrounding rock are obtained, and the values of the asymmetric stress ratio are calculated.

4 Numerical Simulation Results and Discussion

4.1 Asymmetric Stress Ratio of Arch Shoulder

The 256 kinds of numerical models were established: The surrounding rock grades are III, IVRock, IVSoil and V; the burial depths are 5 m, 10 m, 15 m, 20 m, 25 m, 30 m, 35 m and 40 m; and the surface slope angles are 5°, 10°, 15°, 20°, 25°, 30°, 35° and 40°. From the model calculations, the von Mises stresses of the surrounding rock at key points 1 and 2 corresponding to different numerical models were obtained. Then, based on Eq. (8), the asymmetric stress ratios were calculated.

Diagrams of the variation trends of the arch shoulder surrounding rock stress ratio for grades III, IVRock, IVSoil and V are plotted with the changes in the burial depths and surface slope angles, as shown in Fig. 4.

Fig. 4
figure 4

Diagram of the variation trend of the asymmetric load stress ratio of surrounding rock: a grade III surrounding rock; b grade IVRock surrounding rock; c grade IVSoil surrounding rock; d grade V surrounding rock

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As shown in Fig. 4, regardless of whether the surrounding rock is of grade III, IVRock, IVSoil or V, the asymmetric stress ratio of the surrounding rock decreases and tends to stabilize with increasing burial depth and decreasing surface slope angle. When the surrounding rock is of grade III, the values of the stress ratio converge to approximately 1.64; when the surrounding rock is of grade IVRock, the values of the stress ratio converge to approximately 1.59; when the surrounding rock is of grade IVSoil, the values of the stress ratio converge to approximately 1.54; and when the surrounding rock is of grade V, the values of the stress ratio converge to approximately 1.17. The asymmetric stress ratio decreases with decreasing surrounding rock grades.

Furthermore, from Fig. 4, two groups of behavior were clearly classified: Regardless of whether the surrounding rock is of grade III, IVRock, IVSoil or V, slope angles between 5° and 20° show a gradual decrease, and slope angles between 25° and 40° show a rapid decrease when the burial depth is increased. Taking the burial depth of 5 m for example, when the slope angle is between 5° and 20° and the surrounding rock is of grade III, the values of stress ratio range from 1.17 to 2.74; when the surrounding rock is of grade IVRock, the values of stress ratio range from 1.15 to 2.53; when the surrounding rock is of grade IVSoil, the values of stress ratio range from 1.13 to 2.30; when the surrounding rock is of grade V, the values of stress ratio range from 1.12 to 2.11; it can be seen from the above that the values of stress ratio decreases gradually, and the range of variation decreases gradually with the reduction in surrounding rock grades. When the slope angle is between 25° and 40° and the surrounding rock is of grade III, the values of stress ratio range from 4.28 to 8.52; when the surrounding rock is of grade IVRock, the values of stress ratio range from 3.81 to 7.19; when the surrounding rock is of grade IVSoil, the values of stress ratio range from 3.34 to 5.91; when the surrounding rock is of grade V, the values of stress ratio range from 2.93 to 3.89; it can be seen from the above that the values of stress ratio decreases gradually, and the range of variation decreases gradually with the reduction in surrounding rock grades, but the range of stress ratio variation is significantly larger than that at 5°–20°. This phenomenon indicates that the sensitivity of the slope angle to the stress ratio increases after the slope angle exceeds 20°, and the effect of the asymmetric loads on the tunnel will be strengthened gradually.

Diagrams of the variation trends of the arch shoulder surrounding rock stress ratio for grades III, IVRock, IVSoil and V are plotted in three-dimensional coordinates with the changes in the burial depths and surface slope angles, as shown in Fig. 5.

Fig. 5
figure 5

Diagram of the variation trend of the surrounding rock stress ratio in three-dimensional coordinates: a grade III surrounding rock; b grade IVRock surrounding rock; c grade IVSoil surrounding rock; d grade V surrounding rock

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As seen from the diagram above, the stress ratio shows a good curve variation. Along with the decreasing burial depths and increasing surface slope angles, the stress ratio gradually reaches the maximum value; with the increasing burial depths and decreasing surface slope angles, the stress ratio gradually reaches the minimum value. When the surrounding rock is of grade III, the maximum stress ratio is 8.52; when the surrounding rock is of grade IVRock, the maximum stress ratio is 7.20; when the surrounding rock is of grade IVSoil, the maximum stress ratio is 5.92; and when the surrounding rock is of grade V, the maximum stress ratio is 3.89. The values of the maximum stress ratio decrease with decreasing surrounding rock grades, and the values of the minimum stress ratio are all approximately 1.00, with little change.

Using a mathematical function, the three-dimensional surfaces were fitted in Fig. 5 to obtain the variation equation of the stress ratio with burial depths and surface slope angles under different surrounding rock grades. The equation of stress ratio is as Eq. (9).

$$\Delta_{2} = \frac{{A + Bh - C\alpha + D\alpha^{2} - Eh\alpha }}{{1 + Fh - G\alpha + Hh^{2} + I\alpha^{2} - Jh\alpha }}$$
(9)

where h is the burial depth (m), \(\alpha\) is the surface slope angle (°), + is the stress ratio, and A, B, C, D, E, F, G, H, I and J are parameters. The corresponding parameters are different at different grades of surrounding rock, as shown in Table 4.

Table 4 Parameters of different grades of surrounding rock
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When judging whether a tunnel is an asymmetric tunnel, it is only necessary to know the surrounding rock grade, burial depth and slope angle. The corresponding numerical model was established using finite element software, the surrounding rock stress of the arch shoulder was extracted, and the stress ratio \(\Delta_{1}\) was calculated according to Eq. (8). The parameters in Eq. (9) were selected according to Table 4, and the corresponding burial depth and slope angle were entered to calculate the stress ratio \(\Delta_{2}\).

4.2 Asymmetric Critical Stress Ratio of Empirical Values

The data used to judge asymmetric tunnels in the Chinese Code are empirical data obtained from a large number of practical works. In general, determining whether a tunnel is subject to asymmetric loads is based on empirical values that are determined by the maximum burial depths and surface slope angles listed in the Chinese Code for the Design of Railway Tunnel (TB 1003–2019). When the vertical distance from the lateral arch shoulder of the tunnel to the surface is not more than the limit value listed in Table 5, the tunnel should be treated as being subjected to asymmetric loads. However, in real geological conditions, the maximum burial depths and surface slope angles of the tunnel are not necessarily completely consistent with the Chinese Code; in other words, the continuous discrimination of asymmetric tunnels is not realized. Therefore, Eq. (9) for the continuous identification of asymmetric tunnels was proposed, and a comparative analysis was performed with the corresponding empirical values of the Chinese Code.

Table 5 Maximum burial depth above the lateral arch shoulder of a double-lined tunnel under asymmetric loads (m)
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Based on Table 5, numerical models of critical states for different surrounding rock grades are established to further verify the approach proposed above. The values of Von Mises stresses at key points in the surrounding rock are extracted, and the corresponding empirical values of asymmetric stress ratios are calculated according to Eq. (8). Furthermore, according to the critical states for different surrounding rock grades listed in Table 4, Table 5 and Eq. (9) proposed in this paper, the corresponding stress ratio is calculated.

The results are shown in Table 6, and the results were divided into three parts. In the first part, the surrounding rock grades are III and IVRock, and the ground slope is greater than or equal to 1:1.5; in the second part, the surrounding rock grades are III and IVRock, and the ground slope is 1:2; and in the third part, the surrounding rock grades are IVSoil and V, and there is no restriction on the ground slope.

Table 6 Surrounding rock stress ratio for different surrounding rock grades of the tunnel in the critical state
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By comparing the stress ratios of the empirical value and equation value in Table 6, the following information is available. When the surrounding rock is of grade III, the empirical values are greater than the equation values. When the ground slope is greater than or equal to 1:1.5, there is a large difference between the empirical values and equation values. When the ground slope is 1:2, the empirical value is basically the same as the equation value. When the surrounding rock is of grade IVRock, the empirical values are greater than the equation values. When the ground slope is greater than or equal to 1:1.5, there is a large difference between the empirical values and equation values. When the ground slope is 1:2, the difference between the empirical value and equation value is small. When the surrounding rock grades are IVSoil and V, the equation values are in good agreement with the empirical values.

It can be seen from the above analysis that, in general, when the surrounding rock grades are III and IVRock, the empirical values are greater than the equation values, and the proposed approach in this paper is more conservative than the empirical method. When the surrounding rock grades are IVSoil and V, the equation values are in good agreement with the empirical values. The proposed approach in this paper will provide significant insights for the continuous identification of asymmetrical loading tunnels, the design of support structures and the safety risk assessment of tunnels.

5 Summary and Conclusions

In this paper, based on numerical simulation, the asymmetric stress ratio for different slope angles and burial depths was analyzed in the presence of grades III, IVRock, IVSoil and V surrounding rock, and a new approach to achieve the continuous discrimination of asymmetric tunnels was proposed. Based on the achieved results, the following conclusions are drawn:

  1. 1.

    With decreasing surrounding rock grade, the asymmetric stress ratio tends to decrease. Regardless of whether the grade is III, IVRock, IVSoil or V, the asymmetric stress ratios of the surrounding rock tend to converge and stabilize with increasing burial depths and decreasing dip angles, and the effect of the asymmetric loads on the tunnel will be weakened gradually.

  2. 2.

    The sensitivity of the slope angle to the stress ratio increased after the slope angle exceeded 20°, and the effect of the asymmetric loads on the tunnel will be strengthened gradually.

  3. 3.

    When the surrounding rock grades are III and IVRock, the values calculated using the proposed approach are lower than the empirical values, and the approach is more conservative than the empirical method. When the surrounding rock grades are IVSoil and V, the values calculated using the proposed approach are in good agreement with the empirical values.

  4. 4.

    The asymmetric stress ratio of the surrounding rock arch shoulder can be used as an index to ascertain whether a tunnel is subjected to asymmetric loads. When the surrounding rock grades are III and IVRock, and the values of the stress ratio are greater than those calculated by Eq. (9), and when the surrounding rock grades are IVSoil and V, the values of the stress ratio are greater than or approximately equal to those calculated by Eq. (9), and the tunnel can be considered to be in an asymmetrical loading state.