#### Abstract

Large deformation, collapse, and destruction of supporting structure occurred many times due to rainfall in Xiaohegou tunnel, which is located at Taixing Railway in Shanxi Province. In this paper, the Xiaohegou tunnel is taken as an example, and the large deformation numerical simulation method is used to reproduce the collapse process of the slope at the entrance and the internal section of the tunnel. The reasons for the failure of the original initial support of the tunnel are analyzed, and the failure mechanism of the support structure of the expansive loess tunnel was clarified. The results show that the rainfall leads to the destruction of the tunnel support and the decrease of the bearing capacity of the support structure, which results in the collapse of the tunnel.

#### 1. Introduction

China is one of the countries with the largest loess coverage in the world. The loess is widely distributed in North China, Central China, and Northwest China [1]. Expansive loess is a geological soil formed in the process of natural geological changes in the loess area, which is particularly harmful to engineers. It has the characteristics of loess and expansive soil [2]. The expansion-shrinkage and fissure properties of expansive loess have serious harm to engineers. And the more prominent problems are the large deformation, collapse, and landslides during the construction of tunnels located at the expansive loess layer under the continuous rainfall [3].

At present, many scholars have carried out relevant research on the stability of loess tunnel and slope engineering under the rainfall. Wang et al. [4] analyzed the causes of the collapse of the roof of the Bailuyuan tunnel during construction and confirmed that the particular characteristics of loess and the synergy of groundwater were the internal causes of the tunnel’s collapse as well as, to a certain extent, atmospheric precipitation. Zhao et al. [5] simulated the unsaturated seepage process and humidification process of the expansive soil slope under rainfall based on saturated-unsaturated seepage theory and analyzed the effects of strength attenuation, seepage softening, and moistening expansion on the overall stability of the expansive soil slope. Liu and Lai [6] carried out a loading model test to find out the characteristics and laws of lining cracking under the effect of slide surface immersion. And the effects of different types of slide surface and different immersion degrees on the secondary lining are analyzed. Sun et al. [7] presented a review and analysis of previous studies and 27 typical loess tunnels in China, which indicates three different types of water-rich loess surrounding rocks, and the formation mechanism, seepage field characteristics, and consequent stress and deformation characteristics of the surrounding rock are explained in detail. Deng et al. [8] established the plastic constitutive relation of loess with different water contents and proved that the constitutive relation can describe the stress-strain law of loess well. Wang et al. [9] analyzed the correlation between the deformation of surrounding rock and rainfall during the construction of loess tunnels in the rainy season and proved that the deformation development of the surface and surrounding rock can be predicted by the rainfall. Wang et al. [10] conducted a numerical simulation analysis of the mechanical-rheological seepage characteristics of the tunnel slope under rainfall infiltration based on the finite difference method, and the potential sliding zone and damage-affected area of surrounding rock are studied to determine the key reinforcement range of the tunnel. Zeng 3 analyzed the influence of rainfall infiltration on the mechanical properties of surrounding rock, support characteristics, and stability of expansive loess tunnels based on numerical simulation, and the evolution law of surrounding rock failure is revealed. Feng [11] carried out a study on the deformation of the surrounding rock of an expansive loess tunnel with different expansion stresses under rainfall infiltration based on the similarity of the thermal expansion and humidification expansion, in which the humidification expansion is simulated by thermal expansion. In this engineering, the steel arch-grille arch-jointed support technology is adopted. Zhang et al. [12] studied the stress and deformation characteristics of large-thickness collapsible loess tunnels under water immersion or humidification conditions. Wang et al. [13] explored the influences of variate structural properties on the mechanical properties of surrounding rock of shield tunnels by the modified Mohr-Coulomb strength criterion and Fenner formula. Lai et al. [14] combined on-site monitoring, laboratory tests, and numerical simulation methods to reveal the law of moisture migration in surrounding rock during tunnel excavation and unloading at different distances from the overlying soft plastic loess layer to the tunnel vault. Wang et al. [15, 16] resulted that the surrounding rock pressure of deep-buried loess shield tunnels can be solved by deformation pressure through theoretical analysis. Some scholars have also deduced the calculation formula of the surrounding rock pressure of loess tunnels through the limit equilibrium theory and limit analysis method [17, 18]. However, the surrounding rock load of the shield tunnel is in dynamic evolution. For the loess shield tunnels, the evolution law of the stratum after flooding will be more complicated [19, 20]. Han et al. [21] proposed the solution method of surrounding rock pressure of deep-buried loess shield tunnel, combining with the structural constitutive model of loess and the mechanical analysis of deep-buried tunnel and considering construction and seepage effect by using complex variable function.

Overall, the existing research mainly focuses on unsaturated seepage analysis, thermal expansion deformation, moisture content, and the influence of supporting structures on the tunnel under rainfall. The stability of the landslide at loess tunnel entrance and the relationship between the landslide and the stability of the tunnel structure are rarely analyzed. In this paper, the large deformation numerical simulation method is used to reproduce the collapse process of the slope at the entrance and the internal section of the tunnel. The reasons for the failure of the original initial support of the tunnel were analyzed, and the failure mechanism of the support structure of the expansive loess tunnel was clarified.

#### 2. Project Overview

Xiaohegou tunnel is located in Xiaohegou Village, Loufan County, Taiyuan City, Shanxi Province. The starting and ending mileage of the tunnel are from DK73 + 754 to DK75 + 557. It is a double-track railway tunnel with a total length of 1803 m and a maximum buried depth of about 82 m. Figure 1 shows the longitudinal section of the tunnel. The tunnels all pass through the expansive loess strata, which are mainly sandy loess and expansive loess, and the expansive loess has a medium expansion potential classification.

The three-bench seven-step excavation method is used by the Xiaohegou tunnel. Because the tunnel passes through the expansive loess stratum, large-scale landslides have occurred many times under the rainfall during the construction process. There are three major landslides, two inside the tunnel and one at the entrance. One of the most serious landslides occurred in July 2010. A large landslide occurred near DK75 + 190 during construction, where the primary support was crushed, and the slope of the tunnel entrance collapsed. After the rainfall, as the soil expands and deforms, the surrounding rock squeezed the primary support structure, causing the shotcrete at arches and vaults to peel and fall off, and the steel arch also buckled. Ultimately, the landslide buried the entrance of the tunnel. The damaged primary support and collapsed tunnel are shown in Figure 2.

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The original primary support of the tunnel is mainly composed of 20b I-beam steel arches with a spacing of 0.6 m and C25 shotcrete with a thickness of 25 cm. The tunnel section and support structure are shown in Figure 3.

During the construction of the loess tunnel, the primary support was damaged due to the expansion of the soil under the effect of rainfall, and a collapse accident occurred. After the rainfall, as the soil expands and deforms, the surrounding rock deformed, the shotcrete failed, and the steel arch also buckled. The cracks inside the tunnel increased and expanded. The deformation and failure process changes from slow to fast, and the lining structure gradually loses its bearing capacity, and the deformation of the tunnel continues to increase until the tunnel collapses. Figure 4 shows the monitoring curve of tunnel vault settlement.

It can be seen from Figure 4 that the original support method for the tunnel cannot control the large deformation of the loess tunnel under the rainfall so that the combined support method of “steel arch + steel-grid + shotcrete” is adopted in this engineering, as shown in Figure 5.

The main influencing factors of tunnel entrance collapse may be landslides or insufficient tunnel support strength due to the action of rainfall. The collapse of the slope at the entrance of the tunnel will have an impact on the tunnel support, which will lead to the damage of the support. In addition, the damage to the tunnel support will also cause the disturbance of the surrounding rock and induce the landslide at the entrance of the tunnel. Therefore, the collapse of the slope at the entrance of this tunnel may be caused by the longitudinal collapse force of the slope pushing down the tunnel support, or it may be caused by the insufficient bearing capacity of the tunnel support.

In order to explore the reasons for the instability of the tunnel and solve the supporting problems, the numerical simulation methods are used to analyze the slope of the tunnel entrance and the stability of the tunnel, combining with the on-site collapse and monitoring data.

According to the laboratory test results of loess, the surrounding rock parameters of the expansive loess tunnel are shown in Table 1, and the initial support parameters are shown in Tables 2–5.

#### 3. Computational Principles

In order to simulate the failure process of the landslide at the tunnel entrance and internal collapse section of the tunnel under rainfall, the Lagrangian integral point method based on barycentric interpolation is introduced. The basic principle is described as follows.

According to the principle of virtual work, the weak form of integration of the discretization equation can be expressed as follows [22]:

The relationship between the stress and virtual strain rate of a unit can be expressed as follows:where *D*_{υ} is the viscosity matrix.

According to the formula,

The relationship between the velocity of any point in the unit and the nodal velocity can be expressed as the following trial function:

Thus, the following equation can be derived:

According to the arbitrariness of virtual velocity, the following equation can be derived:where and .

Considering the temporal variation of large deformations, a viscoelastic–plastic constitutive model was adopted, which consisted of viscous, elastic, and plastic components connected in series, as shown in Figure 6.

The total strain rate can be expressed as the sum of viscous, elastic, and plastic strain rates:

Dividing the strain rate into two components, partial and spherical strain rates, the following equation can be derived:where is the partial strain rate, and is the trace of the strain rate matrix.

Stress rate is affected by rigid object rotation. The component of the stress rate that is not affected by rigid object rotation is referred to as the Jaumann stress rate, , which is obtained by deducting the effect of rigid object rotation. Thus, the relationship between stress and strain rate can be expressed as follows:where is the Jaumann stress rate and satisfies the following equation:

Designating and and defining the equivalent shear viscosity and the equivalent bulk viscosity , the first line of (9) can be rewritten as follows:

Thus, the following equation can be derived:

Designating as and as , the following equation can then be derived:

Designating , (13) can then be rewritten as follows:

Similarly, the following equation can be derived:

With the inertial force not considered, the following equation can be derived:where *f* consists of three components: (1) external force, (2) the force resulting from the previous time step, and (3) the force resulting from plastic deformation. The three components are designated as **f**_{1}, **f**_{2}, and **f**_{3}, respectively, and satisfy the following equation:

The barycentric interpolation trial functions can be constructed as follows.

The subdivision of an arbitrary polygon, Ω, can be expressed as follows: . Each of the resulting polygonal units, Ω^{e}, can be illustrated in Figure 7. Assume a polygonal unit with *n* sides, Ω^{e}. Subdivide the polygon into *n* triangles by connecting the vertexes of the polygon (*P*_{1}, *P*_{2}, ..., *P*_{n}) with an arbitrary point inside the polygonal element, . Designating the interior angle of the triangles as *α*_{i} and with , where *x* is the coordinate of *P* and *X* = (*x*, *y*), then the following shape function can be constructed for the polygonal unit:where is the weight function for node *P*_{i}.

Assume a polygonal domain, Ω^{e}, and a unit circle with a point in the polygon, *P*, as its center. The circle and line segment *PP*_{i} then intersect at *H*_{i}. A polygon circumscribing the circle and crossing *H*_{i} can then be derived.

The weight function is defined as the ratio of the side length, *Q*_{i}*Q*_{i−1}, of the polygon *Q*_{1}*Q*_{2}...*Q*_{n} to the length of *PP*_{i}:

According to the Gauss divergence theorem, the barycentric interpolation function for the polygonal unit can be derived:

#### 4. Numerical Simulation of Collapse Process of Expansive Loess Tunnel

The calculation model of the tunnel entrance slope is established, as shown in Figure 8. The surrounding rock parameters are shown in Table 1. The boundary conditions are the normal constraints on the left and right sides, and the fixed constraints on the bottom boundary and free on the upper surface. The calculations prove that the slope of the tunnel entrance can stabilize by itself when the surrounding rock is dry.

However, when the effect of rainfall is considered, the slope collapses as the soil viscosity changes. The simulation of the collapse process is shown in Figure 9. Comparing the calculated slope slump process with the phenomena observed in the field, the slope slip surface can be drawn according to Figure 9, combining with the slip surface revealed by the collapse of the tunnel entrance observed on-site. The comparison results are shown in Figure 10.

According to on-site observations, the slip surface exposed by the collapse of the tunnel entrance is shown in Figure 10.

It can be seen from Figure 10 that the slope slip surface of the Xiaohegou tunnel caused by rainfall is different from that observed on-site. It is preliminarily concluded that the tunnel opening was buried not due to the instability of the slope but may be due to the failure of the support of the tunnel under rainfall conditions, resulting in the collapse of the upper slope of the tunnel.

For further verification, the collapse section inside the tunnel is selected to analyze the stability of the tunnel. The surrounding rock parameters are shown in Table 1. And the model boundary is taken as three times the span of the opening. It is calculated that the tunnel can remain stable under dry conditions. After rainfall, the soil strength decreases and the viscosity increases, and the tunnel becomes unstable. The large deformation process of the tunnel under the rainfall is shown in Figure 11.

It can be seen from Figure 11 that the soil peeled off at the side walls of the tunnel first, and then, the inner side of the tunnel was deformed as a whole until it collapsed. This phenomenon is consistent with the field observations. It is further confirmed that the rain caused the damage to the tunnel support, which caused the tunnel collapse.

#### 5. Conclusion

In this paper, the reasons for the collapse of the tunnel and the stability of the tunnel have analyzed by numerical simulation and theoretical analysis, based on the project of the loess tunnel in Taixing, Shanxi Province. The conclusions are as follows:(1)The Lagrangian integral point method based on barycentric interpolation can be used to simulate the collapse process of the tunnel entrance slope and the internal section of the tunnel. On this basis, the article expounds the reasons for the collapse of the tunnel.(2)The slope slip surface at the entrance of the Xiaohegou tunnel caused by rainfall is different from that observed on-site, which confirms that the buried tunnel entrance is not caused by the instability of the slope but may be due to the support of the tunnel under rainfall conditions. The first failure caused the upper slope of the tunnel to collapse, which confirms that the tunnel opening was buried not due to the instability of the slope but may be due to the failure of the support of the tunnel under rainfall conditions, resulting in the collapse of the upper slope of the tunnel.(3)By analyzing the stability of the internal collapse section of the tunnel, the deformation and instability process of the tunnel are consistent with the field observation. It is further confirmed that the tunnel collapse is due to the overloading of the support structure caused by the rainfall, and the support structure is damaged, which further causes the tunnel collapse, rather than the slope collapse.

#### Data Availability

All data, models, or codes that support the findings of this study are available from the first author upon reasonable request.

#### Conflicts of Interest

The authors declare no conflicts of interest.

#### Acknowledgments

This work was partially supported by the Natural Science Foundation of Shanxi Province (No. 202103021223057) and the Fundamental Research Funds for the Central Universities, CHD (No. 300102211503).