Impact of Evaluation of Freeze–Thaw Cycles on Collapse Zone at Entrance and Exit of Loess Tunnel

Abstract

The entrance and exit of loess tunnel are easily affected by freezing and thawing, which leads to collapse in cold regions. Based on the slip line network method, this paper proposed a graphical method and analyzed the impact of evaluation of freeze–thaw cycles on the collapse zone at the entrance and exit of a loess tunnel. Firstly, the slip line network method was improved for a graphical calculation program, and the program was validated by monitoring the data of collapse evolution in an unlined loess tunnel in situ. Then, some triaxial tests of freeze–thaw cycles were carried out on loess, and the attenuation law of loess strength was summarized after freeze–thaw cycles in cold regions. Finally, the graphical calculation program was applied to evaluate the impact of freeze–thaw cycles on collapse range in the unlined loess tunnel. The results show that introducing the classic slip line field into the analysis process of a tunnel collapse zone can provide a feasible and efficient analysis method for a tunnel surrounding rock collapse arch for tunnel engineering, and the collapse zone increases with the growing number of freeze–thaw cycles. Moreover, the attenuation of loess strength which suffered from the freeze–thaw cycles would cause the collapse zone area to increase by a maximum of 16.13%.

1. Introduction

In cold regions, freeze–thaw hazards are common at the entrance and exit of tunnels. These hazards are particularly prominent in high latitudes, bringing immense damages to the entrance and exit of loess tunnel projects [1]. Currently, loess tunnel calculation mainly relies on the finite-element simulation or classic analytic theories [2]. Xing et al. [3] summed up the deformation law of loess highway tunnels through three-dimensional numerical simulation based on field monitoring data. Shao et al. [4] proposed a calculation approach for the collapse deformation of loess foundation of tunnels, and derived a way to determine the surrounding rock pressure of the tunnel. Xue et al. [5] theoretically deduced the correlation function of the surrounding rock in loess tunnels, applied the function to merge the evaluation indices and came up with a grading method of surrounding rock in loess tunnels. The above-mentioned research promoted the related evaluation of tunnels, but faced some problems, as many factors were involved in the complex computing process, and the results were not intuitive. It is worth noting that the soil arching of loess tunnels was also not taken into account in the above-mentioned research; as pointed out by the authors of [6], loess collapse arch is one of the key issues affecting loess tunnel safety and design.

Due to their mechanical properties, degradation of sandstone and marble suffered from freeze–thaw cycles [7,8,9], and loess underwent the same deterioration after the freeze–thaw cycle. Through direct shear tests, Dong et al. [10] disclosed the degradation law of loess shear strength after freeze–thaw cycles. Xu et al. [11,12,13] carried out scanning electron microscopy (SEM) and direct shear tests to study the change law of the microstructure and strength of remolded loess after freeze–thaw cycles. Li et al. [14] designed freeze–thaw tests with different temperature levels, and observed the temperature-specific influence law of freeze–thaw cycles on the shear strength of intact loess. Wang et al. [15] pointed out the initiation and evolution law of microcracks in stabilized loess under cyclic freeze–thaw conditions, as shown in Figure 1. The above scholars not only analyzed the initiation and evolution of microcracks in microstructure under cyclic freeze–thaw conditions, but also evaluated the strength degradation of loess which resulted from the freeze–thaw cycles.

In terms of analytical methods, the elastoplastic slip line field theory is a very good method and is widely used in slope stability [16]. Zhu et al. [17] optimized and popularized the classic slip line field by detailing the numerical simulation of slip line direction field, as well as the tracking of slip lines based on their directions. Li et al. [18] established a full-area slip line field equation suitable for numerical calculation, and applied the equation to solve the stability problem. Gong et al. [19] combined slip line field theory and dynamic finite-element method into a strategy for analyzing soil slope stability under seismic effect. Drawing on slip line field theory and dilatancy effect, Zhang et al. [20] determined the shape of the shear failure surface for concrete-rock mass, and designed a load transfer method for bored cast in situ piles in soft rocks. Following elastoplastic slip line field theory, to realize efficient calculation of slope stability, Fang et al. [21,22] proposed a novel quantification method for slope stability based on the slip line field theory and Monte Carlo framework. Cheng et al. [23] modeled the slip line field of the floor rock layer in the roadway, and then analyzed the asymmetric floor heave mechanism of the roadway. Chen et al. [24] proposed and verified a rapid extraction method for automatic identification of slope slip lines, which can reduce the influence of grid size and avoid the subjectivity of artificially defining slip lines. Keshavarz et al. [25,26,27] evaluated the active lateral earth pressure by stress characteristic method or sliding line method, considered the influence of surface slope, wall angle, adhesion and friction angle of soil–wall interface, evaluated the influence of retaining wall geometry and soil–wall interface parameters. Santhoshkumar et al. [28] proposed a sliding line solution to analyze active earth pressure on cantilever retaining walls supported by layered non-cohesive backfill, which can track adaptive failure surfaces by developing a network of slip lines based on the properties of each layer. At the same time, some researchers have studied the stress path of soil through the finite-difference method and numerical analysis method [29,30,31,32].

Field surveys in Ahong karez of Turpan prove, as An et al. [33] discovered, the failure patterns of loess tunnels under freeze–thaw conditions, including sidewall spalling, cracks, and arch collapse, and believed that the failure of the tunnel was related to the capillary action and strength degradation of the loess after freeze–thaw. To extend the elastoplastic slip line field to evaluating the loess arching stability of the unlined loess tunnel, Zhao et al. [34] put forward a slip line network method based on the collapse zone in tested loess tunnels, and discussed the implementation of basic calculation steps in loess tunnel analysis. This approach lays the basis for analyzing the collapse zone in loess tunnel with slip line network method. Bai et al. [35] improved the slip line network method for the surrounding rock in the full cross-section of unlined loess tunnel, prepared a calculation program for the improved method, and validated the program.

Due to the entrance and exit of loess tunnel suffering from freezing and thawing in cold regions, this paper will propose a geometric graphical analysis method and analyze the impact of evaluation of freeze–thaw cycles on the collapse zone at the entrance and exit of the loess tunnel based on the slip line network method. The research results reveal the attenuation mechanism of the shear strength index of loess under freeze–thaw cycles, and based on this, the attenuation characteristics of the collapse area of unlined loess tunnels under freeze–thaw cycles are obtained. The research results may help to promote the application of impact evaluation of freeze–thaw cycles on the collapse zone of the unlined loess tunnels in cold regions.

2. Principle and Procedure of Network Map

In the elastoplastic plane strain problem, there exist two intersecting shear failure surfaces for any point on the plane [36]. Two sets of slip lines can be obtained by connecting the shear failure surfaces of different points, for example, curves α-α and β-β in Figure 2. The tangential direction of any point on a slip line is the direction of the corresponding slip surface.

The direction of each point on a slip line is related to the direction of principal stress trajectories, that is, two sets of orthogonal curves (1-1 and 2-2 in Figure 2) obtained by connecting the segments indicating the principal stress direction of each point. Taking the trajectory of the maximum principal stress σ₁ as the base line, the curve with an acute angle from the base line in the clockwise direction is called line α, and that with an acute angle from the base line in the counterclockwise direction is called line β. For Mohr–Coulomb (M–C) elastoplastic material, the angle between the two sets of slip lines is 2µ = π/2 − φ, and the angle with principal stress trajectories is µ = π/4 − φ/2, where φ is the internal friction angle of the Mohr–Coulomb elastoplastic material.

When the stress is solved by the classic slip line network method, the two sets of slip lines form a curve coordinate (S_α, S_β). Under the coordinate system of (S_α, S_β), the following equations can be solved:

$$\frac{dy}{dx}=\tan (\theta -\mu )\begin{array}{cc}& \end{array}(line\alpha )$$

(1)

$$\frac{dy}{dx}=\tan (\theta +\mu )\begin{array}{cc}& \end{array}(line\beta )$$

(2)

$$-\sin \left(2\mu \right)\frac{\partial P}{\partial S_{\alpha }}+2R\frac{\partial \theta }{\partial S_{\alpha }}+\gamma \left[\sin \left(2\mu \right)\frac{\partial x}{\partial S_{\alpha }}+\cos \left(2\mu \right)\frac{\partial y}{\partial S_{\alpha }}\right]=0$$

(3)

$$\sin \left(2\mu \right)\frac{\partial P}{\partial S_{\beta }}+2R\frac{\partial \theta }{\partial S_{\beta }}+\gamma \left[-\sin \left(2\mu \right)\frac{\partial x}{\partial S_{\beta }}+\cos \left(2\mu \right)\frac{\partial y}{\partial S_{\beta }}\right]=0$$

(4)

where, P is the mean stress; R is the radius of the stress circle (for M–C elastoplastic material, R = P·sinφ + c·cosφ); γ is bulk density; θ is the angle between the direction of the first principal stress σ₁ and axis x.

Due to the curve coordinates, the difference equation thus obtained is nonlinear. Formulas (1)–(4) can only be solved by numerical solutions through the difference method (Gong, [36]). The slip line field theory has been widely adopted to determine the slip surface of slopes. Figure 3 presents the conventional slip line field of slopes.

For the collapse zone of loess arch in loess arching theory, the collapse zone of an unlined loess tunnel may be plotted according to the slip line theory (Figure 4). The collapse begins from the side walls and arch feet, and gradually expands to the vault.

Since the cross-section of the tunnel is vertically symmetric, using geometric symmetry, the structure of the right half of the unlined loess tunnel was taken and rotated by 90° clockwise, as shown in Figure 5. It is worth noting that the slip line network of the half tunnel is very similar to the slope slip line analysis (Figure 5).

Inspired by the similarity between the half tunnel slip line and slope slip line, a slip line network method for analyzing the collapse zone in a loess tunnel might be proposed, and the construction of a slip line pattern will be discussed in detail, as shown in Figure 6. With slip lines at the collapse zone, the slip line network method simplifies the curved slip lines as the straight lines, according to the collapse contours observed in loess tunnel surveys, such that the solved bearing capacity of the surrounding rock approximates the lower limit of plastic limit analysis.

The implementation steps of the slip line network method for the collapse zone of an unlined loess tunnel are as follows:

Step 1: Drawing a slip line of the collapse zone in the passive stress zone (side wall section of tunnel)

The passive stress zone is located at the side wall section of an unlined loess tunnel. The passive stress zone consists of the area enclosed by points of O₁, O₄, O_5. The linear boundary between the passive stress zone and the elastic zone is obtained with the angle between the slip line and tunnel side wall being 45° − φ/2. The linear boundary between the passive stress zone and the transition zone is obtained with the angle between the slip line and tunnel crown being 45° − φ/2. Then the boundary direction range of the passive stress zone and the transition zone is obtained, and the passive stress zone could be drawn.

Step 2: Drawing a slip line of the collapse zone in the transition zone

The transition zone consists of the area enclosed by points of O₁, O₃, O_4. The shape of the slip line in the transition zone is assumed to be a logarithmic spiral. The linear boundary between the transition zone and the active stress zone is obtained with the vertical direction being 45° + φ/2. Then the boundary direction range of the transition zone and the active stress zone is obtained, and the transition zone could be drawn.

Step 3: Drawing a slip line of the collapse zone in the active stress zone

The active stress zone consists of the area enclosed by points of O₁, O₂, O₃. From the drawn boundary of the transition zone and the active stress zone, the upper boundary of the active stress zone can be calculated by using the triangle theorem, and the active stress zone in the collapse zone can be drawn.

Step 4: Determine the horizontal uniform bearing capacity (P₀) of the vault

The O₁-O₂ surface is the arch section of the soil arch in the active stress zone. The reaction force on the logarithmic spiral surface is the vector R, whose value is unknown, but its vector direction points to the point O₁.

According to the vector relationship of the force, the horizontal uniform bearing capacity P₀ generated on the O₁-O₂ surface by the cohesive force as the uniform load on the soil arch surface is obtained. The horizontal uniform bearing capacity (P₀) on the O₁-O₂ surface needs to subtract the cohesive force of soil which exists on the O₁-O₂ surface.

Step 5: Determine the horizontal tension (P₁) of the vault

Under the condition of considering the loess volume force, the volume force of the passive stress zone is vector Q₁, the volume force of the transition zone is vector Q₂, and the volume force of the active stress zone is vector Q₃. There is a shear friction stress τ on the sliding surface between the collapse zone and the elastic zone.

The force vector polygon is composed of the volume force (Q₁) of the passive stress zone, the volume force (Q₂) of the transition zone, the volume force (Q₃) of the active stress zone, the reaction force (R) on the logarithmic spiral surface, and the shear friction force (τ) on the sliding surface. Then, calculate the load (P₁) generated by the volume force in the vertical plane of the collapse area according to the force vector polygon method. The direction of tension (P₁) is horizontal, triangular on the O₁-O₂ surface, and equal to zero at the point O₁.

Step 6: Determine the ultimate horizontal bearing capacity of the collapse zone

According to the physical action of the bearing capacity P₀ and load P₁ on the surrounding rock, the ultimate horizontal bearing capacity of the soil arch can be obtained:

P = P₀ − P₁.

(5)

where, P₀ is the pressure generated by soil cohesion on the arch crown section; P₁ is the tension generated by the soil volume force on the arch crown section.

If the slip line network method is directly applied, the elastoplastic slip line network will take a long time to plot the whole collapse zone of an unlined loess tunnel. What is worse, the plotting accuracy would be undesirable, because the slip line is drawn manually. Therefore, this method cannot be easily used in actual engineering, although it is feasible for predicting the collapse zone and ultimate bearing capacity of unlined loess tunnels. To solve the problem, this paper realizes the slip line network method for the whole collapse zone of unlined loess tunnels by developing a visual analysis program.

3. Research and Evaluation

The target loess tunnel is a single-lane tunnel of the length 250.0 m, passing through soil layer Q₁. The tunnel lies in Zichang County, in northwestern China’s Shaanxi Province. Zhao et al. analyzed the development process of the collapse zone of the unlined loess tunnel (Zhao et al. [34]). During the five-year monitoring period, the surrounding rock of the unlined tunnel gradually collapsed year by year. Figure 7 shows how the collapse zone of the unlined loess tunnel developed through the monitoring period.

This paper develops a visual analysis program for computing implementation of the slip line network method, which targets the collapse zone of an unlined loess tunnel. The program visualizes the calculation process, and plots the slip lines of the surrounding rock. The outputs of the program help to judge the collapse zone and ultimate bearing capacity.

The calculation process (Figure 8) presents the inputs of the visual analysis program and includes c value, φ value, γ value, and the depth-span ratio of the excavation cross-section of an unlined loess tunnel; the outputs of the program are the collapse range and the ultimate bearing capacity of the loess arch. Figure 9 presents the use of the program interface. The program simulation corresponding to a specific set of parameters and loading conditions required around 30 s on a computer with 2.66 GHz 6-Core processors and 16 GB, 2900 MHz memory.

The loess parameters were determined as follows according to the field data of the collapse zone of the loess tunnel in situ, as well as the indoor test results of basic physical parameters and freeze–thaw responses of soil samples. Bulk density, γ = 19.6 kN/m³; cohesion, c = 64.07 kPa; internal friction angle, φ = 25.11°; depth-span ratio, 5:4. The slip line network of the surrounding rock of the loess tunnel in situ was obtained by applying our analysis program (Figure 10). The predicted lines of the collapse zone were very close to the monitored contours of the collapse development in the unlined loess tunnel. The results confirm the accuracy and feasibility of the developed program.

4. Results of Freeze–Thaw Cycles on Strength Degradation of Loess

Several freeze–thaw tests were carried out indoors on loess collected from Zichang County with the dry densities (1.60 g/cm³). The loess samples had three different initial water contents (16%, 20%, and 24%). During the tests, the freezing temperature and thawing temperature were set to –20 °C and 20 °C (Figure 11), respectively; the freeze–thaw time was set to 8 h; the number of freeze–thaw cycles were set to 0, 2, 4, 6, and 8. After the freeze–thaw cycles, the consolidated undrained (CU) test was performed immediately (Figure 12).

Table 1. Loess strength parameters under different freeze–thaw conditions.

Strength Indices	Dry Density (g/cm3)	Water Content%	Number of Freeze–Thaw Cycles
			0	2	4	6	8
c/kPa	1.60	16	78.56	59.45	56.45	52.33	51.47
		20	61.24	47.64	42.87	42.26	41.76
		24	42.16	35.68	30.29	29.34	28.86
φ/°		16	29.12	27.89	27.13	26.96	26.77
		20	28.78	26.84	26.24	26.12	25.98
		24	28.33	26.79	25.36	24.37	24.13

reported the test results.

The cohesion of the loess declined with the growing number of freeze–thaw cycles, and the declining trend slowed down until reaching a stable level, as shown in Figure 13. Based on the test data, the cohesion of the loess was degraded by 34.48%, 31.81%, and 31.54% under freeze–thaw cycle conditions of 8 times, with different initial water contents, respectively. The results reveal the degradation impact of freeze–thaw cycles on loess cohesion.

The internal friction angle of the loess decreased with the growing number of freeze–thaw cycles, but at a very slow rate, as shown in Figure 14. Based on the test data, the internal friction angle of the loess was degraded by 15.52%, 9.7%, and 14.83% under freeze–thaw cycles conditions of 8 times, with different initial water contents, respectively. The results reveal the degradation effect of freeze–thaw cycles on the internal friction angle of loess. Any change of the angle would cause a shift to the maximum shear surface in the slip line field of the soil mass.

With the increase of freeze–thaw cycle times, the cohesion and the internal friction angle of the loess strength decreased gradually, and this experimental phenomenon of loess strength attenuation is consistent with the analysis by Wang et al. [15], but it is worth noting that the attenuation of cohesion was significantly greater than that of the internal friction angle.

5. Results

The entrance and exit of a loess tunnel are easily affected by freezing and thawing, which leads to collapse in cold regions. According to the results of freeze–thaw tests and the monitoring data on a loess tunnel, the computational parameters are listed in

Table 2. Computational parameters and results.

Case		Loess Parameters	Condition	Collapse Zone Parameters
ρd/(g/cm3)		w/%		Bearing Capacity (P)/kN	Area/m2
Case 1-1	1.6	18	Undisturbed loess tunnel	2599.45	12.52
Case 1-2			After freeze–thaw cycles	1534.97	13.66
Case 2-1		20	Undisturbed loess tunnel	1893.07	12.67
Case 2-2			After freeze–thaw cycles	1123.37	14.00
Case 3-1		24	Undisturbed loess tunnel	1104.46	12.88
Case 3-2			After freeze–thaw cycles	561.74	14.95

.

On this basis, our program computes the area of the collapse zone and the variation of the bearing capacity (P) for the entrance and exit section of a loess tunnel under three different working conditions of Case 1-1, Case 2-1, Case 3-1, and corresponding to the suffering from the freeze–thaw cycles under three different working conditions of Case 1-2, Case 2-2 and Case 3-2. The calculation results are given in Table 2.

Comparative analysis of the bearing capacity (P) of the loess tunnel before and after freezing and thawing is shown in Figure 15. The bearing capacity (P) of the loess tunnel was degraded by 49.13%, 40.66%, and 37.56% in Case 1-2, Case 2-2, Case 3-2, respectively. The results reveal the significant degradation effect of freeze–thaw cycles on the bearing capacity (P) of loess tunnel. With the increase in water content of the surrounding rock, the bearing capacity (P) of the section of loess tunnel decreases. This is mainly the soil mass in the entrance and exit section that witnesses a continued decline in shear strength of loess suffered from the freeze–thaw cycles.

According to the data of Case 3-1 and Case 3-2, in order to vividly reveal the change of the collapse area in the loess tunnel area, we have drawn the schematic diagram of the collapse area before and after freezing and thawing in Figure 16. Comparative analysis of the collapse zone area at the entrance and exit section of the loess tunnel before and after freezing and thawing are shown in Figure 16a,b, and the excess area (Figure 16c) of the collapse zone could be achieved by a subtraction between the collapse zone area Case 3-1 and the collapse zone area Case 3-2. It is worth noting that the ratio of excess area is up to 16.13% in Case 3 during the freeze–thaw cycles. As a result, the effect of the freeze–thaw cycle will significantly increase the area of the collapse zone of the loess tunnel.

6. Conclusions

In order to better evaluate the impact of the evaluation of freeze–thaw cycles on the collapse zone at the entrance and exit of a loess tunnel, a series of laboratory tests and program calculations were carried out under different freezing–thawing conditions. The change mechanism of bearing characteristics on the collapse zone at the entrance and exit of the loess tunnel under freeze–thaw cycles was announced, which has important theoretical and engineering significance for the construction of loess tunnels in cold regions. The conclusions are as follows:

(1)

Based on the monitoring data on the tested loess tunnel, this paper develops a visual analysis program for computing implementation of the slip line network method, which targets the collapse zone of an unlined loess tunnel. The program visualizes the calculation process, and plots the slip lines of the surrounding rock. The outputs of the program help to judge the collapse zone and ultimate bearing capacity.

(2)

With the increase of freeze–thaw cycle times, the cohesion and the internal friction angle of the loess strength decreased gradually in the consolidated undrained (CU) test under different freeze–thaw conditions, the cohesion of loess strength decreased by 34.48%, and the internal friction angle decreased by 15.52%. However, it is worth noting that the attenuation of cohesion was significantly greater than that of the internal friction angle.

(3)

Comparative analysis of the collapse zone area at the entrance and exit section of the loess tunnel before and after freezing and thawing, and the freeze–thaw cycles expanded by 16.13% at the maximum. As a result, the freeze–thaw cycle significantly will increase the area of the collapse zone of the loess tunnel.