Spatiotemporal Mechanical Effects of Framework–Slope Systems Under Frost Heave Conditions

Abstract

To investigate the slope instability caused by differential frost heaving mechanisms from the slope crest to the toe during frost heave processes, this study takes a typical silty clay slope in Xinjiang, China, as the research object. Through indoor triaxial consolidated undrained shear tests, eight sets of natural and frost-heaved specimens were prepared under confining pressure conditions ranging from 100 to 400 kPa. The geotechnical parameters of the soil in both natural and frost-heaved states were obtained, and a spatiotemporal thermo-hydro-mechanical coupled numerical model was established to reveal the dynamic evolution law of anchor rod axial forces and the frost heave response mechanism between the frame and slope soil. The analytical results indicate that (1) the frost heave process is influenced by slope boundaries, resulting in distinct spatial variations in the temperature field response across the slope surface—namely pronounced responses at the crest and toe but a weaker response in the mid-slope. (2) Under the coupled drive of the water potential gradient and gravitational potential gradient, the ice content in the toe area increases significantly, and the horizontal frost heave force exhibits exponential growth, reaching its peak value of 92 kPa at the toe in February. (3) During soil freezing, the reverse stress field generated by soil arching shows consistent temporal variation trends with the temperature field. Along the height of the soil arch, the intensity of the reverse frost heave force field displays a nonlinear distribution characteristic of initial strengthening followed by attenuation. (4) By analyzing the changes in anchor rod axial forces during frost heaving, it was found that axial forces during the frost heave period are approximately 1.3 times those under natural conditions, confirming the frost heave period as the most critical condition for frame anchor design. Furthermore, through comparative analysis with 12 months of on-site anchor rod axial force monitoring data, the reliability and accuracy of the numerical simulation model were validated. These research outcomes provide a theoretical basis for the design of frame anchor support systems in seasonally frozen regions.

1. Introduction

Slope deformation and instability phenomena are objectively noticeable facts in seasonally frozen regions [1,2], particularly in slopes reinforced by frame anchor systems. Influenced by frost heaving, the volumetric expansion of frozen soil induces increased tensile forces in anchor rods. Differential frost heaving between the slope crest and toe triggers spatially variable deformation and structural damage [3], posing severe threats to slope stability and reinforcement integrity. Consequently, it necessitates an exploration into the interaction mechanisms between the frame and slope soil under the combined thermo-hydro-mechanical actions during frost heaving, aiming to reveal frost deformation patterns along the slope surface. This investigation will provide theory-informed supports for ensuring the structural safety of frame systems in seasonally frozen regions.

Previous studies have extensively explored freeze–thaw characteristics and hydrothermal exchange effects on slope stability. Li et al. [4] employed discrete element modeling to simulate randomly fractured slopes, revealing that frost heave forces generate new fractures while destabilizing load-bearing structures. Li et al. [5] demonstrated cyclic variations in frost heave forces through freeze–thaw model tests on steep slopes, identifying coupled frost heave and fissure water pressures as primary destabilizing factors. McRoberts et al. [6] integrated solid–liquid ratios into slope stability analysis, proposing a combined effective stress-thaw consolidation methodology. Weeks et al. [7] attributed rapid pore pressure buildup during thawing to frozen soil barriers impeding drainage under transient thermal fluctuations. Qin et al. [8] developed a reservoir slope THM model linking crest cracking to thermal gradients. Liu et al. [9] formulated a coupled PDE system relaxing continuum assumptions for enhanced numerical convergence. Zhang et al. [10] validated a vapor-inclusive unsaturated soil THM model through controlled freezing experiments. Zhang et al. [11] established a thermodynamically consistent constitutive model for saturated frozen soils, verified via cryogenic triaxial testing. Sanchez et al. [12] advanced a dual-porosity THM framework elucidating macro–micro pore synergies during heterogeneous frost heave. Muhemaier et al. [13] studied a typical high-slope loess along high-speed railway lines, finding through indoor direct shear tests and consolidation tests that under constant dry density and moisture content, the cohesion and compression modulus of the samples gradually decrease with increasing cycles, while the void ratio and compression coefficient increase. Yang et al. [14] investigating the structural expansive soil of railway embankment slopes after freeze–thaw cycles through temperature-controlled dynamic triaxial tests under such cycles, discovered that under deviatoric stress, freeze–thaw cycles induce additional shear deformation in expansive soil, thereby increasing the soil’s dynamic shear modulus.

With respect to reinforcement systems, frame–anchor technology has been shown to effectively mitigate freeze–thaw-induced progressive failure. Notable mechanistic investigations include that by Xiang et al. [15], who deciphered spatiotemporal coordination between slope displacement fields and anchor axial forces, highlighting frost-phase force amplification exceeding thaw-phase magnitudes. Dong [16] innovated thermally regulated frame anchors exhibiting uniform stress distribution under frost loading. Dong et al. [17] derived dynamic equations for frame–anchor–slope systems via Winkler–D’Alembert synthesis, validated through shake-table testing. Li et al. [18] analytically resolved anchor stress distributions post freeze–thaw cycles, corroborated by field instrumentation. Sun et al. [19] integrated thermal–seepage–strength degradation factors into stability assessment protocols. Xiang et al. [20] computationally revealed bidirectional frost deformation–anchor force interdependencies. Yuan et al. [21] quantified a 27% anchor capacity reduction after 30 freeze–thaw cycles via pullout tests. Chang et al. [22] classified axial force evolution into stable, gradual, and accelerated growth modes through in situ THM monitoring. The above computational models fail to account for the influence of slope surface frost heave on changes in anchor rod axial forces or the temporal and spatial mechanical relationships between frameworks and slopes, thus neglecting to address engineering issues such as spatially variable deformation and structural damage caused by differential frost heave between slope crests and toes.

Although significant progress has been made in understanding freeze–thaw effects on slope stability and frame–anchor interactions, few studies have systematically addressed the non-uniform frost heave mechanisms along the slope surface from crest to toe. The spatial heterogeneity of frost-induced deformation and its implications on the design and performance of frame–anchor systems remain insufficiently understood, particularly under fully coupled THM conditions. Existing models often simplify slope geometry or overlook the spatial variability of frost heave forces and anchor responses.

Based on the comprehensive analysis of existing research findings, few studies have investigated the differential frost heave mechanisms from slope crest to toe in frame-anchored slopes during freezing processes to ensure deformation control and stability. This study takes an engineering case in Xinjiang, China, as the research object. Through a triaxial consolidated undrained (CU) shear test on slope geomaterials, the mechanical parameters of soils under both natural and frost heave conditions were obtained. A thermo-hydro-mechanical coupling numerical model was established to analyze the evolution patterns of temperature fields, displacement fields, and stress fields near the slope surface during frost heaving. The research clarifies the dynamic evolution of axial forces in anchors and the frost heave response mechanisms between the frame structure and slope soils, providing theoretical guidance for designing frame-anchored slopes in seasonal frozen soil regions.

2. Experimental

2.1. Engineering Background

The study is based on a slope reinforcement project located in Xinjiang, China. As shown in Figure 1, the slope has a height of 20 m and the local soil freezes at approximately −1 °C. The average surface temperature during the winter half-year is around −20 °C. The initial reinforcement scheme adopts a frame–anchor system using C30-grade concrete and HRB400-grade steel bars for the anchors. The spacing between adjacent anchors is 2 m. The elastic moduli of the concrete, anchor rods, and anchor grout body are 3 × 10⁴ MPa, 3.5 × 10⁵ MPa, and 3 × 10⁴ MPa, respectively. The anchor borehole diameter is 150 mm, and each anchor is 10 m long. The horizontal beams and vertical columns of the frame structure have design widths of 550 mm and 500 mm, respectively, with 2 m spacing between frame units.

This slope, located in the hinterland of the Eurasian continent, is primarily composed of silty clay and features a typical temperate continental semi-humid climate. The landslide area exhibits mid–low mountain topography with distinct climatic variations, with concentrated summer rainfall influenced by monsoons and deep seasonal snow cover in winter driven by the Siberian High. This climatic pattern aligns with regions sharing comparable slope frost heave classifications—including Kazakhstan, Kyrgyzstan in Central Asia adjacent to Xinjiang, and northwestern/northeastern China—making it a widespread engineering concern. Geological surveys confirm that the area belongs to a typical seasonal freeze–thaw cycle zone. The study area spans the Kunes River’s third terrace, the Aiken Daban structural denudation platform, and the Xiaoyuludus Basin transition zone. Quaternary sediments are predominantly fluvio-pluvial silty clay, with in situ tests indicating a natural moisture content of w = 12.6%, frost heave rates of 3–12%, and a Class III frost heave classification [23].

This project employed ZY-FLJ-200 axial force meters to dynamically monitor anchor rod axial forces during slope deformation. Monitoring points were installed on anchor rods near the slope crest and toe. These instruments offer advantages including stable performance, high sensitivity, and excellent waterproofing, with an operating temperature range of −45 to +60 °C. The configuration and installation are shown in Figure 2. The field monitoring equipment is also installed as shown in Figure 2. Axial force on anchor rods can be calculated using Equation (1).

$$p_2=K_2\left[(F_1^{\prime }+F_2^{\prime }+F_3^{\prime })/3-(F_0^{\prime }+F_0^{\prime }+F_0^{\prime })/3\right]+b_1(T_i-T_0)+B_2$$

(1)

where $p_2$ is the axial force acting on the anchor bolt; $K_2$ is the instrument calibration coefficient; $b_1$ and $B_2$ are the temperature correction coefficient and computation correction value of the load cell, respectively; $F_1^{\prime }$, $F_2^{\prime }$, $F_3^{\prime }$, and $F_0^{\prime }$ are the measured values of the load cell and the average value of a single reference-equivalent measurement, respectively; $T_i$ and $T_0$ are the measured temperature and the reference temperature of the load cell, respectively.

2.2. Laboratory Testing

To examine the effects of frost heave on the mechanical properties of the slope surface soil, a series of laboratory tests were conducted to determine the variation in strength under freezing conditions. These tests provide essential geotechnical parameters for subsequent numerical simulations. CU triaxial shear tests were performed on both natural and frozen soil samples.

To investigate the influence of frost heave on the strength of the slope surface soil, laboratory tests were conducted to reveal the characteristics of frozen strength variation and provide geotechnical parameters for numerical simulations. Natural and frozen soil samples were selected for consolidated undrained triaxial tests. The specific preparation procedure [24] includes the following: after crushing undisturbed soil and sieving through a 2 mm standard sieve to remove organic matter and coarse particle impurities, soil moisture content was adjusted to the target value (w = 12.6%) using atomized spraying; then, the samples were sealed in polyethylene bags for 24 h curing in a thermostatic humidistat cabinet (20 ± 1 °C, RH 95 ± 3%) to ensure homogeneous moisture migration; finally standard triaxial specimens were prepared (φ39.1 mm × 80 mm) by layered static compression with dry density controlled at 1.76 g/cm³ (±0.02 g/cm³ precision).

To control experimental dispersion and minimize the random errors associated with small samples, key mechanical indices were repeatedly measured to ensure uniform soil quality in the specimens. All specimens met the mechanical stress–strain testing requirements under different confining pressures. Four sets of specimens were prepared for both natural-state and frozen-state conditions following the parallel testing principle, totaling eight specimens.

3. Results and Discussion

3.1. Stress–Strain Characteristics of Soil After Freezing

The stress–strain behavior of frozen soil was examined using CU triaxial tests under confining pressures of 100 kPa, 200 kPa, 300 kPa, and 400 kPa. The shearing rate was set to 0.5 mm/min, and the termination criterion was either 15% axial strain or the appearance of significant plastic deformation. Testing was performed using a TAS-LF fully automated triaxial apparatus manufactured by China Zhejiang Jio Technology Co., Ltd. (Hangzhou, Zhejiang, China). The device features a digital control system capable of automated data acquisition and output. The displacement measurement accuracy reaches ±0.1%FS, and the force resolution is better than 0.1 kN.

The stress–strain curves obtained under different confining pressures are shown in Figure 3. Here, σ₁ denotes the major principal stress, σ₃ the minor principal stress, and ε the axial strain. The results show that natural soil specimens exhibited strain-softening behavior, while frozen specimens displayed clear strain-hardening characteristics. The stress–strain response followed a three-stage pattern. Initially, deviatoric stress increased linearly with axial strain, indicating elastic deformation. After approximately 3% strain, the curve began to flatten, suggesting a transition from elastic to plastic deformation. The failure strength increased with confining pressure, reaching a maximum at 400 kPa. Notably, under frozen conditions, the failure strength was approximately 5.58 to 6.6 times greater than in the natural state.

3.2. Effect of Frost Heave on Shear Strength of Soils

Based on a triaxial shear testing system, strength parameter tests were conducted on natural and frozen specimens. An axial strain-controlled loading procedure was implemented to obtain peak deviator stress datasets under consolidated undrained conditions for four standard confining pressures. Following the Mohr–Coulomb strength criterion, Mohr’s circles representing four standard stress states were constructed in a two-dimensional stress space. The specimens exhibited strain-softening behavior in their stress–strain curves. By measuring failure stresses at confining pressures of 100 kPa, 200 kPa, 300 kPa, and 400 kPa, the shear strength envelope was plotted using MATLAB R2022b. Figure 4 illustrates the Mohr’s circles and strength envelope for specimens with a moisture content of 12.6% under varying confining pressures.

Table 1. Strength parameters of soil under different working conditions.

Working Condition	Cohesion (kPa)	Angle of Internal Friction (°)
Natural state	41.2	30.9
Frozen state	72.5	42.7

summarizes the strength parameters under both conditions. In the natural state, the soil retained its primary structure, with particle interaction governed mainly by mineral surface friction, yielding a cohesion of 41.2 kPa and an internal friction angle of 30.9°. In the frozen state, the formation of ice crystals enhanced interparticle bonding and densified particle packing. Additionally, ice infill in the pores formed a composite ice–soil cementation network [25]. As a result, the cohesion increased significantly to 72.5 kPa and the peak friction angle rose to 42.7°. These findings confirm that the shear strength of the soil during the frozen phase significantly exceeds that in the natural state. The quantified strength parameters obtained from the triaxial tests serve as key inputs for the multi-field numerical modeling of slope behavior under coupled thermal and hydraulic conditions.

4. Development of Coupled Multi-Field Numerical Model

4.1. Fundamental Assumptions

To investigate the evolution of frost heave processes in seasonally frozen slopes and to understand the internal ice–water phase transition mechanisms that drive anchor force redistribution from the slope crest to the toe, a fully coupled thermo-hydro-mechanical (THM) model was developed using COMSOL Multiphysics v6.2 (COMSOL AB, Stockholm, Sweden). The model represents a frame-anchored slope subjected to freezing and thawing, and the following assumptions were made to simplify the simulation:

(1)

The initial state of the slope is stable, and there is no interaction force between the frame and the slope surface.

(2)

Both frozen and unfrozen soils are assumed to be isotropic.

(3)

Heat loss during the freeze–thaw cycles are neglected.

(4)

The moisture field is considered closed, meaning the external water supply and evaporation are ignored.

(5)

A deformation compatibility condition is applied between the frame anchors and surrounding soil.

These assumptions provide a reasonable approximation for simulating frost heave effects in cold-region slope engineering scenarios.

4.2. Boundary Conditions and Meshing

To investigate the spatiotemporal evolution of temperature and moisture fields under frost heave, appropriate boundary conditions were applied to the numerical model. According to the heat–moisture coupling theory in frozen soil mechanics, thermal boundary conditions are generally classified into three types [26]. In this study, both heating and cooling processes on the lateral boundaries were considered. A first-type (Dirichlet) boundary condition was applied at the upper boundary to simulate surface temperature variation, while the bottom boundary was treated as thermally insulated to reflect adiabatic conditions.

The moisture field was modeled as a closed system, meaning that external water sources and surface evaporation were not considered. This assumption reflects the dry winter conditions commonly observed in cold regions, where the soil system tends to be self-contained in terms of moisture redistribution.

For the mechanical boundary conditions, the bottom boundary of the model was fixed in both vertical and horizontal directions to prevent displacement. The left and right boundaries were constrained horizontally (roller supports), allowing vertical movement but preventing lateral displacement. At the interface between the frame beams and the anchor rods, right-angle constraints were applied to simulate structural integrity and displacement compatibility.

The finite element mesh was generated using COMSOL Multiphysics built-in mapped and free triangular meshing algorithms. To achieve high accuracy in the simulation, mesh refinement was applied. The maximum mesh element size for the general slope domain was set to 0.42 m. In regions of high-stress gradients, particularly between the frames where soil arching effects are prominent, the mesh was further refined to a minimum element size of 0.0055 m. This mesh refinement is critical to ensuring the numerical stability of the coupled thermal–hydraulic–mechanical analysis and the accurate capturing of localized deformation.

The resulting mesh configuration is illustrated in Figure 5.

The air temperature data used in this study were obtained from on-site temperature sensors installed at the slope location, covering the full freeze–thaw cycle of the year 2024. The measured temperature trends are presented in Figure 6.

The temperature data employed in this study were acquired from on-site temperature sensors during 2024, as shown in Figure 6. The statistically regressed dataset presented in Figure 6b was used to analyze the thermo-mechanical characteristics of the frame-anchored slope during freeze–thaw processes. The temporal evolution of the slope temperature fields was characterized using Dirichlet formulations, achieving a determination coefficient R² of 0.97410 for the fitted temperature function, as shown in Equation (2):

$$T=-1.1+21.2\mathrm{s}\mathrm{i}\mathrm{n}({\displaystyle \frac{2\pi t}{365}}+{\displaystyle \frac{\pi }{2}})$$

(2)

4.3. Material Parameter Determination

The test site consists of silty clay commonly found in seasonally frozen regions. Fundamental characteristic parameters were determined through in situ field tests and laboratory triaxial tests supplemented by relevant data. The thermal and hydraulic properties are presented in

Table 2. Model parameters.

Density (kg/m3)	Latent Heat of Phase Change L (J/kg)	Specific Heat Capacity (kJ/(m3·°C))	Thermal Conductivity (W/(m·°C))
1800	360,000	0.84	1.34

and

Table 3. Parameters of frame and anchor rod.

Component	Density (kg/m3)	Elastic Modulus E (MPa)	Poisson’s Ratio µ
Frame	2400	30,000	0.25
Anchor rod	7990	350,000	0.30

, respectively. As freeze–thaw processes progress, the mechanical parameters of the soil exhibit corresponding variations. Accounting for these effects [15], the Young’s modulus and Poisson’s ratio values were established, as given in Equations (3) and (4).

$$E=\left\{\begin{array}{cc}21+{\left|T\right|}^{0.6}\hfill & \hspace{1em}\hspace{1em}\hspace{1em}T\le T_f\hfill \\ 21 \hfill & \hspace{1em}\hspace{1em}\hspace{1em}T>T_f \hfill \end{array}\right.$$

(3)

$$v=\left\{\begin{array}{cc}0.3+{\left|T\right|}^{0.6}\hfill & \hspace{1em}\hspace{1em}\hspace{1em}T\le T_f\hfill \\ 0.3 \hfill & \hspace{1em}\hspace{1em}\hspace{1em}T>T_f \hfill \end{array}\right.$$

(4)

5. Freeze–Thaw Effects on Frames and Slopes

To systematically investigate the frost heave behavior of the frame–slope system, a fully coupled THM model was used to simulate the evolution of the temperature, moisture, and displacement fields. Special attention was given to understanding the interaction between the frost-heave-induced soil arching effects and the structural frame elements under varying freeze–thaw conditions.

5.1. Classification of Frost Heave Types

Based on the spatial distribution of freezing depth and deformation patterns during the frost heave process, the slope surface can be divided into three distinct frost heave types, as illustrated in Figure 7.

Type I frost heave, as shown in Figure 7a, occurs during the early stage of freezing. The slope crest begins to freeze first, followed by the slope surface and toe as the ambient temperature continues to decrease. According to the constraint conditions associated with frost expansion, three sub-regions are identified, named Region A, Region B, and Region C.

Region A is primarily influenced by unidirectional horizontal freezing, leading to dominant vertical expansion due to minimal lateral restraint. Region B experiences dual thermal influence from both the slope crest and surface. It exhibits the deepest freezing depth and is partially constrained by the horizontal frame structure, resulting in the most pronounced vertical heave. Region C is constrained by the frame along the slope surface, as well as by adjacent zones B and the slope toe. As a result, it exhibits the smallest amount of frost-induced deformation.

Type II frost heave, illustrated in Figure 7b, corresponds to the intermediate stage of freezing. As the ambient temperature continues to drop, the freezing depth increases, particularly at the crest and toe. Two additional transitional regions—Region D (on the mid-slope) and Region E (near the slope toe)—begin to emerge.

Region D develops above Region C as a result of deepening frost penetration and forms a dual-direction constrained zone. The temperature isolines in this region run parallel to the slope surface. Being surrounded by constrained soil, Region D generates a substantial horizontal frost heave force acting on the frame. Region E shows temperature isolines with a smaller slope than in Region D and is subject to toe constraints. The freezing depth here remains relatively shallow.

Type III frost heave, as shown in Figure 7c, occurs in the late stage of the freezing process. At this point, the freezing depth reaches its maximum, and the distribution of frozen regions undergoes significant transformation.

Region C is completely replaced by the transitional Region D, and the slope of the isotherms in Region D becomes less steep. Region B, located near the slope crest, reaches its maximum frost penetration depth, indicating intensified frost heave.

5.2. Coupling Mechanism of Freezing and Expansion on Slopes

To examine the deformation mechanisms of the slope induced by moisture migration during freezing, a monitoring system was established along the slope surface to record freezing depth and displacement evolution from the toe to the crest. The moisture migration rate and pathway were considered key factors in the frost-heave-induced deformation. The spatial layout of the monitoring points and the displacement field during the frost heave period are shown in Figure 8.

As illustrated in Figure 9, the frame–anchor support system does not impose vertical rigidity on the slope surface soil. During the frost heave period, the deformation is characterized predominantly by vertical displacement. In contrast, horizontal displacement is significantly restrained by the rigid beam–column frame structure, resulting in much lower horizontal movement compared to vertical displacement. Due to the combined constraint from the frame beam and surrounding soil, the displacements at monitoring points c, d, and e near the slope surface were limited to approximately 3 mm. However, points a and b located at the slope crest, which lack vertical constraint, showed greater vertical displacement. In frost heave Stages I and II, monitoring point b—influenced by both crest-side and surface-side freezing—exhibited vertical displacement 17.1 to 33.8% higher than point a, which is farther from the slope surface.

To further explore the relationship between the temperature field, ice content, and slope deformation, Figure 10 presents the spatiotemporal evolution of these variables. The frost heave process is found to consist of the three following distinct stages:

In Stage I (initial phase transition), surface frost heave deformation is triggered by the phase change in pore water at the slope crest. Due to thermal conduction lag, the freezing front gradually propagates downslope. During this phase, ice content remains relatively uniform across monitoring points.

In Stage II, the freezing depth continues to increase, leading to the formation of dual-direction transitional zone D and dominant freezing zone E. Driven by coupled hydraulic and gravitational potential gradients, unfrozen water from deeper layers migrates upward through the soil pore network and accumulates near the slope toe. As a result, the ice content at the toe increases significantly.

In Stage III, the freezing front reaches the critical freezing depth, and pore water migrates along the thermal gradient toward the front. As the slope toe acts as the terminal accumulation zone, it becomes the area with the highest volumetric ice content—substantially greater than in other regions.

As shown in Figure 11, the evolution of the horizontal frost heave force field exhibits significant spatial variability characteristics. Influenced by the bidirectional freezing coupling effect, monitoring point b recorded higher horizontal frost heave forces than other monitoring points from Stage I to Stage II of frost heave. With changes in thermodynamic conditions, moisture migrated to the slope toe area, forming an ice-rich accumulation zone, resulting in notably pronounced frost heave forces at the slope toe. During the monitoring period of the frost heave stage, the horizontal frost heave force at monitoring point e on the slope toe displayed a distinct exponential growth pattern, peaking at 92 kPa in February and exceeding that of monitoring point b at the slope top. For the design of frame anchor bolt supporting systems in seasonally frozen slopes, it is crucial to significantly enhance the strength indicators of anchor bolt materials at the slope toe. This approach effectively mitigates the risk of structural instability caused by excessive frost heave forces and ensures the overall stability of the slope.

In summary, the slope toe acts as the terminal convergence zone for the advancing freezing front. During Stage III, this region develops a distinct ice-enrichment belt with pronounced spatial heterogeneity. The horizontal frost heave force in this zone increases nonlinearly and is strongly correlated with the increase in ice content. The peak frost heave force of 92 kPa indicates that the slope toe represents a mechanically dominant zone where frost-induced energy is ultimately released. Therefore, frost-resistant design measures must prioritize this critical region for structural reinforcement.

5.3. Analysis of Soil Arch Effect

During the reinforcement of the slope using a frame–anchor system, horizontal and vertical beams are orthogonally arranged along the slope surface, forming a series of quadrilateral frame units. Within these structures, horizontal soil arching develops between adjacent vertical beams, while vertical soil arching forms between adjacent horizontal beams. This section focuses on the evolution of vertical soil arching under frost heave conditions. A conceptual model of the soil arching mechanism is shown in Figure 12.

As illustrated in Figure 13a, during the initial phase of frost heave, vertical soil arches begin to form. With decreasing temperature, ice crystals nucleate near the surface and gradually grow, causing localized volume expansion. Ice lens development initiates, and frost heave forces are transmitted through the soil arch to the surrounding frame structure. As the freezing front progresses deeper, ice lenses enlarge and intensify the frost heave pressure, resulting in increased arch height and the development of a more complete stress arch structure (Figure 13b). Once the soil becomes fully frozen, the formation of new ice lenses ceases and frost heave forces stabilize. At this stage, the soil arch reaches its maximum height and adopts a steady-state configuration (Figure 13c).

5.4. Anchor Axial Force Analysis

The variation pattern of anchor bolt axial forces in supported slopes under frost heave effects is illustrated in Figure 14. A comparative analysis between natural conditions and the frost heave period reveals that the axial force during the frost heave period is approximately 1.3 times that under natural conditions. The significant increase in anchor bolt axial forces during the frost heave period can be primarily attributed to two factors. First, unfrozen water migration during soil freezing induces phase change volume expansion, generating frost heave forces that act on the anchor bolt structure through shear displacement constraints. Due to the rigid connection constraint between the anchor bolt and the frame structure, axial deformation is restricted, leading to the accumulation of internal tensile stress and manifesting as abnormal axial force elevation. Second, as shown in the deformation characteristics of the slope during the frost heave period in Figure 11, soil freezing-induced expansion generates displacements in the free face direction. This tensile action on the anchor bolts results in a notable increase in axial forces.

The monitoring data along the slope profile direction show that the axial force response of the anchors shows a significant elevation effect, and the axial load of the support structure shows gradient-increasing characteristics from the foot of the slope to the top of the slope during the freezing and expansion process. According to Figure 14, it can be seen that the anchorage system in the top area (the first row) produces significant load jump in the freezing and expansion stage, and the axial load of its anchorage section increases by about 215 kN compared with the baseline value in the natural stage; the axial load increment of (the second and the third rows) is about 145 kN and 155 kN, respectively, which confirms that freezing and expansion have significant spatial heterogeneity; according to the change rule of its displacement field, temperature field, and stress field, it is shown that, under the action of freezing and expansion, the support structure axial load response shows a significant elevation effect from the foot to the slope. According to its displacement field, temperature field, and stress field, the region at the top of the slope under the action of freezing and expansion shows that the displacement change is obvious, the depth of freezing and expansion is large, and the axial force of the anchor rod changes greatly.

Combined with Figure 11, it is evident that during the frost heave period, both the axial force of the anchor rods and the horizontal frost heave forces exerted on the slope body exhibit an increasing trend, with the axial force ranging from 145 to 215 kN and the frost heave forces ranging from 20 to 52 kPa, indicating a significantly greater increase in axial force compared to frost heave forces. By the end of the frost heave period, as temperatures rise, the axial force of the anchor rods retracts slightly but remains higher than that under natural conditions. This residual stress arises from two coupled mechanisms, namely the continued alteration of the soil’s internal friction angle and cohesion due to moisture migration during thawing and the irreversible nature of slope displacement induced by the freezing process. Since frost heave deformation cannot be fully recovered, the axial force at the end of the frost heave period consistently exceeds that in the natural state. This indicates that the portion of axial force added to the anchor rods during the frost heave period becomes locked into the anchor system through the permanent deformation of the soil structure.

Field-measured anchoring forces during the frost heaving process are shown in Figure 14. An analysis of dynamic changes in the axial force of anchor bolts from natural to frost heave periods reveals that the measured values are generally slightly higher than the numerical simulation results. Error analysis demonstrates that the deviation between numerical simulations and field monitoring data, calculated using the percentage difference method, ranges from 3.5 to 6.2%. The field monitoring data show close agreement with simulation results, with fundamentally consistent variation trends.

In conclusion, to prevent excessive increases in axial force during freezing—potentially leading to anchor detachment or failure—it is recommended to adopt the frost heave period as the governing design condition for frame–anchor systems. Particular attention should be paid to controlling displacements at the slope crest and monitoring anchor force changes at both the crest and toe. These measures are essential to reduce structural instability risks and ensure the long-term stability of frame-supported slopes in seasonally frozen regions.

6. Conclusions

This study investigates the frost heave evolution process of a silty clay slope in Xinjiang, China. Triaxial consolidated undrained (CU) shear tests were conducted under both natural and frozen conditions to obtain the stress–strain constitutive relationships of the soil. A coupled thermal–hydraulic–mechanical (THM) numerical model was developed using the PDE and solid mechanics modules of COMSOL Multiphysics, enabling a simultaneous solution of temperature, stress, and moisture fields. The model simulates the behavior of a frame–anchor-supported slope under seasonal freezing conditions. The main conclusions are as follows:

(1)

Based on the spatial evolution of freezing depth along the slope surface, three distinct types of frost heave behavior were identified. As the temperature decreases, transitional regions characterized by one- and two-directional freezing develop near the slope crest and toe. These areas eventually evolve into dominant bidirectional freezing zones as freezing progresses.

(2)

The frost heave process is strongly influenced by slope boundary conditions. The spatial temperature response along the slope surface exhibits significant heterogeneity. Due to the bidirectional freezing effect, both the crest and toe regions experience more intense responses than the mid-slope area. The maximum freezing depth reached 2.3 m at the crest. Under prolonged thermodynamic deterioration, water migration leads to the accumulation of ice content at the slope toe, resulting in exponential growth in horizontal frost heave force, which peaked at 92 kPa in February.

(3)

During the freezing process, frost-induced soil arching generates a reverse stress field whose temporal evolution mirrors the temperature field. Along the arch height direction, the reverse frost heave force exhibits a nonlinear trend: it first increases and then gradually decays as freezing stabilizes.

(4)

An analysis of the dynamic response patterns of anchor rod axial forces during frost heave revealed that axial forces during the frost heave period are approximately 1.3 times those under natural conditions, with significant increases particularly observed at the slope crest and toe. Based on the design principle of adopting the most unfavorable condition, it is reasonable to establish frost heave conditions as the design criteria for frame–anchor structures. It is recommended that in the design of slope frame–anchor support structures in seasonally frozen regions, priority must be given to controlling displacement at the slope crest and the variations in axial forces of anchor rods at both the crest and toe. This approach will effectively mitigate structural instability risks and ensure slope stability.

Given the prevalence of frost heave issues in climatically similar regions of Northwest China, Northeast China, and Central Asia, this study investigates a typical silty clay slope in Xinjiang. Through conducting consolidated undrained triaxial shear tests under freeze–thaw cycles and establishing a thermo-hydro-mechanical coupled numerical model, the research reveals the response mechanism of slopes supported by a frame anchor system under frost heave conditions. It specifically analyzes the evolution patterns of the displacement field, stress field, and temperature field within the supported slope and provides insights into the dynamic interaction mechanism between the frame structure and slope soil during the frost heave process. The conclusions offer technical support for anchor reinforcement projects in similar regions.

These findings provide valuable guidance for designing anti-frost slope reinforcement systems in seasonal frozen soil regions. However, certain limitations persist as follows: (1) the numerical model simplifies temperature boundary conditions using local average air temperature as a sinusoidal function, whereas actual engineering scenarios exhibit nonlinear variations in temperature amplitude attenuation and phase lag with depth, affecting the frost heave simulations of frame-anchored slopes; (2) the moisture field simulation disregards external water replenishment conditions, deviating from field engineering realities. Future studies will include conducting scaled outdoor physical model tests to verify observed soil arching effects between frame elements; prioritizing freeze–thaw mechanisms in subsequent research, focusing on soil anisotropy evolution and groundwater impacts on frost-affected slopes; and investigating tripartite synergistic mechanisms (frame–soil–anchor) under repeated freeze–thaw cycles, along with cumulative slope impacts and anchorage interface damage accumulation.