Research on the bearing mechanism and structural design method of embedded foundations for transmission towers under slope conditions

Introduction: Embedded foundations, prized for their high bearing capacity and compact footprint, are a prevalent and innovative solution for transmission tower support. Nevertheless, their bearing mechanism remains insufficiently understood, particularly under complex terrains like slopes, where dedicated design methods are lacking.

Methods: A comprehensive numerical simulation study was conducted to analyze the mechanical response of embedded foundations under uplift loads. The investigation encompassed various scenarios defined by slope gradient, foundation geometry, and size. The study examined the displacement response of foundations and surrounding rock masses under incremental loading, investigated the interaction mechanism on slopes, and traced the progressive evolution of fracture surfaces. By analyzing maximum shear strain nephograms under ultimate load, the spatial distribution characteristics of the fracture surface in the rock mass were determined, leading to the establishment of a predictive model for fracture surface geometry around foundations on slopes.

Results: The findings reveal that plastic zones initially form at the foundation-rock interface upon surpassing the foundation's self-weight load. With increasing load, these zones propagate deeper, expanding and interconnecting. The fracture surface trajectory demonstrates a clear relationship with slope gradient: its angle on the downhill side increases, while on the uphill side it decreases, with the change approximately equal to the slope gradient. Near the slope face, fracture surfaces assume a near-circular shape, whose diameter is proportional to foundation size but remains largely invariant with changes in slope gradient or other parameters. Based on limit equilibrium analysis, a set of calculation formulas for ultimate uplift resistance under slope conditions was derived and validated.

Discussion: The proposed formula successfully reduces to the established flat-ground condition when the slope gradient is zero and has been implemented in practical engineering. This study provides a theoretical foundation and a practical analytical tool for designing embedded foundations in complex slope environments.

1 Introduction

In recent years, the southwest region of China has experienced rapid economic development, leading to a proportional increase in electricity demand. Given the predominantly mountainous terrain in southwestern China, transmission lines inevitably traverse steep slopes. To ensure that the transmission tower foundations can continuously and stably support the overhead transmission lines and systems, these structures must possess uplift strength (Rattley et al., 2008). Since the uplift forces exerted on the transmission towers are ultimately borne by the foundations in various ways, the study of these foundations becomes particularly important. Compared to conventional transmission tower foundations, embedded foundations offer unique advantages in mountainous transmission line projects due to their smaller footprint and significant reduction in mountain excavation.

Therefore, scholars have conducted in-depth studies on the bearing mechanisms of transmission tower foundations using various methods such as theoretical analysis, model testing, field experiments, and numerical simulations. Researchers such as Mors (1964), Stas and Kulhawy (1984) have proposed theories for calculating the ultimate bearing capacity of soils through theoretical and experimental research. Suhendra and Pradita (2024) used the Broms method to analyze the horizontal bearing capacity of foundations with different embedment depths and pile diameters, finding that square piles have higher horizontal bearing capacity than circular piles. Liu and Xu (2023) proposed a calculation method for vertical bearing capacity considering the influence of compaction and stress level on the friction angle, and verified its applicability. Although these studies have provided valuable insights into shallow foundations, most of them focus on flat terrain, and there is limited research on the impact of slope strata on foundation bearing capacity. Nhat et al. (2023) and Bhattacharya (2016) used numerical simulation and finite element analysis to study the bearing capacity of foundations under slope conditions, finding that slope gradients significantly affect foundation bearing capacity, especially under rainfall and undrained conditions. Li et al. (2020) and Chae et al. (2004) further analyzed the bearing capacity of rock foundations near slopes, but still did not conduct in-depth research on the performance of embedded foundations in sloped conditions. Sangseom et al. (2003) studied the response of slope pile foundations under the load transfer method, finding that this method tends to underestimate the slope safety factor. Muthukkumaran (2014) investigated the impact of soil properties, horizontal load direction, and the distance between pile foundations and slope toes on foundation bearing capacity. Therefore, while existing research provides valuable support for understanding foundation bearing capacity, there is still a significant gap in the study of embedded foundations under complex slope conditions.

With the continuous increase in building heights and the increasing scarcity of construction land, pile foundations, rock-socketed foundations, and other forms have been increasingly used in high-rise buildings and large-scale infrastructure projects. Among these, uplift caused by wind loads has become an important research topic. Uplift refers to the upward displacement of foundation soil under the action of foundation loads, which may lead to foundation settlement and tilting, thus affecting the stability and safety of buildings. Therefore, scholars have conducted in-depth studies on the bearing and deformation characteristics of foundations under uplift loads. Zhang et al. (2024) studied the eccentric uplift of cushion foundations, finding that both eccentric distance and foundation shape affect its uplift capacity. Huang et al. (2023) used three-dimensional finite element analysis to study the effect of helical plate spacing and soil stiffness on uplift bearing capacity, finding that increasing the spacing of helical plates and anchor rod diameters significantly reduces uplift capacity. Inder et al. (2023) used finite element analysis to study the impact of burial depth and friction angle on the uplift capacity of helical piles and established a predictive model using artificial neural networks. Wang et al. (2023) adopted finite element limit analysis to investigate the impact of different burial depths and the diameter ratio of the blade and pile body on the uplift capacity of helical piles, proposing a failure mechanism for helical piles. Researchers such as Arya and Author Anonymous, 2023 and Moayedi and Mosallanezhad (2017) also studied the uplift capacity of bored cast-in-place piles and pile foundations with enlarged heads, finding that increasing burial depth significantly improves uplift performance. Wang et al. (2024) conducted experimental studies on the uplift performance of belled piles in horizontal and inclined ground, showing that inclined ground worsens the uplift performance, thus increasing the displacement of the pile cap and ground surface. However, most of these studies focus on traditional pile foundations and consider either slope conditions or uplift loads individually, neglecting the combined effect of both on foundation bearing capacity. Additionally, while these studies provide important data for traditional pile foundations, insufficient attention has been given to the performance of embedded foundations under complex slope conditions. Therefore, it is necessary to further study the bearing characteristics of embedded foundations under multiple complex factors to fill this gap.

A significant amount of research has been conducted on the design methods for transmission tower foundations, primarily focusing on pile foundations in flat terrain. Zuo et al. (2024) calculated the three-dimensional ultimate uplift bearing capacity of anchor cables based on the Hoek-Brown failure criterion and variational analysis, and derived numerical solutions for the fracture surface curve function through finite difference theory, which better reflects the failure state of anchor cables compared to predefined assumptions. Liu et al. (2023) constructed a curved uplift failure mechanism under the overall failure conditions of surrounding rock for piles in rock strata and derived theoretical formulas for the ultimate uplift bearing capacity of the rock fracture surface and pile body. Gu et al. (2023) proposed a new semi-analytical solution to accurately evaluate the uplift bearing capacity of spread foundations in sandy soils and validated its applicability through existing experimental results. Hu et al. (2023) developed a theoretical method to calculate the uplift capacity of horizontal anchor plates at any depth in sandy soils, without distinguishing between shallow and deep burial types. Chattopadhyay and Pise (1986) proposed an analytical method to predict the uplift capacity of piles in sandy soils, considering the pile’s length-to-diameter ratio, surface properties, and soil properties. Although these studies have provided rich theoretical support for pile foundation design, most of the research has focused on flat conditions. There is a lack of relevant research and design standards for embedded foundations, especially for embedded foundations under complex slope conditions. These deficiencies indicate that existing design methods cannot be directly applied to embedded foundations under complex slope conditions, highlighting the urgent need for in-depth research on the bearing characteristics of such foundations under complex conditions.

Based on the above research gaps, this study considers various working conditions, such as different slope gradients, embedded foundation structural forms, and dimensions, through numerical simulations. The aim is to reveal the interaction mechanism between embedded foundations and rock masses. Furthermore, the development pattern of plastic zones around embedded foundations under graded uplift loading and different slope conditions will be analyzed, and the fracture surface generated in the surrounding rock mass under uplift load conditions will be studied. Based on the fracture surface morphology, a three-dimensional calculation model for rock mass fracture surfaces under slope conditions will be established, and a design calculation formula for the uplift capacity of embedded foundations under slope conditions will be derived and validated.

2 Analysis of basic mechanical characteristics of embedded foundations

Owing to their excellent bearing capacity and stability, embedded foundations—as a relatively new structural form—can effectively address issues of settlement and deformation, thus guaranteeing the long-term stability of buildings. This article addresses their structural configurations, characterizes their load distributions and primary loading conditions across different scenarios, and investigates the associated failure modes.

2.1 Model construction

Embedded foundations, as a novel foundation construction technique, possess unique advantages in transmission tower engineering. Common embedded foundations, which include jar-shaped and Y-shaped types, are depicted in Figure 1. In practice, traditional foundations require significant excavation in mountainous areas, which can lead to collapses, especially in unstable geological conditions. In contrast, embedded foundations can utilize blasting, mechanization, and manual methods to excavate the foundation pit shape and install a steel reinforcement cage within, significantly reducing the amount of mountain excavation and protecting the mountainous environment. During construction, embedded foundations adopt the method of layered concrete pouring and follow a scientific curing procedure to ensure the solidity and stability of the foundation. Compared to traditional transmission tower foundations, embedded foundations have a smaller plan size and higher bearing capacity. Additionally, the construction technique for embedded foundations is simple, allowing for rapid completion of foundation construction work with good economic benefits, indicating broad application prospects in transmission tower engineering.

Figure 1

Figure 1. Schematic diagram of embedded foundation. (a) Jar-shaped foundation (b) Y-shaped foundation.

2.2 Load characteristics of embedded foundations

Figure 2 illustrates the load-bearing mechanism of transmission tower foundations under wind loads. When faced with wind loads, the windward side of the transmission tower foundation needs to bear uplift loads and horizontal loads, while the leeward side primarily bears compressive loads and horizontal loads. Given the variability of natural loads, transmission tower foundations are exposed to diverse stress environments. The transmission tower foundation is subjected to alternating combinations of periodic compression, uplift, and horizontal forces. Owing to the high bearing capacity of the rock mass, the foundation’s stability under downward pressure possesses a high safety factor that meets design specifications. The main controlling factor for foundation stability is the uplift resistance (Turner and Kulhawy, 1990). Therefore, this article focuses on studying the uplift force to ensure the stability of transmission towers in mountainous areas.The specific magnitude of the design wind load, expressed as an ultimate uplift force, is derived from engineering practice and used for validation, as detailed in the subsequent section on engineering application. Current research on the uplift bearing characteristics of embedded foundations, especially under complex conditions in mountainous areas, is still incomplete. Consequently, a correct understanding of the uplift behavior of these foundations is crucial for optimizing their design, carrying significant theoretical and practical value.

Figure 2

Figure 2. Stress mechanism of embedded foundation under wind load.

2.3 Failure modes of embedded foundations

Existing literature has described the failure modes of embedded foundations under uplift loads, which can be summarized into the following categories:

1. The foundation cracks due to tension;

2. Vertical cylindrical cracks appear in the soil on the side of the foundation, with the enlarged head as the endpoint;

3. The soil on the side of the foundation cracks into inverted circular truncated cone-shaped (or funnel-shaped) cracks, connecting to the bottom of the foundation;

4. Composite failure, manifesting as cylindrical cracks at the lower part of the foundation, while through cracks occur in the soil above the foundation.

The main failure modes of embedded foundations are illustrated in Figure 3. The jar-shaped and Y-shaped embedded foundations studied in this paper belong to variable cross-section structures. Their failure patterns in the foundation are extremely complex and may vary significantly due to changes in construction techniques, foundation depth, and soil and rock properties. The failure patterns shown in Figures 3b,c are more common in hard clay, while the failure patterns shown in Figures 3a,d may occur when the foundation size is small and the embedment depth is shallow.

Figure 3

Figure 3. Failure mode of embedded foundation. (a) Destruction Ⅰ (b) Destruction Ⅱ (c) Destruction Ⅲ (d) Destruction Ⅳ.

The uplift capacity of jar-shaped and Y-shaped embedded foundations stems from the complex role of the enlarged base. Initially, under low loads, the foundation behaves like a constant cross-section pile. With increasing load, the rock mass yields according to the Mohr-Coulomb criterion, forming initial shear bands, while elastic foundation elongation transfers load via shaft resistance (Figure 4a). Once the load is fully mobilized in the shaft, sub-parallel slip surfaces develop and propagate downwards to engage the enlarged head.

Figure 4

Figure 4. Development process of shear failure of surrounding rock and soil around the foundation. (a) Initial stage (b) Progressive engagement (c) Slip surface development (d) Composite failure.

Subsequent load increases are resisted by the enlarged head, which compresses the surrounding rock mass, potentially leading to interface detachment depending on rock properties and load magnitude (Figure 4b). Under ultimate loads, irreversible plastic deformation occurs, and slip surfaces around the shaft and base coalesce (Figure 4c). A composite failure surface—combining a lower cylindrical shear zone and an upper continuous failure surface—forms if the initial shear band reaches the ground surface prior to the circumferential connection of slip surfaces (Figure 4d).

3 Numerical calculation scheme

To investigate the displacement and failure conditions of embedded foundations of different structural types (jar-shaped and Y-shaped) and sizes under various slopes subject to uplift load, a series of numerical simulations were conducted using the finite difference code FLAC3D, as outlined in Table 1. To minimize boundary effects, the dimensions of the numerical model in all directions were set to approximately 20 times the foundation diameter. A generalized numerical model of the embedded foundation was established in FLAC3D (Itasca Consulting GroupInc, 1997), as shown in Figure 5, where the embedded foundation height (h) is 3 m, with top and bottom surface radii of 0.6 m and 0.9 m, respectively. A loading area (radius: 0.6 m, height: 0.5 m) and a 1.6 m × 1.6 m working platform were incorporated atop the foundation to facilitate uplift load application. The wind-induced uplift load was simulated as a uniformly distributed pressure applied vertically upward on the loading area atop the foundation. The load was applied in a graded manner, with incremental steps, to accurately capture the non-linear load-displacement response and the progressive failure of the rock mass.Given the negligible impact of internal reinforcement and grouting on the overall response, the foundation was modeled as a homogeneous, linearly elastic body with material properties corresponding to C30 concrete. The surrounding rock mass was simulated as a Class III rock using the Mohr-Coulomb constitutive model. The parameters for both the foundation and rock mass are summarized in Table 2.

Table 1

Table 1. Calculation plan.

Figure 5

Figure 5. Numerical model.

Table 2

Table 2. Physical and mechanical parameters of materials.

4 Bearing mechanism of slope-embedded foundations under uplift load

4.1 Evolution of displacement between embedded foundation and rock mass

Figure 6 depicts the load-displacement curve of a Y-shaped embedded foundation, which was simulated by gradually applying loads to the foundation. The loads were equivalent to uniformly distributed forces acting on the embedded foundation of the transmission tower, and the displacement of the embedded foundation corresponding to each load level was recorded. It can be observed from the figure that the uplift bearing capacity of the embedded foundation is negatively correlated with the slope angle, meaning that the greater the slope angle, the lower the uplift force the foundation can withstand. The overall trend of the uplift load-displacement curve under various slope conditions is relatively gentle, and the curve changes can be divided into three stages: ① In the initial stage, the curve is approximately linear, with a stable deformation rate, indicating that the rock mass exhibits linear elastic deformation characteristics. ② In the mid-stage, the curve is nonlinear, indicating that the rock mass enters a local plastic state. ③ In the final stage, the curve tends to be horizontal, indicating that the rock mass deformation increases significantly, resulting in overall shear failure.

Figure 6

Figure 6. Load displacement curve of Y-shaped foundation.

4.2 Evolution of plastic zones in rock mass

Figure 7 shows the development of plastic zones in the rock mass around jar-shaped and Y-shaped embedded foundations, illustrating the failure behavior of the surrounding rock under loading. In the initial loading stage, no plastic zone forms around the foundation, as the applied load is primarily balanced by the foundation’s self-weight, with no excess load transmitted to the rock mass. As the load gradually exceeds the foundation weight, the additional force is transferred to the surrounding rock through lateral friction, leading to the emergence of plastic zones at the foundation–rock interface. With further load increase, the stress propagates deeper into the rock mass, where shear resistance between rock layers inhibits mutual displacement. Localized plastic zones begin to coalesce, resulting in shear failure, and the plastic region continues to expand deeper into the rock. Once the load exceeds the ultimate bearing capacity of the rock mass, the plastic zones fully connect, causing ultimate uplift failure.

Figure 7

Figure 7. Plastic zones of Jar-shaped and Y-shaped foundations under different load conditions. (a) Jar-shaped foundation, load P = 6 MPa. (b) Jar-shaped foundation, load P = 12 MPa. (c) Jar-shaped foundation, load P = 20 MPa. (d) Jar-shaped foundation, load P = 32 MPa. (e) Y-shaped foundation, load P = 6 MPa. (f) Y-shaped foundation, load P = 12 MPa. (g) Y-shaped foundation, load P = 20 MPa. (h) Y-shaped foundation, load P = 32 MPa.

The asymmetry of the rock mass failure surface is driven by the slope geometry. The uphill side, under compressive stress, develops a limited plastic zone adjacent to the failure surface. Conversely, the downhill side, with its free face, exhibits an extensive plastic zone spanning from the foundation interface to the slope crest. This contrast occurs because the uphill rock experiences reduced confining pressure, inhibiting deformation, whereas the downhill rock is subject to greater gravitational stress and confinement, facilitating more significant deformation and broader plastic zone propagation.

4.3 Evolution of failure at the foundation-rock mass interface

Figure 8 illustrates the contact and slip behavior between the embedded foundation and the surrounding rock mass under uplift loading. “Contact and slip conditions” refer to the mechanical state of interaction and relative displacement between the two bodies. Initially, the foundation remains entirely stationary. As the uplift load increases and exceeds the self-weight of the foundation, the frictional resistance along the foundation–rock interface begins to mobilize. With further loading, the enlarged base of the foundation engages to carry the uplift force until the ultimate capacity is reached, resulting in shear failure of the rock mass.

Figure 8

Figure 8. Failure of the contact surface between the foundation and surrounding rock. (a) Jar-shaped foundation I. (b) Jar-shaped foundation II. (c) Y-shaped foundation I. (d) Y-shaped foundation II.

As shown in the figure, the slip and contact patterns are generally consistent across foundations of different sizes but vary significantly with foundation shape. Under high uplift loads, cracking occurs in the rock mass at the interface. In the case of the jar-shaped foundation, which tapers uniformly with a gradually increasing radius from top to bottom, upward movement causes the wider lower sections to compress and fill the cracks generated in the upper rock regions. This leads to full contact along the foundation length, as depicted in the diagram.

In contrast, the Y-shaped foundation comprises three distinct structural segments: an upper cylinder of constant radius, a middle jar-shaped transition zone with a progressively flaring radius, and a lower cylinder of larger radius. During uplift, the foundation’s upward movement induces cracking and fracturing in the surrounding rock. This leads to a loss of contact along much of the upper cylindrical segment, causing the middle jar-shaped zone to become the primary load-bearing region through maintained rock contact. Consequently, this critical transition segment warrants specific reinforcement in design to enhance its compressive and shear resistance.

5 Analysis of rock mass fracture surface around embedded foundations

5.1 Location information of rock mass fracture surface

In numerical simulations, the shape of the fracture surface is typically represented by the maximum shear strain contour map. Shear strain is one of the most sensitive parameters reflecting the failure and deformation of rock mass. Therefore, maximum shear strain is one of the critical indicators for the formation and development of the failure surface. Additionally, it can effectively quantify local failure or displacement concentration, providing the morphology and extent of the failure surface. By representing the shear strain values in the model in the form of colors or contours, the location, morphology, and development trend of the failure surface can be observed and analyzed intuitively. Therefore, this paper determines the shape of the failure surface of the uplifted rock mass by analyzing the shear strain condition of the rock mass around the foundation.

Figure 9 depicts the shear strain contour plot of the Y-shaped embedded foundation under extreme uplift force conditions, clearly showing the shape of the fracture surface of the surrounding rock around the foundation. To quantitatively describe the trajectory of the fracture surface for subsequent calculations, the fracture surfaces under different foundations and different slopes are plotted using curves, resulting in the fracture surface trajectories under various conditions as shown in Figure 10. It can be observed from the figure that the fracture surface trajectories of the jar-shaped and Y-shaped foundations under the same slope are similar, and the general trend is that the angle of the fracture surface on the downslope side increases with the increase in slope, while the angle on the upslope side decreases with the increase in slope, and the change in angle is approximately equal to the slope.

Figure 9

Figure 9. Maximum shear strain cloud at different slopes. (a) Y-shaped foundation. (b) Y-shaped foundation, (c) Y-shaped foundation, (d) Jar-shaped foundation, (e) Jar-shaped foundation, (f) Jar-shaped foundation.

Figure 10

Figure 10. Fracture surfaces of rock masses at different slopes. (a) Jar-shaped foundation. (b) Y-shaped foundation.

5.2 Shape of rock mass fracture surface

After analyzing the trend of the fracture surface of the deep rock mass with slope, the shape of the fracture surface at the slope surface is further explored to obtain a theoretical calculation model. Under flat conditions, due to the symmetry of the rock mass, the shape of the fracture surface at the slope surface is obviously a symmetrical circle. To investigate the change in the shape of the slope fracture surface with the slope, the displacement contour plot of the slope surface under slopes ranging from 0° to 50° is obtained as shown in Figure 11. It can be seen from the figure that the fracture surfaces at the slope surfaces of different slopes are approximately circular, indicating that the shape of the fracture surface at the slope surface is not affected by the slope gradient and remains constant as a circle.

Figure 11

Figure 11. Displacement cloud of fracture surface at different slopes. (a) Slope = 0° (b) Slope = 10° (c) Slope = 20° (d) Slope = 30° (e) Slope = 40° (f) Slope = 50°.

Since the parameters of rock mass vary with different engineering projects in practical engineering, it is essential to consider the influence of rock mass parameters on the shape of the fracture surface at the slope surface. Therefore, an analysis of the diameter of the fracture surface at the slope surface under different rock mass parameters for jar-shaped and Y-shaped embedded foundations with a height of 3 m was conducted (Table 3). As indicated in the table, the diameter of the fracture surface for jar-shaped and Y-shaped foundations with a height of 3 m does not change with variations in rock mass parameters. The diameter of the fracture surface at the slope surface under different parameters is approximately 5 m.

Table 3

Table 3. Diameter of surface fracture surface under different rock parameters.

In practice, the dimensions of an embedded foundation are determined by its design loads. To investigate how this scaling affects the failure mechanism, we examined the relationship between foundation height and the diameter of the resultant fracture surface at the slope face. Numerical simulations were performed to obtain the displacement contour maps of the rock mass for different foundation sizes (Figure 12). These contours reveal that the vertical angle of the maximum displacement curve remains largely invariant with increasing foundation size. This observation suggests a proportional relationship between the foundation size and the diameter of the circular fracture surface on the slope. A statistical analysis of the fracture surface diameter versus foundation size was conducted (Figure 13), the results of which confirm a direct proportionality, thereby validating our hypothesis.

Figure 12

Figure 12. Displacement contour maps of foundations with different sizes. (a) h = 1.5 m (b) h = 3 m (c) h = 4.5 m (d) h = 6 m.

Figure 13

Figure 13. Changes in diameter of fracture surface with basic dimensions.

Based on the results from Table 3 and Figure 13, the diameter of the fracture surface remains nearly constant at approximately 5 m, unaffected by variations in rock mass parameters. This result indicates that, although the slope has little impact on the shape of the fracture surface, changes in rock mass parameters do not significantly affect the size of the fracture surface, further validating the effectiveness of the cone model. The cone model predicts that the fracture surface on the slope is a circular area, with the diameter directly related to the foundation size and independent of the slope gradient or rock mass parameters, which aligns with the actual results of this study.

From the results obtained above, it can be seen that the fracture surface shape generated in the deep rock mass under the uplift force is an irregular curve, which is difficult to express with a formula (Duan et al., 2024; Zheng et al., 2016). Therefore, to facilitate the subsequent calculation of the ultimate uplift capacity of embedded foundations, the shape of the fracture surface in the deep rock mass is simplified into a calculable straight line, as shown in Figure 14. The relationship between the vertical angle ${\omega }_1$, ${\omega }_2$ of the fracture surface and the slope gradient $\alpha$ is given in formulas 1, 2.

${\omega }_1=\frac{\pi }{3}+\alpha (1)$

${\omega }_2=\frac{\pi }{3}-\alpha (2)$

Figure 14

Figure 14. Schematic diagram of fracture surface calculation.

Figure 15 is a schematic diagram of an oblique cone. According to its definition, any cross-section parallel to the base of the oblique cone is circular. Through numerous numerical simulations, this paper found that the arbitrary cross-sections of the three-dimensional fracture surface of the rock mass, parallel to the slope surface, are circular under different slope gradients, parameters, foundation types, and sizes. Therefore, the shape of the rock mass fracture surface is a part of an oblique cone. To obtain the surface area of the rock mass fracture surface, it is necessary to subtract the area of the missing part of the oblique cone from the area of the complete oblique cone. The calculation schematic is shown in Figure 16.

Figure 15

Figure 15. Oblique cone.

Figure 16

Figure 16. Schematic diagram for calculating the surface area of the fracture surface.

6 Research on the design method of embedded foundations under slope conditions

Taking the jar-shaped embedded foundation as an example, a cross-section is made through the axis of the foundation, perpendicular to the z-axis, to obtain the force analysis diagram of the foundation, as shown in Figure 17. The vertical angles of the fracture surfaces at both ends are also indicated. From the figure, it is clear that the ultimate uplift capacity of the foundation is jointly determined by the shear strength of the rock mass and the gravity of both the foundation and the surrounding rock mass.

Figure 17

Figure 17. Force analysis of embedded foundation.

Therefore, in the subsequent calculations, we will determine the fracture surface area using geometric relationships and coordinate transformations, derive the shear strength of the rock mass based on Mohr-Coulomb theory, and compute the gravity of the foundation and surrounding rock mass. Figure 18 provides a clear framework for these calculations, helping to clarify the interrelationships between the various steps and establishing the theoretical foundation for the final design method.

Figure 18

Figure 18. Calculation flowchart for fracture surface area, shear strength, and gravity of embedded foundations.

6.1 Calculation of fracture surface area

Let S₁ denote the area of the desired rock mass fracture surface, and S₂ denote the area of the missing part of the oblique cone. The geometric relationship is shown in Figure 19, where S is the sum of the two parts' areas, and its expression is given in Formula 3

$S=S_1+S_2(3)$

Figure 19

Figure 19. Schematic diagram for calculating the surface area of the fracture surface.

For convenience of calculation, two sets of coordinate axes are used to calculate the area of the fracture surface. The first coordinate axis is used to calculate the surface area S of the entire oblique cone, while the second coordinate axis is used to calculate the surface area of the missing part of the oblique cone S, as shown in Figure 20. The second coordinate axis is obtained by rotating the first coordinate axis counterclockwise at an angle θ, which can be derived from the geometric relationship that the rotation angle θ is equal to the slope angle α of the slope. The schematic diagram of coordinate axis rotation is shown in Figure 20, and the coordinate transformation formula is given in formula 4.

$\left\{\begin{array}{l}{x}_1=x\cos \alpha -y\sin \alpha \\ y_1=y\cos \alpha +x\sin \alpha \end{array}\right.(4)$

Figure 20

Figure 20. Schematic diagram of coordinate system.

Figure 21 shows the two coordinate systems used to calculate the surface area of the missing part of the oblique cone ${\omega }_1$ and the total area S. In the known coordinate system 1, the coordinates of point A are (2R, 0), the foundation height is h, the working platform height is L, and the vertical angles of the fracture surfaces at both ends of the rock mass are ${\omega }_1$ and ${\omega }_2$. Based on mathematical relationships and the coordinate transformation formula, the coordinates of other points and the expressions of various lines can be obtained.

Figure 21

Figure 21. Calculation schematic diagram. (a) Axis 1. (b) Axis 2.

Based on the coordinates of the above points and the expressions of the lines, the area of the oblique cone $S_1$ can be obtained through integration. Figure 22a is a schematic diagram of the integration of the oblique cone $S_2$, where $\Delta S$ is an arbitrary small curved surface section obtained by sectioning perpendicular to the y-axis, and its expression is given in Formula 5

$\Delta S=\pi \left(f_2\left(y\right)-f_1\left(y\right)\right)ds(5)$

Figure 22

Figure 22. Integral diagram. (a) Axis 1. (b) Axis 2.

Since the hypotenuse $ds$ of any segment of the curved surface of the oblique cone $\Delta S$ varies after being unfolded into a plane, it is approximated by Formula 6

$ds=\frac{1}{2}\left(d{s}_1+d{s}_2\right)(6)$

The approximate curved surface $\Delta S$ expression is given in Formula 7

$\Delta S=\pi \left(f_2\left(y\right)-f_1\left(y\right)\right)\frac{1}{2}\left(d{s}_1+d{s}_2\right)(7)$

From the geometric relationship, the following Formulas 8, 9 can be obtained

$d{s}_1=\sqrt{d{f_1}^2\left(\mathrm{y}\right)+d^{2}y}(8)$

$d{s}_2=\sqrt{d{f_2}^2\left(\mathrm{y}\right)+d^{2}y}(9)$

The surface area $S_2$ expression is given in Formula 10

$S_2=\int \Delta S(10)$

By substituting expressions (7), (8), and (9) into expression (10), the expression for $S_2$ is obtained. Similarly, the expression for $S$ can be obtained.

6.2 Calculation of shear strength of surrounding rock

According to the Mohr-Coulomb theory, the shear strength of the surrounding rock can be derived and calculated. The shear strength calculation formula on the failure surface is as follows (Formula 11):

${\tau }_f=c+\sigma \tan \phi (11)$

Where c is cohesion of the surrounding rock, kPa, $\phi$ is internal friction angle of the surrounding rock, σ is normal stress on the surrounding rock.

In the simplified calculation process, it is assumed that the normal stress on the failure surface is uniformly distributed. Therefore, the normal stress at the middle height is taken as the representative value of the uniform stress on the failure surface, as shown in Figure 23. The normal stress on the failure surface can be calculated using the following formula.

$\sigma =\frac{1}{2}{\gamma }_{2}h\sin \omega (12)$

Figure 23

Figure 23. Calculation of normal stress.

Where ${\gamma }_2$ is unit weight of the surrounding rock, $\omega$ is vertical angle of the fracture surface.

Due to the irregular shape of the fracture surface, the values of $h$ and $\omega$ vary at different locations. For convenience of calculation, the calculations of $h$ and $\omega$ are simplified to the following formulas

$h=\frac{1}{2}\left(h_1+h_2\right)(13)$

$\omega =\frac{1}{2}\left({\omega }_1+{\omega }_2\right)(14)$

By substituting Formulas 12–14 into Formula 11, the calculation formula 15 for the shear strength of the surrounding rock is obtained

${\tau }_f=c+\frac{1}{2}{\gamma }_2\left(\frac{1}{2}{h}_1+\frac{1}{2}{h}_2\right)\sin \left(\frac{1}{2}{\omega }_1+\frac{1}{2}{\omega }_2\right)\tan \phi (15)$

6.3 Gravity of the foundation and rock mass in the failure zone

As shown in Figure 24, $V_1$ is the volume of the embedded foundation, $V_2$ is the volume of the failed rock mass, and $V_3$ is the imaginary oblique cone created for convenience of calculation. $V$ is the total volume of the three. According to the principle of Zu Geng, combined with the coordinates of each point under the two coordinate axes, the calculation formulas for $V$ and $V_3$ are obtained as follows

$\begin{array}{l}V=\frac{1}{3}\pi {\left(\frac{\left(b_2-b\right)\cos \alpha +\left(b_{2}k-b{k}_2\right)\sin \alpha }{k-k_2}-\frac{b\left(\cos \alpha +k_1\sin \alpha \right)}{k_1-k}\right)}^2\\ \left(\frac{b_2\left(k_1\cos \alpha -\sin \alpha \right)}{k_1-k_2}-\frac{b\left(k_1\cos \alpha -\sin \alpha \right)}{k_1-k}\right)\end{array}(16)$

$V_3=\frac{1}{3}\pi R^2\frac{k_1{b}_2}{k_1-k_2}(17)$

Figure 24

Figure 24. Schematic diagram of volume proportion.

Since the shape of this embedded foundation is a standard circular truncated cone, the expression $V_1$ obtained using the volume calculation formula for a circular truncated cone is

$V_1=\frac{1}{3}\pi h\left(R^2+r^2+Rr\right)(18)$

Based on the mathematical relationships and Equations 16–18, the following expression $V_2$ can be derived

$V_2=V-V_1-V_3(19)$

According to the assumptions, the total gravity of the embedded foundation and the surrounding rock mass experiencing failure forms the total gravity of the foundation and failure zone, expressed as follows

$W=W_1+W_2(20)$

Where $W_1$ is foundation gravity, $W_2$ is gravity of the rock mass in the failure zone.

The gravity of the foundation and surrounding rock can be calculated by

$W_1={\gamma }_1{V}_1(21)$

Where ${\gamma }_1$ is unit weight of the foundation, $V_1$ is volume of the foundation.

$W_2={\gamma }_2{V}_2(22)$

Where ${\gamma }_2$ is unit weight of the rock mass in the failure zone, $V_2$ is volume of the rock mass in the failure zone.

By substituting Equations 18, 19, 21, 22 into Equation 20, the formula for calculating the total gravity of the foundation and rock mass in the failure zone can be obtained.

6.4 Ultimate pullout capacity formula

Under the action of uplift force, the rock-embedded foundation relies on the vertical components of the shear strength of the failure region, as well as the self-weight of the foundation and the failure plane, to jointly resist the uplift effect. The mechanical balance relationship yields the following equation

$T=W+{\tau }_f{S}_1\cos \omega (23)$

Where $W$ is gravity of the rock mass in the failure region, ${\tau }_f$ is shear strength on the failure plane, $S_1$ is area of the rock mass in the failure region.

By substituting the respective equations into (23), the formula for calculating the ultimate pullout capacity of the jar-shaped foundation is obtained. It can be seen that the basic parameters required for the formula are:

1. Slope gradient $\alpha$

2. Rock mass parameters: cohesion c, friction angle $\phi$, gravity density ${\gamma }_1$

3. Foundation parameters: gravity density ${\gamma }_2$, top radius $r$, bottom radius $R$, height $h$, and bottom angle of the foundation $\beta$

4. Excavation height of the working platform $L$

5. Diameter of the rupture plane $R^{\prime }$

6.5 Verification of the design method

Under flat ground conditions, the rupture surface of the rock mass surrounding the foundation is symmetrical along the axis of the foundation:

${\omega }_1={\omega }_2=\omega (24)$

By substituting Formula 24 into Equation 23, we obtain the expression for the ultimate pullout capacity of the foundation under flat ground conditions (Formula 25):

$\begin{array}{l}T={\gamma }_1\frac{1}{3}\pi h\left(R^2+r^2+Rr\right)+{\gamma }_1\frac{1}{3}\pi h\left[R^2+{\left(h\tan \omega +R\right)}^2+R\left(h\tan \omega +R\right)\right]\\ +\left(c+\frac{1}{2}{\gamma }_{2}h\sin \omega \tan \phi \right)\pi h\left(2R+h\tan \omega \right)\end{array}(25)$

This is consistent with the existing formulas for the ultimate tensile resistance capacity of foundations under flat ground conditions (Nhat et al., 2023; Bhattacharya, 2016), demonstrating the validity of the formula when the slope angle is equal to 0.

To verify the effectiveness of this formula, twelve scenarios with different foundation sizes, slopes, and parameters were selected for calculation and numbered from one to 12 (as shown in Table 4). The results of numerical simulations and formula calculations are shown in Figure 25. It can be seen from the figure that the results obtained from numerical simulations and formula calculations are relatively consistent, verifying the validity of the formula.

Table 4

Table 4. Number of calculation plan.

Figure 25

Figure 25. Ultimate uplift load curves for the two calculation methods.

6.6 Engineering application

This project involves a dedicated power supply line for a railway traction substation, with a tower importance factor of 1.0. The new transmission line originates from the “Nanxi River Traction” gas-insulated switchgear (GIS) terminal at the 220 kV Nanjiang Substation and terminates at the overhead line entrance of the Nanxi River Traction Substation. Along its route, the line crosses Provincial Highway S223, the Nanxi River, the S26 Zhuyong Expressway, the 110 kV Qinghe Line, the 220 kV Pannan and Panjiang Lines, and the Hangzhou-Wenzhou Railway tunnel. The line generally follows a south-to-north alignment through mountainous terrain.

The total length of the new line is 29.134 km, constructed as a single-circuit line throughout. This includes 28.858 km of overhead lines and 0.276 km of cable sections. A total of 65 new single-circuit steel towers were erected. The terrain along the route comprises 5% riverine areas, 5% flat land, 50% mountainous areas, and 40% high-altitude mountainous areas, with elevations ranging from 0 to 590 m.

In flat terrain conditions, the subsurface stratigraphy consists of silty clay down to 3 m depth, underlain by gravelly soil, with a tuff bedrock surface encountered at 18–20 m depth. In mountainous areas, the subsurface comprises silty clay mixed with crushed stone down to 3 m depth, underlain by highly weathered and moderately weathered tuff. As the line predominantly traverses mountainous and high-altitude terrain, this study focuses on verifying whether the ultimate bearing capacity of the embedded foundations under mountainous conditions meets the design requirements.

Given the heterogeneous nature of rock strata in mountainous conditions, a conservative approach was adopted by assuming homogeneous rock properties equivalent to silty clay mixed with crushed stone for the foundation bearing capacity calculations.

Figure 26 presents the dimensional design of the embedded foundation. The slope rock mass consists of highly weathered and moderately weathered tuff, with a maximum slope angle of 50°. The foundation was cast using C25 concrete, with parameters detailed in Table 5.

Figure 26

Figure 26. The dimension design drawing of the embedded foundation.

Table 5

Table 5. Physical and mechanical parameters of materials.

Analysis of three typical loading conditions during tower operation yielded the design force values for the tower foundation, as summarized in Table 6. The results indicate that the maximum uplift force on the foundation occurs under the maximum wind speed condition, representing the most critical loading scenario. The foundation dimensions must satisfy the requirement that the ultimate uplift bearing capacity exceeds the uplift load under maximum wind conditions.

Table 6

Table 6. The design values of the tower foundation forces.

According to Formula 23, the calculated ultimate uplift bearing capacity of the foundation under maximum wind conditions is 348.2 kN, which surpasses the corresponding design load. Therefore, the design dimensions of the embedded foundation meet the required specifications.

7 Conclusion

Numerical simulations were employed to investigate the interaction mechanism between embedded foundations and rock masses under slope conditions, leading to the development of a corresponding design methodology. The principal findings and implications are summarized as follows:

1. Failure Mechanism and Design Implications: Under uplift loading, foundation failure primarily results from shear failure induced by the progressive development and interconnection of plastic zones within the surrounding rock mass. This failure mechanism underscores the importance of controlling the propagation of shear slip surfaces in the rock mass during uplift design. For foundations embedded in strongly weathered slope rock masses, engineering measures aimed at enhancing the integrity and shear strength of the surrounding rock are recommended.

2. Rupture Surface Characteristics and Slope Effects: Both jar-shaped and Y-shaped foundations exhibit similar rupture surface trajectories, characterized by significant slope effects. Specifically, the rupture surface angle on the downslope side increases with the slope angle, whereas the angle on the upslope side decreases correspondingly. Designers should note that the resistance provided by the downslope rock mass diminishes as the slope angle increases, which is a critical factor governing the uplift capacity of foundations on slopes. The rupture surface at the slope face approximates a circular shape, with its diameter determined mainly by the foundation size and remaining largely unaffected by variations in slope angle or rock mass parameters. This observation provides a basis for simplifying design calculations.

3. Validation and Limitations of the Proposed Method: The proposed formula for calculating the ultimate uplift capacity, derived based on limit equilibrium theory, demonstrates good agreement with numerical simulation results across various foundation sizes, slope angles, and rock mass parameters. Thus, it is suitable for preliminary design of embedded foundations in similar slope engineering applications. However, it should be noted that the method assumes a homogeneous rock mass; consequently, its applicability to non-homogeneous rock masses featuring significant jointing, fissures, or lithological variations requires further validation.

This study establishes a theoretical framework for designing embedded foundations on sloped terrain, clarifying the influential roles of slope gradient and rock mass properties. The methodology paves the way for future extensions to more complex geological conditions—such as rock mass heterogeneity and varying soil properties—and dynamic loading scenarios including seismic events, thereby enhancing foundation robustness in challenging environments.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author contributions

MW: Conceptualization, Formal Analysis, Methodology, Writing – original draft, Writing – review and editing. YZ: Conceptualization, Formal Analysis, Methodology, Supervision, Writing – original draft, Writing – review and editing. ZC: Writing – review and editing, Formal Analysis, Methodology, Supervision, Conceptualization. CW: Conceptualization, Methodology, Supervision, Writing – original draft, Writing – review and editing. XW: Conceptualization, Formal Analysis, Methodology, Writing – original draft, Writing – review and editing. AY: Conceptualization, Formal Analysis, Methodology, Writing – original draft, Writing – review and editing. ZZ: Conceptualization, Formal Analysis, Methodology, Writing – original draft, Writing – review and editing. XF: Conceptualization, Methodology, Writing – original draft, Writing – review and editing.

Funding

The authors declare that financial support was received for the research and/or publication of this article. The Key R&D Program of Yunnan Province (202303AA080010), the Research and Development Plan of Railway Siyuan Survey and Design Institute Group Co., Ltd. (2022K085), the International Partnership Program of Chinese Academy of Sciences (Grant No.131551KYSB20180042) Hubei Provincial Natural Science Foundation (No. 2023 AFB802). The authors declare that this study received funding from Railway Siyuan Survey and Design Institute Group Co., Ltd. The funder was not involved in the study design, collection, analysis, interpretation of data, the writing of this article, or the decision to submit it for publication.

Acknowledgements

This research was supported by the Key Research and Development Program of Yunnan Province (Grant No. 202303AA080010), the Research and Development Plan of Railway Siyuan Survey and Design Institute Group Co., Ltd. (Grant No. 2022K085), the International Partnership Program of Chinese Academy of Sciences (Grant No. 131551KYSB20180042), and the Hubei Provincial Natural Science Foundation (Grant No. 2023AFB802). The authors are thankful for the supports.

Conflict of interest

Author ZC was employed by Powerchina Kunming Engineering Corporation Limited.

Authors CW, XW, and AY were employed by China Railway Siyuan Survey and Design Group Co., Ltd.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/feart.2025.1730498/full#supplementary-material

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