Model Test Study on Rock Rolling Characteristics

Abstract

In order to study the influence of falling rock shapes on their rolling characteristics and to determine the optimization of falling rock protection design, a series of research experiments were conducted. Model experiments were designed to explore the rolling characteristics of rockfalls with different shapes. Based on the experimental results, it was found that the slenderness ratio, center of gravity, and rotational inertia of the rockfalls can affect their rolling characteristics, leading to swaying and changing the rolling axis during the rolling process, thereby affecting their rolling speed. Building upon these findings, an analysis of the formation mechanism of rolling resistance was conducted. It was determined that the primary cause of energy loss was the rolling resistance arm formed with the rolling surface during rockfall motion. A shape parameter was proposed to quantify the rolling resistance. These parameters were incorporated into a kinematic formula that considered the influence of rockfall shape, slope, and slope roughness on the rolling speed. Combined with the offset and initial position of the rockfall, the formula could be used to calculate the rolling speed and impact energy in the rolling region at any position in the region. The calculation formula was validated using model experimental data, and the results showed that the error between the experimental and calculated values was small. The error was corrected based on the experimental data. After on-site testing and verification, it could provide reference for the management of rockfall disasters.

1. Introduction

Rocks involved in collapses and landslides exhibit characteristics such as high destructiveness, suddenness, scattered distribution, and complex and unpredictable trajectories [1,2]. They represent a primary geological hazard in the mountainous regions of China. In 2022 alone, China witnessed 1366 incidents of rockfall disasters, constituting 24.14% of the total geological disasters in the country and posing a severe threat to the safety of people’s lives and property. Figure 1 and Figure 2 illustrate the devastation caused by rockfalls.

In the design of protection against rockfall, it is crucial to accurately calculate the characteristics of rockfall movement. The shape of the falling rock is a pivotal factor in determining its rotation and collision behavior, playing a key role in energy consumption and trajectory alterations during the rolling process. Different rockfall shapes result in varying moments of inertia, impacting the rotation behavior of the rockfall upon contact with the slope [3]. These dynamics significantly influence the extent of the hazard and contribute to the design of effective protection measures against rockfall. Data related to the rockfall’s position, velocity, rotation, energy, and impact force are utilized to ascertain the influence range and hazard degree of the rockfall. Consequently, this enables the determination of optimized designs for rockfall protection.

The current research methods related to the characteristics of rockfall movement mainly include theoretical derivation [4], model experiments [5,6,7,8], numerical simulations [9,10,11,12], and mathematical probability models for data analysis and statistics of historical rock collapse events [13,14,15]. Theoretical derivation typically relies on kinematic principles and contact mechanics to derive the trajectory of a falling rock. For instance, Zhou [16] formulated equations for calculating motion characteristics such as rockfall velocity and displacement. Cui et al. [17] established a model for the oblique throwing and impact of falling rocks, allowing for the calculation of movement distance and impact force. Yang et al. [18] considered the influence of the rotational inertia of rockfalls, dividing the motion into five stages and providing calculation formulas for each stage. On the basis of considering geometric similarity, physical similarity, and material similarity, indoor model experiments will scale down the actual site in the model device. Through designing indoor experiments, the motion of falling rocks was simulated using test blocks of different shapes, assuming that there were no collisions or fractures during the rolling process. The rolling characteristics of test blocks with different shapes were observed, and the influence of shape on their rolling characteristics was studied. Among these studies, Oda et al. [19] highlighted, based on indoor test results, that rolling resistance is not solely a result of contact behavior during the rolling process, but is also influenced by the shape of rockfalls. Ai et al. [20] emphasized that the shape effect serves as a significant source of rolling resistance for rockfalls. Gao et al. [21] clarified, based on model test results, that the sphericity of rockfalls is a crucial shape parameter affecting rolling characteristics. Giani et al. [22] pointed out that the rotational speed of a block is a function of its rotational inertia on the cross-section where it moves. Ushiro et al. [23] discovered through experiments that the rolling distance and velocity of rockfalls are positively correlated with their volume. Cui et al. [24] used the ratio of rock block volume to its minimum circumscribed circle volume as a parameter to describe its shape characteristics, and proposed a rolling speed calculation model including the shape parameter of rockfall based on indoor experimental data. From this, it can be seen that conducting physical experiments on rock rolling using laboratory equipment is an effective approach to studying the influence of shape factors on their rolling characteristics. Lü et al. [25] investigated the fracture mechanism of rock blocks and emphasized that the degree of fracture is closely linked to the mechanical properties of the rockfall, rather than being significantly associated with the size of the rockfall. Therefore, in the case of rockfalls with poor mechanical properties in karst areas [26,27], calculations should consider the potential occurrence of fractures during the rolling process. Numerical simulation can be categorized into rockfall software simulation and code for developing rockfall simulation. Leine et al. [28] proposed a rock collapse trajectory analysis model based on the framework of multi-body dynamics and non-smooth contact dynamics, capable of simulating rock collapse and rockfall events involving rock blocks of various shapes on different characteristic slopes in full 3D. The trajectory analysis model suggested by Yan et al. [29] can simulate rockfalls with different sphericity and convexity. Bourrier et al. [30] calibrated existing 2D rock collapse prediction models, achieving accurate predictions of the farthest impact range of rockfall. Zhang et al. [31] imported a three-dimensional geological model into simulation software (RAMMS) to obtain the motion trajectory of rockfalls. Xu et al. [32] utilized the energy tracking method (ETM) to study the impact of rock fragmentation on its motion trajectory, pointing out that the fragmentation process is accompanied by energy loss, resulting in a decrease in rolling speed.

In summary, the shape and size of a rockfall are key factors influencing its rolling speed and determining its moment of inertia. These factors, in turn, affect its rotation behavior in the air and upon contact with the ground. The rolling resistance of a rockfall arises from mutual contact with the slope during the rolling process, and the nature of this contact varies with different shapes of rockfalls and the slope. Therefore, different shapes of falling rocks exhibit significant differences in overcoming rolling resistance during the rolling process. The rolling motion of falling rocks is one of their movement characteristics on the slope, primarily controlled by the slope gradient—rolling dominates on steeper slopes, while sliding dominates on gentler slopes [33]. Through investigation and analysis of the slope gradient in historical rock collapse events, it was found that the slopes in areas prone to rockfall disasters are mainly concentrated above 45° [34]. It can be seen that it is necessary to study and quantify the relationship between rockfall shape and rolling resistance, to explore the rolling characteristics of falling rocks on slopes, to supplement the quantitative parameters of the influence of rockfall shape on rolling velocity, on the basis of the existing calculation formula.

It is difficult to establish an ideal calculation model with a single research method when considering the influence of the shape of a falling rock on its rolling speed. Therefore, in the follow-up research, model test research and theoretical analysis can be combined to obtain the movement of different shapes of fallen rocks on the slope. Finally, according to the test results, the influence mechanism of rockfall shape on its rolling characteristics is analyzed and, on this basis, the influence of other factors is combined to establish a calculation model.

Currently, there are two commonly used methods for establishing models of the motion speed of rolling rocks: the concentrated mass method and the rigid body method. The concentrated mass method concentrates the mass of a continuously distributed object to a single point, assuming the object only has mass without stiffness parameters. This simplifies the analysis of the object’s motion and is suitable for cases with simple shapes and uniformly distributed mass [35]. The rigid body method considers the shape and size of the object but neglects its deformation, making it a common method for building rockfall models [36]. Since the concentrated mass method treats falling rocks as an infinitely small point, it cannot reflect the impact of shape and size on rolling characteristics. In contrast, the rigid body method accurately represents the irregularity of the rock’s shape [29]. Therefore, in theoretical analysis, this paper adopts the rigid body method, assuming the falling rock behaves as an ideal rigid body with no changes in shape and size during the rolling process. This allows for the investigation of how the shape of falling rocks influences their translation and rotation.

To investigate the impact of shape on the movement characteristics of falling rocks prone to rockfall disasters, the selected rock shapes underwent model tests. Test blocks of varying shapes were released from slopes with different heights and inclinations. The rolling characteristics and changes in rolling speed of each test block were observed at different release heights, slope roughness levels, and slopes. The experimental results were then utilized to analyze the relationship between the shape of falling rocks and their rolling resistance. Proposed shape parameters were introduced into the kinematic formula, accounting for the influence of specimen shape, size, release height, slope, and slope roughness. This integration resulted in a formula for calculating the rolling speed and energy of rockfall. Subsequently, the accuracy and applicability of the formula were verified using data obtained from model experiments. By combining site characteristics with the shape of falling rocks, it becomes possible to infer the impact range of rolling falling rocks, providing valuable insights for the prevention and control of rockfall disasters.

2. Experimental Introduction

2.1. Experimental Apparatus

To investigate the rolling characteristics of rocks with varying shapes and sizes on slopes, an experimental model apparatus made of an aluminum alloy was designed, as depicted in Figure 3. Before designing the experimental setup, considerations were given to the proportional relationship between the size of the test blocks and the slope, as well as the similarity scale between the setup and actual rockfall disasters [37]. This ensured that the dimensions of both the test setup and the actual rockfall disaster were proportionally similar, allowing for a better representation of the dimensions of the slopes and falling rocks in real rockfall events [38]. Combining previous research on the characteristics of terrain in historical rockfall disaster events, the similarity ratio of the setup was set to fifty, and the similarity ratios for gravity and time were both set to one.

The experimental device comprises two symmetrically arranged support rods, an upper frame for placing the rolling surface material of the test block, and a bottom fixed frame to maintain device stability. The upper frame is constructed from aluminum alloy square tubes and connectors, while the bottom frame is connected by four aluminum alloy components of the same length arranged longitudinally, with horizontally placed angle steel, secured using bolts. Preliminary adjustment of the angle between the upper and lower frames is achieved by reserving holes for bolts on the support rod. After reaching the required slope for the test, the reserved interface of the horizontally placed angle steel at the bottom is used to fine-tune the slope and fix the support rod.

The rolling slope dimensions are 1.3 m × 1.7 m, and single or composite slope materials can be placed according to experimental requirements. Notably, there are no barriers on either side of the slope, to avoid energy loss caused by collision or friction between the test block and the side walls of the slope during rolling. The slope adjustment range is 25–75°, ensuring that the slope of the test block is greater than its rolling friction angle when released, allowing the test block placed on the slope to tip under gravity and develop into continuous rolling.

The experimental shooting equipment utilized was a high-speed camera produced by SONY, model AX-700 (Tokyo, Japan), with a selected frame rate of 1000 frames/s. To enhance contrast during rockfall movement and achieve a better shooting effect, grid lines were drawn on the slope material as the camera background.

2.2. Experimental Materials

2.2.1. Rock Materials and Shape Parameters

When selecting the shapes of rockfalls for the test, reference was made to the three typical shape features of fallen rocks in mountainous areas that are prone to induce rockfall disasters, proposed by Fityus [39], including three shape types: sphere type, cylinder type, and disc type.

Shapes were defined based on their three-dimensional dimensions (d_max—maximum dimension, d_mid—middle dimension, d_min—minimum dimension), where d_max represents the longest dimension of the falling rock, d_min is measured perpendicular to d_max, and d_mid is obtained by measuring a plane perpendicular to both d_max and d_min, as illustrated in Figure 4.

For rocks with nearly equal three-dimensional dimensions (d_max ≈ d_mid ≈ d_min), they were classified as spherical in shape, including spheres and regular polyhedra. For rocks where the maximum dimension was significantly larger than the other two dimensions (d_max > (d_mid ≈ d_min)), they were classified as cylindrical, including cylinders and prisms. For rocks where the two larger dimensions were approximately equal ((d_max ≈ d_mid) > d_min), they were classified as disc-shaped, including discs and hemispheres.

When selecting the test block material, the density and hardness of the material was considered as it may affect its rolling characteristics. In order to avoid the interference of other influencing factors in the model test, C60 concrete was used to make test blocks. The mix design of C60 concrete is shown in

Table 1. Mix design of C60 concrete grouting materials.

Materials	Po52.5 Cement	8–20 Mesh Quartz Stone	20–80 Mesh Quartz Stone	Fly Ash	Silica Fume	Water
Material Usage (%)	32.5	8	39	5	3	12.5
Additive	ZY808 Defoamer	ZY1100 Plastic Expansion Agent	ZY8121 Water Reducing Agent	MK400 Cellulose
Dosage per 100 kg (g)	30	20	130	15

. The mold can be used to cast any shape test block, and the high hardness can ensure that it does not break during the rolling process. The test block is shown in Figure 5, and the specific shape parameters are shown in

Table 2. Shape parameters of test blocks.

Test Block	Shape	Size	Test Block	Mass (g)	Volume (cm3)
		Characteristic Length
A	Sphere	Diameter	70	449	179.5
B	Regular Hexahedron	Edge Length	56	432	175.6
C	Regular Octahedron	Edge Length	72	432	176
D	Cylinder	Diameter/Height	50/90	434	176.6
E	Four-sided Prism ①	Length/Width/Height	45/45/88	437	178.2
F	Four-sided Prism ②	Length/Width/Height	50/55/64	440	176
G	Octagonal Prism	Edge Length/Height	20/90	427	173.8
H	Dodecagonal Prism	Edge Length/Height	13/91	423	172.2
I	Hexagonal Disc	Edge Length/Height	50/27	432	175.4
J	Decagonal Disc	Edge Length/Height	28/29	429	175
K	Dodecagonal Disc	Edge Length/Height	23/30	436	174.9
L	Circular Disc	Diameter/Height	90/28	437	178

.

2.2.2. Slope Surface Material Parameters

The back side of ceramic tiles was chosen as the slope surface material for the experiments to ensure uniform roughness, as illustrated in Figure 6. Before conducting the experiments, the sliding friction coefficient between the test blocks and the ceramic tile surface was determined using the proposed method for calculating slope surface friction coefficients [40]. It was ensured that the sliding friction coefficient of the test blocks exceeded the rolling friction coefficient, ensuring that the blocks would only undergo rolling after being released from the slope.

2.2.3. The Calculation Method for the Rolling Friction Coefficient of Slope Materials

Block A is a sphere, and the resistance it needs to overcome when rolling on the slope can be determined by Formula (1):

$$W_f=F_{f}L=\frac{d}{r}mg\cos \theta L$$

(1)

d—the distance from the instant center to the center of mass of the ball when it rolls on the rolling surface.

r—the radius of the ball.

L—the rolling distance of the ball on the slope surface.

θ—the slope.

The total energy of Block A comes from gravitational potential energy E_h, and combining Formula (2) yields the proportion C_f between the overcome resistance and the total energy:

$$C_f=\frac{W_f}{E_h}=\frac{E_h-\frac{1}{2}m{v}^2}{E_h}=\frac{d}{r}\cot \theta$$

(2)

Here, C_f is in a linear relationship with the tangent of the slope and, neglecting the deformation during the rolling process of Block A, the slope represents the rolling friction coefficient μ derived from the slope characteristics; v is the rolling speed of Block A (m/s). Thus, the measured rolling friction coefficient μ on the back side of the ceramic tile is 0.09, as shown in Figure 7.

2.3. Experimental Procedure

When determining the slope range for the model experiment, we referred to the analysis results of multiple historical rock collapse disaster events by Wei [41]. Test slopes of 30°, 45°, and 60° were selected based on this analysis. The distance from the test block on the slope to the bottom of the slope is defined as the release height h. Within the range of 0.3 m to 0.9 m, one height was selected every 0.15 m as the release height for the test.

Initially, the slope was adjusted to the desired angle using support rods to simulate the single-slope rolling of rocks. The slope angle at the time of release should exceed the rolling friction angle to ensure the continuous tilting of the test blocks. Secondly, the high-speed camera was set up in advance to fully capture the motion plane of the entire experimental apparatus. Lastly, it was ensured that the test blocks were in a stationary state at the time of release, ensuring that the reason for the blocks tilting and subsequently rolling was that the torque-resisting tilting was smaller than the tilting torque.

To ensure that the offset areas on both sides of the test blocks during their rolling on the slope were of the same size, the test blocks were released from the midpoint of the slope during the experiment. The specific position of the release point could be determined by the distance h from the release point to the slope bottom.

The following methods were employed for video processing based on the grid lines drawn on the apparatus:

The corresponding proportionality relationship was determined between the actual dimensions and the dimensions in the video: L_actual:L_image = k:1, for converting experimental data.

A coordinate system on the rolling surface was established using the grid lines and determines the test block’s rolling trajectory based on the coordinates. The initial position coordinates are (x₀, y₀), and after continuous rolling, the coordinates of the test block are (x_n, y_n). To calculate the distance between the two points, which, after scaling, represents the actual length, denoted as L [42], the below formulas are used:

$$L=k\sqrt{{(x_n-x_0)}^2+{(y_n-y_0)}^2}$$

(3)

$$\begin{array}{l}v=\frac{k\sqrt{{(x_n-x_{n-1})}^2+{(y_n-y_{n-1})}^2}}{2\Delta t}\\ +\frac{k\sqrt{{(x_{n+1}-x_n)}^2+{(y_{n+1}-y_n)}^2}}{2\Delta t}\end{array}$$

(4)

The entire rolling process was analyzed frame by frame, recording the test block’s rolling trajectory from the moment of release until it comes to a stop, thus obtaining the actual rolling distance S on the slope surface.

2.4. Analysis of Experimental Data Dispersion

Due to the influence of various factors on the model experiment, the obtained data exhibit strong randomness. Before using the experimental data for theoretical analysis, it is necessary to examine the dispersion of the data. In order to comprehensively assess the data’s dispersion, the coefficient of variation for each set of data is analyzed. Specifically, the ratio of the mean value to the standard deviation of each experimental dataset is taken as the standard for examining its dispersion. The specific data are shown in Figure 8, Figure 9 and Figure 10, and the letters represent the coefficient of variation for the test block.

The average coefficient of variation for all experimental data in this experiment is 5.5%, with a maximum coefficient of variation of 14.1%. Combining the dispersion of the experimental data with the observed situations during the experiment, it is found that the inconsistency in data obtained under the same experimental conditions is normal and reasonable. This is because the model experiment cannot consider all the influencing factors, controlling only a few key factors. Additionally, various factors interact during the block’s rolling process, resulting in the final results having a certain degree of randomness. An analysis was conducted on the data with a coefficient of variation greater than 10%, revealing that excessive dispersion is due to significant lateral displacement and sliding of the blocks during the rolling process. Excessive displacement leads to an overestimation of the rolling distance on the slope, accompanied by an increase in energy loss during the offset process. When the slope is relatively low, blocks with higher rolling resistance may exhibit sliding during the rolling process, altering their initial motion state and increasing their energy consumption.

From Figure 11, it can be seen that only 5.7% of the experimental data have a relatively large degree of dispersion, with a coefficient of variation exceeding 10%. The remaining experimental results have a small degree of dispersion, indicating that the quality of the model experimental data is relatively reliable and can be used to analyze the impact of the block shape on its rolling speed.

2.5. Analysis of Experimental Results

For a more intuitive observation of the experimental data, the experimental results are plotted as scatter plots, as shown in Figure 12, Figure 13 and Figure 14. At the same time, error bars are used to depict the errors between each experimental data point and the mean value, with the error range indicated.

2.5.1. Influence of Block Shape on Rolling Speed

As evident from Figure 12, Figure 13 and Figure 14, under identical conditions, spherical block A exhibits a significantly higher rolling speed compared to blocks B and E. Analysis suggests that the rolling resistance experienced by spherical block A during rolling depends on the roughness of the rolling slope. In contrast, blocks B and E primarily experience rolling resistance based on their contact with the rolling surface, specifically the contact area between the block and the surface after each roll. Blocks B and E have much larger contact areas with the slope during rolling compared to block A, substantially increasing the rolling resistance torque they need to overcome. From the perspective of energy consumption, for blocks with the same mass, the larger the rolling resistance torque, the more work that is required for each roll, leading to a greater energy consumption. Hence, for blocks rolling on a hard slope surface, their shape is the primary factor leading to their energy consumption.

From Figure 14, it can be observed that on a 60 degree slope, for blocks E (roundness of 0.9) and G (roundness of 0.97) with equal mass, released from a height of 90 cm, the actual speeds are 2.71 m/s and 3.08 m/s, respectively. Block E’s speed is 13.65% lower than that of block G, and this phenomenon is consistently observed in experiments with blocks of different shapes. This indicates that the rolling speed of the blocks is somewhat related to the two-dimensional cross-sectional roundness of their rolling surface and is positively correlated with the roundness perpendicular to the rolling surface. The fundamental reason for this phenomenon is that, for blocks with the same volume, sections with higher roundness have a smaller contact area with the slope during rolling compared to sections with lower roundness. Consequently, blocks with higher roundness experience less resistance to overcome, resulting in faster rolling speeds.

During the test, it was found that when the test block rolled downwards from the slope, it tended to roll around the rolling axis with the least rolling resistance. This aligns with the conclusion drawn by Azzoni [43] through the analysis of field rockfall test data. A parameter called the length-to-width ratio (λ) was defined, where λ = l_min/l_max, with l_max being the maximum length of the block and l_min being the minimum length. Analysis of the experimental data revealed that when λ < 0.33, the blocks typically roll around the axis with minimum rolling resistance and rarely change their rolling axis, as shown in Figure 15. The lateral offset of the test block in the rolling process is small, attributed to the low rolling resistance of the test block itself and the slope surface. This enables the test block to maintain a consistent rolling direction. When 1 > λ > 0.33, the test block usually undergoes a transformation of the rolling axis during the rolling process, as illustrated in Figure 16. The frequency of rolling axis transformations correlates positively with the magnitude of λ. For test block E, its rolling resistance is relatively high, and the difference in rolling resistance between any two-dimensional sections is not significant. Consequently, during the rolling process, the rolling axis easily undergoes transformations, leading to an increased energy loss and a reduction in the rolling speed of the test block. In contrast, for test block D, despite having a slenderness ratio within the range of 1 > λ > 0.33, the small resistance of the initial rolling section upon release prevents a change in the rolling axis, resulting in minimal lateral offset, similar to test block G. This observation indicates that when rolling on a slope without obstacles, the transformation of the rolling axis of the test block is positively correlated with both the slenderness ratio and the rolling resistance originating from the shape of the test block itself.

From Figure 14, it is evident that at a release height of 45 cm, despite block D having slightly greater roundness compared to block K, its rolling speed (2.23 m/s) is slightly lower than that of block K (2.26 m/s). The reason for this phenomenon is that during the rolling process, block K’s moment of inertia is greater than that of block D, allowing block K to possess more rotational kinetic energy when rolling down the slope, as depicted in Figure 17 and Figure 18. Consequently, block K overcomes some of the rolling resistance caused by its shape, leading to its rolling speed approaching that of block D, and in some cases, even surpassing it. Although the contact area between test block K and the rolling surface is smaller than that of test block D, the center of gravity of test block K is higher than that of test block D, resulting in the left and right shaking of test block K during the rolling process. However, under the action of rotational inertia, it can still maintain the initial rolling axis. The shaking of test block K during the rolling process consumes some of the rolling kinetic energy, resulting in a decrease in the rolling speed of the test block.

2.5.2. Influence of Slope and Release Height on Rolling Speed of Test Blocks with Different Shapes

For the spherical test block A, which can ignore the rolling resistance caused by its own shape, under the same release height, its rolling distance and slope offset will decrease with the increase of the slope. Additionally, the rolling resistance required to overcome the slope characteristics will also decrease. It can be observed that for spherical test blocks, under the same conditions as other influencing factors, the rolling speed is positively correlated with the slope angle. In other words, for test blocks with high rolling resistance, a decrease in slope at the same height will increase the rolling distance of the test block on the slope, thereby increasing the energy consumption of the test block. When the slope is gentle and the rolling distance is too long, there may be a situation where the rolling speed of the test block continues to decrease and eventually stops on the slope.

As seen in Figure 12, Figure 13 and Figure 14, for the test block released at rest, with the increase in the release height, the test block has greater gravitational potential energy when released, and the rolling speed of the test block with all shapes increases with the increase in the release height. However, the increasing amplitude of the rolling speed of different test blocks with the increase in the release height is very different. The most noticeable difference is the change in the rolling speed of test block A and test block B with the increase in height in Figure 12. The increase in rolling speed of test block A is much greater than that of test block B. It can be observed that for a test block with high rolling resistance, increasing its release height on the slope has no obvious effect on its rolling speed. However, this phenomenon mainly occurs when the slope is small. As shown in Figure 13, when the slope is 45°, the increase in the rolling speed of all test blocks can be clearly observed with the increase in release height. The reason for this may be that the overturning moment of the test block is smaller when the slope is smaller, the resistance needed to overcome each roll is larger, and the distance required to leave the slope is longer. Therefore, even if the release height increases, for a test block with high rolling resistance, a large part of the increased height potential energy is still consumed during rolling, resulting in an insignificant increase in rolling speed.

To sum up, it can be seen that the rolling resistance in the rolling process of rockfall comes from the roughness of the slope and the shape of the rockfall. The difference in shape makes the rockfall have a different roundness, moment of inertia, slenderness ratio, and contact area with the slope during the rolling process, thus affecting its rolling speed. The rolling speed of different shapes of rockfalls on the slope is also affected by the release height and slope, and these factors should be integrated in the calculation formula.

2.6. Sensitivity Analysis of Experimental Data

The rolling speed of rockfalls is influenced by various factors. While analyzing the impact of individual factors through orthogonal experiments provides insights into the changing trends of rockfall rolling under each factor, it fails to capture the comprehensive impact of multiple factors. To assess the influence of various factors on the rolling speed of rockfalls, the range analysis method is employed. This method identifies the most significant factors and ranks their impact, offering valuable insights for the prevention and mitigation of rockfall disasters.

From

Table 3. Rolling speed range analysis results.

Item	ψ	Release Height	Mass	Slope	Roundness	R′
K1	78.33	58.944	39.84	146.999	115.99	39.84
K2	83.05	71.608	129.043	130.714	113.42	27.64
K3	37.66	82.419	37.66	129.353	34.51	37.66
R	0.765	1.183	0.791	0.294	0.687	0.827
convert coefficient d	0.32	0.4	0.34	0.52	0.37	0.31
R′	1.095	2.84	1.275	1.185	1.393	1.088
Factor primary and secondary order	Release height—roundness -Mass-Slope-ψ- R′

and Figure 19, it can be observed that release height has the most significant impact on rolling speed. The sensitivity ranking of factors affecting speed is as follows: release height > roundness > mass > slope > ψ > R′. Consequently, the shape of the rockfall is identified as one of the key factors influencing its rolling speed. When establishing a model to calculate the rolling speed of rockfalls, parameters accurately describing their shape characteristics should be included.

3. Theoretical Analysis

Model Derivation

The current study has clarified that the shape of falling rock will affect its rolling characteristics and has pointed out that falling rocks with different shapes will experience a different rolling resistance during rolling. However, no clear definition of rolling resistance has been provided. To accurately describe the influence of rockfall shape on its rolling characteristics, two concepts related to rolling—rolling resistance and rolling resistance distance—need to be established [44]. For the same test block placed stationary on the slope, its rolling resistance arm δ is determined by the size of the contact surface between the test block and the slope, as shown in Figure 20.

Through the analysis of data obtained from model experiments, the mechanisms through which the rock shape influences its rolling speed have been identified. Consequently, this paper defines a shape parameter that reflects the overall shape and local convex–concave characteristics of a rolling stone. This parameter is based on the difference between the two-dimensional cross-sectional area perpendicular to the shortest and longest dimensions of the falling stone and the area of its minimum circumscribed circle, as seen in Figure 21. The rolling resistance coefficient ψ, stemming from the shape of the rolling stone, is defined as follows.

$$\psi =\frac{u_1-u_2}{u_1}$$

(5)

u₁—The area of the smallest circumscribed circle of the 2D cross-section of the rolling stone perpendicular to the slope.

u₂—The area of the 2D cross-section of the rolling stone perpendicular to the slope.

Figure 21. Illustration of three categories of rock scales [45].

Figure 21. Illustration of three categories of rock scales [45].

Additionally, the rolling resistance of a rock also arises from factors such as slope angle and roughness. Taking into account the rolling friction coefficient related to the roughness of the slope [46], the rolling resistance f for rock rolling is defined as:

$$f=(\mu +\psi )N$$

(6)

μ—Rolling friction coefficient related to the roughness of the slope.

The assumptions made during this derivation are as follows:

(1) The rolling stone experiences rolling but not sliding on the slope. The slope angle θ must satisfy the following condition:

$$arc\tan \frac{H}{\delta }<\theta <arc\tan \mu$$

(7)

where: δ—Rolling resistance arm of the rolling stone.

H—Distance from the center of gravity of the rolling stone to the slope.

μ—Rolling friction coefficient related to the roughness of the slope.

(2) The rolling stone and the slope do not deform during the rolling process.

(3) The rolling stone remains intact while rolling on the slope.

According to the principle of conservation of energy, the energy calculation formula for the rolling stone on a rigid slope can be derived as ((A1), refer to Appendix A):

In experiments, the actual rolling distance of the test block can be determined by analyzing the entire process of the block’s rolling, captured by a high-speed camera during model tests. In practical applications, combined with Formulas (A2) and (A3), the shortest and longest rolling distances S₁ and S₂ of the rolling stone on the slope can be estimated based on the stone’s height and the corresponding maximum offset η [47]. The actual rolling distance and the extent of damage can then be estimated based on the specific shape of the falling stone and slope characteristics, as shown in Figure 22.

In the calculation of the rolling speed of blocks with different length-to-width ratios (λ), it is necessary to consider the situation where the rolling axis changes during the rolling process. For falling stones with 0.33 < λ < 1, it is necessary to calculate the moments of inertia for both the minimum and maximum rolling axes, along with the corresponding rolling friction coefficients and minimum circumscribed circle radii of the two-dimensional cross-sectional shape. These values are used to determine the upper and lower bounds of the rock’s rolling speed. For falling stones with λ < 0.33, only the parameters corresponding to the rolling axis with the minimum rolling resistance need to be determined as the theoretical values for calculating the rock’s rolling speed.

The formula for calculating the angular velocity of a rolling stone is as follows:

$${\omega }^{\prime }=\frac{\sqrt{v_x^2+v_y^2}}{R^{\prime }}=\frac{v}{R^{\prime }}$$

(8)

v_x, v_y—Velocities of the rolling stone in the x and y directions (m/s).

v—Rolling speed of the rolling stone (m/s).

R′—Minimum circumscribed circle radius of the two-dimensional cross-section perpendicular to the slope (m).

For non-circular cross-sectional shapes of rolling stones, the following modification can be made:

$$\omega =\frac{{\omega }^{\prime }}{d_k}$$

(9)

d_k—Circularity of the cross-section, d_k = 2πR′/c.

c—Circumference of the minimum circumscribed circle of the cross-section (m).

By combining Formulas (6) and (A3) to solve Formula (A1), the range of the rolling speed of the rolling stone (v₁ < v < v₂) can be obtained, where v₁ and v₂ represent the upper and lower bounds of the rolling stone’s rolling speed, respectively:

$$v_1=\sqrt{\frac{2\left[mgH-S({\psi }_1+\mu )N\right]}{m+\frac{I_1}{{\left({R^{\prime }}_1{d}_{k1}\right)}^2}}}$$

(10)

$$v_2=\sqrt{\frac{2\left[mgH-S({\psi }_2+\mu )N\right]}{m+\frac{I_2}{{\left({R^{\prime }}_2{d}_{k2}\right)}^2}}}$$

(11)

d_k1, d_k2—Maximum and minimum circularities of the two-dimensional cross-sectional shape perpendicular to the rolling axis.

ψ₁, ψ₂—Maximum and minimum rolling friction coefficients corresponding to the shape.

I₁, I₂—Maximum and minimum moments of inertia corresponding to the shape (kg/m²).

R₁′, R₂′—Minimum circumscribed circle radii corresponding to the maximum and minimum circularities of the shape (m).

Similarly, the maximum and minimum rolling energies of the rolling stone (E₁ and E₂) can be obtained by Formulas (A4) and (A5).

Specific calculation parameters for the test blocks are provided in

Table 4. Calculation parameters for test blocks.

Test Block	λ	R1′/R2′ (mm)	dk1/dk2	I1/I2 (×10−4)	ψ1/ψ2
A	1	35	1	2.2	0
B	1	40	0.9	2.29	0.36
C	1	50	0.9	2.28	0.34
D	0.56	25/51.5	1/0.87	1.38/5.74	0/0.44
E	0.51	31.8/49.4	0.90/0.86	1.5/3.63	0.36/0.48
F	0.86	37/40.6	0.9/0.89	2.03/2.61	0.36/0.38
G	0.59	26.1/52.3	0.97/0.71	1.2/5.9	0.1/0.45
H	0.55	25.1/52	0.99/0.69	1.12/5.78	0.06/0.44
I	0.27	50	0.96	4.11	0.17
J	0.31	47	0.95	4.06	0.13
K	0.34	44	0.99	4.09	0.04
L	0.31	45	1	4.51	0

.

4. Experimental Data Comparative Analysis

Calculation of Speed v₁, v₂, and Comparison with Experimental Values

For blocks with λ < 0.33, their rolling speeds can be calculated using Formula (10), and the calculation results are compared with experimental results, denoted as the experimental speed values and theoretical speed values for each block. The calculation results are denoted as the block with index’, and the connecting lines are dashed lines. The experimental results are denoted as the letters of the test block, and the connecting lines are solid lines.

In Figure 23, Figure 24 and Figure 25, the maximum error between the theoretical values and experimental values for blocks of different shapes is 10%, with an average error of 5.126%. Through the analysis of experimental data, the reasons for the occurrence of errors are summarized as follows:

(1)

In the process of formula derivation, sliding friction losses during the rolling process were not considered. However, it was observed in the experiments that as the slope angle increases, some blocks with relatively high rolling friction coefficients (e.g., the error for Block B at a release height of 0.45 m and a slope of 45° is 3.39%. When the slope increases to 60°, the error becomes 7.87%.) tend to experience a small amount of sliding friction when rolling on the slope. These blocks have higher overturning moments, and when the slope angle approaches the sliding friction angle, the forces acting on the slope surface after rolling can result in a small amount of sliding friction. This phenomenon gradually decreases with increasing rotational kinetic energy.

(2)

For some blocks with relatively low rolling friction coefficients, during the process of shifting direction until stabilization, the two-dimensional cross-section used for calculation is not exactly perpendicular to the actual rolling axis, leading to a slightly larger actual rolling resistance than the calculated value. For example, the error for Block D at a release height of 0.6 m and a slope of 45° is 9.4%. For Block L under the same conditions, the error is 4.59%. And during the directional shift in the rolling process, the two-dimensional shape that is approximately perpendicular to the rolling axis resembles an ellipse, as shown in Figure 26. However, calculations are still performed assuming a circular shape, resulting in errors. In contrast, for spherical blocks, the roundness of any cross-section is one, so even if there is a directional shift during rolling, the two-dimensional cross-section perpendicular to the rolling axis remains consistent with the one used for calculation, resulting in smaller errors between the calculated and theoretical values.

(3)

For some blocks with small contact areas with the slope surface and a high center of gravity, rocking may occur during the rolling process, consuming some energy. For example, blocks I, J, K, and L exhibit this phenomenon, and it becomes more pronounced as the contact area with the slope surface decreases. The position of the rolling axis oscillates with the change in the center of gravity, causing the entire rolling process to be unstable. However, for blocks rolling on either side, the rotational inertia keeps them in a balanced state, as shown in Figure 27. In contrast, blocks J and K, which have the same circularity and volume as blocks G and H, do not exhibit rocking during rolling because they have larger contact areas with the slope surface and a lower center of gravity. As shown in Figure 28, no energy loss occurs due to rocking during rolling, reducing the error between the calculated and experimental values.

Figure 26. Deviation of the rolling axis during block rolling.

Figure 26. Deviation of the rolling axis during block rolling.

Figure 27. Test block sways left and right during rolling.

Figure 27. Test block sways left and right during rolling.

Figure 28. Block rolling without swinging, maintaining the initial rolling axis.

Figure 28. Block rolling without swinging, maintaining the initial rolling axis.

For blocks with 0.33 < λ < 1, it is necessary to consider the situation where the rolling axis changes during the rolling process. Therefore, for such blocks, both the high and low values of the calculated speeds are given using Formulas (10) and (11), denoted as the block with index ′ and the block with index ″, as shown in the figures below.

In Figure 29, the experimental speed of Block D closely matches the theoretical high-speed value, and the average error between the experimental values and the theoretical high values is 4.92%. This is because Block D has a significantly different rolling resistance arm, moment of inertia, and minimum circumscribed circle radius for its two rolling cross-sections. When such-shaped blocks roll on a flat slope, a longer rolling distance is required to change the rolling axis, so the entire process mainly involves rolling around the axis with a lower rolling resistance. The actual speed from a 0.6 m height release is situated between the high and low theoretical values. During the actual rolling process of the model experiment, there was a slight deviation in the rolling axis, as shown in Figure 26, but the deviation was small, and the block did not complete the change of the rolling axis after rolling off the slope. However, the process of changing the rolling axis involves an increase in the rolling resistance arm, which increases the energy consumption during rolling, leading to the rolling speed leaning towards the lower calculated value.

For the cases where the experimental speed is less than the minimum theoretical speed in Figure 30 and Figure 31, the reason is the energy loss caused by the slight jumping and rocking of Blocks E and F during rolling. This characteristic is similar to that of disc-shaped blocks. However, due to the similar moment of inertia and rolling resistance arm for rolling around the different axes, the rotational inertia required for rolling around the initial rolling axis cannot be maintained. During the rolling process, these blocks go through several consecutive bounces and swings, completing the axis change process at a position before reaching the slope bottom, as shown in Figure 32. With the change in the rolling axis and the axis itself, the rolling resistance arm, deviation, and rolling speed of the block vary to different extents. In addition, comparing the errors between the theoretical and experimental values for Block B and Blocks E and F under the same conditions, it is observed that as the block’s aspect ratio increases, the number of axis changes during the rolling process also increases, showing a positive correlation. Therefore, for such shaped blocks, the actual rolling speed may lean towards the lower calculated value or even slightly below it. However, this was not the case for test Blocks G and H, as shown in Figure 33 and Figure 34.

For these types of blocks, it is possible to determine the two-dimensional cross-sectional shape perpendicular to the slope surface during the process of changing the rolling axis and to calculate the corresponding rolling friction coefficient. This coefficient can then be substituted into the calculation formula to obtain the corresponding rolling speed. The calculation results are shown in Figure 35 and Figure 36, where the experimental values for Blocks E and F fall within the range of the theoretical high and low values. Among them, the average difference between the actual value of the test block and the high and low values is E (6.7%, 14%) and F (6.3%, 14.3%), respectively.

Based on the above analysis, for blocks with 0.33 < λ < 1, the experimental speeds obtained from the model tests are close to the high and low theoretical speeds calculated by the formulas. Experimental data were compared with the results from the theoretical formulas, revealing a maximum error of 10% and an average error of 5.126%. Discrepancies between the experimental and theoretical values were analyzed, and corrections were made to address these errors. The experimental results fell within the theoretical formula range, confirming the validity of the theoretical calculation formulas for describing block motion characteristics.

5. Discuss

5.1. Countermeasures When There Is Randomness in the Calculation Parameters

When encountering discontinuity in the slope roughness and slope in practical applications, the rolling situation of the test block on the slope can be segmented according to the changes in roughness and slope (n), where n is taken based on the changes in roughness and slope. As shown in Figure 37, Formulas (A2) and (A3) are used to calculate the shortest and longest rolling distances S_1n and S_2n of fallen rocks on different rolling slopes, and the rolling distances and corresponding slope θ_n and roughness μ_n are substituted into Formulas (10) and (11), and can be obtained using:

$$v_1=\sqrt{\frac{2\left[mgh-S_{1a}({\psi }_1+{\mu }_a)mg\cos {\theta }_a-S_{1b}({\psi }_1+{\mu }_b)mg\cos {\theta }_b-\dots -S_{1n}({\psi }_1+{\mu }_n)mg\cos {\theta }_n\right]}{m+\frac{I_1}{{\left({R^{\prime }}_1{d}_{k1}\right)}^2}}}$$

(12)

$$v_2=\sqrt{\frac{2\left[mgh-S_{2a}({\psi }_2+{\mu }_a)mg\cos {\theta }_a-S_{2b}({\psi }_2+{\mu }_b)mg\cos {\theta }_b-\dots -S_{2n}({\psi }_2+{\mu }_n)mg\cos {\theta }_n\right]}{m+\frac{I_2}{{\left({R^{\prime }}_2{d}_{k2}\right)}^2}}}$$

(13)

5.2. Countermeasures When There Is Randomness in the Calculation Parameters

The calculation formula can be used to calculate the speed and energy of rockfalls in the continuous rolling stage but is not applicable to the stages of continuous sliding, jumping, and free falling, as well as the situation where rockfalls break during the rolling process. For potential threats of rockfall on unobstructed and pot-holed flat sloping surfaces, after mastering the location, three-dimensional size, density, slope roughness, and other information of the rockfall, the formula can calculate the speed and impact energy range of the rockfall as it rolls off the slope. At the same time, combined with the maximum offset of the rockfall, the rolling range of the rockfall and the rolling speed of the falling stone at any position within this range can be predicted.

5.3. Comparison of Rockfall Rolling Speed Calculation Models

Existing rockfall calculation models can be broadly categorized into empirical formula methods, kinematic principles methods, and experimental data analysis fitting methods. Among them, Hu [47] proposed resistance characteristic parameters based on field experience and obtained the rockfall’s motion speed based on the release height. Zhao et al. [48] provided a calculation formula for the rolling speed of rockfall on a slope based on functional principles. However, they simplified the rockfall to a rigid and uniform sphere, without considering the influence of rotational inertia. The formula takes into account the impact of rockfall shape on the rolling speed and is suitable for spherical or nearly spherical rockfalls, providing their rolling speed on the slope. However, it is not applicable to non-spherical test blocks. Sardana et al. [49] conducted regression analysis on the data obtained from indoor experiments and fitted a formula for the falling speed of rockfall. The formula includes release height and slope but does not consider the influence of rockfall shape. While the mentioned calculation models can determine the motion speed of rockfall, compared to the proposed model in this paper, they do not quantify the influence of rock block shape on the rolling speed, and the considered influencing factors in the formulas are not comprehensive enough.

5.4. Prospects, Limitations and Future Work

During the derivation of the formula, it is assumed that rigid test blocks experience pure rolling on a flat slope, without considering the possibility of collisions, jumping, and fragmentation during the rolling process. It also does not account for the presence of significant irregularities and obstacles on the slope, limiting its applicability. When estimating the rolling speed of rockfalls in practical engineering applications, the formula’s utility is constrained by the difficulty in ensuring the smoothness of natural slope surfaces. Collisions, jumps, and fragmentation on the slope can lead to actual rolling speeds leaning towards or even below the calculated low values. Additionally, for smaller-sized rockfalls, the rolling friction coefficient of the slope in the formula may not accurately quantify the impact of the slope on the rockfall’s rolling speed. In future research, this study will explore the mechanisms through which different shaped test blocks are influenced by collisions, jumps, and fragmentation, providing a formula that quantifies the relationship between rockfall shape and these motion characteristics.

6. Conclusions

(1) Through model experiments, the influence of the shape and size of rockfalls on their rolling speed was analyzed, and it was found that the shape of rockfalls is the main factor affecting their rolling speed. Under the same conditions, the rolling speed of rockfalls is positively correlated with the roundness of their two-dimensional rolling cross-section shape. For circular rockfall, the initial rolling state can be maintained during the rolling process under the action of rotational inertia, indicating that the rotational inertia of the rockfall also affects its rolling characteristics. At the same time, the influence of both the degree of the slope and the release height on different shapes of rockfalls is different.

(2) The relationship between rockfall shape and rolling resistance was analyzed. It was concluded that the magnitude of the rolling resistance depends on the rolling resistance moment in contact with the rolling surface during rolling, and the rolling resistance arm δ is determined by the size of the contact surface between the rockfall and the slope. By introducing the shape parameter ψ into the kinematic equation, the formulas for calculating the rolling velocity and kinetic energy of falling rocks with different shapes and sizes on different roughness slopes are derived.