Experimental Investigation of the Normal Coefficient of Restitution in Rockfall Collisions: Influence and Interaction of Controlling Factors

Abstract

Rockfalls pose significant threats to infrastructure, transportation routes, and human safety in mountainous regions, making them a critical concern in natural hazard and risk management. Accurate prediction of rockfall behavior is essential for designing effective mitigation strategies. The normal coefficient of restitution (R_n) is a key kinematic parameter for modeling falling rock dynamics, specifically quantifying the energy retained after collision between a rock and a slope surface. While this parameter is not directly used in prevention design, it is crucial for predicting the movement and trajectory of falling rocks and can indirectly support the development of more effective hazard mitigation strategies. However, R_n is influenced by multiple factors, including slope angle, surface material, falling rock shape, and initial velocity. The interactions among these factors make a precise prediction of R_n particularly challenging. Existing theoretical and empirical formulas typically consider individual factors in isolation, often neglecting their interactions, which leads to significant discrepancies in the results. To address this gap, we conducted a series of laboratory physical model tests to investigate the interactions among highly sensitive controlling factors and improve the accuracy of R_n prediction. A self-designed release apparatus, coupled with a high-speed recording and analysis system, was used to capture full kinematic data during rockfall collisions on slopes. This study not only examined how the main controlling factors and their interactions affect R_n but also developed a multi-factor interaction regression model, which was verified using on-site test data. The results show that the effect of the main controlling factors decreases in the following order: falling rock shape, slope surface material, initial velocity, and slope angle. Considering that falling rock shape and slope surface material cannot be quantitatively evaluated, the shape factor (η) and material factor (A_slope) are proposed to represent two controlling factors, respectively. Specifically, increases in η, A_slope, initial velocity, and slope angle are negatively correlated with R_n. Highly significant interactions were observed among falling rock shape–slope surface material, falling rock shape–initial velocity, falling rock shape–slope angle, slope surface material–initial velocity, and falling rock shape–slope surface material–initial velocity. These interactions mitigate the R_n reduction, resulting in a weaker effect than the stacking effect of the individual factors. The phenomenon is primarily attributed to the fact that high-level η, A_slope, initial velocity, and slope angle diminish the effect of intersecting factors. Finally, a comparison of the multi-factor interaction model with on-site tests and empirical formulas revealed the accuracy of the proposed model.

1. Introduction

Rockfalls, along with other large-scale mass movements such as landslides and debris flows, represent significant geohazards that pose considerable risks to infrastructure, transportation networks, and human safety in mountainous regions [1,2]. The movements refer to events involving extensive area coverage and high volume of displaced material, typically affecting multiple structures or large sections of land. These types of hazards are often characterized by their high energy release and rapid movement, making their accurate prediction critical for disaster management. Effective risk management for these hazards requires accurate prediction models that can simulate rockfall behavior, and R_n plays a key role in such models. R_n is a key parameter in rockfall dynamics, which quantifies the amount of kinetic energy retained after collision between falling rock and slope surface [3,4,5]. Specifically, R_n is defined as the ratio of the relative velocity along the normal direction before and after impact. R_n reflects how elastic or inelastic the collision is, and it plays an essential role in modeling the rebound and trajectory prediction of the falling rock after collision. While R_n is influenced by various factors, its accurate estimation is crucial for improving hazard risk assessments related to rockfall events. In this study, the term “rockfall” is used in two distinct contexts: (1) “rockfall” refers to “falling rock” itself, which is the object of study in terms of its movement and behavior during impact; (2) “rockfall” can also refer to the event or the area affected by the rockfall, including the slope surface over which the rock moves or the region where the collision takes place. In these contexts, we use terms like “rockfall” or “slope surface” to distinguish the process from the rock block itself. R_n is widely applied in the framework of the lumped mass method for simulating block dynamics, but its accurate estimation remains challenging [6,7]. R_n is influenced by several factors, each of which affects the energy dissipation and rebound behavior during rockfall collisions. These factors include (1) slope angle: the angle of the slope influences the impact velocity and the directional forces acting on the falling rock, affecting its kinetic energy retention; (2) slope surface material: different slope surface materials (e.g., concrete and turf) have varying frictional properties and energy absorption capacities, influencing how much energy is dissipated during collision; (3) falling rock shape: the geometry and roughness of the rock affect its contact area with the surface, which influences the energy dissipation and how the rock rebounds; and (4) initial velocity: the speed at which the rock is released influences the impact force and the subsequent energy dissipation. High initial velocities generally result in more inelastic collisions, leading to greater energy loss. The interactions among these factors complicate the estimation of R_n, making it difficult to predict precisely. The rockfall collision mechanisms refer to the physical processes and energy exchange that occur when falling rock collides with the slope surface. These mechanisms involve elastic and inelastic interactions, where the rock’s kinetic energy is partially conserved (elastic) or dissipated (inelastic) through friction, deformation, and compression. In the context of this study, the term also encompasses the contact dynamics between the rockfall and slope surface material, which are influenced by factors such as surface roughness, material hardness, and the angle of impact. These mechanisms are essential for understanding how the rock will rebound after collision, which in turn helps predict the rock’s post-impact trajectory and kinetic behavior. These challenges highlight the need for a more comprehensive approach to modeling R_n, which this study aims to address by investigating the interactions between multiple controlling factors and developing a more accurate prediction model.

Numerous studies, including theoretical analyses [8,9,10,11,12,13,14], laboratory experiments [15,16,17,18,19,20,21,22], and field tests [23,24,25,26,27,28], have been conducted to estimate R_n under various conditions. In theoretical analyses, several scholars have developed models for R_n based on classical theories. For instance, Johnson [8] and Thornton [9] respectively derived the formulas for estimating R_n under ideal elastic-plastic conditions based on Hertz contact theory. Similarly, He et al. [10] derived the normal impact velocity in the initial yield stage considering the mechanical properties of slope soil and proposed an estimation method for R_n. Moreover, Yang and Zhou [11] approximated the falling rock as an ellipsoid, derived velocity expressions of various falling rock kinematics and the corresponding R_n. In addition, Zhang et al. [12] employed the Logistic equation to fit the normal falling rock velocity curve, deriving R_n based on quasi-static contact mechanics theory. It is important to note that while this theory provides useful insights into rockfall collisions, our study does not directly apply this approach. Instead, our research focuses on experimental modeling of rockfall dynamics, primarily investigating the interactions between key parameters (such as slope surface material, falling rock shape, and initial velocity) to estimate R_n. Ye et al. [13,14] derived the R_n formula for spheres using viscoelastic contact theory and developed a viscoelasticoplastic contact model. While these theoretical models are valuable for understanding rockfall behavior, we have not adopted these models in our study. Instead, we focus on direct laboratory measurements of R_n and the interactions among main controlling factors, aiming to refine prediction methods and improve the accuracy of rockfall hazard assessment.

In laboratory tests, many researchers have investigated the influence of various factors on R_n focusing on the characteristics of falling rock, kinematics, and slope. For example, Peng [15] developed an empirical formula, incorporating Schmidt hardness and slope angle, to estimate R_n based on laboratory spherical falling rock free-fall tests. In the tests, rocks are released freely from a specified height to impact a flat surface or cushioning material, and the resulting rebound velocities are measured to calculate R_n. Meanwhile, Labiouse and Heidenreich [16] conducted half-scale falling rock free-fall tests on sandy slopes, which investigated the effects of slope angle, falling rock mass, shape, and falling height on R_n. The tests provided controlled conditions to study how different parameters influence R_n during the collision. Additionally, Aryaei et al. [17] examined the effects of falling rock size on R_n through laboratory spherical falling rock free-fall tests. The tests were designed to capture the behavior of the rockfall’s collision with the surface and its energy dissipation characteristics. Moreover, Asteriou and Tsiambaos [18] analyzed the effects of impact velocity, falling rock mass, and material hardness on R_n, and they also proposed a new semi-empirical formula based on the correlative factors. Similarly, Ji et al. [19] examined the interactions among seven factors and their nonlinear correlations with R_n. Liao et al. [20] performed laboratory falling rock free-fall tests to investigate the effects of slope angle, kinetic energy, and falling rock shape on R_n. The tests replicated the free-fall movement and impact behavior, allowing for more accurate modeling of energy dissipation during collisions. Tang et al. [21] studied the effects of sandy soil thickness, impact angle, falling rock mass, and falling height on R_n. Recently, Zhao et al. [22] conducted laboratory falling rock free-fall tests to investigate the effects of impact velocity, incident angle, and slope surface material on R_n. These studies have collectively advanced our understanding of how various factors influence R_n.

In on-site tests, Zhang et al. [26] validated the numerical simulation method based on on-site rockfall collision tests. Giacomini et al. [24] obtained an empirical formula for R_n related to the impact angle by fitting the on-site polyhedral falling rock test data. Wyllie [25] collected several on-site spherical rockfall collision tests to fit the empirical formula between impact angle and R_n. Zhang et al. [23] verified the accuracy of the numerical simulation method by using the results of the on-site rockfall collision tests. Duan and Sun [27] calibrated the theoretical model calculation results by taking the on-site rockfall collision tests. Ji et al. [28] determined R_n required in the numerical simulation of the falling rock disaster by using the results of the on-site rockfall collision tests.

It is generally recognized that R_n is closely related to falling rock and slope surface characteristics. However, the factors affecting R_n vary significantly across previous research. Most existing models for estimating R_n predominantly focus on individual factors, leaving the underlying mechanisms of the main controlling factors poorly understood. In this study, falling rock refers to the rock that is released from a height and collides with the slope surface. Its behavior, such as velocity and energy dissipation during collision, is influenced by several physical parameters, including density, shape, surface roughness, and material composition. Slope surface characteristics encompass factors such as slope angle, surface roughness, and irregularities in the surface. These characteristics influence how the rockfall interacts with the slope surface during collision, affecting R_n by altering the energy loss during the collision. Most existing models for estimating R_n focus on individual factors in isolation, neglecting the interactions between these factors, which leads to significant deviations and contradictions in the results. This study addresses these gaps by investigating the interactions among the main controlling factors and improving the accuracy of R_n prediction. For example, Asteriou and Tsiambaos [18] employed three empirical formulas to calculate R_n with errors of up to 65%, and Zhao et al. [22] estimated R_n empirical formulas to derive error results of up to 80%. Therefore, it remains a critical challenge to develop a practical and accurate estimation method that accounts for multi-factor interactions and can be validated in both laboratory and field conditions. This study aims to address this gap. This research employs laboratory physical model tests to better quantify the effect of the falling rock, slope, and kinematics characteristics on R_n. Seven relevant factors are identified from the aforementioned research results. Initial falling rock conditions are generated using a self-designed release apparatus, while kinematic data during rockfall–slope collisions are captured and analyzed with a high-speed recording and analysis system. The main controlling factors were identified, and their interaction mechanisms were elucidated. A new estimation model for R_n, incorporating multi-factor interactions, is proposed and validated through field tests. The proposed model is expected to enhance the reliability of R_n estimation and provide a solid foundation for designing falling rock prevention and control strategies on slopes.

2. Materials and Methods

The laboratory experiments were designed to replicate the full rockfall movement process, starting from the controlled release of the falling rock to its final resting state. The system includes a release apparatus, variable slope configurations, and high-speed movement capture to track every stage of the rockfall.

Although the data acquisition covers the entire movement sequence, our primary analytical interest lies in the impact event—specifically, the moment of collision between the rock block and the slope surface. By analyzing the kinematic behavior during this phase, we determine the R_n and investigate the influence of interacting factors such as slope surface material, slope angle, falling rock shape, and initial velocity.

In this study, the term “rockfall kinematics” refers to the movement of an individual falling rock, particularly its pre-impact trajectory, velocity, and post-impact rebound path and velocity. These parameters are used to calculate R_n, which reflects the energy retention in the normal direction. The experiments do not involve multi-block interactions or full run-out simulations but rather isolate the rockfall–slope collision under controlled conditions to investigate energy dissipation and movement characteristics.

2.1. Experimental Apparatus

To investigate the influence of multiple interacting factors on R_n, a physical falling rock model system was developed. This section describes the experimental apparatus used to carry out the laboratory tests. The apparatus was designed to simulate rockfall collisions under controlled conditions, enabling the adjustment of key parameters such as initial velocity, slope angle, and slope surface material. The setup provided a reliable and repeatable platform for capturing high-resolution kinematic data during rockfall–slope collisions.

The experiment utilized a falling rock model system. The system included several components: a test bench, a fill slope, a release apparatus, a high-speed camera, and a data analysis system, as shown in Figure 1. The model bench measured 3.5 m in length, 2.0 m in width, and 2.5 m in height. The model slope can be adjusted to a predetermined angle by artificially cutting and compacting, with the surface covered with slope surface material. Both the front and rear panels were made of acrylic. To enhance visibility during recording, the panel was painted white to differentiate the slope surface from the falling rocks in the footage.

A falling rock release apparatus, designed specifically for the experiment, is shown in Figure 2. The apparatus allows for adjustments to both the initial speed and angle of the falling rock. The initial velocity was controlled by rotating the handwheel to adjust the spring compression, which was converted into the initial kinetic energy of the falling rock. The impact angle was set by adjusting the bracket and angle adjustment lever.

2.2. Selection of the Related Factors

R_n is affected by the characteristics of falling rock, slope, and kinematics [29]. This study chose seven key factors related to R_n based on previous research [7,30,31].

(1)

Shape factor of falling rock

In this study, the term “shape factor” refers specifically to the quantification of the rock’s physical shape, particularly its degree of angularity or roundness [19]. η helps describe how the shape of the rock influences its kinematic behavior during the collision with the slope. η is used to classify rockfalls based on their shape characteristics, which are defined in terms of surface area distribution and geometric irregularity. η directly affects R_n, as it influences the amount of energy retained during impact. The four typical rock shapes selected in this study were based on different sphericity degrees, ranging from highly rounded blocks to more angular blocks, as shown in Figure 3. R_n was then used to effectively characterize how these different η affect the rockfall’s collision with the slope surface. The specific estimation formula is provided in Equation (1).

$$\eta ={\displaystyle \frac{1}{\psi \gamma }}={\displaystyle \frac{3S_0}{S\times \left\{\frac{d_{\min }}{d_{\max }}+\frac{d_{\min }}{d_{\mathrm{mid}}}+\frac{d_{\mathrm{mid}}}{d_{\max }}\right\}},}$$

(1)

where ψ is the degree of true sphericity, γ is the abnormity index, S₀ is the actual surface area of the falling rock, S is the surface area of a sphere of the same volume as the falling rock, d_max is the longest dimension of the falling rock, d_min is the smallest dimension of the falling rock perpendicular to d_max, and d_mid is the dimension of the falling rock both perpendicular to d_min and d_max.

(2)

Slope surface material

The four slope surface materials selected for this study were concrete, wood, sand, and artificial turf, each representing a different category of slope surface types, as shown in Figure 4. These materials were chosen to simulate various natural slope conditions, including both hard and soft surfaces. Specifically, concrete was selected as a rigid, non-deformable surface that simulates highly compacted surfaces. Wood was used as a medium-rigidity, smooth-textured material, representing moderately soft and smooth surfaces often found in engineered slopes. Sand was chosen for its loose, granular texture, which simulates the frictional interaction in loose material environments. Turf was selected to mimic soft, energy-absorbing natural surfaces, such as those found in vegetated areas. Unlike rigid materials like plastic, turf was chosen because it provides better energy dissipation during collision, thereby more accurately simulating rockfall interactions with natural surfaces.

To better quantify the differences in material behavior, a material factor (A) was introduced to reflect the mechanical properties of each surface. The formulation and corresponding parameters are provided in Equation (2) and

Table 1. The physical parameters and material factor.

Material	Type	E/MPa	SH	υ	A
Slope surface	Concrete	50.0	17.2	0.17	0.1295
	Wood	35.0	16.0	0.20	0.1996
	Sand	15.0	11.3	0.25	0.3454
	Turf	7.8	6.6	0.24	0.4340
Falling rock	Granite	94.6	44.7	0.15	0.7149
	Marble	62.4	38.2	0.17	0.7979
	Limestone	37.3	36.1	0.22	0.8695
	Sandstone	19.6	30.7	0.25	1.0279

.

It is worth noting that the same material factor introduced in Table 1 is applied to both the slope surface material and the falling rock type. This shared parameter reflects the mechanical interaction between the rockfall and the slope during collision and is used to ensure consistency in evaluating surface-related effects across different experimental conditions.

$$\begin{array}{c}A=0.4173-0.0952\times \lambda +0.0442\times \mathit{SH},\\ \lambda =\upsilon E\times {\left[\left(1-2\upsilon \right)\left(1+\upsilon \right)\right]}^{-1}\end{array}$$

(2)

where λ is the Lamé constant, υ is the Poisson’s ratio, E is the elastic modulus, and SH is the Schmidt hardness.

(3)

Initial velocity

To investigate the effect of initial velocity on v, four velocities (0 m∙s⁻¹, 2 m∙s⁻¹, 4 m∙s⁻¹, and 6 m∙s⁻¹) were selected considering the initiating velocity of perilous rock in mountainous areas [13]. The initial velocity was controlled using a self-designed release apparatus to ensure the accuracy and reliability of the experiment.

(4)

Slope angle

The field investigation of the high-frequency falling rock hazard area led to the generalization of the slope angle range from 20° to 80°. Four horizontal slope angles (30°, 45°, 60°, 75°) were selected for the test [32], as shown in Figure 5.

(5)

Falling rock type

Four rock types—marble, granite, limestone, and sandstone—were selected based on their geological structure characteristics, with material factor A used to characterize falling rock type. To differentiate between the slope surface material and the falling rock type, A_slope represents the former, while A_rock represents the latter. Table 1 presents the physical parameters and material factors for each falling rock type.

(6)

Falling height

Different falling heights (h) affect falling rock velocity and energy dissipation [21]. Four falling heights of 0.625 m (1/4 height of bench), 1.25 m (1/2 height of bench), 1.875 m (3/4 height of bench), and 2.5 m (The height of bench) were selected, respectively.

(7)

Falling rock mass

Falling rock mass (m) is a main factor in disaster prevention and mitigation design [7]. Considering the scale effect of the medium-scale model test, four samples with different masses (0.44 kg, 0.65 kg, 0.93 kg, and 1.27 kg) were selected for the experiment.

Therefore, the seven influencing factors and the actual levels are shown in

Table 2. Seven influencing factors and their actual levels used in this study to model rockfall behavior. The actual levels represent the quantitative settings of each factor, indicating the range of values tested in the experiment. These levels were chosen to evaluate the effects of individual and interacting factors on R_n.

Type	Actual Levels *
Shape factor of falling rock (η)	1	1.167	1.242	1.390
Material factor (Aslope), slope surface	0.1295	0.1996	0.3454	0.4340
Initial velocity (v/m·s−1)	0	2	4	6
Slope angle (θ/rad)	0.5236	0.7853	1.0472	1.3090
Material factor (Arock), rock type	0.7149	0.7979	0.8695	1.0279
Falling height (h/m)	0.625	1.25	1.875	2.5
Falling rock mass (m/kg)	0.44	0.65	0.93	1.27

* Shape factor of falling rock (η): the index of the falling rock, with higher values corresponding to more angular rocks and lower values to rounder rocks. Material factor (A_slope): reflects the rigidity and deformability of the slope surface material. Higher values represent softer, energy-absorbing materials like turf, while lower values correspond to rigid surfaces like concrete. Initial velocity (v): the speed at which the rock is released, representing kinetic energy at the moment of impact. Slope angle (θ): the angle of the slope with respect to the horizontal, affecting the impact velocity and directional forces. Material factor (A_rock): characterizes the rigidity and hardness of the falling rock. Higher values represent denser and harder rocks. Falling height (h): the vertical distance from which the rock is released, affecting the initial impact velocity. Falling rock mass (m): the mass of the falling rock, influencing the magnitude of the forces involved in the collision.

, where the slope angles are converted to radians.

2.3. Testing Design

Conducting full-factor tests is challenging due to the weak influence of low-sensitivity factors and their interaction on R_n. Based on this, sensitivity analyses of R_n for seven typical factors were conducted using the control variable method, requiring 28 test groups. The main and secondary controlling factors were identified, and each independent analysis was performed for the main controlling factors. The interaction analyses were then conducted, and orthogonal theory was applied to design the experiments [33,34,35], using the L16(4⁴) orthogonal design table shown in

Table 3. Orthogonal experimental design.

Test Number	η	Aslope	v/m·s−1	θ/rad
1	1	0.1295	0	0.5236
2	1	0.3454	2	1.3090
3	1	0.4340	4	0.7853
4	1	0.1996	6	1.0472
5	1.167	0.1996	0	0.7853
6	1.167	0.4340	2	1.0472
7	1.167	0.3454	4	0.5236
8	1.167	0.1295	6	1.3090
9	1.242	0.3454	0	1.0472
10	1.242	0.1295	2	0.7853
11	1.242	0.1996	4	1.3090
12	1.242	0.4340	6	0.5236
13	1.390	0.4340	0	1.3090
14	1.390	0.1996	2	0.5236
15	1.390	0.1295	4	1.0472
16	1.390	0.3454	6	0.7853

. In total, 37 experimental groups were conducted, excluding 7 replicate sets. To ensure comparability, each experimental group started from the same reference point and start mode. Additionally, any damage to the falling rock specimen or slope surface material, along with changes in test factors and levels, was promptly addressed.

To clarify the interaction response of the main controlling factors on R_n, a third-order polynomial regression model was adopted, considering the existing interactions among the factors [36], as follows:

$$y=a_0+{\displaystyle \sum _{i=1}^n{a}_i{X}_i}+{\displaystyle \sum _{i=1}^{n-1}{\displaystyle \sum _{j=2}^n{a}_{\mathit{ij}}{X}_i{X}_j}}+ {\displaystyle \sum _{i=1}^{n-2}{\displaystyle \sum _{j=2}^{n-1}{\displaystyle \sum _{k=3}^n{a}_{\mathit{ijk}}{X}_i{X}_j{X}_k}}}+\epsilon ,$$

(3)

where X_i, X_j,…,X_k are the main controlling factors, y is the predicted value, n is the number of factors, i, j, and k are the index number of factor variables, a₀ is a constant, a_i is the first-order main effect, a_ij is the second-order interaction effect, a_ijk is the third-order interaction effect, and ε is the random error between predicted and measured variables [37].

To reveal the dimensionless characteristics of each main controlling factor within the variation range, the following transformation is applied to the factors:

$$x_i=(X_i-X_0)/\mathsf{\Delta }{X}_i,$$

(4)

where x_i is the conversion factor, X_i is the non-conversion factor, X₀ is the minimal value, and ΔX_i is the change factor. The main and interaction effects of the established model were determined using mathematical and statistical analysis software (RStudio, version 2023.03.0+386) [38].

2.4. Image Analysis and Estimations

A high-speed video system was used to capture the full movement trajectory of the falling rock during the experiments. The system employed a Photron FASTCAM UX50 high-speed camera (manufactured by Photron Inc., in Tokyo, Japan), with a maximum frame rate of 2000 frames per second and a resolution of 1280 × 1024 pixels. The minimum time interval between recorded frames was 5 × 10⁻⁴ s, and the maximum recording capacity of the device can reach up to 160,000 fps.

The camera was connected to and controlled via Photron FASTCAM Viewer (PFV, version 3.6.9.1) software, which enabled real-time video capture and playback of the entire movement process, as shown in Figure 6a. Prior to formal testing, the camera was calibrated to ensure horizontal alignment with the slope model. A trial recording was performed to verify the camera parameters and confirm the image clarity. During experiments, PFV enabled live monitoring of the falling rock movement, with zoom-in and zoom-out tools enabling on-site inspection of image quality. To ensure uniform lighting and minimize movement blur, a flicker-free LED illumination system was installed over the falling rock movement area. The videos were stored and later analyzed using Photron FASTCAM Analysis (PFA, version 1.4.3.0) software, as shown in Figure 6b. The movement analysis process involved grid-based segmentation and spatial calibration using fixed reference markers along the slope. A static coordinate system was established, and tracking points were manually defined on the falling rock specimens. Both PFV and PFA are commercial software [39]. PFV is used to control the camera and view the captured footage, and PFA enables detailed movement analysis, including calculations of displacement, velocity, and acceleration.

The centroid of each falling rock block was tracked semi-automatically using PFA software. From this, the displacement, velocity, and acceleration at each time step were calculated. Impact and rebound points were identified frame by frame based on the contact event and change in direction. All movement parameters were exported using the software’s built-in data output functions for subsequent analysis.

R_n was calculated based on the ratio of the normal components of the rebound and impact velocities, derived from the tracked movement data. An uncertainty analysis was also conducted: based on repeated trials, the positional uncertainty was estimated at ±2 pixels (approximately ±0.5 cm), velocity uncertainty at ±0.05 m/s, and the resulting R_n uncertainty at ±0.03. These uncertainties were considered in the data interpretation and comparative analysis. R_n was estimated as follows:

$$R_{\mathrm{n}}={\displaystyle \frac{v_{\mathrm{r},\mathrm{n}}}{v_{\mathrm{i},\mathrm{n}}},}$$

(5)

where v_r,n and v_i,n are the incident and rebound velocities in normal movement before and after the collision, respectively.

In addition to basic motion parameters, the image analysis was also used to calculate the impact angle and rebound angle of the falling rock. These angles were determined by analyzing the trajectory vectors immediately before and after the moment of collision, based on frame-by-frame displacement data. The angles, along with velocity components, were used to evaluate the energy dissipation characteristics of different slope materials.

Combined with the velocity vectors, these parameters contributed to a detailed understanding of the rock–slope interaction dynamics. All extracted data, including trajectories, velocity vectors, impact/rebound angles, and contact points, were used as inputs for calculating R_n and establishing the multi-factor interaction regression model.

3. Results

3.1. Determination of the Main Controlling Factors Influencing R_n

Figure 7 presents the influence of seven controlling factors on R_n, highlighting the sensitivity differences among them. The results demonstrate that R_n is significantly more sensitive to four controlling factors—falling rock shape, slope surface material, initial velocity, and slope angle—as reflected by larger ΔR_n values. These are categorized as the main controlling factors. The other three—falling rock type, falling height, and falling rock mass—exhibited relatively minor influence and are therefore classified as minor controlling factors. The visual contrast in ΔR_n across the bars emphasizes the hierarchy of sensitivity, forming the basis for selecting the main controlling factors for the subsequent multi-factor interaction analysis.

3.2. The Influence of Main Controlling Factors on R_n

Figure 8 shows that R_n decreases as η increases, with a 53.75% reduction. To facilitate interpretation, η is classified into two levels: Low-level η refers to rocks that are more rounded, near-spherical, and symmetrical in form, typically exhibiting smoother surface characteristics. High-level η, in contrast, corresponds to angular, irregular, and polyhedral shapes with more pronounced edges and surface complexity. This classification helps capture the influence of shape geometry on the rock’s impact behavior, particularly in terms of energy dissipation and the resulting normal coefficient of restitution (R_n). Notably, R_n for low-level η is significantly higher than that for high-level η. The collision behaviors captured by the camera reveal that the low-level η mainly exhibits point–face contact with the slope. In contrast, high-level η predominantly exhibits line–face and face–face contact, reducing the probability of point–face contact. Falling rock shape affects energy dissipation by altering the contact mode and deformation. As the contact areas between the rockfall and the slope surface increase, and the friction energy dissipation rises, this leads to high energy dissipation and a decrease in R_n.

Figure 9 demonstrates a negative correlation between slope surface material and R_n. A_slope is classified into two levels based on deformability and collision behavior of the slope surface material: Low-level A_slope refers to rigid, non-deformable materials, such as concrete, which simulate highly compacted, engineered surfaces with limited energy absorption capacity. These materials result in higher rebound velocities and less energy dissipation during collision. High-level A_slope refers to soft, deformable surfaces, such as turf, which simulate natural, energy-absorbing slope conditions, typically found in vegetated or loose soil environments. These materials provide greater energy dissipation during rockfall collisions, leading to lower rebound velocities. This classification helps characterize how different types of slope surface materials affect energy dissipation during rockfall collisions and how these effects subsequently influence R_n. When both the rockfall and slope surface interact, their combined effect determines R_n, with the final R_n being influenced by the interaction. R_n has a 45.57% decrease. The low-level A_slope has high strength, hardness, and deformation recovery, resulting in high R_n. However, as A_slope increases, collision contact areas increase (indicated by the dotted line in the plot) and deformation recovery decreases, resulting in low R_n. The mechanical properties of the slope surface material influence energy transformation and dissipation. Materials with high strength and hardness exhibit strong deformation recovery, reducing plastic deformation energy dissipation and enhancing R_n. However, materials with low strength and hardness have weak deformation recovery, resulting in increased plastic deformation energy dissipation and lower R_n.

Figure 10 shows that R_n decreases as v increases, with a 35.47% reduction. The falling rock trajectory reveals that at low-level v, despite multiple collisions, the energy dissipation per collision remains low. As v increases, the higher impact energy leads to greater plastic deformation energy dissipation, resulting in a decrease in R_n. The increase in v directly raises impact energy, which leads to greater deformation, higher energy dissipation, and a significant decrease in R_n.

As shown in Figure 11, the slope angle is negatively correlated with R_n, with an 11.46% decrease. At the same height, low-level θ has a small normal force component, with energy dissipation primarily occurring through friction. At high-level θ, the falling rock movement distance is shortened, the increased normal force component leads to greater impact energy dissipation, resulting in a decrease in R_n. High-level θ increases the normal velocity component, converting more energy into normal impact energy or other unrecoverable forms, thereby increasing energy dissipation. Consequently, R_n decreases as θ increases.

3.3. Establishment of R_n Estimation Model

According to the aforementioned research, considering only the effect of individual factors on R_n, a non-interaction regression model was developed in RStudio based on the main controlling factors as follows:

$$R_{\mathrm{n}}=1.3316-0.6002X_1-0.4583X_2-0.0323X_3+0.0009{\mathrm{X}}_4,$$

(6)

where X₁ is the η value, X₂ is the A_slope value, X₃ is the v value, and X₄ is the θ value.

In the non-interaction model analysis, falling rock shape, slope surface material, and initial velocity were found to have a much greater influence on R_n than slope angle. However, this contrasts with the sensitivity analysis results. The R_n prediction relying only on individual factors was inaccurate, and significant interactions among main controlling factors may exist. The non-interaction regression model, which neglects the effects of interaction on R_n, amplifies the effect of individual factors, failing to accurately reflect the actual influence of the main controlling factors. Therefore, it is essential to study the interactions among the main controlling factors and establish a comprehensive quantitative model.

Combined with the theoretical basis of interaction analysis, a polynomial regression model was established. The model analysis results are shown in

Table 4. The analysis of the regression model.

Explanatory Variables	Estimating Coefficients	Standard Error	t Values	p Values (Pr > |t|)	Significance
Intercept	0.7286	0.0099	73.352	<2 × 10−16	***
Falling rock shape	−0.4630	0.0247	−18.713	<2 × 10−16	***
Slope surface material	−0.3962	0.0453	−8.748	7.68 × 10−13	***
Initial velocity	−0.3723	0.0296	−12.571	<2 × 10−16	***
Slope angle	0.0783	0.0257	3.045	0.00328	**
Falling rock shape–slope surface material	0.4945	0.0701	7.050	1.01 × 10−9	***
Falling rock shape–initial velocity	0.3987	0.0648	6.144	4.36 × 10−8	***
Falling rock shape–slope angle	−0.1824	0.0418	−4.367	4.27 × 10−5	***
Slope surface material–initial velocity	0.2705	0.0809	3.345	0.001326	**
Falling rock shape–slope surface material–initial velocity	−0.4841	0.1258	−3.847	0.000261	***
R2 = 0.9736

Asterisk is the significance level: 0 < ‘***’ < 0.001 < ‘**’ < 0.01.

.

From Table 4, the p-values of the one-factor explanatory variables (falling rock shape, slope surface material, initial velocity, and slope angle) are lower than 0.01, indicating that the one-factor explanatory variables significantly affect the response variables. For the interaction parameters, the two-factor explanatory variables (falling rock shape–slope surface material, falling rock shape–initial velocity, falling rock shape–slope angle, and slope surface material–initial velocity) are as significant as the three-factor explanatory variable (falling rock shape–slope surface material–initial velocity). Substituting Equation (4) to Equation (3), the transformation model that considers the interaction is derived as follows:

$$\begin{array}{l}{R}_{\mathrm{n}}=0.7286-0.4630x_1-0.3962x_2-0.3723x_3+0.0783x_4+,\\ 0.4945x_1{x}_2+0.3987x_1{x}_3+0.2705x_2{x}_3-0.1824x_1{x}_4-0.4841x_1{x}_2{x}_3\end{array}$$

(7)

where x₁ is the η value after transformation, x₂ is the A_slope value after transformation, x₃ is the v value after transformation, and x₄ is the θ value after transformation.

3.4. The Interactive Effects of Main Controlling Factors on R_n

3.4.1. The Interaction of Falling Rock Shape and Slope Surface Material

Figure 12 shows R_n decreases with η and A_slope increase. This decrease is 61.96%, which is lower than the combined independent effect of the two factors (74.83%). The interaction between falling rock shape and slope surface material is significantly weakened.

The effect of falling rock shape on R_n varies with A_slope. For low-level A_slope, high bounces occur, resulting in a 62.96% decrease in R_n (from blue to red) as η decreases. However, with high-level A_slope, low bounces occur, leading to a 25.89% reduction in R_n (from orange to red). In this case, the effect of slope surface material on R_n is less than half that of low-level A_slope. Low-level A_slope exhibits high deformation recovery, effectively restoring deformation energy dissipation and converting more kinetic energy into rebound kinetic energy. Combined with low-level η, energy dissipation is reduced, resulting in higher R_n. With high-level η, the increased contact area leads to higher friction energy dissipation, causing a significant decrease in R_n. However, the weak deformation recovery of high-level A_slope results in greater plastic deformation energy dissipation and lower R_n. Consequently, the influence of falling rock shape on R_n diminishes.

Additionally, the correlation between slope surface material and R_n varies with η. For low-level η, fewer contact areas cause a 48.67% reduction in R_n (from blue to orange) as A_slope changes. In contrast, for high-level η, more contact areas leads to only a 2.69% reduction in R_n (red remains constant), making the effect of slope surface material on R_n negligible. Low-level η has concentrated impact force and small contact area, and the ratio of friction energy dissipation is small. The material deformation recovery is weakened as A_slope rises, resulting in energy dissipation increases and R_n decreases. In contrast, high-level η has dispersed impact force and a large contact area, and the ratio of friction energy consumption is large. Falling rock shape dominates the energy dissipation, diminishing the influence of slope surface material on R_n. Therefore, the change in A_slope under high-level η has less effect on R_n.

In summary, high-level η significantly reduces the effect of slope surface material on R_n. However, high-level A_slope also diminishes the effect of falling rock shape on the coefficient. The interaction suggests that the effect of falling rock shape on R_n (−48.67%~2.69%) is more pronounced than that of slope surface material (−62.96%~−25.89%). When interacting with high-level η, the slope surface material has a little significant effect on R_n.

3.4.2. The Interaction of Falling Rock Shape and Initial Velocity

Figure 13 shows R_n decreases with η and v increase. This decrease is 69.07%, which is lower than the combined independent effect of the two factors (70.15%). The interaction between falling rock shape and initial velocity is slightly weaker than the independent superimposed effects.

The effect of falling rock shape on R_n varies significantly with v. At low-level v, weak collisions occur, causing a 58.33% decrease in R_n (from blue to orange) as η decreases. In contrast, at high-level v, heavy collisions occur, resulting in a smaller 35.02% reduction in R_n (from yellow to red). In this case, the effect of falling rock shape on R_n is less pronounced than low-level v. At low-level v, the collision kinetic energy is small, and changes in η substantially affect the contact area and friction force, influencing the energy dissipation. Under low-level η, rockfall has a smaller contact area, lower friction force, less energy dissipation, and higher R_n. However, under high-level η, rockfall has a larger contact area, increased friction force, and significantly increased deformation energy dissipation, resulting in a decrease in R_n. At high-level v, the collision kinetic energy increases significantly, leading to greater impact energy dissipation and diminishing the effect of falling rock shape on R_n as η rises. This occurs because high-level v causes more collision kinetic energy to be converted into deformation and impact energy, attenuating the effect of falling rock shape on R_n.

In addition, the correlation between initial velocity and R_n varies significantly depending on η. For low-level η, fewer contact areas occur, leading to a 52.40% reduction in R_n (from blue to yellow) as v increases. However, for high-level η, more contact areas occur, resulting in a smaller 25.77% reduction in R_n (from orange to red). In this case, the effect of falling rock shape on R_n is approximately half of that observed for low-level η. Under low-level η, there is less contact area, smaller local deformation, and smaller collision kinetic energy dissipation. Low-level v results in higher R_n. However, as v rises, the collision kinetic energy increases significantly, and the ratio of plastic deformation energy dissipation in the collision process increases significantly, leading to a decrease in R_n, and the high-level η has more contact area, resulting in larger local deformation energy dissipation. As v increases, the rising collision kinetic energy causes local deformation and friction energy dissipation, leading to a decrease in R_n. However, the effect of initial velocity on R_n is relatively small because the high-level η already causes large energy dissipation.

In conclusion, high-level η significantly reduces the effect of initial velocity on R_n. However, high-level v also diminishes the effect of falling rock shape on R_n. The interaction indicates that falling rock shape has a stronger impact on R_n (−52.40%~−25.77%) than initial velocity (−58.33%~−35.02%). The significant energy consumption caused by high-level η and v collectively reduces R_n when their effects are combined.

3.4.3. The Interaction of Falling Rock Shape and Slope Angle

Figure 14 shows that R_n decreases with increasing η and θ, but the reduction (51.58%) is less than the combined independent effect of the two factors (59.05%). This indicates that the interaction between falling rock shape and slope angle is significantly weakened.

The effect of falling rock shape on R_n significantly relies on θ. With low-level θ, weak collisions occur, resulting in a 64.25% decrease in R_n (from blue to red) as η decreases. However, with high-level θ, heavy collisions occur, leading to a 32.11% reduction in R_n (from green to orange). In this case, the effect of slope angle on R_n is less than that observed with low-level θ. At low-level θ, the normal velocity component is small, and the falling rock movement distance is long. Low-level η has low energy dissipation and high R_n due to the small contact area. In contrast, high-level η disperses the impact force, leading to increased plastic deformation energy dissipation and a decrease in R_n. At high-level θ, the normal velocity component increases, and the energy dissipation becomes complicated as η increases. However, the energy dissipation decreases due to the shortened falling rock movement distance. Therefore, the effect of falling rock shape on R_n is smaller than that of the low-level θ.

In addition, the correlation between slope angle and R_n varies significantly with η. With low-level η, few contact areas occur, resulting in a 28.68% decrease in R_n (from blue to green) as θ increases. In contrast, with high-level η, more contact areas occur, leading to a 35.42% decrease in R_n (from red to orange). In this case, the effect of falling rock shape on R_n is more than that observed with low-level η. Low-level η has fewer contact areas and concentrated impact force. The collision energy mainly exists in the elastic recovery form. Therefore, with θ increases, the normal velocity component gradually rises and R_n decreases due to increased normal impact energy dissipation. In contrast, high-level η has larger contact areas and dispersed impact force. The energy dissipation mainly occurs through local deformation and contact friction. When θ increases, the rising normal velocity component increases the ratio of plastic deformation energy dissipation. However, the shortened movement distance reduces energy dissipation and increases R_n.

To sum up, high-level η significantly enhances the effect of slope angle on R_n. Conversely, high-level θ diminishes the effect of falling rock shape on R_n. The interaction suggests that the effect of falling rock shape on R_n (−28.68%~35.42%) is more pronounced than that of slope angle (−62.25%~−32.11%). When interacting with high-level η, the influence of slope angle on R_n has changed, exhibiting the opposite effect compared to low-level η.

3.4.4. The Interaction of Slope Surface Material and Initial Velocity

Figure 15 shows R_n decreases with A_slope and v increase. This decrease is 62.74%, which is lower than the combined independent effect of the two factors (64.87%). The interaction between slope surface material and initial velocity was slightly weaker than the independent superimposed effects.

The effect of initial velocity on R_n significantly depends on A_slope. For low-level A_slope, high bounces occur, causing a 58.56% reduction in R_n (from blue to red) as v decreases. However, high-level A_slope results in low bounces, leading to a 27.14% reduction in R_n (from orange to red). In this case, the effect of slope surface material on R_n is less than half of that observed for low-level A_slope. Low-level A_slope exhibits strong deformation recovery, reducing the collision kinetic energy dissipation. Therefore, R_n is higher at low-level v. However, as v rises, the impact energy dissipation increases, resulting in a significant decrease in R_n. High-level A_slope has weak deformation recovery, leading to higher collision kinetic energy dissipation and lower R_n. With v increases, more collision kinetic energy is converted into plastic deformation energy dissipation, reducing the effect of the initial velocity on R_n.

Similarly, the correlation between slope surface material and R_n changes significantly with v. At low-level v, weak collisions occur, leading to a 48.86% decrease in R_n (from blue to orange) as A_slope increases. In contrast, at high-level v, heavy collisions occur, causing only a small 10.09% reduction in R_n (red remains constant). In this case, the effect of initial velocity on R_n is negligible. Under low-level v, the collision kinetic energy is small, and the slope surface material determines the energy dissipation. The deformation recovery of low-level A_slope reduces the energy dissipation, resulting in high R_n. The deformation recovery of high-level A_slope is weakened, and the increase in plastic deformation energy dissipation leads to a decrease in R_n. Under high-level v, the collision kinetic energy increases, and the ratio of unrecoverable energy dissipation, such as impact and friction energy, rises. Initial velocity dominates the energy dissipation, reducing the influence of slope surface material on R_n.

Given the above, high-level v significantly reduces the effect of slope surface material on R_n. Conversely, high-level A_slope also diminishes the effect of initial velocity on R_n. The interaction suggests that the effect of initial velocity on R_n (−48.86%~−10.09%) is more pronounced than that of slope surface material (−58.56%~−27.14%). This is because high-level v and A_slope lead to significant energy consumption, which minimizes R_n under their combined effects.

The two-factor interaction analysis reveals significant effects among falling rock shape–slope surface material, falling rock shape–initial velocity, falling rock shape–slope angle, and slope surface material–initial velocity, highlighting the interactions among the main controlling factors: falling rock shape, slope surface material, and initial velocity. However, the analysis alone fails to fully capture the more complex synergy effects among these factors. The multi-factor interaction model analysis further reveals a significant three-factor interaction among falling rock shape, slope surface material, and initial velocity. Consequently, the following section will focus on the three-factor interaction among these factors.

3.4.5. The Interaction of Falling Rock Shape, Slope Surface Material, and Initial Velocity

The study of the interaction mechanisms among the main controlling factors reveals significant interaction among falling rock shape, slope surface material, and initial velocity. As shown in Figure 16, the contour decreases with v increases under the interaction between falling rock shape and slope surface material. The interaction is consistent with the results of the two-factor interaction analysis, with the effect of falling rock shape being stronger than that of slope surface material. High-level v increases collision kinetic energy and reduces the kinematic behaviors, such as rolling, sliding and collision. When v is 6 m∙s⁻¹, the collisions between high-level η and A_slope result in R_n remaining constant. The local contact deformation areas increase, while the impact force decreases. At this point, the energy dissipation is mainly dominated by the falling rock shape and the initial velocity, while the effect of slope surface material on R_n is weakened, as shown in Figure 17.

By analyzing the effect of multi-factor interactions on R_n, it is found that falling rock shape influences the contact pattern and energy dissipation. The falling rock with low-level η, which is round or smooth, has a smaller contact area when it hits the slope. This means it slides more easily with less friction, allowing more of its kinetic energy to be preserved, resulting in a higher R_n. In addition, the falling rock with high-level η, with sharp edges or irregularities, creates a larger contact area with the slope, leading to more friction and greater energy dissipation during the collision. As a result, the rock retains less kinetic energy, resulting in a lower R_n. In natural geological contexts, this means that smoother rocks (like those that have undergone natural weathering) will rebound more efficiently when they hit the ground, while rougher, angular rocks will experience more energy loss and bounce less. This behavior is important when designing rockfall mitigation strategies, as the shape of the falling rocks and the type of slope surface they collide with play a key role in how far they travel after collision. The strength and hardness discrepancies in slope surface materials affect R_n. However, R_n is less sensitive to A_slope under high-level η. As v increases, the increase in impact energy dissipation reduces R_n, which reaches its minimum for high-level η. The normal velocity component increases with the angle, while R_n decreases for low-level η. In contrast, for high-level η, the movement distance is shortened, reducing plastic deformation energy dissipation and increasing R_n. The three-factor interaction analysis reveals the synergistic mechanism among the three main controlling factors and provides an accurate basis for predicting and controlling falling rock kinematic behavior. However, the model’s applicability and reliability need to be verified. Therefore, to assess the predictive ability of the estimation model in engineering, this study validates the proposed multi-factor interaction estimation model combined with on-site experiments, considering multi-factor conditions during falling rock movement.

3.5. Verification of On-Site Falling Rock Tests

An on-site rockfall collision test was carried out on a highway slope in Linwei District, Weinan City. This test aimed to evaluate the reliability of the multi-factor interaction regression model, as shown in Figure 18. The areas are typical for frequent falling rock occurrences and are influenced by gravity and geological conditions year-round. The slope height is about 8 m, with a slope angle of approximately 45°. The natural soil slope consists of sand material. The falling rock is released under the influence of gravity without initial velocity, and its collision process is recorded using the high-speed camera.

To validate the established multi-factor interaction model, the on-site test results for different falling rock shapes were compared with the estimated values of R_n from an empirical model developed by previous scholars, as shown in

Table 5. The estimation and comparison.

Estimation Methods	Falling Rock Shape		The Error Compared with the Experiment/%
	Sphere	Prism
on-site experiment	0.459	0.413	0	0
multi-factor interaction model	0.474	0.399	3.27	3.39
non-interaction model	0.574	0.474	25.05	14.77
Peng [13]	0.275	/	40.09	/
Giacomini [22]	/	0.116	/	71.91
Wyllie [23]	0.387	/	15.68	/

.

The results show that the estimated values of the multi-factor interaction model under different falling rock shapes exhibit small errors and high reliability. In contrast, the errors of the non-interaction model and the other three empirical models are large [15,24,25], especially the Giacomini empirical model, where errors are significant and deviate considerably from the on-site test results. The multi-factor regression interaction model is relatively reliable and improves accuracy by approximately 68.52% compared to the Giacomini empirical model, making it suitable for widespread application in engineering.

4. Discussion

Analysis of the experimental test results shows that falling rock shape is a main controlling factor affecting R_n. Low-level η experiences minimal local deformation due to the small contact areas and concentrated impact force. As A_slope, v, and θ increase, energy dissipation increases, leading to a decrease in R_n. High-level η involves large contact areas and significant local deformation energy dissipation. The energy dissipation is primarily controlled by falling rock shape. The influence of slope surface material and initial velocity on R_n weakens. As θ increases, the normal velocity component increases, raising the ratio of local deformation energy dissipation. However, energy dissipation decreases, and R_n increases due to the shorter movement distance. As η increases, frictional and plastic deformation energy dissipation rises, weakening the effects of slope surface material, initial velocity, and slope angle on R_n. This suggests that falling rock shape dominates the changes in R_n as η increases.

The effect of falling rock shape on R_n is influenced by its regulation of contact patterns and post-collision dynamic behavior: (1) Difference in contact pattern: Regular shape (sphere) primarily exhibits point–face contact, where normal energy is concentrated and dissipated in a controllably, promoting high R_n. Irregular shapes (cube, prism, and triangular block) lead to increased normal energy dissipation due to mixed line/face–face contacts, inducing tangential movements (sliding or rolling) that significantly reduce R_n. The discrepancies weaken the R_n sensitivity to other main controlling factors by altering the contact stress distribution. (2) Nonlinear dynamic behavior: Regular falling rock movement is dominated by bouncing, and its energy dissipation is approximately linear with respect to the initial conditions. Irregular rockfall triggers uneven energy dissipation and random trajectories due to complex movements such as sliding and rolling. This nonlinear behavior amplifies the shape effect in multi-factor interaction.

In falling rock disaster prevention and control engineering, falling rock shape plays a critical role in determining the contact areas between the rockfall and the slope surface. Based on geological feature analysis and potential kinetic risk coefficient assessment, active energy dissipation and intervention strategies, such as on-site fragmentation or migration, should be considered. The adoption of flexible protection network–slope shape synergistic control technology could alleviate the local stress concentration and energy concentration effects induced by irregular falling rocks.

5. Conclusions

(1)

The four main controlling factors—falling rock shape, slope surface material, initial velocity, and slope angle—exhibit decreasing effects on R_n. The analysis revealed a significant two-factor interaction: falling rock shape–slope surface material, falling rock shape–initial velocity, slope surface material–initial velocity, and falling rock shape–slope angle. Notably, a significant three-factor interaction was observed among falling rock shape, slope surface material, and initial velocity.

(2)

At high-level η, the falling rock shape interacts with other main controlling factors, the line/face–face collision energy consumption increases. This effect on R_n is weaker than that of low-level η interacting with slope surface material and initial velocity, but stronger than its interaction with slope angle. At high-level A_slope, the slope surface material interacts with falling rock shape and initial velocity is weakened due to a decrease in slope surface material strength and deformation recovery. The collision strength increases at high-level v, which reduces the effect of falling rock shape and slope surface material on R_n. The energy consumption discrepancy is mainly attributed to falling rock shape at high-level θ, while the effect of slope angle on R_n diminishes.

(3)

In three-factor interaction, the R_n sensitivity on A_slope becomes negligible as η and v increase. High-level η reduces the effect of slope surface material on R_n at high-level v.

(4)

The maximum error between the estimated R_n from the multi-factor interaction model and the field value is 0.014, corresponding to a 3.39% error. This indicates high accuracy compared to the results from the non-interaction model and other empirical formulas. The accuracy of R_n can be improved by up to 68.52% compared to the estimated results by the non-interaction model and other empirical formulas.