Numerical Analysis of the Dynamic Response of Concrete Bridge Piers under the Impact of Rock Debris Flow

Abstract

The impact and damage caused by debris flow on concrete bridges have become a typical disaster scenario. However, the impact disaster mechanism of debris flow on bridge structures remains unclear. This study focused on investigating the impact mechanism of debris avalanches on concrete bridge piers. By employing the discrete element numerical simulation method to examine the effect of debris on concrete bridge piers, the analysis explored the influence of three significant factors: the pier’s section shape, the impact distance, and the slope angle of the sliding chute. The discussions included the accumulation pattern of rock debris, the impact force on the pier, and the shear force and bending moment at the pier’s bottom, as well as the displacement and velocity response laws at the pier’s top. The results demonstrate that rectangularly shaped piers have a high efficiency in obstructing debris, leading to higher impact forces and internal forces on piers. Arched-shaped piers exhibit a short-duration, high-peak instantaneous impact from debris. Increasing the impact distance of the piers can significantly reduce the impact force of debris. The accumulation height of debris, pier impact force, and the pier’s bottom internal forces decrease and then increase with the increase in slope angles, with a 45° slope angle being the critical point for the transition of debris impact on piers. The results can provide references for the disaster prevention design of concrete bridge structures in hazardous mountainous areas.

1. Introduction

In the ecologically fragile and geologically complex mountainous areas of southwest China, unscientific engineering activities and extreme natural factors (such as heavy rainfall and earthquakes) are prone to triggering landslides in deep gullies and steep slopes. The high dispersion, high fragmentation, and high mobility [1,2] of rock debris can cause destructive damage to various types of engineering structures along the migration path [3], posing a significant threat to the safe operation of a large number of concrete bridges in such challenging mountainous regions.

The model for rock debris avalanche impact force can effectively represent the load characteristics of the impact force. Albaba et al. [4] studied the impact force of dry granular flow on rigid retaining walls using numerical analysis and the Discrete Element Method (DEM) and discussed the range of slope angles and the applicability of average pressures of debris flow. Jiang et al. [1] proposed a semi-empirical formula for the impact force of dry granular flow on retaining walls and validated the effectiveness of the formula through model experiments. Ashwood et al. [5] quantitatively analyzed the impact force of debris flow on rigid and flexible barriers through model experiments and proposed a load model using active earth pressure to simplify the calculation of debris flow impact force. Sui et al. [2] conducted model experiments on the impact of debris flow on rigid retaining walls, studying the normal and tangential impact forces of debris flow, as well as the impact height index. They established a load model for debris flow impact force and investigated the influence of different friction angles on debris flow impact force. The factors influencing the impact force of debris flow are numerous and complex. Jiang et al. [6] utilized the calibrated DEM to investigate the effect of disintegration of dry granular flow with various particle sizes on the transport characteristics of debris flow, revealing the energy dissipation mechanism during the movement of debris flow. Zhang et al. [7] conducted a parametric analysis using the DEM with debris flow landslide site conditions as variables. They explored the effects of different slope types, including linear, concave, and convex slopes, on the movement of debris bodies by varying the slope angle, accumulation area slope, and site geometric characteristics. Luo et al. [8] studied the effects of submerged sills of different shapes on the consumption of volume, velocity, and impact energy of debris flow. The results indicate that arc-shaped submerged sills have the highest energy dissipation. Kim et al. [9] conducted model experiments to study the dynamic impact of debris flows on cylindrical rigid baffles. Liu et al. [10] used the DEM–FEM coupled analysis method to quantitatively analyze the blocking effect of flexible protective nets on the impact of debris flows and proposed a three-stage analysis model for the impact of debris flow.

The high-speed motion of rock debris poses a significant impact hazard to downstream bridge structures [11]. Luo et al. [12] employed the LS-DYNA explicit dynamic analysis method to simulate the impact damage process of debris avalanches on reinforced concrete buildings. They evaluated the landslide impact risk of buildings, revealing the mechanism of rock debris impact disasters. Cheng et al. [13] conducted a debris flow impact model analysis on dual-column bridge piers in V-shaped gully areas, investigating the influence of pier layout and impact angle on the debris flow impact action. Chen et al. [14] replicated the initiation, acceleration, impact, and deposition processes of debris flows through scale model experiments, studying the law of impact action of debris flow distance and volume on bridge pier impacts. They established the relationship between maximum impact force and impact distance, as well as debris flow volume. Zhao et al. [15] conducted impact tests on reinforced concrete hollow piers to study the impact force, internal forces, and failure modes. They analyzed the effects of parameters including rock diameter, impact velocity, and pier reinforcement ratio on the dynamic response and damage modes of the piers. Xie et al. [16] investigated the dynamic behavior and vulnerability of piers with different cross-sections under the impact of rockfall using the Finite Element Method (FEM). Zhang et al. [17] obtained the probability distribution of rockfall impact parameters through Monte Carlo simulations and evaluated the safety of bridge structures under rockfall disasters by utilizing three-dimensional nonlinear dynamic analysis combined with rock movement characteristics. Li et al. [18] analyzed the impact force and damage response characteristics of hollow thin-walled piers under rockfall impact using LS-DYNA and proposed a structural damage assessment method based on a response surface model.

In order to ensure the structural safety of bridges in mountainous areas under rock debris impact, He et al. [19] conducted research on the impact protection of the Chediguan Bridge. They introduced a novel flexible energy-absorbing bridge pier protection device and utilized the non-linear analysis method to assess the device’s effectiveness. Zhong et al. [20,21] conducted full-scale concrete pier tests under rockfall impact to investigate the residual strength and damage evolution of concrete piers. They employed a DEM–FEM coupled numerical model to study the interaction between debris flow and bridge piers. Additionally, they conducted reliability analysis on the protective performance of the proposed steel–sand composite protection structure. Yan et al. [22] proposed a modular bridge pier collision protection device, which consists of a closed-cell aluminum foam-filled composite structure, and they evaluated the mechanical behavior and energy absorption characteristics of the protection device through quasi-static compression tests. Yu et al. [23] studied the effectiveness of protective measures including densely spaced stirrups, thickened plain concrete, and externally wrapped steel tubes against rockfall impact on concrete bridge piers based on the Holmquist–Johnson–Cook damage model, and the results indicated that the protective effect of externally wrapped steel tubes was the most effective. Gu et al. [24] analyzed the damage and failure modes of mountain bridges under rockfall impact using LS-DYNA. Liu et al. [25] investigated the impact resistance of strengthened piers by studying the composite reinforcement method of using a fiber-reinforced plastic grid and steel-reinforced ultra-high-performance concrete and conducted a parameter analysis.

Research on the impact of debris flows on piers primarily focuses on the development of rigid and flexible protective nets. However, there is a scarcity of studies on the load characteristics of debris flow impact on bridge piers, dynamic response laws of bridge piers, disaster-causing mechanisms, and others. Based on this, the present study focuses on the impact of debris flow on concrete bridge piers, using Particle Flow Code (PFC-3D 6.0) numerical simulations which were conducted on the chute experiments of rock debris impact concrete bridge piers. The study explores the influence of pier cross-sectional shape, impact distance, and slope angle on the debris flow, in addition to analyzing debris flow accumulation patterns, impact forces on the piers, internal forces at the bottom of the piers, and dynamic movement responses at the tops of the piers. The research results can provide references for the prevention and safety assessment of impact disasters on concrete bridge structures in mountainous areas.

2. Debris Flow Model

2.1. Contact Model

2.1.1. Linear Parallel Bond Model

To reflect the mechanical behavior of concrete bridge pier materials, the linear parallel bond model is used to simulate the bonding effect between discrete elements [26]. Figure 1 illustrates the spatial configuration and mechanical model of the linear parallel bond model, where pieces i and j refer only to the concrete balls that exhibit bonding effects. When the bonding effects are activated, a bonding material of certain dimensions forms between the balls. The interaction between the elements generates normal linear elastic pressure $F_{\mathrm{n}}^{\mathrm{l}}$, normal damping force $F_{\mathrm{n}}^{\mathrm{d}}$, normal bonding force ${\overline{F}}_{\mathrm{n}}$, tangential linear elastic shear force $F_{\mathrm{s}}^{\mathrm{l}}$, tangential damping force, $F_{\mathrm{s}}^{\mathrm{d}}$ tangential bonding force ${\overline{F}}_{\mathrm{s}}$, and bonding moments $\overline{M}$ that can withstand rotational deformation. However, it should be noted that the bonding component between the elements and the linear spring and damping components act mechanically at the same time and are not related to each other.

2.1.2. Linear Model

For the contact behavior between discrete rock debris and various components, a linear elastic contact model can be utilized [26]. Figure 2 illustrates the linear elastic contact model, where the pieces i and j represent the pebbles of the rock debris, facets of the chute and the base plate, and the balls of the concrete pier. When the linear elastic contact model is activated, normal elastic force $F_{\mathrm{n}}^{\mathrm{l}}$ and normal damping force $F_{\mathrm{n}}^{\mathrm{d}}$ are generated between the contact surfaces, while tangential elastic force $F_{\mathrm{s}}^{\mathrm{l}}$ and tangential damping force $F_{\mathrm{s}}^{\mathrm{d}}$ are simultaneously formed. Compared with the linear parallel bond model, it can be seen that the linear elastic contact model is essentially a degeneration of the linear parallel bond model, neglecting the bonding forces between the contact surfaces of the pieces.

2.2. Model Parameter Selection

The discrete element model of debris flow impacting piers mainly involves the materials of concrete and rock debris. The basic physical parameters of rock debris and concrete determine their mechanical contact behavior, which has an important influence on the motion behavior laws of discrete elements.

The concrete density of the bridge pier is 2500 kg/m³, with a strength grade of C30. The elastic modulus E_c is 3.0 × 10⁴ MPa, the shear modulus G_c is 1.26 × 10⁴ MPa, the compressive strength f_c is 20.1 MPa, the tensile strength ft is 2.01 MPa, and the material Poisson’s ratio v_c is 0.2 [27,28]. The selection of parameters for the rock debris material is referenced in the literature [8,29,30]. According to the design of the chute experiment conditions, the density is 2800 kg/m³, the elastic modulus E_d is 5.0 × 10⁴ MPa, the internal friction angle is 35°, and the Poisson’s ratio v_d is 0.2.

Table 1. Discrete element model parameters.

Material	Element Type	Contact Model	Contact Type	Parameters	Value
Concrete	ball	Linear parallel bond model	Ball-ball	Normal stiffness kn (MPa/m)	60
				Shear stiffness ks (MPa/m)	40
				Frictional coefficient μ	1.0
				$\overline{\sigma }$ (MPa)	60
				$\overline{c}$ (MPa)	60
				$\overline{\phi }$ (°)	50
		/	/	Damping ratio β	0.20
Debris	clump	Linear model	Pebble-ball	Normal stiffness kn (MPa/m)	100
				Shear stiffness ks (MPa/m)	75
				Frictional coefficient μ	0.25
			Pebble-pebble	Normal stiffness kn (MPa/m)	100
				Shear stiffness ks (MPa/m)	50
				Frictional coefficient μ	0.30
			Pebble-facet	Normal stiffness kn (MPa/m)	100
				Shear stiffness ks (MPa/m)	20
				Frictional coefficient μ	0.20 (0.00)
		/	/	Damping ratio β	0.20

Note: (1) The concrete “ball” material and the base plate “facet” are connected and fixed, ensuring that the position of the pier remains unchanged during the impact process. (2) The values in parentheses represent the friction coefficient between the pebbles and the side boards of the chute.

provides the specific material properties and contact parameters for the discrete elements.

2.3. Model Establishment

2.3.1. Rock Debris Clumps

Based on the investigation of rock debris landslides in mountainous areas, actual debris bodies are characterized by obvious angular shapes. Therefore, in the simulation experiment of debris impact on bridge piers, the flake debris and pyramid debris clumps shown in Figure 3 are utilized to simulate the angular shapes of the debris bodies. The flake debris and pyramid debris clumps are both assembled from four equally sized spherical particles, mixed in a 1:1 volume ratio, and the particle clumps have a Gaussian distribution of particle sizes, ranging from 4.5 cm to 7.5 cm in diameter. The effectiveness of this method has been confirmed by studies [7,31].

2.3.2. Concrete Bridge Piers

The concrete of the bridge piers is simulated using spherical particles, and the contact behavior between particles is modeled using a parallel bond model. The particle size distribution of the concrete material ranges from 1.2 cm to 8.0 cm. The model height of the concrete bridge is 1.5 m, with cross-sectional shapes including round-end, circular, rectangular, and square. Figure 4 shows the specific cross-sectional shapes and dimensions. Analyzing the impact force distribution patterns of bridge piers with different cross-sectional shapes, this study aims to investigate the influence of pier cross-sectional shapes on the impact effect of rock debris.

2.3.3. Experimental Conditions

The study focuses on the influence of the bridge pier cross-sectional shape, impact distance, and chute slope angle parameters on the impact effect of rock debris.

Table 2. Controlling parameters of the PFC model.

Parameter Name	Unit	Value
Cross-sectional shape	/	round-ended, circular, rectangular, square
Impact distance/l0	m	0.6, 0.8, 1.0
Slope angle/θ	Degree (°)	30, 45, 60

provides the loading conditions and values of different analysis variables.

2.3.4. Model of Rock Debris Impact on Bridge Piers

The schematic diagram of a chute model for testing the impact of rock debris avalanches on bridge piers is shown in Figure 5. The rock debris are concentrated in the material source area at the rear of the chute. By removing the constraint of the baffle components, the rock debris forms a downward flow along the chute under its own weight, creating an impact on the bridge pier components placed on the base plate.

3. Results Analysis

3.1. Cross-Section Shape of Piers

3.1.1. Accumulation Form of Rock Debris

The cross-sectional shape of bridge piers significantly influences the transport characteristics of debris flow. Moreover, the accumulation form of rock debris particles varies under the different cross-sectional shapes of bridge piers. Figure 6 shows the plan view of the debris accumulation for bridge piers with round-ended, circular, rectangular, and square cross-sectional shapes under the working conditions of l₀ = 0.6 m and θ = 45°.

From the analysis of Figure 6, it can be observed that due to the circular arc tip of the impact surface on round-ended and circular piers, the arc-shaped surface has a deflecting and accumulating effect on the debris flow, causing the rock debris to tend to spread towards both sides of the pier. Therefore, the plan view width of the debris accumulation is greater in Figure 6a,b, while it is smaller in rectangular and square piers. Furthermore, for rectangular and square piers, the debris flow encounters the pier face at an orthogonal angle, causing the debris to easily accumulate and stagnate at the front of the pier. This results in a relatively smaller planar accumulation area for the debris but increases the impact force of the debris on the pier.

For the circular pier in Figure 6b, due to the smaller cross-sectional area of the pier and lower obstruction to the debris flow, some debris may flow around the pier after impact, leading to localized accumulation on the rear side of the pier. However, it should be noted that the contact between the accumulated debris behind the pier has no influence on the force applied to the pier.

The cross-sectional shape of the piers effectively changes the trajectory of debris movement and the accumulation mode, while also having a significant influence on the accumulation height of debris in front of the pier. Figure 7 shows the accumulation height of debris in front of piers with different cross-sections under the condition of l₀ = 0.6 m and θ = 45°.

The rock debris accumulation heights in front of the piers are calculated at 41.3 cm for rounded-ended piers, 33.7 cm for circular piers, 44.0 cm for rectangular piers, and 37.7 cm for square piers based on the scale in Figure 7. The results indicate that the accumulation heights differ, with rounded-ended and rectangular cross-sections showing greater heights compared to circular and square shapes. This discrepancy can be attributed to the oversight of impact distance changes in the design of the impact model for various cross-sectional shapes. The round-ended and rectangular piers have a section length of 60 cm, while the circular and square piers have a section length of 30 cm. Under the premise of keeping the center position of the pier unchanged, the impact distance of the round-ended and rectangular piers is shortened by 15 cm, resulting in a greater accumulation height of debris in front of the round-ended and rectangular piers.

Additionally, the accumulation height of debris in front of piers with different cross-sections reveals that rectangular piers are 6.5% higher than round-ended piers, while square piers are 11.9% higher than circular piers. This is mainly due to the wider impact face of rectangular and square piers, resulting in a significant hindering effect on the movement of rock debris.

3.1.2. Impact Force on Piers

The impact force of debris on bridge piers is influenced by the shape of the pier’s cross-section, which is reflected in modifications to the impact face shape. Figure 8 illustrates both the theoretical analysis model and the discrete element simulation model for the arc-shaped impact face (representing a circular bridge pier) and the rectangular impact face (representing a square bridge pier). The impact faces are symmetrical in shape and can be represented by the impact forces in different regions of the impact face as p_Left, p_Middle, and p_Right. By placing impact force measuring balls on the impact face of the bridge pier, the impact forces of debris on various regions of the pier can be measured.

For simplification of analysis, this section focuses solely on analyzing and discussing the impact force results of the arc-shaped impact face (circular pier) and the rectangular impact face (square pier).

(1)

Horizontal distribution of impact force

The cross-section form of piers effectively changes the sliding path and accumulation pattern of rock debris, while also significantly influencing the impact force on piers. Figure 9 presents the time–history curves of impact forces within the height range of 0 to 0.1 m at the bottom of the circular and square piers under the condition of l₀ = 0.6 m and θ = 45°.

From Figure 9, it can be observed that the impact force on the bridge piers exhibits characteristics of short duration and high peak value in the early stage of debris impact (highlighted by the red dashed box). This is mainly due to the high impact velocity of particles at the front edge of the debris flow, resulting in a strong instantaneous impact force on the bridge piers. As time progresses, the impact force of debris on the piers rapidly increases. The peak impact force regions for both types of piers mainly persist in the range of 7 to 15 s, after which they gradually decrease. Piers with different cross-sections exhibit variations in impact force behavior. For instance, circular piers display a prolonged impact force duration in the central region (7 to 20 s), while square piers demonstrate a pattern of an initial increase in impact force, followed by a decrease in the central region before gradually rising again. This phenomenon can be attributed to the gradual increase in the height of debris accumulation in front of the square pier, where the lateral pressure from the accumulated mass becomes the predominant force on the piers during the later stages of impact. In contrast, circular piers with arc-shaped impact surfaces exhibit a decreasing trend in lateral pressure.

Comparing the impact forces at various measuring points on circular and square piers reveals that the trends of impact pressure variations at the left and right monitoring balls of circular piers are generally consistent. This consistency suggests that the impact forces of debris on circular piers typically follow a pattern of symmetrical and stable impacts. Conversely, there are notable disparities in the impact forces at the left and right monitoring balls of square piers, indicating a more random behavior in the impact of debris.

(2)

Vertical distribution of impact force

Figure 10 shows the vertical variation of the debris impact force time–history curves in circular and square piers under the condition of l₀ = 0.6 m and θ = 45°. Based on Figure 10, it can be observed that the impact force of debris on the pier decreases with increasing impact height, and the impact force is the greatest within the range of 0.1 to 0.2 m above the bottom of the pier. Furthermore, comparing the impact forces on circular and square piers, it can be observed that the impact force on circular piers exhibits a distinct peak effect with short duration and significant fluctuations, while the impact force on square piers mainly shows a gradual increase and later stabilizes.

3.1.3. Pier Internal Force Response

(1)

Method for calculating internal forces

The schematic diagram in Figure 11 illustrates the impact mode of rock debris on the bridge pier. According to the principle of superposition in structural force calculation, the shear force at the bottom of the pier under the impact of the rock debris can be expressed as:

$$Q_0=F_1+F_2+\cdot \cdot \cdot +F_n={\displaystyle \sum _{i=1}^n{F}_i}$$

(1)

The bending moment at the pier’s bottom can be expressed as follows:

$$M_0=F_1\cdot h_1+F_2\cdot h_2+\cdot \cdot \cdot +F_n\cdot h_n={\displaystyle \sum _{i=1}^n{F}_i\cdot h_i}$$

(2)

where Q₀ and M₀ are the shear force and bending moment at the pier’s bottom, F_i represents the impact force on the i-th segment of the pier, and h_i represents the central height of the i-th segment of the pier.

(2)

Internal forces at the pier’s bottom

Figure 12 shows the time–history curves of the shear force and bending moment internal forces at the bottom of circular and square piers under the condition of l₀ = 0.6 m and θ = 45°. Analysis of Figure 12 reveals that in the initial stage (approximately 0–7 s) of the rock debris avalanche event, the shear force and bending moment internal forces at the bottom of the square pier are smaller than those of the circular pier. As the impact time continues, the square pier with a wider front face forms a stable debris accumulation body. Subsequently, the rock debris indirectly impacts the pier through the accumulation body, leading to a larger shear force and bending moment at the bottom of the pier in the later stage of the impact event. Furthermore, although the internal forces of the square pier are generally larger than those of the circular pier, attention should be paid to the significant instantaneous internal force peaks of the circular pier in the early stages of the impact event when designing the anti-impact scheme for bridge pier structures.

3.1.4. Movement Response of the Piers

Under the impact of rock debris, the concrete pier exhibits deformation behavior, leading to movement responses at the pier’s top. Figure 13 illustrates the response curves for displacements at the pier’s top and the velocities for various cross-sections under the conditions of l₀ = 0.6 m and θ = 45°.

The analysis of Figure 13a shows that there are significant differences in displacement responses at the pier’s top for piers with different cross-sectional shapes. Circular piers exhibit the largest displacement response at the pier’s top, while rectangular piers show the smallest. This is mainly due to the differences in the inertia moments of the pier sections and the coupled effect of random impact forces from debris. In Figure 13, except for the circular pier, the top displacement of the piers gradually stabilizes in the later stages of impact events, indicating that the lateral pressure from the accumulated debris is the main force acting on the piers in the later stages of the impact event, and its variation shows a relatively small change over time. For circular piers, due to the relatively limited constraint of the impact face toward the movement of debris, the debris accumulation is unstable, leading to significant changes in impact force and displacement at the pier’s top in the later stages of the impact event. This conclusion is consistent with the analysis results of the impact force on the piers. Furthermore, for the velocity responses of the pier’s top with different cross-sections in Figure 13b, it can be observed that the circular pier has the largest velocity response at the pier’s top, while the rectangular pier has the smallest response. This conclusion is consistent with the displacement response results at the pier’s top.

The analysis of the impact of debris on piers with various cross-sections reveals that square and circular piers have distinct characteristics with rectangular and arc-shaped impact surfaces. Subsequently, the analysis primarily concentrates on the individual influence of distance and slope angle for square piers and circular piers.

3.2. Impact Distance

3.2.1. Accumulation Form of Rock Debris

The position of piers influences the patterns of movement and the accumulation of debris. Figure 14 shows the accumulation forms of debris for square piers under the conditions of θ = 45° and impact distances of l₀ = 0.6 m, l₀ = 0.8 m, and l₀ = 1.0 m, respectively.

According to Figure 14, the differences in the accumulation forms of debris under various impact distances are subtle but notable. With the increase in impact distance, the width of the planar debris accumulation decreases, resulting in a linear accumulation along the direction of fluid sliding. This is mainly attributed to the reduced obstruction effect of the bridge piers on debris movement when they are positioned further from the outlet of the sliding chute, consequently leading to a strip distribution of debris accumulation along the sliding direction. Furthermore, upon comparing the accumulation ridge lines of debris at different impact distances, it is evident that when the impact distances are 0.6 m and 0.8 m, there is a noticeable bulge of debris accumulation in front of the piers. However, at an impact distance of 1.0 m, the accumulation ridge line of debris tends toward a straight line, suggesting that when the impact distance from the bridge piers is greater, the debris accumulation appears in a state of natural sliding and rest.

Enhancing the impact distance reduces the obstruction effect of the piers on the movement of debris, leading to a decrease in the accumulation height of debris in front of the piers. Figure 15 illustrates the relationship between impact distances of 0.6 m, 0.8 m, and 1.0 m under the condition of θ = 45° and the corresponding debris accumulation height. The data in Figure 15 demonstrate a significant decrease in debris accumulation height as the impact distance increases. Notably, the highest accumulation height is observed in front of the rectangular pier, while the lowest is in front of the circular pier. This finding is consistent with the observations made in Figure 7 regarding the influence of the bridge pier’s cross-sectional shape on debris accumulation height.

3.2.2. Impact Force on Piers

Figure 16 shows the vertical distribution of impact forces on square piers at various impact distances under the condition of θ = 45°.

From Figure 16, it can be observed that the impact force of debris on the piers shows a distribution pattern of decreasing from bottom to top along the vertical direction. Additionally, the magnitude of the impact force decreases as the impact distance increases. Under constant slope angle and impact surface shape conditions, the debris first dissipates energy through friction on the platform base plate after falling along the chute, before impacting the piers. Therefore, the impact energy of debris on the piers decreases as the impact distance increases, and the distribution pattern of impact forces on the piers aligns with the characteristics of debris movement.

3.2.3. Internal Forces of Piers

According to Equations (1) and (2), the calculation method for the shear force and bending moment at the pier’s bottom, Figure 17 provides the time–history curves of the square pier’s shear force and bending moment at the pier’s bottom when the slope angle is 45°. From Figure 17, it is evident that the response of the internal forces at the pier’s bottom varies significantly at different impact distances. As the impact distance increases, the shear force and bending moment responses decrease significantly. Therefore, in designing anti-impact measures for piers in rugged mountainous areas, the operational safety of piers can be greatly enhanced by selecting appropriate avoidance distances.

3.2.4. Movement Response of Piers

Figure 18 illustrates the displacement and velocity responses at the pier’s top under various impact distances for the condition of θ = 45°. The analysis indicates that the displacement and velocity responses at the pier’s top are more pronounced when the impact distance is smaller. This effect is mainly attributed to the substantial reduction in the impact force of debris on the pier as the impact distance increases, consequently leading to a notable decrease in the motion response at the pier’s top. This finding aligns with the conclusion drawn from the response of the internal forces of the piers shown in Figure 17.

3.3. Slope Angle of the Chute

3.3.1. Accumulation Form of Rock Debris

For the circular piers with an impact distance of l₀ = 1 m, the accumulation of rock debris under different slope angles is presented in Figure 19. The analysis of Figure 19 reveals a decreasing trend in the length of debris accumulation as the slope angle increases, while the width of accumulation demonstrates an increasing pattern with the slope angle increase. Additionally, when comparing the accumulation front shapes under different slope angles, a symmetrical profile is evident for θ = 30° and θ = 45°, whereas, under θ = 60°, an asymmetric shape is pronounced. This asymmetry is mainly attributed to the higher slope angle that boosts the impact kinetic energy of the rock debris, resulting in a significant non-uniform flow state during collisions with the bridge pier. The trend indicates that with a steeper slope angle, the randomness in the debris flow migration pattern becomes more visible.

Figure 20 shows the variation curve of debris accumulation height in front of the pier with l₀ = 1.0 m under different slope angles. Upon analysis of Figure 20, it becomes apparent that the debris accumulation height follows a pattern of decreasing initially and then increasing as the slope angle increases. When the horizontal distance of the chute is a detained constant, the potential energy of the debris mass increases with a larger slope angle. As the debris mass slides downward, it gains higher impact energy. Upon impacting the platform, the debris mass undergoes oblique impact and frictional energy dissipation, reducing its kinetic energy. Finally, after colliding with the pier, the rock debris gradually accumulates near the pier. Therefore, the variation in the accumulation height of rock debris in front of piers should consider factors such as the friction coefficient, stiffness parameters, and the slope angle of the platform. In conducting impact mitigation analysis for mountain piers, it is possible to effectively utilize the impact distance segment to dissipate the kinetic energy of the rock debris and reduce the impact risk on the piers.

3.3.2. Impact Force on Piers

In Figure 21, the vertical distribution of impact force on the circular piers is illustrated under various slope angles with a condition of l₀ = 1.0 m. Analysis of the Figure reveals that the impact force of rock debris on the piers is predominantly concentrated within the range of 0 to 0.2 m. Furthermore, comparing the magnitude of impact forces under different slope angles further reveals that the impact force of rock debris initially decreases before increasing with the slope angle’s increase. By combining the analysis of the accumulation height of rock debris in front of the pier, a slope angle of 45° for the chute is identified as the transitional slope angle for the impact of rock debris on the bridge pier. Therefore, in the design analysis for anti-impact measures on piers in mountainous regions, adjusting the slope angle of the mountainside can alleviate the impact of rock debris on the pier.

3.3.3. Internal Forces of Piers

Figure 22 shows the time–history curves of the shear force and bending moment at the circular pier bottom under various slope angle conditions with l₀ = 1.0 m. Analysis of Figure 22 indicates that for the θ = 30° and θ = 60° conditions, the internal forces generally increase over time, whereas, under the θ = 45° condition, the internal forces remain relatively steady with smaller magnitudes. In addition, under the condition of θ = 45°, the peak internal force at the pier bottom occurs around 4.8 s, while under the condition of θ = 60°, the peak internal force at the pier bottom occurs around 4.4 s. There is no significant short-duration peak internal force for the condition of θ = 30°. The variation in internal forces at the pier bottom is primarily influenced by the increasing potential energy of the debris as the slope angle increases. Although the distance of the sliding chute increases with the slope angle, the diminishing friction force leads to less significant dissipation of frictional energy on the debris within the chute. However, when the slope angle is substantial, the immense instantaneous impact energy of the debris generates a peak impact force on the bridge pier. In the design of pier impact resistance, special consideration should be given to the damaging impact of short-duration peak internal forces on the structure.

3.3.4. Movement Response of Piers

Figure 23 shows the displacement and velocity response curves of the circular pier’s top under different slope angles when l₀ = 1.0 m. A comparison of the displacement and velocity responses under different slope angles reveals that the movement response of the pier’s top is significant during the mid-impact period for the condition of θ = 45°, while for the conditions of θ = 30° and θ = 60°, the pier’s top movement response is more pronounced in the post-impact period. In addition, by comprehensively comparing the effects of pier cross-sectional shape, impact distance, and slope angle on the pier’s top movement response, it is known that under the condition of l₀ = 1.0 m, the displacement time–history curves of the pier’s top show a small amount of reverse displacement. This is mainly due to the forward impact force at the pier bottom causing an inertial force on the pier, resulting in a reverse motion trend. The small magnitude of the overall displacement response amplifies the characteristics of reverse motion.

4. Discussion

Field surveys on debris flow disasters have revealed that as rock debris slides and rolls downhill, rock fragmentation characteristics are present [1]. Various factors such as the varying slope angles of mountainous terrain [7], frictional resistance [32], the properties of debris particles [33], and the sorting effect based on particle size [6] have a significant influence on the mobility features of debris. The impact disaster mechanism of debris on bridge structures is complicated. The study solely carried out numerical simulation experiments on reduced-scale concrete bridge piers for rock debris flow impact, which holds certain limitations. Specifically, the material parameters and contact parameters of the discrete element model utilized in the study were based on previous research [28,34], with some parameter values being simplified and adjusted. Nevertheless, the characteristics of the bridge pier’s impact force time–history curves aligned with existing research [4,7], affirming the reliability of the research findings.

The impact disaster modes of rock debris flow on bridge piers are diverse, with a complex dynamic action mechanism. The main research approach in the subsequent stages of this research project will involve conducting large-scale physical experiments on debris flow impacting bridge piers. Through a multi-factorial analysis of debris flow impacting bridge structures, exploring the energy dissipation mechanism of debris flow, extracting the typical damage modes of bridge piers, and deducing the impact disaster process of bridge structures, this research aims to establish a foundation for preventing impact disasters and assessing the safety performance of concrete bridge structures in steep mountainous areas.

5. Conclusions

This study conducted chute experiments on the impact of rock debris on concrete bridge piers using PFC. The research involved variable analysis of the cross-sectional shapes of the piers, impact distances, and slope angles of the chute. It investigated the accumulation height and planform of the rock debris and discussed the distribution patterns of impact forces from the rock debris on the piers in horizontal and vertical directions. Additionally, the study analyzed the shear forces and bending moment at the pier bottom, as well as the displacement and velocity responses at the pier top, and the following conclusions have been drawn:

(1)

The influence of the pier’s cross-sectional shape on the transportation and accumulation patterns of debris is significant, with different impact forces operating on piers with arch-shaped and rectangular impact surfaces. Rectangular piers obstruct debris movement and exert a greater accumulation height on rock debris than arch-shaped face piers. Nonetheless, arch-shaped face piers exhibit short-duration, high-peak internal force responses, which will be crucial in the internal force design of piers.

(2)

The impact distance is a sensitive factor in the impact of rock debris on piers. Increasing the impact distance can significantly reduce the internal forces and motion response of the piers. Increasing the impact distance can reduce the obstruction of debris movement by the piers, lower the accumulation height of debris, dissipate debris energy through the impact distance segment, and reduce the risk of impact disasters to pier structures.

(3)

The influence of the slope angle of the chute on the impact of rock debris on piers exhibits a transition angle, and reasonably controlling the slope angle can mitigate the impact effect. The accumulation height of debris in front of the piers follows a pattern of decreasing first and then increasing with the increase in the slope angle. The impact force on the piers, the internal forces at the pier’s bottom, and the movement response at the pier’s top are relatively small under the condition of θ = 45°. Therefore, when designing piers in rugged mountainous areas, optimizing the slope angle of the terrain could be considered to improve the impact of rock debris on the piers.

(4)

The shape of the pier’s cross-section, the impact distance of rock debris, and the slope angle of the mountainous terrain all have a certain influence on the impact force on the pier. When conducting anti-impact design for concrete bridge piers in mountainous areas, it is necessary to comprehensively consider the effects of different design factors on the impact on the pier. Prioritizing economically efficient and environmentally friendly anti-collision design methods is recommended while ensuring safety and reliability.