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In the design of rock sheds for the mitigation of risk due to rapid and long landslides, a crucial role is played by the evaluation of the impact force exerted by the flowing mass on the rock sheds. This paper is focused on the influencing factors of the impact force of dry granular flow onto rock shed and in particular on the evaluation of the maximum impact force. The coupled DEM-FEM model calibrated with small-scale physical experiment is used to simulate the movement of dry granular flow coupled with impact forces on the rock-shed. Based on the numerical results, three key stages were identified of impact process, namely startup streams slippery, impact and pile-up. The maximum impact force increases linearly with bulk density, and the maximum impact force exhibits a power law dependence on the impact height and slop angle respectively. The sensitivities of bulk density, impact height, and slope angle on the maximum impact force are: 1.0, 0.496, and 2.32 respectively in the benchmark model. The parameters with high sensitivity should be given priority in the design of the rock shed. The results obtained from this study are useful for facilitating design of shed against dry granular flow.

The mountainous areas in Southwest China, where the topography is complex, feature seriously weathered rock masses and have experienced a large number of landslides. These landslides provide abundant materials for the initiation of dry granular flow. When the earthquake or heavy rainfall occurs, the landslide will collide with the mountain and slide down the slope, leading to a dry granular flow (Chen & Zhang, 1994; Zhu, Wang, & Tang, 2000). The formed dry granular flow, with high speed and large displacement, can greatly threaten the safety of the residents and the smooth traffic flow. For instance, the Wenchuan earthquake caused more than 15,000 geo-hazards in the form of dry granular flows which resulted in about 20,000 deaths in 2008 (Yin, Wang, & Sun, 2009). Countermeasures have been made to minimize the dry granular flow’s risk to downstream residential areas or transportation routes. There are mainly two kinds of protection structures that used to minimize this hazard: active ones (like nets) and passive ones. As active ones is hard to carry out because of avalanches’ potential source area is difficult to figure out, engineers and researchers usually choose the passive ones. Rock sheds are regarded as passive protection structures and are widely used to protect against mountain hazards such as dry granular flow, due to its unique edge in terms of low construction cost and strong constructability in complex areas (Pei, Liu, & Wang, 2016; Kawahara & Muro, 2006; Mommessin, Perrotin, & Ma, 2012). Most rock sheds are made of concrete, and have a shock-absorbing layer such as sands on top of the structure (Montani, Descoeudres, & Labiouse, 1996; Kishi & Konno, 2003).

Up to now, the design of such structures takes into account only the impact of an individual rock block (Kishi & Konno, 2003; Calvetti, 2011; Delhomme, Mommessin, & Mougin, 2005; Wang, Zhou, & Luo, 2017). Though the standardized design of rock sheds under a single block impact has accumulated rich engineering experience (Montani, Descoeudres, & Labiouse, 1996; Kawahara & Muro, 2006), it cannot be applied to the design of rock sheds impacted by dry granular flow, due to totally different dynamic mechanical characteristics. To date, No firm guidelines built upon sounded theoretical basis are available for the design of rock shed impacted by dry granular flow. Therefore, further researches on the dynamic behavior of a rock shed impacted by a dry granular flow are urgently required. However, due to the estimation of the impact force exerted by dry granular flow is a prerequisite parameter for shed design, so it is necessary to study the influencing factors of the impact force, so as to provide references for facilitating design of shed against dry granular flow.

Physical modeling has been widely used in geotechnical engineering research because of its excellent controllability in testing conditions and good reliability of testing results. For instance, using indoor experimental methods, Jiang et al. investigated the impact of dry granular flow against a rigid retaining wall by calculating the impact force (Jiang & Towhata, 2013; Jiang, Zhao, & Towhata, 2015). Jiang designed a set of experiments to investigate the impact mechanism of dry granular flow against a curved rock shed (Jiang, Wang, & Son, 2018). Thus, the research results can serve as a significant reference for practice engineering. A quantitative analysis of impact force is eagerly needed in the design. In engineering practices, several semi-empirical methods have been used to estimate the maximum impact force of debris flows acting on a rigid barrier, such as hydrostatic approach, shock wave approach and hydrodynamic approach (Shen, Zhao, & Zhao, 2018). These studies are useful for providing us with ideas. Nevertheless, these available methods still have the difficulties in estimating the impact force of dry granular flow on rock sheds. This is because each method was obtained in specific impacting and boundary conditions with strong assumptions, such that they cannot be generalized for wider applications. In addition, these methods fail to consider the influence of dry granular flow-rock shed coupled interaction. In the present study, a robust numerical tool is a good choice. As the dry granular flow is a collection composed of a large number of discrete particles, DEM is an effective method for studying dry granular flow. Lo et al. used the PFC-3D software to study the maximum impact of the rock shed suffered by the dry granular flow (Lo, Lee, & Lin, 2016). Bi et al. studied the optimization of buffer layer under the impact of dry granular flow by two-dimensional discrete element software, and obtained the optimal thickness of the buffer layer (Bi, He, & Li, 2016). However, the discrete element method is not suitable for investigations of disaster-structure coupled interaction.

The above researches generally focus on the research of a single factor of uncoupled impact force. Unfortunately, the coupled dynamic interaction between dry granular flow and a rock shed is very complicated because it depends on the kinematics of dry granular flow (like solid mass and velocity), the stiffness and geometrical characteristics of the rock shed. So a quantitative analysis of these conditions on impact force is eagerly needed in the design.

The DEM has been widely used for numerical modeling of rock avalanches (Lo, Lee, & Lin, 2016; Bi, He, & Li, 2016; Cundall, 2008). It is an appropriate tool for modeling rock avalanches because of the discrete nature of materials involved in these phenomena. On the other hand, the FEM, based on continuum mechanics theory, has been well developed. Stress-strain development path and failures of elements are easy to simulate by FEM. Therefore FEM is a highly suitable method to model the rock shed (Albaba, Lambert, & Kneib, 2017). The coupled DEM-FEM method can well consider the coupled interaction between dry granular flow-rock sheds.

In this paper, a coupled DEM-FEM method was introduced for addressing the coupled response of rock shed impacted by dry granular flow, which combines advantages of both finite element and discrete element methods (Section 2). A coupled DEM-FEM model was built. A set of spherical discrete particles were used to model the dry granular flow. The model of barrier was simulated by FEM. The numerical model was validated by comparing the numerical results with the tests (Section 3). The coupled model was naturally employed to investigate the impact process of dry granular flow on rock sheds. The model was further employed to examine the effect of bulk density of dry granular flow, impact height of dry granular flow and slope angle on the coupled impact force (Section 4).

The DEM is employed to model the particle system in a dry granular flow. It is assumed that the particles are all elastic soft spheres of different sizes. The use of spherical particles in DEM simulations will inevitably lead to a soil structure different from that of real natural soils with a reduced granular internal friction. However, through careful model calibrations, an assembly of spherical particles with proper mechanical and physical properties can still be used to simulate the behavior of debris flows. This setting simplifies the complexity of dry granular flow but is also able to deliver a realistic simulation of the interaction between obstacle and flow (Bi, He, & Li, 2016; Cundall, 2008; Karajan, Han, & Teng).

The motion of discrete elements is governed by the second Newton’s law, and there are one or more forces acting on each element. The distribution and evolution of the system are described through the motion and state change of each element in the system (Cundall, 2008; Albaba, Lambert, & Kneib, 2017; Karajan, Han, & Teng, 2014). For element i:

{ m i u ¨ i = m i g + ∑ k = 1 m ( f n , i k + f t , i k ) I i θ ¨ i = ∑ k = 1 m T i k (1)

where g is the gravitational acceleration. m i , u ¨ i , I i and θ ¨ i are the mass, translational acceleration, rotary inertia and rotational acceleration of element i respectively. f n , i k , f t , i k and T i k are the normal contact force, the tangential contact force and the torques of element i acted by its neighboring element k respectively. T i k can be obtained by formula T i k = l i k × ( f n , i k + f t , i k ) , and l i k is the arm vector of the force to the center of element i.

Particles in the simulations are interacting with a linear spring-dashpot contact (LSD) law with Coulomb failure criterion, which is simple and computationally efficient compared to Hertz contact model (Karajan, Han, & Teng, 2014). The contact model of two particles is shown in

The overlap δ of two particles is calculated as follows:

δ = r i + r k − | x i − x k | (2)

where r i and r k are the radius of particles i and k respectively. x i and x k are the position vector of particles i and k respectively.

The normal contact force f n , i k between interacting particles is calculated as follows:

f n , i k =(- k n δ + c n δ ˙ ) n (3)

where k n , c n , δ ˙ , and n are the normal spring stiffness, normal damping coefficient, the relative normal velocity and the unit normal displacement vector respectively.

The tangential contact forces f t , i k between interacting particles are calculated as follows:

f t , i k = { (- k t δ t + c t δ ˙ t ) ; if | f n , i k | μ > | - k n δ t + c t δ ˙ t | ∑ k = 1 m T i k (- k t δ t + c t δ ˙ t ) | - k t δ t + c t δ ˙ t | | f n , i k | μ ; otherwise (4)

where k t , c t , δ t and μ are the tangential spring stiffness, tangential damping coefficient, the incremental tangential displacement, and the friction coefficient respectively. k t is taken as 2/7 k n (Albaba, Lambert, & Kneib, 2017; Karajan, Han, & Teng, 2014). k n is calculated as follows (Karajan, Han, & Teng, 2014):

k n = n k κ i r i κ k r k κ i r i + κ k r k ， and κ = E 3 ( 1 − 2 ν ) (5)

where n k is a stiffness proportionality constant. κ i and κ k are the bulk modulus of particle i and k respectively. E and ν are the elastic modulus and poisson ratio of particle respectively.

c n and c t are calculated as follows (Karajan, Han, & Teng, 2014):

c n =2 η n m i m k m i + m k k t , c t =2 η t m i m k m i + m k k n (6)

where η n and η t are the normal damping ratio and tangential damping ratio of particles respectively.

The coupled governing equations are given by Equation (7). The first and second conditions refer to the governing equations of DEM. The final condition gives the governing equation of FEM.

{ m i u ¨ i = m i g + ∑ k = 1 m ( f n , i k + f t , i k ) + ∑ j = 1 l ( f n , i j + f t , i j ) I i θ ¨ i = ∑ k = 1 m T i k + ∑ j = 1 l T i j M X ¨ + C X ˙ + K X = f a + f b (7)

where f n , i j , f t , i j and T i j are the normal contact force, tangential contact force and the torques of discrete element i acted by its neighboring finite element j respectively. T i j can be obtained by formula T i j = l i j × ( f n , i j + f t , i j ) , and l i j is the arm vector of the force. M, C, and K are the mass matrix, damping matrix and stiffness matrix of system respectively. X is the displacement of finite element node. f a and f b are the external force vector of finite elements and the contact force vector of between finite elements and discrete elements respectively.

The interaction between contact surfaces is handled following the penalty method. As before-mentioned, the combined finite-discrete element method proposed in this paper is focused at dynamic simulation, and the Central Difference Method (CDM) is employed to solve Equation (7). Since CDM is conditional convergence, the step must satisfy the numerical stability conditions. Both DEM and FEM adopt the conditional stable central difference method, and their coupling requires that their integrals must be synchronized, which requires both to adopt the same time step under the same calculation framework. The time step Δ t DEM-FEM takes the smaller value of both (Karajan, Han, & Teng, 2014).

Δ t DEM-FEM = min ( Δ t DEM , Δ t FEM ) (8)

where Δ t D E M = β 0.2 π m / K spring and Δ t F E M ≤ L min / c . c is the material sound speed. β is the scaling coefficient of time step length. m is the particle mass. K spring is the contact spring stiffness of particles, and L min is the minimum finite element size.

The flume, which measured 2.93 m in length, 0.35 m in height, and 0.3 m in width, was constructed to reproduce the flow environment of dry particles as shown in

polyethylene sheets for protection. The base of the flume was covered by a type of acrylic board to produce base friction. A retaining wall instrumented with 6 load cells was installed at the bottom end of the flume, perpendicular to the flume base, and the impact force in the normal direction was measured. The summation of these six force fractions of the load cells is the total force exerted on the retaining wall. A trigger gate was used to instigate the flow of the sliding mass. The length of the initial deposition of the sliding mass is 0.44 m. The height H of the initial deposition is 0.15 m. The distance between the trigger gate and the retaining wall model is 2.19 m. The width of retaining wall is 300 mm, the same as that of the flume. The tilt angle of the flume is 40˚. The particle sizes range from 10 mm to 25 mm. The specific parameters are shown in

Due to the existence of inter-particle porosity, the density of sand is set to 2800 kg/m^{3} through the numerical volume test (Adrian Jensen, Kirk Fraser & George Laird, 2014), so that the bulk density of the initial debris deposition could be guaranteed to be 1350 kg/m^{3}. This test allows the analyst to adjust the bulk density. The normal damping ratio between particles is set to 0.7. The tangential damping ratio is set to 0.4. The stiffness proportionality constant is set to 0.01. These three values were obtained by trial and error, so that the overall numerical results of debris dynamics can match the experimental observations in the model validation process. The particle Young's modulus and Poisson's ratio are set according to the commonly used values in numerical simulations of granular medium, as listed in

As a channel, the side wall and bottom wall have little influence on the test, and so they are modeled as rigid wall. In the experiment, the load cells upon impact have a very small normal strain, and so the retaining wall is simulated by elastic wall. The material parameters are shown in

The friction coefficients of the particles ( μ 1 ), the flume base ( μ 2 ) and the barrier ( μ 3 ) are chosen according to the experimental observations (Jiang YJ & Towhata I, 2013). In all the simulations, the flow is initiated by instantaneous removal of the top trigger gate. Then, the granular mass would slide under gravity downwards the flume with confined motions by the two side walls. At the bottom end of the flume, the granular mass is arrested by the barrier.

Dry bulk density | 1350 kg/m^{3} | Angle of repose | 53˚ |
---|---|---|---|

D_{50} | 14.1 mm | Friction angle of side wall-particles | 25˚ |

Uniformity coefficient, C_{u} | 1.5 | Friction angle of bottom wall-particles | 21˚ |

Friction angle of retaining wall-particles | 15˚ |

Density | 2800 kg/m^{3} | Density | 2000 kg/m^{3} | |||
---|---|---|---|---|---|---|

Young’s modulus | 30 Gpa | Wall (Rigid) | Young’s modulus | 30 Gpa | ||

Poisson’s ratio | 0.3 | Poisson’s ratio | 0.3 | |||

Normal damping ratio | 0.7 | Density | 7850 kg/m^{3} | |||

Granular | Tangential damping ratio | 0.4 | Barrier (Elastic) | Young’s modulus | 200 Gpa | |

Stiffness proportionality constant | 0.01 | Poisson’s ratio | 0.3 | |||

Particle-particle friction coefficient μ_{1} | 1.38 | |||||

particle-flume friction coefficient μ_{2} | 0.47 | Gravitational acceleration | 9.8 m/s^{2} | |||

particle-barrier friction coefficient μ_{3} | 0.38 | |||||

It can be observed that the numerical results can match well the experimental measurements (see

A comparison between the experimental and the numerical results indicates that the coupled DEM-FEM experiment adopted in this study can well simulate the

laboratory experiment. In this paper, based on the indoor model of dry granular flow against a retaining wall, a model of rock shed impacted by a dry granular flow is established by placing the retaining wall at the horizontal plane. Moreover, it is advisable to implement a numerical experiment for investigation of influencing factors of the avalanche-structure interaction.

The model of rock shed impacted by a dry granular flow is established by placing the retaining wall at the horizontal plane. For the simplified model, the rectangularly shaped debris flow material may not have the same impact energy compared with the actual case; however, this paper aims to study the regular variation of impact energy for qualitative analysis rather than quantitative examination. Simplifying the model makes the analysis simpler and easier. The geometric scheme adopted for this study is shown in

ρ (kg/m^{3}) | 1060, 1205, 1350, 1495, 1640 | H = 3.0 m, θ = 60˚ |
---|---|---|

H (m) | 2.0, 2.5, 3.0, 3.5, 4.0 | ρ = 1350 kg/m^{3}, ρ = 60˚ |

θ (˚) | 44, 52, 60, 68, 76 | ρ = 1350 kg/m^{3}, ρ = 3.0 m |

are shown in

In this section, the general features of granular flow impacting on a rock shed of ρ = 1350 kg/m^{3}, H = 2.5 m and θ = 60˚ are illustrated.

The sensitivity analysis method in the system analysis can be used to determine the main influence parameters. The relationship between the parameters of the dry granular flow and the maximum impact force is respectively fitted, and Equation (9) is obtained.

{ F max = 0.589 ρ ; R 2 = 0.966 F max = 457.3 H 0.496 ; R 2 = 0.940 F max = 0.06 θ 2.32 ; R 2 = 0.986 (9)

According to the sensitivity calculation formula 10 (Zhang & Zhun, 1993):

S a k ∗ = | ( d φ k ( a k ) d a k ) a k = a k ∗ | a k ∗ U ∗ (10)

where S a k * is sensitivity value. φ k ( a k ) is the sensitivity function of sensitivity parameter a k , which is the fitting function in this paper. a k * is the reference value of sensitivity parameters. U * is the value of sensitivity function when a k = a k * .

The reference values of parameters of the benchmark model for sensitivity analysis in this paper are: ρ 0 = 1350 kg/m^{3}, H 0 = 3.0 m and θ 0 = 60˚. The corresponding maximum impact force F max - 0 is 790 N. Through Equation (10), the sensitivity value of each parameter in the benchmark model is shown in

The impact of a dry granular flow on a rock shed has been investigated by a novel numerical framework. A coupled DEM-FEM approach is employed in this framework. The coupled DEM-FEM model was successfully verified by indoor test results. As such, the validated model was then employed to investigate the evaluation of the maximum impact force of dry granular flow against rock shed under different influencing factors. The key findings from this study are summarized as follows:

Based on the numerical modeling, three key stages during impact process, namely the startup streams slippery, impact and pile-up were identified.

Certain outcomes were discussed with particular emphasis on the influences of bulk density, impact height, and slop angle on the impact forces exerted on the rock shed. The maximum impact force increases linearly with bulk density. The maximum impact force increases in the form of a power law with the increases of the impact height, and its power index is less than 1. The maximum impact force increases in the form of a power law with the increases of the slope angle, and the power index is greater than 1. The sensitivities of bulk density, impact height, and slope angle on the maximum impact force are: 1.0, 0.496,

Sensitive parameters | S ρ ∗ | S H ∗ | S θ ∗ |
---|---|---|---|

Sensitivity value | 1.01 | 0.49 | 2.36 |

and 2.32 respectively in the benchmark model. The parameters with high sensitivity should be given priority in the design of the rock shed.

However, this paper remains a rather preliminary pilot study. Only the maximum impact force of the dry granular flow on the rock shed is taken as the dependent variable, and the response to the internal force of rock shed needs to be further modeled and analyzed. On the other hand, the effects of the buffer layer and initial shape of granular flow on the maximum impact force need to be further study.

This research was supported by the National Natural Science Foundation of China under Grant No. 51678504 and No. 51408498.

The authors declare no conflicts of interest regarding this article.

Liu, C., Yu, Z. X., & Huang, J. F. (2019). Evaluation of the Impact Force of Dry Granular Flow onto Rock Shed. Journal of Geoscience and Environment Protection, 7, 1-15. https://doi.org/10.4236/gep.2019.75001