# New Insights into the Estimation of Impact Load of Granular Flow against a Rigid Wall

Authors: Xiao Siyou; Yang Junwei; Wei Zhongju
Journal: Shock and Vibration
 (2022)

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## Abstract

The installation of a rigid wall is a main approach to mitigate a granular flow. The methods of the impact load against a rigid wall still remains a challenge. In this study, the progresses in the estimation of the impact load of granular flow against a rigid wall are introduced in detail. The influences of the slope angle, the installed angle of rigid wall, the volume of granular flow are not fully considered into the present models. The discrete element method is employed to reveal the effects of these factors on the impact load. The numerical results show that the volume of granular flow is a key factor to determine the evolution of the impact load on a rigid wall. The dynamic impact load of granular flow at the initial stage is decided by the angle between the rigid wall and slope surface. If the installation angle is same, the slope angle mainly determines the static load of granular flow. Based on these findings, we propose simplified model that the relationship between the impact load and the multiple physical parameters, such as slope angle, installation angle, repose angle, flow height and flow velocity, basal friction angle, and the volume of granular flow, is taken into account.


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## Body

## 1. Introduction

Granular flow is one of the common geological disasters in mountainous areas [1, 2]. Rigid walls are usually used for intercepting granular flows in the field [3–8]. After the location to install the rigid wall is determined, it is an important work to estimate the impact load of a granular flow against the rigid wall [9]. There are at present many methods to estimate the maximum impact loads.


### 1.1. The Earth Pressure Theory

When a granular flow impacts on the rigid wall, parts of particles pile up in front of the rigid wall. The region is called dead zone, as shown in Figure1 [5]. The static Earth pressure of the dead zone on the rigid wall could be used for estimating the maximum impact load. If Rankine Earth pressure theory is used for estimating the Earth pressure, the slope angle θ should be 0, and the interface between the rigid wall and the granular flow should be a smooth face. In the engineering practice, the slope angle θ and the friction angle between the rigid wall and the granular flow should be considered into the design of rigid wall. Hence, Coulomb Earth pressure theory is usually used for estimating the Earth pressure Fp as follow [5, 11]:(1)Fp=Wdeadsinθ−δ1sinπ−θ−θ1+δ1+δ2,where Wdead is the weight of dead zone, δ1 is the friction angle between the slope surface and the granular flow, δ2 is the friction angle between the rigid wall and the granular flow, and θ is slope angle. θ1 is installed angle of rigid wall. This model is applicable to estimate the impact load of granular flow whose fluidity is low. The maximum impact load is close to the residual load.Figure 1
Division of the dead zone and the flowing layer [10].


### 1.2. Estimation Based on the Relationship between Impact Load and Dimensionless Numbers

Froude number, Savage number, and Bagnold number are usually used for describing the fluidity of granular flow [12]. However, it still lacks the study on how to use Savage number and Bagnold number for quantitatively estimating the maximum impact load. Hübl et al. pointed out that the hydrostatic model can be used for estimating the maximum impact load when Fr is less than 2 [3]. If Fr is greater than 2, the hydrodynamic model is better to calculate the maximum impact load. The hydrostatic model and the hydrodynamic model could be written as [10, 13]:(2)Fmax=12kρghdeposit2wsinθ1,Fmax=CDρv02h0wsinθ+θ1,where k is the empirical hydrostatic coefficient. CD is the empirical hydrodynamic coefficient. A large number of experimental results indicated that the dynamic impact load coefficient CD and Fr presents a negative exponential relationship [5, 6, 14]:(3)CD=acFr−n,where ac and n are constant values.Li et al. found thatk also has a relationship as follows [15]:(4)k=aFr+b,where a and b are constant values.From the insight of the engineering application, Kwan pointed out that equation (2) can be applied to estimate the maximum impact load of viscosity debris flow [4]. However, the influences of the installed angle of rigid wall, the volume of granular flow, and its fluidity on the impact load are not considered into equation (3). Hence, the empirical hydrodynamic coefficient is usually determined based on the engineering experiences of the engineers. In this study, we discussed the main progresses and deficiencies in estimation of the impact load of granular flow impacting a rigid wall. Then, we introduced a potential method to estimate the impact load of granular flow against rigid wall.


## 1.1. The Earth Pressure Theory

When a granular flow impacts on the rigid wall, parts of particles pile up in front of the rigid wall. The region is called dead zone, as shown in Figure1 [5]. The static Earth pressure of the dead zone on the rigid wall could be used for estimating the maximum impact load. If Rankine Earth pressure theory is used for estimating the Earth pressure, the slope angle θ should be 0, and the interface between the rigid wall and the granular flow should be a smooth face. In the engineering practice, the slope angle θ and the friction angle between the rigid wall and the granular flow should be considered into the design of rigid wall. Hence, Coulomb Earth pressure theory is usually used for estimating the Earth pressure Fp as follow [5, 11]:(1)Fp=Wdeadsinθ−δ1sinπ−θ−θ1+δ1+δ2,where Wdead is the weight of dead zone, δ1 is the friction angle between the slope surface and the granular flow, δ2 is the friction angle between the rigid wall and the granular flow, and θ is slope angle. θ1 is installed angle of rigid wall. This model is applicable to estimate the impact load of granular flow whose fluidity is low. The maximum impact load is close to the residual load.Figure 1
Division of the dead zone and the flowing layer [10].


## 1.2. Estimation Based on the Relationship between Impact Load and Dimensionless Numbers

Froude number, Savage number, and Bagnold number are usually used for describing the fluidity of granular flow [12]. However, it still lacks the study on how to use Savage number and Bagnold number for quantitatively estimating the maximum impact load. Hübl et al. pointed out that the hydrostatic model can be used for estimating the maximum impact load when Fr is less than 2 [3]. If Fr is greater than 2, the hydrodynamic model is better to calculate the maximum impact load. The hydrostatic model and the hydrodynamic model could be written as [10, 13]:(2)Fmax=12kρghdeposit2wsinθ1,Fmax=CDρv02h0wsinθ+θ1,where k is the empirical hydrostatic coefficient. CD is the empirical hydrodynamic coefficient. A large number of experimental results indicated that the dynamic impact load coefficient CD and Fr presents a negative exponential relationship [5, 6, 14]:(3)CD=acFr−n,where ac and n are constant values.Li et al. found thatk also has a relationship as follows [15]:(4)k=aFr+b,where a and b are constant values.From the insight of the engineering application, Kwan pointed out that equation (2) can be applied to estimate the maximum impact load of viscosity debris flow [4]. However, the influences of the installed angle of rigid wall, the volume of granular flow, and its fluidity on the impact load are not considered into equation (3). Hence, the empirical hydrodynamic coefficient is usually determined based on the engineering experiences of the engineers. In this study, we discussed the main progresses and deficiencies in estimation of the impact load of granular flow impacting a rigid wall. Then, we introduced a potential method to estimate the impact load of granular flow against rigid wall.


## 2. Estimating the Maximum Impact Load

In the introduction section, we have introduced the methods to estimate of the maximum impact load with no consideration of the composition of the impact load. In order to improve the precision of the estimated results, some geotechnical researchers suggested that the maximum impact load should be divided into two parts [12, 16, 17]: the static load and the dynamic load. Accordingly, the maximum impact force can be written as follows:(5)Fmax=12khydroρgαh02wsinθ1+CDfρβ0v02h0wsinθ1+θ−θd,(6)Fmax=Wdeadsinθ−δ1cosδ1sinπ−θ−θ1+δ1+δ2+CDfρβ0v02h0wsinθ+θ1,where khydro is the empirical static-hydro coefficient. CDf is the empirical dynamic-hydro coefficient. β0 is the reduction factor of velocity. θd is the pileup angle, as shown in Figure 1. The differences of aforementioned equations are as follows:(1)
The definition of the dead zone: If the Earth pressure theory is used for calculating the static load, it assumes that the flowing layer never impacts on rigid wall. The particles are assumed to be stationary after impacting the dead zone. On the contrary, the flowing layer can impact on rigid wall after it impacts on the dead zone. The flowing layer can impact on rigid wall after it flows along the top of the dead zone.(2)
The impact direction of the flowing layer: Owing to the differences in the definition of the dead zone, the impact direction of flowing layer is different. For equation (6), the impact direction of the flowing layer is decided by the interface of the flowing layer and the dead zone. The impact direction is along the slope direction for equation (7).Although the maximum impact load divided into static load and dynamic load, it is more consistent with the actual process. There are still some problems to develop an accurate model to estimate the maximum impact load. The main reasons are as follows.(A) It is hard to estimate the static Earth pressure of dead zone accurately. The static load of the granular matter against a rigid wall is regarded as being close to the active Earth pressure [5, 16, 18]. The static load calculated by Coulomb Earth pressure theory may overestimate the active Earth pressure because the distribution of the Earth pressure is supposed to be linear [19]. In fact, the Earth pressure is nonlinear, as shown in Figure 2.Figure 2
Active Earth pressure against rigid wall of limited soil mass (revised from Fan et al. [19]).(B) The interaction between the flowing layer and the dead zone. Koo et al. [17] and Xiao et al. [20] pointed out that the flowing layer has an obvious influence on the evolution of the dead zone. The range of the dead zone is hard to determine. Hence, it is hard to calculate the static load. Moreover, the buffering and friction of the dead zone on the flowing layer is also difficult to be calculated quantitatively. The reasons are as follows:(1)
Which friction model should be chosen to calculate the friction force between flowing layer and dead zone? For a static granular matter, the Coulomb friction model is usually used for calculating the friction force. However, the Coulomb friction model cannot be not applied to a dynamic granular matter.(2)
How the mobilised friction varies spatially and temporally during impact?(3)
How to quantitatively calculate the influences of the buffering effect on the impact load?These issues lead to be difficult to propose an accurate model of the impact load against a rigid wall. For the engineering practice, it may be a key to develop a simplified model to estimate the peak impact load against a rigid wall instead of an accurate model with many parameters because many parameters in field are very hard to determine. Xiao et al. [20] proposed an estimation model of the maximum impact load based on the pileup characteristics. There is a critical pileup height hb when the granular flow with a given flow height h0 and flow velocity vo impacts on rigid wall. After the critical pileup height reaches, the pileup height does not increase any more. Whether the pileup height reaches to the critical value, hb is decided by the volume of granular flow. At beginning of the impact process, the flowing layer can impact on the rigid wall. This phase is defined as the continuous impact phase. As shown in Figure 3(a). The impact load is mainly composed of the dynamic impact load. The pileup length OA keeps increasing, and the pileup height OD remains a fix value after the pileup height reaches to the critical value. It also indicates that the critical value is the maximum pileup height corresponding to a given flow velocity and a flow height. This phase is defined as the slope pileup phase. In this phase, the load against the rigid wall is mainly composed of the static load because the flowing layer cannot impact on the rigid wall.Figure 3
Impact of debris flow on rigid wall: (a) continuous phase and (b) slope pileup phase.
(a)(b)


## 3. The Distribution of Impact Load of Granular Flow

### 3.1. The Spatial Distribution of the Impact Load

The spatial and temporal distribution of impact load is the key to design a rigid wall. A contact surface pressure gauge sensor was applied to measure the distribution of the impact load of granular flow against the rigid wall [21]. The results show that the impact load of granular flow is close to uniform in the horizontal direction. The impact load of granular flow with higher density presents linear attenuation characteristics from bottom to top along the wall, while the impact load of granular flow with lower density presents the attenuation characteristics of logarithmic function from bottom to top along the wall. Moreover, Jiang and Towhata [5] and Song et al. found the distribution of the impact load approaches to a triangle [12, 22]. The distribution of the impact load also may presents obvious nonlinearity because of the arching effect.


### 3.2. The Temporal Distribution of the Impact Load

According to the present experimental achievements [5, 9, 12, 16, 22], it can be reported that there are two characteristics of the temporal distribution of impact load: (1) The impact load remains unchanged after the impact load reaches to the peak load, as shown by the red line in Figure 4. (2) The impact load decreases after the impact load reaches to the peak load, as shown as the blue line in Figure 4. Although the impact load may keep increasing when viscosity debris flows or water impact on rigid wall. The calculated impact load of dry granular flow will increase constantly if the interaction between the dead zone and the flowing layer is not considered into the estimation of impact load. This is not in agreement with the experimental results and the field monitoring results [5, 9, 12, 16, 22, 23].Figure 4
Time history of impact loaf.


## 3.1. The Spatial Distribution of the Impact Load

The spatial and temporal distribution of impact load is the key to design a rigid wall. A contact surface pressure gauge sensor was applied to measure the distribution of the impact load of granular flow against the rigid wall [21]. The results show that the impact load of granular flow is close to uniform in the horizontal direction. The impact load of granular flow with higher density presents linear attenuation characteristics from bottom to top along the wall, while the impact load of granular flow with lower density presents the attenuation characteristics of logarithmic function from bottom to top along the wall. Moreover, Jiang and Towhata [5] and Song et al. found the distribution of the impact load approaches to a triangle [12, 22]. The distribution of the impact load also may presents obvious nonlinearity because of the arching effect.


## 3.2. The Temporal Distribution of the Impact Load

According to the present experimental achievements [5, 9, 12, 16, 22], it can be reported that there are two characteristics of the temporal distribution of impact load: (1) The impact load remains unchanged after the impact load reaches to the peak load, as shown by the red line in Figure 4. (2) The impact load decreases after the impact load reaches to the peak load, as shown as the blue line in Figure 4. Although the impact load may keep increasing when viscosity debris flows or water impact on rigid wall. The calculated impact load of dry granular flow will increase constantly if the interaction between the dead zone and the flowing layer is not considered into the estimation of impact load. This is not in agreement with the experimental results and the field monitoring results [5, 9, 12, 16, 22, 23].Figure 4
Time history of impact loaf.


## 4. New Insights into the Estimation of Impact Load

The installation angle, slope angle, volume of granular flow have influences on the impact load of granular flow on the rigid wall [20]. Here, we intend to discuss how these factors make influences on the impact load using discrete element method. The numerical geometry is same as the physical test conducted by Jiang and Towhata [5], as shown in Figure 5. The length L1 and width w of the flume is 2.93 m and 0.3 m, respectively. The flow distance is 2.19 m. The size of the material box is 15 cm × 44 cm × 30 cm. The volume of the test materials is about 0.02 m3. The slope angles θ range from 30° to 50°. The installed angle is 90° and 90°−θ, respectively.Figure 5
Sketch of the experimental flume: (a) tilted rigid wall and (b) vertical rigid wall.As this experiment has been reported widely, we do not introduce the experimental setup in detail [5, 9, 20]. The physical parameters of the granular flow and the flume are listed in Table 1. The DEM tool used in this study was developed by DEM Solutions, Ltd. The contact model is the Hertz–Mindlin (no-slip) model [24]. The model validation has been discussed by Xiao et al. [20]. Here, we just list the contact parameters in Table 2.Table 1
Boundary material parameters.
BoundaryDensity (kg/m3)Shear modulusG (MPa)Poisson ratioBasal790038500.25Rigid wall790012300.25Side wall12006000.25Table 2
Contact parameters of DEM model.
Contact parameterParticle-particleParticle-basalParticle-rigid wallParticle-side wallRestitution coefficient0.50.50.50.4Static friction coefficient1.330.4530.3840.27Rolling friction coefficient0.050.050.050.04


### 4.1. Influences of Installation Angle and Slope Angle on Impact Load

In many studies [5, 9, 11–13, 15, 18], the rigid walls are vertically installed to the slope surface. However, the walls are usually erect if the height is less than 5 m, as shown in Figure 6. It is worthy to point out that the effects of slope angle on the impact load are different when rigid walls are installed in the gullies and on the slopes. If the rigid wall is installed on a slope, the slope angle should be considered into the estimation of the impact load usually, as shown in Figure 6(a). However, it does not need to consider the effects of slope angle on the impact load in general. The main reason is that the rigid wall is usually installed at the flat location.Figure 6
The rigid wall installed in the gully and on the slope: (a) slope and (b) Gully (Xiao et al. 2019).
(a)(b)Figure7 is the time history of the impact loads against the titled rigid wall and the vertical rigid wall as the slope angles increase from 30° to 50°. The ratio of the impact loads against the tilted walls to the impact loads against the vertical rigid wall changes from 0.84 to 0.66 as the slope angle increases from 30° to 50°. It indicates that the installation angle has little influence on the maximum impact load when the slope angles are less than 40°. With the increase in the slope angle, the impact load against the vertical rigid wall is obviously greater than the impact load against the tilted rigid wall. Figure 8 is the distribution of the impact load against the titled rigid wall and the vertical rigid wall as the slope angles increase from 30° to 50°. It indicates that the distribution of the impact load is linear when the granular flows impact on the titled rigid wall. The distribution of the impact load approaches to a triangle. It is same as the previous experimental results of the impact of the debris flow or granular flow against the titled rigid wall [11, 12, 21, 25]. Different from the titled rigid wall, the distribution of the impact load presents obvious nonlinear when the granular flow impacts on the vertical rigid wall. It indicates that the angle between the slope surface and the rigid wall has an important influence on the distribution of the impact load.Figure 7
Evolution of the normal impact loads on the rigid wall: (a) tilted wall and (b) vertical wall.
(a)(b)Figure 8
The distribution of the normal impact loads at the moment of maximum normal impact loads: (a) tilted wall and (b) vertical wall.
(a)(b)According to the aforementioned analysis, the effects of the slope angle and the installation angle of the rigid wall on the impact load are different with the impact phases. We conducted a simplified numerical experiment to reveal the effects of the slope angle and the installation angle on the impact load at different impact phases. The experimental steps are as follows:


#### 4.1.1. Determination of the Flow Velocity and the Flow Height

The flow velocity and flow height are obtained according to the aforementioned numerical flume test. The flow height and flow velocity are determined at the moment of the maximum impact load. When the slope angle is 50° and the installation angle is 90°, the flow velocity and flow height is 4.97 m/s and 0.051 m. Three tests are conducted. The slope angle and the installation angle of each test are listed in Table3.Table 3
The slope angle and installation angle of each test.
Test IDSlope angle (°)Installation angle (°)10902509035040


#### 4.1.2. The Impact of the Steady Flow against Rigid Wall

As shown in Figure9, two sections are used for generating particles. The volume of each section is equal to the volume of the physical test material V0. The height of each section is 0.051 m. The length of each section is calculated using the following formula:(7)Ls=V0wh0.Figure 9
The geometry of the numerical experiments: (a) titled rigid wall and (b) vertical rigid wall.
(a)(b)Based on equation (7), the length of each section is about 1.8 m.


#### 4.1.3. The Generation of Particles

The particles are full with each section. The mass of each section is equal to the mass of the test materials of the physical test. The gradation of the particles is also same as the test materials. The velocity of each particles is 4.97 m/s. After all particles in Section2 flow into Section 1, the particles are regenerated in Section 2. The gravity field is only applied in the dead zone to ensure the flow velocity of the granular flow is close to 4.97 m/s. Figure 10 indicates that the installation angle has obvious influences on the impact load at the preliminary stage of the impact. The impact load against the vertical wall is about 500 N more than the load against the titled wall as the slope angle is 50°. However, the impact load against the rigid wall, whose installation angle is 90°−θ, is very close. It illustrates that the installation angle instead of the slope angle is the main cause of the differences between the impact load against titled wall and the vertical wall at the beginning of the impact. The impact load against the wall is different with the slope angle if the installation angle is same (in this case, the installation angle is 90°), as shown in Figure 11. It indicates that the slope angle is the main cause of the difference between the static impact load against the rigid wall when the installation angle is the same.Figure 10
History curve of impact force produced by dry granular flow against under different slope angles and different rigid wall types.Figure 11
The history curve of dry granular flow with different volume on the rigid wall.


### 4.2. The Effects of the Volume of Granular Flow on Impact Load

To reveal the effects of volume of granular flow on impact load, the different volume of granular flows are adopted to impact the vertical wall. The flow velocity and flow height is still 4.97 m/s and 0.051 m. The test volumes are ranged from 1 to 10 times of the volume in the physical experiment.The history curves of the impact loads are plotted in Figure11. It indicates that the impact load increases about to 1000 N at preliminary stage. The hydrodynamic coefficient CD is about 1.7. Based on Figure 7, the dynamic coefficient is about 0.7 when the installation angle of the rigid wall is 40°. It illustrates that the dynamic coefficient should be determined based on the installation angle of the rigid wall. Then, the impact load increases linearly. At the moment t, the active Earth pressure of the dead zone against the rigid wall calculated by the Coulomb Earth pressure theory can be written as follows:(8)Fpt=v0ht0ρwg⋅sinθ−δ1sinπ−θ−θ1+δ1+δ2cosδ1.The impact loadF of the flowing layer against the rigid wall can be written as follows:(9)F=CDρvd2wh0.The active pressure calculated by the Coulomb Earth pressure theory is plotted in the gray line. This line is agreement with other lines in the first 2 seconds generally. It shows that the active pressure of the dead zone against the rigid wall can be calculated by Coulomb Earth pressure theory. As the increase in the volume of the dead zone, the impact load remains unchanged. It illustrates that the dead zone has a buffering effect on the flowing layer when the flowing layer impacts on the dead zone. Compared with the residual load against the wall, it indicates that the difference between the maximum impact load and the residual load decreases with the increase in the volume of the granular flow. The impact load of the flowing layer decreases after it reaches to the peak value if the volume of the granular flow is less than 6 times of the volume of the test materials in the physical experiment. It can be concluded that the volume of the granular flow has significant influences on the estimation of the impact load against rigid wall.


### 4.3. Simplified Estimation Model of Impact Load

Based on the previous analysis, install angle, slope angle, volume of granular flow, and the collision and friction between the flowing layer and the dead zone should be all considered together into the estimation model. Here we introduce the model in detail.Suppose the pileup height OD, the pileup length OA and the length between the dead zone and the flowing layer AD in Figure3 are L(t), S(t), and h(t) at the moment t. The relationship between L(t), S(t), and h(t) as follows [20]:(10)Ltsinθ+θ1=htsinθd=Stsinθ+θ1−θd.The impact volumeVf is vot at moment t. The relation between the volume of the dead zone and flowing layer can be written as follows:(11)V=Vf+Va−Vb,where V is the volume of the dead zone and Va is the volume of the section OA of the flowing layer at the beginning of the impact. Vb is the volume of the flowing layer on the top of the dead zone (Section AD). Substituting equation (13) into equation (14) yields the following equation:(12)v0th0+Sth0=12S2tsinθ+θ1sinθ+θ1−θdsinθd+Stsinθ+θ1sinθ+θ1−θdha.By solving equation (12),S(t) can be expressed as follows:(13)St=hasinθ+θ1−hasinθ+θ1−θd+hasinθ+θ1−θd−hasinθ+θ12+2v0th0sinθ+θ1sinθdsinθ+θ1−θdsinθdsinθ+θ1.According to equations (10) and (13), h(t) can be written as follows:(14)ht=sinθ+θ1−θdsinθ+θ1,hasinθ+θ1−hasinθ+θ1−θd+hasinθ+θ1−θd−hasinθ+θ12+2v0th0sinθ+θ1sinθdsinθ+θ1−θd.Suppose the height of the flowing layer on the top of the dead zone keeps being equal to the flow heighth0, the pileup height h(t) at any moment can be obtained based on equation (15). The impact velocity Vd of the flowing layer against the rigid wall decreases with increasing in the volume of the dead zone. The friction effect and buffering effect of the dead zone on the flowing layer are the main reasons which lead to the decrease in flow velocity. The flow velocity drops to va when the flowing layer impacts on the dead zone. Then, the flow velocity of the flowing layer on the top of the dead zone further decreases because of the friction between the dead zone and the flowing layer. According to the kinetic energy theorem, the relationship between the impact velocity Vd on the rigid wall and va can be written as follows:(15)12va2=12vd2+gLtsinθd−θ+gtanφcosθd−θLt.Supposeva is the flow velocity after the flowing layer impacts on the dead zone. The coefficient of restitution is used for calculating va. In this study, the restitution coefficient κ is the ratio of the kinetic energy of particle before and after the collision. The flow velocity va after the collision can be written as follows [20]:(16)va=κv0.Substituting equation (15) into equation (16), the impact velocity vd of the flowing layer is expressed as follows:(17)vd=κv02−2gcosθhttanφcosθd−θ+sinθd−θsinθd.If the impact velocityVd is greater than 0, it is the continuous phase. If the impact velocity vd is equal to 0, it is the slope pileup phase. The pileup height reaches to critical value hcri at this moment. The critical pileup height hcri can be written as follows:(18)hcri=κv02sinθd2gcosθtanφcosθd−θ+sinθd−θ.The criterion for the flow phase as follows:(1)
IfV0 > Vc = 0.5 hcri2/sinθdsinθ1+θsinθ+θ1−θd, the impact is end up in the slope pileup phase. The maximum static load is close to the active Earth pressure of the critical part of the dead zone instead of the total of the dead zone.(2)
IfV0 >Vc = 0.5 hcri2/sinθdsinθ1+θsinθ+θ1−θd, the impact is end up in the continuous impact phase. The maximum static load is the active Earth pressure of the total of the dead zone.


## 4.1. Influences of Installation Angle and Slope Angle on Impact Load

In many studies [5, 9, 11–13, 15, 18], the rigid walls are vertically installed to the slope surface. However, the walls are usually erect if the height is less than 5 m, as shown in Figure 6. It is worthy to point out that the effects of slope angle on the impact load are different when rigid walls are installed in the gullies and on the slopes. If the rigid wall is installed on a slope, the slope angle should be considered into the estimation of the impact load usually, as shown in Figure 6(a). However, it does not need to consider the effects of slope angle on the impact load in general. The main reason is that the rigid wall is usually installed at the flat location.Figure 6
The rigid wall installed in the gully and on the slope: (a) slope and (b) Gully (Xiao et al. 2019).
(a)(b)Figure7 is the time history of the impact loads against the titled rigid wall and the vertical rigid wall as the slope angles increase from 30° to 50°. The ratio of the impact loads against the tilted walls to the impact loads against the vertical rigid wall changes from 0.84 to 0.66 as the slope angle increases from 30° to 50°. It indicates that the installation angle has little influence on the maximum impact load when the slope angles are less than 40°. With the increase in the slope angle, the impact load against the vertical rigid wall is obviously greater than the impact load against the tilted rigid wall. Figure 8 is the distribution of the impact load against the titled rigid wall and the vertical rigid wall as the slope angles increase from 30° to 50°. It indicates that the distribution of the impact load is linear when the granular flows impact on the titled rigid wall. The distribution of the impact load approaches to a triangle. It is same as the previous experimental results of the impact of the debris flow or granular flow against the titled rigid wall [11, 12, 21, 25]. Different from the titled rigid wall, the distribution of the impact load presents obvious nonlinear when the granular flow impacts on the vertical rigid wall. It indicates that the angle between the slope surface and the rigid wall has an important influence on the distribution of the impact load.Figure 7
Evolution of the normal impact loads on the rigid wall: (a) tilted wall and (b) vertical wall.
(a)(b)Figure 8
The distribution of the normal impact loads at the moment of maximum normal impact loads: (a) tilted wall and (b) vertical wall.
(a)(b)According to the aforementioned analysis, the effects of the slope angle and the installation angle of the rigid wall on the impact load are different with the impact phases. We conducted a simplified numerical experiment to reveal the effects of the slope angle and the installation angle on the impact load at different impact phases. The experimental steps are as follows:


### 4.1.1. Determination of the Flow Velocity and the Flow Height

The flow velocity and flow height are obtained according to the aforementioned numerical flume test. The flow height and flow velocity are determined at the moment of the maximum impact load. When the slope angle is 50° and the installation angle is 90°, the flow velocity and flow height is 4.97 m/s and 0.051 m. Three tests are conducted. The slope angle and the installation angle of each test are listed in Table3.Table 3
The slope angle and installation angle of each test.
Test IDSlope angle (°)Installation angle (°)10902509035040


### 4.1.2. The Impact of the Steady Flow against Rigid Wall

As shown in Figure9, two sections are used for generating particles. The volume of each section is equal to the volume of the physical test material V0. The height of each section is 0.051 m. The length of each section is calculated using the following formula:(7)Ls=V0wh0.Figure 9
The geometry of the numerical experiments: (a) titled rigid wall and (b) vertical rigid wall.
(a)(b)Based on equation (7), the length of each section is about 1.8 m.


### 4.1.3. The Generation of Particles

The particles are full with each section. The mass of each section is equal to the mass of the test materials of the physical test. The gradation of the particles is also same as the test materials. The velocity of each particles is 4.97 m/s. After all particles in Section2 flow into Section 1, the particles are regenerated in Section 2. The gravity field is only applied in the dead zone to ensure the flow velocity of the granular flow is close to 4.97 m/s. Figure 10 indicates that the installation angle has obvious influences on the impact load at the preliminary stage of the impact. The impact load against the vertical wall is about 500 N more than the load against the titled wall as the slope angle is 50°. However, the impact load against the rigid wall, whose installation angle is 90°−θ, is very close. It illustrates that the installation angle instead of the slope angle is the main cause of the differences between the impact load against titled wall and the vertical wall at the beginning of the impact. The impact load against the wall is different with the slope angle if the installation angle is same (in this case, the installation angle is 90°), as shown in Figure 11. It indicates that the slope angle is the main cause of the difference between the static impact load against the rigid wall when the installation angle is the same.Figure 10
History curve of impact force produced by dry granular flow against under different slope angles and different rigid wall types.Figure 11
The history curve of dry granular flow with different volume on the rigid wall.


## 4.1.1. Determination of the Flow Velocity and the Flow Height

The flow velocity and flow height are obtained according to the aforementioned numerical flume test. The flow height and flow velocity are determined at the moment of the maximum impact load. When the slope angle is 50° and the installation angle is 90°, the flow velocity and flow height is 4.97 m/s and 0.051 m. Three tests are conducted. The slope angle and the installation angle of each test are listed in Table3.Table 3
The slope angle and installation angle of each test.
Test IDSlope angle (°)Installation angle (°)10902509035040


## 4.1.2. The Impact of the Steady Flow against Rigid Wall

As shown in Figure9, two sections are used for generating particles. The volume of each section is equal to the volume of the physical test material V0. The height of each section is 0.051 m. The length of each section is calculated using the following formula:(7)Ls=V0wh0.Figure 9
The geometry of the numerical experiments: (a) titled rigid wall and (b) vertical rigid wall.
(a)(b)Based on equation (7), the length of each section is about 1.8 m.


## 4.1.3. The Generation of Particles

The particles are full with each section. The mass of each section is equal to the mass of the test materials of the physical test. The gradation of the particles is also same as the test materials. The velocity of each particles is 4.97 m/s. After all particles in Section2 flow into Section 1, the particles are regenerated in Section 2. The gravity field is only applied in the dead zone to ensure the flow velocity of the granular flow is close to 4.97 m/s. Figure 10 indicates that the installation angle has obvious influences on the impact load at the preliminary stage of the impact. The impact load against the vertical wall is about 500 N more than the load against the titled wall as the slope angle is 50°. However, the impact load against the rigid wall, whose installation angle is 90°−θ, is very close. It illustrates that the installation angle instead of the slope angle is the main cause of the differences between the impact load against titled wall and the vertical wall at the beginning of the impact. The impact load against the wall is different with the slope angle if the installation angle is same (in this case, the installation angle is 90°), as shown in Figure 11. It indicates that the slope angle is the main cause of the difference between the static impact load against the rigid wall when the installation angle is the same.Figure 10
History curve of impact force produced by dry granular flow against under different slope angles and different rigid wall types.Figure 11
The history curve of dry granular flow with different volume on the rigid wall.


## 4.2. The Effects of the Volume of Granular Flow on Impact Load

To reveal the effects of volume of granular flow on impact load, the different volume of granular flows are adopted to impact the vertical wall. The flow velocity and flow height is still 4.97 m/s and 0.051 m. The test volumes are ranged from 1 to 10 times of the volume in the physical experiment.The history curves of the impact loads are plotted in Figure11. It indicates that the impact load increases about to 1000 N at preliminary stage. The hydrodynamic coefficient CD is about 1.7. Based on Figure 7, the dynamic coefficient is about 0.7 when the installation angle of the rigid wall is 40°. It illustrates that the dynamic coefficient should be determined based on the installation angle of the rigid wall. Then, the impact load increases linearly. At the moment t, the active Earth pressure of the dead zone against the rigid wall calculated by the Coulomb Earth pressure theory can be written as follows:(8)Fpt=v0ht0ρwg⋅sinθ−δ1sinπ−θ−θ1+δ1+δ2cosδ1.The impact loadF of the flowing layer against the rigid wall can be written as follows:(9)F=CDρvd2wh0.The active pressure calculated by the Coulomb Earth pressure theory is plotted in the gray line. This line is agreement with other lines in the first 2 seconds generally. It shows that the active pressure of the dead zone against the rigid wall can be calculated by Coulomb Earth pressure theory. As the increase in the volume of the dead zone, the impact load remains unchanged. It illustrates that the dead zone has a buffering effect on the flowing layer when the flowing layer impacts on the dead zone. Compared with the residual load against the wall, it indicates that the difference between the maximum impact load and the residual load decreases with the increase in the volume of the granular flow. The impact load of the flowing layer decreases after it reaches to the peak value if the volume of the granular flow is less than 6 times of the volume of the test materials in the physical experiment. It can be concluded that the volume of the granular flow has significant influences on the estimation of the impact load against rigid wall.


## 4.3. Simplified Estimation Model of Impact Load

Based on the previous analysis, install angle, slope angle, volume of granular flow, and the collision and friction between the flowing layer and the dead zone should be all considered together into the estimation model. Here we introduce the model in detail.Suppose the pileup height OD, the pileup length OA and the length between the dead zone and the flowing layer AD in Figure3 are L(t), S(t), and h(t) at the moment t. The relationship between L(t), S(t), and h(t) as follows [20]:(10)Ltsinθ+θ1=htsinθd=Stsinθ+θ1−θd.The impact volumeVf is vot at moment t. The relation between the volume of the dead zone and flowing layer can be written as follows:(11)V=Vf+Va−Vb,where V is the volume of the dead zone and Va is the volume of the section OA of the flowing layer at the beginning of the impact. Vb is the volume of the flowing layer on the top of the dead zone (Section AD). Substituting equation (13) into equation (14) yields the following equation:(12)v0th0+Sth0=12S2tsinθ+θ1sinθ+θ1−θdsinθd+Stsinθ+θ1sinθ+θ1−θdha.By solving equation (12),S(t) can be expressed as follows:(13)St=hasinθ+θ1−hasinθ+θ1−θd+hasinθ+θ1−θd−hasinθ+θ12+2v0th0sinθ+θ1sinθdsinθ+θ1−θdsinθdsinθ+θ1.According to equations (10) and (13), h(t) can be written as follows:(14)ht=sinθ+θ1−θdsinθ+θ1,hasinθ+θ1−hasinθ+θ1−θd+hasinθ+θ1−θd−hasinθ+θ12+2v0th0sinθ+θ1sinθdsinθ+θ1−θd.Suppose the height of the flowing layer on the top of the dead zone keeps being equal to the flow heighth0, the pileup height h(t) at any moment can be obtained based on equation (15). The impact velocity Vd of the flowing layer against the rigid wall decreases with increasing in the volume of the dead zone. The friction effect and buffering effect of the dead zone on the flowing layer are the main reasons which lead to the decrease in flow velocity. The flow velocity drops to va when the flowing layer impacts on the dead zone. Then, the flow velocity of the flowing layer on the top of the dead zone further decreases because of the friction between the dead zone and the flowing layer. According to the kinetic energy theorem, the relationship between the impact velocity Vd on the rigid wall and va can be written as follows:(15)12va2=12vd2+gLtsinθd−θ+gtanφcosθd−θLt.Supposeva is the flow velocity after the flowing layer impacts on the dead zone. The coefficient of restitution is used for calculating va. In this study, the restitution coefficient κ is the ratio of the kinetic energy of particle before and after the collision. The flow velocity va after the collision can be written as follows [20]:(16)va=κv0.Substituting equation (15) into equation (16), the impact velocity vd of the flowing layer is expressed as follows:(17)vd=κv02−2gcosθhttanφcosθd−θ+sinθd−θsinθd.If the impact velocityVd is greater than 0, it is the continuous phase. If the impact velocity vd is equal to 0, it is the slope pileup phase. The pileup height reaches to critical value hcri at this moment. The critical pileup height hcri can be written as follows:(18)hcri=κv02sinθd2gcosθtanφcosθd−θ+sinθd−θ.The criterion for the flow phase as follows:(1)
IfV0 > Vc = 0.5 hcri2/sinθdsinθ1+θsinθ+θ1−θd, the impact is end up in the slope pileup phase. The maximum static load is close to the active Earth pressure of the critical part of the dead zone instead of the total of the dead zone.(2)
IfV0 >Vc = 0.5 hcri2/sinθdsinθ1+θsinθ+θ1−θd, the impact is end up in the continuous impact phase. The maximum static load is the active Earth pressure of the total of the dead zone.


## 5. Discussion

Faug [18] pointed out the flow state can be divided to 3 states based on the Froude number. The three states are dilute-dense flow, dense flow, and quasi-static flow, respectively. In field, the granular flows is usually dilute-dense flow or dense flow. If the Froude number is greater than 10, it is dilute-dense flow. Otherwise, it is dense flow. In this study, Froude is less than 10. The flow state of the granular flow is the dense flow. The effects of the flow state on the impact load are not considered into the analysis results. If the grade of the test materials is considered into the estimation results, the relationship between the average particle size D50 and the flow height may be a criteria of the flow state as follows:(1)
Ifh0 > D50, and Fr number < 10, dense flow(2)
Ifh0 < D50, dilute-dense flow(3)
Ifh0 > D50, and Fr number > 10, dilute-dense flowIf the granular flow is the dilute-dense flow, the stronger impulse load will impact on the rigid wall. It cannot be assumed that the granular flow is a continuous flow to estimate the impact load [12]. The impulse load should be estimated based on the Hertz theory [26]. It has been concluded that the particle size segregation can benefit to the fluidity of the flowing layer. It can lead to the increase in impact load [27]. Hence, how to quantitatively estimate the effect of the particle size segregation on impact load remains a challenge.


## 6. Conclusion

At present, it is still hard to accurately estimate the impact load of granular flow against a rigid wall. Hence, there is a technological issue what is the requirement of the estimation precision. For instants, the dynamic coefficient or the static coefficient can be determined up to 2 to 3 if the hydrodynamic model and hydrostatic model is used to estimate the maximum impact load respectively. In this study, we propose some suggestions to improve the estimation precision:(1)
The angle between the slope surface and rigid wall controls the dynamic impact load of the flowing layer at the preliminary stage. If the angle between the slope surface and the rigid wall is the same, the slope angle has an influence on the static load of dead zone against rigid wall. It means that the relationship between Froude number and hydrodynamic coefficient or hydrostatic coefficient may be different with the slope angle and installation angle. Whether the Froude number presents a negative exponential relationship with the hydrodynamic coefficient or presents a liner relationship with the hydrostatic coefficient needs further study.(2)
The Coulomb Earth pressure theory appropriates to calculate the static load. The dynamic impact load could be calculated using the hydrodynamic model. To conservatively estimate the dynamic impact load, the dynamic coefficient increases from 1 to 2 with the installation angle of rigid wall increasing from 90°−θ to 90°.(3)
The impact process can be divided into two phases according to whether the flowing layer has an impact on the rigid wall directly. The volume of granular flow controls the impact process at the end of which impact phase.(4)
The effects of volume of granular flow should be considered into the estimation of the maximum impact load. The impact load of the granular flow with a given flow velocityV0 and flow height h0 has a critical value. If the impact load reaches to this value, the impact load will not increase anymore because the dead zone has the friction and the buffering effects on the flowing layer.



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*Source: 2777308-2022-03-25.xml*