Abstract
The maximum impact force of granular flow and its action point on a rigid barrier are the key indices of the anti-slip and the anti-overturning calculation, respectively. On the basis of the present impact force models and the observation in model experiments, this paper proposes a new semi-empirical impact force model focusing on the normal impact force and its point of action. By comparing the analytical solution and the experimental results, the new semi-empirical analytical model can estimate the normal impact force composed of both dynamic and static components with an error margin within ± 20% compared with the experimental results and so is for the corresponding point of action and tangential force. To calculate the residual force generated only by the static dead zone, the static friction angle between the dead zone and the chute base should be less than what was measured under kinetic conditions. Based on a large number of tests, it was found that assuming that the reaction force generated by the chute base operates at 2/3 of the deposition length makes the estimated normal impact force best suited to the experimental results.
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Abbreviations
- a :
-
Empirical constant a = 10.8
- b :
-
Width of chute (m)
- C d :
-
Empirical drag coefficient
- C u :
-
Uniformity coefficient
- D 50 :
-
Mean particle diameter (m)
- e oy :
-
y coordinate of centroid O0 (m)
- e ox :
-
x coordinate of centroid O0 (m)
- F :
-
Experimental total normal impact force (N)
- F m :
-
Experimental maximum total normal impact force (N)
- F r :
-
Experimental residual total normal impact force (N)
- F i :
-
Experimental sub-normal impact force (N), i is the number of the load cell
- F n :
-
Analytical total normal impact force (N)
- F d :
-
Dynamic impact force (N)
- F s :
-
Static force (N)
- F gf :
-
Gravity- and friction-induced force (N)
- F p :
-
Passive soil force (N)
- Fr :
-
Froude number
- G :
-
Weight of the dead zone (N)
- G cri :
-
Critical weight of the dead zone (N)
- g :
-
Gravitational acceleration (m s−2)
- H :
-
Height of granular deposition (m)
- h :
-
Experimental point of action of total normal impact force (m)
- h m :
-
Experimental point of action of the maximum total normal impact force (m)
- h r :
-
Experimental point of action of the residual total normal impact force (m)
- h 0 :
-
Thickness of the flowing layer (m)
- h 1 :
-
Depth measured vertically down from the front of the free surface (m)
- h a1 :
-
Distance between the point of action of R1 and O (m)
- h a2 :
-
Distance between the point of action of R2 and O (m)
- h i :
-
Distance from the center of each load cell to the bottom of the rigid barrier (m), i is the number of the load cell
- k :
-
Earth pressure coefficient
- L :
-
Length of granular deposition (m)
- l d1 :
-
Dead zone deposition length on the chute base (m)
- l d2 :
-
Dead zone deposition length on the rigid barrier (m)
- l cri :
-
Critical dead zone deposition length on the rigid barrier (m)
- M :
-
Bending moment (N m)
- n i :
-
Normal line of the chute base surface or rigid barrier surface, i = 1, 2
- n :
-
Empirical constant n = 1.3
- O :
-
Origin of the coordinate system
- O 0 :
-
Centroid of the dead zone mass
- R i :
-
Reaction force provided by the chute base surface or rigid barrier surface, i = 1, 2 (N)
- T :
-
Experimental total tangential force (N)
- T m :
-
Experimental total tangential force corresponding to the maximum total normal impact force (N)
- T r :
-
Experimental residual total tangential force (N)
- T i :
-
Experimental sub-tangential force (N), i is the number of the load cell
- T n :
-
Analytical tangential force (N)
- v :
-
Depth-averaged velocity (m/s)
- v n :
-
Shock wave traveling speed (m/s)
- x :
-
x coordinate (m)
- y :
-
y coordinate (m)
- α :
-
Slope angle (°)
- β :
-
Polar angle from 0° (°)
- Ө :
-
Angle of repose (°)
- Ө d :
-
Pileup angle (°)
- γ max :
-
Maximum dry unit weight (N/m3)
- γ min :
-
Minimum dry unit weight (N/m3)
- ρ d :
-
Dynamic density (kg/m3)
- ρ s :
-
Static density (kg/m3)
- δ 1 :
-
Interface friction angle of the chute base (°)
- δ 2 :
-
Interface friction angle of the rigid barrier (°)
- δ 3 :
-
Interface friction angle of the sidewall (°)
- φ :
-
Dynamic angle of repose (°)
- φ 0 :
-
Volume fraction of the flowing layer
- φ 1 :
-
Volume fraction of the dead zone
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Acknowledgments
A deep appreciation should be given to the CAS Pioneer Hundred Talents Program for making possible the completion of this research. The authors wish to acknowledge the financial contributions to this research and express their appreciation to the funding organizations for supporting this fundamental research.
Funding
The authors received joint support provided by the Strategic Priority Research Program of Chinese Academy of Sciences (XDA 23090202), the National Natural Science Foundation of China (Grant Nos. 41761144077, 41877524), and the opening fund of Shock and Vibration of Engineering Materials and Structures Key Laboratory of Sichuan Province (19kfgk01, 18kfgk10).
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Jiang, YJ., Fan, XY., Su, LJ. et al. Experimental validation of a new semi-empirical impact force model of the dry granular flow impact against a rigid barrier. Landslides 18, 1387–1402 (2021). https://doi.org/10.1007/s10346-020-01555-8
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DOI: https://doi.org/10.1007/s10346-020-01555-8