angle of repose

File:TalusConesIsfjorden.jpg, Svalbard, Norway, showing angle of repose for coarse sediment]]

The angle of repose is sometimes used in the design of equipment for the processing of particulate solids. For example, it may be used to design an appropriate hopper or silo to store the material, or to size a conveyor belt for transporting the material. It can also be used in determining whether or not a slope (of a stockpile, or uncompacted gravel bank, for example) would likely collapse; the talus slope is derived from angle of repose and represents the steepest slope a pile of granular material can take. This angle of repose is also crucial in correctly calculating stability in vessels.

It is also commonly used by mountaineers as a factor in analysing avalanche danger in mountainous areas.{{citation needed |date=December 2016}}

If the coefficient of static friction μ_s is known of a material, then a good approximation of the angle of repose can be made with the following function. This function is somewhat accurate for piles where individual objects in the pile are minuscule and piled in random order.

{{cite book

|last1=Nichols |first1=E. L.

|last2=Franklin |first2=W. S.

|year=1898

|title=The Elements of Physics

|url=https://books.google.com/books?id=8IlCAAAAIAAJ

|volume=1 |page=101

|publisher=Macmillan

|lccn=03027633

}}

:$\backslash tan\{(\backslash theta)\}\; \backslash approx\; \backslash mu\_\backslash mathrm\{s\}\backslash ,$

where $\backslash theta$ is the angle of repose.

File:Free Body Diagram (Angle of Repose).png

A simple free body diagram can be used to understand the relationship between the angle of repose and the stability of the material on the slope. For the heaped material to resist collapse, the frictional forces must be equivalent to the horizontal component of the gravitational force $m\; g\; \backslash sin\backslash theta$, where $m$ is the mass of the material, $g$ is the gravitational acceleration and $\backslash theta$ is the slope angle:

:$m\; g\; \backslash sin\backslash theta\; =\; f$

The frictional force $f$ is equivalent to the multiplication product of the coefficient of static friction $\backslash mu$ and the Normal Force $N$ or $mg\backslash cos\backslash theta$:

:$m\; g\; \backslash sin\backslash theta\; =\; N\; \backslash mu$

:$m\; g\; \backslash sin\backslash theta\; =\; \backslash mu\; m\; g\; \backslash cos\backslash theta$

:$\backslash left\; (\; \backslash frac\{\backslash sin\; \backslash theta\}\{cos\backslash theta\}\; \backslash right\; )\; =\; \backslash mu$

:$\backslash theta\_R\; =\; \backslash arctan(\backslash mu)$

Where $\backslash theta\_R$ is the angle of repose, or the angle at which the slope fails under regular conditions, and $\backslash mu$ is the coefficient of static friction of the material on the slope.

There are numerous methods for measuring angle of repose and each produces slightly different results. Results are also sensitive to the exact methodology of the experimenter. As a result, data from different labs are not always comparable. One method is the triaxial shear test, another is the direct shear test.

The measured angle of repose may vary with the method used, as described below.

This method is appropriate for fine-grained, non-cohesive materials with individual particle size less than 10 mm. The material is placed within a box with a transparent side to observe the granular test material. It should initially be level and parallel to the base of the box. The box is slowly tilted until the material begins to slide in bulk, and the angle of the tilt is measured.

The material is poured through a funnel to form a cone. The tip of the funnel should be held close to the growing cone and slowly raised as the pile grows, to minimize the impact of falling particles. Stop pouring the material when the pile reaches a predetermined height or the base a predetermined width. Rather than attempt to measure the angle of the resulting cone directly, divide the height by half the width of the base of the cone. The inverse tangent of this ratio is the angle of repose.

The material is placed within a cylinder with at least one transparent end. The cylinder is rotated at a fixed speed, and the observer watches the material move within it. The effect is similar to watching clothes tumble over one another in a slowly rotating clothes dryer. The granular material assumes a certain angle as it flows within the rotating cylinder. This method is recommended for obtaining the dynamic angle of repose, which may vary from the static angle of repose measured by other methods.

File:Cornpiled2017.jpg has a low angle of repose]]

Here is a list of various materials and their angle of repose.

{{cite book

|last=Glover |first=T. J.

|year=1995

|title=Pocket Ref

|publisher=Sequoia Publishing

|isbn=978-1885071002

}} All measurements are approximated.

Different supports modify the shape of the pile, in the illustrations below sand piles, although angles of repose remain the same.Ileleji, K. E.. (2008-10-28). "The angle of repose of bulk corn stover particles". Powder Technology 187 (2): 110–118. {{doi|10.1016/j.powtec.2008.01.029}}.Lobo-Guerrero, Sebastian. (2007-03-23). "Influence of pile shape and pile interaction on the crushable behavior of granular materials around driven piles: DEM analyses" (em en). Granular Matter 9 (3–4): 241. {{doi|10.1007/s10035-007-0037-3}}. {{ISSN|1434-5021}}.

{{Multiple image|width=110px|align=center|image1=Sandpile Matemateca (1).webm|image2=Sandpile Matemateca (2).webm|image3=Sandpile Matemateca (4).webm|image4=Sandpile Matemateca (5).webm}}

File:Antlion trap.jpg

The larvae of the antlions and the unrelated wormlions Vermileonidae trap small insects such as ants by digging conical pits in loose sand, such that the slope of the walls is effectively at the critical angle of repose for the sand.

{{cite journal

|last1=Botz |first1=J. T.

|last2=Loudon |first2=C.

|last3=Barger |first3=J. B.

|last4=Olafsen |first4=J. S.

|last5=Steeples |first5=D. W.

|year=2003

|title=Effects of slope and particle size on ant locomotion: Implications for choice of substrate by antlions

|journal=Journal of the Kansas Entomological Society

|volume=76 |issue=3 |pages=426–435

}} They achieve this by flinging the loose sand out of the pit and permitting the sand to settle at its critical angle of repose as it falls back. Thus, when a small insect, commonly an ant, blunders into the pit, its weight causes the sand to collapse below it, drawing the victim toward the center where the predator that dug the pit lies in wait under a thin layer of loose sand. The larva assists this process by vigorously flicking sand out from the center of the pit when it detects a disturbance. This undermines the pit walls and causes them to collapse toward the center. The sand that the larva flings also pelts the prey with loose rolling material that prevents it from getting any foothold on the easier slopes that the initial collapse of the slope has presented. The combined effect is to bring the prey down to within grasp of the larva, which then can inject venom and digestive fluids.

{{Excerpt|Slope stability#Angle of repose}}

The angle of repose plays a part in several topics of technology and science, including:

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• Aeolian processes
• Barchan
• Bulk cargo
• Concrete slump test
• Grade (slope)
• Mass wasting
• Oceanic trench
• Retaining wall
• Rotary kiln
• Sand volcano

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Category:Particulates

Category:Shear strength

Category:Soil mechanics

Category:Engineering concepts