Angle of repose

# Angle of repose

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The angle of repose or, more precisely, the critical angle of repose, of a granular material
Granular material
A granular material is a conglomeration of discrete solid, macroscopic particles characterized by a loss of energy whenever the particles interact . The constituents that compose granular material must be large enough such that they are not subject to thermal motion fluctuations...

is the steepest angle of descent or dip
Strike and dip
Strike and dip refer to the orientation or attitude of a geologic feature. The strike line of a bed, fault, or other planar feature is a line representing the intersection of that feature with a horizontal plane. On a geologic map, this is represented with a short straight line segment oriented...

of the slope relative to the horizontal plane when material on the slope face is on the verge of sliding. This angle is in the range 0°–90°.

When bulk granular materials are poured onto a horizontal surface, a conical
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

pile will form. The internal angle between the surface of the pile and the horizontal surface is known as the angle of repose and is related to the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

, surface area
Surface area
Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces...

and shapes of the particles, and the coefficient of friction of the material. Material with a low angle of repose forms flatter piles than material with a high angle of repose.

The term has a related usage in mechanics
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....

, where it refers to the maximum angle at which an object can rest on an inclined plane
Inclined plane
The inclined plane is one of the original six simple machines; as the name suggests, it is a flat surface whose endpoints are at different heights. By moving an object up an inclined plane rather than completely vertical, the amount of force required is reduced, at the expense of increasing the...

without sliding down. This angle is equal to the arctangent of the coefficient of static friction μs between the surfaces.

## Applications of theory

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
Silo
A silo is a structure for storing bulk materials.Silo may also refer to:* Silo , a 3D modeling software* Silo , a defunct chain of retail electronics stores* SILO , used in Linux...

to store the material, or to size a conveyor belt
Conveyor belt
A conveyor belt consists of two or more pulleys, with a continuous loop of material - the conveyor belt - that rotates about them. One or both of the pulleys are powered, moving the belt and the material on the belt forward. The powered pulley is called the drive pulley while the unpowered pulley...

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) will likely collapse; the talus
Scree
Scree, also called talus, is a term given to an accumulation of broken rock fragments at the base of crags, mountain cliffs, or valley shoulders. Landforms associated with these materials are sometimes called scree slopes or talus piles...

slope is derived from angle of repose and represents the steepest slope a pile of granular material will take. This angle of repose is also crucial in correctly calculating stability
Slope stability
The field of slope stability encompasses the analysis of static and dynamic stability of slopes of earth and rock-fill dams, slopes of other types of embankments, excavated slopes, and natural slopes in soil and soft rock...

in vessels.

It is also commonly used by mountaineers
Mountaineering
Mountaineering or mountain climbing is the sport, hobby or profession of hiking, skiing, and climbing mountains. While mountaineering began as attempts to reach the highest point of unclimbed mountains it has branched into specialisations that address different aspects of the mountain and consists...

as a factor in analysing avalanche
Avalanche
An avalanche is a sudden rapid flow of snow down a slope, occurring when either natural triggers or human activity causes a critical escalating transition from the slow equilibrium evolution of the snow pack. Typically occurring in mountainous terrain, an avalanche can mix air and water with the...

danger in mountainous areas.

## Measurement

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
Direct Shear Test
A direct shear test also known as shearbox test is a laboratory or field test used by geotechnical engineers to measure the shear strength properties of soil or rock material, or of discontinuties in soil or rock masses.- Soil :...

.

If the coefficient of static friction 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.

Where:
• μs is the coefficient of static friction
• θ is the angle of repose

Procedure to Measure and Calculate the Angle of Repose
1. Pour the dry sand into a pile on a level surface allowing it to build a pile from the top. This will result in a pile with a relatively circular base, making measurement easier.
2. Using the ruler and a tape measure, measure the height (h) of the pile of sand from the peak to the ground. Stand the ruler next to the pile so it can be read easily. Extend the tape measure carefully to the top of the pile without disturbing the pile and allow the other end of the tape measure to intersect the ruler. While keeping the tape measure level, observe the intersection of the tape measure with the ruler. Write the value on the paper. (For example: if the height of the pile is 12 units, write h = 12)
3. Using the tape measure, measure the horizontal distance (d) from the middle of the pile to the edge. Place the tape measure on the ground beside the pile. Line up one end with one side of the pile and extend the tape measure to the other end of the pile. Write the value on the paper and divide by 2. This will give you the distance from the centre of the pile to the edge. (For example: If the total distance on tape measure from one end of the pile to the other is 30 units, divide by 2 to get 15. d= 15)
4. The equation for calculating the angle of repose is tan-1(h/d) . Using your scientific calculator, multiply height (h calculated in Step 2) by 2 and divide this value by the distance (d calculated in Step 3). Then, hit the inverse tangent key (or tan-1) and the answer just calculated. This will give you the angle of repose, α.
5. Place the protractor on the level surface next to the pile of sand. Using the ruler, create a straight line from the peak of the sand pile down the slope. Read the angle of repose value (α) and write the value on the paper.
6. Compare the calculated angle of repose from Step 4 and the measured angle of repose from Step 5. If the values are not within 1 degree of each other, repeat Step 5.

## Exploitation by antlion and wormlion (Vermileonidae) larvae

The larvae of the antlion
Antlion
Antlions are a group of insects in the family Myrmeleontidae . The most well-known genus is Myrmeleo. There are about 2,000 species...

s and the unrelated wormlions Vermileonidae
Vermileonidae
The Brachyceran family Vermileonidae is a small family of uncertain affinities and unusual biology, containing fewer than 80 rare species in 10 genera. Historically the Vermileonids had been regarded as belonging to the family Rhagionidae, possibly in a subfamily Vermileoninae...

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. 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 so much loose, rolling material as to prevent 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.

## Methods in Determining the Angle of Repose

Tilting Box Method
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 at a rate of approximately .3 degrees/second. Tilting is stopped when the material begins to slide in bulk, and the angle of the tilt is measured.

Fixed Funnel Method
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.

Revolving Cylinder Method
The material is placed within a cylinder with at least one transparent face. The cylinder is rotated at a fixed speed and the observer watches the material moving within the rotating cylinder. The effect is similar to watching clothes tumble over one another in a slowly rotating clothes dryer. The granular material will assume a certain angle as it flows within the rotating cylinder. This method is recommended for obtaining the dynamic angle of repose, and may vary from the static angle of repose measured by other methods. When describing the angle of repose for a substance, always specify the method used.

## Angle of Repose of various materials

Here is a list of various materials and their angle of repose. All measurements are approximated and not exact.
Material (condition) Angle of Repose (degrees)
Ash
Wood ash
Wood ash is the residue powder left after the combustion of wood. Main producers of wood ash are wood industries and power plants.-Composition:...

es
40°
Asphalt
Asphalt
Asphalt or , also known as bitumen, is a sticky, black and highly viscous liquid or semi-solid that is present in most crude petroleums and in some natural deposits, it is a substance classed as a pitch...

(crushed)
30–45°
Bark
Bark
Bark is the outermost layers of stems and roots of woody plants. Plants with bark include trees, woody vines and shrubs. Bark refers to all the tissues outside of the vascular cambium and is a nontechnical term. It overlays the wood and consists of the inner bark and the outer bark. The inner...

(wood refuse)
45°
Bran
Bran
Bran is the hard outer layer of grain and consists of combined aleurone and pericarp. Along with germ, it is an integral part of whole grains, and is often produced as a by-product of milling in the production of refined grains. When bran is removed from grains, the grains lose a portion of their...

30–45°
Chalk
Chalk
Chalk is a soft, white, porous sedimentary rock, a form of limestone composed of the mineral calcite. Calcite is calcium carbonate or CaCO3. It forms under reasonably deep marine conditions from the gradual accumulation of minute calcite plates shed from micro-organisms called coccolithophores....

45°
Clay
Clay
Clay is a general term including many combinations of one or more clay minerals with traces of metal oxides and organic matter. Geologic clay deposits are mostly composed of phyllosilicate minerals containing variable amounts of water trapped in the mineral structure.- Formation :Clay minerals...

(dry lump)
25–40°
Clay
Clay
Clay is a general term including many combinations of one or more clay minerals with traces of metal oxides and organic matter. Geologic clay deposits are mostly composed of phyllosilicate minerals containing variable amounts of water trapped in the mineral structure.- Formation :Clay minerals...

(wet excavated)
15°
Clover seed
Clover
Clover , or trefoil, is a genus of about 300 species of plants in the leguminous pea family Fabaceae. The genus has a cosmopolitan distribution; the highest diversity is found in the temperate Northern Hemisphere, but many species also occur in South America and Africa, including at high altitudes...

28°
Coconut
Coconut
The coconut palm, Cocos nucifera, is a member of the family Arecaceae . It is the only accepted species in the genus Cocos. The term coconut can refer to the entire coconut palm, the seed, or the fruit, which is not a botanical nut. The spelling cocoanut is an old-fashioned form of the word...

(shredded)
45°
Coffee bean
Coffee bean
A coffee bean is a seed of a coffee plant. It is the pit inside the red or purple fruit often referred to as a cherry. Even though they are seeds, they are referred to as 'beans' because of their resemblance to true beans. The fruits - coffee cherries or coffee berries - most commonly contain two...

(fresh)
35–45°
Earth
Soil
Soil is a natural body consisting of layers of mineral constituents of variable thicknesses, which differ from the parent materials in their morphological, physical, chemical, and mineralogical characteristics...

30–45°
Flour (wheat) 45°
Granite
Granite
Granite is a common and widely occurring type of intrusive, felsic, igneous rock. Granite usually has a medium- to coarse-grained texture. Occasionally some individual crystals are larger than the groundmass, in which case the texture is known as porphyritic. A granitic rock with a porphyritic...

35–40°
Gravel
Gravel
Gravel is composed of unconsolidated rock fragments that have a general particle size range and include size classes from granule- to boulder-sized fragments. Gravel can be sub-categorized into granule and cobble...

(loose dry)
30–45°
Gravel
Gravel
Gravel is composed of unconsolidated rock fragments that have a general particle size range and include size classes from granule- to boulder-sized fragments. Gravel can be sub-categorized into granule and cobble...

(natural w/ sand)
25–30°
Malt
Malt
Malt is germinated cereal grains that have been dried in a process known as "malting". The grains are made to germinate by soaking in water, and are then halted from germinating further by drying with hot air...

30–45°
Sand
Sand
Sand is a naturally occurring granular material composed of finely divided rock and mineral particles.The composition of sand is highly variable, depending on the local rock sources and conditions, but the most common constituent of sand in inland continental settings and non-tropical coastal...

(dry)
34°
Sand
Sand
Sand is a naturally occurring granular material composed of finely divided rock and mineral particles.The composition of sand is highly variable, depending on the local rock sources and conditions, but the most common constituent of sand in inland continental settings and non-tropical coastal...

(water filled)
15–30°
Sand
Sand
Sand is a naturally occurring granular material composed of finely divided rock and mineral particles.The composition of sand is highly variable, depending on the local rock sources and conditions, but the most common constituent of sand in inland continental settings and non-tropical coastal...

(wet)
45°
Wheat
Wheat
Wheat is a cereal grain, originally from the Levant region of the Near East, but now cultivated worldwide. In 2007 world production of wheat was 607 million tons, making it the third most-produced cereal after maize and rice...

28°

## See also

The angle of repose plays a part in several topics of technology and science, including:
• Aeolian processes
• Avalanche
Avalanche
An avalanche is a sudden rapid flow of snow down a slope, occurring when either natural triggers or human activity causes a critical escalating transition from the slow equilibrium evolution of the snow pack. Typically occurring in mountainous terrain, an avalanche can mix air and water with the...

• Barchan
Barchan
A barchan dune, also barkhan is an arc-shaped sand ridge, comprising well-sorted sand. This type of dune possesses two "horns" that face downwind, with the slip face at the angle of repose of sand, or approximately 35 degrees . The upwind side is packed by the wind, and stands at about 15 degrees...

• Bulk cargo
Bulk cargo
Bulk cargo is commodity cargo that is transported unpackaged in large quantities. This cargo is usually dropped or poured, with a spout or shovel bucket, as a liquid or as a mass of relatively small solids , into a bulk carrier ship's hold, railroad car, or tanker truck/trailer/semi-trailer body...

• Concrete slump test
Concrete slump test
The concrete slump test is used for the measurement of a property of fresh concrete.The test is an emprical test that measures the workability of fresh concrete. More specifically, it measures the consistency of the concrete in that specific batch. It is also used to determine consistency between...

• Mass wasting
Mass wasting
Mass wasting, also known as slope movement or mass movement, is the geomorphic process by which soil, regolith, and rock move downslope under the force of gravity. Types of mass wasting include creep, slides, flows, topples, and falls, each with its own characteristic features, and taking place...

• Oceanic trench
Oceanic trench
The oceanic trenches are hemispheric-scale long but narrow topographic depressions of the sea floor. They are also the deepest parts of the ocean floor....

• Sand volcano
Sand volcano
A sand volcano or sand blow is a cone of sand formed by the ejection of sand onto a surface from a central point. The sand builds up as a cone with slopes at the sand's angle of repose. A crater is commonly seen at the summit...

• Scree
Scree
Scree, also called talus, is a term given to an accumulation of broken rock fragments at the base of crags, mountain cliffs, or valley shoulders. Landforms associated with these materials are sometimes called scree slopes or talus piles...