The steepest angle at which a sloping surface formed of loose material can be maintained without slumping is called the angle of repose. This is an important consideration in fields like geology, construction, and physics when dealing with granular materials like sand, gravel, or soil. The angle of repose depends on the shape, size, surface area, and friction coefficients of the material particles. In this article, we will discuss what factors affect the angle of repose, how it is measured, and provide some examples of typical angles of repose for common materials.

## What Factors Affect the Angle of Repose?

There are several factors that influence the angle of repose for a granular material:

### Particle Shape

Angular, jagged particles can interlock and form steeper slopes than rounded, smooth particles. Materials like crushed stone tend to have high angles of repose.

### Particle Size

Smaller particles stack more tightly and have higher angles of repose. Larger particles are more likely to roll down a sloping pile. Fine sands can form steeper slopes than coarse sands.

### Particle Size Distribution

A well-graded mix of different particle sizes allows for dense packing and high angles of repose. A uniform size leads to looser packing and shallower slopes.

### Surface Area

Materials with higher surface area, like clays, tend to have higher cohesion and friction, increasing the angle of repose. Smooth, low surface area particles like glass beads have lower angles.

### Moisture Content

Adding moisture tends to increase cohesion and capillary forces between particles, allowing steeper slopes. Too much water can lead to slumping.

### Compaction

Compacting material increases inter-particle friction and allows higher angle of repose. Loose, unpacked materials tend to have lower angles.

### Temperature

Increasing temperature can expand granular materials, decreasing friction and cohesion, leading to lower angles of repose. Cooler temperatures densify and strengthen particle contacts.

### Vibration

Vibrating or disturbing a material causes particles to settle and pack together densely, increasing the angle of repose.

So in summary, angular, small, varied particles with high surface area and cohesion allow the steepest stable slopes. Factors like moisture, compaction, and vibration also play a role.

## Measuring the Angle of Repose

There are a couple common methods used to measure the angle of repose in the lab or field:

### Tilting Box Method

The granular material is placed in a box with one transparent side. The box is gradually tilted until the material begins to slide in bulk, and the maximum angle is measured.

### Pile Method

The material is poured from a funnel or orifice onto a flat surface, forming a conical pile. The height and diameter of the pile are measured and used to calculate the angle of repose based on the geometry of the cone.

### Inclined Plane Method

The material is placed on an inclined metal or wood plate. The plate angle is slowly increased until the material begins to slide down. This maximum angle is the angle of repose.

The angle of repose measured depends somewhat on the method used and the standardization, but values are generally consistent. Multiple tests should be run and averaged to get accurate, reliable results.

## Typical Angles of Repose

Here are some examples of approximate angle of repose values for common granular materials:

Material | Angle of Repose |
---|---|

Talc, graphite | 10-15° |

Sand, gypsum, limestone | 30-35° |

Gravel | 35-40° |

Pebbles | 40-45° |

Crushed stone | 45-50° |

Materials like talc and graphite have very low cohesion and smooth, lubricated surfaces, leading to shallow angles around 10-15 degrees.

On the other hand, coarse, interlocking materials like gravel and crushed stone can sustain much steeper slopes approaching or even exceeding 45 degrees.

Most ordinary loose soils and sands fall somewhere in the 30-35 degree range. Finer cohesive powders like clay can get up to 40 degrees when dry.

Moist sands and clays may reach 32-33 degrees, as the water provides some cohesion between the particles. But too much moisture causes the materials to slump.

Well-graded mixes of particle sizes, like soil with rocks or pebbles mixed in, can pack together well and achieve higher angles around 40 degrees due to granular interlock between the different particle sizes.

For any material, the angle of repose depends significantly on the density or compaction. Loose initial packing leads to shallower angles, while dense, compacted materials can sustain much steeper slopes. Proper moisture conditioning and compaction is important for engineered fills like embankments and dams to achieve stable slopes.

## Geological Examples

Understanding angles of repose helps predict stable slopes in geological formations:

### Talus Slopes

The angle of repose for coarse rocky debris at the base of cliffs is typically 35-45 degrees. This matches the observed slope angles of talus deposits.

### Sand Dunes

The leeward slopes of sand dunes where sand avalanches down can reach the angles of repose of sand, around 30-35 degrees.

### Loess Deposits

Loess is an unconsolidated silt-sized sediment deposited by wind. It can stand at high angles up to 60-80 degrees when dry due to cohesion from clay minerals. But it is prone to landslides when saturated.

### Mine Tailings

Mine waste dumps of loose tailings are designed based on the angle of repose to prevent material sliding and maintain stability.

## Construction and Engineering Applications

The angle of repose is important for many engineering applications with granular materials:

### Excavation Slopes

Temporary excavation side slopes are limited to angles less than the angle of repose for safety against collapses.

### Levees and Dams

Earth embankments like levees and dams require proper compaction to achieve stable side slopes matching the angle of repose.

### Foundations

The stability of shallow foundations can be affected by undermining the soil beneath them, which depends on the soil’s angle of repose.

### Piling Granular Materials

Knowing the angle of repose dictates the maximum stable steepness when piling or storing bulk solid materials like ores, aggregates, and powders.

### Chutes and Hoppers

Design of chutes and hoppers for material handling depends on the angle of repose to avoid plugging and ensure smooth gravity flow.

### Friction and Wear

The angle of repose is related to the internal friction angle, a key design parameter for granular materials. The lower the angle of repose, the lower the shear resistance.

## Physics Experiments

Measuring the angle of repose is a common experiment in physics and engineering labs to study granular materials:

### Granular Flow

Flow of powders in hoppers can be modeled using angles of repose. Segregation of particle sizes can be studied as the piles form.

### Effect of Moisture

Adding controlled amounts of liquid and measuring angle changes analyzes capillary forces between particles.

### Effect of Packing

Pouring methods or compaction can vary initial density, providing insight into how packing affects inter-particle friction.

### Particle Shape Analysis

The angle of repose can characterize differences in shape between specimens by their slope stability. Angular particles have higher angles than rounded ones.

### Surface Area Effects

Using uniform sized particles of different materials reveals the influence of surface smoothness and area on cohesion and angles.

## Conclusion

The steepest stable angle for a loose granular pile is called the angle of repose, an important property for bulk solid materials. Particle shape, size distribution, compaction, and moisture content all affect the magnitude of the angle of repose, along with material properties like surface area. Typical values range from around 10-15 degrees for smooth, rounded particles up to 45 degrees or more for coarse, angular, interlocking materials. The angle of repose applies to critical engineering problems like the stability of slopes, foundations, and embankments. It also provides insight into physics of granular systems and is a simple experiment to study granular material properties and behavior. Understanding what factors control the maximum stable slopes of loose granular piles has broad implications across geology, construction, and physics.