When a soil mass gets deposited naturally or artificially, the coefficient of earth pressure attains a value known as coefficient of earth pressure at rest (K_{0}) which is intermediate between coefficient of active earth pressure (K_{a}) and coefficient of passive earth pressure (K_{p}) values. In this case the horizontal stress is given by,

—————————–(1)

Where, _{ }= Vertical Stress at depth ‘z’

= Unit weight of soil

z = Depth below ground surface

The value of K_{0} depends upon the relative density of sand, and the process by which the process of deposition has taken place. If the process does not involve artificial tamping, the value of K_{0} ranges from about 0.4 for loose sand to 0.6 for dense sand. Artificial tamping increases the value of K_{0} which is about 0.8.

### Derivation of earth pressure at rest (K_{0})

The value of K_{0} can be obtained by the theory of elasticity. Consider a cylindrical sample of soil which is acted upon by stresses , in three principal directions. Let be the strain induced in the direction of . Then we can write,

———————(2)

But in this case,

Hence,

Or,

Or,

Or, ——————————(2.a)

Therefore,

where, —————————-(2-b)

Jacky in 1948 has suggested an approximate relationship for K_{0}. The relation is:

——————(3)

For normally consolidated clay, the value of K_{0} is less than 1. For pre-consolidated soils with over-consolidation ratio more than 4.5, K_{0} could be 1 or even more. For normally consolidated clays,

Brooke and Ireland (1965) suggested relation for K_{0} as:

——————(4)

Fraser (1957) suggested relation for K_{0} as:

———————-(5)

Kezdi (1962) suggested relation for K_{0} as:

———–(6)

For normally consolidated clay, Alpan in 1967 suggested relation for K_{0} as :

——————-(7)

Where, PI = plasticity index.

For over-consolidated clay, Alpan in 1967 suggested relation for K_{0} as:

———————–(8)

Where, ——————(9)