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, Ïƒ_{v }= Vertical Stress at depth ‘z’

Î³ = 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 Ïƒ_{x}, Ïƒ_{y}, and Ïƒ_{z}, in three principal directions. Let Æ_{x} be the strain induced in the direction of Ïƒ_{x}. Then we can write,

———————(2)

But in this case, Ïƒ_{x} = Ïƒ_{y} and Æ_{x} = 0, hence,

Ïƒ_{x }– µ(Ïƒ_{x} + Ïƒ_{z}) = 0

Or, Ïƒ_{x}(1- µ)- µÏƒ_{z} = 0

Or,

Or, ——————————(2.a)

Therefore, Ïƒ_{x }= K_{0}Î³z

where, —————————-(2-b)

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

K_{0} = 1 – sinÉ¸ ——————(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:

K_{0} = 0.95 – sinÉ¸ ——————(4)

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

K_{0} = 0.9 (1 – sinÉ¸) ———————-(5)

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

———–(6)

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

K_{0} = 0.19 + 0.233Log_{10}(PI) ——————-(7)

Where, PI = plasticity index.

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

K_{0(OC)} = K_{0(NC)} Ã— OCR^{n} ———————–(8)

Where, n = 0.54 Ã— 10^{(- Ip/281)} ——————(9)