Lateral earth pressure coefficients are considered when earth retaining structures are designed. Depending on the lateral movement of the soil
and the structure, they are categorized into three types. The three categories of earth coefficients are as follows:

• Coefficient of at-rest earth pressure
• Coefficient of active earth pressure
• Coefficient of passive earth pressure

At- rest earth pressure coefficient for cohesionless soil is calculated using Jaky's formula, 1944:

\(K_{0(\text{NC})} = \frac{1 + \frac{2}{3} \sin \phi'}{1 + \sin \phi'} (1 - \sin \phi')\), or as a simplification:\([ K_0 = 1 - \sin(\phi) ]\)

For cohesive soil the following equation can also be used:

\(K_0 = 0.44 + 0.42\left(\frac{\text{PI}(\%)}{100}\right)\)

Active and passive earth pressure coefficients are calculated using Rankine's theory, 1857. The theory assumes that the soil is cohesionless, the wall is frictionless, the soil-wall interface is vertical, the failure surface on which the soil moves is planar and, and the resultant force is angled parallel to the backfill surface. Note that '\(\phi\)' is the angle of shearing resistance of the soil and the backfill is
inclined at angle '\(\beta\)'to the horizontal. The equations for active and passive lateral earth pressure coefficients are given below.

Active Earth Pressure:

\(K_a = \frac{\cos \beta - (\cos^2 \beta - \cos^2 \phi)^{1/2}}{\cos \beta + (\cos^2 \beta - \cos^2 \phi)^{1/2}} *\cos \beta \quad \text{when} \quad \beta = \tan^2 \left( 45^\circ - \frac{\phi}{2} \right)\)

Passive Earth Pressure:

\(K_p = \frac{\cos \beta + (\cos^2 \beta - \cos^2 \phi)^{1/2}}{\cos \beta - (\cos^2 \beta - \cos^2 \phi)^{1/2}} *\cos \beta \quad \text{when} \quad \beta = \tan^2 \left( 45^\circ + \frac{\phi}{2} \right)\)

1. Rankine,W.(1856)On the stability of loose earth. Philosophica lTransactions of the Royal Society of London, Vol. 147.
2. Jaky, J. The coefficient of earth pressure ate rest. J. Soc. Hung. Archit. Eng. 1944, 78, 355-388.
3. Jaky, J. Pressure in silos. In Proceeding of the 2nd International Conference on Soil Mechanics and Foundation Engineering ICSMFE, London, UK, 21-30 June 1948; pp. 103-107.