The lateral earth pressure is the pressure that soil exerts in the horizontal direction. It is important because it affects the consolidation behavior and strength of the soil and because it is considered in the design of geotechnical engineering structures such as retaining walls, basements, tunnels, deep foundations and braced excavations. The earth pressure problem dates from the beginning of the 18th century, when Gautier listed five areas requiring research, one of which was the dimensions of gravity-retaining walls needed to hold back soil. However, the first major contribution to the field of earth pressures was made several decades later by Coulomb, who considered a rigid mass of soil sliding upon a shear surface. Rankine extended earth pressure theory by deriving a solution for a complete soil mass in a state of failure, as compared with Coulomb's solution which had considered a soil mass bounded by a single failure surface. Originally, Rankine's theory considered the case of only cohesionless soils, with Bell subsequently extending it to cover the case of soils possessing both cohesion and friction. Caquot and Kerisel modified Muller-Breslau's equations to account for a nonplanar rupture surface.
The coefficient of lateral earth pressure The coefficient of lateral earth pressure, K, is defined as the ratio of the horizontal effective stress, σ’h, to the vertical effective stress, σ’v. The effective stress is the intergranular stress calculated by subtracting the pore water pressure from the total stress as described in soil mechanics. K for a particular soil deposit is a function of the soil properties and stress history. The minimum stable value of K is called the active earth pressure coefficient, Ka; the active earth pressure is obtained, for example, when a retaining wall moves away from the soil. The maximum stable value of K is called the passive earth pressure coefficient, Kp; the passive earth pressure would develop, for example against a vertical plow that is pushing soil horizontally. For a level ground deposit with zero lateral strain in the soil, the "at-rest" coefficient of lateral earth pressure, K0 is obtained. There are many theories for predicting lateral earth pressure; some are empirically based, and some are analytically derived.
Symbols definitions In this article, the following variables in the equations are defined as follows:
OCR Overconsolidation ratio β Angle of the backslope measured to the horizontal δ Wall friction angle θ Angle of the wall measured to the vertical φ Soil stress friction angle φ' Effective soil stress friction angle φ'cs Effective soil stress friction angle at critical state
At rest pressure The in situ lateral pressure of soil is called earth pressure at rest and it is generally calculated by the product of the overburden stress times the coefficient K0; the latter is called the coefficient of earth pressure at rest. K0 can be obtained directly in the field based on e.g. the dilatometer test (DMT) or a borehole pressuremeter test (PMT), although it is more commonly calculated using the well-known Jaky's formula. For loosely deposited sands at rest, Jaky showed analytically that K0 deviates from unity with downward trend as the sinusoidal term of the internal friction angle of material increases, i.e.
K 0 ( N C ) = 1 − sin ϕ ′ {\displaystyle K_{0(NC)}=1-\sin \phi '}
Jaky's coefficient has been proved later to be also valid for normally consolidated granular deposits and normally consolidated clays. From a purely theoretical point of view, the very simple 1 − sin ϕ ′ {\displaystyle 1-\sin \phi '} formula works ideally for the two extreme values of ϕ ′ {\displaystyle \phi '} , where for ϕ ′ {\displaystyle \phi '} = 0o it gives K 0 = 1 {\displaystyle K_{0}=1} referring to hydrostatic conditions and for ϕ ′ {\displaystyle \phi '} = 90o (theoretical value) it gives K 0 = 0 {\displaystyle K_{0}=0} referring to a frictional material that can stand vertically without support, thus, exerting no lateral pressure. These extreme cases are enough evidence that the correct expression for the coefficient of earth pressure at rest is the K 0 = 1 − sin ϕ ′ {\displaystyle K_{0}=1-\sin \phi '} . There is a general impression that Jaky's (1944) coefficient of earth pressure at rest is empirical and, indeed, the K 0 = 1 − sin ϕ ′ {\displaystyle K_{0}=1-\sin \phi '} expression is just a simplification of the expression below:
K 0 ( N C ) = 1 + 2 / 3 sin ϕ ′ 1 + sin ϕ ′ ( 1 − sin ϕ ′ ) {\displaystyle K_{0(NC)}={\frac {1+2/3\sin \phi '}{1+\sin \phi '}}(1-\sin \phi ')}
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