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Lateral earth pressure
Lateral earth pressure is the pressure that soil exerts in the horizontal direction. The lateral earth pressure 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 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 pressure from the total stress as described in soil mechanics. K for a particular soil deposit is a function of the soil properties and the stress history. The minimum stable value of K is called the active earth pressure coefficient, K_{a}; 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, K_{p}; 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 "atrest" coefficient of lateral earth pressure, K_{0} is obtained.
There are many theories for predicting lateral earth pressure; some are empirically based, and some are analytically derived.
Contents
At rest pressure
At rest lateral earth pressure, represented as K_{0}, is the in situ lateral pressure. It can be measured directly by a dilatometer test (DMT) or a borehole pressuremeter test (PMT). As these are rather expensive tests, empirical relations have been created in order to predict at rest pressure with less involved soil testing, and relate to the angle of shearing resistance. Two of the more commonly used are presented below.
Jaky (1948)^{[1]} for normally consolidated soils:
 <math> K_{0(NC)} = 1  \sin \phi ' \ </math>
Mayne & Kulhawy (1982)^{[2]} for overconsolidated soils:
 <math> K_{0(OC)} = K_{0(NC)} * OCR^{(\sin \phi ')} \ </math>
The latter requires the OCR profile with depth to be determined. OCR is the overconsolidation ratio and <math>\phi '</math> is the effective stress friction angle.
To estimate K_{0} due to compaction pressures, refer Ingold (1979)^{[3]}
Soil Lateral Active Pressure and Passive Resistance
The active state occurs when a retained soil mass is allowed to relax or deform laterally and outward (away from the soil mass) to the point of mobilizing its available full shear resistance (or engaged its shear strength) in trying to resist lateral deformation. That is, the soil is at the point of incipient failure by shearing due to unloading in the lateral direction. It is the minimum theoretical lateral pressure that a given soil mass will exert on a retaining that will move or rotate away from the soil until the soil active state is reached (not necessarily the actual inservice lateral pressure on walls that do not move when subjected to soil lateral pressures higher than the active pressure). The passive state occurs when a soil mass is externally forced laterally and inward (towards the soil mass) to the point of mobilizing its available full shear resistance in trying to resist further lateral deformation. That is, the soil mass is at the point of incipient failure by shearing due to loading in the lateral direction. It is the maximum lateral resistance that a given soil mass can offer to a retaining wall that is being pushed towards the soil mass. That is, the soil is at the point of incipient failure by shearing, but this time due to loading in the lateral direction. Thus active pressure and passive resistance define the minimum lateral pressure and the maximum lateral resistance possible from a give mass of soil.
Rankine theory
Rankine's theory, developed in 1857,^{[4]} is a stress field solution that predicts active and passive earth pressure. It assumes that the soil is cohesionless, the wall is frictionless, the soilwall interface is vertical, the failure surface on which the soil moves is planar, and the resultant force is angled parallel to the backfill surface. The equations for active and passive lateral earth pressure coefficients are given below. Note that φ' is the angle of shearing resistance of the soil and the backfill is inclined at angle β to the horizontal
 <math> K_a = \cos\beta \frac{\cos \beta  \left(\cos ^2 \beta  \cos ^2 \phi \right)^{1/2}}{\cos \beta + \left(\cos ^2 \beta  \cos ^2 \phi \right)^{1/2}}</math>
 <math> K_p = \cos\beta \frac{\cos \beta + \left(\cos ^2 \beta  \cos ^2 \phi \right)^{1/2}}{\cos \beta  \left(\cos ^2 \beta  \cos ^2 \phi \right)^{1/2}}</math>
For the case where β is 0, the above equations simplify to
 <math> K_a = \tan ^2 \left( 45  \frac{\phi}{2} \right) = \frac{ 1  \sin(\phi) }{ 1 + \sin(\phi) }</math>
 <math> K_p = \tan ^2 \left( 45 + \frac{\phi}{2} \right) = \frac{ 1 + \sin(\phi) }{ 1  \sin(\phi) } </math>
Coulomb theory
Coulomb (1776)^{[5]} first studied the problem of lateral earth pressures on retaining structures. He used limit equilibrium theory, which considers the failing soil block as a free body in order to determine the limiting horizontal earth pressure. The limiting horizontal pressures at failure in extension or compression are used to determine the K_{a} and K_{p} respectively. Since the problem is indeterminate,^{[6]} a number of potential failure surfaces must be analysed to identify the critical failure surface (i.e. the surface that produces the maximum or minimum thrust on the wall). Mayniel (1908)^{[7]} later extended Coulomb's equations to account for wall friction, symbolized by δ. MüllerBreslau (1906)^{[8]} further generalized Mayniel's equations for a nonhorizontal backfill and a nonvertical soilwall interface (represented by angle θ from the vertical).
 <math> K_a = \frac{ \cos ^2 \left( \phi  \theta \right)}{\cos ^2 \theta \cos \left( \delta + \theta \right) \left( 1 + \sqrt{ \frac{ \sin \left( \delta + \phi \right) \sin \left( \phi  \beta \right)}{\cos \left( \delta + \theta \right) \cos \left( \beta  \theta \right)}} \ \right) ^2}</math>
 <math> K_p = \frac{ \cos ^2 \left( \phi + \theta \right)}{\cos ^2 \theta \cos \left( \delta  \theta \right) \left( 1  \sqrt{ \frac{ \sin \left( \delta + \phi \right) \sin \left( \phi + \beta \right)}{\cos \left( \delta  \theta \right) \cos \left( \beta  \theta \right)}} \ \right) ^2}</math>
Instead of evaluating the above equations or using commercial software applications for this, books of tables for the most common cases can be used. Generally instead of K_{a}, the horizontal part K_{ah} is tabulated. It is the same as K_{a} times cos(δ+θ).
The actual earth pressure force E_{a} is the sum of a part E_{ag} due to the weight of the earth, a part E_{ap} due to extra loads such as traffic, minus a part E_{ac} due to any cohesion present.
E_{ag} is the integral of the pressure over the height of the wall, which equates to K_{a} times the specific gravity of the earth, times one half the wall height squared.
In the case of a uniform pressure loading on a terrace above a retaining wall, E_{ap} equates to this pressure times K_{a} times the height of the wall. This applies if the terrace is horizontal or the wall vertical. Otherwise, E_{ap} must be multiplied by cosθ cosβ / cos(θ − β).
E_{ac} is generally assumed to be zero unless a value of cohesion can be maintained permanently.
E_{ag} acts on the wall's surface at one third of its height from the bottom and at an angle δ relative to a right angle at the wall. E_{ap} acts at the same angle, but at one half the height.
Caquot and Kerisel
In 1948, Albert Caquot (1881–1976) and Jean Kerisel (1908–2005) developed an advanced theory that modified MullerBreslau's equations to account for a nonplanar rupture surface. They used a logarithmic spiral to represent the rupture surface instead. This modification is extremely important for passive earth pressure where there is soilwall friction. Mayniel and MullerBreslau's equations are unconservative in this situation and are dangerous to apply. For the active pressure coefficient, the logarithmic spiral rupture surface provides a negligible difference compared to MullerBreslau. These equations are too complex to use, so tables or computers are used instead.
Equivalent fluid pressure
Terzaghi and Peck, in 1948, developed empirical charts for predicting lateral pressures. Only the soil's classification and backfill slope angle are necessary to use the charts.
Bell's relationship
For soils with cohesion, Bell developed an analytical solution that uses the square root of the pressure coefficient to predict the cohesion's contribution to the overall resulting pressure. These equations represent the total lateral earth pressure. The first term represents the noncohesive contribution and the second term the cohesive contribution. The first equation is for an active situation and the second for passive situations.
 <math> \sigma_h = K_a \sigma_v  2c \sqrt{K_a} \ </math>
 <math> \sigma_h = K_p \sigma_v + 2c \sqrt{K_p} \ </math>
Coefficients of earth pressure
Coefficient of active earth pressure at rest
Coefficient of active earth pressure
Coefficient of passive earth pressure
See also
References
 Coduto, Donald (2001), Foundation Design, PrenticeHall, ISBN 0135897068
 California Department of Transportation Material on Lateral Earth Pressure
Notes
 ^ Jaky J. (1948) Pressure in silos, 2nd ICSMFE, London, Vol. 1, pp 103107.
 ^ Mayne, P.W. and Kulhawy, F.H. (1982). “K0OCR relationships in soil”. Journal of Geotechnical Engineering, Vol. 108 (GT6), 851872.
 ^ Ingold, T.S., (1979) The effects of compaction on retaining walls, Gèotechnique, 29, p265283.
 ^ Rankine, W. (1857) On the stability of loose earth. Philosophical Transactions of the Royal Society of London, Vol. 147.
 ^ Coulomb C.A., (1776). Essai sur une application des regles des maximis et minimis a quelques problemes de statique relatifs a l'architecture. Memoires de l'Academie Royale pres Divers Savants, Vol. 7
 ^ Kramer S.L. (1996) Earthquake Geotechnical Engineering, Prentice Hall, New Jersey
 ^ Mayniel K., (1808), Traité expérimental, analytique et preatique de la poussée des terres et des murs de revêtement, Paris.
 ^ MüllerBreslau H., (1906) Erddruck auf Stutzmauern, Alfred Kroner, Stuttgart.

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