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| [[Image:Earth Pressure In Action.jpg|thumb|right|An example of lateral earth pressure overturning a [[retaining wall]]]]
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| '''Lateral earth pressure''' is the [[pressure]] that [[soil]] exerts against a structure in a sideways, mainly horizontal direction. The common applications of lateral earth pressure theory are for the design of ground [[engineering]] structures such as [[retaining wall]]s, [[basement]]s, [[tunnel]]s, and to determine the [[friction]] on the sides of [[deep foundation]]s.
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| To describe the pressure a soil will exert, an earth pressure [[coefficient]], K, is used. K is a function of the [[soil mechanics|soil properties]] and has a horizontal component K<sub>h</sub> and a smaller vertical component K<sub>v</sub>. K<sub>h</sub> has a value between 0 (completely solid) and 1 (completely liquid). Horizontal earth pressure is assumed to be [[Proportionality (mathematics)|directly proportional]] to the vertical pressure at any given point in the soil profile. K can also depend on the stress history of the soil. Lateral earth pressure coefficients are broken up into three categories: at-rest, active, and passive.
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| The pressure coefficient used in [[geotechnical engineering]] analyses depends on the characteristics of its application. There are many theories for predicting lateral earth pressure; some are [[empirically]] based, and some are analytically derived.
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| == At rest pressure ==
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| At rest lateral earth pressure, represented as K<sub>0</sub>, 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 [[shear strength (soil)|shearing resistance]]. Two of the more commonly used are presented below.
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| Jaky (1948)<ref>Jaky J. (1948) Pressure in silos, 2nd ICSMFE, London, Vol. 1, pp 103-107.</ref> for normally consolidated soils:
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| : <math> K_{0(NC)} = 1 - \sin \phi ' \ </math>
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| Mayne & Kulhawy (1982)<ref>Mayne, P.W. and Kulhawy, F.H. (1982). “K0-OCR relationships in soil”. Journal of Geotechnical
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| Engineering, Vol. 108 (GT6), 851-872.</ref> for overconsolidated soils:
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| : <math> K_{0(OC)} = K_{0(NC)} * OCR^{(\sin \phi ')} \ </math>
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| 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.
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| To estimate K<sub>0</sub> due to [[compaction]] pressures, refer Ingold (1979)<ref>Ingold, T.S., (1979) The effects of compaction on retaining walls, Gèotechnique, 29, p265-283.</ref>
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| == Active and passive pressure ==
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| [[Image:Retaining Wall Type Function.jpg|thumb|right|Different [[Retaining wall#Types of Retaining Wall|types of wall structures]] can be designed to resist earth pressure.]]
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| The active state occurs when a soil mass is allowed to relax or move outward to the point of reaching the limiting strength of the soil; that is, the soil is at the failure condition in extension. Thus it is the minimum lateral soil pressure that may be exerted. Conversely, the passive state occurs when a soil mass is externally forced to the limiting strength (that is, failure) of the soil in compression. It is the maximum lateral soil pressure that may be exerted. Thus active and passive pressures define the minimum and maximum possible pressures respectively that may be exerted in a horizontal plane.
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| === Rankine theory ===
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| [[Rankine theory|Rankine's theory]], developed in 1857,<ref>Rankine, W. (1857) On the stability of loose earth. Philosophical Transactions of the Royal Society of London, Vol. 147.</ref> is a stress field solution that predicts active and passive earth pressure. It 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 [[plane (geometry)|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 [[shear strength (soil)|shearing resistance]] of the soil and the backfill is inclined at angle β to the horizontal
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| : <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>
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| : <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>
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| For the case where β is 0, the above equations simplify to
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| :<math> K_a = \tan ^2 \left( 45 - \frac{\phi}{2} \right) = \frac{ 1 - \sin(\phi) }{ 1 + \sin(\phi) }</math>
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| :<math> K_p = \tan ^2 \left( 45 + \frac{\phi}{2} \right) = \frac{ 1 + \sin(\phi) }{ 1 - \sin(\phi) } </math>
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| === Coulomb theory ===
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| [[Charles-Augustin de Coulomb|Coulomb]] (1776)<ref>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</ref> 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''<sub>a</sub> and ''K''<sub>p</sub> respectively. Since the problem is [[Statically indeterminate|indeterminate]],<ref>Kramer S.L. (1996) Earthquake Geotechnical Engineering, Prentice Hall, New Jersey</ref> 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)<ref>Mayniel K., (1808), Traité expérimental, analytique et preatique de la poussée des terres et des murs de revêtement, Paris.</ref> later extended Coulomb's equations to account for wall friction, symbolized by ''δ''. Müller-Breslau (1906)<ref>Müller-Breslau H., (1906) Erddruck auf Stutzmauern, Alfred Kroner, Stuttgart.</ref> further generalized Mayniel's equations for a non-horizontal backfill and a non-vertical soil-wall interface (represented by angle θ from the vertical).
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| : <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>
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| : <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>
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| 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<sub>a</sub>, the horizontal part K<sub>ah</sub> is tabulated. It is the same as K<sub>a</sub> times cos(δ+θ).
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| The actual earth pressure force E<sub>a</sub> is the sum of a part E<sub>ag</sub> due to the weight of the earth, a part E<sub>ap</sub> due to extra loads such as traffic, minus a part E<sub>ac</sub> due to any cohesion present.
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| E<sub>ag</sub> is the integral of the pressure over the height of the wall, which equates to K<sub>a</sub> times the specific gravity of the earth, times one half the wall height squared.
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| In the case of a uniform pressure loading on a terrace above a retaining wall, E<sub>ap</sub> equates to this pressure times K<sub>a</sub> times the height of the wall. This applies if the terrace is horizontal or the wall vertical. Otherwise, E<sub>ap</sub> must be multiplied by cosθ cosβ / cos(θ − β).
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| E<sub>ac</sub> is generally assumed to be zero unless a value of cohesion can be maintained permanently.
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| E<sub>ag</sub> 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<sub>ap</sub> acts at the same angle, but at one half the height.
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| === Caquot and Kerisel ===
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| In 1948, [[Albert Caquot]] (1881–1976) and [[Jean Kerisel]] (1908–2005) developed an advanced theory that modified Muller-Breslau's equations to account for a non-planar 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 soil-wall friction. Mayniel and Muller-Breslau'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 Muller-Breslau. These equations are too complex to use, so tables or computers are used instead.
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| === Equivalent fluid pressure ===
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| [[Karl von Terzaghi|Terzaghi]] and [[Ralph Brazelton Peck|Peck]], in 1948, developed empirical charts for predicting lateral pressures. Only the [[Unified Soil Classification System|soil's classification]] and backfill slope angle are necessary to use the charts.
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| == Bell's relationship ==
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| 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 non-cohesive contribution and the second term the cohesive contribution. The first equation is for an active situation and the second for passive situations.
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| : <math> \sigma_h = K_a \sigma_v - 2c \sqrt{K_a} \ </math>
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| : <math> \sigma_h = K_p \sigma_v + 2c \sqrt{K_p} \ </math>
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| == Coefficients of earth pressure ==
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| Coefficient of active earth pressure at rest
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| Coefficient of active earth pressure
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| Coefficient of passive earth pressure
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| == See also ==
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| * [[Mohr-Coulomb theory]]
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| * [[Soil mechanics]]
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| == References ==
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| *{{Citation|surname=Coduto|given=Donald|author-link=|year=2001|title=Foundation Design|Place=|publisher=Prentice-Hall|isbn=0-13-589706-8|url=}}
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| * [http://www.dot.ca.gov/hq/esc/construction/manuals/OSCCompleteManuals/TrenchingandShoringManual2011.pdf California Department of Transportation Material on Lateral Earth Pressure]
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| ==Notes== | |
| {{reflist}}
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| {{Geotechnical engineering}}
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| [[Category:Soil mechanics]]
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