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 GautierGautier, H. Dissertation sur L’epaisseur des Culées des Ponts, sur la Largeur des Piles, sur la Portée des Voussoirs, sur L’effort Et la Pesanteur des Arches À Differens Surbaissemens, Et sur Les Profils de Maçonnerie Qui Doivent Supporter des Chaussées, des Terrasse; Chez André Cailleau: Paris, France, 1717; . 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,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 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 BellBell, A.L. The lateral pressure and resistance of clay and the supporting power of clay foundations. Minutes Proc. Inst. Civ. Eng. 1915, 199, 233–272. 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, JakyJaky, J. The coefficient of earth pressure at rest. J. Soc. Hung. Archit. Eng. 1944, 78, 355–388.Jaky, J. Pressure in silos. In Proceedings of the 2nd International Conference on Soil Mechanics and Foundation Engineering ICSMFE, London, UK, 21–30 June 1948; pp. 103–107. showed analytically that K0 deviates from unity with downward trend as the sinusoidal term of the internal friction angle of material increases, i.e.
Jaky's coefficient has been proved later to be also valid for normally consolidated granular depositsBishop, A.W.; Eldin, A.K.G. The effect of stress history on the relation between and porosity in sand. In Proceedings of the 3rd International Conference Soil Mechanics, Zurich, Switzerland, 16–27 August 1953; pp. 100–105.Hendron, A.J., Jr. The Behavior of Sand in One-Dimensional Compression. Ph.D. Thesis, University of Illinois, Urbana, IL, USA, 1963.Saglamer, A. Soil parameters affecting coefficient of earth pressure at rest of cohesionless soils. In Proceedings of the Istanbul conference on soil/mechanics and Foundation Engineering, Istanbul, Turkey, 31 March–4 April 1975; pp. 9–16. and normally consolidated clays.Brooker, E.W.; Ireland, H.O. Earth pressures at rest related to stress history. Can. Geotech. J. 1965, 2, 1–15.Abdelhamid, M.S.; Krizek, R.J. At-rest lateral earth pressure of a consolidating clay. J. Geotech. Geoenviron. Eng. 1976, 102, 721–738.Abdelhamid, M.S.; Krizek, R.J. At Rest Lateral Earth Pressures of a Consolidating Clay. J. Geotech. Geoenviron. Eng. 1977, 103, 820–821.
From a purely theoretical point of view, the very simple formula works ideally for the two extreme values of , where for = 0o it gives referring to hydrostatic conditions and for = 90o (theoretical value) it gives 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 .
There is a general impression that Jaky's (1944) coefficient of earth pressure at rest is empirical and, indeed, the expression is just a simplification of the expression below:
However, the latter derives from a fully analytical procedure and corresponds to an intermediate state between the state at rest and the active state (for more information see Pantelidis).
As mentioned earlier, according to the literature, Jaky's equation matches very well the experimental data for both normally consolidated sands and clays. Some researchers state, however, that slightly modified forms of Jaky's equation show better fit to their data. However, although some of these modifications gained great popularity, they do not provide better estimations for . For example, Brooker and Ireland's has been based on the laboratory determination of of only five samples, whilst the effective angle of shearing resistance of three of them was obtained from the literature, having no control on them. Moreover, refinements in the order of a few percentage points rather support the validity of the expression than the superiority of the refined expression.
For overconsolidated soils, Mayne & KulhawyMayne, P.W. and Kulhawy, F.H. (1982). "K0-OCR relationships in soil". Journal of Geotechnical Engineering, Vol. 108 (GT6), 851-872. suggested the following expression:
The latter requires the OCR profile with depth to be determined. OCR is the overconsolidation ratio and is the effective stress friction angle.
To estimate K0 due to compaction pressures, refer to Ingold (1979)Ingold, T.S., (1979) The effects of compaction on retaining walls, Gèotechnique, 29, p265-283.
Pantelidis offered an analytical expression for the coefficient of earth pressure at rest, applicable to cohesive-frictional soils and both horizontal and vertical pseudo-static conditions, which is part of a unified continuum mechanics approach (the expression in question is given in the section below).
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 engaging 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 wall that will move or rotate away from the soil until the soil active state is reached (not necessarily the actual in-service 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 given mass of soil.
Coulomb's earth pressure coefficients Coulomb (1776) first studied the problem of lateral earth pressures on retaining structures. He used the 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 and respectively. Since the problem is indeterminate,Kramer S.L. (1996) Earthquake Geotechnical Engineering, Prentice Hall, New Jersey 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). Coulombs main assumption is that the failure surface is planar. In simpler terms, Coulomb's theory imagines a wedge of soil behind the wall that is just on the verge of sliding. By comparing the forces trying to push the wedge forward with those resisting movement, the maximum or minimum pressure on the wall can be estimated.
Mayniel (1808)Mayniel K., (1808), Traité expérimental, analytique et preatique de la poussée des terres et des murs de revêtement, Paris. later extended Coulomb's equations to account for wall friction, denoted by . Müller-Breslau (1906)Müller-Breslau H., (1906) Erddruck auf Stutzmauern, Alfred Kroner, Stuttgart. further generalized Mayniel's equations for a non-horizontal backfill and a non-vertical soil-wall interface (represented by an angle from the vertical). The symbols and represent dimensionless "earth pressure coefficients": for active pressure (when the wall moves away from the soil and the soil expands slightly) and for passive pressure (when the wall moves into the soil and the soil is compressed). These coefficients scale the weight of the soil to give the actual horizontal pressure on the wall.
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 , the horizontal part is tabulated. It is the same as times . [Note that under certain conditions, the equation for "blows up". For example, if and , then .]
The actual earth pressure force is the sum of the part due to the weight of the earth, a part due to extra loads such as traffic, minus a part due to any cohesion present.
is the integral of the pressure over the height of the wall, which equates to times the specific gravity of the earth, times one half the wall height squared.