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 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, the Rankine's theory considered the case of only cohesionless soils. However, this theory has subsequently been extended by Bell 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’, to the vertical effective stress, σ’. 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; 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; 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, K 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: 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 K; the latter is called the coefficient of earth pressure at rest. K 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 Ko deviates from unity with downward trend as the sinusoidal term of the internal friction angle of material increases, i.e. formula_1 Jaky 's coefficient has been proved later to be also valid for normally consolidated granular deposits and normally consolidated clays. From the purely theoretical point of view, the very simple formula_2 formula, works ideally for the two extreme values of formula_3, where for formula_3 = 0 it gives formula_5 referring to hydrostatic conditions and for formula_3 = 90 (theoretical value) it gives formula_7 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 formula_8. There  is a general impression that Jaky's (1944) coefficient of earth pressure at rest is empirical and, indeed, the formula_8 expression is just a simplification of the expression below: formula_10 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 formula_8 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 shows better fit to their  data. However, although some of these modifications gained great popularity, they do not provide better estimation for formula_12. For example, Brooker and Ireland's formula_13 has been based on the laboratory determination of formula_12 of only five samples, whilst the effective angle of shearing resistance of the 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 formula_8 expression than the superiority of the refined expression. For overconsolidated soils Mayne & Kulhawy suggest the following expression: The latter requires the OCR profile  with depth to be determined. OCR is the overconsolidation ratio and formula_17 is the effective stress friction angle. To estimate K due to compaction pressures, refer Ingold (1979) 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 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. Rankine's earth pressure coefficients and Bell's extension for cohesive soils. Rankine's theory, developed in 1857, is a stress field solution that predicts active and passive earth pressure. It assumes that the soil is cohesionless, the wall is non-battered and frictionless whilst the backfill is horizontal. The failure surface on which the soil moves is planar. The expressions for the active and passive lateral earth pressure coefficients are  given below. formula_25 formula_26 where, β is the backfill inclination angle. Coulomb's earth pressure coefficients. Coulomb (1776) 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" and "K" respectively. Since the problem is indeterminate, 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. Mayniel (1908) later extended Coulomb's equations to account for wall friction, denoted by "δ". Müller-Breslau (1906) further generalized Mayniel's equations for a non-horizontal backfill and a non-vertical soil-wall interface (represented by angle θ from the vertical). 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, the horizontal part K is tabulated. It is the same as K times cos(δ+θ). The  actual earth pressure force E is the sum of the part E due to the weight of the earth, a part E due to extra loads such as traffic, minus a part E due to any cohesion present. E is the integral of the pressure over the height of the wall, which equates to K 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 equates to this pressure times K times the height of the wall. This applies if  the terrace is horizontal or the wall vertical. Otherwise, E must be multiplied by cos θ cosβ / cos(θ − β). E is generally assumed to be zero unless a value of cohesion can be maintained permanently. E 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 acts at the same angle, but at one half the height. Caquot and Kerisel's analysis for log-spiral failure surfaces. 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. Mononobe-Okabe's and Kapilla's earth pressure coefficients for dynamic conditions . Mononobe-Okabe's and Kapilla's earth pressure coefficients for dynamic active and passive conditions respectively have been obtained on the same basis as Coulomb's solution. These coefficients are given below: formula_29formula_30with horizontal components of earth pressure: formula_25 formula_26 where, formula_33 and formula_34 are the seismic coefficients of horizontal and vertical acceleration respectively, formula_35, formula_36 is the back face inclination angle of the structure with respect to vertical, formula_37 is the angle of friction between structure and soil and formula_38 is the back slope inclination. The above coefficients are included in numerous seismic design codes worldwide (e.g., EN1998-5 , AASHTO), since being suggested as standard methods by Seed and Whitman. The problems with these two solutions are known (e.g., see Anderson]) with the most important one being the square root of negative number for formula_39 (the minus sign stands for the active case whilst the plus sign stands for the passive case). The various design codes recognize the problem with these coefficients and they either attempt an interpretation, dictate a modification of these equations, or propose alternatives. In this respect: It is noted that the above empirical corrections on formula_33 made by AASHTO and the Building Seismic Safety Council  return coefficients of earth pressure very close to those derived by the analytical solution proposed by Pantelidis (see below). Mazindrani and Ganjale's approach for cohesive-frictional soils with inclined surface. Mazindrani and Ganjale presented an analytical solution to the problem of earth pressures exerted on a frictionless, non-battered wall by a cohesive-frictional soil with inclined surface. The derived equations are given below for both the active and passive states: formula_44 formula_45 with horizontal components for the active and passive earth pressure are: formula_25 formula_26 ka and kp coefficients for various values of formula_48, formula_36, and formula_50 can  be found in tabular form in Mazindrani and Ganjale. Based on a similar analytical procedure, Gnanapragasam gave a different expression for ka. It is noted, however, that both Mazindrani and Ganjale's and Gnanapragasam's expressions lead to identical active earth pressure values. Following either approach for the active earth pressure, the depth of tension crack appears to be the same as in the case of zero backfill inclination (see Bell's extension of Rankine's theory). Pantelidis' unified approach: the generalized coefficients of earth pressure. Pantelidis offered a unified fully analytical continuum mechanics approach (based on Cauchy's first  law of motion) for deriving earth pressure coefficients for all soil states, applicable to cohesive-frictional soils and both horizontal and vertical pseudo-static conditions. The following symbols are used: formula_33 and formula_34 are the seismic coefficients of horizontal and vertical acceleration respectively formula_53, formula_48 and formula_55 are the effective cohesion, effective internal friction angle (peak values) and unit weight of soil respectively formula_56 is the mobilized cohesion of soil (the mobilized shear strength of soil, i.e. the formula_56and formula_58 parameters, can be obtained either analytically or through relevant charts; see Pantelidis) formula_59 and formula_60 are the effective elastic constants  of soil (i.e. the Young modulus and Poisson's ratio respectively) formula_61 is the wall height formula_62 is the depth where the earth pressure is calculated The coefficient of earth pressure "at rest". formula_63with formula_64 The coefficient of "active" earth pressure. formula_65with formula_66 The coefficient of "passive" earth pressure. formula_67with formula_68 The coefficient of "intermediate" earth pressure on the active "side". formula_69with formula_70 The coefficient of "intermediate" earth pressure on the passive "side". formula_71where, formula_72, formula_73, formula_74 and formula_75 with formula_76 formula_77 and formula_78 are parameters related to the transition from the soil wedge of the state at rest to  the soil wedge of the passive state (i.e., inclination angle of soil wedge changing from formula_79 to formula_80. Also, formula_81 and formula_82 are the lateral displacement of the wall and the lateral (maximum) displacement of the wall corresponding to the active or passive state (both at depth formula_62). The latter is given below. The lateral maximum displacement of the wall corresponding to the active or passive state. formula_84for smooth retaining wall andformula_85for rough retaining wall with formula_86 or formula_87 for the active and passive "side" respectively. The depth of tension crack (active state) or neutral zone (state at rest). The  depth of "neutral zone" in the state at rest is:formula_88whilst the depth of "tension crack" in the active state is:formula_89Under static conditions ( formula_90=formula_91=0), where the mobilized cohesion, formula_92, is equal to the cohesion value at the critical state, formula_93, the above expression is transformed to the well-known:formula_22 Derivation of the earth pressure at rest by the active earth pressure coefficient. Actually, this has been foreseen in EM1110-2-2502 with the application of a Strength Mobilization Factor (SMF) to c′ and tanφ′. According to this Engineer Manual, an appropriate SMF value allows calculation of  greater-than-active earth pressures using Coulomb's active force equation. Assuming an average SMF value equal to 2/3 along Coulomb's failure surface, it has been shown that for purely frictional soils the derived coefficient value of earth pressure matches quite well with the respective one derived from Jaky's formula_8 equation. In the solution proposed by Pantelidis, the SMF factor is the formula_96 ratio and what has been foreseen by EM1110-2-2502, it can be calculated exactly. Example #1: For formula_93=20 kPa, formula_3=30, γ=18 kN/m, formula_90=formula_91=0, and formula_101=2 m, for the state at rest  formula_102=0.211, formula_92=9.00 kPa and formula_104=14.57. Using this (formula_92, formula_104) pair of values in place of the (formula_93, formula_3) pair of values in the coefficient of active earth pressure (formula_109) given previously, the latter returns a coefficient of earth pressure equal to 0.211, that is, the coefficient of earth pressure at rest. Example #2: For formula_93=0kPa, formula_3=30, γ=18 kN/m, formula_90=0.3, formula_91=0.15, and formula_101=2 m, for the state at rest formula_115=0.602, formula_92=0 kPa and formula_104=14.39. Using this (formula_92, formula_104) pair of values in place of the (formula_93, formula_3) pair of values and formula_90=formula_91=0 in the coefficient of active earth  pressure (formula_109) given previously, the latter returns a coefficient of earth pressure equal to 0.602, that is, again the coefficient of earth pressure at rest.