# 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

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, 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 stress friction angle at critical state

## At rest pressure

At rest lateral earth pressure, represented as K0, 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. Some of these relations are presented below.

Jaky (1948) for normally consolidated soils:

$K_{0(NC)}=1-\sin \phi '\$ Although Jaky derived the above result using a theoretical model, the associated assumptions are not related to the physical problem. In this light, the good predictions provided by his solution are often viewed as a coincidence.

According to Llano-Serna et al., a slight change to Jaky's expression have been proven effective to determine K0(NC) for fine-grained soils at critical states (See critical state soil mechanics):

$K_{0(NC)}=1.15-\sin \phi _{cs}'\$ However, caution should be exercised when dealing with high frictional angles and high Poisson ratios.

Mayne & Kulhawy (1982) for overconsolidated soils:

$K_{0(OC)}=K_{0(NC)}*OCR^{(\sin \phi ')}\$ The latter requires the OCR profile with depth to be determined. OCR is the overconsolidation ratio and $\phi '$ is the effective stress friction angle.

To estimate K0 due to compaction pressures, refer Ingold (1979)

## 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 theory

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 friction angle of the wall is equal to the inclination of the backfill (i.e. the wall is frictionless when the backfill is horizontal), the soil-wall 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

$K_{a}=\cos \beta {\frac {\cos \beta -\left(\cos ^{2}\beta -\cos ^{2}\phi \right)^{1/2}}{\cos \beta +\left(\cos ^{2}\beta -\cos ^{2}\phi \right)^{1/2}}}$ $K_{p}=\cos \beta {\frac {\cos \beta +\left(\cos ^{2}\beta -\cos ^{2}\phi \right)^{1/2}}{\cos \beta -\left(\cos ^{2}\beta -\cos ^{2}\phi \right)^{1/2}}}$ For the case where β is 0, the above equations simplify to

$K_{a}=\tan ^{2}\left(45-{\frac {\phi }{2}}\right)={\frac {1-\sin(\phi )}{1+\sin(\phi )}}$ $K_{p}=\tan ^{2}\left(45+{\frac {\phi }{2}}\right)={\frac {1+\sin(\phi )}{1-\sin(\phi )}}$ ### Coulomb theory

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 Ka and Kp 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).

$K_{a}={\frac {\cos ^{2}\left(\phi -\theta \right)}{\cos ^{2}\theta \cos \left(\delta +\theta \right)\left(1+{\sqrt {\frac {\sin \left(\delta +\phi \right)\sin \left(\phi -\beta \right)}{\cos \left(\delta +\theta \right)\cos \left(\beta -\theta \right)}}}\ \right)^{2}}}$ $K_{p}={\frac {\cos ^{2}\left(\phi +\theta \right)}{\cos ^{2}\theta \cos \left(\delta -\theta \right)\left(1-{\sqrt {\frac {\sin \left(\delta +\phi \right)\sin \left(\phi +\beta \right)}{\cos \left(\delta -\theta \right)\cos \left(\beta -\theta \right)}}}\ \right)^{2}}}$ 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 Ka, the horizontal part Kah is tabulated. It is the same as Ka times cos(δ+θ).

The actual earth pressure force Ea is the sum of the part Eag due to the weight of the earth, a part Eap due to extra loads such as traffic, minus a part Eac due to any cohesion present.

Eag is the integral of the pressure over the height of the wall, which equates to Ka 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, Eap equates to this pressure times Ka times the height of the wall. This applies if the terrace is horizontal or the wall vertical. Otherwise, Eap must be multiplied by cosθ cosβ / cos(θ − β).

Eac is generally assumed to be zero unless a value of cohesion can be maintained permanently.

Eag 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. Eap 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 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.

### 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. [citation needed]

## 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 non-cohesive contribution and the second term the cohesive contribution. The first equation is for the active earth pressure condition and the second for the passive earth pressure condition.

$\sigma _{h}=K_{a}\sigma _{v}-2c{\sqrt {K_{a}}}\$ $\sigma _{h}=K_{p}\sigma _{v}+2c{\sqrt {K_{p}}}\$ ## Coefficients of earth pressure

Coefficient of active earth pressure at rest

Coefficient of active earth pressure

Coefficient of passive earth pressure