# Lateral earth pressure

An example of lateral earth pressure overturning a retaining wall

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 walls, basements, tunnels, and to determine the friction on the sides of deep foundations.

To describe the pressure a soil will exert, an earth pressure coefficient, K, is used. K is a function of the soil properties and has a horizontal component Kh and a smaller vertical component Kv. Kh has a value between 0 (completely solid) and 1 (completely liquid). Horizontal earth pressure is assumed to be 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.

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.

## 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. Two of the more commonly used are presented below.

Jaky (1948)[1] for normally consolidated soils:

$K_{0(NC)} = 1 - \sin \phi ' \$

Mayne & Kulhawy (1982)[2] for overconsolidated soils:

$K_{0(OC)} = K_{0(NC)} * OCR^{(\sin \phi ')} \$

The latter requires the OCR profile with depth to be determined.

To estimate K0 due to compaction pressures, refer Ingold (1979)[3]

## Active and passive pressure

Different types of wall structures can be designed to resist earth pressure.

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.

### 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 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)[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 Ka and Kp 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. Müller-Breslau (1906)[8] 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 a 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.

## 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 an active situation and the second for passive situations.

$\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