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.

Earth Pressure In Action
An example of lateral earth pressure overturning a retaining wall

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:

Overconsolidation ratio
Angle of the backslope measured to the horizontal
Wall friction angle
Angle of the wall measured to the vertical
Soil stress friction angle
Effective soil stress friction angle
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)[1] for normally consolidated soils:

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.[2], 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):

However, caution should be exercised when dealing with high frictional angles and high Poisson ratios[2].

Mayne & Kulhawy (1982)[3] for overconsolidated soils:

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 Ingold (1979)[4]

Soil lateral active pressure and passive resistance

Retaining Wall Type Function
Different types of wall structures can be designed to resist earth pressure.

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,[5] 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

For the case where β is 0, the above equations simplify to

Coulomb theory

Coulomb (1776)[6] 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,[7] 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)[8] later extended Coulomb's equations to account for wall friction, denoted by δ. Müller-Breslau (1906)[9] 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 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.

Coefficients of earth pressure

Coefficient of active earth pressure at rest

Coefficient of active earth pressure

Coefficient of passive earth pressure

See also


  1. ^ Jaky J. (1948) Pressure in silos, 2nd ICSMFE, London, Vol. 1, pp 103-107.
  2. ^ a b Llano-Serna, M. A.; Farias, M. M.; Pedroso, D. M.; Williams, D. J.; Sheng, D. (2018). "An assessment of statistically based relationships between critical state parameters". Géotechnique. 68 (6): 556–560. doi:10.1680/jgeot.16.T.012. ISSN 0016-8505.
  3. ^ Mayne, P.W. and Kulhawy, F.H. (1982). “K0-OCR relationships in soil”. Journal of Geotechnical Engineering, Vol. 108 (GT6), 851-872.
  4. ^ Ingold, T.S., (1979) The effects of compaction on retaining walls, Gèotechnique, 29, p265-283.
  5. ^ Rankine, W. (1857) On the stability of loose earth. Philosophical Transactions of the Royal Society of London, Vol. 147.
  6. ^ 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
  7. ^ Kramer S.L. (1996) Earthquake Geotechnical Engineering, Prentice Hall, New Jersey
  8. ^ Mayniel K., (1808), Traité expérimental, analytique et preatique de la poussée des terres et des murs de revêtement, Paris.
  9. ^ Müller-Breslau H., (1906) Erddruck auf Stutzmauern, Alfred Kroner, Stuttgart.



A borehole is a narrow shaft bored in the ground, either vertically or horizontally. A borehole may be constructed for many different purposes, including the extraction of water, other liquids (such as petroleum) or gases (such as natural gas), as part of a geotechnical investigation, environmental site assessment, mineral exploration, temperature measurement, as a pilot hole for installing piers or underground utilities, for geothermal installations, or for underground storage of unwanted substances, e.g. in carbon capture and storage.


Clay is a finely-grained natural rock or soil material that combines one or more clay minerals with possible traces of quartz (SiO2), metal oxides (Al2O3 , MgO etc.) and organic matter. Geologic clay deposits are mostly composed of phyllosilicate minerals containing variable amounts of water trapped in the mineral structure. Clays are plastic due to particle size and geometry as well as water content, and become hard, brittle and non–plastic upon drying or firing. Depending on the soil's content in which it is found, clay can appear in various colours from white to dull grey or brown to deep orange-red.

Although many naturally occurring deposits include both silts and clay, clays are distinguished from other fine-grained soils by differences in size and mineralogy. Silts, which are fine-grained soils that do not include clay minerals, tend to have larger particle sizes than clays. There is, however, some overlap in particle size and other physical properties. The distinction between silt and clay varies by discipline. Geologists and soil scientists usually consider the separation to occur at a particle size of 2 µm (clays being finer than silts), sedimentologists often use 4–5 μm, and colloid chemists use 1 μm. Geotechnical engineers distinguish between silts and clays based on the plasticity properties of the soil, as measured by the soils' Atterberg limits. ISO 14688 grades clay particles as being smaller than 2 μm and silt particles as being larger.

Mixtures of sand, silt and less than 40% clay are called loam. Loam makes good soil and is used as a building material.

Diaphragm (structural system)

In structural engineering, a diaphragm is a structural element that transmits lateral loads to the vertical resisting elements of a structure (such as shear walls or frames). Diaphragms are typically horizontal, but can be sloped such as in a gable roof on a wood structure or concrete ramp in a parking garage. The diaphragm forces tend to be transferred to the vertical resisting elements primarily through in-plane shear stress. The most common lateral loads to be resisted are those resulting from wind and earthquake actions, but other lateral loads such as lateral earth pressure or hydrostatic pressure can also be resisted by diaphragm action.

The diaphragm of a structure often does double duty as the floor system or roof system in a building, or the deck of a bridge, which simultaneously supports gravity loads.

Diaphragms are usually constructed of plywood or oriented strand board in timber construction; metal deck or composite metal deck in steel construction; or a concrete slab in concrete construction.

The two primary types of diaphragm are flexible and rigid. Flexible diaphragms resist lateral forces depending on the tributary area, irrespective of the flexibility of the members that they are transferring force to. On the other hand, rigid diaphragms transfer load to frames or shear walls depending on their flexibility and their location in the structure. The flexibility of a diaphragm affects the distribution of lateral forces to the vertical components of the lateral force resisting elements in a structure. An Investigation of the Influence of Diaphragm Flexibility on Building Design through a Comparison of Forced Vibration Testing and Computational Analysis

Parts of a diaphragm include:

the membrane, used as a shear panel to carry in-plane shear

the drag strut member, used to transfer the load to the shear walls or frames

the chord, used to resist the tension and compression forces that develop in the diaphragm, since the membrane is usually incapable of handling these loads alone.


Gravel is a loose aggregation of rock fragments. Gravel is classified by particle size range and includes size classes from granule- to boulder-sized fragments. In the Udden-Wentworth scale gravel is categorized into granular gravel (2 to 4 mm or 0.079 to 0.157 in) and pebble gravel (4 to 64 mm or 0.2 to 2.5 in). ISO 14688 grades gravels as fine, medium, and coarse with ranges 2 mm to 6.3 mm to 20 mm to 63 mm. One cubic metre of gravel typically weighs about 1,800 kg (or a cubic yard weighs about 3,000 pounds).

Gravel is an important commercial product, with a number of applications. Many roadways are surfaced with gravel, especially in rural areas where there is little traffic. Globally, far more roads are surfaced with gravel than with concrete or asphalt; Russia alone has over 400,000 km (250,000 mi) of gravel roads. Both sand and small gravel are also important for the manufacture of concrete.

Index of soil-related articles

This is an index of articles relating to soil.


K0 may refer to:

Spectral class K0, a star spectral class

the 1965 first model of the Honda CB450 motorbike

the Grothendieck group in abstract algebra

the Lateral earth pressure at rest

the neutral Kaon, a strange meson with no charge in nuclear physics

K0 may refer to Khinchin's constant

K0 the order-zero graph

Natchez silt loam

In 1988, the Professional Soil Classifiers Association of Mississippi selected Natchez silt loam soil to represent the soil resources of the State. These soils exist on 171,559 acres (0.56% of state) of landscape in Mississippi.

Overburden pressure

Overburden pressure, also called lithostatic pressure, confining pressure or vertical stress, is the pressure or stress imposed on a layer of soil or rock by the weight of overlying material.

The Oxford Dictionary of Earth Sciences describes 'confining pressure' as "the combined hydrostatic stress and lithostatic stress; i.e. the total weight of the interstitial pore water and rock above a specified depth." Confining pressure might influence ductile behavior of rocks as well. Ductile behavior is enhanced where high confining pressures are combined with high temperatures and low rates of strain, conditions characteristic of deeper crustal levels.

The overburden pressure at a depth z is given by

where ρ(z) is the density of the overlying rock at depth z and g is the acceleration due to gravity. p0 is the datum pressure, the pressure at the surface.

In deriving the above equation it is assumed that gravitational acceleration g is a constant over z, since it is placed outside the integral. In reality, g is a (non-constant) function of z and should appear inside the integral. But since g varies little over depths which are a very small fraction of the Earth's radius, it is placed outside the integral in practice for most near-surface applications which require an assessment of lithostatic pressure. In deep-earth geophysics/geodynamics, gravitational acceleration varies significantly over depth and g may not be assumed to be constant.

This should be compared with the equivalent concept of hydrostatic pressure in hydrodynamics.

Rankine theory

Rankine's theory (maximum-normal stress theory), developed in 1857 by William John Macquorn Rankine, 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.

For the case where β is 0, the above equations simplify to

Response spectrum

A response spectrum is a plot of the peak or steady-state response (displacement, velocity or acceleration) of a series of oscillators of varying natural frequency, that are forced into motion by the same base vibration or shock. The resulting plot can then be used to pick off the response of any linear system, given its natural frequency of oscillation. One such use is in assessing the peak response of buildings to earthquakes. The science of strong ground motion may use some values from the ground response spectrum (calculated from recordings of surface ground motion from seismographs) for correlation with seismic damage.

If the input used in calculating a response spectrum is steady-state periodic, then the steady-state result is recorded. Damping must be present, or else the response will be infinite. For transient input (such as seismic ground motion), the peak response is reported. Some level of damping is generally assumed, but a value will be obtained even with no damping.

Response spectra can also be used in assessing the response of linear systems with multiple modes of oscillation (multi-degree of freedom systems), although they are only accurate for low levels of damping. Modal analysis is performed to identify the modes, and the response in that mode can be picked from the response spectrum. These peak responses are then combined to estimate a total response. A typical combination method is the square root of the sum of the squares (SRSS) if the modal frequencies are not close. The result is typically different from that which would be calculated directly from an input, since phase information is lost in the process of generating the response spectrum.

The main limitation of response spectra is that they are only universally applicable for linear systems. Response spectra can be generated for non-linear systems, but are only applicable to systems with the same non-linearity, although attempts have been made to develop non-linear seismic design spectra with wider structural application. The results of this cannot be directly combined for multi-mode response.

Retaining wall

Retaining walls are relatively rigid walls used for supporting soil laterally so that it can be retained at different levels on the two sides.

Retaining walls are structures designed to restrain soil to a slope that it would not naturally keep to (typically a steep, near-vertical or vertical slope). They are used to bound soils between two different elevations often in areas of terrain possessing undesirable slopes or in areas where the landscape needs to be shaped severely and engineered for more specific purposes like hillside farming or roadway overpasses. A retaining wall that retains soil on the backside and water on the frontside is called a seawall or a bulkhead.


Sand is a granular material composed of finely divided rock and mineral particles. It is defined by size, being finer than gravel and coarser than silt. Sand can also refer to a textural class of soil or soil type; i.e., a soil containing more than 85 percent sand-sized particles by mass.The composition of sand varies, depending on the local rock sources and conditions, but the most common constituent of sand in inland continental settings and non-tropical coastal settings is silica (silicon dioxide, or SiO2), usually in the form of quartz. The second most common type of sand is calcium carbonate, for example, aragonite, which has mostly been created, over the past half billion years, by various forms of life, like coral and shellfish. For example, it is the primary form of sand apparent in areas where reefs have dominated the ecosystem for millions of years like the Caribbean.

Sand is a non-renewable resource over human timescales, and sand suitable for making concrete is in high demand. Desert sand, although plentiful, is not suitable for concrete. 50 billion tons of beach sand and fossil sand is used each year for construction.


Silt is granular material of a size between sand and clay, whose mineral origin is quartz and feldspar. Silt may occur as a soil (often mixed with sand or clay) or as sediment mixed in suspension with water (also known as a suspended load) and soil in a body of water such as a river. It may also exist as soil deposited at the bottom of a water body, like mudflows from landslides. Silt has a moderate specific area with a typically non-sticky, plastic feel. Silt usually has a floury feel when dry, and a slippery feel when wet. Silt can be visually observed with a hand lens, exhibiting a sparkly appearance. It also can be felt by the tongue as granular when placed on the front teeth (even when mixed with clay particles).

Soil classification

Soil classification deals with the systematic categorization of soils based on distinguishing characteristics as well as criteria that dictate choices in use.

Soil mechanics

Soil mechanics is a branch of soil physics and applied mechanics that describes the behavior of soils. It differs from fluid mechanics and solid mechanics in the sense that soils consist of a heterogeneous mixture of fluids (usually air and water) and particles (usually clay, silt, sand, and gravel) but soil may also contain organic solids and other matter. Along with rock mechanics, soil mechanics provides the theoretical basis for analysis in geotechnical engineering, a subdiscipline of civil engineering, and engineering geology, a subdiscipline of geology. Soil mechanics is used to analyze the deformations of and flow of fluids within natural and man-made structures that are supported on or made of soil, or structures that are buried in soils. Example applications are building and bridge foundations, retaining walls, dams, and buried pipeline systems. Principles of soil mechanics are also used in related disciplines such as engineering geology, geophysical engineering, coastal engineering, agricultural engineering, hydrology and soil physics.

This article describes the genesis and composition of soil, the distinction between pore water pressure and inter-granular effective stress, capillary action of fluids in the soil pore spaces, soil classification, seepage and permeability, time dependent change of volume due to squeezing water out of tiny pore spaces, also known as consolidation, shear strength and stiffness of soils. The shear strength of soils is primarily derived from friction between the particles and interlocking, which are very sensitive to the effective stress. The article concludes with some examples of applications of the principles of soil mechanics such as slope stability, lateral earth pressure on retaining walls, and bearing capacity of foundations.

Specific storage

In the field of hydrogeology, storage properties are physical properties that characterize the capacity of an aquifer to release groundwater. These properties are storativity (S), specific storage (Ss) and specific yield (Sy).

They are often determined using some combination of field tests (e.g., aquifer tests) and laboratory tests on aquifer material samples. Recently, these properties have been also determined using remote sensing data derived from Interferometric synthetic-aperture radar.


Thixotropy is a time-dependent shear thinning property. Certain gels or fluids that are thick or viscous under static conditions will flow (become thin, less viscous) over time when shaken, agitated, sheared or otherwise stressed (time dependent viscosity). They then take a fixed time to return to a more viscous state.

Some non-Newtonian pseudoplastic fluids show a time-dependent change in viscosity; the longer the fluid undergoes shear stress, the lower its viscosity. A thixotropic fluid is a fluid which takes a finite time to attain equilibrium viscosity when introduced to a steep change in shear rate. Some thixotropic fluids return to a gel state almost instantly, such as ketchup, and are called pseudoplastic fluids. Others such as yogurt take much longer and can become nearly solid. Many gels and colloids are thixotropic materials, exhibiting a stable form at rest but becoming fluid when agitated. Thixotropy arises because particles or structured solutes require time to organize. An excellent overview of thixotropy has been provided by Mewis and Wagner.Some fluids are anti-thixotropic: constant shear stress for a time causes an increase in viscosity or even solidification. Fluids which exhibit this property are sometimes called rheopectic. Anti-thixotropic fluids are less well documented than thixotropic fluids.


A trench is a type of excavation or depression in the ground that is generally deeper than it is wide (as opposed to a wider gully, or ditch), and narrow compared with its length (as opposed to a simple hole).In geology, trenches are created as a result of erosion by rivers or by geological movement of tectonic plates. In the civil engineering field, trenches are often created to install underground infrastructure or utilities (such as gas mains, water mains or telephone lines), or later to access these installations. Trenches have also often been dug for military defensive purposes. In archaeology, the "trench method" is used for searching and excavating ancient ruins or to dig into strata of sedimented material.

Void ratio

The void ratio of a mixture is the ratio of the volume of voids to volume of solids.

It is a dimensionless quantity in materials science, and is closely related to porosity as follows:


where is void ratio, is porosity, VV is the volume of void-space (such as fluids), VS is the volume of solids, and VT is the total or bulk volume. This figure is relevant in composites, in mining (particular with regard to the properties of tailings), and in soil science. In geotechnical engineering, it is considered as one of the state variables of soils and represented by the symbol e.

Note that in geotechnical engineering, the symbol usually represents the angle of shearing resistance, a shear strength (soil) parameter. Because of this, the equation is usually rewritten using for porosity:


where is void ratio, is porosity, VV is the volume of void-space (air and water), VS is the volume of solids, and VT is the total or bulk volume.

Retaining walls
Numerical analysis


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