Lateral earth pressure

Lateral earth pressure theory is the theory predicting the amount of pressure or stress that soil exerts in the horizontal plane (i.e. perpendicular to gravity). The common applications of lateral earth pressure theory are for the design of ground engineering structures such as retaining walls, deep basements, tunnels, and also for determining the friction on the sides of piles for deep foundations. In fact, most geotechnical engineering design requires an understanding of the horizontal earth pressure for the given problem being considered, including bearing capacity and slope stability.

Systematics

To describe the pressure a soil will exert, a lateral earth pressure coefficient, K is used. K is the ratio of lateral (horizontal) pressure to vertical pressure (K = σ_h/σ_v). Thus horizontal earth pressure is assumed to be a direct proportion of the vertical pressure at any given point in the soil profile. Determining K is difficult to do exactly, as it depends on the soil material properties, which typically vary, and the stress history of the soil.

Lateral earth pressure coefficients are broken up into three different categories: "At-rest", "active", and "passive". The At-rest state is the in-situ horizontal pressure (i.e. as it is in the ground prior to engineering works). The Active state occurs when a soil mass is allowed to relax/ move outward to the point of reaching the limiting strength of the soil; i.e. 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 (i.e. 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, with the At-Rest pressure being somewhere between these two extents and dependant on the stress history of the particular soil.

The pressure coefficient used in geotechnical engineering analysis 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.

Discussion

Imagine a soil deposit with a horizontal surface and with a vertical membrane of zero thickness and infinite stiffness inserted into it without disturbing the soil. The stress on each side of the membrane is called the at-rest lateral stress σ_o. Its value at any depth z is given by K_o.γ.z = K_o.σ_v where σ_v is vertical stress at depth z and K_o is the coefficient of at-rest lateral stress.

The value of K_o cannot be calculated as it depends on many factors, including the geological history of the soil. There are some empirical expressions for estimating its value, and attempts can be made to measure it but, of course, the insertion of a measuring instrument into the ground immediately disturbs the soil and the measured value may be greater or less than the actual value. Typically, in a normally-consolidated soil its value may be around 0.5, and in an over-consolidated soil it may be up to 2 or 3.

Imagine the membrane being held in place while soil is excavated on one side of it to the base of the membrane over a large horizontal distance. Then imagine the membrane being allowed to move away from the soil it is retaining by rotation about the base of the excavation, or by lateral translation. The lateral stress at any depth will steadily reduce until a certain amount of movement has occurred, when it will remain constant even though further movement of the membrane is allowed. This stress is called the active lateral stress and is equal to K_a.σ_v where K_a is the coefficient of active lateral stress. The amount of movement of the top of the membrane necessary to reduce the lateral stress to the active value is quite small (perhaps less than 0.1% of the excavation depth). The name 'active' is given to this stress because the soil actively follows the membrane as it is moved away.

Instead of allowing the membrane to move away from the soil, imagine it being pushed towards the soil, rotating about the base of the excavation or moving laterally. The stress required to move it will increase until a maximum value is reached, which will not increase with further movement. This stress is called the passive lateral stress and is equal to K_p.σ_v where K_p is the coefficient of passive lateral stress. The amount of membrane movement at the ground surface required to reach the passive value is much greater than that needed to reduce the stress to the active value (perhaps 10% of the excavation depth). The name 'passive' is given to this stress because the soil passively resists the membrane as it is pushed towards it.

The lateral stress values in the ground can be expressed in terms of total or effective stresses. In the case of a total stress analysis in a saturated, fine-grained soil, K_a = K_p = 1.0. In an effective stress analysis, the effective active lateral stress σ_a' = K_a'.σ_v' and the effective passive lateral stress σ_p' = K_p'.σ_v', where σ_v' is the effective vertical stress at depth z. Typically, the value of K_a' is around 0.3 and the value of K_p' will be similar that that of K_o in heavily-overconsolidated soils i.e. 2 to 3.

In design, no safety factor is applied to the active stress. However, a reduction factor is applied to the passive stress because the amount of lateral movement required to generate the full value would be unacceptable in most soil retaining situations. A factor of 3 to 4 is commonly used.

Water pressure and shrinkage cracks

Static water in the soil will exert a pressure on the wall that is hydrostatic beneath the water table. Although it might be thought that negative water pressure above the water table in fine-grained soils would result in suction on the wall back, any tendency for this to occur ceases rapidly as the soil shrinks away from the soil back, leading to what is called a 'tension crack' from the ground surface down to the water table. Such a crack can fill with rain water or other run-off and lead to hydrostatic pressure behind the top of the wall. This must be avoided by grading the ground surface away from the wall back, and sealing the surface of the backfill.

Gravity retaining walls

The theoretical methods of estimating K_a and K_p in gravity retaining wall design involve certain assumptions. The simplest assumes a horizontal ground surface, a vertical wall back, a cohesionless soil, and zero relative movement between the wall back and the soil. This latter assumption is often called 'zero wall friction', but this is misleading. It may happen that, as the wall moves away from the soil the ground beneath it compresses just enough to match the tendency of a wedge of soil to slip down behind the wall. It may happen that, as the wall moves towards the soil, it is pushed upwards and moves together with the wedge of soil being displaced behind it. An analysis of the compressibility of the soil beneath the wall base in the case of active stress, or the direction of the overall passive force being applied to the wall, is needed in order to estimate the likely amount of relative movement between wall back and retained soil and, therefore, whether a frictional component of stress is likely to be generated on the wall back.

Obviously, if the wall is founded on rock it is highly likely that the maximum relative movement between wall and soil will occur in the active situation, and that the coefficient of wall friction μ will be that of the soil on the wall material - timber, steel or concrete. This is also possible in the passive case, but each situation must be evaluated on its own merits. Usually, some proportion of the maximum possible wall friction is used in design.

In many cases the ground surface is not horizontal and the wall back is not vertical. Also, the soil may have a cohesive component of strength as well as a frictional one, or the design may be done in terms of total stress or in terms of effective stress. The common methods of estimating Ka and Kp allow for these types of situations.

Propped retaining walls

When a retaining wall is not free to rotate about its base or to move laterally, the simple triangular or near-triangular active and passive stress distributions no longer apply. Calculation of the likely stress distribution is not possible, and design methods rely on measurements of stress distributions in actual construction situations. The best-known of these is due to Terzaghi and Peck who produced envelopes of likely maximum stress distributions obtained from strut load measurements in deep trenches in several soil types.

Basement walls, bridge abutments and reinforced soil are common cases of propped retaining walls. They continue to represent a great challenge to geotechnical engineers in producing safe and economical designs.

Drainage behind retaining walls

Because the word 'drainage' is used, it is a common misconception that the principal purpose of weep holes through the stems of walls, or gravel layers or geo-synthetic drains down the backs of walls, is to remove water. Their purpose is actually to introduce zones of air at atmospheric pressure into the soil immediately behind the wall. When this is done, a hydraulic gradient is set up between the water some distance behind the wall and the water in the air spaces in the drain, such that seepage occurs towards the wall. At the wall back, the water pressure is zero, and a simple flow net may be used to verify that, as seepage approaches the wall, the water table slopes rapidly downwards and meets the wall at or near its base. Of course, some water seeps out of the soil and is removed by the drains, but the quantity is generally very small.

When the force applied on the back of a retaining wall due to a hydrostatic stress is estimated and compared with the effective active stress, it will be seen to be very large compared with the soil force. It is much cheaper to add drainage than to resist water pressure.

At rest pressure

At rest lateral earth pressure, represented as K₀, 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) for normally consolidated soils:

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

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.

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

Active and passive pressure

Rankine

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 frictionless, the soil-wall interface is vertical, and the failure surface on which the soil moves is planar. 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.

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

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

For the case where coehsionless backfill is inclined at angle β to the horizontal, infinite slope solutions can be used (Terzaghi, 1943; Taylor, 1948) based on the assumption that the pressure resultant is angled parallel to the backfill surface, to compute Ka and Kp as:

$K_{a}=\cos \beta {\frac {\cos \beta -\left(\cos ^{2}\beta -\cos ^{2}\phi \right)^{1/2}}{\cos \beta +\left(\cos ^{2}\beta -\cos ^{2}\phi \right)^{1/2}}}$

$K_{p}=\cos \beta {\frac {\cos \beta +\left(\cos ^{2}\beta -\cos ^{2}\phi \right)^{1/2}}{\cos \beta -\left(\cos ^{2}\beta -\cos ^{2}\phi \right)^{1/2}}}$

Coulomb

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). Mayniel (1908) later extended Coulomb's equations to account for wall friction, symbolized 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 α to the horizontal):

$K_{a}={\frac {1}{\sin \alpha \cos \delta }}{\frac {\sin ^{2}\left(\alpha +\phi \right)\cos \delta }{\sin \alpha \sin \left(\alpha -\delta \right)\left(1+{\sqrt {\frac {\sin \left(\phi +\delta \right)\sin \left(\phi -\beta \right)}{\sin \left(\alpha -\delta \right)\sin \left(\alpha +\beta \right)}}}\ \right)^{2}}}$

$K_{p}={\frac {1}{\sin \alpha \cos \delta }}{\frac {\sin ^{2}\left(\alpha -\phi \right)\cos \delta }{\sin \alpha \sin \left(\alpha +\delta \right)\left(1-{\sqrt {\frac {\sin \left(\phi +\delta \right)\sin \left(\phi +\beta \right)}{\sin \left(\alpha +\delta \right)\sin \left(\alpha +\beta \right)}}}\ \right)^{2}}}$

A more common expression of the Coulomb equation with backslope angle, angle of wall backface (θ to vertical), and the wall friction is presented below. Note that when wall friction and slope angle are set to zero, the formulae revert to the Rankine coefficients.

$K_{a}={\frac {\cos ^{2}\left(\phi -\theta \right)}{\cos ^{2}\theta \cos \left(\delta +\theta \right)\left(1+{\sqrt {\frac {\sin \left(\delta +\phi \right)\sin \left(\phi -\beta \right)}{\cos \left(\delta +\theta \right)\cos \left(\beta -\theta \right)}}}\ \right)^{2}}}$

$K_{p}={\frac {\cos ^{2}\left(\phi +\theta \right)}{\cos ^{2}\theta \cos \left(\delta -\theta \right)\left(1-{\sqrt {\frac {\sin \left(\delta +\phi \right)\sin \left(\phi +\beta \right)}{\cos \left(\delta -\theta \right)\cos \left(\beta -\theta \right)}}}\ \right)^{2}}}$

Caquot and Kerisel

In 1948, Caquot and Kerisel 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. In order for this method to be safe to implement, the values given are conservative. This method was developed with Rankine theory.

Bell's relation

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}}}\$

See also

• Soil mechanics

References

• Lateral Earth Support Systems, DeNatale, 2006
• Department of Transportation Material on Lateral Earth Pressure