Lateral Earth Pressure

Lateral Earth Pressure

At-rest, active, and passive earth pressures, Rankine's and Coulomb's theories.

Overview

Lateral earth pressure is the pressure that a soil mass exerts in the horizontal direction against a retaining structure. Accurate determination of this pressure is fundamental to the design of retaining walls, sheet piling, basement walls, and underground structures. The magnitude of this pressure is profoundly influenced by the displacement of the retaining structure relative to the soil backfill.

At-Rest, Active, and Passive Pressures

The relationship between the horizontal effective stress (

\sigma_h'

) and the vertical effective stress (

\sigma_v'

) at any point in a soil mass is defined by the earth pressure coefficient,

K

K = \frac{\sigma_h'}{\sigma_v'}

There are three primary states of lateral earth pressure, depending on the movement of the retaining wall:

Procedure

STEP-BY-STEP

At-Rest State ( $K_0$ ): The soil mass is completely static. The retaining wall is rigid and unyielding (e.g., heavily braced basement walls). The soil experiences zero lateral strain. For normally consolidated soils, Jaky's empirical equation is often used: $K_0 \approx 1 - \sin \phi'$ . For overconsolidated soils, the at-rest pressure is higher: $K_0 = (1 - \sin \phi') \cdot OCR^{\sin \phi'}$ .
Active State ( $K_a$ ): The retaining wall yields or moves slightly away from the soil backfill. The soil mass stretches horizontally, leading to a decrease in lateral pressure until shear failure occurs along a slip plane. This is the minimum possible lateral earth pressure.
Passive State ( $K_p$ ): The retaining wall is forcefully pushed into the soil mass. The soil is horizontally compressed, causing a significant increase in lateral pressure until shear failure occurs along an upward slip plane. This is the maximum possible lateral earth pressure. It requires substantially more wall movement to mobilize than the active state.

Rankine's Earth Pressure Theory (1857)

William John Macquorn Rankine developed a classical theory to estimate lateral earth pressures. His theory relies on several key, albeit simplifying, assumptions.

Rankine's Assumptions

The soil mass is isotropic, homogeneous, and cohesionless ( $c'=0$ ).
The backface of the wall in contact with the soil is perfectly smooth (wall friction angle, $\delta = 0^\circ$ ).
The backface of the wall is perfectly vertical.
The rupture surface (slip plane) is a straight line.

For a horizontal backfill surface, Rankine's active and passive earth pressure coefficients are derived from Mohr's circle at failure:

K_a

K_a = \tan^2\left(45^\circ - \frac{\phi'}{2}\right) = \frac{1 - \sin \phi'}{1 + \sin \phi'}

K_p

K_p = \tan^2\left(45^\circ + \frac{\phi'}{2}\right) = \frac{1 + \sin \phi'}{1 - \sin \phi'}

Where

\phi'

is the effective angle of internal friction of the soil.

Total Force Calculation

The total lateral thrust force (

P

) per unit length of wall acts at a height of

H/3

from the base of the wall (for a triangular pressure distribution).

P_a = \frac{1}{2} \gamma H^2 K_a

P_p = \frac{1}{2} \gamma H^2 K_p

Where:

$\gamma$ is the unit weight of the soil
$H$ is the wall height

Coulomb's Earth Pressure Theory (1776)

Charles-Augustin de Coulomb developed an earlier, yet often more versatile, theory. Unlike Rankine, Coulomb considered the friction between the wall and the soil and did not require the wall backface to be vertical.

Coulomb's Assumptions

The soil is isotropic, homogeneous, and has internal friction ( $\phi'$ ).
There is friction between the wall and the soil (wall friction angle, $\delta > 0^\circ$ ). This is a significant improvement over Rankine.
The backface of the wall can be inclined at an angle $\theta$ to the horizontal.
The backfill surface can be sloped at an angle $\alpha$ .
The rupture surface is assumed to be a straight plane, defining a sliding wedge of soil. Coulomb's method involves finding the critical wedge that maximizes active force or minimizes passive force (trial wedge method).

Coulomb's analytical equations for the coefficients, considering all variables (

\phi', \delta, \theta, \alpha

), are substantially more complex.

K_a

K_a = \frac{\sin^2(\theta + \phi')}{\sin^2 \theta \sin(\theta - \delta) \left[1 + \sqrt{\frac{\sin(\phi' + \delta) \sin(\phi' - \alpha)}{\sin(\theta - \delta) \sin(\theta + \alpha)}}\right]^2}

K_p

K_p = \frac{\sin^2(\theta - \phi')}{\sin^2 \theta \sin(\theta + \delta) \left[1 - \sqrt{\frac{\sin(\phi' + \delta) \sin(\phi' + \alpha)}{\sin(\theta + \delta) \sin(\theta + \alpha)}}\right]^2}

Curved Rupture Surfaces

Coulomb's assumption of a planar rupture surface is reasonably accurate for active pressure. However, for passive pressure, especially when the wall friction

\delta

is high (e.g.,

\delta > \phi'/3

), the actual rupture surface is significantly curved (logarithmic spiral). Using Coulomb's planar assumption for passive pressure under these conditions will dangerously overestimate the passive resistance, leading to unsafe designs. In such cases, Log-Spiral methods are preferred.

Active Earth Pressure with Cohesion and Tension Cracks

Effect of Cohesion

In cohesive soils (

c' > 0

), cohesion significantly reduces the active earth pressure. The Rankine active pressure equation becomes:

\sigma_a' = \sigma_v' K_a - 2c'\sqrt{K_a}

Because of the negative term, the active pressure can become negative (tension) at shallow depths. Soil cannot sustain tension, leading to the formation of Tension Cracks. The theoretical depth of the tension crack (

z_c

) where

\sigma_a' = 0

is:

z_c = \frac{2c'}{\gamma \sqrt{K_a}}

Before the formation of a tension crack, the total active force (

P_a

) integrates the pressure profile from

z=0

. However, for design, it's safer to assume the tension crack has formed and no tension is transferred to the wall. Water filling the crack adds hydrostatic pressure, which must be considered by adding

\gamma_w \cdot z_c

to the active thrust calculation if the crack is expected to fill with water (e.g., during rainfall).

Culmann's Graphical Method

Graphical Approach to Coulomb's Theory

Culmann's method is a versatile graphical technique based on Coulomb's wedge theory. It allows for the determination of the active earth pressure force (

P_a

) for complex geometries, such as irregular backfill surfaces, concentrated line loads, and varying soil stratigraphy, where analytical equations become overly cumbersome. It involves plotting trial wedges and a pressure locus to find the critical slip surface graphically.

Effect of Groundwater

Hydrostatic Pressure on Walls

The presence of groundwater behind a retaining wall drastically changes the lateral pressure distribution. Water exerts a hydrostatic pressure that acts independently of the soil's lateral earth pressure.

The soil below the water table is submerged, so its effective unit weight ( $\gamma' = \gamma_{sat} - \gamma_w$ ) must be used to calculate effective vertical stress ( $\sigma_v'$ ).
The lateral earth pressure is calculated using effective stress: $\sigma_h' = K \cdot \sigma_v'$ .
Hydrostatic pressure ( $u = \gamma_w \cdot z_w$ ) must be added to the lateral effective soil pressure to get the total lateral pressure: $\sigma_h = \sigma_h' + u$ .
Because water pressure is fully hydrostatic ( $K=1$ ), a fully saturated, undrained backfill exerts significantly more total lateral force than a dry backfill. Proper drainage systems (weep holes, drains) are critical to alleviate this water pressure.

Effect of Surcharge Loads

Surcharge loads (uniform, point, line, or strip loads) applied to the backfill surface significantly increase the lateral earth pressure on the retaining wall.

Procedure

STEP-BY-STEP

Uniform Surcharge ( $q$ ): A uniform load extending indefinitely behind the wall adds a constant rectangular pressure distribution down the entire height of the wall. The additional lateral pressure is $\Delta \sigma_h = K \cdot q$ . The total lateral thrust increases by $\Delta P = K \cdot q \cdot H$ , acting at $H/2$ from the base.
Point and Line Loads: Boussinesq's theory, modified by empirical equations (like those from Terzaghi), is used to calculate the lateral pressure distribution from concentrated or line loads (e.g., footings or railways near the wall). The pressure distribution is non-linear and bulb-shaped, peaking at some depth below the surface.
Strip Loads: Used for loads like highways running parallel to the wall. The pressure increase on the wall is calculated using elastic theory (e.g., integrating Boussinesq's point load solution over a strip area), and varies non-linearly with depth.

Seismic Earth Pressures: Mononobe-Okabe Theory

Mononobe-Okabe (M-O) Method

The Mononobe-Okabe method is an extension of Coulomb's theory that accounts for earthquake-induced inertial forces. It adds pseudo-static horizontal (

k_h

) and vertical (

k_v

) seismic coefficients to the sliding wedge analysis. The total active earth pressure (

P_{ae}

) under seismic conditions is:

P_{ae} = \frac{1}{2} \gamma H^2 (1 - k_v) K_{ae}

Where

K_{ae}

is the seismic active earth pressure coefficient, which depends on

\phi'

\delta

\beta

(slope angle),

\theta

(wall angle), and the seismic inertia angle

\psi = \tan^{-1}(k_h / (1 - k_v))

. The dynamic increment (

\Delta P_{ae} = P_{ae} - P_a

) acts higher up the wall, typically around

0.6H

from the base.

Key Takeaways

Lateral earth pressure states—At-Rest ( $K_0$ ), Active ( $K_a$ ), and Passive ( $K_p$ )—depend on the relative movement between the wall and the soil mass.
Rankine's theory assumes a smooth, vertical wall backface and planar slip surfaces, resulting in simplified equations for $K_a$ and $K_p$ .
Coulomb's theory accounts for wall friction ( $\delta$ ) and sloping backfills/wall faces, utilizing a trial wedge approach.
While Coulomb's theory is more realistic for active pressures with wall friction, it can dangerously overestimate passive resistance if wall friction is high, necessitating the use of Log-Spiral methods.
The Mononobe-Okabe theory extends Coulomb's approach to include pseudo-static seismic forces for earthquake design.