Lateral Earth Pressure

Lateral Earth Pressure

Retaining structures such as retaining walls, sheet piles, and basement walls must resist the lateral pressure exerted by the soil backfill. Understanding these pressures is fundamental for structural stability.

States of Stress

The magnitude of lateral pressure depends heavily on the movement of the wall relative to the soil mass.

1. At-Rest State ( $K_{o}$ )

At-Rest Condition: The wall does not move at all (e.g., heavily braced basement walls). The soil is in its natural state of consolidation, neither compressed nor expanded.

At-Rest Earth Pressure

Lateral earth pressure coefficient for undisturbed soil with no horizontal movement; used for rigid basement walls and braced excavations.

\sigma_h = K_o \sigma_v

Variables

Symbol	Description	Unit
$\sigma_h$	Horizontal at-rest earth pressure	-
$K_{o}$	Coefficient of at-rest earth pressure	-
$\sigma_v$	Vertical effective stress	-

Normally Consolidated Soil (Jaky's empirical formula): $K_o = 1 - \sin \phi'$
Overconsolidated Soil: $K_o = (1 - \sin \phi') \sqrt{OCR}$

2. Active State ( $K_{a}$ )

Active Condition: The wall moves away from the soil mass (expansion). The lateral pressure decreases to a theoretical minimum value as the soil shears and mobilizes its full internal friction.

Requires very small movement to mobilize (approx. $H/1000$ for dense sand).
The failure plane develops at an angle of $45^\circ + \phi/2$ from the horizontal.

3. Passive State ( $K_{p}$ )

Passive Condition: The wall moves into the soil mass (compression). The lateral pressure increases to a theoretical maximum value as the soil resists failure.

Requires significant movement to mobilize fully (approx. $H/100$ to $H/10$ ).
The failure plane develops at an angle of $45^\circ - \phi/2$ from the horizontal.

Important

Active pressure is the minimum possible lateral pressure, while passive pressure is the maximum possible resistance. Walls are typically designed to withstand Active pressure.

Interactive Wall Stability Lab

Explore how wall height and soil properties affect the active earth pressure and overall stability.

Gravity Retaining Wall Stability

Wall Height, H (m): 5

Base Width, B (m): 3

Soil Friction,

\\phi

(deg): 30

FS Overturning

0.00

Target: ≥ 2.0

FS Sliding

0.00

Target: ≥ 1.5

The red dashed line represents the Rankine failure plane. The red arrow represents the active earth pressure ( $P_a$ ), which tries to overturn and slide the wall. The blue arrow represents the weight ( $W$ ), which provides resistance.

Rankine's Earth Pressure Theory

Developed by W.J.M. Rankine (1857), this theory is a simplification that evaluates the stress state of a soil element in plastic equilibrium.

Rankine Assumptions

Rankine's theory is widely used due to its simplicity, but relies on specific assumptions:

The back of the retaining wall is perfectly smooth ( $\delta = 0$ , no wall friction).
The wall is perfectly vertical ( $\beta = 90^\circ$ ).
The failure surface is a straight plane.

Rankine Coefficients

Rankine Active Earth Pressure Coefficient

Minimum lateral pressure coefficient when a retaining wall moves away from the soil, mobilizing full shear strength; for frictionless walls only.

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

Variables

Symbol	Description	Unit
$K_{a}$	Active earth pressure coefficient	-
$\phi$	Angle of internal friction	-

Rankine Passive Earth Pressure Coefficient

Maximum lateral pressure coefficient when a wall is pushed into the soil; significantly larger than the active coefficient.

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

Variables

Symbol	Description	Unit
$K_{p}$	Passive earth pressure coefficient	-
$\phi$	Angle of internal friction	-

Relationship: $K_a \times K_p = 1$ .

Pressure Distribution

The pressure increases linearly with depth, forming a triangular distribution.

Rankine Active Earth Pressure

Total lateral active pressure on a wall from cohesive-frictional backfill; cohesion reduces pressure and creates tension cracks near the surface.

p_a = \gamma z K_a - 2c \sqrt{K_a}

Variables

Symbol	Description	Unit
$p_{a}$	Active earth pressure at depth z	-
$\gamma$	Unit weight of soil	-
$z$	Depth	-
$K_{a}$	Active earth pressure coefficient	-
$c$	Cohesion of the soil	-

Rankine Passive Earth Pressure

Total lateral passive pressure on a wall pushed into cohesive-frictional soil; cohesion increases passive resistance.

p_p = \gamma z K_p + 2c \sqrt{K_p}

Variables

Symbol	Description	Unit
$p_{p}$	Passive earth pressure at depth z	-
$\gamma$	Unit weight of soil	-
$z$	Depth	-
$K_{p}$	Passive earth pressure coefficient	-
$c$	Cohesion of the soil	-

Note on Cohesion: For cohesive soils ( $c > 0$ ), the active pressure equation produces a negative value near the surface. This implies tension in the soil (tension cracks). Since soil cannot reliably hold tension, this negative pressure is typically ignored (set to zero) in structural design.

Depth of Tension Crack ( $z_{c}$ )

In cohesive soils, the depth at which the active pressure is exactly zero is known as the depth of the tension crack. Above this depth, the soil theoretically pulls away from the wall.

Depth of Tension Crack

Depth to which active tension cracks can develop in cohesive backfill; water-filled tension cracks significantly increase the total active force on the wall.

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

Variables

Symbol	Description	Unit
$z_{c}$	Depth of tension crack	-
$c$	Cohesion of the soil	-
$\gamma$	Unit weight of soil	-
$K_{a}$	Active earth pressure coefficient	-

Design Implication: Assume tension cracks fill with rainwater, creating full hydrostatic pressure acting on the back of the wall from the surface down to $z_c$ .

Coulomb's Earth Pressure Theory

Developed earlier by C.A. Coulomb (1776), this theory considers the equilibrium of a sliding wedge of soil and is more versatile than Rankine's.

Coulomb vs. Rankine

Coulomb's theory explicitly accounts for friction between the wall and the soil ( $\delta$ ) and can easily handle non-vertical walls and sloped backfills. It assumes a planar failure surface.

Coulomb Active Coefficient ( $K_{a}$ )

Coulomb Active Earth Pressure Coefficient

Active pressure coefficient accounting for wall-backfill friction and sloping ground; more accurate than Rankine for rough walls.

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

Variables

Symbol	Description	Unit
$K_{a}$	Coulomb active earth pressure coefficient	-
$\phi$	Angle of internal friction of the soil	-
$\beta$	Angle of the wall back face with the horizontal (usually 90^\circ for vertical)	-
$\alpha$	Angle of the backfill slope	-
$\delta$	Wall friction angle (typically assumed as 2/3 \phi to \phi for rough concrete)	-

Culmann's Graphical Method

Because Coulomb's equation is cumbersome and cannot easily account for irregular backfill surfaces or non-uniform point/line surcharge loads, Culmann's Graphical Method is employed.

It visually constructs a force polygon for a series of assumed trial failure wedges.
The maximum active force ( $P_a$ ) required for equilibrium is graphically determined from the culmination point of the resulting Culmann line.
It is a highly powerful tool for analyzing retaining walls subjected to complex train loads or building foundations sitting directly on the backfill slope.

Effect of Surcharge and Water

Additional loads on the surface and the presence of groundwater drastically alter lateral pressures.

Surcharge Loads ( $q$ )

A uniform surcharge load $q$ ( $kPa$ ) acting on the ground surface adds a constant, rectangular lateral pressure distribution over the entire height of the wall.

Lateral Pressure from Surcharge

Additional lateral pressure on a retaining wall caused by a uniform surcharge load at the backfill surface.

\Delta \sigma_h = q K_a

Variables

Symbol	Description	Unit
$\Delta \sigma_h$	Increase in horizontal stress	-
$q$	Uniform surcharge load	-
$K_{a}$	Active earth pressure coefficient	-

Resultant Force: $P_q = q \cdot H \cdot K_a$ (acting at $H/2$ from the base).

Water Pressure

Water exerts hydrostatic pressure ( $u = \gamma_w z$ ) which acts equally in all directions (isotropic, so $K_w = 1$ ). Water affects the wall in two major ways:

It reduces the effective vertical stress of the soil ( $\sigma'_v = \sigma_v - u$ ), which reduces the soil's frictional contribution to lateral pressure.
It adds full hydrostatic water pressure directly to the wall face.

Total Horizontal Stress with Water

Total horizontal pressure acting on a submerged retaining wall, including both effective lateral earth pressure and hydrostatic water pressure.

\sigma_h = (\sigma'_v K_a) + u

Variables

Symbol	Description	Unit
$\sigma_h$	Total horizontal stress on the wall	-
$\sigma'_v$	Effective vertical stress (\sigma_v - u)	-
$K_{a}$	Active earth pressure coefficient	-
$u$	Pore water pressure (\gamma_w z)	-

Important

Uncontrolled water buildup is the leading cause of retaining wall failures. Proper drainage (weep holes, perforated pipes) is absolutely crucial to eliminate hydrostatic pressure.

Key Takeaways

Lateral earth pressure states depend strictly on wall movement: At-Rest ( $K_o$ ) for no movement, Active ( $K_a$ ) for expansion (minimum pressure), and Passive ( $K_p$ ) for compression (maximum resistance).
Rankine's Theory is widely used for its simplicity but assumes a smooth, vertical wall and horizontal backfill.
Coulomb's Theory is more robust for complex geometries as it accounts for wall friction ( $\delta$ ) and backfill slope angles ( $\alpha$ ).
Culmann's Graphical Method provides a practical solution for finding active earth pressure under complex surcharge loading conditions that Coulomb's equation cannot easily handle.
Water Pressure is purely hydrostatic ( $K=1$ ) and adds massive unresisted load to the wall. Providing adequate drainage is the most critical aspect of retaining wall design.

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States of Stress

1. At-Rest State (KoK_oKo​)

At-Rest Earth Pressure

2. Active State (KaK_aKa​)

3. Passive State (KpK_pKp​)

Important

Interactive Wall Stability Lab

Gravity Retaining Wall Stability

Rankine's Earth Pressure Theory

Rankine Assumptions

Rankine Coefficients

Rankine Active Earth Pressure Coefficient

Rankine Passive Earth Pressure Coefficient

Pressure Distribution

Rankine Active Earth Pressure

Rankine Passive Earth Pressure

Depth of Tension Crack (zcz_czc​)

Depth of Tension Crack

Coulomb's Earth Pressure Theory

Coulomb vs. Rankine

Coulomb Active Coefficient (KaK_aKa​)

Coulomb Active Earth Pressure Coefficient

Culmann's Graphical Method

Effect of Surcharge and Water

Surcharge Loads (qqq)

Lateral Pressure from Surcharge

Water Pressure

Total Horizontal Stress with Water

Important

1. At-Rest State ( $K_{o}$ )

2. Active State ( $K_{a}$ )

3. Passive State ( $K_{p}$ )

Depth of Tension Crack ( $z_{c}$ )

Coulomb Active Coefficient ( $K_{a}$ )

Surcharge Loads ( $q$ )