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 ()
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
$$
\sigma_h = K_o \sigma_v
$$2. Active State ()
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. for dense sand).
- The failure plane develops at an angle of from the horizontal.
3. Passive State ()
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. to ).
- The failure plane develops at an angle of 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
FS Overturning
0.00
Target: ≥ 2.0
FS Sliding
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Target: ≥ 1.5
The red dashed line represents the Rankine failure plane. The red arrow represents the active earth pressure (), which tries to overturn and slide the wall. The blue arrow represents the weight (), 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 (, no wall friction).
- The wall is perfectly vertical ().
- The failure surface is a straight plane.
Rankine Coefficients
Rankine Active Earth Pressure Coefficient
$$
K_a = \tan^2(45^\circ - \phi/2) = \frac{1 - \sin \phi}{1 + \sin \phi}
$$Rankine Passive Earth Pressure Coefficient
$$
K_p = \tan^2(45^\circ + \phi/2) = \frac{1 + \sin \phi}{1 - \sin \phi}
$$Pressure Distribution
The pressure increases linearly with depth, forming a triangular distribution.
Rankine Active Earth Pressure
$$
p_a = \gamma z K_a - 2c \sqrt{K_a}
$$Rankine Passive Earth Pressure
$$
p_p = \gamma z K_p + 2c \sqrt{K_p}
$$Note on Cohesion: For cohesive soils (), 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 ()
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
$$
z_c = \frac{2c}{\gamma \sqrt{K_a}}
$$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 () and can easily handle non-vertical walls and sloped backfills. It assumes a planar failure surface.
Coulomb Active Coefficient ()
Coulomb Active Earth Pressure Coefficient
$$
K_a = \frac{\sin^2(\beta + \phi)}{\sin^2 \beta \sin(\beta - \delta) \left[ 1 + \sqrt{\frac{\sin(\phi + \delta) \sin(\phi - \alpha)}{\sin(\beta - \delta) \sin(\alpha + \beta)}} \right]^2}
$$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 () 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 ()
A uniform surcharge load () acting on the ground surface adds a constant, rectangular lateral pressure distribution over the entire height of the wall.
Lateral Pressure from Surcharge
$$
\Delta \sigma_h = q K_a
$$Water Pressure
Water exerts hydrostatic pressure () which acts equally in all directions (isotropic, so ).
Water affects the wall in two major ways:
- It reduces the effective vertical stress of the soil (), 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
$$
\sigma_h = (\sigma'_v K_a) + u
$$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 () for no movement, Active () for expansion (minimum pressure), and Passive () 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 () and backfill slope angles ().
- 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 () and adds massive unresisted load to the wall. Providing adequate drainage is the most critical aspect of retaining wall design.