Lateral Earth Pressure: Rankine and Coulomb Theories

Understanding how soil pushes against structures is fundamental to designing safe and economical retaining walls, basements, and foundations. The two primary classical theories—Rankine and Coulomb—are used by engineers to calculate these lateral earth pressures. Mastering these concepts allows you to determine the forces that could cause a wall to slide, overturn, or fail, ensuring your designs remain stable under all conditions.

The Starting Point: At-Rest Earth Pressure

Before a wall moves, the soil is in a state of at-rest pressure. Imagine a stack of books between two bookends that are glued in place. The books push sideways, but the bookends don't budge. Similarly, when a retaining structure is perfectly rigid and does not yield (like a massive basement wall braced at the top and bottom), the soil behind it experiences no lateral strain. The lateral pressure in this state is called the at-rest earth pressure.

The coefficient of at-rest earth pressure, $K_0$, relates the effective horizontal stress to the effective vertical stress: $${\sigma }_h^{\prime }=K_0\cdot {\sigma }_v^{\prime }$$ For normally consolidated cohesionless soils, a common empirical estimate is $K_0=1-\sin {\varphi }^{\prime }$, where ${\varphi }^{\prime }$ is the soil's effective angle of internal friction. This condition represents the upper bound for active pressure and the lower bound for passive pressure, serving as a critical benchmark.

Rankine's Theory: A Stress State Approach

Rankine's theory (1857) analyzes stress states within a soil mass assuming it is homogenous, isotropic, and has a frictionless interface with the wall. It defines two limiting equilibrium states: active and passive.

Active Pressure occurs when the wall moves away from the soil mass, allowing the soil to expand laterally. This reduces the horizontal stress to a minimum. For a cohesionless soil with a horizontal backfill, the Rankine active earth pressure coefficient, $K_a$, is: $$K_a=\frac{1-\sin \varphi }{1+\sin \varphi }={\tan }^2(45^{\circ }-\varphi /2)$$ The resulting horizontal pressure at any depth $z$ is ${\sigma }_h=K_a\cdot \gamma z$, where $\gamma$ is the soil unit weight. This produces a triangular pressure distribution. The total active force per unit length of wall is the area of the triangle: $P_a=\frac{1}{2}{K}_a\gamma H^2$, acting at one-third of the wall height from the base.

Passive Pressure is mobilized when the wall is pushed into the soil, causing lateral compression. The horizontal stress increases to a maximum. The Rankine passive earth pressure coefficient, $K_p$, is: $$K_p=\frac{1+\sin \varphi }{1-\sin \varphi }={\tan }^2(45^{\circ }+\varphi /2)$$ The force is $P_p=\frac{1}{2}{K}_p\gamma H^2$. Note that $K_p$ is significantly larger than $K_a$ (e.g., for $\varphi =30^{\circ }$, $K_a=0.33$ and $K_p=3.0$), meaning mobilizing full passive resistance requires substantial wall movement.

For Cohesive Soils, Rankine's equations incorporate cohesion ($c$). The active and passive pressures at depth $z$ become:

• Active: ${\sigma }_a=K_a\gamma z-2c\sqrt{K_a}$
• Passive: ${\sigma }_p=K_p\gamma z+2c\sqrt{K_p}$

The tensile component in the active case creates a zone of tension near the surface, which is often neglected due to potential cracking. The pressure distribution is therefore considered trapezoidal, starting from a depth $z_0=2c/(\gamma \sqrt{K_a})$.

Coulomb's Theory: A Wedge Mechanism

Coulomb's theory (1776) takes a more geometric approach by considering the equilibrium of a failing soil wedge. Its major advantage is that it accounts for wall friction ($\delta$), the angle between the soil and the wall, and a sloping backfill ($\beta$).

Coulomb analyzes the forces acting on a trial wedge of soil as it slides along a failure plane and the back of the wall. By applying static equilibrium (force polygon), the theory finds the wedge that produces the maximum active thrust or the minimum passive resistance. The coefficients are more complex but account for more realistic conditions.

For active pressure with wall friction, the soil wedge slides down, causing the resultant force on the wall to be inclined upward. This is beneficial, as it slightly reduces the horizontal component of the thrust. For passive pressure, the soil wedge moves up, and the wall force is inclined downward. However, with high wall friction, Coulomb's method can overestimate passive resistance, as it assumes a planar failure surface when the actual surface may be curved.

Earth Pressure Distributions for Different Conditions

The shape of the pressure diagram changes with backfill and loading conditions.

• Uniform Surcharge: A load ($q$) on the backfill adds a constant lateral pressure of $K\cdot q$ over the entire wall height, resulting in a rectangular pressure block atop the triangular diagram.
• Stratified Soil: Layers with different $\varphi$ or $\gamma$ create a pressure diagram with slope changes at layer interfaces. The pressure at any depth is ${\sigma }_h=K_{(layer)}\cdot (\sum \gamma z)$ for that layer.
• Cohesive Backfill: As derived, the active diagram shows tension down to depth $z_0$, requiring careful consideration in design to avoid cracks that would fill with water.
• Water Pressure: The presence of water is critical. Lateral earth pressure is calculated using effective stresses (${\sigma }^{\prime }=K\cdot {\sigma }_v^{\prime }$). Hydrostatic water pressure (${\gamma }_{w}z$) is added separately to the total lateral pressure. Poor drainage is a leading cause of retaining wall failure.

Common Pitfalls

Ignoring Drainage and Water Pressure. This is the most critical error. Pore water pressure is not multiplied by $K_a$ or $K_p$. Engineers must always compute effective stress for soil pressure and add full hydrostatic pressure separately. Always design for adequate drainage behind the wall.

Misapplying Passive Pressure. Achieving the full theoretical passive resistance requires significant wall movement (often 2-5% of wall height), which may be unacceptable. For permanent structures, it's common to use only a fraction of $K_p$ or to rely on at-rest pressure for stability.

Confusing Soil Parameters. Using total stress parameters ($c$, $\varphi$) for drained, long-term conditions, or vice-versa, leads to grossly incorrect pressures. Effective stress parameters ($c^{\prime }$, ${\varphi }^{\prime }$) must be used for granular soils and long-term stability of fine-grained soils.

Overlooking Tension Cracks in Clays. In active pressure calculations for cohesive soils, the theoretical tension zone is often assumed to be a crack that can fill with water. The prudent approach is to take the active pressure as zero from the surface to $z_0$ and add a hydrostatic pressure from that depth, significantly increasing the design load.

Summary

• At-rest pressure ($K_0$) applies to completely non-yielding structures and is the baseline state of soil stress.
• Rankine's theory provides earth pressure coefficients based on soil stress states, handling both cohesionless and cohesive soils, but assumes a smooth, vertical wall.
• Coulomb's theory uses a failing wedge analysis, which is more versatile as it incorporates wall friction and sloping backfills, making it the preferred method for most practical retaining wall designs.
• Pressure distributions are not always triangular; they are modified by layered soils, surface loads, water, and cohesion, each requiring careful diagram construction.
• Always design for drainage. Uncontrolled water pressure is the primary cause of retaining wall failures, and its calculation is separate from effective soil pressure.