1.1 Lateral Earth Pressure

Introduction to Lateral Earth Pressure

Lateral earth pressure is the pressure that soil exerts in the horizontal direction against a retaining structure. For PE Construction candidates, mastering this topic is absolutely critical. In construction engineering, you will constantly encounter scenarios involving retaining walls, sheet piling, temporary shoring systems, trench boxes, and deep excavations. If a retaining structure fails, whether it's a temporary cofferdam or a permanent concrete wall, it is often due to an underestimation of lateral earth pressures, failure to account for groundwater, or a fundamental misunderstanding of the soil state.

There are three primary states of lateral earth pressure that dictate how we calculate the loads on these structures: at-rest, active, and passive.

At-Rest Earth Pressure ($K_o$)

The at-rest condition occurs when the soil mass yields neither away from nor toward the retaining structure. The wall is essentially rigid and unyielding. Typical examples include basement walls braced at the top by floor framing and at the bottom by a foundation slab, or heavy bridge abutments. Because the wall does not move, the soil remains in its original state of stress. The lateral earth pressure coefficient at rest, $K_o$, for normally consolidated soils can be approximated using Jaky's empirical relationship:

$K_o = 1 - \sin(\phi')$

where $\phi'$ is the effective angle of internal friction of the soil. For overconsolidated soils, $K_o$ is higher and must be adjusted using the overconsolidation ratio (OCR).

Active Earth Pressure ($K_a$)

The active state occurs when the retaining wall moves or yields away from the backfill soil. This outward movement allows the soil to expand laterally, mobilizing its shear strength and reducing the lateral pressure to a theoretical minimum value. For granular soils, it only takes a very small movement (typically 0.1% to 0.4% of the wall height) to fully mobilize the active state. The active earth pressure coefficient, $K_a$, using Rankine's theory for a horizontal backfill is:

$K_a = \tan^2(45^\circ - \frac{\phi'}{2})$

Passive Earth Pressure ($K_p$)

The passive state occurs when the retaining wall is pushed into the soil mass, such as the soil resisting the movement of an anchor block or the embedded toe of a sheet pile wall. The soil is compressed horizontally, mobilizing its shear strength to resist the movement. This results in the maximum possible lateral pressure. The passive earth pressure coefficient, $K_p$, using Rankine's theory for a horizontal backfill is:

$K_p = \tan^2(45^\circ + \frac{\phi'}{2})$

Note that $K_p$ is typically much larger than $K_a$. However, it requires significantly more wall movement (typically 1% to 4% of the wall height) to fully mobilize passive pressure. Because such large movements may not be tolerable for the structure, engineers often apply a factor of safety (e.g., 2.0 or 3.0) to the passive resistance in design.

Rankine vs. Coulomb Earth Pressure Theories

Two classical theories are used to calculate lateral earth pressures: Rankine and Coulomb. Understanding their assumptions is key for the PE exam.

Rankine Theory (1857):

• Assumes a frictionless wall (wall friction angle $\delta = 0$).
• Assumes a strictly vertical wall surface.
• Assumes the failure surface in the soil is a plane.
• Resultant force is parallel to the backfill surface.
• Generally provides a more conservative (higher) estimate for active pressure and a highly conservative (lower) estimate for passive pressure.

Coulomb Theory (1776):

• Accounts for wall friction ($\delta > 0$).
• Can accommodate non-vertical (battered) walls.
• Considers a sliding wedge of soil and uses principles of static equilibrium.
• Resultant force acts at an angle $\delta$ to the normal of the wall.
• Yields more realistic results when wall friction is significant, especially for calculating passive resistance where Rankine's assumption of zero friction can be excessively conservative.

In PE Construction problems, Rankine is typically used for simplified calculations unless the problem specifically provides a wall friction angle or dictates Coulomb's theory.

Thrust Calculation Formulas and Cohesive Soils

The lateral earth pressure $\sigma_h$ at any depth $z$ is the product of the vertical effective stress $\sigma_v'$ and the appropriate lateral earth pressure coefficient ($K$), plus any pore water pressure $u$. For cohesive soils with cohesion $c$, the active pressure includes a negative term:

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

Because of the cohesion term, the theoretical active pressure near the surface of a clay backfill is negative (in tension). The depth where the active pressure is exactly zero is called the depth of the tension crack, $z_c$:

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

For a dry, homogeneous cohesionless soil retained by a vertical wall of height $H$, the total lateral thrust force $P$ per unit length of wall is the area of the triangular pressure distribution:

$P = \frac{1}{2} K \gamma H^2$

where $\gamma$ is the unit weight of the soil. The resultant force acts at a distance of $H/3$ from the base of the wall.

Typical Soil Properties

Understanding typical soil properties is vital when specific data is missing in an exam scenario or for sanity-checking your calculated values.

Worked Example: Calculating Total Active Lateral Force

Problem: A 15-ft high cantilever retaining wall retains a dry backfill of dense sand with a unit weight of 120 pcf and an effective friction angle of 34 degrees. The backfill surface is horizontal. Using Rankine's theory, calculate the total active lateral force per foot of wall and determine its point of application.

Solution:

1. Calculate the active earth pressure coefficient ($K_a$): $K_a = \tan^2(45^\circ - 34^\circ/2) = \tan^2(28^\circ) \approx 0.2827$

2. Calculate the vertical effective stress at the base ($z = 15$ ft): $\sigma_v' = \gamma H = 120 \text{ pcf} \times 15 \text{ ft} = 1800 \text{ psf}$

3. Calculate the active lateral earth pressure at the base: $\sigma_a = K_a \sigma_v' = 0.2827 \times 1800 \text{ psf} \approx 508.9 \text{ psf}$

4. Calculate the total active lateral force ($P_a$) per foot of wall: $P_a = \frac{1}{2} K_a \gamma H^2 = \frac{1}{2} (0.2827)(120)(15)^2 = 3816.5 \text{ lbs/ft}$

5. Point of application: The pressure distribution is triangular, so the resultant acts at $H/3$ from the base: $y = 15 / 3 = 5 \text{ ft}$

Answer: The total active force is approximately 3,817 lbs/ft acting 5 ft above the base of the wall.

Exam Traps and Tips

• Water Table: This is the most common trap. If groundwater is present, you must compute the effective vertical stress using the buoyant unit weight ($\gamma' = \gamma_{sat} - \gamma_w$). Calculate the soil's lateral earth pressure using this effective stress, and then ADD the hydrostatic water pressure ($u = \gamma_w z_w$) to find the total lateral pressure. Remember, water pressure is isotropic; it acts with a coefficient of 1.0, not $K_a$ or $K_p$.
• Surcharge Loads: A uniform surface surcharge $q$ (like construction equipment or traffic) adds a uniform rectangular pressure block to the wall. This increases the lateral force by $P_q = K q H$, and this rectangular resultant acts at $H/2$ from the base.
• Displacement Verification: Remember that at-rest pressure applies to rigidly braced walls (like basement walls). Active pressure requires outward wall movement. Ensure you are applying the correct coefficient based on the structural condition described in the problem.