5.1 Lateral Earth Pressures and Retaining Structures

Fundamentals of Lateral Earth Pressures

In geotechnical engineering, lateral earth pressure refers to the horizontal force exerted by a soil mass against a retaining structure. Designing a retaining wall requires a precise understanding of the magnitude, direction, and distribution of these pressures. Depending on the movement of the retaining structure relative to the soil mass, earth pressures are classified into three distinct states:

1. At-rest earth pressure: This state occurs when the wall is completely rigid and undergoes zero horizontal displacement. The soil remains in an elastic state of equilibrium. The ratio of horizontal effective stress to vertical effective stress is defined as the coefficient of earth pressure at-rest (K_0). For normally consolidated soils, Jaky's empirical relationship is widely used:

   K_0 = 1 - sin(phi')

   where phi' is the angle of internal friction of the soil.

2. Active earth pressure: When the retaining wall moves away from the soil mass, the soil expands laterally, causing the horizontal pressure to decrease. This movement continues until the soil shear strength is fully mobilized along a failure surface. The minimum lateral pressure reached during this process is the active earth pressure. The ratio of horizontal to vertical effective stress in this state is the active earth pressure coefficient (K_a).

3. Passive earth pressure: Conversely, if the wall is pushed into the soil mass, the soil is compressed laterally, increasing horizontal resistance. The wall continues to push until a failure wedge is forced upward. The maximum lateral pressure achieved is the passive earth pressure. The ratio of horizontal to vertical effective stress is the passive earth pressure coefficient (K_p).

It is vital for the PE exam to recognize that active pressure requires very small wall rotation/displacement (typically 0.001 * H to 0.002 * H, where H is wall height) to develop, whereas passive pressure requires significantly larger movement (0.01 * H to 0.05 * H) to be fully mobilized.

Classical Earth Pressure Theories

Geotechnical engineers rely on two classical theories to calculate active and passive earth pressures: Rankine theory and Coulomb theory.

Rankine's Earth Pressure Theory

Developed by William Rankine in 1857, this theory assumes a state of plastic equilibrium in the soil mass. The fundamental assumptions of Rankine's theory are:

• The soil is homogeneous, isotropic, and semi-infinite.
• The backfill surface is planar (either horizontal or inclined).
• The face of the retaining wall in contact with the soil is vertical and perfectly smooth. This assumption means there is no shear stress or wall friction (delta = 0) between the soil and the wall.
• The failure wedge acts as a rigid body, and the shear stress at failure follows the Mohr-Coulomb criterion.

Under these assumptions, the principal stress directions are vertical and horizontal. The Rankine coefficients for a horizontal backfill surface are:

K_a = tan²(45° - phi'/2) = (1 - sin(phi')) / (1 + sin(phi'))

K_p = tan²(45° + phi'/2) = (1 + sin(phi')) / (1 - sin(phi'))

Note that K_a * K_p = 1. For an inclined backfill at an angle beta to the horizontal, the coefficients become:

K_a = cos(beta) * [cos(beta) - sqrt(cos²(beta) - cos²(phi'))] / [cos(beta) + sqrt(cos²(beta) - cos²(phi'))]

K_p = cos(beta) * [cos(beta) + sqrt(cos²(beta) - cos²(phi'))] / [cos(beta) - sqrt(cos²(beta) - cos²(phi'))]

Coulomb's Earth Pressure Theory

Proposed by Charles-Augustin de Coulomb in 1776, this theory pre-dates Rankine's and is based on a limit equilibrium wedge analysis. Unlike Rankine, Coulomb accounts for:

• Friction between the wall and the soil, denoted by the wall friction angle delta (typically assumed to be between phi'/2 and 2/3 * phi').
• A sloped or inclined back face of the wall (inclined at angle theta from the vertical).
• An inclined backfill surface (angle beta).

Coulomb's theory assumes that the failure surface is a plane passing through the heel of the wall. The active and passive earth pressure coefficients are calculated using the following equations:

K_a = cos²(phi' - theta) / [cos²(theta) * cos(theta + delta) * (1 + sqrt(sin(phi' + delta)*sin(phi' - beta) / (cos(theta + delta)*cos(theta - beta))))²]

K_p = cos²(phi' + theta) / [cos²(theta) * cos(theta - delta) * (1 - sqrt(sin(phi' + delta)*sin(phi' + beta) / (cos(theta - delta)*cos(theta - beta))))²]

When the wall is vertical (theta = 0), the backfill is horizontal (beta = 0), and the wall is smooth (delta = 0), Coulomb’s equations simplify exactly to Rankine's equations. On the PE exam, Coulomb's active theory is widely used because wall friction reduces the calculated active pressure, which is more realistic. However, Coulomb's passive theory can significantly overestimate passive pressure when delta > phi'/3 due to the assumption of a planar failure surface (in reality, it is curved). Therefore, alternative methods like the Log-Spiral method are preferred for passive pressures with high friction.

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Earth Pressure Distributions

To compute the total thrust against a wall, we must integrate the lateral pressures over the wall height, representing them as pressure diagrams.

Cohesionless Soils (c' = 0)

For dry or moist sand backfills, the lateral active pressure p_a at any depth z is:

p_a = K_a * sigma'_v = K_a * gamma * z

where gamma is the soil unit weight. The pressure distribution is triangular, starting at 0 at the surface and reaching K_a * gamma * H at the base. The total active thrust per unit length of the wall, P_a, is the area of this triangle:

P_a = 0.5 * K_a * gamma * H²

The line of action for this resultant force passes through the centroid of the triangle, which is at a height of H/3 from the base.

Cohesive Soils (c' > 0)

For soils exhibiting cohesion (c'), the Mohr-Coulomb shear strength adds a tensile component to the active state and a compressive component to the passive state. The lateral pressures are:

p_a = K_a * sigma'_v - 2 * c' * sqrt(K_a)

p_p = K_p * sigma'_v + 2 * c' * sqrt(K_p)

In the active state, at shallow depths, the negative term 2 * c' * sqrt(K_a) exceeds the vertical stress term, resulting in negative lateral pressure (tension). Since soil has negligible tensile strength, a tension crack will develop. The depth of this tension crack, z_c, is the depth where p_a = 0:

K_a * gamma * z_c - 2 * c' * sqrt(K_a) = 0 => z_c = (2 * c') / (gamma * sqrt(K_a))

For stability design, engineers typically assume that the soil down to depth z_c cannot support any active thrust. Thus, the active force is integrated only from z_c to H:

P_a = 0.5 * (H - z_c) * (K_a * gamma * H - 2 * c' * sqrt(K_a))

Alternatively, if the tension crack is assumed to fill with water during a storm, hydrostatic water pressure must be added down to z_c, which is a common and conservative design practice.

Hydrostatic and Surcharge Pressures

If a water table is present within the backfill, lateral pressures must account for both effective soil stress and pore water pressure:

p = K * (sigma_v - u) + u = K * gamma' * z + gamma_w * z

where gamma' is the submerged (buoyant) soil unit weight (gamma' = gamma_sat - gamma_w) and gamma_w is the unit weight of water (62.4 lb/ft³ or 9.81 kN/m³).

Additionally, if a uniform surcharge load q is applied at the backfill surface, it transmits a constant horizontal pressure over the entire wall height:

p_surcharge = K * q

The total force due to surcharge is P_surcharge = K * q * H, acting at a height of H/2 from the base.

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Retaining Wall Design and Stability Analysis

Retaining walls must be checked for safety against three primary external failure modes: overturning, sliding, and bearing capacity failure.

1. Overturning Stability

The lateral earth pressures and surcharges act to tip the wall forward about its toe. The factor of safety against overturning (FS_overturning) is the ratio of the resisting moment (Sum(M_R)) to the overturning moment (Sum(M_O)), both evaluated about the toe of the wall footing:

FS_overturning = Sum(M_R) / Sum(M_O) >= 2.0

• The overturning moments (Sum(M_O)) are generated by the horizontal components of the active earth force (P_aH) and surcharge force.
• The resisting moments (Sum(M_R)) are generated by the vertical forces, including the weight of the concrete wall components, the weight of the soil block resting on the footing's heel, and the vertical component of active earth force (P_aV) if the force is inclined.

2. Sliding Stability

The horizontal thrust forces try to push the wall laterally away from the backfill. The factor of safety against sliding (FS_sliding) is the ratio of resisting shear forces along the base to the driving horizontal force:

FS_sliding = Sum(F_R) / Sum(F_d) >= 1.5

The resisting shear force along the base of the footing is:

F_R = V * tan(delta_b) + c_b * B + P_p

where:

• V is the sum of all vertical forces acting on the base.
• delta_b is the friction angle between the soil and the concrete base (often taken as 2/3 * phi').
• c_b is the adhesion (cohesion) between the concrete base and the foundation soil.
• B is the width of the footing base.
• P_p is the passive earth resistance at the toe of the wall. Because soil in front of the toe can be eroded or excavated, many design codes recommend neglecting P_p or applying a high reduction factor to it.

3. Bearing Capacity and Eccentricity Analysis

The combination of vertical loads (V) and horizontal forces (H) creates an eccentric loading condition on the footing. To prevent structural instability and localized soil failure:

• We must compute the eccentricity (e) of the resultant vertical force relative to the center of the footing base:

  e = B/2 - (Sum(M_R) - Sum(M_O)) / V

• To ensure that no tension develops between the footing base and the soil (which would lead to separation and loss of contact), the resultant must lie within the middle third of the footing base. This requires:

  e <= B/6

• If e <= B/6, the contact pressure distribution is trapezoidal, with the maximum and minimum bearing pressures calculated as:

  q_max = (V/B) * (1 + 6 * e / B)

  q_min = (V/B) * (1 - 6 * e / B)

• The maximum bearing pressure (q_max) must not exceed the allowable soil bearing capacity:

  FS_bearing = q_ult / q_max >= 3.0

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Detailed Worked Example

Problem Statement: A cantilever retaining wall is 18 ft tall with a vertical back face. The backfill soil is a dry sand with a unit weight of gamma = 115 pcf and a friction angle of phi' = 32°. The backfill is horizontal and carries a uniform surcharge of q = 250 psf. Using Rankine theory:

1. Calculate the active earth pressure coefficient (K_a).
2. Determine the total active thrust force (P_a) and surcharge force (P_q) per foot of wall.
3. Calculate the total overturning moment (M_O) about the base of the wall.

Step-by-Step Solution:

1. Calculate K_a:

   K_a = tan²(45° - 32°/2) = tan²(29°) = 0.307

2. Calculate Active Thrust Force (P_a):

   The soil active thrust is:

   P_a = 0.5 * K_a * gamma * H² = 0.5 * (0.307) * (115 pcf) * (18 ft)² = 0.5 * 0.307 * 115 * 324 = 5,716 lb/ft

   This force acts at H/3 = 18/3 = 6 ft from the base.

3. Calculate Surcharge Force (P_q):

   The lateral force due to the surcharge is:

   P_q = K_a * q * H = (0.307) * (250 psf) * (18 ft) = 1,381.5 lb/ft

   This force acts at H/2 = 18/2 = 9 ft from the base.

4. Calculate Total Overturning Moment (M_O):

   M_O = P_a * (H/3) + P_q * (H/2) = (5,716 lb/ft * 6 ft) + (1,381.5 lb/ft * 9 ft)

   M_O = 34,296 lb·ft/ft + 12,433.5 lb·ft/ft = 46,729.5 lb·ft/ft