Earth Pressure Theories - Soil Mechanics - Civil Engineering (CE) PDF Download

Rankine's Earth Pressure Theory

Rankine's theory assumes that the backfill is cohesionless or cohesive homogeneous soil, the wall is smooth (no wall friction, δ = 0), and the failure surfaces are planar. The resultant lateral force on the wall is assumed to act parallel to the backfill surface. When a retaining wall is rigid and does not move, the soil behind it remains in a state of elastic equilibrium; when the wall moves sufficiently, a plastic wedge of soil develops and the state becomes one of limiting (active or passive) equilibrium.

Consider a prismatic element of backfill at depth z. The vertical stress at that depth is σ_v = γ z, where γ is the effective unit weight of the soil. If lateral stress is proportional to vertical stress at every depth, the ratio

K = σ_h / σ_v = σ_h / (γ z)

is constant with depth. Different states (at-rest, active, passive) correspond to different values of K.

Earth Pressure at Rest

The coefficient of earth pressure at rest (often denoted K₀ or K_ϕ in some texts) is the lateral pressure coefficient when the soil mass is laterally unstrained (no lateral deformation). It is commonly used when the wall does not move (for example, uniformly supported excavations or strutted systems).

Two frequently used relations for the at-rest coefficient are:

• K₀ = ν / (1 - ν) from elastic theory, where ν is Poisson's ratio.
• K₀ = 1 - sin φ (Jaky). This empirical relation is widely used in practice (the input reference given: Jaky (1994)).

Typical ranges for K₀ depend on material and deposit history: for naturally deposited sands without compaction it varies roughly from 0.4 (loose sand) to 0.6 (dense sand); compaction (tamping) can increase it up to about 0.8.

For a vertical wall of height H with soil at rest, the lateral stress at the base is σ_h = K₀ γ H, and the total lateral resultant per unit length of the wall is

P₀ = 0.5 K₀ γ H².

Rankine's Earth Pressures for Cohesionless Backfill (Vertical Wall, Horizontal Surface)

1. Active earth pressure (Rankine) - cohesionless soil

   In Rankine's active case the soil wedge above the potential rupture plane moves downwards and outwards relative to the retained mass. Rankine assumes the failure plane is planar and that the active state corresponds to the minimum lateral stress consistent with plastic equilibrium.

   The active earth pressure coefficient is

   K_a = tan²(45° - φ/2)

   Derivation (compact, stepwise):

   Assume principal stress orientation from Mohr-Coulomb relations and geometry of failure plane.

   Express K_a in terms of φ using trigonometric identities.

   Obtain an alternate form:

   K_a = (1 - sin φ) / (1 + sin φ)

   The lateral active stress at depth z is

   σ_h,a = K_a γ z

   For a wall of height H, the lateral pressure at the base is K_a γ H, the total resultant per unit length is triangular:

   P_a = 0.5 K_a γ H²

   The line of action of this resultant on the vertical wall lies at H/3 above the base and the pressure acts normal to the wall.



2. Passive earth pressure (Rankine) - cohesionless soil

   In the passive case the wall moves towards the backfill, compressing the soil wedge. The passive earth pressure coefficient is the reciprocal of the active coefficient:

   K_p = tan²(45° + φ/2)

   or

   K_p = (1 + sin φ) / (1 - sin φ)

   The lateral passive stress at depth z is

   σ_h,p = K_p γ z

   The total resultant per unit length for a wall of height H is

   P_p = 0.5 K_p γ H²

   The resultant acts at H/3 above the base on the wall and the pressure distribution is linear with depth.



3. Rankine active earth pressure with a sloping cohesionless backfill

   If the backfill surface slopes, the Rankine active wedge geometry changes but the approach is similar: an active wedge bounded by a planar rupture surface is assumed and the active coefficient is modified by the slope angle. The pressure at depth H measured normal to the wall can still be expressed in the form

   σ_h,a = K_a γ z (with K_a evaluated for the given slope and φ)

   The total resultant per unit length becomes

   P_a = 0.5 K_a γ H²

   The resultant acts at a height H/3 from the base and parallel to the sloping surface of the backfill.

Remarks on Rankine theory

• Assumption of zero wall friction: Rankine assumes δ = 0. If wall friction is significant, Rankine's predictions may be non-conservative for passive pressure or may not represent the actual distribution acting on a wall surface.
• Applicability: Rankine theory is best used to determine earth pressures on a vertical plane within a soil mass (idealised conditions) or for smooth walls. It gives simple closed-form expressions for K_a and K_p for horizontal backfill and is commonly used for preliminary design.
• Limitations: It assumes planar rupture surfaces, homogeneous soil, and that the resultant acts parallel to the backfill surface. For sloping walls, non-zero wall friction, layered soils or non-planar failure, other methods (e.g., Coulomb) are preferred.

Rankine's Active Pressure for Cohesive Soils (Horizontal Backfill, Smooth Vertical Wall)

When cohesion c is present, the active lateral stress at depth z under Rankine assumptions becomes

σ_h,a(z) = K_a γ z - 2 c √K_a

Thus the total resultant lateral thrust per unit length on a wall of height H is

P_a = 0.5 K_a γ H² - 2 c √K_a H

Setting the lateral stress equal to zero gives the depth at which the tensile (or zero-pressure) condition occurs:

Set σ_h,a(z) = 0

Solve for z:

0 = K_a γ z - 2 c √K_a

z = 2 c √K_a / (K_a γ) = 2 c / (γ √K_a)

This z is often called the depth of the tensile crack (z_ϕ).

An often-quoted measure of the maximum unsupported height of a cohesive slope or wall (critical height) derived from Rankine reasoning is

H_c = 4 c / (γ √K_a)

In practice Terzaghi and other field observations have shown that the actual critical height for cohesive materials is often lower than the simple theoretical value, because of factors such as time-dependent behaviour (creep), imperfect geometry, layered soils and disturbance.

Coulomb's Wedge Theory

Coulomb (1776) developed a general limiting-equilibrium method to determine earth pressures by considering the forces acting on a potential sliding wedge that separates from the backfill when the wall moves. The analysis treats the sliding wedge as a rigid body and enforces equilibrium of forces.

The lateral pressure on the wall equals the reaction force that the wall must provide to keep the sliding wedge in equilibrium. Coulomb's method allows for non-zero wall friction (δ) and for wall inclination and backfill slopes; for these reasons Coulomb theory is more general than Rankine's and is often used for practical wall design when friction at the wall or sloping backfill is important.

Common assumptions in Coulomb's wedge method

• The backfill is dry (or drained for effective stress analysis), homogeneous, isotropic, ideally plastic and rigid-perfectly plastic (soil deformability is neglected).
• The slip surface is assumed planar and passes through the heel of the wall (for the classical form of the method).
• The wall surface may be rough; the resultant earth pressure on the wall is inclined at an angle δ to the normal to the wall, where δ is the wall-soil friction angle.
• The sliding wedge acts as a rigid body and equilibrium (sum of forces) of the wedge is used to obtain the lateral force.
• The position and inclination of the resultant are taken at the limiting position (commonly assumed to act at one-third height from the base if the pressure distribution is assumed linear), but the general Coulomb solution gives the magnitude and line of action through the limiting geometry.

Practical use and solution methods

Coulomb's formula for active or passive thrust can be derived by writing equilibrium of the sliding wedge and minimising the resultant with respect to the unknown wedge inclination; the final expression depends on the wall inclination, backfill slope, φ and δ. Exact algebraic forms are lengthy; therefore in practice graphical and numerical methods are commonly used.

Some graphical and trial methods used to obtain earth pressures are:

• Culmann's graphical solution
• The trial-wedge (or trial plane) method
• Logarithmic spiral methods for non-planar failure surfaces (for cases where curved rupture surfaces are more realistic)

When wall friction (δ) is significant or when the backfill slope is non-zero, Coulomb's solution usually gives more realistic estimates of lateral thrust than Rankine's simple formulae. For design, it is common to compare results from both theories and, when in doubt, use the more conservative value or perform a stability check with limit equilibrium (including factors of safety).

Summary remarks (practical guidance)

• Use Rankine formulae for preliminary design with vertical, smooth walls and horizontal backfill or when simplicity is required.
• Use Coulomb (or numerical limit-equilibrium / FEM) approaches when wall friction, wall inclination, sloping backfill, layered soils or cohesion are important and a more accurate estimate is required.
• For cohesive soils include the cohesion term: lateral stress = K γ z - 2 c √K, and evaluate the possibility of tensile cracking near the surface.
• For at-rest conditions use K₀ = 1 - sin φ (Jaky) or the elastic relation K₀ = ν/(1 - ν) as appropriate.
• Always check assumptions: field conditions (drained/undrained), wall movement, wall-soil interface properties, layering and groundwater conditions can markedly change the required earth pressure model and numerical values.