Retaining Wall & Earth Pressure Theories | Civil Engineering SSC JE (Technical) - Civil Engineering (CE) PDF Download

Retaining Wall / Earth Pressure Theories

Contrasting Points of Active and Passive Pressures

Earth Pressure at Rest (K_o)

Definition. Earth pressure at rest is the horizontal pressure acting on a rigid structure when the soil mass is not allowed to deform laterally (i.e., no lateral strain).

For elastic isotropic soils an expression based on Poisson's ratio is used (Terzaghi/elastic formulation):

where ν = Poisson's ratio.

For cohesionless soils a commonly used empirical expression is the Jaky relation:

where φ = angle of internal friction of soil.

For overconsolidated soils an expression proposed by Schmertmann is often used. This expression introduces the overconsolidation ratio (OCR) into the calculation of K_o:

where OCR = Over Consolidation Ratio.

Typical values of K_o (approximate):

• Dense sand: 0.40 - 0.45
• Loose sand: 0.45 - 0.50
• Mechanically compacted sand: 0.8 - 1.5
• Normally consolidated clay: 0.50 - 0.60
• Over consolidated clay: 1.0 - 4.0

Active Earth Pressure

mobilisation of active pressure requires small horizontal movement of the wall relative to the backfill.

Typical values of wall movement ΔH required to reach active state:

• ΔH ≈ 0.2% of H for dense sand
• ΔH ≈ 0.5% of H for loose sand
• ΔH ≈ 0.4% of H for clays (approx.)
• Length of failure block (top projection) = H · cot(45° + φ/2)

Standard expression for the coefficient of active earth pressure (for cohesionless soil):

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

Passive Earth Pressure

mobilisation of passive pressure requires larger movement of the wall into the soil mass.

Typical values of wall movement ΔH to mobilise passive resistance:

• ΔH ≈ 2% of H for dense sand
• ΔH ≈ 15% of H for loose sand (large displacement required)
• Length of failure block (top projection) = H · cot(45° - φ/2)

K_p = coefficient of passive earth pressure.

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

Note. K_a · K_p = 1 (for the idealised Rankine case with same backfill).

Ordering of pressures for the same depth: P_a < P_o < P_p, where

• P_a = active earth pressure
• P_o = earth pressure at rest
• P_p = passive earth pressure

Active Earth Pressure by Rankine Theory

Assumptions (Rankine): soil is homogeneous, isotropic, cohesionless (unless modified), wall is frictionless and vertical (in the basic form), backfill surface may be horizontal or inclined, failure planes are planar.

(i) Cohesionless soil on a vertical smooth wall

The resultant active force acts at H/3 from the base (for a triangular pressure distribution). Let P_a denote the active force per unit length of wall.

(ii) Submerged cohesionless soil on a vertical smooth wall

For submerged conditions the submerged unit weight γ′ is used in place of γ.

The resultant still acts at H/3 from the base.

Layered backfill and surcharge effects (schematic)

When backfill has layers or surcharge, the resulting active pressure is the algebraic sum of contributions from each layer or surcharge. Example notation used in many schematics:

P_a2 = K_a γ1 H₂² acts at H₂/2 from base.

Other contributions and their points of action are shown by the sequence of figures below; each elemental pressure has its own centroidal distance from the base and should be summed vectorially to get the total.

Total Active Earth Pressure (P_a)

The total active force and its point of application are obtained by summing the individual contributions from the backfill weight, soil pressure triangles/trapezia and any surcharge pressures.

The resultant acts at the distance shown in the figure below from the base.

Typical formulae for surcharge and distributed loads are presented in design references. Example terms and derivations are represented in the figures that follow.

P_a1 = K_a · q · H acts at the distance shown in the figure below from base.

Other elemental results and lines of action:

acts at the distance shown below.

acts at

from base.

(v)

• acts at (H₂ + H₁) from base.
• acts at the distance shown in the figure below from base.
• acts at H₂ from base.

(vi)

The resultant active force from triangular distributions acts at H/3 from base but the line of action is parallel to the backfill surface when the backfill is inclined.

Acts at H/3 from base where the pressure triangle is measured normal to the backfill.

(vii)

where W = weight of the soil block of unit length corresponding to the imaginary vertical back.

P_av = active earth pressure on imaginary vertical back.

p_a = resultant active earth pressure on the wall.

a = angle of inclination of resultant P_a with the horizontal.

(viii)

where P_av = active earth pressure on the imaginary vertical back of wall (acts parallel to backfill), and W = weight of soil block of unit length.

Active Earth Pressure for Cohesive Soil

For cohesive soils, cohesion introduces an additional component to the earth pressure. In many formulations an influence factor N_f appears in the expression for cohesion contribution.

where N_f = influence factor related to φ and wall geometry.

• Active earth pressure at any depth z is given by the distribution shown in the figure below.

• Active earth pressure at the surface (z = 0) is given by the expression shown in the figure.

• At z = z_c → P_a = 0 (tension crack depth).

where z_c = depth of tension crack.

where H_c = critical depth; this is the maximum unsupported depth (depends on c, γ and φ in the standard expression for tension crack depth).

Note: For cohesive soils the depth of tension crack (z_c) can be obtained from equilibrium expressions and depends on cohesion c, unit weight γ and internal angle φ (see standard derivations).

Passive Earth Pressure for Cohesive Soil

• Passive earth pressure at any depth z is shown in the figure below.

• Total P_p on unit length is obtained by integrating the pressure distribution (diagram below).

Coulomb's Wedge Theory (General Form)

In the Coulomb wedge analysis the following angles are used:

• a = angle of back of wall with horizontal
• b = angle of sloping ground (backfill slope)
• f = angle of interface friction between soil and wall (f = 0° for smooth walls)

• When tension cracks are not developed the pressure diagram is a trapezoid whose centroid gives the line of action of the resultant.

The resultant acts at the centroid of the trapezoidal distribution.

• When tension cracks are developed the pressure distribution changes as shown and the resultant location is that of the centroid of the new trapezoid/triangular region.

The resultant acts at the centroid of the trapezoid.

• When tension cracks are developed the expressions become those indicated in the figures below.

or

acts at

Remember: Cohesion decreases the active earth pressure while it increases the passive earth pressure (all other parameters equal).

Acts at H/3 from base on a triangular distribution; the line of action is normal to the backfill or parallel to the backfill depending on the assumed diagram.