Earth Pressure Theories

Understanding the pressure exerted by soil on retaining structures like walls, abutments, and foundations is crucial in civil engineering. This pressure, known as earth pressure, depends on the type of soil, its moisture content, the presence of groundwater, and the movement of the retaining structure.

Types of Earth Pressure

Earth pressure can be categorized into three main types based on the state of stress in the soil mass and the movement of the retaining structure:

1. At-Rest Earth Pressure (K₀)

This pressure occurs when the retaining wall is built and the soil is placed behind it without any lateral movement. The soil mass is in a state of equilibrium, neither expanding nor compressing laterally. The coefficient of earth pressure at rest (K₀) is a function of the soil's Poisson's ratio (ν) and the effective angle of internal friction (φ'). For normally consolidated soils, K₀ can be approximated by:

K₀ = 1 - sin(φ')

In granular soils, K₀ typically ranges from 0.4 to 0.6. For cohesive soils, the value can be higher.

2. Active Earth Pressure (K_a)

This pressure develops when the retaining wall moves away from the soil mass, causing the soil to expand laterally. This expansion reduces the lateral stress in the soil. The soil reaches a state of plastic equilibrium, and the lateral pressure is at its minimum value. The coefficient of active earth pressure (K_a) is given by:

K_a = tan²(45° - φ'/2)

For cohesive soils, the active earth pressure is modified to account for the tensile strength of the soil. The pressure distribution is not linear from the top. There is a depth of "tension crack" (z₀) from which the soil can theoretically pull away:

z₀ = 2c' / (γK_a)

where c' is the effective cohesion and γ is the unit weight of the soil.

The resultant active force (P_a) acts at a height of H/3 from the base of the wall, where H is the height of the wall. The pressure profile has a negative component due to cohesion up to z₀, and from z₀ downwards, it becomes positive.

3. Passive Earth Pressure (K_p)

This pressure develops when the retaining wall moves into the soil mass, causing the soil to compress laterally. This compression increases the lateral stress in the soil. The soil reaches a state of plastic equilibrium, and the lateral pressure is at its maximum value. The coefficient of passive earth pressure (K_p) is given by:

K_p = tan²(45° + φ'/2)

Similar to active pressure, cohesive soils also affect the passive pressure. The passive pressure increases from zero at the surface and develops a resultant passive force (P_p) acting at a height of H/3 from the base.

Rankine's Theory of Earth Pressure

Rankine's theory assumes that the soil is homogeneous, isotropic, and cohesionless. It also assumes that the back of the wall is vertical and there is no friction between the soil and the wall. The theory considers different wall backfill conditions:

1. Vertical Wall with Horizontal Backfill

This is the basic case where the wall is vertical, and the soil surface behind it is horizontal. The coefficients K_a and K_p are as derived above.

2. Vertical Wall with Inclined Backfill

If the backfill surface is inclined at an angle β to the horizontal, the active and passive pressure coefficients are modified:

K_a = cos(β) * [cos(β) - sqrt(cos²(β) - cos²(φ'))] / [cos(β) + sqrt(cos²(β) - cos²(φ'))]

K_p = cos(β) * [cos(β) + sqrt(cos²(β) - cos²(φ'))] / [cos(β) - sqrt(cos²(β) - cos²(φ'))]

The direction of the resultant active force is at an angle β to the normal of the wall. The resultant passive force is also at an angle β to the normal.

3. Inclined Wall with Horizontal Backfill

If the wall face is inclined at an angle α to the vertical, and the backfill is horizontal, the formulas become more complex. The direction of the resultant force is also affected.

Coulomb's Theory of Earth Pressure

Coulomb's theory is more general than Rankine's. It considers the friction between the soil and the retaining wall and can handle cohesive soils. It assumes that the failure surface is a plane and the active wedge is a rigid body. The theory uses a trial-and-error approach or graphical methods (like the slip line method) to determine the active earth pressure. The Coulomb equation for the coefficient of active earth pressure is:

K_a = [sin²(φ' + ψ)] / [sin²(φ') * sin(φ' - δ) * (1 + sqrt(sin(φ' + δ) * sin(φ' - β) / (sin(φ' - δ) * sin(β + α))))²]

where:

• ψ is the angle of the backfill slope with the horizontal.
• α is the angle of the wall face with the vertical.
• δ is the angle of friction between the soil and the wall.
• β is the angle of the backfill slope with the horizontal.

For cohesive soils (c'), the resultant active force (P_a) is given by:

P_a = 0.5 * γ * H² * K_a - 2 * c' * sqrt(K_a)

The term 2 * c' * sqrt(K_a) represents the reduction in force due to cohesion. The point of application is still typically at H/3 from the base, but this can vary depending on the cohesion term.

Earth Pressure Diagrams

The earth pressure at any depth 'z' is given by p = K * γ * z (for cohesionless soils). The pressure distribution is linear, starting from zero at the surface and reaching a maximum at the base of the wall. The total force is the area of this triangular diagram.

For cohesive soils, the diagram is modified. In active pressure, there's a zone of tension, leading to a diagram that is zero at the surface, negative (tension) up to a certain depth z₀, and then positive (compression) increasing linearly to the base.

Bearing Capacity of Soil

Bearing capacity refers to the maximum pressure that the soil can withstand without shear failure or excessive settlement. It's a critical factor in designing foundations, ensuring that the load from a structure is safely transferred to the ground.

Factors Affecting Bearing Capacity

Several factors influence the bearing capacity of soil:

• Type of soil: Cohesive soils (clays) and cohesionless soils (sands, gravels) behave differently.
• Shear strength parameters: Cohesion (c) and angle of internal friction (φ).
• Water table depth: A high water table reduces the effective stress and thus the bearing capacity.
• Foundation type and geometry: Shape (strip, square, circular), depth (Df), and width (B) of the foundation.
• Loading conditions: Eccentric or inclined loads reduce the effective area and increase pressure.
• Soil layering: The presence of different soil layers can significantly alter the bearing capacity.
• Compressibility: While shear failure is primary, settlement is also a concern, especially in compressible soils.

Terzaghi's Bearing Capacity Theory

Karl Terzaghi developed the first comprehensive theory for shallow foundations. It's based on the assumption of a general shear failure in a soil mass and considers the foundation as a strip footing (width B) extending infinitely.

The ultimate bearing capacity (q_u) for a shallow foundation is given by:

q_u = c'N_c + qN_q + 0.5γBN_γ

where:

• c' = effective cohesion of the soil
• q = surcharge pressure at the foundation level = γD_f (D_f is the depth of foundation)
• γ = unit weight of the soil below the foundation
• B = width of the foundation
• N_c, N_q, N_γ = bearing capacity factors, which are functions of the effective angle of internal friction (φ')

Terzaghi's factors are:

N_q = e^{(π tan φ')} * tan²(45° + φ'/2)

N_c = (N_q - 1) * cot(φ')

N_γ = 1.5 * (N_q - 1) * tan(φ')

Modification for Different Foundation Shapes and Depths (Meyerhof's Equation)

Meyerhof extended Terzaghi's theory to include factors for foundation shape (s), depth (d), and load inclination (i). The general bearing capacity equation becomes:

q_u = c'N_cs_cd_ci_c + qN_qs_qd_qi_q + 0.5γBN_γs_γd_γi_γ

The factors s_c, d_c, etc., depend on the specific geometry and soil properties. For vertical loads and standard shapes, these factors adjust the contributions of cohesion, surcharge, and soil weight.

Effect of Water Table

The presence of a water table below the foundation level reduces the effective unit weight of the soil, thereby decreasing the bearing capacity. The reduction factor (R_w) is applied to the third term (0.5γBN_γ) and sometimes to the second term (qN_q) depending on the water table depth:

• If the water table is at or above the base of the foundation (D_f + B/2), R_w = 0.5 for the third term.
• If the water table is deep below the foundation (below D_f + B), R_w = 1.0 (no effect).
• For intermediate depths, linear interpolation is used.

Allowable Bearing Capacity

The ultimate bearing capacity (q_u) is the load that causes shear failure. However, foundations must be designed to limit settlement. The allowable bearing capacity (q_a) is determined by applying a factor of safety (FS) to the ultimate bearing capacity and considering settlement criteria:

q_a = q_u / FS

A typical factor of safety is 3. For cohesive soils, settlement often governs the design, and allowable bearing pressure might be determined empirically based on soil type and settlement limits.

Plate Load Test

The Plate Load Test (PLT) is a field test conducted to determine the ultimate bearing capacity and settlement characteristics of a soil deposit at a specific depth. It is particularly useful for assessing the load-bearing capacity of shallow foundations.

Procedure

The test involves loading a rigid plate (typically circular, with diameters ranging from 300 mm to 1200 mm) placed on the ground surface or at the foundation level. The load is applied incrementally, and the corresponding settlement is measured using dial gauges or settlement plates.

1. Excavation: Excavate a pit to the desired foundation depth. The size of the pit should be at least 5 times the diameter of the plate.
2. Plate Placement: Place the loading plate centrally on the prepared soil surface. Ensure uniform contact.
3. Loading: Apply load to the plate in increments, typically doubling the load at each step until the ultimate bearing capacity is reached or the settlement becomes excessive.
4. Settlement Measurement: Record the settlement of the plate at each load increment. Readings are taken until the rate of settlement becomes negligible.
5. Unloading: After the test, the load is released in decrements, and the rebound settlement is recorded.

Interpretation of Results

The results are plotted as a load-settlement curve (Load vs. Settlement). The ultimate bearing capacity can be determined from this curve:

• For granular soils: The point where the load-settlement curve becomes steep or shows a sudden drop.
• For cohesive soils: The load corresponding to a specific settlement (e.g., 25 mm) or the load at which the settlement is twice the diameter of the plate.

The ultimate bearing capacity of a prototype foundation (B_f) can be estimated from the plate load test results (B_p) using empirical relationships:

For granular soils:

q_u(f) / q_u(p) = B_f / B_p

For cohesive soils:

q_u(f) = q_u(p)

Note: These are simplified relationships. More complex methods account for settlement differences.

Advantages of Plate Load Test

• Provides direct measurement of load-settlement behavior.
• Useful for determining allowable bearing pressure and settlement.
• Can be performed at the actual foundation level.

Limitations of Plate Load Test

• Only tests the soil up to a depth of approximately 2 times the plate diameter.
• Results might not be representative of the entire soil mass if there is significant layering or variability.
• The relationship between plate bearing capacity and foundation bearing capacity is empirical and can be unreliable, especially for large foundations.
• Test is expensive and time-consuming.

Standard Penetration Test (SPT)

The Standard Penetration Test (SPT) is one of the most widely used in-situ tests for determining the engineering properties of soil, particularly in granular soils. It provides a measure of the soil's resistance to penetration, which is correlated with its density, strength, and bearing capacity.

Procedure

The test involves driving a standard split-spoon sampler into the soil at the bottom of a borehole using a standardized hammer.

1. Borehole Drilling: A borehole is drilled to the desired depth.
2. Sampler Assembly: A standard split-spoon sampler (length 600 mm, outer diameter 50.8 mm, inner diameter 34.9 mm) is attached to drill rods.
3. Hammer Drop: A standard hammer weighing 63.5 kg (140 lb) is dropped from a height of 760 mm (30 inches).
4. Penetration Recording: The number of blows required to drive the sampler through three successive 150 mm (6 inches) intervals is recorded.
5. SPT N-Value: The sum of the blows for the second and third 150 mm intervals is known as the Standard Penetration Resistance or N-value. The first 150 mm penetration is considered a seating drive.

Corrections to SPT N-Value

The raw N-value needs to be corrected for various factors to obtain a standard N₆₀ value, which accounts for variations in hammer energy, drill rod length, and borehole diameter.

N₆₀ = N * (E_m / E_std)

where E_m is the actual hammer energy and E_std is the standard hammer energy (typically 60% for a safety hammer).

Other important corrections include:

• Overburden Pressure Correction: N-values tend to decrease with depth due to increasing effective stress.
• Groundwater Correction: If the water table is above the sampler level, the N-value should be corrected.

Correlation of SPT N-Value

The SPT N-value is empirically correlated with various soil properties:

• Relative Density (D_r) for sands: Higher N-values indicate denser sands.
• Angle of Internal Friction (φ') for sands: Higher N-values correlate with higher φ'.
• Consistency (Cohesion) for clays: Higher N-values indicate stiffer clays.
• Bearing Capacity: N-values are used to estimate allowable bearing capacity, often through charts or empirical formulas (e.g., Meyerhof's method).
• Settlement: N-values can also be used to estimate settlement.

Advantages of SPT

• Relatively inexpensive and easy to perform.
• Provides a disturbed soil sample (split-spoon) that can be visually classified.
• Widely used and accepted globally, with extensive empirical correlations available.
• Effective in a wide range of soil types, especially granular soils.

Limitations of SPT

• The N-value is influenced by the drilling method, hammer type, and operator skill.
• The soil sample obtained is disturbed, limiting its use for detailed laboratory testing.
• Less reliable in very soft clays or very dense gravels/cobbles.
• The "standard" conditions are not always strictly followed in practice.