2.6 Bearing Capacity and Foundation Settlements

2.6 Bearing Capacity and Foundation Settlements

Introduction to Foundation Engineering

• The primary functions of a foundation are to safely transfer structural loads to the underlying soil or rock and to limit settlements to acceptable levels.

• Bearing capacity defines the maximum pressure a soil can withstand without undergoing shear failure.

• Settlement defines the vertical deformation of soil under applied load, which must be controlled to prevent structural damage.

• This section systematically explores the theories, calculations, and practical considerations for ensuring foundation stability and serviceability.

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1. Bearing Capacity: Concepts and Influencing Factors

1.1 Definition and Types of Bearing Capacity

• Ultimate Bearing Capacity ($\boldsymbol{q_u}$): The maximum gross pressure at the base of the foundation at which the soil fails in shear.

• Net Ultimate Bearing Capacity ($\boldsymbol{q_{nu}}$): The net increase in pressure at the base of the foundation that causes shear failure.

  $\boldsymbol{q_{nu} = q_u - \gamma D_f}$

  where, $\boldsymbol{\gamma}$ is the unit weight of soil above the footing base and $\boldsymbol{D_f}$ is the depth of foundation.

• Gross Safe Bearing Capacity ($\boldsymbol{q_s}$): The maximum gross pressure the soil can carry without shear failure, incorporating a factor of safety.

  $\boldsymbol{q_s = \frac{q_u}{F}}$

  where, $\boldsymbol{F}$ is the factor of safety (typically 2.5 to 4).

• Net Safe Bearing Capacity ($\boldsymbol{q_{ns}}$): The net pressure that can be safely applied to the soil, considering shear failure only.

  $\boldsymbol{q_{ns} = \frac{q_{nu}}{F} = q_s - \gamma D_f}$

• Allowable Bearing Pressure ($\boldsymbol{q_a}$): The net loading intensity at which the soil can carry the load safely, considering both shear failure and settlement criteria. It is the design value used in practice.

1.2 Factors Affecting Bearing Capacity

• Soil Characteristics:

  • Shear strength parameters ($\boldsymbol{c}$, $\boldsymbol{\phi}$).

  • Unit weight ($\boldsymbol{\gamma}$).

  • Density and relative density (for sands).

  • Drainage conditions.

• Foundation Parameters:

  • Depth of embedment ($\boldsymbol{D_f}$): Increases bearing capacity.

  • Width ($\boldsymbol{B}$) and shape (square, rectangular, circular, strip): Wider footings generally have higher capacity in cohesive soils but a slightly reduced capacity in cohesionless soils due to the scale effect.

  • Inclination of load: Reduces capacity.

• Groundwater Table Position:

  • If the water table is at or above the base of the footing, the submerged unit weight ($\boldsymbol{\gamma_{sub}}$) is used for the soil below the base, reducing bearing capacity.

  • If the water table is between the base and a depth $\boldsymbol{B}$ below the base, a weighted average unit weight is used.

• Eccentricity of Load: Eccentric loads reduce the effective area ($\boldsymbol{B'}$, $\boldsymbol{L'}$) of the footing, significantly lowering capacity.

• Slope of Ground: Foundations near slopes have reduced capacity.

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2. Modes of Foundation Failure

• Foundations can fail in three primary modes, depending on soil type and foundation geometry.

2.1 General Shear Failure

• Characteristics:

  • Occurs in dense sands and stiff clays.

  • Well-defined, continuous failure surface develops from the edge of the footing to the ground surface.

  • Sudden, catastrophic failure accompanied by heaving of soil on sides.

  • Clear ultimate load is reached on the load-settlement curve, followed by a sharp drop.

• Load-Settlement Curve: Shows a distinct peak (ultimate load) and then a significant reduction in resistance.

2.2 Local Shear Failure

• Characteristics:

  • Occurs in medium-dense sands and clay of medium consistency.

  • Failure surface initiates below the footing but does not propagate fully to the ground surface.

  • Significant vertical settlement occurs, with some heaving on the sides.

  • Failure is gradual, and the ultimate load is not well-defined.

• Load-Settlement Curve: Curve is continuous and steep, without a pronounced peak.

2.3 Punching Shear Failure

• Characteristics:

  • Occurs in loose sands and very soft clays.

  • Failure surface is primarily vertical, directly beneath the footing.

  • The footing "punches" into the ground with large settlements.

  • Little to no heaving of soil on the sides.

  • No distinct ultimate load is observed.

• Load-Settlement Curve: Shows a continuously steepening curve, with settlement increasing rapidly without a clear failure point.

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3. Terzaghi’s Bearing Capacity Theory

• Terzaghi (1943) proposed the first comprehensive theory for calculating the ultimate bearing capacity of shallow foundations, assuming a general shear failure mode.

3.1 Assumptions

• The base of the footing is rough.

• The soil above the footing base is considered as a surcharge load ($\boldsymbol{q = \gamma D_f}$).

• The soil is homogeneous and isotropic.

• The failure surface consists of a wedge zone (Zone I), radial shear zones (Zone II), and Rankine passive zones (Zone III).

• The effect of soil above the footing base is replaced by an equivalent surcharge.

3.2 General Bearing Capacity Equation

• The ultimate bearing capacity for a strip footing (long continuous footing) is given by:

  $\boldsymbol{q_u = c N_c + q N_q + 0.5 \gamma B N_\gamma}$

  where,

  • $\boldsymbol{c}$ is the cohesion of the soil.

  • $\boldsymbol{q = \gamma D_f}$ is the effective surcharge at the foundation level.

  • $\boldsymbol{\gamma}$ is the effective unit weight of soil below the footing base.

  • $\boldsymbol{B}$ is the width of the footing.

  • $\boldsymbol{N_c, N_q, N_\gamma}$ are the bearing capacity factors, which are functions of the angle of internal friction ($\boldsymbol{\phi}$).

3.3 Bearing Capacity Factors

• Terzaghi derived the following factors:

  • $\boldsymbol{N_q = \frac{e^{2(3\pi/4 - \phi/2)\tan\phi}}{2 \cos^2(45^\circ + \phi/2)}}$

  • $\boldsymbol{N_c = (N_q - 1) \cot \phi}$ (for $\boldsymbol{\phi > 0}$); $\boldsymbol{N_c = 5.7}$ for $\boldsymbol{\phi = 0}$ (pure clay).

  • $\boldsymbol{N_\gamma = \frac{\tan \phi}{2} \left( \frac{K_{p\gamma}}{\cos^2\phi} - 1 \right)}$, where $\boldsymbol{K_{p\gamma}}$ is a passive earth pressure coefficient.

• These factors are typically obtained from published tables or charts.

3.4 Modifications for Footing Shape

• For square, circular, and rectangular footings, Terzaghi introduced shape factors. The general equation becomes:

  • Square Footing: $\boldsymbol{q_u = 1.3 c N_c + q N_q + 0.4 \gamma B N_\gamma}$

  • Circular Footing: $\boldsymbol{q_u = 1.3 c N_c + q N_q + 0.3 \gamma B N_\gamma}$

  • Rectangular Footing ($\boldsymbol{B \times L}$): Shape factors $\boldsymbol{s_c}$, $\boldsymbol{s_q}$, $\boldsymbol{s_\gamma}$ are used, leading to: $\boldsymbol{q_u = c N_c s_c + q N_q s_q + 0.5 \gamma B N_\gamma s_\gamma}$

3.5 Limitations

• Primarily valid for shallow foundations ($\boldsymbol{D_f / B \le 1}$).

• Assumes general shear failure; corrections are needed for local shear failure by using reduced strength parameters: $\boldsymbol{c^* = \frac{2}{3}c}$ and $\boldsymbol{\phi^* = \tan^{-1}(\frac{2}{3}\tan\phi)}$.

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4. Ultimate Bearing Capacity of Different Soils

4.1 Cohesionless Soils ($\boldsymbol{c=0}$)

• For sands and gravels, the bearing capacity is derived from friction and the unit weight of the soil.

• For a strip footing:

  $\boldsymbol{q_u = q N_q + 0.5 \gamma B N_\gamma}$

• Key Characteristics:

  • Bearing capacity increases with depth ($\boldsymbol{D_f}$) and width ($\boldsymbol{B}$).

  • Very sensitive to the position of the water table.

  • Relative density (and hence $\boldsymbol{\phi}$) is the most critical parameter.

4.2 Cohesive Soils ($\boldsymbol{\phi=0}$ Concept)

• For saturated clays under undrained conditions (short-term analysis), $\boldsymbol{\phi_u = 0}$.

• The bearing capacity equation simplifies significantly since $\boldsymbol{N_q = 1}$ and $\boldsymbol{N_\gamma = 0}$.

• For a strip footing:

  $\boldsymbol{q_u = c_u N_c + \gamma D_f}$

  where, $\boldsymbol{c_u}$ is the undrained cohesion and $\boldsymbol{N_c = 5.7}$.

• For a square or circular footing:

  $\boldsymbol{q_u = 1.3 c_u N_c + \gamma D_f = (1.3 \times 5.7)c_u + \gamma D_f \approx 7.4 c_u + \gamma D_f}$

• Key Characteristics:

  • Bearing capacity is independent of the footing width ($\boldsymbol{B}$).

  • Increases linearly with depth of embedment.

  • Governed by the undrained shear strength ($\boldsymbol{c_u}$).

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5. Consolidation: Concept and Mechanics

5.1 Basic Concept

• Consolidation is the time-dependent process of volume reduction in saturated cohesive soils due to the expulsion of pore water under an applied static load.

• It results from the dissipation of excess pore water pressure and the gradual transfer of load from the pore water to the soil skeleton (effective stress increase).

5.2 Types of Consolidation

• Primary Consolidation: The time-dependent volume change resulting from the expulsion of pore water and the corresponding increase in effective stress. This is the main phase analyzed by Terzaghi's theory.

• Secondary Compression (Creep): The continued volume change that occurs at constant effective stress after the excess pore pressure has fully dissipated. It is due to the plastic readjustment of the soil skeleton.

5.3 Terzaghi's One-Dimensional Consolidation Theory

• Assumptions: The soil is homogeneous and fully saturated, drainage and compression are one-dimensional, Darcy's law is valid, soil particles and water are incompressible, and the coefficient of consolidation ($\boldsymbol{c_v}$) is constant.

• Governing Differential Equation:

  $\boldsymbol{\frac{\partial u}{\partial t} = c_v \frac{\partial^2 u}{\partial z^2}}$

  where, $\boldsymbol{u}$ is the excess pore water pressure, $\boldsymbol{t}$ is the time, and $\boldsymbol{z}$ is the depth.

5.4 Consolidation Tests and Parameters

• The Oedometer Test is used to determine consolidation parameters.

• Key Parameters:

  • Compression Index ($\boldsymbol{C_c}$): Slope of the virgin compression curve on an e-log σ' plot.

  • Recompression Index ($\boldsymbol{C_r}$): Slope of the reloading/unloading curve.

  • Coefficient of Consolidation ($\boldsymbol{c_v}$): Determines the rate of settlement, calculated from time-settlement data using Casagrande's log-time or Taylor's root-time method.

  • Coefficient of Volume Compressibility ($\boldsymbol{m_v}$):

    $\boldsymbol{m_v = \frac{\Delta e / (1 + e_0)}{\Delta \sigma'}}$

  • Preconsolidation Pressure ($\boldsymbol{\sigma_p'}$): The maximum past effective stress experienced by the soil. It is determined using Casagrande's graphical construction.

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6. Settlement: Analysis and Calculation

6.1 Types of Settlement

• Immediate (Elastic) Settlement ($\boldsymbol{S_e}$): Occurs almost instantly upon load application due to the elastic distortion of the soil, without change in water content. Significant in sands and unsaturated clays.

• Primary Consolidation Settlement ($\boldsymbol{S_c}$): The major time-dependent settlement in saturated clays, resulting from the dissipation of excess pore pressure.

• Secondary Compression Settlement ($\boldsymbol{S_s}$): The long-term, slow settlement that continues after primary consolidation is complete.

6.2 Nature and Effects of Settlement

• Total Settlement: The sum of all settlement components ($\boldsymbol{S_{total} = S_e + S_c + S_s}$).

• Differential Settlement: The uneven settlement between different parts of a structure. This is more critical than total settlement as it causes tilting, cracking, and structural distress.

• Effects: Cracking in walls and floors, jamming of doors/windows, tilting of structures, and in extreme cases, structural collapse.

6.3 Settlement Calculations

6.3.1 Immediate Settlement ($\boldsymbol{S_e}$)

• Commonly calculated using elastic theory. For a flexible surface footing on saturated clay ($\boldsymbol{\phi_u=0}$):

  $\boldsymbol{S_e = \frac{q B (1 - \mu^2) I_f}{E}}$

  where,

  • $\boldsymbol{q}$ is the net applied pressure.

  • $\boldsymbol{B}$ is the width of footing.

  • $\boldsymbol{\mu}$ is the Poisson's ratio of soil.

  • $\boldsymbol{E}$ is the Young's modulus of soil.

  • $\boldsymbol{I_f}$ is the influence factor dependent on footing shape and rigidity.

6.3.2 Primary Consolidation Settlement ($\boldsymbol{S_c}$)

• Calculated from oedometer test data.

• For normally consolidated clays ($\boldsymbol{\sigma_0' \approx \sigma_p'}$):

  $\boldsymbol{S_c = \frac{H_0}{1 + e_0} C_c \log_{10} \left( \frac{\sigma_0' + \Delta \sigma}{\sigma_0'} \right)}$

• For overconsolidated clays:

  • If $\boldsymbol{\sigma_0' + \Delta \sigma \le \sigma_p'}$:

    $\boldsymbol{S_c = \frac{H_0}{1 + e_0} C_r \log_{10} \left( \frac{\sigma_0' + \Delta \sigma}{\sigma_0'} \right)}$

  • If $\boldsymbol{\sigma_0' + \Delta \sigma > \sigma_p'}$:

    $\boldsymbol{S_c = \frac{H_0}{1 + e_0} \left[ C_r \log_{10} \left( \frac{\sigma_p'}{\sigma_0'} \right) + C_c \log_{10} \left( \frac{\sigma_0' + \Delta \sigma}{\sigma_p'} \right) \right]}$

  where, $\boldsymbol{H_0}$ is the initial thickness of the compressible layer, $\boldsymbol{e_0}$ is the initial void ratio, and $\boldsymbol{\sigma_0'}$ is the initial effective overburden pressure.

6.3.3 Secondary Compression Settlement ($\boldsymbol{S_s}$)

• Calculated as:

  $\boldsymbol{S_s = \frac{H_p}{1 + e_p} C_\alpha \log_{10} \left( \frac{t}{t_p} \right)}$

  where,

  • $\boldsymbol{H_p}$ is the layer thickness at the end of primary consolidation.

  • $\boldsymbol{e_p}$ is the void ratio at the end of primary consolidation.

  • $\boldsymbol{C_\alpha}$ is the secondary compression index.

  • $\boldsymbol{t}$ is the time since loading.

  • $\boldsymbol{t_p}$ is the time to end of primary consolidation.

6.4 Rate of Consolidation Settlement

• The time factor ($\boldsymbol{T_v}$) relates time and the coefficient of consolidation.

  $\boldsymbol{T_v = \frac{c_v t}{H_{dr}^2}}$

  where, $\boldsymbol{H_{dr}}$ is the drainage path length (half the layer thickness for two-way drainage, full thickness for one-way drainage).

• Average Degree of Consolidation ($\boldsymbol{U}$): The ratio of settlement at time t to the ultimate primary settlement.

  $\boldsymbol{S_t = U \cdot S_c}$

  For $\boldsymbol{U \le 60\%}$, $\boldsymbol{T_v \approx \frac{\pi}{4} U^2}$. For higher U, the relationship is more complex.

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7. Practical Considerations and Design

7.1 Bearing Capacity from Field Tests

• Standard Penetration Test (SPT): Correlations are used to estimate $\boldsymbol{\phi}$ for sands and $\boldsymbol{c_u}$ for clays, which are then used in bearing capacity equations.

• Plate Load Test (PLT): Provides direct empirical measurement of ultimate bearing capacity and settlement characteristics for the specific soil at a particular depth.

7.2 Settlement Criteria

• The allowable bearing pressure ($\boldsymbol{q_a}$) is the lesser of the value based on bearing capacity (with a factor of safety) and the value based on an allowable settlement limit.

• Typical allowable total settlement limits for isolated footings range from 25 mm to 50 mm, with differential settlement limits being much stricter (e.g., $\boldsymbol{\Delta / L \le 1/300}$).

7.3 Mitigation of Settlement Problems

• Preloading/Surcharging: Applying a temporary load to induce settlement before construction.

• Vibro-compaction or Dynamic Compaction: For cohesionless soils.

• Use of Deep Foundations (Piles, Piers): To transfer loads to deeper, more competent strata.

• Soil Improvement Techniques: Lime/cement stabilization, stone columns, vertical drains (wick drains) to accelerate consolidation.

This systematic understanding of bearing capacity and settlement is fundamental to designing safe, stable, and serviceable foundations. Accurate prediction requires careful soil investigation, appropriate selection of strength and compressibility parameters, and the application of sound theoretical principles adjusted for real-world conditions.

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