2.2 Stresses on Soil and Seepage

2.2 Stresses on Soil and Seepage

Introduction to Stresses and Seepage

• Soil, as a particulate and porous medium, responds to forces and water flow in complex ways that are fundamental to geotechnical engineering.

• Understanding how stresses are distributed within a soil mass and how water moves through its pores is critical for assessing stability, deformation, and failure.

• This unit focuses on the principle of effective stress—the cornerstone of soil mechanics—which governs soil strength and compressibility.

• We then explore seepage phenomena, including the analysis of water flow using flow nets and dangerous conditions like quicksand.

• Finally, we examine soil compressibility and the process of compaction, linking the theoretical concepts to practical applications in foundation design, slope stability, and earthworks.

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1. Effective Stress

1.1 Concept and Principle

• Definition: The stress carried by the solid skeleton of the soil particles. It is the key parameter controlling soil behavior, including shear strength, volume change, and compressibility.

• Effective Stress Principle (Terzaghi, 1925):

  • The total stress ($\boldsymbol{\sigma}$) at any point in a saturated soil mass is the sum of the effective stress ($\boldsymbol{\sigma'}$) carried by the soil grains and the neutral stress or pore water pressure ($\boldsymbol{u}$).

    $\boldsymbol{\sigma = \sigma' + u}$

  • All measurable effects of a change in stress, such as compression, distortion, and a change in shear strength, are due exclusively to changes in the effective stress.

• Significance: Effective stress is the fundamental principle distinguishing soil mechanics from continuum mechanics of solids, as it accounts for the biphasic (solid + fluid) nature of soil.

1.2 Factors Affecting Effective Stress

• Change in Total Stress ($\boldsymbol{\Delta \sigma}$):

  • Due to surface loads (building, embankment), excavation (unloading), or changes in overburden depth.

  • Immediate effect is a change in pore water pressure if drainage is restricted (undrained condition).

• Change in Pore Water Pressure ($\boldsymbol{\Delta u}$):

  • Rise in Water Table: Decreases effective stress ($\boldsymbol{u}$ increases, $\boldsymbol{\sigma'}$ decreases). This can lead to reduced bearing capacity or slope instability.

  • Lowering of Water Table: Increases effective stress ($\boldsymbol{u}$ decreases, $\boldsymbol{\sigma'}$ increases), causing consolidation settlement.

  • Seepage Forces: Upward or downward flow alters pore pressure distribution.

  • Capillary Action: In unsaturated soils above the water table, negative pore water pressure (suction) develops, which increases the effective stress.

1.3 Capillary Rise in Soils

• Mechanism: In fine-grained soils (silts, clays), water rises above the groundwater table due to surface tension in the small interconnected pores. This creates a zone of capillary saturation.

• Capillary Height ($\boldsymbol{h_c}$):

  • Approximate height of rise is inversely proportional to the pore size (effective diameter $\boldsymbol{D_{10}}$).

    $\boldsymbol{h_c \approx \frac{C}{e D_{10}}}$

    where, $\boldsymbol{C}$ is an empirical constant (0.1 to 0.5 cm²), $\boldsymbol{e}$ is void ratio.

  • Fine sands/silts: $\boldsymbol{h_c}$ = 0.3 to 1 m. Clays: Can be >10 m.

• Effect on Effective Stress:

  • In the capillary zone, pore water pressure is negative (less than atmospheric, $\boldsymbol{u < 0}$).

  • This increases the effective stress: $\boldsymbol{\sigma' = \sigma - (-|u|) = \sigma + |u|}$.

  • The apparent cohesion due to capillary suction gives dry strength to silty soils.

• Significance: Important for frost heave (ice lenses form in capillary zone), swelling of clays upon wetting (loss of suction), and stability of shallow foundations.

1.4 Quick Sand Condition

• Definition: A condition where a cohesionless soil (like fine sand or silt) loses its strength and behaves like a viscous liquid. It is not a special type of sand, but a condition induced by seepage.

• Mechanism: Occurs when the upward seepage force becomes equal to the submerged weight of the soil particles.

• Critical Hydraulic Gradient ($\boldsymbol{i_{cr}}$):

  • The hydraulic gradient at which effective stress becomes zero.

    $\boldsymbol{i_{cr} = \frac{\gamma'}{\gamma_w}}$

    where, $\boldsymbol{\gamma'}$ = submerged unit weight of soil, $\boldsymbol{\gamma_w}$ = unit weight of water.

    $\boldsymbol{i_{cr} = \frac{G-1}{1+e}}$

    where, $\boldsymbol{G}$ = specific gravity, $\boldsymbol{e}$ = void ratio.

  • Typically, $\boldsymbol{i_{cr} \approx 1}$ (ranges from 0.9 to 1.1 for most sands).

• Effective Stress at Quicksand:

  $\boldsymbol{\sigma' = 0}$

• Practical Situations:

  • Upward flow from artesian aquifer.

  • In front of sheet piles during excavation.

  • Near the bottom of hydraulic structures (dams, weirs).

• Prevention:

  • Providing adequate seepage path length to reduce gradient.

  • Using filters or gravel blankets.

  • Installing relief wells.

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2. Seepage Analysis

2.1 Seepage Pressure and Force

• Seepage Pressure ($\boldsymbol{p_s}$):

  • The pressure exerted by percolating water on soil grains due to frictional drag.

  • It is a body force per unit volume.

    $\boldsymbol{p_s = i \cdot \gamma_w}$

    where, $\boldsymbol{i}$ = hydraulic gradient.

• Seepage Force ($\boldsymbol{J}$):

  • The total force exerted by seepage on the soil mass.

    $\boldsymbol{J = p_s \cdot V = i \cdot \gamma_w \cdot V}$

    where, $\boldsymbol{V}$ is the volume of soil.

• Direction: Acts in the direction of flow. Upward seepage force reduces effective stress (can cause heave or boiling); downward seepage force increases effective stress.

2.2 Flow Nets and Applications

• What is a Flow Net?

  • A graphical solution to Laplace's equation for two-dimensional, steady-state seepage through a porous medium.

  • Consists of two orthogonal families of curves:

    • Flow Lines: Paths followed by water particles.

    • Equipotential Lines: Lines along which the total head is constant.

• Properties of a Valid Flow Net:

  • Flow lines and equipential lines intersect at right angles.

  • The field formed between adjacent flow/equipotential lines is approximately a "square" (curvilinear square).

  • Each flow channel carries the same quantity of flow.

• Construction Rules:

  • Boundary Conditions:

    • Impervious boundary = Flow line.

    • Constant head boundary = Equipotential line.

    • Phreatic line (top flow line in an earth dam) = Flow line with atmospheric pressure.

  • The number of flow channels ($\boldsymbol{N_f}$) and equipotential drops ($\boldsymbol{N_d}$) are integers.

• Applications and Calculations from a Flow Net:

  • Total Discharge ($\boldsymbol{Q}$):

    $\boldsymbol{Q = k \cdot H \cdot \frac{N_f}{N_d}}$

    where, $\boldsymbol{k}$ = coefficient of permeability, $\boldsymbol{H}$ = total head loss.

  • Pore Water Pressure ($\boldsymbol{u}$) at any Point: $\boldsymbol{u = h_p \cdot \gamma_w}$ where $\boldsymbol{h_p}$ = pressure head at the point = total head - elevation head.

  • Exit Gradient ($\boldsymbol{i_{exit}}$):

    $\boldsymbol{i_{exit} = \frac{\Delta h}{\Delta l}}$

    where, $\boldsymbol{\Delta h}$ = head loss in last equipotential drop, $\boldsymbol{\Delta l}$ = length of the last field. This must be < $\boldsymbol{i_{cr}}$ to prevent piping.

  • Uplift Pressure under hydraulic structures.

• Significance: Flow nets provide a visual and quantitative tool for analyzing seepage quantity, pressure distribution, and potential failure zones (e.g., piping at downstream toe of a dam).

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3. Soil Compressibility

3.1 Concept

• Definition: The property of soil to decrease in volume when subjected to an increase in effective stress. It is a time-dependent process in saturated fine-grained soils due to the slow expulsion of pore water.

• Main Mechanism: Consolidation – the gradual compression of saturated clayey soils as pore water drains out under sustained load.

• Significance: Predicts the magnitude and rate of settlement of structures founded on compressible soils.

3.2 Various Indices of Compressibility

• Compression Index ($\boldsymbol{C_c}$):

  • The slope of the linear portion of the void ratio ($\boldsymbol{e}$) vs. log effective stress ($\boldsymbol{\sigma_v'}$) plot from a consolidation test.

    $\boldsymbol{C_c = \frac{-\Delta e}{\Delta \log \sigma_v'}}$

  • Significance: Used to calculate primary consolidation settlement for normally consolidated clays.

  • Empirical Relation: $\boldsymbol{C_c \approx 0.009(LL - 10)}$ for remolded clays, where LL is Liquid Limit (%).

• Swelling/Recompression Index ($\boldsymbol{C_s}$ or $\boldsymbol{C_r}$):

  • The slope of the unloading/reloading curve on the e-log σ' plot.

  • Much smaller than $\boldsymbol{C_c}$ (typically $\boldsymbol{\frac{1}{10} \text{ to } \frac{1}{5}}$ of $\boldsymbol{C_c}$).

  • Used for calculating settlement when stress increase is within the preconsolidation range.

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

  • The compression per unit thickness per unit increase in effective stress.

    $\boldsymbol{m_v = \frac{a_v}{1 + e_0}}$

    where, $\boldsymbol{a_v}$ = coefficient of compressibility = $\boldsymbol{-\Delta e / \Delta \sigma_v'}$.

  • Units: m²/kN or m²/MN.

  • Used in the Terzaghi consolidation theory to find the coefficient of consolidation ($\boldsymbol{c_v}$).

• Coefficient of Consolidation ($\boldsymbol{c_v}$):

  • A measure of the rate of consolidation.

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

    where, $\boldsymbol{k}$ = permeability, $\boldsymbol{T_v}$ = time factor, $\boldsymbol{H_{dr}}$ = drainage path length, $\boldsymbol{t}$ = time.

  • Determined from laboratory consolidation test using Casagrande's or Taylor's method.

• Preconsolidation Pressure ($\boldsymbol{\sigma_p'}$):

  • The maximum past effective stress the soil has ever been subjected to.

  • Determined from the e-log σ' plot using Casagrande's construction method.

  • If current effective stress ($\boldsymbol{\sigma_0'}$) < $\boldsymbol{\sigma_p'}$ → Overconsolidated (stiffer, less compressible).

  • If $\boldsymbol{\sigma_0' \approx \sigma_p'}$ → Normally Consolidated.

• Compression Ratio ($\boldsymbol{CR}$):

  $\boldsymbol{CR = \frac{C_c}{1 + e_0}}$

  An alternative parameter for settlement calculation.

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4. Compaction

4.1 Definition

• Compaction: The process of densifying a soil by mechanical means (rolling, tamping, vibration) by expelling air from the void spaces. It is an instantaneous process applied to unsaturated soils.

• Purpose:

  • Increase shear strength (by increasing density and interlocking).

  • Reduce compressibility.

  • Reduce permeability.

  • Decrease future settlement.

  • Increase stability of slopes and embankments.

4.2 Laboratory Compaction Tests

• Standard Proctor Test (IS 2720 Part VII):

  • Mould Volume: 1000 cm³.

  • Hammer: 2.6 kg, 310 mm drop.

  • Layers: 3 layers, 25 blows per layer.

  • Energy: $\boldsymbol{0.595 \text{ kJ/m}^3}$.

• Modified Proctor Test (IS 2720 Part VIII):

  • Higher compactive effort to simulate heavy compaction in the field.

  • Mould Volume: 1000 cm³.

  • Hammer: 4.9 kg, 450 mm drop.

  • Layers: 5 layers, 25 blows per layer.

  • Energy: $\boldsymbol{2.7 \text{ kJ/m}^3}$.

• Compaction Curve:

  • A plot of Dry Density ($\boldsymbol{\rho_d}$) vs. Water Content ($\boldsymbol{w}$).

  • Shows a peak, defining:

    • Maximum Dry Density (MDD or $\boldsymbol{\rho_{d_{max}}}$)

    • Optimum Moisture Content (OMC)

4.3 Factors Affecting Compaction

• Water Content:

  • Below OMC: Soils are stiff, difficult to compact (low density).

  • At OMC: Water acts as a lubricant, allowing particles to rearrange into denser packing.

  • Above OMC: Water occupies pore space, pushing particles apart, reducing dry density.

• Compactive Effort (Energy):

  • Increased compactive effort (more passes, heavier roller) increases MDD and decreases OMC.

  • The compaction curve shifts up and left.

• Soil Type:

  • Coarse-grained (Sands & Gravels): Well-defined peak, achieve high MDD at low OMC.

  • Cohesive Soils (Clays): Broad peak, lower MDD, higher OMC. Sensitive to water content.

  • Uniformly Graded Soils: Lower MDD than well-graded soils due to less particle interlocking.

• Method of Compaction (Type of Roller):

  • Sheepsfoot/Pad-foot: Best for cohesive soils.

  • Smooth-wheeled: For granular soils and finishing.

  • Pneumatic-tyred: Good for most soils.

  • Vibratory: Most effective for granular soils.

• Lift Thickness: Thinner lifts allow more uniform compaction throughout the layer.

• The Zero Air Voids Curve (ZAV):

  • Represents the theoretical maximum density for a given water content at full saturation ($\boldsymbol{S=1}$).

    $\boldsymbol{\rho_d = \frac{G \rho_w}{1 + w G}}$

  • The actual compaction curve always lies below the ZAV curve because 100% saturation is not achievable with standard compaction.

4.4 Field Control of Compaction

• Specification: Usually a percentage of laboratory MDD (e.g., 95% of Modified Proctor MDD for embankments).

• Field Measurement:

  • Sand Replacement Method (IS 2720 Part XXVIII): To determine field density by volume displacement.

  • Core Cutter Method: For cohesive soils.

  • Nuclear Density Gauge: Fast, non-destructive.

• Compaction in Saturated or Underwater Conditions: Requires special techniques like vibroflotation for sands or using stone columns.

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Conclusion

The interlinked concepts of effective stress, seepage, compressibility, and compaction form the analytical core for solving geotechnical problems. From predicting the settlement of a building on clay to ensuring the stability of an earth dam against seepage forces, these principles allow engineers to model and manipulate soil behavior to create safe and enduring foundations for infrastructure.

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