UNIT 3: SOIL BEHAVIOR, PERMEABILITY, CONSOLIDATION, CLASSIFICATION & STRENGTH

I. STRESS DISTRIBUTION IN SOIL MASS

A. Boussinesq's Theory

• Assumptions:

  • Soil is homogeneous, isotropic, elastic, and semi-infinite.

  • Soil obeys Hooke's law.

  • No shear strength (only normal stresses considered).

  • Point load applied on ground surface.

• Vertical Stress beneath a Point Load (P):

$$ \sigma_z = \frac{3P}{2\pi z^2} \frac{1}{\left[1 + \left(\frac{r}{z}\right)^2\right]^{5/2}} $$

Where, `z` = depth, `r` = radial distance from load axis.

• Vertical Stress beneath a Uniformly Loaded Area:

  • Obtained by integrating Boussinesq's equation over the loaded area.

  • Equivalent Point Load Method: For rectangular areas, stress at a point can be computed using Influence Factors (I_f) from tables or charts.

$$ \Delta \sigma_z = q \cdot I_f $$

Where `q` = uniform load intensity.

• Isobars: Lines connecting points of equal vertical stress increase. Plotted using Influence Charts (e.g., Newmark's).

B. Westergaard's Theory

• Assumptions:

  • Soil is anisotropic (non-erodible, layered).

  • Incompressible vertical, elastic horizontal.

  • No vertical strain (σ_x = 0).

• Key Difference from Boussinesq:

  • Westergaard's vertical stress is less than Boussinesq's at the same depth and radial distance, especially for r/z > 0.5.

  • More realistic for layered soils or soils with thin, stiff strata.

• Formula (for point load):

$$ \sigma_z = \frac{P}{\pi z^2} \frac{1}{(1 + 2\frac{r^2}{z^2})^{3/2}} $$

[!TIP] Exam Focus: Numerical problems often combine stresses from multiple point loads or use I_f for rectangular footings. Remember: Boussinesq for homogeneous, Westergaard for layered/stratified soils.

II. PERMEABILITY & SEEPAGE ANALYSIS

A. Fundamentals & Measurement

• Darcy's Law:

$$ q = k \cdot i \cdot A $$

*   `q` = discharge (volume/time), `k` = coefficient of permeability (length/time), `i` = hydraulic gradient (Δh/L), `A` = cross-sectional area.

*   **Limitations:** Valid for **laminar flow** in **saturated soils**, low gradients (i < 1). Not for coarse gravels or high gradients.

• Factors Affecting k:

  • Grain size (D₁₀), void ratio, soil structure, temperature, viscosity of fluid, degree of saturation.

• Constant Head Test:

  • Procedure: Maintain constant head h across sample length L. Measure discharge Q over time t.

  • Calculation:

$$ k = \frac{Q \cdot L}{A \cdot h \cdot t} $$

*   Suitable for **coarse-grained soils** (high `k`).

• Falling Head Test:

  • Procedure: Initial head h₁ falls to h₂ in time t. Standpipe area a, sample area A.

  • Calculation:

$$ k = \frac{2.303 \cdot a \cdot L}{A \cdot t} \log_{10} \frac{h_1}{h_2} $$

*   Suitable for **fine-grained soils** (low `k`).

B. Stratified Soils & Equivalent Permeability

• Horizontal Flow (Parallel to Stratification):

  • Total head loss (Δh) is same across all layers.

  • Total discharge (q) is sum of discharges through each layer.

  • Derivation:

$$ q = \sum_{i=1}^{n} k_i \cdot \frac{Δh}{L_i} \cdot b \cdot z_i = \frac{Δh}{L} \cdot b \cdot \sum_{i=1}^{n} k_i z_i $$

*   **Equivalent `k_h` (for total thickness H = Σz_i):**

$$ \boxed{k_h = \frac{\sum_{i=1}^{n} k_i z_i}{H}} \quad \text{(Arithmetic mean weighted by thickness)} $$

• Vertical Flow (Normal to Stratification):

  • Discharge (q) is same through all layers.

  • Total head loss (Δh) is sum of losses in each layer.

  • Derivation:

$$ q = k_i \cdot \frac{Δh_i}{z_i} \cdot b \cdot B \Rightarrow Δh_i = \frac{q \cdot z_i}{k_i \cdot b \cdot B} $$

$$ Δh = \sum Δh_i = \frac{q \cdot B}{b} \sum \frac{z_i}{k_i} $$

*   **Equivalent `k_v` (for total thickness H):**

$$ \boxed{k_v = \frac{H}{\sum_{i=1}^{n} \frac{z_i}{k_i}}} \quad \text{(Harmonic mean)} $$

C. Seepage Analysis & Flow Nets

• Flow Net: Graphical representation of flow lines (path of water particles) and equipotential lines (equal total head).

• Characteristics:

  1. Flow lines & equipotentials intersect at right angles.

  2. Equipotentials are continuous from upstream to downstream.

  3. Flow channels are bounded by flow lines.

  4. Curves are smooth.

• Applications:

  • Seepage Quantity:

$$ q = k \cdot H \cdot \frac{n_f}{n_d} \quad \text{(per unit length)} $$

    *   `n_f` = number of flow channels, `n_d` = number of equipotential drops.

*   **Uplift Pressure:** Pressure at any point = `γ_w × (number of equipotential drops above that point)`.

*   **Exit Gradient (i_exit):** Hydraulic gradient at exit point.

$$ i_{exit} = \frac{Δh \text{ (across last cell)}}{Δl \text{ (length of last flow channel)}} $$

    *   **Factor of Safety against Piping/Boiling:** 

$$ FS = \frac{i_c}{i_{exit}} > 3 \text{ to } 4 $$

D. Seepage Forces & Critical Conditions

• Seepage Pressure (j): Force per unit volume exerted by flowing water on soil skeleton.

$$ j = i \cdot \gamma_w \quad \text{(Direction: same as flow)} $$

• Quick Sand Condition (Boiling):

  • Definition: Loss of shear strength in cohesionless soil due to upward seepage, causing soil particles to float and behave like a liquid.

  • Critical Hydraulic Gradient (i_c): Gradient at which effective stress becomes zero.

  • Derivation:

    At boiling: σ' = σ - u = 0 → u = σ = γ_sat × z

    Also, u = γ_w × i_c × z

    Equating: γ_w × i_c × z = γ_sat × z

$$ \boxed{i_c = \frac{\gamma_{sat}}{\gamma_w} = \frac{(G_s + e)\gamma_w}{(1+e)\gamma_w} = \frac{G_s + e}{1 + e}} $$

    For **normally consolidated saturated soil**, `γ_sat ≈ (G_s + e)γ_w/(1+e)`. Often simplified to:

$$ \boxed{i_c = \frac{G_s - 1}{1 + e}} \quad \text{(if submerged unit weight used)} $$

*   **Factor of Safety:** 

$$ FS = \frac{i_c}{i} > 3 \text{ to } 4 $$

III. CONSOLIDATION & SETTLEMENT

A. Fundamental Concepts

• Compaction vs. Consolidation:

  | Aspect | Compaction | Consolidation | |------------------|------------------------------------|------------------------------------| | Process | Mechanical expulsion of air | Expulsion of water under load | | Soil Type | All soils (unsaturated) | Primarily saturated clays/silts | | Time | Immediate (during loading) | Time-dependent (primary: months/years) | | Volume Change| Due to air expulsion | Due to water expulsion |

• Primary vs. Secondary Consolidation:

  • Primary: Due to dissipation of excess pore water pressure (Terzaghi's theory). Governs rate of settlement.

  • Secondary (Creep): Due to plastic rearrangement of soil particles after primary consolidation. Occurs under constant effective stress.

• Drainage Conditions:

  • One-way drainage: Water escapes from one surface only (e.g., clay layer under impermeable layer). Drainage path d = H (total thickness).

  • Two-way drainage: Water escapes from both top and bottom. Drainage path d = H/2.

  • Significance: Drainage path d directly affects time for consolidation (t ∝ d²).

B. One-Dimensional Consolidation Theory (Terzaghi)

• Assumptions:

  1. Soil is homogeneous, isotropic, saturated.

  2. Water & soil particles are incompressible.

  3. Flow & compression are one-dimensional (vertical).

  4. Darcy's law valid.

  5. Small strains, constant k & m_v during consolidation.

  6. Load is applied instantaneously and remains constant.

• Differential Equation:

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

Where `u` = excess pore water pressure, `c_v` = coefficient of consolidation.

• Solution & Time Factor (T_v):

$$ T_v = \frac{c_v \cdot t}{d^2} $$

Where `d` = longest drainage path.

• Degree of Consolidation (U) vs. T_v:

  • For U ≤ 60%:

$$ T_v = \frac{\pi}{4} U^2 $$

*   For **U = 60%**: `T_v = 0.197` (exact)

*   For **U ≥ 60%**: 

$$ T_v = 0.933 \log_{10} \frac{1}{1-U} - 0.085 $$

• Time for Consolidation:

$$ \boxed{t = \frac{T_v \cdot d^2}{c_v}} $$

C. Coefficients & Parameters

• Coefficient of Consolidation (c_v):

$$ c_v = \frac{k}{m_v \gamma_w} \quad \text{(Units: m²/s or cm²/s)} $$

Where `k` = permeability, `m_v` = coefficient of volume compressibility.

• Methods to Determine c_v (from Oedometer Test):

  1. Square Root of Time Fitting Method: For U ≤ 60%. Plot √t vs. d (dial gauge reading). t_{50} corresponds to U=50%.

$$ T_v = \frac{\pi}{4} U^2 \Rightarrow \text{At } U=50\%, T_v = 0.049 $$

2.  **Logarithm of Time Fitting Method:** For `U ≥ 60%`. Plot `log t` vs. `d`. Find `t_{60}` from curve.

$$ \text{At } U=60\%, T_v = 0.197 \quad \Rightarrow \quad c_v = \frac{0.197 \cdot d^2}{t_{60}} $$

• Compressibility Parameters (from e vs. log σ' plot):

  • Compression Index (C_c): Slope of virgin compression curve (normally consolidated).

$$ C_c = \frac{\Delta e}{\log \sigma'_f - \log \sigma'_0} = -\frac{\Delta e}{\Delta \log \sigma'} $$

*   **Coefficient of Compressibility (`a_v`):** Slope of **recompression/expansion curve**.

$$ a_v = -\frac{\Delta e}{\Delta \sigma'} \quad \text{(Units: m²/kN)} $$

*   **Coefficient of Volume Compressibility (`m_v`):**

$$ m_v = \frac{a_v}{1 + e_0} = \frac{C_c}{1 + e_0} \cdot \frac{1}{\Delta \sigma'} \quad \text{(Units: m²/kN)} $$

D. Settlement Calculations

• Primary Consolidation Settlement (S_c):

  • Normally Consolidated Clay:

$$ \boxed{S_c = \frac{C_c}{1 + e_0} H \log \frac{\sigma'_f}{\sigma'_0}} $$

*   **Overconsolidated Clay:**

    *   If `σ'_f ≤ σ'_p` (preconsolidation pressure): Use **recompression index (`C_r`)**.

$$ S_c = \frac{C_r}{1 + e_0} H \log \frac{\sigma'_f}{\sigma'_0} $$

    *   If `σ'_f > σ'_p`: Use `C_c` for stress increment above `σ'_p`, and `C_r` for increment from `σ'_0` to `σ'_p`.

$$ S_c = \frac{C_r}{1 + e_0} H \log \frac{\sigma'_p}{\sigma'_0} + \frac{C_c}{1 + e_0} H \log \frac{\sigma'_f}{\sigma'_p} $$

*   `H` = initial thickness of clay layer.

• Immediate Settlement (S_i): For saturated clays, often estimated using elastic theory (S_i = \frac{q B (1-\mu^2)}{E_u} I_s), but not a primary focus in these papers.

E. Stress Concepts in Layered Systems

• Terzaghi's Effective Stress Principle:

$$ \boxed{\sigma' = \sigma - u} $$

*   `σ'` = effective stress (governs strength & deformation).

*   `σ` = total stress (from overburden & loads).

*   `u` = pore water pressure (neutral stress).

• Total, Neutral, Effective Stress Diagrams:

  • Water Table Below G.S. with Partial Saturation (Sr < 100%):

    • Above WT: σ = γ_d × depth + γ_sat × (depth below top of sat. zone). u = 0 (capillary pressure negative, often ignored in basic analysis). σ' ≈ σ.

    • Below WT: σ increases with γ_sat. u = γ_w × (depth below WT). σ' = σ - u.

  • Water Table at G.S.:

    • σ increases with γ_sat from surface.

    • u = γ_w × depth (from surface).

    • σ' = (γ_sat - γ_w) × depth = γ' × depth (submerged unit weight).

IV. SOIL CLASSIFICATION (Grain Size & Plasticity)

A. Grain Size Analysis

• Sieve Analysis: For coarse-grained soils (gravel & sand). Results plotted on semi-log graph (particle size vs. % finer).

• Hydrometer Analysis: For fine-grained soils (silt & clay). Based on Stokes' Law (terminal velocity of settling particles).

• Gradation Parameters:

  • Uniformity Coefficient:

$$ \boxed{C_u = \frac{D_{60}}{D_{10}}} $$

*   **Coefficient of Curvature:** 

$$ \boxed{C_c = \frac{(D_{30})^2}{D_{60} \times D_{10}}} $$

• Interpretation (for coarse-grained soils):

  • Well-graded (GW, SW): C_u ≥ 4 (gravel) or ≥ 6 (sand) AND C_c between 1 and 3.

  • Poorly-graded (GP, SP): Fails either C_u or C_c criteria.

  • Gap-graded: Missing intermediate sizes (not captured by C_u, C_c).

B. Consistency & Plasticity of Fine-Grained Soils

• Atterberg Limits:

  • Liquid Limit (LL): Water content at which soil changes from plastic to liquid state (Casagrande cup, 25 blows).

  • Plastic Limit (PL): Water content at which soil changes from semi-solid to plastic state (thread rolling).

  • Plasticity Index (PI or I_p):

$$ \boxed{I_p = LL - PL} $$

• Consistency Terms (based on natural water content w relative to LL & PL):

  • Liquid: w > LL

  • Plastic: PL < w < LL

  • Semi-solid: PL > w > Shrinkage Limit (SL)

  • Solid: w < SL

C. IS Soil Classification System (Detailed)

• Coarse-Grained (Gravels & Sands): Classified by grain size (gravel > 4.75mm, sand 4.75-0.075mm) and gradation.

  • Well-graded: W (e.g., GW, SW).

  • Poorly-graded: P (e.g., GP, SP).

  • Silty/Clayey: M (silty), C (clayey) if fines > 12% (by weight). Dual symbols (e.g., SC-SM) if fines 5-12%.

• Fine-Grained (Silts & Clays): Classified using Plasticity Chart.

  • A-line: I_p = 0.73 (LL - 20)

  • U-line: I_p = 0.9 (LL - 8) (upper bound of plasticity).

  • Classification:

    | Symbol | Description | Region on Chart | |------------|-------------------------------------|----------------------------------| | CL | Low plasticity clay | Below A-line, LL ≥ 35? (Check) | | CI | Intermediate plasticity clay | On A-line? (Rare) | | CH | High plasticity clay | Above A-line | | ML | Low plasticity silt (inorganic) | Below A-line, LL < 50 | | MI | Intermediate plasticity silt | On A-line? (Rare) | | MH | High plasticity silt (inorganic) | Above A-line, LL < 50 |

  • Organic Soils: Pt (Peat) – high organic content, dark color, fibrous.

• Procedure: 1) % Gravel/Sand/Silt/Clay (sieve + hydrometer). 2) Check coarse fraction gradation (C_u, C_c). 3) For fines > 12%, plot on plasticity chart.

D. Other Classification Systems

• Textural Classification (Triangular Diagram):

  • Based on relative proportions of sand, silt, and clay fractions (by weight).

  • Soil falls into one of 12 textural classes (e.g., sandy clay loam, silty clay).

  • Used primarily in agricultural soil science.

• AASHTO Classification:

  • Groups A-1 to A-8 based on grain size & plasticity.

  • Group Index (GI) quantifies plasticity & fines content:

$$ GI = 0.2a + 0.005ac + 0.01bd $$

    Where `a = (LL - 40)`, `b = (LL - 40)`, `c = (PI - 10)`, `d = (PI - 10)` (use positive values only).

V. SHEAR STRENGTH OF SOILS

A. Fundamentals

• How Soils Attain Shear Strength:

  • Cohesionless ( Sands): Frictional resistance from inter-particle contact & interlocking.

  • Cohesive (Clays): True cohesion (electro-chemical bonding) + friction.

• Mohr-Coulomb Failure Envelope:

$$ \boxed{\tau = c + \sigma' \tan \phi} $$

*   `τ` = shear strength, `σ'` = effective normal stress on failure plane.

*   `c` = **cohesion** (total stress param for UU tests), `c'` = **effective cohesion**.

*   `φ` = **angle of internal friction** (total stress param for UU tests), `φ'` = **effective friction angle**.

• Total Stress vs. Effective Stress Parameters:

  • Total Stress Parameters (c, φ): Used for undrained conditions (short-term, saturated clays). Pore water pressure not measured/controlled.

  • Effective Stress Parameters (c', φ'): Used for drained conditions (long-term) or when pore pressure is measured/controlled. Fundamental for stability analysis.

B. Laboratory Shear Strength Tests

• Direct Shear Test:

  • Procedure: Soil sample in a split box. Apply vertical load (σ), then shear horizontally.

  • Advantages: Simple, fast, good for friction angle of sands, residual strength.

  • Disadvantages: Plane of weakness (box interface), non-uniform stress distribution, cannot measure pore pressure.

  • Failure Plane: Predetermined (horizontal plane at box joint).

• Triaxial Compression Test (Three Standard Types):

  1. CD (Consolidated Drained):

     • Procedure: Sample consolidated under cell pressure (σ₃), then sheared slowly with drainage open (pore pressure dissipates).

     • Measures: c', φ' (effective parameters).

     • Use: Long-term stability (dams, slopes).

  2. CU (Consolidated Undrained):

     • Procedure: Sample consolidated under σ₃ (drainage open), then sheared quickly with drainage closed (undrained).

     • With Pore Pressure Measurement: Measures c', φ' (effective). Pore pressure parameter A determined.

     • Without Pore Pressure Measurement: Measures undrained shear strength c_u and φ_u (often ≈ 0 for saturated clays). Total stress parameters.

     • Use: Short-term stability (saturated clays).

  3. UU (Unconsolidated Undrained):

     • Procedure: No prior consolidation. Sample sheared quickly from initial state with drainage closed.

     • Measures: Undrained shear strength c_u (depends on water content). For saturated clays, φ_u = 0.

     • Use: Quick assessment of c_u for saturated clays.

• Unconfined Compression Test (UCS):

  • Special case of UU test with σ₃ = 0.

  • Cylindrical sample compressed axially until failure.

  • For saturated clays: φ_u = 0, so failure envelope is horizontal.

  • Undrained Shear Strength:

$$ \boxed{c_u = \frac{\text{UCS}}{2}} $$

*   **Suitable for:** **Saturated, cohesive soils** (clays, silts).

C. Strength Parameters from Test Data

• Plotting Mohr Circles (Triaxial Test):

  • For each test at confining pressure σ₃, plot Mohr circle with:

    • Center: (σ₁ + σ₃)/2

    • Radius: (σ₁ - σ₃)/2

  • Failure Envelope: Tangent to all Mohr circles at failure.

  • Effective Stress Analysis: Use effective principal stresses (σ₁' = σ₁ - u_f, σ₃' = σ₃ - u). Envelope gives c', φ'.

  • Total Stress Analysis (UU): Use total σ₁, σ₃. Horizontal envelope gives c_u, φ_u = 0.

• Numerical Problem: Given σ₃ and failure σ₁ (or axial stress at failure), find c, φ.

  • For CU/UU (total stress): Use c = (σ₁ - σ₃)/2 if φ_u = 0. Otherwise, solve from two circles.

  • For CD/CU (effective): Need pore pressure u_f to get σ₁', σ₃'.

VI. COMPACTION

A. Definition & Purpose

• Compaction: Mechanical process of densifying soil by expelling air from voids (for unsaturated soils). Increases dry density.

• Objectives:

  • Increase shear strength & bearing capacity.

  • Decrease compressibility & settlement.

  • Decrease permeability.

  • Control swelling/shrinkage (clays).

• Compaction vs. Consolidation: (See Table in III.A)

B. Standard Laboratory Tests

• Standard Proctor Test (IS Light Compaction):

  • Mold volume: 944 cm³.

  • Hammer: 2.5 kg, drop: 305 mm.

  • Layers: 3, Blows per layer: 25.

  • Energy: 600 kN-m/m³.

• Modified Proctor Test (IS Heavy Compaction):

  • Mold volume: 944 cm³.

  • Hammer: 4.9 kg, drop: 457 mm.

  • Layers: 5, Blows per layer: 25.

  • Energy: 2700 kN-m/m³.

• Comparison:

  | Feature | Standard Proctor | Modified Proctor | |-------------------|----------------------|----------------------| | Energy | 600 kN-m/m³ | 2700 kN-m/m³ | | OMC | Higher | Lower | | MDD | Lower | Higher | | Use | Low-energy fills, subgrade | Highways, dams, heavy fills |

C. Compaction Curve & Concepts

• Compaction Curve: Plot of Dry Density (ρ_d) vs. Water Content (w).

  • Optimum Moisture Content (OMC): Water content at maximum dry density (MDD).

  • MDD: Maximum dry density achievable with given compaction effort.

• Zero Air Void Line (100% Saturation):

  • Equation for dry density at full saturation:

$$ \boxed{\rho_d = \frac{G_s \rho_w}{1 + w}} \quad \text{or} \quad \boxed{w = \frac{G_s \gamma_w}{\gamma_d} - 1} $$

*   Plotted on compaction curve. Shows theoretical limit; actual compaction occurs **below** this line.

• Degree of Saturation on Compaction Curve:

$$ S_r = \frac{w G_s}{e} \quad \text{and} \quad e = \frac{G_s \gamma_w}{\gamma_d} - 1 $$

Can compute `S_r` at any point on curve.

D. Field Compaction

• Equipment:

  • Smooth-wheel rollers: Granular soils, final sealing.

  • Sheepsfoot rollers: Cohesive soils, deep compaction.

  • Pneumatic tyred rollers: Flexible kneading action, all soils.

  • Vibratory rollers: Granular soils, high density.

• Field Density Tests:

  • Core Cutter Method: For cohesive soils. Drive cylinder, weigh wet & dry.

  • Sand Replacement Method: For cohesive & granular soils. Excavate hole, replace with known volume of sand.

VII. SOIL IMPROVEMENT & GEOSYNTHETICS

A. Methods of Soil Stabilization

• Mechanical Stabilization: Blending soils (e.g., sand with clay) + compaction.

• Chemical Stabilization:

  • Cement Stabilization:

    • Mechanism: Cement reacts with soil silicates/aluminates to form cementing compounds (C-S-H, C-A-H).

    • Applications: Road subgrades, airfields, foundations in weak soils.

    • Mix Design Factors: Soil type (fines content, plasticity), cement content, water content, curing period.

  • Lime Stabilization: For high-plasticity clays. Reduces plasticity, increases strength.

  • Bitumen: For waterproofing & binding (road bases).

  • Fly Ash: Pozzolanic reaction, fills voids.

B. Geosynthetics

• Types:

  • Geotextiles: Woven/non-woven fabrics (separation, filtration).

  • Geogrids: Grid-like (high tensile strength, reinforcement).

  • Geomembranes: Impervious sheets (barrier).

  • Geocells: 3D cellular structure (confinement).

  • Geocomposites: Combinations (e.g., geonet + geotextile = drainage composite).

• Functions & Applications:

  | Function | Description | Example Application | |----------------|------------------------------------------|---------------------------------------------| | Separation | Prevent mixing of dissimilar soils | Road subgrade over soft clay | | Reinforcement | Tensile strength to resist loads | Reinforced retaining walls, slopes | | Filtration | Allow flow but retain soil particles | Behind retaining walls, drainage trenches | | Drainage | Collect & convey seepage | Landfill leachate collection, roof drainage| | Protection | Protect geomembranes from puncture | Landfill liners | | Barrier | Prevent fluid migration | Landfill liners, pond liners |

VIII. SPECIAL TOPICS & CONCEPTS

A. Liquefaction

• Phenomenon: Loss of shear strength & stiffness in saturated, loose, fine sandy soils due to cyclic loading (e.g., earthquakes).

• Mechanism: Cyclic loading increases pore water pressure (u), decreases effective stress (σ'), leading to flow-like behavior.

• Susceptibility Factors: Loose saturation, fine sand/silty sand, shallow water table, high seismic intensity.

B. Quick Condition (Boiling)

• Definition: Upward seepage force equals submerged unit weight, causing effective stress to become zero. Soil behaves like a liquid.

• Critical Hydraulic Gradient:

$$ i_c = \frac{G_s - 1}{1 + e} $$

• Factor of Safety:

$$ FS = \frac{i_c}{i} \quad (\text{Required } FS > 3) $$

C. Seepage Pressure

• Definition: The drag force exerted by flowing water on the soil skeleton.

• Expression:

$$ j = i \cdot \gamma_w \quad (\text{Units: kN/m³}) $$

• Direction: Same as flow direction.

• Effect on Effective Stress:

  • Downward flow: σ' = σ - (u - j) → increases effective stress.

  • Upward flow: σ' = σ - (u + j) → decreases effective stress (can cause boiling).

\boxed{\text{END OF UNIT 3 NOTES}}