Geotechnical Engg (CE-701) - Unit 5 Short Notes

UNIT 5: GEOTECHNICAL ENGINEERING

I. Fundamental Soil Properties and Relationships

Phase Diagram: Represents soil as a 3-phase system: solids, water, air.

Key Relationships & Definitions:

• Void Ratio (e): \( e = \frac{V_v}{V_s} \)

• Porosity (n): \( n = \frac{V_v}{V} = \frac{e}{1+e} \)

• Water Content (w): \( w = \frac{M_w}{M_s} \times 100\% \)

• Degree of Saturation (S_r): \( S_r = \frac{V_w}{V_v} \times 100\% \)

• Specific Gravity (G_s): \( G_s = \frac{\rho_s}{\rho_w} \)

Fundamental Relationship (for saturated soil, S_r=1):

$$ e = wG_s \quad \text{or} \quad S_r = \frac{wG_s}{e} $$

[!TIP] Derive this from \( G_s \cdot \gamma_w = \frac{\gamma_d}{1+e} \) and \( \gamma_d = \frac{\gamma}{1+w} \).

Unit Weights:

Problem-Solving Tip: Given \( \gamma, w, G_s \), find \( \gamma_d, e, n, S_r \). Use \( \gamma_d = \gamma/(1+w) \) first, then \( e = (G_s \gamma_w / \gamma_d) - 1 \), then \( n = e/(1+e) \), finally \( S_r = (wG_s)/e \).

II. Soil Classification and Grain Size Analysis

Grain Size Distribution (GSD) Curve: Plot of % finer (y-axis, log scale) vs. particle size (x-axis, log scale).

Coefficients:

• Uniformity Coefficient: \( C_u = \frac{D_{60}}{D_{10}} \)

• Coefficient of Curvature: \( C_c = \frac{(D_{30})^2}{D_{60} \cdot D_{10}} \)

Gradation Types:

Plasticity Characteristics (Fine-grained soils):

• Liquid Limit (LL): Water content at 25 blows in Casagrande cup.

• Plastic Limit (PL): Water content when soil crumbles.

• Plasticity Index (PI): \( I_p = LL - PL \)

IS Classification (IS 1498):

1. Coarse-grained (>50% > 0.075 mm): Divisions based on \( D_{10} \) (gravel/sand) and gradation (W/P).

2. Fine-grained (≥50% < 0.075 mm): Use Plasticity Chart (LL vs. I_p).

   • CL: Low plasticity clay (below A-line, LL<50)

   • CH: High plasticity clay (above A-line, LL≥50)

   • ML: Low plasticity silt (below A-line)

   • MH: High plasticity silt (above A-line)

   • OL/OI: Organic soils (below A-line, LL<50 with organics)

Textural Classification (Triangular Diagram): For coarse-grained soils, based on % gravel, % sand, % fines.

Consistency (based on PI for clays):

III. Permeability and Seepage

Permeability (k): Measure of ease of water flow through soil. Factors: Particle size, void ratio, fabric, viscosity of fluid, temperature.

Lab Determination:

1. Constant Head Test: Suitable for coarse-grained soils (high k).

$$ k = \frac{QL}{A h t} $$

Where: Q = discharge, L = specimen length, A = area, h = head loss, t = time.

2. Falling Head Test: Suitable for fine-grained soils (low k).

$$ k = \frac{2.3 L a}{A t} \log_{10} \frac{h_1}{h_2} $$

Where: a = area of standpipe, h₁/h₂ = heads.

Permeability of Stratified Soils:

• Horizontal Flow (parallel): \( k_H = \frac{\sum k_i H_i}{\sum H_i} \) (Weighted arithmetic mean)

• Vertical Flow (normal): \( k_V = \frac{\sum H_i}{\sum \frac{H_i}{k_i}} \) (Weighted harmonic mean)

[!TIP] \( k_V \) is always less than \( k_H \) for the same deposit.

Flow Nets:

• Definition: Graphical representation of 2D steady seepage (flow lines & equipotentials).

• Characteristics: Orthogonal, curved squares (same Δq, Δh), tangent boundaries.

• Application - Discharge: \( q = k H \frac{N_f}{N_d} \) (per unit length)

  Where: \( N_f \) = flow channels, \( N_d \) = equipotential drops.

• Seepage Pressure: \( p_s = i \gamma_w z \) (acts in direction of flow).

• Critical Hydraulic Gradient (Quick Sand):

  Derivation: Boiling occurs when \( \sigma' = 0 \). \( \sigma' = \gamma_{sub} z - i \gamma_w z = 0 \).

$$ i_{crit} = \frac{\gamma_{sub}}{\gamma_w} = \frac{G_s - 1}{1+e} $$

**Factor of Safety (FS) against boiling:** \( FS = \frac{i_{crit}}{i_{actual}} \)

IV. Effective Stress and Pore Water Pressure

Terzaghi’s Principle: \( \sigma' = \sigma - u \)

• Total Stress (σ): Weight of everything above.

• Pore Water Pressure (u): Pressure of water in voids.

  • Below WT (saturated): \( u = \gamma_w \cdot h \) (h = depth below WT)

  • Above WT (unsaturated): Usually zero unless artesian.

• Effective Stress (σ'): Stress carried by soil skeleton. Governs strength & deformation.

Stress Profile Plotting Steps:

1. Calculate total stress (σ) at each layer interface.

2. Determine pore pressure (u) based on water table & saturation.

3. Compute effective stress (σ') = σ - u.

[!TIP] For partially saturated layers (S_r < 100%), pore pressure is negative (suction), so σ' > σ.

V. Consolidation and Settlement

Primary Consolidation: Expulsion of water from saturated clay under load, volume decrease.

Settlement Calculation (Normally Consolidated Clay):

$$ \Delta H = \frac{C_c}{1+e_0} H \log_{10} \frac{\sigma'_f}{\sigma'_i} $$

Where:

• \( \Delta H \) = Consolidation settlement

• \( C_c \) = Compression index (from e-log σ' curve)

• \( e_0 \) = Initial void ratio

• \( H \) = Initial thickness of clay layer

• \( \sigma'_i \) = Initial effective stress

• \( \sigma'_f \) = Final effective stress after loading

Key Parameters:

• Recompression Index (C_r): For unloading/reloading (slope of recompression curve).

• Coefficient of Compressibility (m_v): \( m_v = \frac{\Delta e}{\Delta \sigma' (1+e_0)} \) (Compressibility per unit stress).

• Coefficient of Consolidation (c_v): \( c_v = \frac{k}{m_v \gamma_w} \). Measures rate of consolidation.

Determination of c_v (Time Factor Method):

1. Square Root of Time Method (√t): \( T_v = \frac{\pi}{4} U^2 \) for U ≤ 60%. Plot \( \sqrt{t} \) vs. settlement.

2. Logarithm of Time Method (log t): \( T_v = 0.197 \) for U = 90%. Plot \( \log t \) vs. settlement.

Time for Consolidation:

$$ t = \frac{T_v H^2}{c_v} $$

Crucial: \( H \) = Longest drainage path.

• Double drainage: \( H = H_{layer}/2 \)

• Single drainage: \( H = H_{layer} \)

Degree of Consolidation (U) & Time Factor (T_v):

• U = 50% → \( T_v \approx 0.197 \)

• U = 90% → \( T_v \approx 0.848 \)

• U = 100% → \( T_v \to \infty \)

Assumptions in 1D Consolidation Theory:

1. Homogeneous, fully saturated soil.

2. Small strains, constant k and m_v.

3. Darcy’s law valid.

4. Instantaneous, uniform load application.

5. One-dimensional flow & compression.

6. Water is incompressible; soil grains are rigid.

VI. Compaction

Definition: Mechanical densification of soil by reducing air voids (not water expulsion).

Compaction Curve: Dry density (\( \gamma_d \)) vs. Moisture content (w). Shows peak (OMC) and max dry density (MDD).

Zero Air Void Line: Theoretical curve for 100% saturation (\( \gamma_d = \frac{G_s \gamma_w}{1+w} \)). Compaction curve approaches but never crosses it.

Proctor Tests Comparison:

IS Standards: IS 2720 (Part 7) for Light, IS 2720 (Part 8) for Heavy.

Field Compaction Methods: Smooth-wheel, sheepsfoot, pneumatic-tired rollers; vibratory plates; hand tampers.

VII. Shear Strength

Mohr-Coulomb Failure Criterion:

$$ \tau = c' + \sigma' \tan \phi' $$

Where:

• \( \tau \) = Shear strength

• \( c' \) = Effective cohesion

• \( \sigma' \) = Effective normal stress on failure plane

• \( \phi' \) = Effective angle of internal friction

Triaxial Shear Tests:

Unconfined Compression Test (UCT): Special UU test with \( \sigma_3 = 0 \). For saturated clays: \( c_u = \frac{\sigma_f}{2} \), \( \phi_u = 0° \).

Shear Strength Envelope: Plot of failure shear stress (τ) vs. normal stress (σ) from Mohr circles at failure. Slope = tan φ', intercept = c'.

Liquefaction: Loss of shear strength in saturated, loose, fine sands/silts due to rapid loading (e.g., earthquake) causing pore pressure build-up (u → σ) and σ' → 0.

VIII. Stress Distribution in Soils

Boussinesq’s Theory (Point Load):

• Assumptions: Homogeneous, isotropic, elastic half-space; load is point load; gravity neglected.

• Vertical Stress Increase at depth z, radial distance r:

$$ \Delta \sigma_z = \frac{3P}{2\pi} \cdot \frac{z^3}{(r^2 + z^2)^{5/2}} $$

• Below center of loaded area (r=0): \( \Delta \sigma_z = \frac{3P}{2\pi z^2} \)

Westergaard’s Theory (for Layered Soils):

• Assumptions: Soil is infinitely stiff in horizontal direction (no lateral strain), like columns in a rigid layer.

• Key Difference: Stress distribution is confined to a vertical column (vertical "bulb"). Gives higher vertical stress at depth near load, lower at large r compared to Boussinesq.

• Formula (below center of strip load): \( \Delta \sigma_z = \frac{q}{\pi} \left[ \beta - \sin \beta \cos \beta \right] \), where \( \beta = \tan^{-1}(b/z) \).

Equivalent Point Load Method: For uniformly loaded areas (rectangular, circular, strip), replace area with equivalent point load at depth z. Use Boussinesq’s equation with \( P = q \times \text{area} \). More accurate for small \( z/B \) ratios.

IX. Soil Stabilization and Geosynthetics

Soil Stabilization Methods:

1. Mechanical: Compaction, blending with better soils.

2. Chemical: Lime, Cement, Bitumen, Fly ash, Chemicals (e.g., calcium chloride).

3. Geosynthetics: Reinforcement, separation.

Cement Stabilization: Mixes soil with cement & water. Effects: Increases strength, reduces compressibility & permeability, controls shrinkage/swell. Best for well-graded soils with some fines.

Geosynthetics: Synthetic polymers in sheet form.

Functions Explained:

• Separation: Prevents mixing of dissimilar soils (e.g., soft subgrade and aggregate base).

• Reinforcement: Tensile element to resist deformation (like steel in concrete).

• Filtration: Allows water flow while retaining soil particles (needs proper AOS).

• Drainage: Conveys water within plane (geonets, geocomposites).

• Protection: Prevents puncture of geomembranes (thick non-wovens).