UNIT 2: GEOTECHNICAL ENGINEERING - EXAM-FOCUSED NOTES

1. FUNDAMENTAL SOIL PROPERTIES & RELATIONSHIPS

Three-Phase Diagram & Volume-Mass Relationships

A soil mass consists of solids, water, and air. The three-phase diagram visually represents volume and mass relationships.

Key Definitions:

• Unit Weights:

  • Moist unit weight, $$\displaystyle \gamma = \frac{W}{V} $$

  • Dry unit weight, $$\displaystyle \gamma_d = \frac{W_s}{V} $$

  • Saturated unit weight, $$\displaystyle \gamma_{sat} = \frac{W_{sat}}{V} $$

  • Submerged unit weight, $$\displaystyle \gamma' = \gamma_{sat} - \gamma_w $$

• Water Content: $$\displaystyle w = \frac{W_w}{W_s} \times 100\% $$

• Void Ratio: $$\displaystyle e = \frac{V_v}{V_s} $$

• Porosity: $$\displaystyle n = \frac{V_v}{V} \times 100\% $$, $$\displaystyle n = \frac{e}{1+e} \times 100\% $$

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

Fundamental Relationship:

$$e = \frac{w G_s}{S_r}$$

Derivation: From $$\displaystyle w = \frac{W_w}{W_s} $$, $$\displaystyle S_r = \frac{V_w}{V_v} = \frac{W_w/\gamma_w}{e V_s} = \frac{w W_s / \gamma_w}{e V_s} $$. But $$\displaystyle G_s = \frac{\rho_s}{\rho_w} = \frac{W_s/V_s}{\gamma_w} \Rightarrow W_s = G_s \gamma_w V_s $$. Substituting gives $$\displaystyle S_r = \frac{w G_s}{e} \Rightarrow e = \frac{w G_s}{S_r} $$.

Interrelationships Table:

Problem-Solving Approach:

1. Draw a three-phase diagram with knowns.

2. Use $$\displaystyle G_s = \frac{\rho_s}{\rho_w} $$ (or $$\displaystyle \gamma_w $$).

3. Apply $$\displaystyle e = \frac{w G_s}{S_r} $$ and $$\displaystyle n = \frac{e}{1+e} $$.

4. For mass of water to saturate: $$\displaystyle W_{w,add} = W_s \left( \frac{e_{final} - e_{initial}}{G_s} \right) $$, where $$\displaystyle e_{final} $$ corresponds to $$\displaystyle S_r = 100\% $$.

Exam Tip: Always check units: $w$ in decimal, $$\displaystyle \gamma_w = 9.81 $$ kN/m³ or 9.81×10⁻⁶ kN/mm³. For saturation, $$\displaystyle S_r $$ must be in decimal in formula.

2. SOIL CLASSIFICATION

Grain Size Analysis

• Sieve Analysis: For coarse-grained soils (>0.075 mm). Results plotted on semi-log graph.

• Hydrometer Analysis: For fine-grained soils (<0.075 mm). Based on Stokes' law.

• Grain Size Distribution Curve: $$\displaystyle D_{10}, D_{30}, D_{60} $$ are diameters at 10%, 30%, 60% finer by weight.

Gradation Parameters:

• Uniformity Coefficient: $$\displaystyle C_u = \frac{D_{60}}{D_{10}} $$

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

Criteria for Well-Graded Soils:

Plasticity Characteristics

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

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

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

• Liquidity Index (LI): $$\displaystyle I_L = \frac{w - PL}{I_p} $$ (indicates consistency)

Unified Soil Classification System (USCS)

Major Divisions:

• Coarse-Grained: >50% retained on No. 200 sieve. Classified by $$\displaystyle C_u $$, $$\displaystyle C_c $$, and grain shape.

  • Gravels (G): $$\displaystyle D_{10} > 0.075 $$ mm

  • Sands (S): $$\displaystyle 0.075 < D_{10} < 4.75 $$ mm

  • Well-graded (W), Poorly-graded (P)

• Fine-Grained: >50% passes No. 200 sieve. Classified by Plasticity Chart (LL vs $$\displaystyle I_p $$).

  • Inorganic: ML, CL, MH, CH (A-line: $$\displaystyle I_p = 0.73(LL-20) $$)

  • Organic: OL, OH (dark color, odor)

  • Silts (M): Low plasticity, $$\displaystyle I_p < 0.4(LL-20) $$ below A-line? Actually silts are ML/MH.

  • Clays (C): High plasticity, above A-line.

• Highly Organic: Pt (peat).

Dual Notation: e.g., SC-SM (silty clay with sand).

IS Classification (Brief): Similar to USCS but uses group symbols like GW, SP, CL, CH, MI, CI, etc.

Textural Classification (Triangular Diagram)

Used for fine-grained soils (<0.075 mm) based on % sand, silt, clay. Boundaries separate sand, loam, silt, clay.

Exam Tip: For classification, first check coarse vs fine fraction. If coarse, use $$\displaystyle C_u $$, $$\displaystyle C_c $$. If fine, plot LL vs $$\displaystyle I_p $$ on plasticity chart. Remember A-line equation.

3. PERMEABILITY & SEEPAGE

Darcy's Law

$$\displaystyle Q = k i A = k \frac{h}{L} A $$

• $k$: coefficient of permeability (cm/s, m/day)

• $i$: hydraulic gradient

• Valid for laminar flow (Reynolds number < 1).

Factors Affecting $k$:

• Pore size (largest $k$ for gravels)

• Viscosity (decreases with temperature)

• Soil structure (layered vs isotropic)

• Entrapped air (reduces $k$)

• Degree of saturation (unsaturated $k$ << saturated)

Laboratory Determination

Constant Head Test (Coarse-grained):

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

• Procedure: Maintain constant head $h$, measure $Q$ in time $t$.

Falling Head Test (Fine-grained):

• $$\displaystyle k = \frac{aL}{A t} \ln \frac{h_1}{h_2} $$

• $a$ = area of standpipe, $A$ = cross-section of specimen.

• Procedure: Record head drop from $$\displaystyle h_1 $$ to $$\displaystyle h_2 $$ in time $t$.

Permeability of Stratified Deposits

Parallel to Flow (Horizontal):

$$k_{avg} = \frac{\sum k_i H_i}{\sum H_i}$$

Derivation: Total discharge $$\displaystyle Q = \sum q_i = \sum k_i i H_i = i \sum k_i H_i $$. But $$\displaystyle Q = k_{avg} i \sum H_i $$, so $$\displaystyle k_{avg} = \frac{\sum k_i H_i}{\sum H_i} $$.

Normal to Flow (Vertical):

$$k_{avg} = \frac{\sum H_i}{\sum \frac{H_i}{k_i}}$$

Derivation: Total head loss $$\displaystyle h = \sum h_i = \sum \frac{q H_i}{k_i} = q \sum \frac{H_i}{k_i} $$. But $$\displaystyle h = \frac{q}{k_{avg}} \sum H_i $$, so $$\displaystyle k_{avg} = \frac{\sum H_i}{\sum \frac{H_i}{k_i}} $$.

Seepage Pressure & Quick Sand Condition

• Seepage Pressure: $$\displaystyle p_s = \gamma_w i $$ (force per unit volume)

• Effective Stress with Seepage: $$\displaystyle \sigma' = \sigma - u + p_s $$ (downward flow increases $\sigma'$, upward decreases).

• Critical Hydraulic Gradient:

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

Derivation: At boiling, $$\displaystyle \sigma'_{critical} = 0 $$. $$\displaystyle \sigma' = \gamma_{sub} z - \gamma_w i z = 0 \Rightarrow i = \gamma_{sub}/\gamma_w = (G_s-1)/(1+e) $$.

• Factor of Safety against Boiling: $$\displaystyle FS = \frac{i_c}{i_{actual}} $$

Exam Tip: Quick sand is a phenomenon, not a soil type. Occurs in saturated loose fine sands/silts under upward seepage.

Flow Nets

• Flow lines & equipotentials are orthogonal.

• Each flow channel carries equal discharge.

• Each drop between equipotentials is equal head loss.

• Curvilinear squares (aspect ratio ~1).

Discharge Calculation (per unit width):

$$q = k H \frac{n_f}{n_d}$$

• $$\displaystyle n_f $$: number of flow channels

• $$\displaystyle n_d $$: number of equipotential drops

• $H$: total head loss.

Effective Stress under Seepage

• Total Stress, $\sigma$: Weight of all material above.

• Neutral Stress, $u$: Pore water pressure.

• Effective Stress, $\sigma'$: $$\displaystyle \sigma' = \sigma - u $$.

• With Seepage: $$\displaystyle \sigma' = \sigma - u \pm \gamma_w i z $$ (plus for downward flow, minus for upward).

Common Pitfall: In upward seepage, effective stress decreases, can reach zero (quick condition). Always check direction of flow.

4. CONSOLIDATION

Concepts

• Primary Consolidation: Expulsion of water from pores under sustained load (time-dependent).

• Secondary Consolidation: Rearrangement of soil skeleton after primary consolidation (plastic deformation).

• Preconsolidation Pressure ($$\displaystyle \sigma'_p $$): Maximum past effective stress.

• Overconsolidation Ratio (OCR): $$\displaystyle OCR = \frac{\sigma'_p}{\sigma'_0} $$ (normally consolidated if OCR=1).

One-Dimensional Consolidation (Terzaghi)

Assumptions:

• Soil is homogeneous, fully saturated.

• Water & solids are incompressible.

• Flow is one-dimensional (vertical).

• Darcy's law valid.

• Small strains, constant $k$ and $$\displaystyle m_v $$.

• Load is instantaneous and uniform.

Differential Equation:

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

where $$\displaystyle c_v = \frac{k}{m_v \gamma_w} $$ (coefficient of consolidation).

Ultimate Settlement:

$$S = \frac{C_c}{1 + e_0} H \log \frac{\sigma'_0 + \Delta \sigma'}{\sigma'_0}$$

• $$\displaystyle C_c $$: compression index (from $e$-log$\sigma'$ curve, normally consolidated range).

• $$\displaystyle C_r $$: recompression index (reloading/unloading).

• $H$: thickness of compressible layer.

• $$\displaystyle \sigma'_0 $$: initial effective stress.

• $\Delta \sigma'$: increase in effective stress.

Compression Characteristics:

• $$\displaystyle C_c = \frac{\Delta e}{\log \sigma'_f - \log \sigma'_i} $$ (slope of virgin curve)

• $$\displaystyle m_v = \frac{\Delta e}{1 + e_0} \frac{1}{\Delta \sigma'} $$ (coefficient of compressibility)

Coefficient of Consolidation ($$\displaystyle c_v $$)

Determined from laboratory oedometer test using:

1. Square-root of Time Method: $$\displaystyle T_v = \frac{c_v t}{H_{dr}^2} $$, plot $\sqrt{t}$ vs. $U$ or settlement.

   • $$\displaystyle T_v = 0.197 $$ for $$\displaystyle U=50\% $$ (double drainage), $$\displaystyle H_{dr} = H/2 $$.

2. Log-time Method: $$\displaystyle T_v = \frac{c_v t}{H_{dr}^2} $$, plot $\log t$ vs. $U$ or settlement.

   • $$\displaystyle T_v = 0.197 $$ for $$\displaystyle U=50\% $$.

Time Factor ($$\displaystyle T_v $$) for Various U:

Time for Consolidation:

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

• $$\displaystyle H_{dr} $$: longest drainage path (single drainage: $$\displaystyle H_{dr}=H $$, double drainage: $$\displaystyle H_{dr}=H/2 $$).

Exam Tip: Always identify drainage conditions first. Use appropriate $$\displaystyle H_{dr} $$. For $$\displaystyle U=50\% $$, $$\displaystyle T_v=0.197 $$ is standard.

5. STRESS DISTRIBUTION IN SOILS

Theories

Boussinesq's Theory (1885):

• Assumptions: Homogeneous, isotropic, semi-infinite elastic medium; point load applied at surface; no shear strength.

• Vertical Stress at Depth $z$:

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

• $r$: radial distance from axis.

• Directly below load ($$\displaystyle r=0 $$): $$\displaystyle \sigma_z = \frac{3P}{2\pi z^2} $$.

Westergaard's Theory (1938):

• Assumptions: Soil with vertical cracks (like clay), incompressible, no lateral strain.

• Vertical Stress:

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

• More appropriate for clays (layered, anisotropic).

Comparison:

Stress Beneath Loaded Areas

Point Load: Use Boussinesq equation directly. Uniformly Loaded Area:

• Equivalent Point Load Method: Replace area with point load at center, use Boussinesq for $$\displaystyle \sigma_z $$ at depth $z$.

• 2:1 Distribution Method: $$\displaystyle \sigma_z = \frac{q \cdot B \cdot L}{(B+z)(L+z)} $$ (assumes 2:1 spread).

• Influence Charts: (e.g., Newmark's) for irregular shapes.

Total, Neutral, and Effective Stress

• Total Stress ($\sigma$): Sum of all weights above.

• Neutral Stress ($u$): Pore water pressure.

• Effective Stress ($\sigma'$): $$\displaystyle \sigma' = \sigma - u $$.

• With Water Table:

  • Below WT: $$\displaystyle u = \gamma_w \cdot \text{depth below WT} $$ (if WT at surface).

  • Above WT (unsaturated): $u$ may be negative (suction) or zero; if $$\displaystyle S_r < 100\% $$, $u$ is less than $$\displaystyle \gamma_w h $$.

Exam Tip: For stress distribution problems, sketch soil profile, mark loads, and use superposition for multiple loads (as in Jun 2025 Q1).

6. SHEAR STRENGTH

Mohr-Coulomb Failure Criterion

$$\tau = c' + \sigma' \tan \phi' \quad \text{(effective stress)}$$

$$\tau = c_u \quad \text{(undrained, $$\displaystyle \phi_u=0 $$ for saturated clays)}$$

• $c'$: effective cohesion

• $\phi'$: effective friction angle

• $$\displaystyle c_u $$: undrained cohesion

Triaxial Shear Tests

Significance:

• UU: Quick test, no consolidation, measures undrained strength.

• CU: Consolidation allowed, undrained failure; with pore pressure measurement gives effective parameters.

• CD: Most realistic for long-term stability, but time-consuming.

Determination of Strength Parameters

Given confining pressures $$\displaystyle \sigma_3 $$ and failure deviator stresses $$\displaystyle (\sigma_1 - \sigma_3) $$:

1. Compute $$\displaystyle \sigma_1 = \sigma_3 + (\sigma_1 - \sigma_3) $$.

2. Plot Mohr's circles (center at $$\displaystyle \frac{\sigma_1+\sigma_3}{2} $$, radius $$\displaystyle \frac{\sigma_1-\sigma_3}{2} $$).

3. Draw failure envelope tangent to circles.

   • For UU: envelope horizontal at $$\displaystyle \tau = c_u $$.

   • For CU/CD: envelope slope = $\tan \phi'$, intercept = $c'$.

Example (Nov 2023):

• $$\displaystyle \sigma_3 = 150 $$ kN/m², failure load (deviator) = 500 kN/m² → $$\displaystyle \sigma_1 = 650 $$ kN/m².

• $$\displaystyle \sigma_3 = 300 $$ kN/m², failure load = 800 kN/m² → $$\displaystyle \sigma_1 = 1100 $$ kN/m².

• Plot circles, find envelope: $$\displaystyle c'_{app} $$ and $$\displaystyle \phi'_{app} $$ (since CU without pore pressure gives apparent values).

Liquefaction

• Definition: Loss of shear strength in saturated, loose, fine-grained soils (sands/silts) due to cyclic or rapid loading, causing pore pressure buildup and effective stress reduction to zero.

• Occurs When: $$\displaystyle S_r \approx 100\% $$, loose structure ($e$ high), cyclic loading (earthquakes), or rapid loading.

• Resistance: Increases with density, confining pressure, fines content.

Exam Tip: For triaxial tests, remember drainage conditions. UU: no drainage at all stages. CU: consolidation with drainage, then undrained failure. CD: drainage throughout.

7. COMPACTION

Definition vs Consolidation

• Compaction: Mechanical densification by expulsion of air (dry/wet). Immediate, volume change due to air compression/expulsion.

• Consolidation: Volume change due to expulsion of water under sustained load. Time-dependent.

Compaction Tests

Compaction Curve

Plot dry density $$\displaystyle \rho_d $$ vs water content $w$.

• Optimum Moisture Content (OMC): $w$ at $$\displaystyle \rho_{dmax} $$.

• Zero Air Void Line: Theoretical $$\displaystyle \rho_d $$ at $$\displaystyle S_r=100\% $$: $$\displaystyle \rho_d = \frac{G_s \gamma_w}{1 + w \cdot G_s / 100} $$.

• Relative Compaction: $\%$ of $$\displaystyle \rho_{dmax} $$ achieved in field.

Field Compaction Methods

• Sheepsfoot Roller: For clays (kneading action).

• Smooth-wheel Roller: For sands/gravels.

• Pneumatic-tire Roller: For uniform compaction.

• Vibratory Roller: For granular soils.

Exam Tip: Modified Proctor gives higher $$\displaystyle \rho_{dmax} $$ and lower $$\displaystyle w_{opt} $$ due to higher energy.

8. SOIL STABILIZATION & GEOSYNTHETICS

Soil Stabilization Methods

1. Mechanical: Compaction, blending with better soils.

2. Chemical:

   • Cement: Works for silts/clays/sands. Hydration products bind particles. Increases strength, reduces swell.

   • Lime: For clayey soils. Reduces plasticity, increases strength via pozzolanic reactions.

   • Bitumen: For waterproofing and binding (roads).

   • Fly Ash: Pozzolanic, improves strength and reduces swelling.

Geosynthetics

Types:

• Geotextiles: Woven (high strength) or non-woven (filtration/drainage).

• Geomembranes: Impermeable liners (HDPE, PVC).

• Geogrids: High tensile strength for reinforcement.

• Geocells: 3D honeycomb for confinement.

Functions:

1. Separation: Prevent mixing of layers (e.g., soft soil and aggregate).

2. Reinforcement: Tensile strength to resist loads (slopes, walls).

3. Filtration: Allow water flow but retain fines.

4. Drainage: Transmit water (geocomposites).

5. Protection: Prevent puncture (geomembranes).

Applications:

• Slopes: Reinforcement with geogrids.

• Retaining Walls: Reinforcement in MSE walls.

• Pavements: Separation/filtration layers.

• Landfills: Liners (geomembranes) and leachate collection (geocomposites).

9. MISCELLANEOUS DEFINITIONS & COMPARISONS

Quick Sand Condition vs Boiling:

• Quick Sand: Upward seepage reduces effective stress to zero, soil loses shear strength, behaves like fluid.

• Boiling: Appearance of water at ground surface due to upward seepage exceeding $$\displaystyle i_c $$. Quick sand is the mechanism, boiling is the symptom.

Seepage Pressure: $$\displaystyle p_s = \gamma_w i $$ (force per unit volume). Acts in direction of flow.

Stresses:

• Total Stress ($\sigma$): Weight of everything above.

• Neutral Stress ($u$): Pore water pressure.

• Effective Stress ($\sigma'$): $$\displaystyle \sigma' = \sigma - u $$ (governs strength & deformation).

Coefficient of Consolidation ($$\displaystyle c_v $$) vs Coefficient of Compressibility ($$\displaystyle m_v $$):

• $$\displaystyle c_v = \frac{k}{m_v \gamma_w} $$ (rate of consolidation, m²/s).

• $$\displaystyle m_v = \frac{\Delta e}{(1+e_0)\Delta \sigma'} $$ (volume compressibility, m²/kN).

Coefficient of Compression ($$\displaystyle C_c $$) vs Compression Index: Same parameter; $$\displaystyle C_c $$ is slope of $e$-log$\sigma'$ curve for normally consolidated range.

Shear Strength vs Shear Stress:

• Shear Stress: Applied stress causing deformation.

• Shear Strength: Maximum shear stress soil can resist (governed by $c'$, $\phi'$ or $$\displaystyle c_u $$).

Assumptions in Terzaghi’s Consolidation Theory:

1. Homogeneous, fully saturated soil.

2. Water & solids incompressible.

3. One-dimensional flow (vertical).

4. Darcy's law valid.

5. Small strains, constant $k$ & $$\displaystyle m_v $$.

6. Instantaneous, uniform load.

Assumptions in Boussinesq’s Theory:

1. Homogeneous, isotropic, elastic, semi-infinite medium.

2. Point load at surface.

3. No shear strength.

4. Gravity stresses initially absent.

Unconfined Compression Test:

• Procedure: Apply axial load to cylindrical specimen (no confining pressure) until failure. Measure unconfined compressive strength (UCS) = $$\displaystyle q_u $$.

• Suitable for: Cohesive soils (clays) with $\phi \approx 0$. $$\displaystyle c_u = q_u/2 $$ for undrained condition.

• Limitation: Not for sands or coarse soils.

Exam Tip: Know key formulas: $$\displaystyle i_c = (G_s-1)/(1+e) $$, $$\displaystyle S = \frac{C_c}{1+e_0} H \log \frac{\sigma'_0+\Delta \sigma'}{\sigma'_0} $$, $$\displaystyle q = k H n_f/n_d $$, $$\displaystyle c_v = \frac{k}{m_v \gamma_w} $$.

END OF UNIT 2 NOTES
Always cross-check with problem-solving approaches from past papers. Practice numericals on permeability, consolidation time, stress distribution, and classification.