Geotechnical Engg (CE-701) - Unit 1 Short Notes

UNIT 1: SOIL PROPERTIES, CLASSIFICATION, PERMEABILITY, SEEPAGE, AND EFFECTIVE STRESS

I. Introduction to Soils

Soil Formation Processes:

• Weathering: Physical (freeze-thaw, thermal expansion) and chemical (hydrolysis, oxidation) disintegration of parent rock.

• Transportation: By water, wind, ice, or gravity.

• Deposition: Sedimentation in environments like rivers, lakes, oceans, or deserts.

Soil Composition:

A three-phase system:

1. Solid particles (minerals/organic matter)

2. Water (liquid phase)

3. Air (gas phase)

Soil Structure Types:

• Single-grained: Coarse-grained soils (gravel/sand), particles touch at points.

• Honeycombed: Fine sand/silt, arch-like structure with large voids.

• Flocculent: Clay particles in a loose, open arrangement (attractive forces dominate).

• Dispersed: Clay particles oriented and packed closely (repulsive forces dominate).

Phase Diagram & Significance:

A visual tool showing volumes/masses of solids, water, air. Used to derive fundamental relationships between index properties (e, n, w, Sᵣ, γ).

II. Phase Relationships and Soil Properties

Key Parameters & Definitions

Fundamental Relationships

1. Basic Definition:

$$e = \frac{V_v}{V_s}, \quad n = \frac{V_v}{V} = \frac{e}{1+e}$$

\boxed{n = \frac{e}{1+e}}

2. Water Content - Void Ratio - Saturation:

$$w = \frac{M_w}{M_s} = \frac{\rho_w V_w}{\rho_s V_s} = \frac{e \cdot S_r \cdot \rho_w}{G_s \cdot \rho_w}$$

\boxed{e = \frac{G_s \cdot w}{S_r}} \quad \text{(For } S_r = 100\% \text{, } e = G_s w\text{)}

3. Moist Unit Weight:

$$\gamma = \frac{W_s + W_w}{V} = \frac{G_s \gamma_w + e \cdot S_r \cdot \gamma_w}{1+e}$$

\boxed{\gamma = \frac{G_s + S_r e}{1+e} \gamma_w}

4. Dry Unit Weight:

$$\gamma_d = \frac{W_s}{V} = \frac{G_s \gamma_w}{1+e}$$

\boxed{\gamma_d = \frac{G_s \gamma_w}{1+e}}

Problem-Solving Strategy

Given any 3 independent parameters (from e, n, w, Sᵣ, γ, γ_d, Gₛ), you can solve for the rest. Use the fundamental relationships as equations. Always check units (γ in kN/m³, γ_w ≈ 9.81 kN/m³ or 9.81×10⁻⁶ kN/mm³).

Laboratory Determination

III. Soil Classification

A. Grain Size Analysis

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

• Hydrometer Analysis: For fine-grained soils (silt & clay). Based on Stokes' Law (settling velocity). Results plotted on semi-log graph.

• Particle Size Distribution (PSD) Curve: X-axis: particle size (log scale), Y-axis: % finer by weight.

  • Shape: Well-graded (flat S-curve), Poorly-graded (steep curve), Gap-graded (flat middle section).

B. Gradation Characteristics

\boxed{C_u = \frac{D_{60}}{D_{10}}} \quad \text{(Uniformity Coefficient)}

\boxed{C_c = \frac{(D_{30})^2}{D_{60} \cdot D_{10}}} \quad \text{(Coefficient of Curvature)}

C. Atterberg Limits (Consistency 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 plastic to semi-solid state (thread rolling).

• Shrinkage Limit (SL): Water content at which soil changes from semi-solid to solid state (volume constant).

• Plasticity Index (PI):

  \boxed{PI = LL - PL}

  • Indicates range of water content where soil exhibits plastic behavior.

  • High PI: Clayey soils, high compressibility, low strength.

  • Low PI: Silty soils, non-plastic.

D. IS Soil Classification System (IS 2800: Part 5)

Step-by-Step Procedure:

1. Coarse-Grained ( >50% retained on 75µm sieve):

   • Gravel (G) if >50% retained on 4.75mm sieve. Sand (S) if >50% passes 4.75mm.

   • Gradation Suffix: W (well-graded, meets Cᵤ & C꜀), P (poorly-graded).

   • Fines Content: If 5-12% fines, use dual symbol (e.g., SW-SM). If >12%, classify based on Plasticity Chart (see below).

2. Fine-Grained ( >50% passes 75µm sieve):

   • Plot on Plasticity Chart (LL vs. PI).

   • A-line: $$\displaystyle PI = 0.73(LL - 20) $$.

   • Above A-line: Clay (C).

   • Below A-line: Silt (M).

   • LL = 50% line: Organic soils (Pt, Pe) if organic content > specified limits.

3. Highly Organic Soils: Pt (Peat), Pe (Peat with high decay).

E. Textural Classification (Triangular Diagrams)

• Used for fine-grained soils (silt & clay).

• USDA Triangle: Based on % sand, silt, clay.

• AASHTO Triangle: Based on % clay & silt (minus clay).

• Key: Determines textural class (e.g., Clay, Silty Clay, Sandy Clay Loam).

Exam Tip: In IS classification, always check % fines first. For coarse-grained soils with >12% fines, ignore gradation (W/P) and classify solely by plasticity chart position.

IV. Permeability and Seepage

A. Darcy's Law

Statement: Flow rate is proportional to hydraulic gradient and cross-sectional area.

\boxed{q = k \cdot i \cdot A}

• q: Discharge (volume/time)

• k: Coefficient of permeability (length/time, e.g., cm/s, m/day)

• i: Hydraulic gradient ($\Delta h / L$, dimensionless)

• A: Cross-sectional area normal to flow

Assumptions:

1. Laminar flow (low velocity, Re < 1).

2. Homogeneous, isotropic soil.

3. Steady-state flow (constant q, i with time).

4. Water is incompressible.

5. Flow is parallel to flow lines (no separation).

Factors Affecting k:

• Particle size & gradation: $$\displaystyle k \propto D_{10}^2 $$ (for sands).

• Void ratio: $$\displaystyle k \propto \frac{e^3}{1+e} $$ (Kozeny-Carman equation).

• Temperature: k increases with temperature (viscosity decreases).

• Soil structure: Flocculent vs. dispersed clays.

B. Laboratory Determination of k

C. Permeability of Stratified Deposits

1. Flow Parallel to Bedding (Horizontal Flow):

Total discharge is sum of discharges through each layer.

\[ q = \sum q_i = \sum (k_i \cdot i \cdot A_i) \]

Since i and A are common (total area A = ΣAᵢ, total head loss h = Σhᵢ, so i = h/L constant):

\[ k_h = \frac{\sum (k_i \cdot t_i)}{\sum t_i} \]

\boxed{k_h = \frac{k_1 t_1 + k_2 t_2 + ...}{t_1 + t_2 + ...}}

2. Flow Normal to Bedding (Vertical Flow):

Total head loss is sum of losses in each layer, discharge q is same through all.

\[ q = \frac{k_1 A \cdot h_1}{t_1} = \frac{k_2 A \cdot h_2}{t_2} = ... = \frac{k_v A \cdot h}{L} \]

\[ \frac{h}{k_v L} = \frac{h_1}{k_1 t_1} + \frac{h_2}{k_2 t_2} + ... \quad \text{and} \quad h = \sum h_i, \quad L = \sum t_i \]

\boxed{k_v = \frac{\sum t_i}{\sum \frac{t_i}{k_i}}}

Critical Insight: $$\displaystyle k_h $$ is always greater than $$\displaystyle k_v $$ for the same stratified system. The arithmetic mean ($$\displaystyle k_h $$) is weighted by thickness, the harmonic mean ($$\displaystyle k_v $$) is weighted by reciprocal of k.

D. Seepage Analysis & Flow Nets

Flow Net Construction:

1. Draw flow lines (path of water particles, tangent to q-vector).

2. Draw equipotential lines (lines of equal total head h, perpendicular to flow lines).

3. Ensure orthogonality ($$\displaystyle \angle = 90^\circ $$) and curvilinear squares (equal size, aspect ratio ≈ 1).

Characteristics:

• Orthogonal network.

• Between two adjacent flow lines, discharge is constant.

• Between two adjacent equipotential lines, head loss is constant ($$\displaystyle \Delta h = \frac{H}{N_d} $$).

• Flow through each "square" is: $$\displaystyle q_{square} = k \cdot \frac{\Delta h}{l} \cdot l = k \cdot \Delta h $$.

Applications:

1. Discharge Calculation:

   \[ Q = k \cdot H \cdot \frac{N_f}{N_d} \quad \text{(per unit width, for 2D flow)} \]

   \boxed{Q = k H \frac{N_f}{N_d}}

2. Seepage Force & Piping:

   • Seepage force per unit volume: $$\displaystyle j = i \cdot \gamma_w $$ (direction of flow).

   • Exit Gradient: $$\displaystyle i_{exit} = \frac{\Delta h_{last}}{l_{last}} $$.

   • Piping occurs if $$\displaystyle i_{exit} \geq i_c $$ (critical gradient).

E. Seepage Forces & Quick Sand Condition

Seepage Pressure (uᵢ):

Force exerted by water on soil skeleton due to flow.

\[ j = i \cdot \gamma_w \quad \text{(kN/m³)} \]

\[ u_s = j \cdot V = i \cdot \gamma_w \cdot V \quad \text{(kN)} \]

Critical Hydraulic Gradient (i꜀):

Condition when effective stress becomes zero at a point (boiling/quicksand).

\[ \sigma' = \sigma - u = \sigma - (i \cdot \gamma_w \cdot z) \]

At boiling: $$\displaystyle \sigma' = 0 \Rightarrow u = \sigma $$.

For submerged soil: $$\displaystyle \sigma = \gamma' \cdot z $$.

\[ i_c \cdot \gamma_w \cdot z = \gamma' \cdot z \Rightarrow i_c = \frac{\gamma'}{\gamma_w} = \frac{G_s - 1}{1 + e} \]

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

Quick Sand Condition:

• Occurs when upward seepage gradient $$\displaystyle i \geq i_c $$.

• Effective stress becomes zero → soil loses shear strength → behaves like a viscous liquid.

• Factor of Safety (FOS) against boiling:

  \[ FOS = \frac{i_c}{i_{applied}} \]

Exam Tip: For quick sand, always consider upward flow. $$\displaystyle i_c $$ depends only on $$\displaystyle G_s $$ and $e$. For clean sand with $$\displaystyle G_s=2.7 $$, $$\displaystyle e=0.7 $$, $$\displaystyle i_c \approx 1 $$.

V. Effective Stress

Terzaghi's Principle:

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

• $\sigma$: Total stress (from overburden/external loads).

• $u$: Pore water pressure (neutral stress, does not shear strength).

• $\sigma'$: Effective stress (governs shear strength, compressibility, volume change).

Calculation of Stresses:

1. Below Water Table (Saturated Soil, $$\displaystyle S_r = 100\% $$):

• Total stress: $$\displaystyle \sigma = \gamma_{sat} \cdot z $$

• Pore pressure: $$\displaystyle u = \gamma_w \cdot (z - z_{WT}) $$ (if WT at surface, $$\displaystyle u = \gamma_w \cdot z $$)

• Effective stress: $$\displaystyle \sigma' = (\gamma_{sat} - \gamma_w) \cdot z = \gamma' \cdot z $$

2. Above Water Table (Unsaturated Soil, $$\displaystyle S_r < 100\% $$):

• Total stress: $$\displaystyle \sigma = \gamma \cdot z $$ (moist unit weight)

• Pore pressure: Negative (suction). $$\displaystyle u = -\chi \cdot \gamma_w \cdot (z_{WT} - z) $$, where $$\displaystyle \chi \approx S_r $$ for low clays.

  • Simplified (for exams): Often take $$\displaystyle u = 0 $$ above WT unless specified.

• Effective stress: $$\displaystyle \sigma' = \gamma \cdot z + |u| $$ (since u is negative).

3. With Water Table at Depth:

• For a layer from $$\displaystyle z_1 $$ to $$\displaystyle z_2 $$:

  • If entirely above WT: $$\displaystyle \sigma' = \gamma \cdot (z_2 - z_1) $$

  • If entirely below WT: $$\displaystyle \sigma' = \gamma' \cdot (z_2 - z_1) $$

  • If straddles WT: Calculate in segments.

Stress Distribution Plotting:

1. Calculate total stress (σ) at key depths (layer boundaries).

2. Calculate pore pressure (u) at same depths.

3. Effective stress (σ') = σ - u.

4. Plot σ, u, σ' vs. depth. u line is linear with slope γ_w below WT. σ' line has slope γ above WT and γ' below WT.

Significance:

• Shear Strength: $$\displaystyle \tau = c' + \sigma' \tan \phi' $$ (effective stress parameters).

• Compression: Settlement depends on increase in $\sigma'$.

• Volume Change: Swelling/shrinkage governed by changes in $\sigma'$.

Common Pitfall: Forgetting to use saturated unit weight for total stress calculation below WT. Always check the saturation condition of each layer.