UNIT 2: FOUNDATION ENGINEERING - EXAM-FOCUSED SHORT NOTES

I. SUBSURFACE INVESTIGATION AND SOIL SAMPLING

A. Significance and Objectives of Site Exploration

• Primary Objective: To determine the stratigraphy (sequence of soil/rock layers), engineering properties (strength, compressibility), and groundwater conditions at a site to design a safe and economical foundation.

• Key Information Gathered:

  • Nature, thickness, and extent of soil/rock strata.

  • Strength parameters (c, φ) and compressibility (E, mv, Cc).

  • Groundwater table depth and fluctuations.

  • Presence of problematic soils (expansive, collapsible, organic).

• [!TIP] Exam Focus: Always link exploration objectives directly to foundation design requirements (bearing capacity, settlement, durability).

B. Methods of Boring/Hole Advancement

1. Rotary Drilling (Most Common for Deep Exploration)

• Procedure: A rotating drill bit (diamond-tipped for rock, drag/roller bits for soil) cuts the formation. Circulation of drilling fluid (mud or water) cools the bit, carries cuttings to surface, and stabilizes the borehole.

• Equipment: Drill rig, drill string, drill bit, mud pumps, slurry pit.

• Advantages:

  • Fast in most soils and rock.

  • Allows for continuous sampling (using Shelby tubes) and in-situ testing (SPT, CPT).

  • Minimal vibration compared to percussion methods.

  • Suitable for deep holes (>50m).

• [!TIP] Exam Focus: Be prepared to contrast rotary drilling with auger boring (fast, cheap, but disturbed samples, no water control) and wash boring (uses water jet, poor sample quality).

C. Soil Sampling Techniques

1. Disturbed vs. Undisturbed Samples

2. Sampling Tube Parameters (Critical Calculation)

Given: Inside diameter of cutting edge (ID_c), Inside diameter of tube (ID_t), Outside diameter of tube (OD_t), Thickness of cutting edge (t_c).

• Inside Clearance (C_i): Space inside tube for soil expansion.

$$C_i = \frac{ID_t - ID_c}{ID_t} \times 100\%$$

• Outside Clearance (C_o): Space between tube OD and borehole wall.

$$C_o = \frac{OD_t - ID_c}{OD_t} \times 100\%$$

• Area Ratio (A_r): Ratio of annular area (metal) to core area. Must be < 10% for good sample.

$$A_r = \frac{OD_t^2 - ID_t^2}{ID_t^2} \times 100\%$$

• Sample Quality Interpretation:

  • C_i ~ 0-3% (too low → sample compression).

  • C_i ~ 1-2% (optimal for cohesive soils).

  • C_o ~ 0-2% (too low → soil dragged, too high → sample disturbance).

  • A_r < 6% (excellent), 6-10% (good), >13% (poor).

D. In-Situ Testing

1. Standard Penetration Test (SPT)

• Procedure: Drive a split spoon sampler (OD=50.8mm, ID=35mm) with a 65.5 kg hammer falling 0.76 m. Count blows for each 15 cm penetration. First 15 cm is "seating drive".

• Output: N-value = blows for last 30 cm (from 15-45 cm). Standardized N-value (N₆₀) is corrected to 60% energy ratio.

• Corrections to N-value (Why?): Raw N is affected by overburden pressure, dilatancy, and hammer energy. Corrected N (N₁₀₀ or N₆₀) allows comparison across sites/depths.

  • Overburden Pressure Correction (Normalization): For sands/silts, N increases with σ'ᵥ₀. Use Begg & Miller (1984) or Liao & Whitman (1986) charts/formulas. Common simplified form:

$$N_{corrected} = N_{observed} \times \sqrt{\frac{100}{\sigma'_{v0}}} \quad (\sigma'_{v0} \text{ in kPa})$$

*   **Dilatancy Correction (Dense Sands/Overconsolidated Clays):** For N > 15 in saturated dense sands/OC clays, apply:

$$N_{corrected} = 15 + 0.5(N_{observed} - 15)$$

*   **Energy Ratio Correction:** Convert raw N to **N₆₀** (60% energy) or **N₁₀₀** (100% energy) using:

$$N_{60} = N_{observed} \times \frac{ER_{observed}}{60}$$

    where ER = (Measured hammer energy / Theoretical max energy) × 100%.

• Uses of Corrected N: Correlation with φ, relative density, settlement, liquefaction potential.

2. Cone Penetration Test (CPT)

• Procedure: Push a standard cone (10 cm² area, 60° apex) into ground at 20 mm/s. Measure:

  • Cone Resistance (q_c): Force on cone / basal area.

  • Sleeve Friction (f_s): Force on friction sleeve / surface area.

  • Friction Ratio (R_f): $$\displaystyle R_f = \frac{f_s}{q_c} \times 100\% $$.

• CPTu (Piezocone): Adds pore pressure (u) measurement behind cone.

• Advantages over SPT:

  • Continuous profile (no discrete sampling).

  • More repeatable, less operator-dependent.

  • Direct measurement of q_c and f_s; better for stratification.

  • CPTu provides u₂ for pore pressure dissipation tests.

3. Plate Load Test

• Setup: Load a rigid plate (300-750mm sq/circ) at foundation depth. Apply increments of load, measure settlement.

• Output: Load-settlement curve. Ultimate bearing capacity (q_u) is where settlement increases rapidly (or use 0.05B settlement criterion).

• Settlement Extrapolation (Terzaghi & Peck): To predict settlement (S_f) of a full-size footing (B_f) from plate test (B_p) at same net pressure (q_net):

$$\frac{S_f}{S_p} = \frac{2}{3} \frac{B_f}{B_p} \quad \text{for cohesive soils (clay)}$$

$$\frac{S_f}{S_p} = \frac{B_f}{B_p} \quad \text{for cohesionless soils (sand)}$$

> [!TIP] **Exam Focus:** This is a **high-frequency question**. Remember: Clay → 2/3 factor, Sand → 1.0 factor. Settlement is proportional to footing width.

E. Geophysical Methods (Brief)

• Seismic Refraction: Measures P-wave velocity from hammer blows/explosives. Used to map bedrock depth, soil layer thickness, and relative density.

• Electrical Resistivity: Measures soil resistivity (ρ). Used to detect soil type changes, groundwater, contamination.

• Ground Penetrating Radar (GPR): Uses electromagnetic pulses. Excellent for shallow depth (<10m), utility mapping, stratigraphy in dry/sandy soils.

F. Bore-log Preparation and Reporting

• Components (IS 1892/1893):

  1. Project details, borehole ID, coordinates.

  2. Stratigraphic column: Depth, soil/rock description (color, consistency, composition), Standard Penetration N-value, sample type & depth.

  3. Groundwater table (depth, date measured).

  4. Laboratory test results (moisture, density, strength, compressibility).

  5. In-situ test profiles (SPT, CPT).

• IS Code Criteria for Depth & Spacing (IS 1892):

  • Depth: Should extend to firm stratum or at least width of foundation (B) below base for shallow foundations. For piles, extend to 1.5-2.0 times pile length or to refusal.

  • Spacing: Typically 10-15m for uniform soil. Reduce to 5m for highly variable strata or large projects.

G. Other Exploration Aspects

• Significant Depth of Exploration: Depth where stress increase from proposed foundation is ≤ 10-20% of initial effective overburden stress. For a square footing, this is approx. 1.5B to 2B below base.

• Selection of Exploration Method: Depends on site access, soil type, depth required, budget, and required sample quality.

II. SHALLOW FOUNDATIONS

A. Types and Selection Criteria

• Types: Isolated Spread, Combined, Strip, Raft/Mat, Floating (Raft with basements).

• Basic Criteria for Satisfactory Performance:

  1. Adequate Bearing Capacity (q_u > applied pressure with FS).

  2. Acceptable Total & Differential Settlement (S_total, S_diff < permissible limits).

  3. Structural Adequacy (shear, moment in footing).

  4. Durability (against scour, frost, chemicals).

B. Bearing Capacity

1. Key Definitions

• Gross Pressure (q): Total load / area (includes weight of soil above footing).

• Net Pressure (q_net): q - γD_f (excludes overburden).

• Ultimate Bearing Capacity (q_u): Max gross pressure before shear failure.

• Net Ultimate Bearing Capacity (q_net,u): q_u - γD_f.

• Net Safe Bearing Capacity (q_net,s): q_net,u / Factor of Safety (FS).

• Allowable Bearing Pressure (q_a): Usually taken as q_net,s (for settlements < 25mm) or based on settlement criteria.

2. Modes of Shear Failure

3. Analytical Methods

• Terzaghi's Theory (1943): For strip, square, circular footings. Assumptions: Depth = 0 (D_f=0), rough base, φ > 0, horizontal ground surface, no shear above base.

  • Strip Footing: $$\displaystyle q_u = cN_c + γD_f N_q + 0.5γBN_γ $$

  • Square Footing: $$\displaystyle q_u = 1.3cN_c + γD_f N_q + 0.4γBN_γ $$

  • Circular Footing: $$\displaystyle q_u = 1.3cN_c + γD_f N_q + 0.3γBN_γ $$

• BIS (IS 6403) Method / General Equation: Incorporates shape (s_c, s_q, s_γ), depth (d_c, d_q, d_γ), inclination (i_c, i_q, i_γ), and ground water (w_c, w_q, w_γ) factors.

$$q_u = cN_c s_c d_c i_c w_c + γD_f N_q s_q d_q i_q w_q + 0.5γBN_γ s_γ d_γ i_γ w_γ$$

Where `s_c = 1 + 0.2(B/L)` for rectangular footing, etc. (Refer IS 6403 tables).

4. Bearing Capacity Factors (N_c, N_q, N_γ)

• Depend only on φ (effective friction angle).

• For φ = 0° (pure clay): $$\displaystyle N_c = 5.7 $$, $$\displaystyle N_q = 1.0 $$, $$\displaystyle N_γ = 0 $$.

• For φ > 0°: Use Terzaghi's or BIS/IS 6403 tables. Generally, $$\displaystyle N_q = e^{\pi \tan \phi} \tan^2(45° + \phi/2) $$, $$\displaystyle N_c = (N_q - 1) \cot \phi $$.

• [!TIP] Exam Focus: Memorize N_c=5.7 for φ=0°. For φ=30°, common values: N_c≈37.2, N_q≈18.4, N_γ≈22.4 (Terzaghi).

5. Water Table Correction

• Effect: Replaces soil unit weight (γ) with submerged unit weight (γ_sub = γ_sat - γ_w) below water table. Also affects N_q (use effective φ).

• Application in General Equation: w_q and w_γ factors.

  • If water table at base (D_f + B): $$\displaystyle w_q = 1 $$, $$\displaystyle w_γ = 1 - 0.1(1 - \frac{D_f}{B}) $$ (approx).

  • If water table above base: Use submerged unit weight for soil below WT, and apply w_q = 0.5 + 0.5 \frac{D_f}{B} (simplified). Best practice: Calculate using effective stresses and submerged weights directly.

6. Special Cases

• Layered Soils (Strong over Weak): Failure may occur in weaker layer. Bearing capacity controlled by weaker layer if its thickness > B/2. Use Hansen's modification or consider as two-layer system.

• Immediate vs. Long-term (Undrained vs. Drained):

  • Immediate (Short-term): Use undrained parameters (φ_u = 0°, c_u) for saturated clays. Total stress analysis.

  • Long-term (Consolidated): Use effective parameters (c', φ') after excess pore pressure dissipation. Effective stress analysis.

C. Settlement of Foundations

1. Components of Total Settlement (S_total)

$$S_{total} = S_i + S_c + S_s$$

• S_i: Immediate (Elastic) Settlement - occurs instantaneously.

• S_c: Primary Consolidation Settlement - due to expulsion of pore water (time-dependent).

• S_s: Secondary Compression (Creep) - after primary consolidation.

2. Immediate Settlement (S_i) Calculation

• Theory: Elastic half-space (Boussinesq). Settlement is proportional to pressure and inversely proportional to modulus.

$$S_i = \frac{q B (1 - \mu^2)}{E_s} I_f$$

Where:

*   `q` = net contact pressure.

*   `B` = footing width.

*   `μ` = Poisson's ratio.

*   `E_s` = **Modulus of elasticity of soil** (from lab/empirical correlation).

*   `I_f` = **Influence factor** (from charts like **Steinbrenner's** or **2:1 distribution** approximation).

• 2:1 Distribution (Approximate): Assumes pressure spreads at 2V:1H.

$$S_i = \frac{q B (1 - \mu^2)}{E_s} \times 1.0 \quad \text{(for square footing, 2:1 approx)}$$

> [!TIP] **Exam Focus:** If `I_f` is given (like 1.06 in May 2022 paper), use it directly. Otherwise, you may need to estimate from charts.

D. Proportioning of Raft Foundations

• Concept of Floating Foundation: Excavation and replacement with compacted fill such that net increase in vertical stress on soil is zero. Load of structure = weight of excavated soil.

• Criteria for Raft Design:

  1. Pressure Distribution: Should be as uniform as possible to limit differential settlement.

  2. Bearing Capacity: Net pressure < net safe bearing capacity.

  3. Settlement: Total & differential < permissible.

  4. Structural Strength: Shear, moment, punching shear checks.

• Typical Proportions: Thickness often 0.5m to 2m. Depth of excavation usually 1.5 to 2 times raft thickness.

III. DEEP FOUNDATIONS (PILES)

A. Pile Classification and Functions

• By Material: Concrete (precast/cast-in-situ), Steel (H-piles, pipe), Timber.

• By Action:

  • End Bearing Pile: Derives capacity from hard stratum (rock/dense sand) at tip.

  • Friction Pile: Derives capacity from skin friction along shaft (soft clay, loose sand).

  • Combined Pile: Both end bearing and friction.

• By Installation:

  • Driven Piles: Precast, displaced soil (crowding effect).

  • Bored/Cast-in-situ: Minimal displacement, can be longer, no hammer noise.

  • Screw Piles, Jet Piles, etc.

B. Pile Load Capacity Estimation

1. Static Load Carrying Capacity (Ultimate)

$$Q_u = Q_b + Q_s$$

• Q_b = Ultimate base resistance.

• Q_s = Ultimate shaft resistance (skin friction).

2. Static Formulae

• For Cohesionless Soils (Sand):

$$Q_b = A_b \cdot q_D \cdot N_q \quad \text{(q_D = vertical stress at base)}$$

$$Q_s = \sum (f_s \cdot A_s) \quad \text{or} \quad Q_s = \sum (K \cdot \sigma'_{v0} \cdot \tan \delta \cdot A_s)$$

Where `K` = earth pressure coefficient (0.5-1.0), `δ` = interface friction angle (≈ φ - 5° to φ).

• For Cohesive Soils (Clay):

  • End Bearing (Bored Piles): $$\displaystyle Q_b = A_b \cdot N_c \cdot c_u $$ (N_c ≈ 9 for bored, 12 for driven).

  • Shaft Friction: $$\displaystyle Q_s = \sum (\alpha \cdot \bar{c}_u \cdot A_s) $$

    • α = Adhesion factor (0.5-1.0, decreases with pile roughness/soil sensitivity).

    • \bar{c}_u = Average undrained cohesion along shaft.

• Effect of Water Table: Use submerged unit weight for sand above WT. For clay, c_u is total stress parameter (less affected).

3. Dynamic Methods (Drop Hammer)

• Engineering News (EN) Formula:

$$Q_{safe} = \frac{W \cdot H}{S + C} \cdot \frac{W + n W_p}{W + W_p}$$

Where:

*   `W` = Hammer weight, `H` = Fall, `S` = Final set (penetration per blow, m), `C` = Constant (25.4 mm for EN).

*   `W_p` = Pile weight, `n` = Coefficient of restitution (0.25-0.4).

*   **FS** applied to get safe load.

• Limitations: Empirical, depends on pile/soil damping.

4. Pile Load Tests

• Routine Test: To verify capacity during installation (maintained load test).

• Maintained Load Test: Cyclic loading to determine settlement behavior and ultimate capacity.

C. Single Pile Capacity in Specific Soils (Recap)

• Sandy Soils: Use φ, N_q, γ. Q_b = A_b * (γ * D_f * N_q) if D_f > 0. Q_s = Σ (K * σ'ᵥ₀ * tanδ * π * D * ΔL).

• Clayey Soils: Use c_u, α. Q_b = A_b * 9 * c_u (bored), Q_s = Σ (α * c_u * π * D * ΔL).

• Water Table: For sand, use γ_sub below WT. For clay, c_u is total stress (if undrained), but long-term may use effective c'.

D. Pile Groups

1. Group Efficiency (η_g):

$$\eta_g = \frac{Q_{ug}}{n \cdot Q_{us}}$$

Where Q_ug = Ultimate group capacity, Q_us = Ultimate single pile capacity (at same spacing).

• η_g > 1: Due to group effect (compaction, overlapping stresses) - rare.

• η_g < 1: Due to shadowing/overlap of stress bulbs - common in clay.

• η_g ≈ 1: For widely spaced piles in sand.

2. Group Capacity Calculation Methods

• Individual Pile Method: $$\displaystyle Q_{ug} = \eta_g \times n \times Q_{us} $$ (when spacing > 3-4D).

• Block Failure Method (for closely spaced groups in clay): Treat group as a single large footing of size (B_g x L_g) at base of group.

$$Q_{ug} = c_u \cdot N_c \cdot (B_g \cdot L_g) + \gamma \cdot D_{group} \cdot N_q \cdot (B_g \cdot L_g) + 0.5 \gamma \cdot B_g \cdot N_γ \cdot (B_g \cdot L_g)$$

(Often simplified: $$\displaystyle Q_{ug} = 9 \cdot c_u \cdot (B_g \cdot L_g) $$ if D_f=0, φ=0).

• Case Study (3x3 Group in Clay, Neglect End Bearing):

  • Spacing = S, Pile diameter = D.

  • Group dimensions: $$\displaystyle B_g = 2D + (3-1)S $$, $$\displaystyle L_g = 2D + (3-1)S $$ (for square group).

  • Use Block Failure if S < 3D. Use Individual Pile Method if S > 4-5D. Intermediate spacing → interpolation.

  • Example (Jun 2025): 3x3 group, D=0.3m, L=10m, S=0.9m, c_u=70 kN/m², α=0.6, FS=2.5, neglect Q_b.

    • S/D = 3 → closely spaced → use Block Failure.

    • B_g = 20.3 + 20.9 = 2.4m.

    • Q_ug = 9 * c_u * (B_g * L_g) = 9 * 70 * (2.4 * 2.4) = 3628.8 kN.

    • Q_safe = Q_ug / FS = 3628.8 / 2.5 = 1451.5 kN.

E. Negative Skin Friction (NSF)

• Definition: Downward drag force on pile due to downward movement of soil relative to pile.

• Causes: Fill placement, lowering water table, consolidation of soft clay.

• Calculation for Single Pile in Cohesive Soil:

$$F_{NSF} = \alpha \cdot \bar{c}_u \cdot \pi \cdot D \cdot L_{eff}$$

Where `L_eff` = Length of pile in compressible soil layer.

• Calculation in Cohesionless Soil:

$$F_{NSF} = K \cdot \sigma'_{v0} \cdot \tan \delta \cdot \pi \cdot D \cdot L_{eff}$$

Where `K` = lateral earth pressure coefficient (≈ 1.0 for fill), `σ'ᵥ₀` = effective overburden at depth.

• Effect: Reduces net pile capacity: $$\displaystyle Q_{net} = Q_{up} - F_{NSF} $$.

F. Special Pile Types

1. Under-reamed Piles

• Concept: Single or multi-belled piles with enlarged bases (under-reams) at one or more levels.

• Components: Shaft, bulb (reversed cone), neck (between bulb and shaft).

• Suitability: Expansive soils (black cotton soil). Bulbs provide uplift resistance and anchor against swelling pressure.

• Ultimate Tensile Capacity (Uplift):

$$Q_{tu} = \sum (A_b \cdot q_u \text{ at bulb}) + \sum (\alpha \cdot c_u \cdot A_s \text{ along shaft})$$

(Neglect suction, adhesion above bulbs).

• Ultimate Compressive Capacity: Similar, but consider bulb bearing in weaker soil.

2. Well Foundations (Caissons)

• Components (Neat Sketch Required):

  • Well curb: Cutting edge at bottom.

  • Well steining: Masonry/concrete ring above curb, provides weight for sinking.

  • Cutting edge: Steel angle to cut soil.

  • Lining: Wall segments (concrete/masonry).

  • Bottom plug: Concrete plug at bottom after reaching depth.

  • Top plug: Concrete fill above bottom plug.

  • Well cap: RCC beam at top to distribute pier load.

• Sinking Process: Excavation inside, self-weight/ballast causes sinking. Resistance components: Skin friction, base suction (if plugged), edge resistance.

IV. EARTH RETAINING STRUCTURES

A. Types of Lateral Earth Pressure

• Active (K_a): Wall moves away from soil → minimum pressure. Soil expands.

• Passive (K_p): Wall moves into soil → maximum pressure. Soil compressed.

• Earth Pressure at Rest (K₀): Wall rigid, no movement. $$\displaystyle K_0 = 1 - \sin \phi' $$ for normally consolidated clays/sands (Jaky's formula).

B. Earth Pressure Theories

1. Rankine's Theory (1875)

• Assumptions:

  • Wall is smooth, vertical.

  • Backfill is horizontal, cohesionless (c=0) or homogeneous cohesive.

  • Failure plane is inclined at (45°+φ/2) to horizontal.

• For Cohesionless Soil (c=0):

$$K_a = \tan^2(45° - \phi/2) = \frac{1 - \sin \phi}{1 + \sin \phi}$$

$$K_p = \tan^2(45° + \phi/2) = \frac{1 + \sin \phi}{1 - \sin \phi}$$

*   Pressure distribution: **Linear** with depth.

*   At depth z: $$\displaystyle \sigma_h = K_a \cdot \gamma \cdot z $$ (active).

• For Cohesive Soil (c>0, φ=0):

$$K_a = 1 - \frac{2c}{\gamma z} \quad \text{(not constant)}$$

*   Pressure distribution: **Parabolic** with intercept `2c√K_a` at surface.

*   **Tension Crack Depth (z_tc):** Depth where σ_h=0.

$$z_{tc} = \frac{2c}{\gamma \sqrt{K_a}}$$

• For Cohesive Soil (c>0, φ>0): Use Coulomb or Modified Rankine.

2. Coulomb's Theory (1776) - Wedge Theory

• Assumptions:

  • Failure plane is inclined at angle θ from horizontal.

  • Wall is rough → wall friction angle (δ) develops.

  • Backfill can be inclined (β).

  • Surface can have surcharge (q).

• Active Earth Pressure Coefficient (K_a):

$$K_a = \frac{\cos^2(\phi - \delta)}{\cos^2 \delta \cos(\delta + \beta) \left[1 + \sqrt{\frac{\sin(\phi + \delta) \sin(\phi - \beta)}{\cos(\delta + \beta)}}\right]^2}$$

*   More **general** than Rankine.

*   **Merits over Rankine:**

    1.  Accounts for **wall friction (δ)** → reduces K_a.

    2.  Allows **inclined backfill (β)**.

    3.  Applicable to **cohesive soils** with c>0, φ>0.

*   **Resultant force** acts at angle δ to the normal of wall.

3. Graphical Methods - Culmann's Method

• Used for active pressure with inclined, broken, or multilayered backfill.

• Procedure:

  1. Plot weight of soil wedge (W) vs. failure plane angle (θ) on graph.

  2. Draw line from origin at angle (45°+φ/2) to intersect W curve.

  3. From intersection, draw line parallel to wall to find K_a.

• Gives minimum K_a for given slope.

C. Earth Pressure Calculations and Diagrams

1. Distribution with Depth:

• Active/Passive (Rankine, c=0): Linear from zero at surface (active) or from 2c√K_a (cohesive) to γHK at base.

• With Water Table: Replace γ with γ_sub below WT. Add hydrostatic pressure (uplift) separately. Seepage adds seepage force.

• With Surcharge (q): Adds uniform pressure of q * K_a (or K_p) over entire height.

2. Total Thrust (P_a) and Point of Application

• For Homogeneous, c=0, Horizontal Backfill (Rankine):

$$P_a = \frac{1}{2} K_a \gamma H^2$$

Acts at **H/3 from base**.

• For Cohesive Soil (c>0, φ=0):

$$P_a = \frac{2}{3} c \sqrt{K_a} H + \frac{1}{2} K_a \gamma H^2$$

Acts **above H/3** due to parabolic component.

• Case Study (Stratified Backfill): Calculate K_a for each layer using appropriate φ, γ. Compute thrust for each layer. Sum magnitudes and moments about base to find resultant magnitude and height of application.

3. Effect of Water Table:

• Below Base: Use γ_sub below WT in K_aγH² term. No effect on K_a.

• At Base: Same as below.

• Above Base: Use γ for layer above WT, γ_sub below. Add water pressure (γ_w * h) as separate triangular distribution. Resultant is vector sum of soil and water pressures.

• Seepage (Downward): Increases active pressure. Use seepage force concept or effective stress with flow net.

D. Retaining Wall Design and Stability

1. Modes of Failure:

• Overturning: Moment about toe > resisting moment. Check Factor of Safety (FOS) > 1.5.

• Sliding: Horizontal thrust > frictional resistance (μ * N) + shear key. FOS > 1.5.

• Excessive Bearing Pressure: Max pressure < allowable; differential pressure < limit.

• Deep-seated failure: Global slope failure (rare for isolated walls).

2. Sheet Piles vs. Retaining Walls

V. SOIL PROPERTIES, PROBLEMATIC SOILS & IMPROVEMENT

A. Problematic Soils

1. Expansive Soils (Black Cotton Soils)

• Characteristics: High montmorillonite clay content, high shrink-swell potential with moisture change, low strength when wet, hard when dry.

• Problems: Heave (swelling) → lift foundations, crack walls. Shrinkage → cracks, differential movement.

• Preventive Measures:

  • Moisture Control: Maintain constant moisture (impermeable layer, landscaping).

  • Deep Foundations: Piles/under-reamed piles below active zone.

  • Chemical Stabilization: Lime, cement treatment to reduce swell.

  • Lightweight Structures: Reduce imposed load.

  • Raft Foundations: Distribute load, reduce pressure.

2. Collapsible Soils (Loess)

• Characteristics: Low density, high porosity, metastable structure (cemented by salts/clay bridges). Dry but may be moist. Sudden collapse upon wetting.

• Problems: Sudden, large settlement upon saturation (rainfall, leakage).

• Preventive Measures:

  • Pre-wetting: Saturate soil before construction to induce collapse.

  • Compaction: Dynamic compaction, heavy tamping.

  • Replacement: Excavate and replace with granular fill.

  • Pile Foundations: Transfer load through collapsible layer.

  • Chemical Stabilization: Lime, cement.

B. Soil Stabilization

• Need: Improve strength, reduce compressibility/shrink-swell, increase durability for weak soils, fills, expansive soils.

• Methods:

  • Mechanical: Compaction (increase density, reduce voids).

  • Chemical: Lime (for clay), Cement (for sand/clay), Bitumen (waterproofing).

  • Electrical Stabilization (Electro-osmosis): Apply DC current between electrodes in clay. Water migrates to anode, causing consolidation and strength gain. Used for very soft, saturated clays where conventional methods fail.

C. Geosynthetics

• Types & Materials:

  | Type | Material | Key Feature | | :--- | :--- | :--- | | Geotextiles | Woven (polypropylene), Non-woven (felt) | Separation, Filtration, Reinforcement | | Geogrids | HDPE, Polyester | High tensile strength, Reinforcement (soil-interlock) | | Geomembranes | HDPE, LLDPE | Impermeable, Containment (liners) | | Geocomposites | Geotextile + Geonet | Drainage (composite sheets) | | Geocells | HDPE strips, welded | 3D confinement, Reinforcement (slopes, soft soil) |

• Functions & Applications in Foundations:

  • Separation: Prevent mixing of dissimilar soils (e.g., soft clay and granular fill).

  • Reinforcement: Increase bearing capacity, reduce settlement (geogrids/geocells over soft soil).

  • Filtration: Allow water flow but retain soil particles (geotextiles behind retaining walls).

  • Drainage: Transmit water (geonets, geocomposites).

  • Containment: Liners for ponds, landfills (geomembranes).

VI. STRESS DISTRIBUTION IN SOIL MASSES

A. Boussinesq's Theory (1885)

• Assumptions:

  1. Homogeneous, isotropic, elastic half-space (obeys Hooke's law).

  2. Semi-infinite (no rigid boundaries).

  3. Point load applied at surface.

  4. No shear strength (stress only).

• Equation for Vertical Stress (σ_z) under a Point Load (Q):

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

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

• Influence Chart/Isobar Concept: Plot lines of equal σ_z (isobars). For uniformly loaded area, use influence factor (I_f):

$$\sigma_z = q \cdot I_f$$

Where `q` = uniform pressure, `I_f` from **2:1 distribution** or **Boussinesq charts** (e.g., **Fadum's chart**).

B. Westergaard's Theory (1938)

• Assumptions:

  1. Soil is incompressible (ν=0.5) when loaded vertically.

  2. Soil mass contains infinitely many, equally spaced, vertical, incompressible sheets (like clay lenses). Load transfer only through vertical compression.

  3. Semi-infinite, elastic.

• Equation for Vertical Stress:

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

• Key Difference: Stress distribution is more concentrated vertically below load compared to Boussinesq. Stress decreases faster with r.

C. Comparison: Boussinesq vs. Westergaard

[!TIP] Exam Focus: Remember: Boussinesq for sands, Westergaard for clays/laminated soils. Westergaard gives lower vertical stress at a given (r,z) compared to Boussinesq.