UNIT 2: FOUNDATION ENGINEERING

Based on RGPV Past Examination Analysis (2022-2025)

1.0 SOIL EXPLORATION & SITE INVESTIGATION

1.1 Significant Depth & Borehole Planning

• Significant Depth: Depth up to which stress increase due to foundation load is significant (typically where Δσ/σ'₀ ≤ 10%). For design, exploration must reach below this depth.

• IS Criteria (IS: 1892 - 1979) for Depth of Boreholes:

  • For isolated spread footings: Depth ≥ width of footing.

  • For raft foundations: Depth ≥ width of raft.

  • For pile foundations: Depth ≥ length of pile + 3m or up to hard stratum.

  • Minimum depth: 1.5m to 3m (to avoid near-surface disturbances).

• Factors Influencing Depth & Spacing:

  • Type of structure & load intensity.

  • Soil/rock stratification.

  • Presence of weak zones (faults, filled-up areas).

  • Groundwater table fluctuations.

  • Past performance of similar foundations in the area.

[!TIP] Exam Focus: IS 1892 criteria for depth is a direct question. Always state the code and its specific provisions.

1.2 Boring & Sampling Methods

• Rotary Drilling:

  • Technique: A rotating drill bit (diamond or tungsten carbide) attached to a drill string, with circulating drilling fluid (bentonite slurry or water) to bring cuttings to surface and stabilize borehole.

  • Equipment: Drill rig, drill pipes, bit, mud pumps, slurry mixing tank.

  • Advantages over Percussion/Auger:

    • Produces undisturbed samples in cohesive soils (using core barrels).

    • Faster in hard soils/rock.

    • Minimal vibration and disturbance.

    • Can bore through boulders and rock.

    • Better control in water-bearing strata.

• Other Boring Methods:

  • Percussion (Cable-tool): Dropping heavy chisel, suitable for boulders/rock, produces disturbed samples.

  • Auger (Hand/Power): Manual or mechanical, for shallow depths in cohesive soils, highly disturbed samples.

  • Wash Boring: Jet of water loosens soil, suitable for sandy soils, samples are disturbed.

• Soil Sampling:

  • Disturbed Sample: Soil structure altered. Used for classification, moisture content, compaction tests. Obtained from auger, bailer, or cuttings.

  • Undisturbed Sample: Soil structure, moisture, and strength preserved. Essential for consolidation, triaxial, permeability tests. Obtained using thin-walled sampling tubes (piston samplers).

• Sampling Tube Design Parameters:

  • Inside Clearance (Cᵢ): $$\displaystyle (D_i - D_e)/D_e \times 100\% $$. Allows sample to expand into tube (typically 0.5-1.5%). Reduces friction.

  • Outside Clearance (Cₒ): $$\displaystyle (D_o - D_e)/D_e \times 100\% $$. Allows tube to move freely in borehole (typically 0-2%).

  • Area Ratio (Aᵣ): $$\displaystyle (D_o^2 - D_i^2)/D_i^2 \times 100\% $$. Should be < 20% for undisturbed samples. Higher Aᵣ causes more disturbance.

  • Sample Quality: Low Cᵢ, Cₒ, and Aᵣ → better quality. CNS Layer (Constant Normal Stiffness) concept relates to sample disturbance during insertion.

• Bore-log Report: Graphical representation of subsurface profile. Includes: depth, soil description (USCS), sample type & recovery, SPT N-value, water table, lab test results, stratigraphy.

1.3 In-Situ Testing

• Standard Penetration Test (SPT):

  • Procedure: Driving a split-spoon sampler (50.8 mm OD, 35 mm ID) 450 mm into soil at bottom of borehole using a 63.5 kg hammer falling 760 mm (30 blows/ft). First 150 mm is seating drive. N-value = blows for last 300 mm.

  • N-value Definition: Number of blows required to drive sampler 300 mm (12 inches) beyond seating drive.

  • Corrections to N-value (to N₁₆₀):

    1. Overburden Pressure Correction (K₀ or Effective Stress Correction): $$\displaystyle N_{corrected} = N_{observed} \times \frac{100}{\sigma'_{v0}} $$ (for sands). Normalizes N to an effective overburden pressure of 100 kN/m².

    2. Dilatancy Correction (for dense sands/gravels): $$\displaystyle N_{corrected} = 15 + 0.5(N_{observed} - 15) $$ for $$\displaystyle N_{observed} > 15 $$ in saturated dense sands. Corrects for negative pore pressure buildup.

    3. Energy Correction: $$\displaystyle N_{160} = N_{field} \times \frac{ER_{field}}{60\%} $$. Converts field hammer energy (often 30-80%) to standard 60% energy ratio (N₁₆₀).

  • Need: Raw N-values are highly dependent on overburden, energy, and soil type. Corrections allow comparison across sites and correlation with soil properties (φ, relative density, modulus).

• Cone Penetration Test (CPT/CPTu):

  • Principle: Pushing a standard cone (10 cm² area, 60° apex angle) into soil at constant rate (20 mm/s) while measuring continuous resistance.

  • Measured Parameters:

    • $$\displaystyle q_c $$: Cone tip resistance (MPa).

    • $$\displaystyle f_s $$: Sleeve friction (MPa).

    • $u$: Pore water pressure (CPTu only).

  • Comparison with SPT:

    • CPT: Continuous profile, faster, more repeatable, provides $$\displaystyle f_s $$ & $u$, better for soft soils/stratigraphy.

    • SPT: Discontinuous, provides physical sample, more common in granular soils, cheaper equipment.

• Plate Load Test:

  • Test Setup: Load applied incrementally on a rigid plate (0.3m² typical) at foundation level. Settlement measured.

  • Procedure on Clay: Load-settlement curve is used to find ultimate bearing capacity ($$\displaystyle q_u $$) from Terzaghi's bearing capacity failure criterion (settlement = 20% of plate width) or logarithmic curve intersection method.

  • Interpretation:

    • Ultimate Bearing Capacity ($$\displaystyle q_u $$): From failure load.

    • Settlement: At working load.

  • Scale Effect (for larger footings): Ultimate bearing capacity increases with size, settlement increases more.

    • Terzaghi & Peck's Method: $$\displaystyle q_{u, footing} = q_{u, plate} \times \left( \frac{B_{footing}}{B_{plate}} \right)^{n} $$, where $n \approx 0.5$ for clay, 0.25-0.4 for sand.

    • Settlement: $$\displaystyle S_{footing} = S_{plate} \times \left( \frac{B_{footing}}{B_{plate}} \right)^{m} \times \left( \frac{B_{plate}}{B_{footing}} \right) $$, where $m \approx 0.5$ for clay.

1.4 Geophysical Methods

• Seismic Refraction: Measures velocity of compressional waves (P-waves) to delineate strata boundaries and bedrock depth. Based on Snell's Law.

• Electrical Resistivity: Measures soil resistivity by passing current between electrodes. Used to map soil types, groundwater, and contamination.

• Application: Rapid, economical for large areas, preliminary site characterization, locating voids/buried objects.

1.5 Bore-log Report

• Components:

  1. Title block (project, location, date, contractor).

  2. Stratigraphic column with depth, soil description, sample type, SPT N-value.

  3. Water table level.

  4. Laboratory test results (moisture, density, shear strength, consolidation).

  5. Graphical representation of soil profile.

  6. Remarks on drilling method, difficulties, etc.

2.0 BEARING CAPACITY OF SHALLOW FOUNDATIONS

2.1 Fundamental Definitions & Concepts

• Ultimate Bearing Capacity ($$\displaystyle q_u $$): Maximum gross pressure soil can sustain before shear failure.

• Net Ultimate Bearing Capacity ($$\displaystyle q_{nu} $$): $$\displaystyle q_{nu} = q_u - \gamma D_f $$. Pressure at foundation level causing failure.

• Net Safe Bearing Capacity ($$\displaystyle q_{ns} $$): $$\displaystyle q_{ns} = q_{nu} / FOS $$. Allowable net pressure.

• Allowable Bearing Pressure ($$\displaystyle q_a $$): $$\displaystyle q_a = q_{ns} + \gamma D_f $$. Maximum safe gross pressure.

• Gross Pressure: Total vertical stress at foundation base including overburden.

• Modes of Shear Failure:

  1. General Shear: In stiff/dense soils. Well-defined failure surface, large settlements, distinct peak in load-settlement curve.

  2. Local Shear: In medium soils. Failure surface limited to under footing, settlements moderate, no well-defined peak.

  3. Punching Shear: In loose/soft soils. Soil punches into footing, minimal lateral spread, settlements large and progressive.

2.2 Theoretical Analysis

• Terzaghi's Theory (1943):

  • Assumptions: Strip footing, soil is homogeneous, isotropic, weightless ($$\displaystyle \gamma=0 $$), $c-\phi$ soil, failure surface is logarithmic spiral + planar, footing is rigid, base is rough, load is vertical & concentric.

  • Equation (for strip footing): $$\displaystyle q_u = c'N_c + q N_q + 0.5 \gamma B N_\gamma $$

  • For Other Shapes (Terzaghi's factors):

    • Square: $$\displaystyle q_u = 1.3c'N_c + q N_q + 0.4 \gamma B N_\gamma $$

    • Circular: $$\displaystyle q_u = 1.3c'N_c + q N_q + 0.3 \gamma B N_\gamma $$

    • Rectangular: $$\displaystyle q_u = (1 + 0.2B/L)c'N_c + q N_q + (0.5 - 0.2B/L)\gamma B N_\gamma $$

• IS Method (BIS Code IS: 6403 - 1981): Uses shape, depth, and inclination factors. General form:

$$q_u = c'N_c s_c d_c i_c + q N_q s_q d_q i_q + 0.5 \gamma B N_\gamma s_\gamma d_\gamma i_\gamma$$

Where $s, d, i$ are shape, depth, and load inclination factors.

• Factors Affecting Bearing Capacity:

  • Shape Factor ($s$): Increases $$\displaystyle N_c $$, $$\displaystyle N_\gamma $$ for square/circular vs strip.

  • Depth Factor ($d$): Increases capacity with depth (for $$\displaystyle D_f/B \leq 1 $$).

  • Water Table: Reduces effective unit weights and surcharge.

  • Load Inclination ($i$): Reduces capacity for inclined/eccentric loads.

  • Base Inclination ($b$): Reduces capacity for tilted base.

  • Ground Surface Inclination ($g$): Reduces capacity for sloping ground.

2.3 Application & Problem Solving

• Calculation Steps:

  1. Determine $c'$, $\phi'$, $\gamma$ from lab tests.

  2. Compute $$\displaystyle N_c $$, $$\displaystyle N_q $$, $$\displaystyle N_\gamma $$ using Hansen's or Vesic's equations (more accurate than Terzaghi's for $$\displaystyle \phi>10^\circ $$).

  3. Apply shape, depth, water table correction factors.

  4. Compute $$\displaystyle q_u $$ (gross) or $$\displaystyle q_{nu} $$ (net) as required.

• Water Table Correction Factors (IS: 6403):

  • If water table at depth $$\displaystyle D_w $$ below base:

    $$\displaystyle w_\gamma = 1 - 0.5 \frac{D_w}{B} $$ (for $\gamma$ term), $$\displaystyle w_q = 1 $$ (for $q$ term).

  • If water table at base: $$\displaystyle w_\gamma = 0.5 $$, $$\displaystyle w_q = 1 $$.

  • If water table above base: $$\displaystyle w_\gamma = 0.5 - \frac{h_w}{2B} $$ (if $$\displaystyle h_w < B $$), $$\displaystyle w_q = 1 - \frac{h_w}{B} $$.

• Special Cases:

  • Purely Cohesive ($$\displaystyle \phi=0 $$): $$\displaystyle N_c = 5.7 $$ (Terzaghi), $$\displaystyle N_q=1 $$, $$\displaystyle N_\gamma=0 $$. $$\displaystyle q_u = c'N_c + \gamma D_f $$.

  • Cohesionless ($$\displaystyle c=0 $$): $$\displaystyle q_u = q N_q + 0.5 \gamma B N_\gamma $$.

  • Cohesive-Frictional ($c-\phi$): Use full equation.

2.4 Settlement of Shallow Foundations

• Components of Settlement:

  1. Immediate (Elastic) Settlement ($$\displaystyle S_i $$): Occurs during/just after construction in saturated clays (undrained) and all sands. Recoverable.

  2. Primary Consolidation Settlement ($$\displaystyle S_c $$): Due to expulsion of pore water from saturated clays over time. Irrecoverable.

  3. Secondary Consolidation Settlement ($$\displaystyle S_s $$): Due to plastic adjustment of soil skeleton after primary consolidation. Very slow.

• Immediate Settlement ($$\displaystyle S_i $$) for Cohesive Soils:

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

Where:

*   $q$ = net pressure.

*   $B$ = footing width.

*   $\nu$ = Poisson's ratio.

*   $$\displaystyle E_s $$ = Modulus of elasticity.

*   $$\displaystyle I_i $$ = **Influence factor** (from charts/tables, depends on $L/B$ and $$\displaystyle D_f/B $$). For square footing, $$\displaystyle I_i \approx 1.06 $$ (for $$\displaystyle D_f/B=0 $$).

*   For **c-φ soils**, use **Janbu's chart** or **elastic theory** (e.g., $$\displaystyle I_i $$ from Fadum's charts).

3.0 PILE FOUNDATIONS

3.1 Classification & Functions

• By Material: Concrete (RCC, Precast), Timber, Steel (H-piles, pipes), Composite.

• By Function:

  • End-Bearing Pile: Transfers load to hard stratum.

  • Friction Pile: Transfers load via skin friction along shaft.

  • Combined Pile: Uses both end bearing and skin friction.

• By Installation:

  • Driven (Displacement): Precast concrete/steel, timber. Vibratory/hammer impact. Causes soil displacement, densification.

  • Bored (Non-displacement): Cast-in-situ concrete. Less noise/vibration, suitable for sensitive areas.

  • Screw Piles: Helical plates, for light loads.

  • Under-reamed Piles: With enlarged bulbs, for expansive soils.

3.2 Load Carrying Capacity of Single Pile

• Static Formulas:

  • For Cohesive Soils (α-method):

$$Q_u = \alpha c_u A_s + c_u A_b$$

    Where:

    *   $\alpha$ = adhesion factor (0.5-1.0, decreases with depth/soft clay).

    *   $$\displaystyle c_u $$ = undrained cohesion.

    *   $$\displaystyle A_s $$ = surface area of shaft.

    *   $$\displaystyle A_b $$ = area of base (often neglected in soft clays).

*   **For Cohesionless Soils (β-method):**

$$Q_u = \sum (K \sigma'_{v0} \tan \delta \Delta A) + q_u A_b$$

    Where:

    *   $K$ = lateral earth pressure coefficient (0.5-1.0 $$\displaystyle K_0 $$).

    *   $$\displaystyle \sigma'_{v0} $$ = effective overburden at depth.

    *   $\delta$ = interface friction angle ($\approx 0.75\phi$ to $\phi$).

    *   $$\displaystyle q_u $$ = bearing capacity factor for base ($$\displaystyle N_q \sigma'_{v0} $$).

• Dynamic Formulas (Drop Hammer):

  • Engineering News Formula (EN):

$$Q_{safe} = \frac{W h}{s + C} \times \frac{E}{6}$$

    Where $W$=hammer weight, $h$=fall, $s$=final set (penetration per blow), $C$=constant (2.5 cm for concrete piles), $E$=efficiency (often 0.6-0.8).

*   **Hiley's Formula (Improved EN):**

$$Q_{safe} = \frac{\eta W h}{s + C/2} \times \frac{1}{FOS}$$

    Where $\eta$ = efficiency factor accounting for hammer, cap, pile, and soil. More accurate.

3.3 Pile Groups

• Group Capacity: $$\displaystyle Q_{ug} = Q_{ug(block)} \leq n \times Q_{us} $$ (individual pile capacity).

• Block Failure (Cohesive Soils):

$$Q_{ug(block)} = c_u (B_g L_g) + c_u (2(B_g+L_g)D_g)$$

Where $$\displaystyle B_g, L_g $$ = group dimensions, $$\displaystyle D_g $$ = depth of block.

• Group Efficiency ($$\displaystyle \eta_g $$): $$\displaystyle \eta_g = Q_{ug(actual)} / (n \times Q_{us}) $$. Usually < 1 for clays (due to overlapping stress zones), >1 for sands (due to densification).

• Geometrical Properties for Spacing:

  • Spacing (s) to Diameter (d) Ratio ($s/d$):

    • For clay: $s/d \geq 3$ to avoid group failure (block failure). At $$\displaystyle s/d=3 $$, $$\displaystyle \eta_g \approx 0.67 $$.

    • For sand: $s/d \geq 3$ to avoid group settlement exceeding sum of individual settlements. $$\displaystyle \eta_g $$ can be >1 at closer spacing due to densification.

• Numerical Problem (3x3 Group in Clay, Neglect End Bearing):

  • Step 1: Single pile shaft capacity: $$\displaystyle Q_{us} = \alpha c_u A_s $$.

  • Step 2: Group capacity (block failure): $$\displaystyle Q_{ug} = c_u \times (Group\;block\;area) = c_u \times [ (3s)^2 - (3 \times \pi d^2/4) ] $$? No. For clay, group capacity often governed by individual pile friction if spacing adequate ($s/d \geq 4$). If $$\displaystyle s/d < 4 $$, use block failure.

  • Given: 3x3 group, $$\displaystyle d=0.3m $$, $$\displaystyle L=10m $$, $$\displaystyle c_u=70 kN/m^2 $$, $$\displaystyle s=0.9m $$ ($$\displaystyle s/d=3 $$), $$\displaystyle \alpha=0.6 $$, FOS=2.5.

  • Calculation:

    • $$\displaystyle A_s = \pi d L = \pi \times 0.3 \times 10 = 9.42 m^2 $$.

    • $$\displaystyle Q_{us} = 0.6 \times 70 \times 9.42 = 396.2 kN $$.

    • $$\displaystyle n=9 $$, $$\displaystyle n \times Q_{us} = 3566 kN $$.

    • Check Block Failure: Group block area = $$\displaystyle (3s)^2 = (2.7)^2 = 7.29 m^2 $$ (assuming square group). But block area for clay group capacity is plan area of group for skin friction? Correction: For cohesive soils, group capacity by block failure is $$\displaystyle Q_{ug(block)} = c_u \times (B_g \times L_g) + \text{shaft friction of block perimeter} $$. However, at $$\displaystyle s/d=3 $$, group efficiency is low (~0.67). Often, adopt lower of block failure or sum of individual capacities.

    • Standard Approach: For soft clays with $$\displaystyle s/d=3 $$, use group efficiency factor $$\displaystyle \eta_g \approx 0.67 $$. So $$\displaystyle Q_{ug} = \eta_g \times n \times Q_{us} = 0.67 \times 3566 = 2390 kN $$.

    • Allowable Load: $$\displaystyle Q_{allow} = Q_{ug} / FOS = 2390 / 2.5 = 956 kN $$.

  • Note: Some codes suggest for $s/d \geq 4$, $$\displaystyle \eta_g=1 $$. For $$\displaystyle s/d=3 $$, $$\displaystyle \eta_g $$ may be 0.7-0.8. Always state assumption.

3.4 Negative Skin Friction (NSF)

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

• Causes:

  • Fill over soft compressible soil (surcharge causes consolidation).

  • Lowering of groundwater table (increases effective stress, causes settlement).

  • Collapsible soils upon wetting.

• Calculation for Single Pile:

$$Q_{nsf} = \alpha \cdot \bar{c}_u \cdot A_s \quad \text{(clay)}$$

or

$$Q_{nsf} = K \cdot \bar{\sigma}'_{v0} \cdot \tan \delta \cdot A_s \quad \text{(sand)}$$

Where $$\displaystyle \bar{c}_u $$, $$\displaystyle \bar{\sigma}'_{v0} $$ are **average** values along the **critical zone** (from ground surface to neutral plane or depth of compressible layer).

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

3.5 Special Piles

• Under-reamed Piles:

  • Concept: Single or multi-belled bored piles with under-reams (reversed cones) at base and sometimes intermediate levels.

  • Components: Shaft, under-ream bulb (diameter 2-3× shaft), neck (narrow portion between bulb and shaft).

  • Suitability: Expansive soils (black cotton soil), loose sands, zones with uplift forces (towers, bridges). Provide tensile resistance and anchor against swelling.

  • Ultimate Tensile Capacity:

$$Q_{tu} = \sum (c_u \cdot A_{bulb}) + \alpha \cdot \bar{c}_u \cdot A_{shaft}$$

    Where $$\displaystyle A_{bulb} $$ = surface area of under-ream bulb (conical surface + base).

• Well Foundations (Caissons):

  • Components:

    • Well curb: Cutting edge at bottom.

    • Steining: Vertical masonry/concrete wall.

    • Well cap: Top slab.

    • Sand filling: Inside well for stability.

  • Sinking: By removing soil from inside, self-weight or external kentledge.

4.0 LATERAL EARTH PRESSURE & RETAINING STRUCTURES

4.1 Types of Lateral Earth Pressure

• At-Rest ($$\displaystyle K_0 $$): No lateral strain. Wall rigid, backfill undisturbed.

• Active ($$\displaystyle K_a $$): Wall moves away from soil. Minimum pressure. Soil expands, $$\displaystyle \sigma'_h $$ decreases.

• Passive ($$\displaystyle K_p $$): Wall moves into soil. Maximum pressure. Soil compressed, $$\displaystyle \sigma'_h $$ increases.

• Relationship: $$\displaystyle K_a < K_0 < K_p $$.

4.2 Rankine's Earth Pressure Theory

• Assumptions:

  • Soil is homogeneous, isotropic, cohesionless ($$\displaystyle c=0 $$) or purely cohesive ($$\displaystyle \phi=0 $$).

  • Backfill is horizontal, wall is vertical & frictionless ($$\displaystyle \delta=0 $$).

  • Failure surface is planar at $$\displaystyle 45^\circ + \phi/2 $$.

  • No tension crack in cohesive backfill.

• For Cohesionless Soil (dry/submerged):

  • Active: $$\displaystyle \sigma'_h = \gamma z K_a $$, where $$\displaystyle K_a = \tan^2(45^\circ - \phi/2) $$.

  • Passive: $$\displaystyle \sigma'_h = \gamma z K_p $$, where $$\displaystyle K_p = \tan^2(45^\circ + \phi/2) $$.

• For Cohesive Soil ($$\displaystyle \phi=0 $$, $$\displaystyle c>0 $$):

  • Active: $$\displaystyle \sigma_h = \gamma z - 2c \sqrt{K_a} $$ (tension crack depth $$\displaystyle z_c = 2c/(\gamma \sqrt{K_a}) $$).

  • Passive: $$\displaystyle \sigma_h = \gamma z + 2c \sqrt{K_p} $$.

• Pressure Diagrams:

  • Level Backfill: Linear distribution for $$\displaystyle c=0 $$, triangular. For $$\displaystyle c>0 $$, active pressure diagram is trapezoidal (negative near top, zero at tension crack depth).

  • With Water Table: Use submerged unit weight below WT, plus water pressure (hydrostatic).

  • With Surcharge ($q$): Add uniform pressure $$\displaystyle qK_a $$ (active) or $$\displaystyle qK_p $$ (passive).

4.3 Coulomb's Earth Pressure Theory

• Assumptions:

  • Soil is homogeneous, cohesionless ($$\displaystyle c=0 $$) or with cohesion.

  • Backfill surface may be inclined ($\beta$).

  • Wall is rough; wall friction angle ($\delta$) considered.

  • Failure plane is planar, makes angle $\theta$ with horizontal.

• Expression for $$\displaystyle K_a $$ & $$\displaystyle K_p $$ (for $$\displaystyle c=0 $$):

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

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

• Comparison with Rankine:

  • Merits of Coulomb: Accounts for backfill inclination ($\beta$) and wall friction ($\delta$). More realistic for rough walls and sloping backfills.

  • Limitations: Assumes planar failure surface (not always true), requires iterative solution for $\theta$ (angle of failure plane), more complex.

4.4 Graphical Methods

• Culmann's Graphical Method:

  • Used for active pressure with sloping backfill and surcharge.

  • Construction:

    1. Plot backfill surface.

    2. Draw failure planes at various angles from toe.

    3. For each plane, compute weight of wedge ($W$) and its angle ($\alpha$) to vertical.

    4. Plot $W$ vector from a pole, then draw line parallel to failure plane.

    5. Intersection of this line with $$\displaystyle K_a $$ lines (from Coulomb) gives active force $$\displaystyle P_a $$ for that plane.

    6. Maximum $$\displaystyle P_a $$ is the active thrust.

  • Result: Gives magnitude and direction of $$\displaystyle P_a $$.

4.5 Numerical Problems

• Steps for Thrust Calculation:

  1. Determine $$\displaystyle K_a $$ or $$\displaystyle K_p $$ (Rankine for simple cases, Coulomb/Culmann for complex).

  2. Stratified Backfill: Calculate pressure at interface using $$\displaystyle K_a $$ of upper layer, then use $$\displaystyle K_a $$ of lower layer for pressure increment. Plot step diagram.

  3. With Water Table: Use $\gamma'$ below WT, add water pressure separately.

  4. With Surcharge ($q$): Add uniform pressure $$\displaystyle qK_a $$.

  5. Total Thrust ($P$): Area of pressure diagram.

  6. Point of Application: Centroid of diagram (for triangular: $H/3$ from base; for trapezoidal: use composite areas).

• Active Pressure on Rigid Wall with Tension Cracks (c-φ soil):

  • Tension crack depth $$\displaystyle z_t = \frac{2c}{\gamma \sqrt{K_a}} $$ (Rankine).

  • Pressure diagram starts from $$\displaystyle z_t $$, not surface. Total thrust = area of triangle from $$\displaystyle z_t $$ to $H$.

4.6 Retaining Walls

• Modes of Shear Failure:

  1. Overturning: Wall rotates about toe due to moment from lateral earth pressure. Check: $$\displaystyle FOS_{OT} = \frac{\sum Resisting\;Moment}{\sum Overturning\;Moment} \geq 1.5 $$.

  2. Sliding: Wall slides horizontally along base. Check: $$\displaystyle FOS_{SL} = \frac{\mu \sum V}{P_a} \geq 1.5 $$, where $$\displaystyle \mu = \tan \delta_{base} $$.

  3. Bearing Capacity Failure: Soil beneath base fails in shear. Check net pressure distribution: $$\displaystyle q_{max} \leq q_{allow} $$, $$\displaystyle q_{min} \geq 0 $$ (no tension).

• Sheet Piles vs. Retaining Walls:

  • Sheet Piles: Thin, interlocking sections (steel, vinyl, wood). Flexible, cantilever or anchored. Used for temporary excavations, waterfront structures, cofferdams. Bending resistance primary.

  • Retaining Walls: Massive, rigid structures (gravity, cantilever, counterfort). Weight provides stability. Used for permanent slopes, road cuts.

• Design Considerations: Stability checks (OT, SL, bearing capacity), drainage (weep holes), weep holes, frost protection.

5.0 SPECIAL SOILS & SOIL IMPROVEMENT

5.1 Problematic Soils

• Expansive Soils (Black Cotton Soil):

  • Characteristics: High montmorillonite clay content, high shrink-swell potential, low strength when wet, cracks on drying.

  • Problems: Heave/frost damage to foundations, slabs, pavements; differential movement.

  • Preventive Measures:

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

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

    • Replacement: Remove and replace with non-expansive fill.

    • Chemical Stabilization: Lime, cement treatment.

    • Raft Foundations: Spread load to reduce pressure.

• Collapsible Soils:

  • Characteristics: Loose, dry, metastable structure (e.g., loess, deposited fills). Sudden collapse upon wetting or loading.

  • Problems: Sudden settlement, damage to foundations, utilities.

  • Preventive Measures:

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

    • Compaction: Dynamic compaction, roller compaction.

    • Deep Foundations: Piles through collapsible zone.

    • Chemical Stabilization: Lime, cement.

5.2 Geosynthetics

• Types & Functions:

5.3 Soil Stabilization

• Need: Improve strength, reduce compressibility, control swell/shrink, increase durability.

• Methods:

  • Mechanical: Compaction (increases density, reduces voids).

  • Chemical:

    • Lime: Best for high-plasticity clays. Reduces plasticity, increases strength via pozzolanic reactions.

    • Cement: For granular soils & low-plasticity clays. Increases strength, reduces permeability.

    • Bitumen: For waterproofing and binding (road subgrades).

  • Electrical: Electro-osmosis. Direct current passed through saturated clay, causing water migration from anode to cathode. Used for dewatering and consolidation of very soft clays.

5.4 Compaction

• Field Compaction Equipment:

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

  • Sheepsfoot Rollers: Cohesive soils, deep compaction.

  • Pneumatic Rollers: Granular & cohesive, uniform pressure.

  • Vibratory Rollers: Granular soils, high density.

• Proctor Tests:

  • Standard Proctor (Light): 2.5 kg hammer, 305 mm drop, 3 layers, 25 blows/layer. Energy ≈ 600 kN-m/m³.

  • Modified Proctor (Heavy): 4.5 kg hammer, 457 mm drop, 5 layers, 25 blows/layer. Energy ≈ 2700 kN-m/m³.

  • Comparison: Modified gives higher maximum dry density (MDD) and lower optimum moisture content (OMC) due to higher compaction energy.

6.0 SETTLEMENT & MISCELLANEOUS TOPICS

6.1 Floating Foundations & Rafts

• Floating Foundation: Foundation placed at depth where net increase in vertical stress from structure equals weight of excavated soil. Net settlement ideally zero. Used for very soft clays.

• Raft Foundation: Large slab covering entire footprint. Used when:

  • Soil bearing capacity low.

  • Loads heavy/unequal.

  • Strata weak/uneven.

• Proportioning (Raft): Typically, length/width ratio between 1 to 5. Depth determined by shear and bending moment criteria. Often combined with piles (piled raft).

6.2 Types of Footings

• Isolated Spread: For individual columns.

• Combined: For two or more columns.

• Strap: Connects isolated footing to column with high eccentricity.

• Mat/Raft: Covers entire area.

6.3 Essential Differences: Boussinesq vs. Westergaard

• Boussinesq (1885):

  • Assumes isotropic, homogeneous, elastic half-space.

  • Point load at surface.

  • Stress distribution: Continuous, decreases with depth and radial distance.

  • Formula: $$\displaystyle \sigma_z = \frac{3P}{2\pi} \frac{z^3}{(r^2+z^2)^{5/2}} $$.

  • Applicability: General soils, most common.

• Westergaard (1938):

  • Assumes soil is elastic but with vertical, non-intersecting sheets (like stratified rock/clay).

  • Stress distribution: Confined to vertical planes, zero lateral stress.

  • Formula: $$\displaystyle \sigma_z = \frac{P}{\pi z^2} \frac{1}{(1 + 2(r/z)^2)^{3/2}} $$.

  • Applicability: Highly stratified soils (clay laminations). Gives lower vertical stress at depth compared to Boussinesq.

6.4 Well Foundations

• Components with Sketch:

  1. Well Curb: Bottom conical/curved cutting edge (steel/iron). Facilitates sinking.

  2. Steining: Vertical wall (brick/masonry/concrete). Provides weight and structural stability.

  3. Well Cap: Top RCC slab. Transfers load from pier to well.

  4. Sand Filling: Inside well for balance and to reduce differential settlement.

  5. Pneumatic Caisson: (Special type) Working chamber under pressure.

• Sinking Process: Excavation inside, self-weight or loading causes sinking. Trimmed to design level, then sealed and filled.

\boxed{\text{KEY TAKEAWAY: Focus on problem-solving for SPT corrections, bearing capacity (with water table), pile group capacity (3x3, block failure), active earth pressure (stratified backfill), and negative skin friction. These are the most recurring 7-mark questions.}}