Engineering Hydrology (CE-802 (A)) - Unit 4 Short Notes

UNIT 4: FOUNDATION ENGINEERING & SOIL MECHANICS

1.0 SUB-SURFACE INVESTIGATION & SOIL SAMPLING

1.1 Significant Depth of Exploration & IS Criteria

• Definition: Depth up to which soil strata are significantly influenced by foundation loads.

• IS 1892 (Part 1) Recommendations:

  • Depth should be at least equal to the width of the foundation.

  • Should extend up to a competent stratum (e.g., hard rock, dense sand).

  • In case of variable soil, depth should be sufficient to locate weak zones.

  • For tall structures, depth may be 1.5–2 times the width.

• Factors Influencing Depth:

  • Foundation width and load intensity.

  • Soil stratification and properties.

  • Presence of groundwater.

  • Adjacent structures.

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Exam Focus: Remember IS code number (IS 1892) and the primary criterion (depth ≥ width). For layered soils, exploration must penetrate weak layers.

1.2 Boring Methods

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Rotary drilling is most versatile and commonly used for deep exploration with undisturbed sampling.

1.3 Sampling Techniques

• Disturbed Sample: Soil structure altered. Used for classification, water content, density tests.

• Undisturbed Sample: Preserves in-situ structure/moisture. Used for strength, consolidation, permeability tests.

• Sampling Tube Parameters:

  • Inside Clearance (Cᵢ): $$\displaystyle (D_i - d_i)/d_i \times 100\% $$ (typically 0.5–1.5%). Allows sample expansion.

  • Outside Clearance (Cₒ): $$\displaystyle (D_o - d_o)/d_o \times 100\% $$ (typically 0–0.5%). Reduces friction.

  • Area Ratio (AR): $$\displaystyle (D_o^2 - D_i^2)/D_i^2 \times 100\% $$ (should be < 10% for good quality).

  • Sample Quality: Lower AR and appropriate clearances give better undisturbed samples.

• Coring: Use of diamond core barrel for rock/stiff soils; provides high-quality core.

• CNS Layer (Constant Volume Sampling): Layer at tube bottom where soil volume remains constant during sampling; indicates good sample recovery.

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Key Formula: Area Ratio $$\displaystyle AR = \frac{D_o^2 - D_i^2}{D_i^2} \times 100\% $$. For thin-walled tube, $$\displaystyle D_o \approx D_i $$, so AR small.

1.4 In-situ Testing

• Standard Penetration Test (SPT):

  • Procedure: Drive split spoon sampler 450 mm (last 300 mm recorded) by 65 kg hammer falling 750 mm. Count blows for 150 mm intervals.

  • N-value: Blows for last 300 mm penetration (standardized).

  • Corrections:

    1. Overburden Pressure Correction (N₁): $$\displaystyle N_1 = N \times \sqrt{\frac{\sigma'_v}{100}} $$ (for $$\displaystyle \sigma'_v $$ in kPa) or $$\displaystyle N_1 = N \times 0.77 \log_{10}\left(\frac{2.73 + \sigma'_v}{2.73}\right) $$ (Terzaghi & Peck).

    2. Dilatancy Correction (N₂): For saturated fine sands/silts, $$\displaystyle N_2 = 15 + \frac{1}{2}(N_1 - 15) $$ if $$\displaystyle N_1 > 15 $$.

    3. Energy Correction (N₆₀): $$\displaystyle N_{60} = N \times \frac{E_R}{60} $$ where $$\displaystyle E_R $$ is actual hammer energy ratio (%). Standard is 60% energy.

  • Need: To compare N-values from different overburden pressures, energy, and saturation conditions.

• Cone Penetration Test (CPT):

  • Equipment: Pushed at 20 mm/s, measures tip resistance ($$\displaystyle q_c $$), sleeve friction ($$\displaystyle f_s $$), pore pressure ($$\displaystyle u_2 $$).

  • Advantages over SPT: Continuous profile, faster, quantitative, less operator-dependent, measures pore pressure (CPTu).

• Vane Shear Test: For soft clays; measures undrained shear strength ($$\displaystyle c_u $$).

• Plate Load Test:

  • Procedure: Load plate (usually 0.3 m²) incrementally, measure settlement.

  • Settlement-Load Curve: Determine ultimate bearing capacity ($$\displaystyle q_u $$) from failure.

  • Modulus of Elasticity ($$\displaystyle E_s $$): From initial linear portion: $$\displaystyle E_s = \frac{(1-\mu^2) q B}{s} I_f $$ (inverse of immediate settlement equation).

  • Size Effect: Use size effect charts to extrapolate from 0.3 m plate to larger footing: $$\displaystyle s_{footing} = s_{plate} \times \frac{B_{plate}}{B_{footing}} \times \text{chart factor} $$.

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SPT Corrections Order: Always correct N to N₁ (overburden) first, then N₂ (dilatancy), then N₆₀ (energy). For clays, only overburden correction may apply.

1.5 Bore-log Report

Components:

1. Project details, location, borehole number/depth.

2. Soil description (color, consistency, stratification).

3. Water table depth.

4. Sample details (type, depth, recovery).

5. Test results (SPT N-values, lab tests).

6. Graphical representation: depth vs. soil layer, N-value, water content, etc.

2.0 SHALLOW FOUNDATIONS: BEARING CAPACITY & SETTLEMENT

2.1 Key Definitions

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

• Net Ultimate Bearing Capacity ($$\displaystyle q_{nu} $$): $$\displaystyle q_{nu} = q_u - \gamma D_f $$ (excludes overburden).

• Net Safe Bearing Capacity ($$\displaystyle q_{ns} $$): $$\displaystyle q_{ns} = \frac{q_{nu}}{FOS} $$.

• Allowable Bearing Pressure ($$\displaystyle q_a $$): Pressure used for design; includes settlement criteria.

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Boxed Relationship: $$\displaystyle q_a \leq q_{ns} $$ (strength) and $$\displaystyle q_a \leq q_{allow} $$ (settlement).

2.2 Modes of Shear Failure

2.3 Theories of Bearing Capacity

• Terzaghi’s Theory (1943):

  • Assumptions: Strip footing, rough base, $c-\phi$ soil, failure surface as shown.

  • Equation:

$$q_u = c N_c + \gamma D_f N_q + 0.5 \gamma B N_\gamma$$

    Shape factors:  

    - Square: $$\displaystyle q_u = 1.3c N_c + \gamma D_f N_q + 0.4 \gamma B N_\gamma $$  

    - Circular: $$\displaystyle q_u = 1.3c N_c + \gamma D_f N_q + 0.3 \gamma B N_\gamma $$  

*   **Bearing Capacity Factors ($$\displaystyle N_c, N_q, N_\gamma $$):** Functions of $\phi$ (use tables).  

• BIS (IS 6403) Method:

  • General Equation:

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

    where $s$ = shape factor, $d$ = depth factor, $i$ = inclination factor.  

*   **Water Table Correction:**  

    - If water table above base: use submerged unit weight ($\gamma'$) for $$\displaystyle \gamma B N_\gamma $$ term and $$\displaystyle \gamma D_f $$ term.  

    - If within B: apply reduction factor $$\displaystyle r_w $$ (from IS 6403).  

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Common Mistake: Forgetting to apply water table correction to both $$\displaystyle \gamma D_f $$ and $$\displaystyle \gamma B N_\gamma $$ terms when water table is between $$\displaystyle D_f $$ and $$\displaystyle D_f+B $$.

2.4 Bearing Capacity Calculations

• Steps:

  1. Identify soil parameters ($c$, $\phi$, $\gamma$).

  2. Determine $$\displaystyle N_c, N_q, N_\gamma $$ from tables.

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

  4. Compute $$\displaystyle q_u $$, then $$\displaystyle q_{nu} $$, $$\displaystyle q_{ns} $$, $$\displaystyle q_a $$.

• Numerical Focus: Mixed $c-\phi$ soils, water table at various levels, different footing shapes.

2.5 Components of Settlement

1. Immediate (Elastic) Settlement ($$\displaystyle S_i $$): Due to shear strain; occurs rapidly.

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

3. Secondary Compression Settlement ($$\displaystyle S_s $$): Due to rearrangement of soil skeleton after primary consolidation; long-term.

2.6 Immediate Settlement Calculation

• Theory: Elastic half-space, vertical stress increase from loaded area.

• Equation:

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

where $q$ = net pressure, $B$ = footing width, $\mu$ = Poisson’s ratio, $$\displaystyle E_s $$ = modulus of elasticity, $$\displaystyle I_f $$ = influence factor (from charts, depends on $L/B$, $$\displaystyle D_f/B $$).  

• For Cohesive Soils ($$\displaystyle \phi=0 $$): $$\displaystyle I_f $$ can be taken as 1.0 for practical purposes if $$\displaystyle D_f/B \leq 1 $$.

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Boxed Formula: $$\displaystyle S_i = \frac{q B (1 - \mu^2)}{E_s} I_f $$. Use $$\displaystyle I_f $$ from Janbu’s chart or Steinbrenner’s for rectangular areas.

2.7 Types of Shallow Foundations

• Isolated footing, combined footing, strap footing, raft/mat foundation, floating foundation.

• Selection Criteria:

  • Load magnitude and distribution.

  • Soil bearing capacity and settlement potential.

  • Depth to competent stratum.

  • Adjacent structures.

2.8 Raft Foundation

• Proportioning: Usually covers entire building area; thickness designed for shear and bending.

• Bearing Capacity: Check overall $$\displaystyle q_u $$ for raft area; often enhanced due to distributed load.

3.0 DEEP FOUNDATIONS (PILE FOUNDATIONS)

3.1 Classification of Piles

3.2 Load Carrying Capacity of Single Pile

• Static Formulae:

  • For Sandy Soils ($$\displaystyle \phi > 0 $$):

$$Q_u = A_b \cdot \gamma D_f N_q + \gamma D_f K \tan \delta \cdot A_s$$

    or simplified: $$\displaystyle Q_u = A_b q_{ub} + A_s f_s $$ where $$\displaystyle f_s = K \sigma'_{vm} \tan \delta $$.  

*   **For Clayey Soils ($$\displaystyle \phi=0 $$):**  

    - **α-method:** $$\displaystyle Q_u = \alpha c_u A_s + A_b N_c c_u $$ (α depends on $$\displaystyle c_u $$ and pile length).  

    - **c_u method:** $$\displaystyle Q_u = c_u A_s + A_b N_c c_u $$ (for soft clays, α=1).  

• Dynamic Methods (Drop Hammer):

$$Q_d = \frac{W h}{s + e} \cdot \frac{W + n P}{W + P}$$

where $W$ = hammer weight, $h$ = fall, $s$ = final set, $e$ = elastic compression, $n$ = coefficient of restitution (0.25–0.3), $P$ = pile weight.  

Allowable load: $$\displaystyle Q_a = \frac{Q_d}{FOS} $$.  

• Field Methods:

  • Load Test (Compression/Tension): Most reliable. Apply load incrementally, measure settlement. Determine $$\displaystyle Q_u $$ from failure or settlement criteria.

3.3 Pile Group Capacity

• Group Efficiency (η): $$\displaystyle \eta = \frac{Q_{ug}}{n Q_{us}} $$ (usually < 1 for friction piles in clay).

• Clay (Friction Piles, Neglecting End Bearing):

$$Q_{ug} = \gamma D_f B^2 \cdot \frac{N_c}{2} \cdot \tan \theta \cdot \left[1 + \frac{B}{2L \tan \theta}\right]$$

or simpler: $$\displaystyle Q_{ug} = \alpha \cdot \text{area of group} \cdot \text{average } c_u \cdot \text{shape factor} $$.  

• Numerical: For square group, $$\displaystyle Q_{ug} = \alpha \cdot (n \cdot A_s) \cdot \bar{c}_u \cdot \text{adhesion factor} $$ (if clay uniform).

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Key Point: In clay, group capacity < sum of individual capacities due to overlapping stress zones. For end-bearing piles in sand/rock, group capacity ≈ sum.

3.4 Negative Skin Friction (NSF)

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

• Causes:

  • Consolidation of soft clay under fill.

  • Loose soils undergoing wetting/compaction.

  • Lowering of water table.

• Calculation:

  • Single Pile: $$\displaystyle Q_{nsf} = \gamma' \cdot K \cdot \tan \delta \cdot A_s \cdot H_{ns} $$ (for cohesive soils) or $$\displaystyle f_s \cdot A_s $$ where $$\displaystyle f_s $$ from lab tests.

  • Pile Group: Consider group as a block; $$\displaystyle Q_{nsf} = \gamma' \cdot \tan \phi \cdot \text{perimeter} \cdot \text{depth} \cdot \text{width} $$.

3.5 Under-reamed Piles

• Components: Shaft, under-ream bulb(s) (enlarged base).

• Suitability: Expansive soils, loose sands, soft clays; provides uplift resistance.

• Ultimate Tensile Capacity:

$$Q_{tu} = \alpha \cdot c_u \cdot A_s + N_q \cdot c_u \cdot A_b$$

where $$\displaystyle A_b $$ = area of under-ream bulb(s), $\alpha$ = adhesion factor (0.3–0.6), $$\displaystyle N_q $$ = bearing capacity factor for bulb.  

3.6 Well Foundations (Caissons)

Components (with sketch):

1. Well Curb: Bottom cutting edge, conical.

2. Well Steining: Vertical wall above curb.

3. Cutting Edge: Bottom edge for sinking.

4. Bottom Plug: Temporary, removed after sinking.

5. Top Plug: Supports well cap.

6. Well Shaft: Main body.

4.0 LATERAL EARTH PRESSURE & RETAINING STRUCTURES

4.1 Types of Lateral Earth Pressure

• At Rest ($$\displaystyle K_0 $$): No lateral strain.

• Active ($$\displaystyle K_a $$): Wall moves away from soil; minimum pressure.

• Passive ($$\displaystyle K_p $$): Wall moves into soil; maximum pressure.

4.2 Earth Pressure Theories

• Rankine’s Theory (1875):

  • Assumptions:

    • Semi-infinite soil mass.

    • Wall smooth, vertical.

    • Backfill horizontal, cohesionless (extended to cohesive).

    • Failure plane inclined at $$\displaystyle (45^\circ + \phi/2) $$.

  • For Cohesionless Soil ($$\displaystyle c=0 $$):

$$K_a = \tan^2\left(45^\circ - \frac{\phi}{2}\right), \quad K_p = \tan^2\left(45^\circ + \frac{\phi}{2}\right)$$

*   **For Cohesive Soil ($$\displaystyle c>0 $$):**  

$$\sigma_a = \gamma z K_a - 2c \sqrt{K_a} \quad (\text{active})$$

$$\sigma_p = \gamma z K_p + 2c \sqrt{K_p} \quad (\text{passive})$$

*   **Water Table:** Use submerged unit weight below WT; add water pressure separately.  

• Coulomb’s Theory (1776):

  • Assumptions:

    • Rigid failure wedge.

    • Wall friction angle ($\delta$) considered.

    • Backfill inclined at $\beta$.

  • Expression (Active):

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

*   **Comparison with Rankine:**  

    - Coulomb considers wall friction ($\delta$) and inclined backfill ($\beta$).  

    - Rankine assumes $$\displaystyle \delta=0 $$, $$\displaystyle \beta=0 $$.  

    - Coulomb $$\displaystyle K_a $$ < Rankine $$\displaystyle K_a $$ when $$\displaystyle \delta>0 $$.  

• Culmann’s Graphical Method:

  • For non-horizontal backfill with surcharge.

  • Draw failure wedges at various angles, compute weight and lateral force, envelope gives pressure diagram.

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Rankine vs Coulomb: Use Rankine for simple cases (smooth wall, horizontal backfill). Use Coulomb when wall friction or sloping backfill exists. Coulomb is more general but iterative.

4.3 Earth Pressure Calculations

• Numerical Steps:

  1. Determine $$\displaystyle K_a $$ or $$\displaystyle K_p $$ (Rankine/Coulomb).

  2. Compute vertical stress $$\displaystyle \sigma_v = \gamma z + q $$ (surcharge).

  3. Calculate lateral pressure $$\displaystyle \sigma_h = K \sigma_v \pm $$ cohesion term.

  4. For water table: add water pressure separately; use $\gamma'$ below WT.

  5. For seepage: apply flow net or use $$\displaystyle \gamma_{sat} $$ below WT with $u$ term.

4.4 Retaining Wall Design & Analysis

• Types:

  • Gravity: Mass concrete, relies on weight.

  • Cantilever: Base slab + stem; economical up to ~6 m.

  • Sheet Pile: Flexible, used for temporary/excavation; not a retaining wall per se.

• Stability Checks:

  1. Overturning: $$\displaystyle \frac{\text{Resisting Moment}}{\text{Overturning Moment}} \geq 1.5 $$ (typical).

  2. Sliding: $$\displaystyle \frac{\text{Resisting Force}}{\text{Driving Force}} \geq 1.5 $$; include base friction and shear key.

  3. Bearing Capacity: Check max and min pressures at base: $$\displaystyle q_{max/min} = \frac{P}{B} \left(1 \pm \frac{6e}{B}\right) $$; $$\displaystyle q_{max} \leq q_{allow} $$.

• Earth Pressure Distribution:

  • For vertical back with horizontal backfill: triangular (Rankine active).

  • With surcharge: trapezoidal (uniform pressure + triangular).

  • With tension crack (cohesive soils): pressure zero at top up to depth $$\displaystyle z_c = \frac{2c}{\gamma \sqrt{K_a}} $$.

4.5 Modes of Failure of Retaining Walls

1. Overturning: Wall rotates about toe.

2. Sliding: Wall slides along base.

3. Excessive Bearing Pressure: Uneven settlement or bearing capacity failure.

4. Structural Failure: Crushing of masonry, tensile cracking in concrete.

5.0 SPECIAL SOILS & SOIL IMPROVEMENT

5.1 Problematic Soils

• Expansive Soils:

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

  • Problems: Heave in wet season, settlement in dry season; damage to foundations, pavements, pipelines.

  • Preventive Measures:

    • Moisture control (impermeable blanket, drainage).

    • Under-reamed piles (tension capacity).

    • Lightweight materials (filled slabs).

    • Soil replacement/stabilization.

• Collapsible Soils:

  • Characteristics: Loose, metastable structure (e.g., loess), sudden collapse upon wetting or loading.

  • Problems: Sudden settlement, differential settlement.

  • Preventive Measures:

    • Pre-wetting before construction.

    • Compaction (dynamic/static).

    • Replacement with good fill.

    • Pile foundations.

5.2 Soil Stabilization

5.3 Geosynthetics

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Key Application: Geotextiles for separation (prevent mixing of subgrade and ballast), geogrids for reinforcement (increase bearing capacity).

6.0 STRESS DISTRIBUTION & SETTLEMENT THEORIES

6.1 Vertical Stress Distribution

• Boussinesq’s Theory (1885):

  • Assumptions: Homogeneous, isotropic, semi-infinite elastic solid; point load at surface.

  • Equation for Point Load:

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

    where $r$ = radial distance, $z$ = depth.  

*   **Influence Charts:** **Newmark’s influence chart** for rectangular/square loaded areas.  

• Westergaard’s Theory (1938):

  • Assumptions: Soil with vertical cracks/laminae; incompressible material; vertical stress only.

  • Equation:

$$\sigma_z = \frac{P}{\pi z^2} \cdot \frac{1}{(1 + 2(r/z)^2)^{3/2}}$$

*   **Comparison with Boussinesq:**  

    - Westergaard gives **lower vertical stress** at a given point because of vertical cracks restricting lateral strain.  

    - Boussinesq assumes isotropic; Westergaard for stratified soils.  

• 2:1 Stress Distribution Method:

  • Approximate: Stress spreads at 2V:1H from loaded area.

  • $$\displaystyle \sigma_z = \frac{q \cdot B \cdot L}{(B + z)(L + z)} $$ for rectangular area.

6.2 Immediate Settlement Revisited

• Use Boussinesq/Westergaard to compute vertical stress increase ($\Delta \sigma$) at depth of influence (usually $B/2$ for clays).

• Then apply elastic theory with strain influence factor ($$\displaystyle I_s $$):

$$S_i = \frac{\Delta \sigma}{E_s} (1 - \mu^2) I_s B$$

where $$\displaystyle I_s $$ depends on $L/B$, $$\displaystyle D_f/B $$.  

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When to use which theory? Boussinesq for homogeneous, non-layered soils; Westergaard for laminated/clayey soils with vertical cracks.