UNIT 4: Foundation Aspects for Bridge Engineering

1.0 Soil Exploration and Subsurface Investigation

1.1 Significant Depth of Exploration & Borehole Depth Criteria (IS Code)

• Significant Depth: Depth up to which stresses due to foundation load are significant (typically where Δσ/σ'₀ ≤ 10%).

• IS 1892 (2016) Criteria for Borehole Depth:

  • For isolated spread footings: Depth = Width of footing or up to hard stratum.

  • For raft foundations: Depth = Width of raft or up to hard stratum.

  • For pile foundations: Depth = Length of pile + 3m to 5m (to check for pile toe bearing stratum and lateral variation).

  • Minimum depth: 3m to 5m to account for seasonal moisture variation and disturbance.

  [!TIP] Exam Focus: Always state IS code (1892) and relate depth to foundation type and stress influence.

1.2 Methods of Boring/Hole Advancement

[!TIP] Rotary Drilling is the most versatile and preferred for detailed investigation, especially for obtaining undisturbed samples.

1.3 In-Situ Testing

1.3.1 Standard Penetration Test (SPT)

• Procedure: Drive a split spoon sampler (50mm ID, 60mm OD) 450mm into soil at bottom of borehole using a 63.5kg hammer falling 760mm. Count blows for each 150mm penetration. N-value = blows for last 300mm (300-450mm).

• Corrections to N-value:

  1. Overburden Pressure Correction (N₁): Normalize to 1 ton/ft² (≈100 kN/m²) effective overburden pressure.

$$N_1 = N \times \left( \frac{\bar{\sigma}'_v}{100 \text{ kN/m}^2} \right)^{0.5}$$

2.  **Dilatancy Correction (N₂)**: For dense saturated fine sands/gravels (N > 15). Corrects for negative pore pressure.

$$N_2 = 15 + \frac{1}{2}(N_1 - 15) \quad \text{for } N_1 > 15$$

3.  **Energy Correction (N₆₀)**: Corrects to standardized 60% hammer energy (E₆₀). Field energy ratio (ER) is measured.

$$N_{60} = N \times \frac{E_R}{60}$$

**Corrected N-value (N_corr)**: Apply in order: N → N₁ → (N₂ if needed) → N₆₀.

• Significance: Empirical correlations for:

  • Relative density of sands.

  • Unconfined compressive strength of clays (qu ≈ 12N for remolded clay).

  • Settlement & bearing capacity estimates.

  [!TIP] Common Pitfall: Forgetting sequence of corrections. Always correct for overburden first.

1.3.2 Cone Penetration Test (CPT) & Seismic CPT (SCPT)

• CPT: Pushes a 60° cone (10 cm² area) at 20 mm/s. Measures tip resistance (qc) and sleeve friction (fs) continuously. Provides friction ratio (Rf = fs/qc × 100%).

• SCPT: Adds a geophone behind cone to measure shear wave velocity (Vs) → estimates small-strain modulus (Gmax).

• Advantages: Continuous profile, rapid, no soil disturbance, quantitative soil profiling.

1.3.3 Plate Load Test

• Setup: Load a rigid plate (0.3m x 0.3m sq. or 0.3m dia.) at foundation depth. Apply load in increments, measure settlement.

• Interpretation: Plot load-settlement curve. Ultimate bearing capacity (qu) from limit state (large settlement) or log-log method.

• Extrapolation for Different Footing Sizes (Terasaki's Method):

  For cohesionless soils:

$$q_{u2} = q_{u1} \left( \frac{B_2}{B_1} \right)$$

For cohesive soils (clay):

$$s_2 = s_1 \left( \frac{B_1}{B_2} \right) \left( \frac{1 + \frac{B_2}{B_f}}{1 + \frac{B_1}{B_f}} \right)$$

Where, B = footing width, Bf = width of plate (0.3m), s = settlement.

> [!TIP] **Key Formula**: For **clay**, settlement ∝ 1/B. For **sand**, bearing capacity ∝ B.

1.4 Soil Sampling

1.4.1 Disturbed vs. Undisturbed Samples

1.4.2 Sampling Tube Parameters

• Inside Clearance (Ci): (Di - Do)/Do. Allows sample expansion. Optimal: 0.5% - 1.5%.

• Outside Clearance (Co): (Do - Dc)/Dc. Reduces friction during driving. Optimal: 1% - 2%.

• Area Ratio (Ar): (Do² - Di²)/Di² × 100%. Should be < 10% for undisturbed. Lower = less disturbance.

  [!TIP] Exam Question: Given tube dimensions, calculate Ci, Co, Ar. Comment: Low Ar & proper Ci/Co = good undisturbed sample.

1.4.3 CNS (Constant Normal Stiffness) Layer

• Concept from critical state soil mechanics. In a triaxial test with constant confining pressure stiffness (K), the sample simulates behavior in a layer with constant lateral stress increment (like under a flexible footing). Useful for understanding settlement in stiff over soft clay.

1.5 Geophysical Methods of Exploration

• Seismic Refraction/Reflection: Measures seismic wave velocities → estimates stratum depth, rock quality.

• Electrical Resistivity: Measures soil resistivity → identifies stratification, groundwater.

• Magnetic/Gravity: Detect large anomalies (boulders, voids).

• Use: Quick, economical for large areas; supplements boreholes.

1.6 Bore-log Report Preparation & Presentation

• Standard Format (IS 1892):

  1. Project details, borehole location & elevation.

  2. Drilling method, sampler type, hammer details.

  3. Stratigraphic column: Depth, description, classification, SPT N-value, sample type & number.

  4. Groundwater table depth.

  5. Laboratory test results summary.

• Graphical Presentation: Depth on vertical axis, soil layers, SPT N-values, water table, lab test results plotted horizontally.

2.0 Shallow Foundations

2.1 Bearing Capacity

2.1.1 Definitions

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

• Net Ultimate Bearing Capacity (qnu): qu - γDf (Df = foundation depth).

• Net Safe Bearing Capacity (qns): qnu / FOS.

• Allowable Bearing Pressure (qa): qns or pressure limited by settlement criteria.

2.1.2 Modes of Shear Failure

1. General Shear: Dense sand/clay. Continuous failure surface to surface. Sharp peak, large settlement. Most critical.

2. Local Shear: Medium dense sand/medium clay. Failure surfaces develop only under footing. Lower peak, moderate settlement.

3. Punching Shear: Very loose sand/soft clay. Soil punches into footing without distinct failure surface. No peak, large settlement.

2.1.3 Theories & Factors

• Terzaghi's Theory (1943): Assumptions: Strip footing, depth/width ≤ 1, φ>0, rigid base, no shear above base.

$$q_u = c'N_c + q N_q + 0.5 γ B N_γ$$

(For φ=0°: `q_u = 5.7c + γDf`)

• IS Method (IS 6403): Uses shape factors (si), depth factors (di), load inclination factors (qi).

$$q_u = c' N_c s_c d_c i_c + q N_q s_q d_q i_q + 0.5 γ B N_γ s_γ d_γ i_γ$$

**Shape Factors** (for rectangular footing):

$$s_c = 1 + 0.2 \frac{B}{L}, \quad s_q = 1 + 0.1 \frac{B}{L}, \quad s_γ = 1 - 0.3 \frac{B}{L}$$

• Bearing Capacity Factors (Nc, Nq, Nγ): Functions of φ' (effective friction angle). Tabulated or from equations.

  [!TIP] Key Values: For φ'=0°, Nc=5.7 (Terzaghi), Nq=1, Nγ=0. For φ'=30°, Nc≈37, Nq≈22, Nγ≈20.

2.1.4 Factors Affecting Bearing Capacity

• Soil: c', φ', γ (unit weight).

• Foundation: Width B, Depth Df, Shape (B/L), Inclination of load.

• Groundwater: Submergence, seepage.

• Method of construction.

2.1.5 Effect of Water Table

• Case 1 (Far below): Use γ below base = γ_sub (for third term only).

• Case 2 (At ground level): Use γ = γ_sub for all terms above base, q = γDf (use γ_sub if Df submerged).

• Case 3 (At foundation level): Use γ = γ_sub for third term only; q = γDf (use γ_sub if Df submerged).

• Correction Factor (w): Often applied to third term: w = 0.5 if water table at base, w = 1 if far below.

2.1.6 Bearing Capacity for Different Shapes & Load Inclination

• Shape Factors (s_c, s_q, s_γ): As above (IS Method). Circular: s_c=1.3, s_q=1.2, s_γ=0.8.

• Load Inclination Factors (i_c, i_q, i_γ): Reduce capacity for eccentric/horizontal loads.

$$i_q = \left(1 - \frac{H}{V + A' c' \cot \phi'}\right)^m, \quad m = \frac{2 + \frac{B}{L}}{1 + \frac{B}{L}}$$

(H=horizontal, V=vertical, A'=effective area)

2.1.7 Bearing Capacity under Rapid Loading

• c-φ soils (clay): Rapid loading → undrained conditions (φ_u=0°, c_u).

  • Immediate: Use undrained parameters (φ_u=0, c_u). Nc=5.7 (Terzaghi) or 5.14 (Skempton for deep clays).

  • Long-term: Drained conditions (φ', c'). Use effective stress parameters.

• Sands: Rapid loading increases capacity slightly due to negative pore pressure ( dilatancy ).

2.2 Settlement of Foundations

2.2.1 Components

1. Immediate (Elastic) Settlement (Si): Due to shear distortion at constant volume. Occurs during/just after construction in cohesionless & cohesive soils.

2. Primary Consolidation Settlement (Sc): Due to expulsion of pore water from saturated clay under increased load. Time-dependent.

3. Secondary Compression (Creep) Settlement (Ss): Due to plastic adjustment of clay skeleton after primary consolidation. Long-term.

2.2.2 Immediate Settlement Calculation (Cohesive Soils)

• Elastic Theory:

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

Where:

*   q = net contact pressure

*   B = footing width

*   ν = Poisson's ratio

*   E_s = Modulus of elasticity of soil (from lab or correlations)

*   **I_z = Influence factor** (from charts/tables, depends on L/B, Df/B). For square footing, I_z ≈ 1.0 - 1.2.

> [!TIP] **Exam Formula**: Always `S_i = (qB/E_s) * I_z * (1-ν²)`. Use given I_z.

2.2.3 Elastic Settlement (General)

• Same formula as above. For sands, E_s is pressure-dependent (use E_s at working stress).

2.2.4 Settlement from Plate Load Test Extrapolation

• For cohesive soils: Settlement ∝ 1/√(Area) or ∝ 1/B.

$$S_{footing} = S_{plate} \times \frac{B_{plate}}{B_{footing}}$$

• For cohesionless soils: Settlement ∝ log(B) or use Burland's correlation.

2.3 Types of Footings & Proportioning

• Spread/Isolated: Under columns. Proportion: B ≥ (P/qa)^0.5; depth ≥ 0.5m; provide minimum steel.

• Combined: Under two closely spaced columns.

• Raft/Mat: Under multiple columns or weak soil. Proportioning: Thickness based on shear (one-way/two-way) and moment; reinforcement for flexure.

• Floating: Excavated soil weight ≈ structure weight. Net increase in vertical stress ≈ 0.

• Basic Criteria: Adequate bearing capacity, acceptable total & differential settlement, structural integrity, constructability.

2.4 Field Compaction & Proctor Tests

[!TIP] Key Point: Heavy Proctor gives lower OMC and higher MDD than Light Proctor. Bridge foundations typically require heavy compaction for subgrade.

3.0 Deep Foundations (Piles & Wells)

3.1 Pile Foundations: Definition, Types & Functions

3.1.1 Classification

• By Material: Timber, Concrete (RCC/Precast), Steel, Composite.

• By Action:

  • End-bearing: Transfer load to hard stratum.

  • Friction/Skin Friction: Transfer via shaft friction.

  • Combined: Most common.

• By Installation:

  • Driven: Precast (displacement, non-displacement).

  • Cast-in-situ: Bored, simplex, Franki (with bulb).

  • Bored/Drilled: Non-displacement, use casing/mud.

3.1.2 Under-reamed Piles

• Concept: Single/multiple bulb-like enlargements (under-reams) along shaft in expansive soils.

• Suitability Criteria:

  • Expansive soils (high swell potential).

  • Loose/medium dense soils below active zone.

  • Depth of active zone < 3-4m.

• Components: Shaft, under-ream bulb(s), pile cap. Bulb diameter ≈ 2-3× shaft dia.

• Function: Provide tension capacity to resist uplift due to swelling; increase shaft friction & end bearing.

3.2 Pile Load Carrying Capacity

3.2.1 Static Load Carrying Capacity

• Ultimate Capacity (Qu): Qu = Qb + Qs (End bearing + Skin friction).

• Safe Load (Qs): Qs = Qu / FOS (FOS=2.5-3 for static loads).

3.2.2 Methods of Estimation

1. Static Formulae (Based on soil parameters):

   • Clay (α-method): Qs = α c_u A_s (α = adhesion factor, 0.5-1.0). Qb = N_c c_u A_b (N_c=9 for deep).

   • Sand (β-method): Qs = β σ'_v0 K tanδ A_s (β=1 for deep, δ≈φ'/2). Qb = N_q σ'_v0 A_b (N_q from bearing capacity).

2. Dynamic Formulas (From pile driving records):

   • Engineering News Record (ENR):

$$Q_{safe} = \frac{W H}{S + C} \times \frac{W + n W_p}{W + W_p} \times \frac{1}{FOS}$$

    Where W=hammer weight, H=drop, S=settlement/blow, C=constant (2.5cm for drop hammer), W_p=pile weight, n=efficiency (0.6-0.8).

> [!TIP] **Dynamic formulas are empirical & conservative**. Use for **driven piles only** during installation.

3.2.3 Single Pile Capacity in Sand & Clay

• Sand (Given φ, γ, D):

  • Qb = N_q γ D A_b (D = depth to base, N_q from charts)

  • Qs = K γ D A_s (K ≈ 1-2, or use β-method)

• Clay (Given c_u, γ):

  • Qb = 9 c_u A_b (for deep, saturated clay)

  • Qs = α c_u A_s (α from table: soft clay=1.0, stiff=0.5)

3.2.4 Pile Capacity in Layered Soils

• Skin friction in layer i: f_i = α_i c_ui (clay) or f_i = K_i σ'_vi tanδ_i (sand).

• Total Qs = Σ (f_i × perimeter × thickness_i).

• End bearing: Use c_u or N_q of bearing stratum only. Ignore weak layers above.

3.3 Pile Groups

3.3.1 Geometrical Properties & Spacing

• Spacing (s): Center-to-center. Minimum: s ≥ 2D (clay), s ≥ 3D (sand) to avoid group effect. Optimal: s = 3D to 4D.

• Group Efficiency (η): η = (Q_ug) / (n × Q_us).

  • For friction piles in clay: η ≈ 1.0 (if s≥3D).

  • For end-bearing piles: η ≈ 1.0.

  • For friction piles in sand: η < 1.0 (due to overlapping stress bulbs).

3.3.2 Group Capacity vs. Sum of Individual Capacities

• Block Failure (clay, close spacing): Group fails as a single unit. Capacity = c N_c A_block + perimeter × length × α c.

• Individual Failure (wide spacing): Each pile fails individually. Capacity = n × Q_us.

• General: For intermediate spacing, capacity is between block and individual failure.

3.3.3 Calculation of Pile Group Capacity (Neglecting/Considering End Bearing)

• Neglecting End Bearing (Friction piles in thick clay):

  Q_ug = α c_u (A_s)_group (use perimeter of group block × length).

• Considering End Bearing:

  Q_ug = α c_u (A_s)_group + N_c c_u (A_b)_group (A_b = area of group block).

3.3.4 Negative Skin Friction (NSF)

• Concept: Downward drag on pile due to settling soil around it (e.g., fill, consolidating clay, lowering water table).

• Calculation for Single Pile:

  NSF force (P_d) = f_d × A_s (f_d = drag unit friction, often = γ × K × tanδ or use α c_u if clay).

  • Depth of NSF zone: From ground to neutral plane (where pile settlement = soil settlement).

• For Pile Group: Use perimeter of group block for A_s.

• Effect: Increases load on pile → reduces net capacity. Must be added to working load for design.

  [!TIP] Critical: Net Ultimate Capacity = Qu - P_d. Always check NSF in fill over soft clay or drawdown conditions.

3.4 Well Foundations (Caissons)

3.4.1 Components & Construction Stages (Sinking)

• Components:

  1. Well curb (bottom cutting edge, usually concrete/steel).

  2. Well steining (wall above curb, masonry/concrete).

  3. Shoring (inside bracing).

  4. Well cap (top).

• Construction Stages:

  1. Sinking: Excavate inside, allow self-weight to sink. Trim bottom, maintain verticality.

  2. Plugging: Bottom plugged with concrete after reaching final depth.

  3. Capping: Construct pile cap/column base on top.

3.4.2 Design Considerations for Bridge Piers

• Depth of Well: Below maximum scour depth + adequate grip length in firm soil/rock.

• Diameter: Based on load, soil bearing, construction constraints. Typically 3-6m.

• Material: Masonry (traditional), RCC (modern).

• Forces: Vertical load, lateral load (earthquake, wind, water), moments.

• Stability Check: Buoyancy (if submerged), floating during sinking.

• Scour Protection: Riprap around well top.

4.0 Earth Pressure and Retaining Structures

4.1 Lateral Earth Pressure: Types

• At-rest (K₀): Wall does not move laterally. K₀ = 1 - sinφ' (for normally consolidated clay/sand).

• Active (Ka): Wall moves away from soil. Minimum pressure. Wall pressure < at-rest.

• Passive (Kp): Wall moves into soil. Maximum pressure. Wall pressure > at-rest.

  K_a = tan²(45° - φ'/2), K_p = tan²(45° + φ'/2) (Rankine, for φ'>0).

4.2 Earth Pressure Theories

4.2.1 Rankine's Theory (1875)

• Assumptions:

  • Semi-infinite soil mass.

  • Wall frictionless (δ=0).

  • Vertical wall face.

  • Horizontal backfill surface.

  • Soil homogeneous, isotropic, obeys Mohr-Coulomb.

• Active Pressure (c-φ soil):

$$σ_a = γ z K_a - 2c \sqrt{K_a} \quad \text{(at depth z)}$$

**Passive Pressure**:

$$σ_p = γ z K_p + 2c \sqrt{K_p}$$

• With Water Table: Use submerged unit weight below WT + pore pressure (u) separately. Total pressure = effective + u.

4.2.2 Coulomb's Theory (1776)

• Assumptions:

  • Rough wall (δ > 0 possible).

  • Inclined wall face (β) & backfill slope (β).

  • Failure plane is planar, makes angle θ with horizontal.

  • Wedge in limit equilibrium.

• Expression:

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

(For δ=0, β=0 → reduces to Rankine).

• Culmann's Graphical Method:

  1. Draw backfill slope.

  2. From wall toe, draw trial failure planes at angle φ' to horizontal.

  3. For each plane, compute weight (W) of wedge.

  4. Draw line parallel to wall face at angle δ to normal.

  5. Intersection gives lateral force (P) for that plane.

  6. Envelope of all P gives pressure diagram. Max P = active thrust.

4.2.3 Comparison: Rankine vs. Coulomb

[!TIP] Merit of Coulomb: Accounts for wall friction (δ) and inclined backfill → more accurate for bridge abutments with sloping backfill.

4.3 Earth Pressure Calculations for Complex Conditions

4.3.1 Effect of Water Table

• Rise to Ground Level: Use submerged unit weight (γ_sub) for soil above base, add pore water pressure (u = γ_w z) separately.

• Total Thrust = Thrust from submerged soil + Uplift force from water.

• Pressure at depth z: σ = γ_sub z K_a + γ_w z (if WT at surface).

4.3.2 Effect of Surcharge Load (q)

• Uniform surcharge (e.g., traffic, strip load):

  • Active: Additional pressure = q K_a (uniform with depth).

  • Passive: Additional pressure = q K_p (uniform with depth).

• Line load/point load: Use influence charts or 2:1 distribution.

4.3.3 Stratified Backfills

• Calculate pressure at interface from top layer.

• Use this pressure as overburden for bottom layer.

• Compute pressure diagram layer by layer.

4.3.4 Tension Cracks in Clay Backfill

• Occurs in cohesive soils (c>0) when active pressure becomes negative near surface.

• Depth of tension crack (z₀):

$$z_0 = \frac{2c}{\gamma \sqrt{K_a}}$$

• Design implication: Top part may crack → consider reduced cohesion or no tension in analysis.

4.3.5 Passive Earth Pressure Calculation

• Use Coulomb with δ = φ' (full friction) for maximum passive resistance.

• For bridge abutments, passive resistance from approach fill is often ignored or reduced (due to compaction, future excavation).

4.4 Retaining Wall Design & Analysis

4.4.1 Total Thrust, Magnitude, and Point of Application

• Active Thrust (Pa):

  • For c-φ soil with horizontal backfill:

$$P_a = \frac{1}{2} γ H^2 K_a - 2c H \sqrt{K_a}$$

    (H = height of wall)

*   **Point of Application**: From base, `H/3` (triangular) or `H/3(2K_p/K_a - 1)/(K_p/K_a - 1)` for Coulomb.

• Passive Thrust (Pp): At base, Pp = (1/2)γ H^2 K_p + 2c H √K_p. Acts at H/3 from base.

• Water Pressure: Uplift force = (1/2) γ_w H^2 at H/3 from base.

4.4.2 Pressure Distribution Diagrams

• Rankine (c-φ): Linear effective pressure diagram, shifted by 2c√K_a at surface.

• Coulomb: Linear for φ>0, but inclined if wall friction considered. Pressure at surface may not be zero if backfill slope.

4.4.3 Modes of Failure of Retaining Walls

1. Overturning: Moment about toe > stabilizing moment. Check factor of safety against overturning ≥ 1.5.

2. Sliding: Horizontal thrust > frictional resistance. FS_sliding = (μ W) / P_a ≥ 1.5 (μ = tanδ).

3. Bearing Capacity Failure: Excessive pressure on soil. Check max pressure < allowable.

4. Structural Failure: Wall stem/base bending, shear.

4.5 Sheet Piles vs. Retaining Walls

[!TIP] Key Difference: Sheet piles are flexible, interlocking, used for containment; Retaining walls are rigid, massive, used for support.

5.0 Special Soil Types & Ground Improvement

5.1 Problematic Soils

5.1.1 Expansive Soils

• Characteristics: High montmorillonite clay content. High shrink-swell potential with moisture change.

• Swelling Potential: Due to adsorbed water uptake. Measured by free swell index, ** swell pressure**.

• Problems in Foundations:

  • Heave in wet season → cracking of slabs, tilting.

  • Shrink-swell cycles → differential movement.

  • Loss of strength when wet.

5.1.2 Collapsible Soils

• Characteristics: Loess, wind-blown deposits. Porous, cemented (calcite/silica), low moisture.

• Collapse Mechanism: Upon wetting + loading → cement bonds break → sudden reduction in volume.

• Problems: Sudden, large settlements under wetting (e.g., due to rain, pipe leak, groundwater rise).

5.1.3 Problems & Preventive Measures

5.2 Soil Stabilization Techniques

• Need: Improve strength, reduce swell/shrink, increase CBR, reduce permeability.

• Situations: Poor natural soils, expansive/collapsible soils, re-use of excavated material.

• Methods:

  1. Mechanical: Compaction, blending with good soil.

  2. Chemical: Lime, cement, fly ash, bitumen. Mechanism: Cation exchange, pozzolanic reactions.

  3. Electrical: Electro-osmosis (for clay dewatering/consolidation). Apply DC voltage → water moves to anode → consolidation.

5.3 Geosynthetics in Ground Improvement

5.3.1 Types

5.3.2 Functions (Detailed)

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

2. Reinforcement: Tensile element to carry load, improve stability (geogrids in walls).

3. Filtration: Allow water flow but retain soil particles (non-woven geotextiles).

4. Drainage: Transmit water/fluids (geonets, geocomposites).

5. Protection: Protect geomembranes from puncture (geotextile cushion).

6. Erosion Control: Temporary/permanent slope protection (geocells, mats).

5.3.3 Uses in Foundation Engineering

• Raft/Mat foundations over weak soil: Reinforcement (geogrid layers) to distribute load, reduce settlement.

• Pile caps: Separation from weak soil.

• Retaining walls: Reinforcement (geogrid wrap-around), drainage (geocomposite).

• Slope stabilization: Reinforcement (geogrids) to increase factor of safety.

• Construction platforms: Reinforcement + separation over soft ground for equipment.

6.0 Stress Distribution & Settlement Theories

6.1 Stress Distribution in Soil Mass

6.1.1 Boussinesq's Theory (1885)

• Assumptions:

  • Homogeneous, isotropic, elastic half-space.

  • Soil obeys Poisson's ratio (ν).

  • Point load at surface.

  • No shear strength, only normal stresses.

• Vertical Stress Increase (Δσ_z) under point load Q at depth z, radial distance r:

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

• Stress Isobars: Curves of equal Δσ_z. 2:1 Distribution Method: Approximate method where stress spreads at 2V:1H. Simple for rectangular loads.

  [!TIP] 2:1 Method: Δσ = Q / [(B+z)(L+z)] at depth z. Overestimates stress near surface, underestimates at depth.

6.1.2 Westergaard's Theory (1938)

• Assumptions:

  • Soil is incompressible (ν=0) in vertical direction.

  • Layered medium with horizontal, non-communicating layers (like rock strata).

  • Point load.

• Vertical Stress Increase:

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

• Key Difference from Boussinesq:

  • Westergaard: Stress confined to vertical column above load (due to incompressibility). Lower Δσ at large r.

  • Boussinesq: Stress spreads laterally (due to ν>0).

  • Westergaard is more appropriate for stratified soils (clay layers, rock).

6.2 Application in Settlement & Bearing Capacity

6.2.1 Use of Influence Charts (Newmark's)

• Construction: Based on Boussinesq equation. Each chart for specific ν.

• Procedure:

  1. Plot foundation plan to scale on transparent sheet.

  2. Place over influence chart (center at point of interest).

  3. Count squares inside foundation plan.

  4. Δσ = q × I × (number of squares / total squares in chart).

• Advantage: Handles any shape foundation, multiple loads.

6.2.2 Stress Increase under Foundation for Settlement Analysis

• Settlement calculation requires Δσ at various depths.

• Method:

  1. Compute Δσ at midpoint of each compressible layer (using Boussinesq, 2:1, or influence chart).

  2. Use consolidation theory for clays: Sc = (H / (1+e₀)) × (Cc / (1+eo)) × log(σ'₀ + Δσ / σ'₀).

  3. For sands, use elastic settlement formula with E_s at appropriate Δσ.

• Key Point: For clay layers, Δσ must be computed accurately at mid-depth of each layer for consolidation settlement.

[!TIP] Exam Application: Given a footing size, load, soil profile → compute Δσ at clay layer midpoint → compute Sc. Always state method used (Boussinesq/2:1).