Bridge Engineering (CE-802 (C)) - Unit 2 Short Notes

UNIT 2: FOUNDATION ENGINEERING (Bridge Engineering Focus)

I. SUBSURFACE INVESTIGATION AND SOIL SAMPLING

Objectives & Planning

• Significance: Determines soil profile, strength, and compressibility for safe, economical bridge foundation design.

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

  • Depth: Boreholes should penetrate to a significant depth where the stress increase from the foundation is ≤ 10% of the existing overburden pressure. For bridges, often extend to bedrock or a competent stratum.

  • Spacing: Typically 30-50m for bridges, reduced to 10-15m under heavily loaded piers/abutments.

• Significant Depth of Exploration (Dₛ): Depth at which vertical stress increase (Δσ) due to foundation load equals 10% of initial vertical stress (σ₀'). For a square footing:

$$ D_s \approx \frac{B}{2} \text{ to } B \quad (\text{where } B = \text{footing width}) $$

> [!TIP] For bridge piers (large loads), Dₛ is greater; often exploration goes 1.5-2 times the footing width or to refusal.

Boring and Drilling Methods

Soil Sampling Techniques

• Disturbed Sample: Soil structure altered. Used for classification, moisture content, Proctor tests.

• Undisturbed Sample: Preserves in-situ structure & moisture. Essential for strength (UU, CU, CD tests) & consolidation tests.

• Sampling Tube Specifications:

  • Inside Clearance (Cᵢ): Gap between sample & tube inner wall. Allows sample expansion. Cᵢ = (Dᵢ - Dₛ)/Dₛ × 100% (Dᵢ = inside dia., Dₛ = sample dia.). Typical: 0.5-1.5%.

  • Outside Clearance (Cₒ): Gap between tube outer & borehole wall. Cₒ = (Dₕ - Dₒ)/Dₒ × 100% (Dₕ = borehole dia., Dₒ = tube outer dia.). Typical: 1-2%.

  • Area Ratio (Aᵣ): Ratio of annular area to core area. Aᵣ = (Dₒ² - Dᵢ²)/Dᵢ² × 100%. Should be < 10% (preferably < 6%) for undisturbed quality.

  [!TIP] High area ratio or low inside clearance causes sample disturbance. Comment: "Sample disturbed due to high area ratio (Aᵣ = X%)" is a common exam remark.

• CNS Layer: Critical Neutral Stress Layer. Depth where change in vertical effective stress (Δσ') is zero. Important for settlement calculations in layered soils.

In-Situ Testing

• Standard Penetration Test (SPT)

  • Procedure: Drive a split spoon sampler (50mm ID, 65mm OD) 450mm into soil at borehole bottom in three 150mm increments using a 63.5kg hammer falling 760mm. Count blows for last 300mm → N-value (blows/300mm).

  • Significance of N-value: Empirical index of soil density (sands) or consistency (clays). Correlates with φ, relative density, undrained shear strength (cᵤ), modulus.

  • Corrections & Need:

    1. Overburden Pressure Correction (K₀-correction): N-values increase with depth due to σ'₀. Correct to standard 1 ton/ft² overburden. N_corr = N_obs × (σ'₀ / 0.1 MPa)^0.5 (for σ'₀ in MPa).

    2. Dilatancy Correction (for Dense Sands/Stiff Clays): Corrects for excess resistance due to soil heaving in sampler. Applied when uncorrected N > 15 in fine sands/silty sands.

    3. Energy Correction: Modern practice corrects to 60% hammer energy (N₆₀).

    [!TIP] Always mention: "N-value must be corrected for overburden pressure to compare with empirical correlations developed at standard energy & pressure."

  • Sketch: Shows split spoon, hammer, anvil, drill rods, borehole.

• Cone Penetration Test (CPT/SCPT)

  • Test: Push a 60° cone (10 cm² area) with friction sleeve into soil at 20 mm/s. Measures tip resistance (qᶜ) and sleeve friction (fₛ) continuously.

  • Differentiation SPT vs SCPT:

    | Feature | SPT | SCPT | | :--- | :--- | :--- | | Output | Discrete N-value (every 1.5m) | Continuous profile (qᶜ, fₛ, pore pressure u₂) | | Disturbance | High (boring, driving) | Very low (continuous push) | | Data | Empirical correlations | Direct, quantitative soil profiling | | Cost | Low | High |

• Plate Load Test

  • Setup: Load plate (0.3m² typical) at foundation depth, apply load in increments, measure settlement.

  • Interpretation: Plot load-settlement curve. Ultimate bearing capacity (qᵤ) from failure point or defined settlement (e.g., 25mm).

  • Settlement Prediction & Scaling: For cohesive soils, settlement ∝ 1/√(area) or 1/B.

$$ S_{2m} = S_{0.3m} \times \sqrt{\frac{B_{0.3}}{B_{2}}} = S_{0.3m} \times \sqrt{\frac{0.3}{2}} \approx S_{0.3m} \times 0.387 $$

    > [!TIP] **For sands:** Settlement ∝ 1/B. **For clays:** Settlement ∝ 1/√B (or 1/√area).

Reporting

• Bore-log Report Components:

  1. Borehole location & elevation.

  2. Soil strata description (depth, thickness, color, consistency, classification).

  3. Groundwater table depth.

  4. SPT N-values (with corrections noted).

  5. Sample type & depth (U, D).

  6. Laboratory test results (moisture, density, strength).

  7. Graphical Log: Standardized columns with soil symbols, SPT blows, sample depths.

  DiagramSEARCH: "standard borehole log diagram IS code"

II. SHALLOW FOUNDATIONS

Types and Selection

• Types: Isolated, Combined, Strap, Raft (Floating), Mat.

• Basic Criteria for Satisfactory Performance:

  1. Adequate bearing capacity (no shear failure).

  2. Tolerable settlement (total & differential).

  3. Structural integrity of footing itself.

• Factors Affecting Selection: Load magnitude/distribution, soil profile (depth to bedrock, bearing capacity), settlement potential, construction feasibility, cost.

Bearing Capacity

• Key Definitions:

  • Gross Pressure (q): Total load / area.

  • Net Pressure (q_net): q - γD_f (D_f = depth).

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

  • Net Ultimate Bearing Capacity (qₙᵤ): qᵤ - γD_f.

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

  • Allowable Bearing Pressure (qₐₗₗ): Usually = qₛₐfₑ (or includes settlement check).

• Theories & Methods:

  • Terzaghi's Theory (1943): Assumptions: Strip footing, φ > 0, soil above base has same φ, base rough, load vertical, no tension in soil.

    • Equations:

      • Strip: qᵤ = cN_c + γD_f N_q + 0.5 γ B N_γ

      • Square: qᵤ = 1.3cN_c + γD_f N_q + 0.4 γ B N_γ

      • Circular: qᵤ = 1.3cN_c + γD_f N_q + 0.3 γ B N_γ

    • BIS (IS) Method (IS 6403): Uses shape factors (sᵢ), depth factors (dᵢ), inclination factors (iᵢ). More general.

      qᵤ = cN_c s_c d_c i_c + γD_f N_q s_q d_q i_q + 0.5 γ B N_γ s_γ d_γ i_γ

      [!TIP] IS method factors: s_c = 1 + 0.2(B/L) for rectangular footing; d_c, d_q, d_γ from depth ratio D_f/B; i factors for load inclination.

  • Factors Affecting Bearing Capacity (Correction Factors):

    • Shape (sᵢ): Square/rectangular vs strip.

    • Depth (dᵢ): Increases capacity due to confinement.

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

    • Base Inclination (bᵢ): Reduces capacity for sloping base.

    • Water Table (wᵢ): Affects effective unit weights. If WT at base: use γ' below base. If WT above base: use γ' for entire depth, add water pressure term.

• Modes of Shear Failure (with Sketches):

  1. General Shear: In dense sands/ stiff clays. Continuous failure surface to surface. Distinct peak, large settlement.

  2. Local Shear: In medium-dense soils. Failure surfaces develop only under footing. No surface heave, moderate settlement.

  3. Punching Shear: In loose sands/ soft clays. Failure surfaces confined under footing. Very large settlement, no distinct peak.

  DiagramCANVAS: Three sketches showing failure surfaces for general, local, and punching shear under a footing.

Settlement

• Components:

  1. Immediate (Elastic) Settlement (Sᵢ): Due to shear distortion, occurs during/soon after construction. Recoverable partially.

  2. Primary Consolidation Settlement (S_c): Due to expulsion of pore water from saturated clays. Time-dependent, large in clays.

  3. Secondary Consolidation Settlement (Sₛ): Due to plastic adjustment of soil skeleton after primary consolidation. Long-term in clays/peats.

• Immediate Settlement Calculation (Cohesive Soils):

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

Where: q = net pressure, B = footing width, ν = Poisson's ratio, Eₛ = Modulus of elasticity, I_f = **Influence factor** (from charts/tables, e.g., 1.06 for 2m x 3m rect. footing at D_f/B=0.5).

> [!TIP] **For granular soils:** Use elastic theory with Eₛ from correlation with N-value.

Special Cases & Applications

• Floating/Raft Foundation: Foundation placed at depth where net increase in vertical stress = weight of excavated soil. Essentially, foundation "floats" on soil. Used when allowable bearing capacity is very low.

  • Proportioning: Area such that total load = (γ_soil × depth of excavation) × area. Reduces net pressure significantly.

• Effect of Water Table:

  • Far below: Use total unit weight (γ) throughout.

  • At Ground Level: Use submerged unit weight (γ') for soil below G.L., add water pressure term γ_w D_f to qᵤ equation.

  • At Foundation Base: Use γ for soil above base, γ' for soil below base.

III. DEEP FOUNDATIONS (PILES)

Classification and Functions

• By Material: RCC, Steel, Timber, Composite.

• By Installation:

  • Driven: Precast, displaced soil (low permeability soils).

  • Bored/Drilled: Cast-in-situ, minimal disturbance (sensitive soils, near structures).

• By Function:

  • End-bearing: Transfer load to hard stratum.

  • Friction: Transfer load via skin friction along shaft.

  • Combined: Both end-bearing & friction (common).

  • Tension/Uplift: Resist uplift forces (e.g., bridge decks on slopes).

  • Lateral Load: Resist horizontal forces (abutments, seismic).

• Functions in Bridges: Support heavy pier/abutment loads, transfer through weak strata to bearing stratum, resist lateral & uplift forces.

Load Carrying Capacity of Single Pile

• Static Load Approach:

  • Clay (α-method): Qᵤ = α cₐ Aₛ + qₚ Aₚ

    • α = adhesion factor (0.5-1.0, decreases with depth/sensitivity).

    • cₐ = average adhesion along shaft = α × cᵤ (undrained shear strength).

    • Aₛ = shaft area, Aₚ = base area, qₚ = net base resistance (≈ 9 cᵤ for soft clays, N_c cᵤ).

  • Sand (β-method): Qᵤ = γ D f Aₛ + qₚ Aₚ

    • β = factor (≈ K₀ tanφ, 0.2-0.4).

    • f = average effective vertical stress over shaft depth.

    • qₚ = q N_q (effective overburden at base × bearing capacity factor).

• Dynamic Load Approach (ENR Formula):

$$ Q_{all} = \frac{W h}{S + 0.1} \times \frac{W + n W_s}{W + W_s} \times \frac{1}{FOS} $$

Where: W = hammer weight, h = effective fall, S = final set (penetration per blow, cm), Wₛ = pile weight, n = coefficient of restitution (0.3-0.5), FOS = 6-8 for driven piles.

> [!TIP] **ENR is empirical, conservative.** Set (S) is measured after initial driving (elastic compression included).

Pile Groups

• Group Efficiency (η): Ratio of group capacity to sum of individual capacities. η < 1 due to overlap of stress zones.

• Block Failure Concept: For closely spaced piles in clay, failure may occur as a single block with perimeter (n×s) and base (n²×Aₚ). Capacity:

  Q_group(block) = cₐ × (n²Aₚ + n×s×L) + γD_f × (n²Aₚ) (if end-bearing considered).

• Capacity of Pile Group: Usually lesser of:

  1. Sum of individual pile capacities (η × n² × Q_single).

  2. Block failure capacity.

  [!TIP] For soft clays with low adhesion: Group capacity ≈ sum of individual capacities (η ≈ 1). For dense sands/ stiff clays: η < 1, block failure governs.

• Example (3x3 group in clay, no end-bearing):

  Q_ug = α cₐ × (Perimeter of block × L) = α cₐ × (3×3 piles × 3 sides? Wait, block perimeter = 2×(3s+3s) if s=spacing? Actually: For 3x3 square group, block width = 3s (s=spacing). Perimeter = 4 × (3s). But piles only on perimeter? No, block failure considers entire block soil.

  Correct: Q_block = cₐ × (Area_base + Perimeter × L) but if no end-bearing, only skin: Q_block = cₐ × (Perimeter_block × L). For 3x3 with spacing s, block perimeter = 2×(3s + 3s) = 12s. So Q_ug = α cₐ × 12s × L. But individual sum = 9 × (α cₐ × πD × L). Compare.

  Past Paper Example: Given diameter 0.3m, L=10m, c=70 kN/m², α=0.6, spacing=0.9m.

  Individual sum = 9 × (0.6×70 × π×0.3 × 10) = 9 × (0.6×70×9.4248) = 9 × 395.84 = 3562.6 kN.

  Block perimeter = 4 × (3×0.9) = 10.8 m. Q_block = 0.6×70 × 10.8 × 10 = 0.6×70×108 = 4536 kN.

  Group capacity = min(3562.6, 4536) = 3562.6 kN. Allowable = 3562.6 / 2.5 = 1425 kN.

Negative Skin Friction (NSF)

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

• Causes: Fill consolidation, lowering water table, collapsible soils, soft clay consolidation.

• Calculation for Single Pile:

  Q_nsf = γ × L_n × πD × K × tanδ

  Where: L_n = length of pile in compressible layer, K = lateral earth pressure coefficient (≈ K₀), δ = pile-soil friction angle (≈ φ for concrete).

  [!TIP] NSF reduces net upward capacity: Q_net_up = Q_skin_up + Q_base - Q_nsf.

• For Pile Group: Consider group as a block. Drag force acts on block perimeter (not individual piles). Q_nsf(group) = γ × L_n × (Perimeter_block) × K × tanδ.

Special Pile Types for Bridges

• Under-reamed Piles

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

  • Components: Shaft, under-reams (bell-shaped), top bulb (optional).

  • Suitability: Expansive soils (swell-shrink). Under-reams provide tensile & uplift resistance by bearing on stable soil below active zone.

  • Ultimate Tensile Capacity (Uplift):

    Q_t = α cₐ Aₛ + q_b A_b (neglect suction & under-ream adhesion as per problem).

    Where: A_b = area of under-ream base, q_b = bearing capacity factor × cᵤ at under-ream level.

• Well Foundations (Caissons)

  • Components with Sketch:

    1. Well Curb: Bottom cutting edge (usually concrete/steel).

    2. Well Steining: Vertical wall (brick/stone/RCC).

    3. Well Cap: Top slab to distribute load.

    4. Curb/Plug: Bottom plug to seal and provide base bearing.

    5. Sand/Water Filling: For stability during sinking.

  • Relevance to Bridges: Used for deep foundations in rivers/waterlogged areas for piers/abutments. Sunk by excavation inside.

  DiagramSEARCH: "well foundation components diagram bridge"

IV. EARTH PRESSURE AND RETAINING STRUCTURES

Types of Lateral Earth Pressure

• At Rest (K₀): No lateral strain. K₀ = 1 - sinφ (for normally consolidated clays).

• Active (Kₐ): Wall moves away from soil → minimum pressure.

• Passive (Kₚ): Wall pushed into soil → maximum pressure.

• Kₐ and Kₚ (Rankine, φ>0):

  Kₐ = tan²(45° - φ/2)

  Kₚ = tan²(45° + φ/2) = 1/Kₐ

Earth Pressure Theories

• Rankine's Theory (1875)

  • Assumptions: Wall smooth & vertical, backfill horizontal, cohesionless (c=0) or homogeneous cohesive, soil mass semi-infinite, failure plane at 45±φ/2.

  • Active/Passive Diagrams:

    • Cohesionless (c=0): Linear from zero at surface (active) or tensile crack depth (active cohesive) to γH Kₐ/Kₚ at base.

    • Cohesive (c>0): Active pressure = γH Kₐ + 2c√Kₐ (uniformly distributed). Tension crack depth: z_c = 2c / (γ √Kₐ).

  • Effect of Water Table: Use submerged unit weight (γ') below WT, add pore pressure diagram (triangular) to total stress diagram.

• Coulomb's Theory (1776)

  • Assumptions: Wall may be rough & inclined, backfill may be sloping, failure plane is planar at angle θ to horizontal. Wedge equilibrium.

  • General Expression (Active):

    Pₐ = (1/2) γ H² Kₐ where Kₐ = [cos²(φ - δ)] / [cos²δ cos(δ+β) (1 + √(sin(φ+δ) sin(φ-β)/cos²(δ+β)) )²]

    (β = backfill slope, δ = wall friction angle).

  • Merits over Rankine:

    1. Accounts for wall friction (δ > 0) → reduces Kₐ.

    2. Applicable for sloping backfills (β > 0).

    3. More realistic for rough walls.

  [!TIP] For smooth wall (δ=0) & horizontal backfill (β=0), Coulomb Kₐ = Rankine Kₐ.

Earth Pressure Calculations (Key Steps)

1. Determine effective stress parameters (φ', c').

2. Find Kₐ (Rankine or Coulomb).

3. Calculate pressure at depth z: σₕ = Kₐ σᵥ' + 2c'√Kₐ (cohesive) or σₕ = Kₐ γ z (cohesionless).

4. For water table: Calculate pore pressure (u) separately. Total pressure = effective + u.

5. For stratified soils: Calculate Kₐ for each layer using its φ'. Pressure at interface must be continuous.

6. For surcharge (q): Add uniform lateral pressure q Kₐ throughout depth.

7. Total Active Thrust (Pₐ): Area under pressure diagram. For linear diagram: Pₐ = (1/2) × base pressure × height. Point of application from base: H/3 (triangular) or at centroid of composite diagram.

Graphical Methods

• Culmann's Graphical Method: For sloping backfill with Coulomb theory. Construct failure planes at various angles, calculate weight of wedge, plot pressure vs. plane angle. Envelope gives pressure diagram.

  DiagramSEARCH: "Culmann's graphical method earth pressure"

Retaining Walls

• Types: Gravity, Cantilever, Counterfort, Sheet pile.

• Distribution of Active Pressure: Triangular (c=0) or trapezoidal (c>0) from base to surface.

• Resultant Thrust: Magnitude = area under diagram. Point of application from base: For triangular, at H/3; for trapezoidal, at H/3 from base for triangular part + c-component acts at H/2.

• Modes of Failure (with Sketches):

  1. Overturning: Moment about toe > resisting moment.

  2. Sliding: Horizontal thrust > frictional resistance (μ × vertical load) ± cohesion.

  3. Bearing Capacity Failure: Excessive pressure under toe or heel (eccentric loading). Check max pressure < allowable.

  DiagramCANVAS: Three sketches showing overturning, sliding, and bearing capacity failure modes for a cantilever retaining wall.

Sheet Piles

• Differentiation: Flexible structural elements (steel, wood, vinyl) driven into ground, rely on lateral resistance of soil for stability. Retaining walls are rigid structures (concrete, masonry) relying on self-weight for stability.

• Uses in Bridge Foundations:

  • Cofferdams: Enclose area for dry construction of piers/abutments in water.

  • Abutment walls: Where space is limited.

  • Temporary shoring during construction.

  • Slope protection in approaches.

V. SOIL IMPROVEMENT AND GEOSYNTHETICS

Soil Stabilization Techniques

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

  • Equipment: Smooth wheel, sheepsfoot, pneumatic tyred, vibratory rollers.

  • Light vs Heavy Proctor: Different energy (Light: 600 kN-m/m³, Heavy: 2700 kN-m/m³). Heavy gives higher dry density, lower OMC.

• Chemical: Lime, cement, bitumen. Improve strength, reduce swell (expansive soils).

• Electrical: Electro-osmosis

  • Process: Insert electrodes into saturated clay, apply DC current. Water migrates from anode to cathode, dewatering clay.

  • Use: Consolidate/strengthen soft saturated clays, especially where conventional drainage is slow.

  [!TIP] Bridge approaches often require stabilization due to weak natural soils or fill materials.

Geosynthetics

• List & Functions:

  | Type | Primary Function(s) | | :--- | :--- | | Geotextiles | Separation, Filtration, Reinforcement, Protection, Drainage | | Geomembranes | Containment (liners), Barrier | | Geogrids | Reinforcement (high tensile strength, open grid) | | Geocells | Confinement, Reinforcement (3D honeycomb) | | Geocomposites | Drainage (geocomposite drains), Reinforcement |

• Detailed Uses in Foundation Engineering:

  • Separation: Between weak subgrade and granular fill/ballast (prevents mixing).

  • Reinforcement: In embankments on soft soils (tensile strength takes load), in retaining walls (wrap-around), in pile caps.

  • Filtration: Replace graded filter layers (geotextiles allow seepage but retain soil).

  • Drainage: Geocomposite drains (geodrains) for vertical/horizontal drainage, accelerate consolidation.

  • Protection: Cover geomembranes/liners.

VI. PROBLEMATIC SOILS

Expansive Soils

• Characteristics: High montmorillonite clay content, high liquid limit (>50%), high shrink-swell potential, low strength when wet, high when dry.

• Problems for Foundations: Heave during wetting (seasonal), shrinkage during drying → differential movement, cracking of foundations, structural damage.

• Preventive Measures:

  1. Moisture Control: Impermeable barriers, landscaping to prevent water infiltration, maintain constant moisture.

  2. Deep Foundations: Piles (especially under-reamed piles) to bypass active zone.

  3. Chemical Stabilization: Lime/cement treatment to reduce swell potential.

  4. Shallow Foundations: Raft foundations to distribute load and accommodate movement.

  5. Geosynthetics: Reinforcement to bridge cracks.

Collapsible Soils

• Characteristics: Metastable structure (e.g., loess), low moisture content, loose/porous, high void ratio, cemented bonds (carbonates, gypsum).

• Problems: Sudden collapse upon wetting or loading → large, differential settlement, foundation failure.

• Preventive Measures:

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

  2. Dynamic Compaction: Densify soil by dropping weight.

  3. Replacement: Excavate and replace with granular fill.

  4. Chemical Stabilization: Cement, lime.

  5. Deep Foundations: Piles to penetrate collapsible zone.

Sampling Challenges

• Area Ratio (Aᵣ): High Aᵣ (>10%) causes sample disturbance (compression, shear). Leads to underestimated strength in lab tests.

• Inside Clearance (Cᵢ): Too small (<0.5%) → sample jams, gets compressed. Too large (>1.5%) → sample twists, disturbs structure.

• Outside Clearance (Cₒ): Too small → friction, sample compressed. Too large → sample may not enter tube cleanly.

  [!TIP] Comment on sample quality: "Sample is disturbed due to high area ratio (Aᵣ = X%) and inadequate inside clearance (Cᵢ = Y%). Laboratory strength tests will yield conservative (lower) values."

VII. STRESS DISTRIBUTION IN SOIL MASSES

Boussinesq's Theory (1885)

• Assumptions: Homogeneous, isotropic, elastic half-space. Load applied at surface as point load (Q). Soil obeys Hooke's law, Poisson's ratio ν.

• Vertical Stress under Point Load:

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

Where: z = depth, r = radial distance.

• Influence Charts: Based on theory, for various load shapes (e.g., Newmark's chart).

• Applicability: Best for saturated clays (ν ≈ 0.5) and dense sands. Overestimates stress in loose sands.

Westergaard's Theory (1938)

• Assumptions: Soil contains numerous incompressible horizontal sheets (like rock layers). Soil is elastic, but lateral strain prevented (ν' = 0).

• Vertical Stress under Point Load:

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

• Applicability: Better for stratified soils (clay seams in sand) or rock masses. Gives lower stresses than Boussinesq at same (r,z).

Comparison

[!TIP] For bridge foundations in layered soils: Westergaard is often more appropriate. For settlement calculation in clay: Boussinesq with ν=0.5 is standard.