Stress distribution in soils is crucial for understanding how loads affect the ground beneath structures. Boussinesq's and Westergaard's theories provide key insights into this process, helping engineers predict soil behavior under various loading conditions.

These theories use mathematical models to calculate stresses at different points in the soil. While they have limitations, they form the foundation for many geotechnical engineering applications, from foundation design to settlement analysis.

Stress Distribution in Soils

Fundamental Principles of Boussinesq's and Westergaard's Theories

• Boussinesq's theory assumes soil as elastic, homogeneous, isotropic, and semi-infinite medium
  • Enables calculation of stresses at any point within soil mass due to surface loads
  • Utilizes concept of point loads and integrates their effects for complex loading scenarios (distributed loads, foundation pressures)
• Westergaard's theory considers soil as layered elastic medium with horizontal planes of zero extensibility
  • Suitable for analyzing stress distribution in stratified soils
  • Introduces influence factor to account for rigid layers or bedrock at shallow depths
• Both theories based on principles of elasticity and use mathematical equations to predict stress distribution patterns
• Vertical stress distribution typically follows bulb-shaped pattern
  • Stress intensity decreases with depth and lateral distance from load application point
• Theories provide foundation for understanding stress transfer mechanisms in soils
  • Crucial for geotechnical engineering applications (foundation design, settlement analysis)

Mathematical Formulations and Applications

• Boussinesq's equation calculates vertical stress at any point in soil mass
  • Uses load magnitude, depth, and radial distance from load application point
  • Applies principle of superposition for multiple point loads
• Line loads treated as series of closely spaced point loads
  • Integration techniques derive equations for stress distribution beneath infinite and finite line loads
• Uniformly distributed loads over circular or rectangular areas use influence factors and charts
  • Simplifies stress calculations at various depths and lateral distances
• 2:1 method provides simplified approach for estimating stress increase under corner of uniformly loaded rectangular area
• Stress isobars plotted using Boussinesq's equations visualize stress distribution pattern
• Westergaard's theory uses dimensionless influence factor
  • Depends on ratio of foundation width to depth of rigid layer or bedrock
  • Provides equations for vertical stresses beneath center and corner of rigid circular and rectangular foundations
• Westergaard's influence charts determine stress distribution factors for various foundation geometries and depth-to-width ratios

Limitations and Considerations

• Accuracy of Boussinesq's theory decreases with depth, particularly for loads over large areas
• Both theories assume linear elastic behavior of soils
  • May not accurately represent stress-strain relationship of real soils (higher stress levels, highly plastic soils)
• Neither theory accounts for soil-structure interaction effects
  • Can significantly influence stress distribution for large or heavily loaded foundations
• Groundwater and pore water pressure influences on stress distribution neglected
  • Significant in saturated or partially saturated soils
• Accuracy decreases with increasing load intensity and foundation size
  • Assumptions of elasticity and small strains become less valid

Boussinesq's Theory Applications

Point and Line Load Analysis

• Point load stress calculation uses Boussinesq's equation
  • Determines vertical stress at any soil point based on load magnitude, depth, and radial distance
• Multiple point loads analyzed through superposition principle
  • Individual stress contributions summed to find total stress at given point
• Line loads treated as series of closely spaced point loads
  • Integration techniques derive stress distribution equations for infinite and finite line loads
• Examples:
  • Single point load from heavy machinery (crane outrigger)
  • Multiple point loads from building columns
  • Line load from continuous wall footing

Distributed Load Analysis

• Uniformly distributed loads over circular or rectangular areas utilize influence factors and charts
  • Simplifies stress calculations at various depths and lateral distances
• 2:1 method estimates stress increase under corner of uniformly loaded rectangular area
  • Provides quick approximation for preliminary design calculations
• Stress isobars plotted to visualize stress distribution pattern
  • Useful for identifying zones of high stress concentration
• Examples:
  • Raft foundation supporting a large storage tank
  • Parking lot pavement design
  • Embankment loading on soft soil

Westergaard's Theory for Foundations

Rigid Foundation Analysis

• Introduces dimensionless influence factor based on foundation width to rigid layer depth ratio
• Provides equations for vertical stresses beneath center and corner of rigid foundations
  • Applicable to circular and rectangular foundation shapes
• Westergaard's influence charts simplify stress distribution factor determination
  • Accounts for various foundation geometries and depth-to-width ratios
• Examples:
  • Machine foundation on layered soil with shallow bedrock
  • Bridge pier foundation in varved clay deposits
  • Wind turbine foundation on stratified soil profile

Layered Soil Considerations

• Accounts for horizontal planes of zero extensibility
  • Relevant for analyzing stress distribution in varved clays or thinly bedded sedimentary deposits
• Predicts more concentrated stress distribution patterns beneath loaded area
  • Especially noticeable for foundations with large width-to-depth ratios
• Useful for analyzing stress distribution in soils with distinct layering
  • Applicable when rigid stratum present at relatively shallow depth beneath foundation
• Examples:
  • Foundation design in glacial till with alternating clay and sand layers
  • Stress analysis for deep foundation in interbedded shale and limestone
  • Shallow foundation on soil profile with stiff clay layer over soft clay

Boussinesq vs Westergaard Theories

Assumptions and Soil Models

• Boussinesq's theory assumes homogeneous, isotropic, and semi-infinite elastic medium
  • Suitable for deep, uniform soil deposits (thick alluvial deposits)
• Westergaard's theory considers layered elastic medium with horizontal zero extensibility planes
  • Better suited for stratified soils or situations with shallow bedrock (sedimentary rock formations)
• Both theories assume linear elastic soil behavior
  • May not accurately represent real soil stress-strain relationships (highly plastic clays, loose sands)

Stress Distribution Patterns

• Westergaard's theory predicts higher stresses directly beneath loaded area
  • Lower stresses at greater depths compared to Boussinesq's theory
• Boussinesq's theory shows more gradual stress dissipation with depth
  • Wider stress bulb compared to Westergaard's predictions
• Examples:
  • Comparison of stress distributions under a large storage tank foundation in homogeneous sand (Boussinesq) vs layered clay-sand profile (Westergaard)
  • Stress analysis for a highway embankment on uniform soft clay (Boussinesq) vs varved clay deposit (Westergaard)

Practical Limitations

• Neither theory accounts for soil-structure interaction effects
  • Significant for large or heavily loaded foundations (high-rise buildings, bridge abutments)
• Groundwater and pore water pressure influences neglected
  • Important in saturated or partially saturated soils (coastal areas, regions with high water table)
• Accuracy decreases with increasing load intensity and foundation size
  • Assumptions of elasticity and small strains become less valid (heavy industrial structures, large dams)
• Examples:
  • Limitations in predicting stress distribution under a tall building foundation in saturated clay
  • Inaccuracies in stress analysis for a large earth dam on heterogeneous foundation soil