Boussinesq assumed a homogeneous, isotropic, elastic, semi-infinite soil mass. For a point load $Q$ at the ground surface, the vertical stress increment at depth $z$ and horizontal offset $r$ is:
Stress decreases as depth increases and as horizontal distance from the load increases. The expression does not require modulus of elasticity or Poisson's ratio when the ideal assumptions are accepted.
Westergaard Stress Distribution
Westergaard's solution is often used for stratified deposits, such as alternating thin layers of sand and silt-clay. It assumes no lateral strain in the layered medium.
Here $q$ is uniform pressure, $r$ is radius of the circular loaded area, and $z$ is depth below the center.
Rectangular Loaded Area and 2:1 Method
For a flexible rectangular area, influence factors are often read from a chart/table using ratios $B/z$ and $L/z$. For stress below the center, divide the footing into four rectangles and multiply the corner influence result by 4.
$$\Delta\sigma_z = qI_z$$
$$\Delta\sigma_{center}=4qI_{corner}$$
The approximate 2 vertical to 1 horizontal method assumes the load spreads linearly with depth.
$$\Delta\sigma_z = \frac{P}{(B+z)(L+z)}$$
$$\Delta\sigma_z = q\frac{BL}{(B+z)(L+z)}$$
Problem: Square Footing by Boussinesq and 2:1 Method
A square footing 4 m on a side transmits a load of 1.6 MN. A clay layer is from 8 m to 12 m below the base. Determine the bearing pressure, estimate stress below the center at mid-height of clay using Boussinesq chart values, and evaluate stress using the 2:1 method.
A 4 m diameter circular footing transmits a concentrated load of 2000 kN. Evaluate the bearing pressure, vertical stress below the center at a depth equal to its diameter, and depth where pressure is reduced to one-tenth of base pressure.
Answer: $q=159.15$ kPa, stress at 4 m depth $=45.27$ kPa, and depth for one-tenth pressure is about 7.4 m.
Problem: Westergaard Point Load
Using Westergaard theory, evaluate vertical stress for $Q=1500$ kN if the point is 3 m directly below the load, 6 m directly below the load, and 6 m below but 3 m horizontally from the load.
Using Boussinesq theory, evaluate vertical stress for $Q=1500$ kN if the point is 2.5 m directly below the load, 5 m directly below the load, and 2.5 m below but 3 m horizontally from the load.
Concept: Engineering Use of Stress Distribution Theories
Boussinesq theory assumes the soil is elastic, homogeneous, and isotropic — conditions rarely met in real soil. Despite these simplifications, it gives reasonable estimates for stress increases in relatively uniform, soft to medium soils and is widely used for consolidation settlement calculations.
Westergaard theory better represents layered or varved deposits like alternating silt and clay, because it assumes no lateral strain occurs. It predicts lower stress increases than Boussinesq at the same depth and horizontal distance. The 2:1 method is the simplest approximation and is often used for preliminary estimates and for checking computer output. It overestimates stress below the corner of a footing but is reasonably accurate below the center for square and rectangular footings at depths comparable to the footing width.
Problem: Stress Below a Strip Load
A flexible strip footing 2 m wide carries a uniform pressure of 80 kPa. Using the 2:1 approximation adapted for strip loading, estimate the vertical stress increase at 2 m and 4 m directly below the center of the strip. For a strip footing, the 2:1 method gives $\Delta\sigma_z = qB/(B+z)$.
Answer: 40 kPa at 2 m depth and 26.7 kPa at 4 m depth below the strip footing center.
Problem: Rectangular Footing Using the 2:1 Method
A 3 m by 5 m rectangular footing carries a column load of 2000 kN. Using the 2:1 method, find the average stress increase in a 2 m thick clay layer that has its top at 4 m below the footing base and its bottom at 6 m below the footing base.
Answer: The average stress increase in the clay layer is approximately 26 kPa.
Problem: Boussinesq Stress at an Offset Point
A concentrated load $Q = 900$ kN is applied at the ground surface. Using Boussinesq theory, compute the vertical stress at a point 4 m below the surface and 3 m horizontally from the load.
Answer: The Boussinesq stress at that point is 8.80 kPa.
Problem: Superposition of Two Point Loads
Two column loads, each 600 kN, are placed at the ground surface 4 m apart. Using Boussinesq theory, find the total vertical stress increase at a point on the line connecting the two loads, 3 m below the surface and exactly midway between the columns. Use $r=2$ m for each load.
Both loads are equidistant from the midpoint, so by superposition:
Answer: The total vertical stress increase at the midpoint is 25.38 kPa.
Problem: Stress Below the Edge of a Circular Footing
A 3 m diameter circular footing applies a uniform pressure of 120 kPa. Using the Boussinesq formula for a uniformly loaded circular area, estimate the vertical stress below the center at a depth equal to the radius (1.5 m below the base).
