Engineering
Boussinesq Point-Load Vertical Soil Stress Increment & Influence Factor
Calculate foundation subsurface vertical stress increment Delta_sigma_z = (3 * Q / (2 * PI() * z^2)) * (1 / (1 + (r / z)^2))^2.5 and Boussinesq influence factor I_B at depth z and radial offset r.
Map pressure-bulb stress dissipation beneath column footings and crane outriggers!
How rapidly does a concentrated surface column or outrigger load Q (kN) spread out with depth z (m) and radial horizontal distance r (m) inside a soil deposit? Joseph Boussinesq's 1885 elastic half-space solution shows that vertical normal stress increment depends ONLY on load magnitude, depth squared, and radial-to-depth geometry—independent of Young's modulus or Poisson's ratio: Delta_sigma_z = ((3 * Q) / (2 * PI() * z^2)) * ((1 / (1 + (r / z)^2)) ^ 2.5)!
Directly beneath the load centerline (r = 0), the dimensionless Boussinesq Influence Factor peaks at I_B(0) = 3 / (2 * PI()) = 0.4775, giving Delta_sigma_z = 0.4775 * Q / z^2! Stepping radially outward to r = 3.0 m at depth z = 4.0 m (r / z = 0.75) drops I_B to 0.1565—cutting induced settlement stress from 29.84 kPa to 9.78 kPa!
| Subsurface Point (Surface Load Q = 1,000 kN) | Depth z & Offset r (m) | Influence Factor I_B | Vertical Stress Delta_sigma_z (kPa) |
|---|---|---|---|
| Centerline at Depth 4 m | z = 4.0 m, r = 0.0 m | 0.4775 | 29.84 kPa |
| Radially Offset 3 m at Depth 4 m | z = 4.0 m, r = 3.0 m (r/z = 0.75) | 0.1565 | 9.78 kPa (67% lower!) |
| Centerline at Depth 8 m (2x Deeper) | z = 8.0 m, r = 0.0 m | 0.4775 | 7.46 kPa (1/4 by 1/z^2 law!) |
Boussinesq Vertical Soil Stress Increment Delta_sigma_z (kPa)
Evaluates ((3 * Q) / (2 * PI() * z^2)) * ((1 / (1 + (r / z)^2))^2.5) from point load B2 (kN), depth B3 (m > 0), and radial offset B4 (m >= 0).
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0), "Invalid Load, Depth, or Offset", ROUND(((3 * B2) / (2 * PI() * (B3 ^ 2))) * ((1 / (1 + ((B4 / B3) ^ 2))) ^ 2.5), 2))Dimensionless Boussinesq Influence Coefficient I_B
Computes I_B = (3 / (2 * PI())) * (1 / (1 + (r / z)^2))^2.5 so Delta_sigma_z = I_B * Q / z^2.
=ROUND((3 / (2 * PI())) * ((1 / (1 + ((B4 / B3) ^ 2))) ^ 2.5), 4)Try the example
Use the original values or change them to test your understanding.