Vertical Stresses

Understanding how stresses from surface loads (foundations, vehicles) distribute through the soil mass is essential for settlement analysis.

Stresses due to Self-Weight

The geostatic stress at any depth is simply the summation of unit weights times layer thicknesses. $\sigma_v = \sum \gamma z$

Stresses due to Point Loads (Boussinesq)

Boussinesq's Equation

For a point load $P$ at the surface of a semi-infinite, homogeneous, isotropic elastic medium, the vertical stress increase $\Delta \sigma_z$ at depth $z$ and radial distance $r$ is: $\Delta \sigma_z = \frac{3P}{2\pi z^2} \left[ \frac{1}{1 + (r/z)^2} \right]^{5/2}$ Under the load ( $r=0$ ): $\Delta \sigma_z = \frac{0.4775 P}{z^2}$

Stresses due to Area Loads

2:1 Approximation Method

A simplified empirical method assumes the load spreads at a 2 (vertical) to 1 (horizontal) slope. For a rectangular footing ( $B \times L$ ) carrying load $P$ : $\Delta \sigma_z = \frac{P}{(B+z)(L+z)}$

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