Vertical Stresses
In geotechnical engineering, we must calculate the vertical stress () at any depth to predict settlement and bearing capacity. Stresses arise from two sources:
- Geostatic Stresses: Due to the self-weight of the soil.
- Induced Stresses: Due to external loads (foundations, embankments, vehicles).
Stresses due to Self-Weight
The vertical stress increases linearly with depth in a homogeneous soil.
Geostatic Stress Equation
Geostatic Vertical Stress
$$
\sigma_v = \sum \gamma z
$$Stresses due to Point Loads (Boussinesq)
Boussinesq (1885) provided a solution for the stress distribution in a semi-infinite, homogeneous, isotropic, elastic medium due to a point load at the surface.
Boussinesq's Equation
The vertical stress increase at depth and radial distance is:
Boussinesq Point Load
$$
\Delta \sigma_z = \frac{3P}{2\pi z^2} \left[ \frac{1}{1 + (r/z)^2} \right]^{5/2}
$$- Directly under the load ():
Stress Directly Under Load
$$
\Delta \sigma_z = \frac{3P}{2\pi z^2} = \frac{0.4775 P}{z^2}
$$Westergaard's Theory
Assume the soil is reinforced by horizontal inextensible sheets (e.g., layered sedimentary soils like varved clays).
Westergaard's Equation
Westergaard Point Load
$$
\Delta \sigma_z = \frac{P}{\pi z^2} \frac{1}{[1 + 2(r/z)^2]^{3/2}}
$$Stresses due to Area Loads
Foundations apply loads over an area (strip, square, rectangle, circle). We integrate the point load solution over the area.
Fadum's Chart (Corner of a Rectangular Area)
To find the vertical stress increase () exactly under the corner of a flexible rectangular area () loaded with a uniform pressure (), engineers use the mathematical integration provided by Fadum (1948).
Fadum's Stress Influence
$$
\Delta \sigma_z = q \cdot I_z
$$Newmark's Influence Chart
A graphical method to determine vertical stress under any irregularly shaped loaded area.
Newmark's Equation
$$
\Delta \sigma_z = I \cdot N \cdot q
$$Approximate 2:1 Method
A simple and practical approximation used to estimate the vertical stress increase at a depth below a loaded rectangular foundation.
2:1 Method Calculation
- The method assumes the load spreads linearly outwards at a slope of 2 vertical to 1 horizontal (an angle of ).
- Given a foundation of width , length , and applied load (or distributed load ), the stress increase at depth is:
Approximate 2:1 Method
$$
\Delta \sigma_z = \frac{P}{(B + z)(L + z)}
$$Key Takeaways
- Vertical stress () is the sum of geostatic stress (weight of soil) and induced stress (external loads).
- Boussinesq's Theory assumes a homogeneous, isotropic, elastic half-space and is the standard for point-load stress distribution.
- Westergaard's Theory is used for highly stratified soils and predicts lower stresses than Boussinesq.
- Fadum's Chart provides the exact elastic solution for the stress increase beneath the corner of a uniformly loaded rectangular area.
- The 2:1 Method is a simple, conservative approximation where the stress is assumed to spread over an expanding area .