Chapter 8 — Stress Distribution in Soil¶
8.1 The problem¶
A foundation load spreads into the ground, so the stress increase \(\Delta\sigma\) at depth is less than the applied pressure and is distributed over a wider area. Quantifying \(\Delta\sigma\) with depth is essential for settlement calculations (Chapter 9) and for bearing capacity.
8.2 Boussinesq theory¶
Boussinesq's solution gives the stress increase in a homogeneous, elastic, isotropic half-space due to a point load at the surface. From it, solutions are built for:
- uniformly loaded strip footings (plane strain);
- rectangular and circular footings; and
- any shape, by superposition or influence charts.
The vertical stress increase beneath the centre of a flexible circular load of radius \(R\) at depth \(z\) is a standard result, and rectangular-footing solutions are tabulated as influence factors.
8.3 The 2:1 method and approximate rules¶
A simple, widely used approximation is the 2:1 method, which spreads the load at a slope of 2 vertical to 1 horizontal:
It is crude but useful for a first estimate; more rigorous solutions are preferred for important structures.
8.4 Newmark influence charts¶
Newmark's influence chart is a graphical tool: the loaded area is drawn to scale over the chart, and the number of divisions covered gives the influence factor. It handles irregular shapes that have no closed-form solution.
8.5 Stress bulbs and depth of influence¶
The stress bulb (or pressure bulb) is the contour within which \(\Delta\sigma\) exceeds a chosen fraction of the applied pressure (e.g. 10% or 20%). It shows:
- the depth of significant influence — how deep to explore and to which layer settlement must be computed;
- the lateral extent — how far adjacent structures or utilities may be affected.
As a rule, the depth of influence is on the order of 2 to 4 times the footing width.
8.6 Key takeaways¶
- Foundation loads spread and attenuate with depth.
- Boussinesq theory and influence factors give \(\Delta\sigma\) for common shapes.
- The 2:1 method is a crude but useful approximation.
- Newmark charts handle irregular shapes.
- The stress bulb sets the depth of exploration and settlement computation (typically 2–4 B).