Geology and Soil Mechanics, UW-Stout

Soils will settle differently when the water table is sufficiently close to the surface. When soil is underwater (or has significant pore water pressure) the stress acting on the soil particles is the total stress minus the water pressure at a particular depth. Thus, water serves to relieve some of the stress "pushing" soil particles together.

Effective Stress Principle: The total stress is equal to the sum of the effective stress and porewater pressure. Put another way, the effective stress is equal to the total stress minus the porewater pressure. Mathematically this is expressed as

p = p' + u

where p = total stress (or pressure), p'= effective stress, and u = water pressure.

As an example, lets calculate the total expected settlement underneath a square footing 5 ft on edge when this footing is used to support a weight of 250 kips. The general approach is to divide the soil into several layers increasing in depth but decreasing in incremental stress placed upon it by the load. The incremental stress is the additional stress placed upon the soil at that particular depth due to the load on the foundation. The soil density is about 105 pcf and laboratory tests indicate that =0.76, effective vertical stress at 12 ft is 10,000 psf. A sketch of this situation is shown in figure 1.

The first step is to calculate the effective vertical stresses at each layer before the foundation load:

Level 1     p'₁ = 105(2) = 210 psf

Level 2    p'₂ = 105(6) - 62.4(2)  = 505 psf

Level 3    p'₃ = 105(12) - 62.4(8) = 761 psf

Level 4    p'₄ = 105(20) - 62.4(16) = 1102 psf

The second step is to estimate the incremental stress placed upon each soil layer at mid-height. To accomplish this, one can use the 2:1 rule. Such that, every two feet in depth below where the footing makes contact with the soil, one considers that the effective area increases 1 ft further out beyond the footing in the horizontal direction.

The second step is to determine the stress increment due to the footing load:

Level 1   Dp'₁ = 250,000/(7²) = 5102 psf

Level 2   Dp'₂ = 250,000/(11²) = 2066 psf

Level 3   Dp'₃ = 250,000/(17²) = 865 psf

Level 4   Dp'₄ = 250,000/(25²) = 400 psf

Now, the total mid-height stress can be calculated for each layer by the equation (p'₁)_total = p'₁ + Dp'₁ :

Level 1 210 + 5102 = 5312 psf

Level 2 505 + 2066 = 2571 psf

Level 3 761 + 865 = 1626 psf

Level 4 1102 + 400 = 1502 psf

We can now estimate the total expected settlement for each layer by using the laboratory results of and the formula

.

Level 1 h₁ = 48 in, Dh₁ = 1.6 in

Level 2  h₂ = 48 in, Dh₂ = 0.8 in

Level 3 h₃ = 96 in, Dh₃ = 0.8 in

Level 4  h₄ = 96 in, Dh₄ = 0.3 in , Therefore, the total estimated settlement is 3.5 in.