Soil Mechanics – Effective and Total Stress

Effective Stress, \(\sigma ‘\) is in units of pressure (Pa, \(\frac{kg*m}{s^2}\), \(\frac{F}{A}\), etc) and is given by this equation:

$$ \sigma ‘ = \sigma – u$$

Total stress, \(\sigma\), is the total amount of stress due to the soil at the depth you are considering:

$$ \sigma = \rho_s g z $$

\(\rho_s\) is the soil density (e.g. 80 lb/ft^3).

\(g\) is the acceleration of gravity (e.g. 32.2 ft/s^2).

\(z\) is the distance of the surface, or beginning of the soil section, to the point you are considering(e.g. 10 ft).

Pore water pressure, \(u\), is the upward pressure due to the buoyant force of the water in the water table. It is subtracted so that, in most cases, the effective stress is less than the total stress due to the buoyant force counteracting it.

$$ u = \rho_w g z_w$$

\(\rho_w\) is the water density, 62.4 lb/ft^3 or 1000 kg/m^3.

\(z_w\) is the depth from the water table elevation. Pore water pressure is ZERO above the water table and starts to have a value only BELOW the water table line.

It is important to understand that the total stress and effective stress are EQUAL above the water table and that the effective stress is less than the effective stress below the water table (again, in most cases).

I say in most cases because the power water pressure can be negative at times. Essentially this means that there is some kind of suction force acting on the area sucking the water out of the area. When this happens the effective stress will be greater than the total stress due to the suction force. I don’t know if this will ever come up on the breadth portion of the PE exam.

Here are the steps to find the effective stress in the picture at the top:

$$ \sigma ‘ = \sigma_{s1} + \sigma_{s2} – u $$

$$ \sigma ‘ = \rho_{s1} g z_{s1} + \rho_{s2} g z_{s2} – \rho_w g z_w $$

$$ \sigma ‘ = (40)(32.2)(z_{s1}) + (50)(32.2)(z_{s2}) – (62.4)(32.2)(z_w) $$

$$ etc $$