Stress Distribution Trough soil - Part 01

When a footing is loaded, a certain stress will be applied on the soil immediately below the footing. the applied stress in the soil will decrease away from the footing location as the stress is distributed over a large area.

Consider a small soil element as shown below and the stress applied on the element under 'Plain strain condition' (plane strain condition : strain is applied in one plane - Y direction no strain)





Due to the stress applied on the soil from the foundation, the increase in the normal and the shear stress in the soil mass under the foundation may lead to failure of cause deformation of the soil causing settlement of the foundation. therefore, it is very important to study stress distribution under a loaded area.

Evan though the soil is neither elastic nor homogeneous, most of the stress distribution are obtained assuming linear elastic and homogeneous soil medium below the foundation.

Vertical stress distribution in homogeneous isotropic soil medium

For this we can use several methods,
1) Boussinesq equation
2) Newmarks charts
3) Stress isobars

Let's consider about the approximate stress distribution of a load in soil,

Generally alpha is taken as 0.5 with the 'tan theeta = 0.5 - 1', theeta = 26.5


sigma z = stress developed at a depth z

Boussinesq equation 

Boussinesq solved the problem of stress inside a semi - infinite mass sue to a point load acting on the surface. the stresses at point (x,y,z) due to a point load Q.

The practical application of the Boussinesq equation is a difficult task, therefore other charts and tables are prepared to estimate of the stress increased sue to a loaded area.

Considering a circular loaded area with radius r is loaded with a surface stress of qo, the stress variation along the vertical axis through the center may be estimated using the Boussinesq equation.

 

 

where,

delta sigma = stress

qo = load

r = radius 

z = depth

Example :
Circular foundation of 1m radius is loaded with 100 K Pa. what is the vertical stress increment along the center line 2 m below the foundation. find this with boussinesq equation.

delta sigma = 100 (1 - 1/[1+ (1/2)^2]^(3/2)  = 28.45 K Pa

Circular foundation of 1m radius is loaded with 100 K Pa. what is the vertical stress increment along the center line 4 m below the foundation. find this with boussinesq equation.

delta sigma = 100 (1 - 1/[1+ (1/4)^2]^(3/2)  = 8.7 K Pa