Concept of Boussinesq’s Analysis - Stress in Soil due to applied load, Soil Mechanics | Soil Mechanics Notes- Agricultural Engineering PDF Download

Concept of Boussinesq’s Analysis   

In the lesson 5, the stress calculation within the soil due to the overburden pressure (pressure due to the soil above any depth is called overburden pressure) has been discussed. In this lesson, the stress calculation within soil due to the applied load will be discussed. Boussinesq (1885) proposed equations to determine stresses in subgrade materials due to the applied loads. The subgrade material has been considered as weightless, unstressed, semi-infinite, elastic, homogeneous, and isotropic. The concentration load (Q) has been applied normally to the upper surface of the material (as shown in Figure 6.1). 

Module 2 Lesson 6 Fig.6.1

Fig. 6.1. Boussinesq’s stresses.

The vertical stress (σz) at a point ‘O’ within the soil with radial distance ‘R’ from the point of application of the concentration load Q can be determined as:

\[{\sigma _Z}={{3Q} \over {2\pi {z^2}}}{\cos ^5}\alpha\]                       (6.1)

The shear stress (trz) at a point ‘O’ within the soil with radial distance ‘R’ from the point of application of the concentration load Q can be determined as:

\[{\tau _{rz}}={{3Q} \over {2\pi {z^2}}}{\cos ^4}\alpha \sin \alpha\]                (6.2)

where z and r are the depth and horizontal distance of the point ‘O’ from the point of application of the concentration load Q, respectively.

 Now,

\[\cos \alpha=\frac{z}{R}=\frac{z}{{{{\left({{r^2}+{z^2}}\right)}^{\frac{1}{2}}}}}\]              (6.3)

Thus, Eq. (6.1) can be written as:

\[{\sigma _Z}=\frac{{3Q}}{{2\pi }}\frac{{{z^3}}}{{{R^5}}}=\frac{{3Q}}{{2\pi }}\frac{{{z^3}}}{{{{\left({{r^2}+{z^2}}\right)}^{\frac{5}{2}}}}}=\frac{{3Q}}{{2\pi {z^2}}}{\left[{\frac{1}{{1+{{\left({\frac{r}{z}}\right)}^2}}}\right]^{\frac{5}{2}}}\]                 (6.4)

The Eq. (6.4) can be written as:

\[{\sigma _Z}={Q \over {{z^2}}}{K_B}\]                        (6.5)

where

\[{K_B}=\frac{3}{{2\pi }}{\left[{\frac{1}{{1+{{\left({\frac{r}{z}} \right)}^2}}}}\right]^{\frac{5}{2}}}\]                   (6.6)

The KB is called Boussinesq influence factor which is a function of (r/z) ratio. If r = 0, KB = 0.4775. Thus, vertical stress just below the point of application of load ‘Q’ on the axis (at any depth) can be expressed as:

\[{\sigma _Z}=0.4775{Q \over {{z^2}}}\]                        (6.7)

The shear stress can be expressed as:

\[{\tau _{rz}} =\frac{{3Q}}{{2\pi }}\frac{{r{z^2}}}{{{{\left({{r^2}+{z^2}}\right)}^{\frac{5}{2}}}}}\]                 (6.8)