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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

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

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)

The document Concept of Boussinesq’s Analysis - Stress in Soil due to applied load, Soil Mechanics | Soil Mechanics Notes- Agricultural Engineering is a part of the Agricultural Engineering Course Soil Mechanics Notes- Agricultural Engineering.
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FAQs on Concept of Boussinesq’s Analysis - Stress in Soil due to applied load, Soil Mechanics - Soil Mechanics Notes- Agricultural Engineering

1. What is Boussinesq’s analysis and how does it relate to stress in soil due to applied load?
Ans. Boussinesq’s analysis is a method used in soil mechanics to analyze the stress distribution in soil caused by an applied load. It is based on the assumption that the soil is an elastic half-space. The analysis calculates the stress distribution at different depths below the surface of the soil, taking into account the load magnitude and area of application.
2. How is Boussinesq’s analysis applied in agricultural engineering?
Ans. In agricultural engineering, Boussinesq’s analysis is widely used to study the stress distribution in soil caused by various agricultural activities such as machinery operations, vehicle movements, or the weight of agricultural structures. By understanding the stress distribution, engineers can design and plan agricultural activities to minimize soil compaction and potential damage to crops.
3. What factors affect the stress distribution in soil according to Boussinesq’s analysis?
Ans. Boussinesq’s analysis considers several factors that affect the stress distribution in soil, including the load magnitude and area of application, the depth below the surface, and the elastic properties of the soil. The analysis assumes that the soil is homogeneous and isotropic, meaning it has similar properties in all directions.
4. Can Boussinesq’s analysis be used for all types of soil?
Ans. Boussinesq’s analysis is most accurate for soils that exhibit linear elastic behavior, which means the stress-strain relationship is proportional and reversible. However, it can still provide useful insights for a wide range of soil types, as long as the soil is reasonably elastic and the assumptions of the analysis are met. Different soil types may require adjustments or additional factors to be considered in the analysis.
5. What are the limitations of Boussinesq’s analysis?
Ans. Boussinesq’s analysis has some limitations. It assumes that the soil is homogeneous and isotropic, which may not be the case in reality. It also assumes linear elastic behavior, which may not accurately represent the actual stress-strain relationship of some soils. Additionally, the analysis does not consider the effects of soil compaction, soil moisture content, or other factors that may influence the stress distribution in practice. Therefore, while Boussinesq’s analysis provides valuable insights, it should be used as a tool for preliminary analysis rather than a definitive solution.
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