Boussinesq (1842 – 1929) evolved equations that can be used to determine stresses at any point P at a depth z as a result of a surface point load.

Boussinesq's Formula Equation

At point P of above figure due to a point load Q,

\[Vertical\hspace{.2cm}{stress}, \sigma_{z}=\frac{3Q}{2\pi{z^{2}}}\frac{1}{[1+(r/z)^{2}]^{5/2}}=\frac{Q}{z^{2}}I_{B}\]

where,

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• Assumptions of Boussinesq's Formula

Boussinesq's formula, a fundamental concept in soil mechanics, provides valuable insights into the behavior of soil under point loads. However, the application of this formula is based on a set of key assumptions. Understanding these assumptions is crucial for accurate analysis and interpretation of results. In this article, we will explore the assumptions underlying Boussinesq's Formula and their implications in soil mechanics.

Boussinesq's theory formula is based on the following assumptions.

1. The soil mass is semi-infinite, homogeneous, and isotropic: The first assumption considers the soil mass as semi-infinite, meaning that the depth of the soil is significantly larger compared to the dimensions of the load. Additionally, the soil is assumed to be homogeneous, exhibiting uniform properties throughout, and isotropic, having similar mechanical properties in all directions. These assumptions simplify the analysis by providing a consistent soil behavior for calculations.
2. The soil has a linear stress-strain relationship: The second assumption assumes that the stress-strain relationship of the soil is linear. This assumption allows for a simplified analysis of the soil's response to the applied load. However, it's important to note that in reality, soil behavior can be nonlinear, particularly at high-stress levels or in the presence of complex soil compositions.
3. The soil is weightless: The third assumption considers the soil as weightless. This assumption is reasonable when analyzing shallow depths or small loads in relation to the overall weight of the soil mass. The weightless assumption simplifies calculations, focusing on the impact of the applied load on the soil rather than considering the self-weight of the soil.
4. The load is a point load acting on the surface: The final assumption assumes that the load acting on the soil is a point load applied at the surface. This assumption allows for simplified analysis, as point loads are relatively easier to handle mathematically. It's important to note that in practical scenarios, loads may not always be precisely point loads, and their distribution can vary.

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• Boussinesq's Equation for Point Loads

Westergaard is a British scientist. For vertical stress computation, he had proposed a formula in 1938. The formula is presented below. 

Westergaard's Equation for Point Loads

If Q is the point load and σ_z is the vertical stress due to the point load,

\[\sigma_{z}=\frac{Q}{2{\pi}z^{2}}\times \frac{\sqrt{(1-2\mu)/(2-2\mu)}} {[(1-2\mu)/(2-\mu)+(r/z)^{2})]^{3/2}}= \frac{Q}{z^{2}}I_{w}\]

Here, μ= Poisson's Ratio. If Poisson's Ratio is considered zero for all practical purposes, 

\[\sigma_{z}=\frac{Q}{{\pi}z^{2}}\times \frac{1} {[1+2(r/z)^{2})]^{3/2}}= \frac{Q}{z^{2}}I_{w}\]

Where, \[I_{w}= \frac{(1/\pi)} {[1+2(r/z)^{2})]^{3/2}}\]

\[I_{w}\] is known as Westergaard stress coefficient. 

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• Westergaard Analysis: Its Application in Soil Mechanics

Nathan Mortimore Newmark was a structural engineer. In 1942 he developed a chart to determine the vertical pressure of soil at any point for uniformly applied load. The chart is developed from Boussinesq's Equation. 

Following is the Newmark's Influence Chart.

Specific heat is a term that indicates the amount of heat needed to raise the temperature of a substance by one degree Celsius. It is a thermodynamic property of the substance. In SI specifications, we recognize specific heat through symbol 'c' and the unit is Joule per Kelvin.

Specific Heat of Soil

There are two things are vital, when you will go for measuring the specific heat of soil mass.

1. Injecting amount of heat to a known mass of soils and
2. Measuring the change in temperature in the process

Soil is a poor heat conductor, so it is not an easy experiment to perform.

Specific heat of dry soil is 0.2 Cal/g and water’s specific heat is 5 times higher than that, 1 Cal/g. Moist soils are generally cooler because the heat energy gets spent in evaporation of soil moisture.

If C_s is the specific heat of the soil, we can calculate this from the following equation.

C_s= C₁M₁+C₂M₂+C₃M₃……+ C_nM_n [cal/g (°C)]

Here, M is denoting mass weight in grams.

Standards Calculating Specific Heat of Soil

1 kilogram (kg) of soil mass should increase the temperature by 1o. Surrounding temperature and atmospheric pressure during the experiment are significant. The room temperature should be 25o C and pressure should be sea level pressure which is 760 mm in Mercury.

Factors Determining Specific Heat

• Temperature and Pressure

Specific heat is a thermodynamic property that fluctuates due to the temperature and pressure deflections. Inappropriate temperature fails to receive the heat and the uneven pressure makes the process faulty.

• Energetic Degrees of freedom

Larger degrees of freedom create a larger amount of specific heat and vice versa.

Importance of Specific Heat of Soil in Civil Engineering

The Earth which takes all the loads coming from the various forms of superstructures is made of soil. Along with the other properties of soil, the thermal property also contains with equal importance.

Specific Heat is the amount of heat that simply indicates the required heat to change the temperature by unity. This change affects different phenomena including the water content, bulk density of the soil. If the soil is a fine-grained soil, the Atterberg Limits (plastic limit, liquid limit, shrinkage limit, etc) are widely influenced.

All these things change pivotal properties like permeability, compressibility, shear strength, bearing strength, particle sizes and consistency of soil mass.