## What is meant by vertical stress?

The load applied to the soil is transferred to the underground. The presence of water at soil pores and solid grains above the soil are the primary components that are responsible for the effective load transfer. Due to the load acting, the internal component of the soil or the underground formations undergoes deformations or a tectonic shift. The amount of deformation depends upon the amount of load acting on the soil surface, which indeed, depends on the weight of everything present above the soil. Due to this, the underground soil element develops internal stresses which try to resist the tectonic changes.

A typical soil element under deformation is characterized by two principal stress, horizontal stresses, and vertical stresses. Vertical stresses can be maximum vertical stress and minimum vertical stress. The directions of horizontal stresses and vertical stresses depend on the rock failure criteria and degree of tectonic deformations. The magnitude of vertical stresses is proportional to the depth and weight of rocks from the soil surface. Vertical stress is also known as overburdened stress.

## Effective stress principle

The mechanism of the behavior of soil under the application of external loads is completely different from the structural members under the application of external loads as studied in structural mechanics.

Karl Terzaghi was the first who introduced the concept of effective stress concept in soil mechanics. Unlike structural members, soil elements do not induce instantaneous strains under the action of external loadings. Saturated soil, which is entrapped liquid particles in the pores, develops pore pressure when an external load is applied to the soil surface. With due time, this pore pressure gets converted to intergranular pressure or effective pressure, as entrapped water particles get dissipated. This results in compacting of the internal soil structure which is primarily composed of rocks. Contact between the rock particles results in force application on each other, which in turn, initiates stress generation. These stresses are often termed effective stresses. The rate and nature of liquid dissipation depend on the type of soil. A clayey soil responds more to pore pressure than sandy soil.

The above Figure 1, shows rocks that are compacted due to liquid dissipation because of external load application on the soil. X-X is a wavy surface that passes through each of the contact points between the rocks. Because of compacting, forces are induced at every contact point. The severity of external load conditions brings the rock granules more closer, which increases the contact points forming contact areas. Forces acting on these contact areas give rise to contact pressure and corresponding stress distributions.

When the contact area is projected on the horizontal plane, it can be assumed to be a rectangular cross-sectional area having area $A$. The area of the rectangular cross-section is proportional to the amount of compacting.

Hence, the effective stress distribution can be given as,

$\sigma =\frac{F}{A}$

where, $F$ is the force acting at the contact area due to contact between the rocks and $\sigma $ denotes the effective stress. This equation is used to determine the effective stress distribution at each of the contact areas between the rock granules.

## Boussinesq equation

The Boussinesq equation is used to determine stresses at any given point at a certain depth due to surface point load.

The magnitude of vertical stress ${\sigma}_{V}$ at a random point $Q$ at a depth of $H$ from the earth's surface, due to a point load $P$ acting at point $O$, is given by the Boussinesq equation, which is

${\sigma}_{V}=\frac{3P}{2{\mathrm{\pi H}}^{2}}\frac{1}{{\left[1+{\left({\displaystyle \frac{r}{H}}\right)}^{2}\right]}^{{\displaystyle \frac{5}{2}}}}$

Where $r$ is the horizontal distance between the random point $Q$ and the point of intersection of the vertical axis from point $O$.

As the depth of point $Q$ increases, the magnitude of vertical stress decreases and vice-versa. It is thus concluded that vertical stress ${\sigma}_{V}$, varies inversely with depth $H$.

## Vertical stress determination methods

There are various theories associated with the determination of the vertical stresses, the Boussinesq theory is well known as outlined above. Some of the other theories are depicted below.

### Westergaard's theory

In Westergaard's theory, the soil mass is assumed to be anisotropic, unlike in the Boussinesq theory which assumed the soil mass to be isotropic. Thin sand layers remain sandwiched between the clay deposits which bear infinite rigidity and incompressibility. These thin sand layers allow axial movements of the soil mass and do not undergo lateral strains.

The vertical stress at a point given by Westergaard is,

${\sigma}_{Z}=\frac{{\displaystyle \frac{C}{2\mathrm{\pi}}}{\left[{\displaystyle \frac{1}{{C}^{2}+{\left({\displaystyle \frac{r}{Z}}\right)}^{2}}}\right]}^{{\displaystyle \frac{3}{2}}}\times Q}{{Z}^{2}}$

Where $Q$ is the load acting at a depth $Z$.

$r$ is the radial distance.

Here, $C=\frac{\left(1-2\mu \right)}{\left(2-2\mu \right)}$

Where $\mu $ is the Poisson's ratio of the soil.

### 2:1 Distribution method

In this method, the vertical stress is generally obtained by assuming that the external load is distributed at the slope of 2 verticals : 1 horizontal from the base of the footing. Footings are the important constituents of a foundation, spread footing and isolated footings are two of the most important footings.

For a rectangular footing ABCD, under the application of uniform pressure $q$, the vertical stress ${\sigma}_{V}$ at a depth $Z$, according to 2:1 distribution method is given by,

${\sigma}_{V}=\frac{q\times \left(l\times b\right)}{\left(l+z\right)\left(b+z\right)}$

Where $l$ is the length of the rectangular footing.

$b$ is the breadth of the rectangular footing.

## Context and Applications

The following topic is taught in different undergraduate and postgraduate degree courses of

- Bachelors in Civil Engineering
- Masters in Civil Engineering
- Masters in Science in Soil Mechanics
- Masters in Geo-technical Engineering
- Masters in Science in Chemistry

## Practice Problems

1. Which of the following theory is used to determine the vertical stress distribution?

- Boussinesq's theory
- Bernoulli's theory
- Westergaard's theory
- Both a and c

Answer: Option d

Explanation: Both Boussinesq's theory and Westergaard's theory is used to determine vertical stress distribution.

2. Which of the following is the other name of vertical stress?

- Lateral stress
- Overburden stress
- Static stress
- None of these

Answer: Option b

Explanation: The other name of vertical stress is known as overburden stress.

3. Which of the following is the assumption in Westergaard's theory?

- Soil mass is isotropic
- Soil mass is homogeneous
- Soil mass is anisotropic
- Soil mass is non-homogeneous

Answer: Option c

Explanation: In Westergaard's theory to determine vertical stress, the soil mass is assumed to be anisotropic.

4. Who was the first person to introduce the concept of effective stress?

- Boussinesq
- Westergaard
- Karl Terzaghi
- None of these

Answer: Option c

Explanation: Karl Terzaghi was the first person to introduce the concept of effective stress.

5. Which of the following soil type responds more to pore pressure?

- Sandy soil
- Loam soil
- Slit soil
- Clayey soil

Answer: Option d

Explanation: Clayey soil responds more to pore pressures.

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