Just like every other material on earth, soils respond to externally applied load by developing stresses and strains within their material structure. The analysis of stress distribution in soils is concerned with the variation of pressure in soils due to external loads. Two primary factors contribute to stress within soil formations:

(a) the self-weight of the soil itself, often referred to as self-weight or overburden pressure, and

(b) external structural loads applied either at the surface or within the subsurface layers.

A good understanding of how stresses are transmitted and distributed through large soil masses is important in design scenarios involving soils in different engineering applications. Examples include the transmission of wheel loads through embankments to culverts below, the distribution of foundation pressures within soil strata beneath building footings, the transmission of pressures from isolated footings to retaining walls, and the transmission of wheel loads through stabilized pavements to the subgrade layers below. In these scenarios, the stresses propagate both downwards and laterally within the soil mass.

The accurate estimation of vertical stresses caused by external loads on a soil mass is important for predicting settlements in structures like buildings, bridges, and embankments. Traditionally, the theory of elasticity has been used to determine these stresses. This theory focuses on the relationships between stresses (forces acting on a material) and strains (deformations caused by those forces). According to elasticity, these relationships are constant.

However, the key requirement for applying this theory isn’t that the soil itself is perfectly elastic, but rather that stresses and their corresponding strains remain proportional. While soil only exhibits this proportionality at relatively low-stress levels, fortunately, the stresses transmitted from typical structural loads also tend to be low. This fortunate coincidence allows the use of elasticity theory for soil stress distribution to provide reasonably accurate results.

In this article, we are going to consider the increment in stress or stress distribution in soils due to;

(a) point loads,

(b) strip loads, and

(C) rectangular loads

**Common Methods for Calculating Stress Distribution in Soil**

The ** Boussinesq **and

**theories are widely used to analyze how stress is distributed within soil. These theories were initially developed for point loads. Later, the point load solutions were extended to calculate stresses under various other load shapes, including uniform strip loads, uniformly loaded circular areas, and uniformly loaded rectangular areas.**

*Westergaard*In 1885, Boussinesq employed the mathematical theory of elasticity to analyze stresses caused by a point load on a specific type of material. Boussinesq’s theory is the most popular and has been extensively applied to various geotechnical problems. This material exhibits the following characteristics:

**Homogeneity:**Consistent properties throughout its volume.**Elasticity:**Deformation under stress with a return to its original shape upon stress removal.**Isotropy:**Identical properties in all directions at a given point.**Semi-infinite extent:**Infinitely deep with a flat top surface (analogous to soil extending downwards forever).

Boussinesq’s theory is based on the following key assumptions:

**Material Properties:**The soil behaves as an elastic, homogeneous, isotropic, and semi-infinite medium, extending infinitely in all directions from a flat surface. (Homogeneity implies identical properties at all points in the same direction, while isotropy implies identical elastic properties in all directions at a single point.)**Stress-Strain Relationship:**The material follows Hooke’s Law, which describes a linear relationship between stress and strain.**Self-Weight Neglect:**The weight of the soil itself is disregarded.**Initial Stress State:**The soil is initially free of any stresses before the point load is applied.**Volume Change Neglect:**Changes in soil volume due to loading are considered negligible.**Top Surface Conditions:**The top surface is free of shear stresses and experiences only the point load at a specific location.**Stress Continuity:**Stresses within the medium are assumed to be continuous, meaning there are no abrupt jumps in stress values.**Stress Symmetry:**The distribution of stresses is considered symmetrical with respect to a vertical axis (Z-axis).

**Calculating Geostatic Stress**

The vertical stress in soil caused by its own weight, also known as geostatic stress, can be determined using the following equation:

σ_{z} = γz ——- (1)

where:

- σ
_{z}represents the vertical stress in the soil at depth*z*due to self-weight. *γ*represents the unit weight of the soil.

When external structural loads are applied to the soil, the total stress at a point is the sum of the geostatic stress and the stress caused by the structural loads. This can be determined by algebraically adding the two stress values.

**Point Loads**

A point load, or a concentrated load applied at a single point is not a very realistic loading scenario in practice, since all practical loads are distributed over an area no matter how small it may be. However, analyzing point loads proves valuable for most problems in geotechnical engineering.

Several practical scenarios can be idealized as point loads for analysis purposes. Here are some examples:

**Single column on soil:**The pressure exerted by a building column, pole, or a pylon on the underlying soil can be approximated as a point load, especially for slender columns.**Anchor Load:**The pull exerted by a soil anchor on the surrounding soil mass can be idealized as a point load acting at the anchor’s embedded depth.**Pile Tip Load:**The force transferred from a pile (a long, slender foundation element) to the soil at its tip can be simplified as a point load, particularly for short piles.**Cone Penetrometer Test:**This test uses a cone-shaped tip pushed into the soil to measure its resistance. The force applied by the penetrometer can be considered a point load for analysis of soil behaviour around the cone.**Sparse Traffic:**In some situations, the weight of a single vehicle on a large soil area (like a wide embankment) might be simplified as a point load for initial stress distribution analysis.

It’s important to remember that these are idealized representations. Real-life scenarios involve footings with finite areas, distributed anchor forces, and pile shafts transferring load along their length. However, using point load approximations helps engineers understand the fundamental behaviour of soil under concentrated loads and serves as a stepping stone for analyzing more complex loading conditions.

The expression obtained by Boussinesq for computing vertical stress σ_{z}, at a depth *z* (Figure 2) due to a point load *P* is;

**Solved Example 1:**A concentrated load of 2500 kN acts on the surface of a homogeneous soil mass of a large extent. Find the stress intensity at a depth of 8 meters.

(a) directly under the load, and

(b) at a horizontal distance of 5 metres.

**Solution**

(a) Directly under the load:

r = 0, therefore r/z = 0

z = 8 m

P = 2500 kN

∆σ_{z} = 3P/2π × [z^{3}/(r^{2 }+ z^{2})^{5⁄2}]= (3 × 2500)/2π × 8^{3}/(0^{2 }+ 8^{2})^{5⁄2 }= 18.65 kN/m^{2}

(b) At a horizontal distance of 5 metres:

r = 5 m; z = 8 m

∆σ_{z} = 3P/2π × [z^{3}/(r^{2 }+ z^{2})^{5⁄2}]= (3 × 2500)/2π × 8^{3}/(5^{2 }+ 8^{2})^{5⁄2 }= 8.18 kN/m^{2}

**Strip Loads (finite width and infinite length)**

Strip loads, representing long, uniformly distributed loads over a width, are a common scenario for analyzing building foundations, retaining walls, and embankments. This scenario represents a **plane strain condition**, where deformations occur primarily in a single plane. This is typical for elongated structures like strip foundations, retaining wall foundations, embankments, and dams.

In such structures, the stress distribution within a specific section (excluding the end sections within 2-3 times the width from the ends) remains consistent across neighbouring sections. This holds true as long as the load doesn’t change in directions perpendicular to the analyzed plane.

The equation introduced earlier for a point load’s vertical stress increase can be applied to determine the vertical stress caused by a wider, flexible strip load of width B (refer to Figure 3). Imagine the load applied to the strip in Figure 3 is uniformly distributed with intensity *q* per unit area.

Now, let’s consider a tiny slice of this strip with an infinitesimal width *dr.* The load acting on this small section can be represented as *qdr* (load per unit length). Because of its narrow width, we can treat this elemental strip as a line load. To calculate the vertical stress increase, we need to substitute *qdr *for *q *and (*x *– *r*) for *x. *So,

The total increase in the vertical stress (∆σ_{z}) at a depth z* *caused by the entire strip load of width *B *can be determined by integration of the equation above with limits of *r *from +*B*/2 to –*B*/2, or;

The Table below shows the variation of ∆σ_{z}* *with 2*z*/*B *for 2*x*/*B. *This table can be used conveniently for the calculation of vertical stress at a point caused by a flexible strip load.

**Solved Example 2**A strip footing of width B = 3m is subjected to a uniform pressure load

*q*= 250 kN/m

^{2}. At a depth of z = 4 m, determine the vertical stress increase at

*x*= 3, and 0 m.

**Solution**q = 250 kN/m

^{2}

^{}z = 4m

B = 2m

*At x = 3m**2x/B = *(2 × 3)/3 = 2.0*2z/B = *(2 × 4)/3 = 2.67

∆σ_{z} /q = 0.208

Ds* _{z }*= 0.208 × 250 = 52 kN/m

^{2}

*At x = 0 m (at the centre of the footing)**2x/B = *(2 × 0)/3 = 0*2z/B = *(2 × 4)/3 = 2.67

∆σ_{z}* */*q* = 0.442

∆σ_{z}= 0.442 × 250 = 110.5 kN/m^{2}

**Rectangular Loads**

In foundation engineering, rectangular areas are a more common load shape, especially for building foundations. By applying the concept of integration, engineers can determine the vertical stress at a point beneath a uniformly loaded rectangular area. This calculation can be based on either Boussinesq’s or Westergaard’s solutions for a point load.

Newmark (1935) specifically addressed this issue by deriving an expression for the vertical stress at a point located below the corner of a uniformly loaded rectangular area.

The following are the two popular forms of Newmark’s equation for σ* _{z}*:

where *m *= *B*/*z *and *n *= *L*/*z*.

The second term within the brackets is an angle in radians. It is of interest to note that the above expressions do not contain the dimension *z*; thus, for any magnitude of *z*, the underground stress depends only on the ratios *m *and *n *and the surface load intensity. Since these equations are symmetrical in *m *and *n, *the values of *m *and *n *are interchangeable. The Equation for stress due to rectangular load may be written in the form:

σ_{z}* *= *q*. *I*_{σ}

where *I*_{σ }= Influence value

Fadum (1941) built upon Newmark’s formula (for corner stress under a rectangular load) by creating a chart (Figure 5). This chart displays “influence values” for various combinations of parameters (m and n) that define the rectangle’s proportions relative to the depth (z) and width (B).

The principle of superposition allows us to determine the vertical stress at the centre of a uniformly loaded rectangle using the known influence value for a corner. This involves dividing the rectangle into four identical quadrants and applying the influence value for a single quadrant (corner stress divided by four). The same principle can be extended to calculate stress at other points by considering appropriate subdivisions and influence values.

**Solved Example 3**A raft foundation of size 12 m x 16m provides an average pressure of 55 kN/m

^{2}. Determine the vertical stress increment at a point 8 m below the centre of the loaded area, and at the corner of the foundation using Boussinesq’s theory.

**Solution**(a) At the corner of the footing

L = 16 m

B = 12 m

z = 8 m

m = B/Z = 12/8 = 1.5

n = L/z = 16/8 = 2.0

From Fadum’s chart;*I*_{σ} = 0.223

∆σ_{z}= 0.223 × 55 = 12.28 kN/m^{2}

(b) At the centre of the footing (divide the foundation into four equal parts)

L = 8 m

B = 6 m

z = 8 m

m = B/Z = 8/8 = 1.0

n = L/z = 6/8 = 0.75

From Fadum’s chart;*I*_{σ} = 0.154

∆σ_{z}= 4 × 0.154 × 55 = 33.88 kN/m^{2}

Average pressure at 8m depth = (33.88 + 12.28)/2 = 23.08 kN/m^{2}

**Compare stress increment using the famous 1:2 method.**

The average vertical stress at depth *z = 8m *is obtained as:

∆σ_{z},_{average} = (55 × 16 × 12)/[(16 + 8) × (12 + 8)] = 10560/480 = 22 kN/m^{2}

**Conclusion**

Understanding stress distribution in soil due to different types of loading is important for safe and efficient foundation design and geotechnical analysis. Theoretical methods based on Boussinesq’s solutions and influence charts provide valuable tools for engineers. However, recognizing the limitations of these methods and considering the complexities of real soil behaviour is essential for accurate and reliable engineering practices.