Vertical Stresses - Theory & Concepts

Vertical Stresses

Learning Objectives

Differentiate between geostatic and induced vertical stresses.
Calculate geostatic stresses based on soil unit weight and depth.
Calculate induced stresses from point loads using Boussinesq's and Westergaard's theories.
Compute stresses under area loads using Fadum's Chart and Newmark's Influence Chart.
Apply the Approximate 2:1 Method for preliminary stress estimation.

In geotechnical engineering, we must calculate the vertical stress ( $\sigma_v$ ) at any depth to predict settlement and bearing capacity. Stresses arise from two primary sources: geostatic stresses from soil weight, and induced stresses from external loads like foundations.

Geostatic Stresses

In-situ vertical stresses arising from the self-weight of the overlying soil.

Induced Stresses

Additional vertical stresses caused by applied external loads, such as buildings, embankments, or vehicles.

Stresses due to Self-Weight

The geostatic vertical stress increases linearly with depth in a homogeneous soil profile.

Geostatic Stress Equation

The vertical stress is calculated by summing the products of the unit weight and thickness of each overlying soil layer.

Geostatic Vertical Stress

In-situ vertical stress at any depth due to the self-weight of all overlying soil layers; increases linearly with depth.

\sigma_v = \sum \gamma z

Variables

Symbol	Description	Unit
$\sigma_v$	Total vertical stress	-
$\gamma$	Unit weight of the soil layer	-
$z$	Thickness of the layer	-

Effect of Groundwater

Below the water table, total stress uses $\gamma_{sat}$ , while effective stress uses $\gamma'$ .

Stresses due to Point Loads (Boussinesq)

Boussinesq's Theory

An analytical solution that calculates stress distribution in a semi-infinite, homogeneous, isotropic, elastic medium due to a point load at the surface.

Boussinesq (1885) provided a foundational solution for point loads that is still widely used in modern geotechnical analysis.

Boussinesq's Equation

The vertical stress increase $\Delta \sigma_z$ at depth $z$ and radial distance $r$ is:

Boussinesq Point Load

Vertical stress increase in an elastic half-space due to a concentrated surface point load; valid for isotropic, homogeneous soils.

\Delta \sigma_z = \frac{3P}{2\pi z^2} \left[ \frac{1}{1 + (r/z)^2} \right]^{5/2}

Variables

Symbol	Description	Unit
$\Delta \sigma_z$	Vertical stress increase	-
$P$	Point load applied at the surface	-
$z$	Depth below the surface	-
$r$	Radial distance from the point load	-

- Directly under the load ( $r=0$ ):

Stress Directly Under Load

Simplified Boussinesq formula for vertical stress directly on the axis of the applied point load (r = 0).

\Delta \sigma_z = \frac{3P}{2\pi z^2} = \frac{0.4775 P}{z^2}

Variables

Symbol	Description	Unit
$\Delta \sigma_z$	Vertical stress increase	-
$P$	Point load	-
$z$	Depth	-

Stress Dissipation

Stress decreases rapidly with depth (proportional to $1/z^2$ ) and with radial distance from the point of application.

Westergaard's Theory

Westergaard's Theory

An alternative stress distribution theory assuming the soil mass is reinforced by inextensible horizontal layers, limiting horizontal deformation.

Westergaard's approach is more appropriate when the soil is highly stratified (e.g., layered sedimentary soils like varved clays) and restricts horizontal expansion.

Westergaard's Equation

An alternative stress distribution formula for horizontally layered (anisotropic) soils.

Westergaard Point Load

Alternative stress distribution formula for horizontally layered (anisotropic) soils; generally yields lower stresses than Boussinesq.

\Delta \sigma_z = \frac{P}{\pi z^2} \frac{1}{[1 + 2(r/z)^2]^{3/2}}

Variables

Symbol	Description	Unit
$\Delta \sigma_z$	Vertical stress increase	-
$P$	Point load	-
$z$	Depth	-
$r$	Radial distance	-

Comparison with Boussinesq

Westergaard's theory typically yields lower vertical stresses directly beneath the load ( $\approx 2/3$ of Boussinesq).
It is considered more appropriate for highly stratified or horizontally layered soils.

Stresses due to Area Loads

Foundations rarely apply true point loads. Instead, they distribute loads over an area (such as a strip, square, rectangle, or circle). The stress from area loads is found by integrating the point load solution over the foundation's footprint.

Fadum's Chart (Corner of a Rectangular Area)

To find the vertical stress increase ( $\Delta \sigma_z$ ) exactly under the corner of a flexible rectangular area ( $B \times L$ ) loaded with a uniform pressure ( $q$ ), engineers use the mathematical integration provided by Fadum (1948).

Fadum's Stress Influence

Vertical stress increase under the corner of a uniformly loaded rectangular area using the Fadum influence chart factor.

\Delta \sigma_z = q \cdot I_z

Variables

Symbol	Description	Unit
$\Delta \sigma_z$	Vertical stress increase	-
$q$	Uniform pressure	-
$I_z$	Influence factor obtained from Fadum's Chart	-

Using Fadum's Chart

The influence factor $I_z$ is based on the dimensionless ratios $m = B/z$ and $n = L/z$ .
Superposition: To find the stress under the center (or any other point), divide the total area into smaller rectangles such that the point of interest is a corner to all of them, then sum their individual contributions.

Newmark's Influence Chart

A graphical method developed by Nathan Newmark (1942) to determine the vertical stress under any irregularly shaped loaded area. The chart consists of concentric circles and radial lines that divide the area into smaller sectors, each contributing an equal fraction of stress to the center point.

How to Use Newmark's Influence Chart

STEP-BY-STEP

Draw the foundation footprint to scale based on the depth $z$ where the stress is required (the scale is usually indicated on the chart, where a specific length represents $z$ ).
Superimpose the scaled drawing over the center of the influence chart such that the point where you want to find the stress is exactly at the origin (center) of the chart.
Count the total number of sectors ( $N$ ) covered by the footprint of the foundation.

Newmark's Equation

Graphical method for determining vertical stress increase under any irregularly shaped foundation using Newmark's influence chart.

\Delta \sigma_z = I \cdot N \cdot q

Variables

Symbol	Description	Unit
$\Delta \sigma_z$	Vertical stress increase	-
$I$	Influence value per sector (typically 0.005)	-
$N$	Number of sectors covered by the foundation plan	-
$q$	Uniform contact pressure	-

Approximate 2:1 Method

A simple and practical approximation used to estimate the vertical stress increase at a depth $z$ below a loaded rectangular foundation.

2:1 Method Calculation

The 2:1 Method assumes the applied load spreads linearly outwards at a slope of 2 vertical to 1 horizontal (an angle of $\approx 26.6^{\circ}$ ).

It is widely used because it is simple and generally conservative for preliminary design, meaning it tends to overestimate the stress increase compared to exact elastic solutions.

Approximate 2:1 Method

Practical approximation of vertical stress increase assuming loads spread at a 2:1 (vertical:horizontal) ratio; conservative for preliminary design.

\Delta \sigma_z = \frac{P}{(B + z)(L + z)}

Variables

Symbol	Description	Unit
$\Delta \sigma_z$	Vertical stress increase	-
$P$	Total applied load	-
$B$	Width of foundation	-
$L$	Length of foundation	-
$z$	Depth below foundation base	-

2:1 Method Characteristics

The area distributing the load effectively becomes $(B+z) \times (L+z)$ .
This method is reasonably accurate for shallow depths but tends to diverge from precise elastic solutions at greater depths.

Key Takeaways

Vertical stress ( $\sigma_v$ ) is the sum of geostatic stress (weight of soil) and induced stress (external loads).
Boussinesq's Theory assumes a homogeneous, isotropic, elastic half-space and is the standard for point-load stress distribution.
Westergaard's Theory is used for highly stratified soils and predicts lower stresses than Boussinesq.
Fadum's Chart provides the exact elastic solution for the stress increase beneath the corner of a uniformly loaded rectangular area.
The 2:1 Method is a simple, conservative approximation where the stress is assumed to spread over an expanding area $(B+z)(L+z)$ .

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