Vertical Stresses - Theory & Concepts

Vertical Stresses

In geotechnical engineering, we must calculate the vertical stress ( $\sigma_v$ ) at any depth to predict settlement and bearing capacity. Stresses arise from two sources:

Geostatic Stresses: Due to the self-weight of the soil.
Induced Stresses: Due to external loads (foundations, embankments, vehicles).

Stresses due to Self-Weight

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

Geostatic Stress Equation

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	-

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

Stresses due to Point Loads (Boussinesq)

Boussinesq (1885) provided a solution for the stress distribution in a semi-infinite, homogeneous, isotropic, elastic medium due to a point load $P$ at the surface.

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 decreases rapidly with depth ( $1/z^2$ ) and radial distance.

Westergaard's Theory

Assume the soil is reinforced by horizontal inextensible sheets (e.g., layered sedimentary soils like varved clays).

Westergaard's Equation

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	-

Typically yields lower stresses than Boussinesq ( $\approx 2/3$ of Boussinesq directly under load).
More appropriate for highly stratified soils.

Stresses due to Area Loads

Foundations apply loads over an area (strip, square, rectangle, circle). We integrate the point load solution over the area.

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	-

Based on dimensionless ratios $m = B/z$ and $n = L/z$ .
Superposition: Divide the area to find stress at non-corner points.

Newmark's Influence Chart

A graphical method to determine vertical stress under any irregularly shaped loaded area.

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 method assumes the load spreads linearly outwards at a slope of 2 vertical to 1 horizontal (an angle of $\approx 26.6^{\circ}$ ).
Given a foundation of width $B$ , length $L$ , and applied load $P$ (or distributed load $q$ ), the stress increase $\Delta \sigma_z$ at depth $z$ is:

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	-

The area distributing the load becomes $(B+z) \times (L+z)$ .
Robust for shallow depths but diverges 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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