2.3. Stresses due to a load surcharge on the Ground Surface

To solve the problem of the stress distribution in the soil, induced by a surcharge at the surface or within the soil mass, it is necessary to assume a stress-strain relationship for the soil. Commonly, it is assumed that the soil follows the rules of linear elasticity (theory of elasticity). The solutions that are generated by this assumption are known as elastic solutions and some of them are provided below.

Point vertical load on the surface of an elastic half-space

Where: 

Values of P🇿 as a function of r/z are given in the following Table 2.1:

Uniformly distributed load along a strip of infinite length

The strip with a width of 2a extends infinitely in the direction perpendicular to the one depicted in the figure (along the horizontal y direction).

The following tables 2.2, 2.3 and 2.4 are providing values of σ𝗓/P, σₓ/P and τₓ🇿/P for different values of x/a and z/a:

The values for the principal stresses σ₁ and σ₃ beneath the strip are given in the following figure:

Load from embankment with infinite length

The value for normal vertical stress, σ𝗓, at different depths on the z-axis, is given by the diagram below:

Uniform load distributed on a circular surface

The value of the vertical normal stress, σ𝗓, at different depths along the vertical axis passing through the center of the circle.

For the calculation of the value σ🇿 and at points outside the vertical axis, one can use Table 2.5 or the stress contours diagram below:

Finally, the figure below provides the stress contours σ₁/P, σ₃/Ρ and (σ₁-σ₃)/Ρ, for the principal stresses that are developed below the circular surface. 

Uniform load distributed on a rectangular surface

In this case, it’s possible to calculate directly the σ🇿 below one of the corners of the rectangle by using the following diagram. However, by using the principle of superposition it’s possible to calculate σ🇿 below any point located either inside or outside the layout of the rectangle.

Diagram p-q - Stress Path

When the principal stresses σ₁ and σ₃ act at the horizontal and vertical planes (σ₁ = σᵥ and σ₃ = σ_h or σ₁ = σ_h and σ₃ = σᵥ), instead of drawing the entire Mohr Circle, we can draw only point A with coordinates:

The diagram in that case is called diagram p-q. If the stress state at the point that we are analyzing changes, the new stress state will be assigned to a new point A’ at the diagram p-q.

The line AA’ shows the change in the stress state at the point that we are analyzing and is called stress path. The arrow is showing the direction of the change.