Boussinesq problem

This example is used to test the ability of numgeo to simulate elastic deformations for correctness. For selected boundary value problems, which are standard problems in geotechnical engineering, an analytical solution exists and is therefore ideally suited for verification. The same problem is also used by Plaxis as a benchmark.

A cylindrical soil body with a radius of 10 m and a height of 10 m is considered. A circular surface load with a radius of \(r=0.1\) m and a magnitude of \(\Delta \sigma = 10\) kPa is applied to the surface of the cylinder centrally along the axis of symmetry.

The corresponding finite element model is given below:

Left: finite element model of the BVP. The soil is discretized using u8-solid-ax elements (8-noded axisymmetric elements with quadratic interpolation).

Analytical solution

Vertical stress as function of the depth below the load \(z\):

\[ \sigma_v(z) = \Delta \sigma_v \left(1-\left(\dfrac{1}{1+\left(r/z\right)^2}\right)^{3/2}\right) \]

Horizontal stress as function of the depth below the load \(z\):

\[ \sigma_v(z) = \dfrac{\Delta \sigma_v}{2} \left( \left(1+2\nu\right)-\dfrac{2(1+\nu)}{\left(1+\left(r/z\right)^2\right)^{1/2}} + \dfrac{1}{\left(1+\left(r/z\right)^2\right)^{3/2}} \right) \]

Input files

The input files for the benchmark simulation can be downloaded here

Material

For the solid a linear elastic constitutive model is chosen. The Young's modulus is \(20\) MPa and the Poisson's ratio \(0.3\).

Results

The following figure presents the comparison of the simulation results obtained with numgeo and the analytical solution. It can be seen, that the simulation results are in perfect agreement with the reference solution.

Distribution vertical and horizontal stress along the axis of symmetry below an external circular load of $\Delta \sigma_v = 10$ kPa.