2. Benchmark solution

2.1. Calculation method used for the reference solution

The axisymmetric linear static mechanics problem under consideration can be solved analytically. The response to the load (volume force and surface force) is solved independently and then summed up.

Quadratic density force \({F}_{V}(r)=\alpha {r}^{2}\)

We consider the equilibrium equations in cylindrical coordinates:

which are simplified, given the axial symmetry, by:

By using the law of behavior and then the deformation-displacement relationships, we arrive at the following differential equation:

The density force applied is of the type: fV=.r²

The solution to the differential equation is then written as:

eq 2.1-1

The two integration constants \({c}_{1}\) and \({c}_{2}\) are determined using the boundary conditions:

We get:

Surface force type pressure \({F}_{S}({R}_{\text{int}})=P\)

The problem to be solved is of the same nature, but with zero applied volume force: \({f}_{V}=0\) or \(\alpha =0\).

The solution on the go [éq 2.1-1] is then written:

, having to respect the conditions:

This results in:

Eq 2.1-2

2.2. Benchmark results

Digital application:

height	= \(0.5m\);
inner radius	= \(1m\);
outer radius	= \(1.4m\);
\(E\)	= \(10\mathrm{Pa}\);
\(\rho\)	= \(1\mathrm{kg}/{m}^{3}\);
\(\nu\)	= \(0.3\);
\(\alpha\)	= \(1N/{m}^{5}\);
\(P\)	= \(1N/{m}^{2}\).

by injecting the numerical values into the solutions [éq 2.1-1] and [éq 2.1-2] we find after summation:

2.3. Uncertainties about the solution

None (analytical reference solution).