2. Benchmark solutions

2.1. Calculation method used for reference solutions

The beam in figure [Figure 1.1.1-a] verifies the equilibrium equations (plane problem).

\({V}_{y}=\frac{\mathrm{dN}}{d\theta }\), \(N+\frac{{\mathrm{dV}}_{y}}{d\theta }=-pR\), \(\frac{\mathrm{dM}}{d\theta }+{\mathrm{RV}}_{y}=0\)

(\(p\): normal constant distributed load at any point on the beam).

\(N(\theta )\) , \({V}_{y}(\theta )\) , \({M}_{z}(\theta )\) refer to the forces (normal, sharp and bending moment) at a point of the arch expressed in the local coordinate system.

Their integration with the boundary conditions:

\({V}_{y}(0)=0\), \({M}_{z}(0)=0\)

give:

\({V}_{y}(\theta )=0\), \(M(\theta )=0\), \(N(\theta )=-\mathrm{pR}\)

2.2. Benchmark results

Domestic efforts for \(\theta \mathrm{=}0°,6°,42°\) and \(60°\).

2.3. Uncertainty about the solution

Analytical solution.

2.4. Bibliographical references

[1] Report no. 2314/A of the Aerotechnical Institute « Proposal and implementation of new test cases missing the validation of ASTER beams »