2. Benchmark solutions

2.1. Calculation method used for reference solutions

In a local coordinate system, \(x\) along the \(\mathit{OA}\) axis of the beam, the bending moment, at the coordinate plane \(x\), has the expression:

\({M}_{{F}_{y}}(x)=p{\int }_{x}^{L}\left[v(\xi )-v(x)\right]d\xi\).

Arrow \(v(x)\) therefore satisfies the equation:

\(E{I}_{z}=\frac{{d}^{2}v}{{\mathrm{dx}}^{2}}=p{\int }_{x}^{L}\left[v(\xi )-v(x)\right]d\xi =-p\left[{\int }_{x}^{L}v(\xi )d\xi +(L-x)v(x)\right]\)

By differentiating the two members, we get the differential equation:

\(\frac{{d}^{3}v}{{\mathrm{dx}}^{3}}+\frac{p}{E{I}_{z}}(L-x)\frac{\mathrm{dv}}{\mathrm{dx}}=0\)

The function \(v\text{'}(x)=\frac{\mathrm{dv}}{\mathrm{dx}}\) satisfies the linear and homogeneous differential equation of the second order:

\(\frac{{d}^{\mathrm{2v}}\text{'}}{{\mathrm{dx}}^{2}}+\frac{p}{E{I}_{z}}(L-x)v\text{'}=0\),

that can be solved using Bessel functions. We then find the value of the critical linear weight equal to:

\({p}_{c}=\mathrm{7,837}\frac{E{I}_{z}}{{L}^{3}}\).

The analytical solution gives numerically:

\({p}_{c}=\mathrm{7,837}2{10}^{11}\cdot \frac{{10}^{-8}}{12}=\mathrm{1,3061667}{10}^{3}\).

2.2. Benchmark results

The critical value of the multiplier \(\lambda\): \({\lambda }_{c}=\frac{{P}_{c}}{\rho Sg}\)

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 lacking in the validation of Aster beams »