2. Benchmark solution#

2.1. Calculation method used for the reference solution#

The aim is to compare the response of a symmetric system consisting of two mass-spring systems that are identical to the response of a mass-spring system. Both problems, described in detail in reference [bib2], are addressed by the same accelerogram.

First, the natural frequencies \({f}_{i}\), the associated eigenvectors normalized with respect to the modal mass \({\Phi }_{\text{Ni}}\) and the static modes \(\Psi\) of the system (analytical values) are calculated. The generalized response of the multi-supported system is then calculated by analytically solving the Duhamel integral [bib1]. Finally, the relative displacement of the shock nodes is restored on a physical basis, which allows us, after having calculated the field of the training displacements, to calculate the field of the absolute displacements.

We calculate function \(\mathit{diff}\) defined as being the difference between the absolute movement of the shocking node on a mobile obstacle and that of the shocking node on a fixed obstacle. We check that it is really zero for various moments.

For G modeling, only problem 2 is modeled. There are no reference solutions other than material data (discharge criterion and stiffness).

2.2. Benchmark results#

Relative and absolute displacements at shock nodes.

2.3. Uncertainty about the solution#

Comparison between two equivalent models.

2.4. Bibliographical references#

  1. J.S. PRZEMIENIECKI: Theory of matrix structural analysis New York, MacGraw-Hill, 1968, pp. 351-357.

  2. Fe. WAECKEL: Use and validation of developments made to calculate the seismic response of multi-supported structures — HP52 /96.002.