2. Reference solution#

2.1. Calculation method used for the reference solution#

This test is developed in detail in reference [bib1].

The fundamental dynamic equation, in relative motion of the mass-spring system with respect to the ground, is written as: \(\ddot{x}+\frac{k(x)}{m}x=\gamma (t)\).

For a displacement of the form \(x=\mathrm{a.sin}(\omega t)\) and \(\ddot{x}=-a{\omega }^{2}\mathrm{sin}(\omega t)\), we obtain from the equation of motion the shape of the accelerogram:

\(\gamma (t)=a\mathrm{sin}(\omega t)\left[-{\omega }^{2}+\frac{{k}_{0}}{m}(1-\frac{\mid a\mathrm{sin}(\omega t)\mid }{{x}_{0}})\right]\)

The fundamental frequency \({f}_{0}\) of the undamped oscillator is \({f}_{0}=\frac{1}{2\pi }\sqrt{\frac{{k}_{0}}{m}}\).

2.2. Benchmark results#

Fundamental frequency \({f}_{0}\) of the undamped oscillator.

Displacements relating to the moments 2, 6, 10, 10, 14 and 18 seconds.

2.3. Uncertainty about the solution#

None if we calculate the Duhamel integral analytically [bib2].

2.4. Bibliographical references#

    1. LALUQUE, P. LABBE, S. PETETIN and A. TIXIER: Seismic response of a reactor building PWR1300 taking into account the separation between the foundation and the ground. Note SEPTEN TA83 .06 (May 1984).

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