1. Reference problem

1.1. Geometry

The system consists of a pole resting on the ground and subjected to seismic stress. It is modelled by a mass, its connection with the ground by a spring \({k}_{0}\) whose behavior relationship reflects a non-linearity of the effort-displacement type.

Characteristics of the pole:

length: \(L=\mathrm{2 }m\);

section: \(S=\mathrm{0,3}{m}^{2}\).

1.2. Material properties

Pole mass: \(m=\mathrm{450 }\mathrm{kg}\).

Link spring stiffness: \({k}_{\mathrm{0 }}={10}^{5}N/m\).

1.3. Boundary conditions and loads

Boundary conditions

The only authorized movements are translations along the \(X\) axis: \(\mathrm{dy}=\mathrm{dz}=0\).

The corrective force \({F}_{c}\) due to the nonlinearity of the ground is defined by the following relationship:

\({F}_{c}(x)=\frac{f({x}_{\mathrm{seuil}})}{{x}_{\mathrm{seuil}}}-f(x)\) with, if \(x>{x}_{\mathit{seuil}}\), \(f(x)={k}_{0}\left[1–\frac{\mid x\mid }{{x}_{0}}\right]x\)

We take \({x}_{\mathrm{seuil}}={10}^{-6}m\), \({k}_{0}={10}^{5}N/m\), and \({x}_{0}=\mathrm{0,1}m\).

Under the keyword RELA_EFFO_DEPL of the DYNA_VIBRA operator, we therefore impose the function: \({F}_{c}(x)=\frac{{k}_{0}}{{x}_{0}}\mathit{x.}[\mid x\mid -{x}_{\mathit{seuil}}]\) if \(∣x∣>{x}_{\mathit{seuil}}\)

\({F}_{c}(x)=0\) if \(∣x∣\le {x}_{\mathit{seuil}}\)

Loading

The ground is subjected to an acceleration \(\gamma (t)\) in the \(x\) direction, constructed in such a way that the movement of the mass-spring system is sinusoidal \(x=\mathrm{a.sin}(\omega t)\) with \(a=\mathrm{0,01}\) and \(\omega =\pi /4\).

1.4. Initial conditions

In the initial state, the system is released from its equilibrium position with a speed \({v}_{\mathrm{0 }}\): to \(t=0\), \(\mathrm{dx}(0)=0\), \({v}_{0}=\mathrm{dx}/\mathrm{dt}(0)=a\mathrm{.}\omega\) .