1. Reference problem#
This test case is inspired by the report cited in reference [bib1].
1.1. Geometry#
The building studied is composed of 3 floors and 9 columns embedded in the floors.
Figure 1-a: Building diagram.
1.2. Material properties#
In order to obtain the center of mass \(\mathrm{Cm}\), eccentric with respect to the geometric center of \(\mathrm{0,3071}m\), a material of density \({\rho }_{2}=\mathrm{1,848}{\rho }_{1}\) is assigned to \(1/4\) of the surface of each floor (\(\mathrm{PLAN21}\), \(\mathrm{PLAN22}\) and \(\mathrm{PLAN23}\)).
Columns and parts \(\mathrm{PLAN11}\), \(\mathrm{PLAN21}\) and \(\mathrm{PLAN31}\) of the floors:
Young’s module: |
\({E}_{1}=\mathrm{4,0}E+10\mathrm{Pa}\) |
Poisson’s ratio: |
\({\nu }_{1}=\mathrm{0,15}\) |
Density: |
\({\rho }_{1}=2500\mathrm{kg}/{m}^{3}\) |
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Parts \(\mathrm{PLAN12}\), \(\mathrm{PLAN22}\) and \(\mathrm{PLAN32}\) of the floors:
Young’s module: |
\({E}_{2}=\mathrm{4,0}E+10\mathrm{Pa}\) |
Poisson’s ratio: |
\({\nu }_{2}=\mathrm{0,15}\) |
Density: |
\({\rho }_{2}=4620\mathrm{kg}/{m}^{3}\) |
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Figure 1-b: Building diagram.
1.3. Boundary conditions and loading#
Boundary condition
The columns are embedded at the foundation level.
Loading
The earthquake is applied in the \(x\) direction.
The response spectrum of oscillators in motion is obtained by superimposing four displacement spectra. Each of these displacement spectra corresponds to the response of an oscillator to a degree of freedom to the sinusoidal excitations defined in the table below:
\(\mathrm{SD}(f,\xi )=\underset{i=1}{\overset{4}{\Sigma }}\frac{{K}_{i}}{4{\pi }^{2}{f}^{2}\sqrt{{(1-\frac{{f}_{i}^{2}}{{f}^{2}})}^{2}+4{\xi }^{2}\frac{{f}_{i}^{2}}{{f}^{2}}}}\)
In particular, the frequencies and damping selected are close to the first four modes of the structure.
Frequency \({f}_{i}(\mathrm{Hz})\) |
Amplitude \({K}_{i}(m)\) |
Dampening \(\xi\) |
|
sine 1 |
1.51 |
0.15 |
0.05 |
sine 2 |
2.05 |
0.25 |
0.05 |
sine 3 |
2.34 |
0.25 |
0.05 |
sine 4 |
4.86 |
0.30 |
0.05 |
Table 1-a: Characteristics of the excitations used.
Neglected modes are represented by a pseudo-mode.