3. Seismic response by spectral method#
The spectral method is an approximate technique for evaluating the maximum response of the structure from the response maximums of each modal oscillator read from the excitation oscillator spectrum.
3.1. Spectral response of a single-press modal oscillator#
The maximum relative displacement response of a modal oscillator \(({\omega }_{i},{\xi }_{i})\) for a direction \(X\) is determined by reading from an absolute pseudo-acceleration oscillator spectrum giving [§1.4.2] the value of the absolute pseudo-acceleration \({a}_{\mathrm{iX}}=\mathrm{Sro}\ddot{{x}_{X}}(A,{\xi }_{i},{\omega }_{i})\) and by dividing by \({\omega }_{i}^{2}\) by, from where:
\({q}_{\mathit{iXmax}}={p}_{\mathit{iX}}\frac{\mathit{SRO}{\ddot{x}}_{X}(A,{\xi }_{i},{\omega }_{i})}{{\omega }_{i}^{2}}={p}_{\mathit{iX}}\frac{{a}_{\mathit{iX}}}{{\omega }_{i}^{2}}\) eq 3.1-1
The contribution of this oscillator to the relative displacement of the structure for component \({x}^{k}\) depends on the participation factor and on the component \({\varphi }_{i}^{k}\) in physical space:
\({x}_{\mathit{iXmax}}^{k}={\phi }_{i}^{k}{q}_{\mathit{iXmax}}={\phi }_{i}^{k}{p}_{\mathit{iX}}\frac{{a}_{\mathit{iX}}}{{\omega }_{i}^{2}}\) eq 3.1-2
and the contribution to the absolute pseudo-acceleration \({\ddot{x}}^{k}\) is the same \({\ddot{x}}_{\mathit{iXmax}}^{k}={\phi }_{i}^{k}{p}_{\mathit{iX}}{a}_{\mathit{iX}}\).
3.2. Spectral response of a multi-press modal oscillator#
We proceed in the same way to determine, from the value read \({\ddot{S}}_{\mathrm{jX}}\) on the absolute pseudo-acceleration oscillator spectrum associated with \({\ddot{s}}_{j}\), the contribution of the support \(j\) in the direction \(X\):
\({q}_{\mathit{iXmax}j}={p}_{\mathit{ijX}}\frac{\mathit{SRO}\ddot{{s}_{j}}(A,{\xi }_{i},{\omega }_{i})}{{\omega }_{i}^{2}}=\frac{{\ddot{S}}_{\mathit{jXX}}}{{\omega }_{i}^{2}}\) eq 3.2-1
The expression of the contribution of this oscillator to the relative displacement of the structure for component \({x}^{k}\) in physical space and for an imposed movement \(j\) becomes:
\({x}_{\mathit{iXmax}j}^{k}={\phi }_{i}^{k}{q}_{\mathit{iXmax}j}={\phi }_{i}^{k}{p}_{\mathit{ijX}}\frac{{\ddot{S}}_{\mathit{jX}}}{{\omega }_{i}^{2}}\) eq 3.2-2
3.3. Generalization to other quantities#
Note:
The spectral analysis method is strictly limited to quantities that are linearly dependent on linear elasticity displacements: deformations, stresses, generalized forces, nodal forces, support reactions.
In particular, it cannot be applied to equivalent quantities of deformation or stresses (VON MISES) .
For each quantity \({R}^{k}\), component of a field by elements, it is possible to calculate the modal component \({r}_{i}^{k}\) associated with the eigenmode \({\varphi }_{i}\), which leads to:
\({R}_{\mathit{iXmax}}^{k}={r}_{i}^{k}{q}_{\mathit{iXmax}}={r}_{i}^{k}{p}_{\mathit{iX}}\frac{{a}_{\mathit{iX}}}{{\omega }_{i}^{2}}\) eq 3.3-1
or
\({R}_{\mathit{iXmax}j}^{k}={r}_{i}^{k}{q}_{\mathit{iXmax}j}={r}_{i}^{k}{p}_{\mathit{ijX}}\frac{{\ddot{S}}_{\mathit{jX}}}{{\omega }_{i}^{2}}\) eq 3.3-2