5. Direct transient seismic response#

Direct integration is feasible either with hypotheses of linear behavior: operator DYNA_LINE_TRAN [U4.53.02] or with hypotheses of non-linear behavior: operator DYNA_NON_LINE [U4.53.01]. Apart from the way to take into account seismic loading (see [§3.3]), the syntaxes of DYNA_NON_LINE and DYNA_LINE_TRAN are identical.

5.1. Taking into account depreciation equivalent to modal depreciation#

Generally, the most accurate information we have on damping comes from vibration tests which make it possible to determine, for a given resonance frequency \({f}_{i}\), the corresponding resonance width and therefore the damping reduced \({\xi }_{i}\) at this resonance. It is therefore necessary to be able to take into account, in a direct transitory calculation, amortization equivalent to modal dampening.

From the spectral development of the identity matrix:

\(\text{Id}=\sum _{i=1}^{n\text{\_modes}}\frac{{X}_{i}{X}_{i}^{T}K}{{X}_{i}^{T}K{X}_{i}}=\sum _{i=1}^{n\text{\_modes}}\frac{{X}_{i}{X}_{i}^{T}K}{{M}_{G\text{\_}i}\text{.}{\omega }_{i}^{2}}\)

we show:

  • that we can develop the damping matrix of structure \(C\) in a series of natural modes:

\(C=\sum _{i=1}^{n\text{\_modes}}{a}_{i}\text{.}(K\text{.}{\Phi }_{i}){(K\text{.}{\Phi }_{i})}^{T}\)

  • and that, taking into account the definition of the critical depreciation percentage:

\({\Phi }_{i}^{T}\text{.}C\text{.}{\Phi }_{i}=2\text{.}{M}_{G\text{\_}i}\text{.}{\omega }_{i}\text{.}{\xi }_{i}\text{.}{a}_{i}=2\text{.}\frac{{\xi }_{i}}{{K}_{G\text{\_}i}\text{.}{\omega }_{i}}\)

It is therefore recommended that the user specify (the syntaxes of DYNA_NON_LINE and DYNA_LINE_TRAN are identical), the values of the modal damping for each natural frequency using the keyword factor AMOR_MODAL.

This amounts to imposing a damping force that is proportional to the relative speed of the structure:

\({F}_{\text{amo}}=C{\dot{X}}_{r}\) with \(C=\sum _{i=1}^{n\text{\_modes}}2\text{.}\frac{{\xi }_{i}}{{K}_{G\text{\_}i}\text{.}{\omega }_{i}}\text{.}(K\text{.}{\Phi }_{i}){(K\text{.}{\Phi }_{i})}^{T}\)

5.2. Taking into account a multi-support request with the restoration of relative and absolute fields#

By default, quantities are calculated in the relative coordinate system. In DYNA_NON_LINE and DYNA_LINE_TRAN, we use a syntax identical to that of DYNA_TRAN_MODAL (presence of the keywords MODE_STAT and MULT_APPUI = “OUI”) to calculate them in the absolute coordinate system.