2. Order description DYNA_ISS_VARI

The operator DYNA_ISS_VARI [U4.53.31] makes it possible to calculate the response of a structure subjected to a variable seismic movement in space based on a coherence function, the interface impedance matrix and the seismic force. These can be calculated using Code_Aster/MISS3D chaining, cf. [U2.06.07]. At the output of DYNA_ISS_VARI, we obtain, in generalized coordinates, the spectral response density or a temporal response to a temporal excitation.

More precisely, modal response spectral vectors (resulting from a spectral decomposition of the coherence matrix) are constructed by means of a harmonic calculation into generalized components. Then, the power spectral density (DSP) of the modal response or the temporal response in generalized components is determined.

The results that can be obtained using the DYNA_ISS_VARI command (for cases with or without spatial variability) are as follows:

• Calculation of transfer functions between the seismic excitation and the response of the structure (the transfer functions are obtained by choosing an excitation by white noise).

• Calculation of the response spectral density for the case where the seismic excitation is given by a spectral density (the Kanai—Tajimi spectrum is most often used to describe seismic excitation, see also [R4.05.02]).

• Calculation of the temporal response in generalized components. The simulation of a temporal response realization then makes it possible to determine floor spectra.

2.1. Seismic analysis with ISS in the frequency domain

The MISS3D software is based on a frequency resolution on the ground side; it makes it possible to determine the impedance matrices as well as the seismic force at the interface. The soil-structure interaction problem amounts to solving the harmonic dynamics equation at the interface:

(2.1)\[ \ left [\ underset {\ text {Building balance equation}}} {\ underset {\ underbrace {}} {{K} _ {b} +i\ omega {C} _ {b}\ mathrm {-} {-} {\ omega}} {\ omega}} ^ {omega} ^ {2} ^ {2} {M} _ {b}}} +\ underset {\ mathit {b}}} +\ underset {\ mathit {b}}} +\ underset {\ mathit {Impedance} d\ text {'}}\ mathrm {-} -} {\ omega}} {\ omega}} ^ {omega} ^ {2} {M} _ {b}} +\ underset {\ mathit {b}}} +\ underset {\ mathit {\ mathit {sol}} {\ underset {\ underbrace {}} {{K}} {{K} _ {s} (\ omega)}}\ right] q (\ omega)\ mathrm {-}\ underset {\ text {text {text {Seismic force}}} {\ underbrace}} {\ underset {\ underbrace}}} {\ underset {\ underbrace}} {\ underbrace {}} {\ underbrace {}}} {\ underbrace {}} {\ underbrace {}}} {{f} _ {s} (\ omega)}}\ mathrm {=} 0\]

In equation (), \(q(\omega )\in {ℂ}^{m}\) is the vector of generalized unknowns describing the displacement.

In the calculations of ISS with MISS3D, one must provide a free field accelerogram. The Fourier transform of this signal and the impedance matrix calculated by MISS3D make it possible to determine the seismic force in the frequency domain, \({f}_{s}(\omega )\).

For the following, the complex transfer function on the move \(H(\omega )\mathrm{\in }{\text{Mat}}_{\mathrm{ℂ}}(m,m)\) is defined as:

(2.2)\[ H (\ mathrm {\ omega}) {f} _ {s} (\ mathrm {\ omega}) =q (\ mathrm {\ omega})\]

If arousal is assumed to be a Gaussian stationary stochastic process, then the answer is also a Gaussian stationary stochastic process. This is true because the transfer function is a linear filter. Thus, we can write the relationship between the spectral densities of the seismic force and the response:

(2.3)\[ H (\ omega) {S} _ {f} (\ omega) {H} ^ {\ text {*}} (\ omega) = {S} _ {q} (\ omega)\]

Where \({H}^{\text{*}}\) refers to the transposed conjugate complex of \(H\) and \({S}_{f}\) is the seismic force spectral density matrix that can be evaluated from \(L\) realizations of the temporal seismic force \({f}_{s}^{l}(t)\):

\[\]

: label: eq-4

{S} _ {f} (omega) =frac {1} {1} {2pi}frac {1} {L}sum _ {l} {f} _ {s} {f} _ {s} ^ {l} _ {s} ^ {l} _ {s} ^ {l}text {*} {f} _ {f} _ {s} {f} _ {s} {f} _ {s} ^ {f} _ {s} ^ {l} _ {s} ^ {l} _ {s} ^ {l} _ {s} ^ {f} _ {s} ^ {l} _ {s} ^ {l} _ {s}

In this expression, \(T\) refers to the time interval and \(W(t)=1/\sqrt{(T)}\) is the natural window to \(\mathrm{[}\mathrm{0,}T\mathrm{]}\) (cf. [bib10]). It is an unbiased estimator of spectral density [bib10].

Note:

You can also write:

\({S}_{f}(\omega )=\underset{T\to \infty }{\text{lim}}\frac{1}{2\pi T}E({f}_{s}(\omega ){f}_{s}\text{*}(\omega ))\)

where \(E(\cdot )\) refers to the mathematical expectation operator.

2.2. Taking account of spatial variability

DYNA_ISS_VARI is based on a probabilistic description of the incident seismic field by its power spectral density (DSP). The latter is generally constructed using a point spectrum and a spatial coherence function. Thus, the cross-spectral density of ground motion in a free field is written as:

\[\]

: label: eq-5

{S} _ {u} (x, xtext {“,}omega) =gamma (x, xtext {“,}omega) {S} _ {0} (omega)

Where \(\mathrm{\gamma }(x,x\text{',}\mathrm{\omega })\) is the coherence function of the seismic signal between two points \(x\) and \(x\text{'}\) and \({S}_{0}\in ℝ\) is the point spectral density of the seismic movement in free field. It is assumed here that the spectral density is set to the interval \(O=\left[-{\mathrm{\omega }}_{s},+{\mathrm{\omega }}_{s}\right]\) and that it is zero outside of this frequency range. If we discretize with respect to the spatial variable, \(x\) and \(x\text{'}\), we obtain the following spectral density matrix:

(2.4)\[ {S} _ {u} (\ omega) =\ gamma (\ omega) {S} _ {0} (\ omega)\ in {\ text {Mat}}} _ {} (m, m)\]

In this expression, \(\gamma\) (of dimension \(m\times m\)) designates the coherence matrix and its components are denoted by \({\gamma }_{\text{ij}}(\omega )=\gamma ({x}_{i},{x}_{j},\omega )\). The elements of \({S}_{u}(\omega )=\left[{S}_{\text{ij}}(\omega )\right]\in \text{Mat}(m,m)\) are cross spectral densities, \({S}_{\text{ij}}(\mathrm{\omega })=S({x}_{i},{x}_{j},\mathrm{\omega })\),, \(m\) being the number of spatial discretization points. In general, it is assumed that the incident seismic field is homogeneous, i.e. that the stochastic description of the field depends only on the distance, \(d=\mid x-x\text{'}\mid\), but that it is independent of the spatial position.

The calculation of the seismic response is based on the spectral decomposition of the coherence matrix \(\gamma (\omega )\). Note that this is a spectral decomposition in relation to the spatial variable and not in relation to the time variable. Thus, we have:

(2.5)\[ {S} _ {u} (\ omega) =\ Phi (\ omega) =\ Phi (\ omega)\ Lambda (\ omega) {*} (\ omega) {S} _ {0} (\ omega)\]

Where \(\mathrm{\Phi }\) is a matrix containing the eigenvectors \({\varphi }_{k}\) of the coherence matrix \(\gamma\) and \(\Lambda\) is the diagonal matrix containing the eigenvalues, \(\mathrm{\Lambda }=\text{diag}({\mathrm{\lambda }}_{k}),k=\mathrm{1,}\dots m\). Later, we will talk about POD (Proper Orthogonal Decomposition) modes to refer to \({\mathrm{\varphi }}_{k}\). This makes it possible to distinguish them from mechanical modes. From the expression in equation (), we define:

(2.6)\[ {s} _ {u} ^ {k} (\ mathrm {\ omega}) = {\ mathrm {\ omega}} _ {k} (\ mathrm {\ omega})\ sqrt {{\ mathrm {\ mathrm {\ lambda}}} _ {\ lambda}}} _ {0} (\ mathrm {\ lambda}}} _ {0} (\ mathrm {\ lambda}}} _ {0} (\ mathrm {\ lambda}}} _ {0} (\ mathrm {\ lambda}}} _ {0}) omega})},\ forall\ mathrm {\ omega}\ in O\]

Knowing that \({S}_{u}(\omega )=\sum _{k=1}^{m}{s}_{u}^{k}(\omega ){s}_{u}^{k}\text{*}(\omega )\). The spectral density of the seismic force is obtained from the movement at the ground interface by the matrix transfer function \(G(\omega )\):

(2.7)\[ {S} _ {f} (\ omega) =G (\ omega) =G (\ omega) =G (\ omega) =G\ text {*} (\ omega)\]

And so:

(2.8)\[ {s} _ {f} ^ {k} (\ omega) =G (\ omega) =G (\ omega) {\ omega) {\ varphi} _ {k} (\ omega)}\ sqrt {{S} _ {0} (\ omega)}\ sqrt {{S} _ {0} (\ omega)},\ forall\ omega\ in\ Omega\]

In studies of ISS with Code_Aster, the transfer function \(G(\mathrm{\omega })\) is calculated by MISS3D, this is described in more detail in section § 2.2.1.

The expression () is the input for traditional seismic analysis, see equation (). The model is « reduced » if you can truncate expression () to \(K\le m\) modes POD.

As we have just seen, the calculation of seismic forces with spatial variability of the incident field involves a spectral decomposition of the coherence matrix \(\gamma\). For the rest of the calculations, only a small number of POD modes are retained, namely \(K\le m\) modes. The precision parameter gives the portion of the « energy » of the matrix that is conserved by retaining only a reduced number of vectors and eigenvalues of \(\gamma\). If we designate by \(K\) the number of POD modes selected (we retain the \(K\) largest eigenvalues), we have:

(2.9)\[ \ text {precision} =\ frac {\ sum _ {i=1} ^ {i=1} ^ {i=1} ^ {1}} {\ sum _ {i=1} ^ {M} {\ lambda} _ {i} ^ {2}}\]

The value of \(\mathrm{0,999}\) is taken by default for precision.

2.2.1. Coherence functions

The coherence function depends on the separation distance \(d\) between two points \(x\) and \(x\text{'}\) and on the frequency. In general, it is expressed by a module term and a phase term:

(2.10)\[ \ gamma (d,\ omega) =489\ gamma (d,\ omega) ===\ mathrm {exp} (-i\ theta (\ omega, d))\]

The term \(\mathrm{exp}(-i\theta (\omega ,d))\) represents the phase difference due to the different arrival times of the waves. The term amplitude corresponds to « pure incoherence. » It can be evaluated from DSP (autospectra and interspectra) at points \(x\) and \(x\text{'}\):

(2.11)\[ {◆\ gamma (d,\ omega) ===} ^ {2} =\ frac {S {(\ omega, x, x\ text {'})} ^ {2}} {S (\ omega, x) {S (\ omega, x) S (\ omega, x, x\ text {'})} {S (\ omega, x, x, x\ text {'})} {S (\ omega, x, x)}\]

The coherence functions currently available in DYNA_ISS_VARI are the Mita&Luco coherence function and the Abrahamson coherence function (for the rock). A phase term is not introduced.

The consistency function of Mita & Luco [bib5, bib6] is written as:

(2.12)\[ \ gamma (d, f) =\ text {exp}\ left [- {\ left (\ frac {\ alpha\ omega d} {{v} _ {s}}\ right)}}\ right)} ^ {2}\ right)} ^ {2}\ right]\]

In this expression, \({v}_{s}\) is the speed of propagation of the SH wave (typically \(200-\mathrm{1000 }m/s\)) and the parameter \(\alpha\) can vary from \(\mathrm{0,1}\) to \(\mathrm{0,5}\) depending on the case but is generally taken to be equal to \(\mathrm{0,5}\). If we choose \(\alpha =0.0\), then we perform a calculation without spatial variability.

We can see that the correlation length for the coherence function of Mita and Luco [bib5, bib6], is characterized by the expression \((\omega \alpha /{v}_{s}{)}^{-1}\).

The generic Abrahamson consistency function [bib9] is written, for \(\omega =2\pi f\):

(2.13)\[ \begin{align}\begin{aligned} \ gamma (d, f) = {\ left [1+ {\ left (\ frac {f\ mathrm {tanh} ({a} _ {3} d)} {{f} _ {c} {a} {a}} _ {1}}\ right)}} ^ {\ mathit {n 1}}\ right)} ^ {-0.5} {\ left [1+ {\ left (\ frac {f}\ mathrm {tanh} ({a} _ {3} d)} {{f} _ {c} {a} _ {2}}\ right)} ^ {\ mathit {n2}}}\ right]}\ right]} ^ {-0.5}\\ With the following parameter values for horizontal movement:\end{aligned}\end{align} \]

(2.14)\[ {f} _ {c} =-\ mathrm {1,886} +\ mathrm {2,221}\ mathrm {ln} (4000/ (d+1) +\ mathrm {1,886}) +\ mathrm {1,5}) {n} _ {1} =\ mathrm {7,02} {n} _ {2} =\ mathrm {5,1} -\ mathrm {0,51}\ mathrm {ln} (d+10) {a} _ {1} =\ mathrm {1,647} {a} _ {2} =\ mathrm {1.01} {a} _ {3} =\ mathrm {0.4}\]

The Abrahamson coherence function for a rock or an average ground (EPRI 1015110, 2007) is written, for \(\omega =2\pi f\):

(2.15)\[ \begin{align}\begin{aligned} \ gamma (d, f) = {\ left [1+ {\ left (\ frac {f\ mathrm {tanh} ({a} _ {3} d)} {{f} _ {c} (d) {a} (d) {a} _ {1}}}\ right)} ^ {\ mathit {1}}}\ right]} ^ {-0.5} {\ left [1+ {\ left (\ frac {f\ mathrm {tanh} ({a} _ {3} d)} {{a} _ {2}}\ right)} ^ {\ mathit {n2}}}\ right]}\ right]} ^ {-0.5}\\ For the rock, we have the following values of the parameters for the horizontal movement:\end{aligned}\end{align} \]

(2.16)\[ {f} _ {c} =\ mathrm {27.9} +\ mathrm {27.9} +\ mathrm {4.82}\ mathrm {ln} (\ mathrm {ln} ((d+1) -\ mathrm {ln} ((d+1) -\ mathrm {3.6}} (d+1)))} ^ {2} {n} _ {1} =\ mathrm {3.8} -\ mathrm {3.8} -\ mathrm {0.04}\ mathrm {ln} (d+1) +\ mathrm {0.0105} {{(\ mathrm {ln}} {ln}) {ln} (\ mathrm {ln}) {ln} {ln} {{ln} {{ln}} {(\ mathrm {ln}}) {ln} {ln} ((n)} {ln} (\ mathrm {ln}}) {ln} {ln} ((n)} {ln} {ln} ( {n} _ {2} =\ mathrm {16.4} {a} _ {1} =\ mathrm {1.0} {a} _ {2} =40 {a} _ {3} =\ mathrm {0.4}\]

This coherence function is an empirical model based on the 78 earthquakes recorded at Pinyon Flat (USA) by Abrahamson. It is also used for other types of soil for which it is considered to be a preservative.

For the average ground, we have the following values of the parameters for horizontal movement:

(2.17)\[ {f} _ {c} =\ mathrm {14.3} +\ mathrm {2.35}\ mathrm {ln} (d+1) {n} _ {1} (d) =2 {n} _ {2} =15 {a} _ {1} =\ mathrm {1.0} {a} _ {2} =\ mathrm {15.8} -\ mathrm {0.044} d {a} _ {3} =\ mathrm {0.4}\]

2.2.2. Modeling seismic forces with Code_Aster

1. • Case of a rigid foundation

The interface modes are the six rigid body modes. The modal seismic force calculated by MISS3D is then written for a unit excitation (white noise):

(2.18)\[ {f} _ {s} (\ omega) = {K} _ {s} (\ omega) {x} _ {0}\]

Where \({K}_{s}(\omega )\) is the modal impedance matrix and \({x}_{0}=(1\text{.}\mathrm{,0}\text{.}\mathrm{,0}\text{.}\mathrm{,0}\text{.}\mathrm{,0}\text{.}\mathrm{,0}\text{.})\) for a seismic excitation in the \(x\) direction, \({x}_{0}=(0\text{.}\mathrm{,1}\text{.}\mathrm{,0}\text{.}\mathrm{,0}\text{.}\mathrm{,0}\text{.}\mathrm{,0}\text{.})\) for a seismic excitation in \(y\) and \({x}_{0}=(0\text{.}\mathrm{,0}\text{.}\mathrm{,1}\text{.}0\text{.}\mathrm{,0}\text{.}\mathrm{,0}\text{.}\mathrm{,0}\text{.})\) for a vertical earthquake. The seismic force is non-zero only for the modal component in the direction of earthquake \((x,y,z)\). Likewise, the coherence function is constructed only for the degrees of freedom of translation in the direction of the earthquake. The other degrees of freedom are not affected.

To take account of spatial variability, we determine the modal participation for each mode PODcaractérisant the spatial variability:

\[\]

: label: eq-21

{f} _ {s} ^ {k} (omega) = {K} _ {s} (omega) {theta} ^ {T} {s}} _ {u} ^ {k} _ {u} ^ {k}

Where \(\Theta\) is the matrix containing the (mechanical) interface modes.

1. • Case of a soft foundation

Interface modes are the \(d\times 6\) unit modes relating to the \(d\) interface nodes. The modal seismic force calculated by MISS3D is then written as:

(2.19)\[ {f} _ {s} (\ omega) = {K} _ {s} (\ omega) {x} _ {0}\]

\({x}_{0}\) is the modal participation vector with \(1\) for degrees of freedom relating to the direction of the earthquake and zeros for the other directions. Likewise, the seismic force and the coherence function are non-zero only for the degrees of freedom in the direction of the earthquake. We build the coherence matrix and then the vectors \({s}_{u}^{k}\) for the degrees of freedom in the direction of the earthquake:

(2.20)\[ {f} _ {s} ^ {k} (\ omega) = {K} _ {s} (\ omega) {s} _ {u} ^ {k}\]

1. • Case of an « any » foundation

In these cases, corresponding either to a depressed foundation, or to a case of soil-fluid-structure interaction, or to a case where the interface modes are any modes different from unit modes relating to the nodes of the interface, the modal participation vector \({x}_{0}\) is no longer itself unitary, nor independent of the frequency and there is no longer an identity between the physical and modal coordinates of the interface.

The modal seismic force calculated by MISS3D is then written as:

(2.21)\[ {f} _ {s} (\ omega) = {K} _ {s} (\ omega) {x} _ {0} (\ omega)\]

\({x}_{0}(\omega )\) the modal participation vector is therefore obtained by inverting the seismic force \({f}_{s}(\omega )\) with respect to the ground impedance \({K}_{s}(\omega )\); the physical vector on the corresponding interface \({X}_{0}(\omega )\) is then written as: \({X}_{0}(\omega )=\Theta {x}_{0}(\omega )\) where \(\Theta\) is the matrix containing the (mechanical) interface modes.

To take account of spatial variability, we determine the physical contribution on the interface for each mode PODcaractérisant the spatial variability:

(2.22)\[ {X} _ {u} ^ {k} (\ omega) = {X} _ {0} (\ omega) {s} _ {u} ^ {k}\]

The physical vector on the interface \({X}_{u}^{k}(\omega )\) corresponds to a modal participation vector \({S}_{u}^{k}(\omega )\) such as: \({X}_{u}^{k}(\omega )=\Theta {S}_{u}^{k}(\omega )\) where \(\Theta\) is the matrix containing the (mechanical) interface modes. Then the corresponding seismic force will be expressed as follows:

(2.23)\[ {f} _ {s} ^ {k} (\ omega) = {K} _ {s} (\ omega) {S} _ {u} ^ {k} (\ omega)\]

2.2.3. Calculating transfer functions and the DSP response

By linear filtering, we obtain:

\[\]

: label: eq-27

{s} _ {q} ^ {k} (omega) =H (omega) {s} _ {f} ^ {k} (omega)

This makes it possible to reconstruct the response spectral density matrix as:

(2.24)\[ {S} _ {q} (\ omega) =\ sum _ {k=1} ^ {K\ le m} {s} _ {q} ^ {k} ({s} _ {q}} ^ {k})\ text {*}\]

The model is reduced if you can truncate expression () to \(K\le m\) modes POD.

For white noise excitation, DSP of the output transfer function is directly obtained.

Note:

Without spatial variability the seismic field and thus the seismic force are the same for all the nodes of the interface (the foundation) :

\({S}_{u}(x,x\text{',}\omega )={S}_{0}(\omega )\)

Attention: As it stands, DYNA_ISS_VARI can only handle the case of a superficial foundation. As for the interface modes, one can choose between modeling by the 6 rigid body modes (rigid foundation), any interface modes (flexible foundation) and modeling by all EF modes (flexible foundation).

2.2.4. Calculating the temporal response to an earthquake

In general, it is possible to simulate the trajectories of a Gaussian stationary process whose DSP is known using the spectral representation theorem.

We could therefore obtain a realization of the temporal structural response (stochastic process characterized by its spectral density \({S}_{q}(\omega )\)) by the formula (cf. [bib4]):

(2.25)\[ q (t) =\ sum _ {k}\ sum _ {n=0} {n=0} ^ {{N} _ {T}} H ({\ omega} _ {n}) {e} _ {n}) {e} ^ {e} ^ {i {\ omega} ^ {i {\ omega}} ^ {i {\ omega}} ^ {i {\ omega}} ^ {i {\ omega}} ^ {i {\ omega}} ^ {i {\ omega}} _ {n}) {e} ^ {i {\ omega}} ^ {i {\ omega}} ^ {i {\ omega}} _ {n} _ {n}) {e} ^ {i {\ omega}} ^ {i {\ omega}} ^ {i {\ omega}} ^ {i {\ omega}} _ {{k} ({\ omega} _ {n})}\ sqrt {{S} _ {0} ({\ omega} _ {n})} {\ xi} _ {n} _ {n} ^ {n} ^ {k}\ sqrt {\ Delta\ omega}\]

where \({\xi }_{n}^{k}\) are independent complex random variables with a reduced centered normal distribution and where \(\Delta \omega\) designates the frequency step (constant), \({\omega }_{n},n=\mathrm{1,}{N}_{T}\) are the frequencies resulting from discretization. Under the hypothesis that we can approach the point spectral density using a number of accelerograms, we have:

\[\]

: label: eq-30

{S} _ {0} ^ {L} (omega) =frac {1} {1} {2pi}frac {1} {L}sum _ {l=1} ^ {L} {u} {u} _ {u} _ {u} _ {0} _ {0} ^ {l} (omega)mathrm {.} {u} _ {0} ^ {l}ast (omega)

Where \({u}_{0}(\omega )\) is obtained from the accelerogram \(u(t)\) in the open field:

\[\]

: label: eq-31

{u} _ {0} (omega) =frac {1} {frac {1} {sqrt {T}}underset {0} {int}} {u} _ {0} (t) {e} (t) {e} ^ {e} ^ {-iomega t}text {dt}} {u} _ {0} (t) (t) {e} ^ {-iomega t}text {dt}

Here, we do not use this method of generating signals from their DSP because we work with non-stationary processes whose embodiment is known, namely the accelerogram as input to the mechanical calculation. Given this context, we are instead placing ourselves in a deterministic signal filtering framework. Thus, a deterministic filter is introduced that models the effect of spatial incoherence. This filter is given by the matrix \(\Phi (\omega )\Lambda (\omega {)}^{1\mathrm{/}2}\). We can thus calculate the frequency response and obtain the temporal response by FFT inverse. The frequency response is written as:

(2.26)\[ q (\ omega) =\ sum _ {k} H (\ omega) G (\ omega) G (\ omega) {\ phi} _ {k} (\ omega\ sqrt {{\ lambda} _ {k} (\ omega)} {k} (\ omega)} {\ omega)} {u} _ {0} (\ omega)\]

By considering equation (), we can verify that the spectral density of answer \(q\) is that given by equation (). The model is « reduced » if you can truncate the expression () to \(K\le m\) modes POD.

2.3. Results calculated by DYNA_ISS_VARI

In summary, the DYNA_ISS_VARI operator makes it possible to obtain the following results:

• Calculation of the spectral density of the modal response for a unit seismic excitation (while moving). In order to obtain the response to a seismic excitation, we multiply the DSP obtained for a unit excitation by the spectrum modeling the excitation (Kanai—Tajimi or other).

• Calculation of transfer functions for unitary seismic excitation (while moving). The transfer functions are given by the root of the auto-response spectrum module, \(\sqrt{\mid {\left[{S}_{v}(\omega )\right]}_{\text{ii}}\mid }\) (root of the absolute value of the self-spectrum [bib9] in physical coordinates). By comparing transfer functions with and without spatial variability, we can determine margins related to the incoherence of the seismic signal (cf. also [bib8]: « incoherence » margin factor in seismic EPS).

• Calculation of floor spectra using the transient response (FONC_SIGNAL must be entered). DYNA_ISS_VARIpermet to calculate the transient acceleration response. The oscillator response spectra (SRO) for a floor can be obtained by post-processing.