3. Operands#

3.1. Keyword FRAGILITE#

FRAGILITE =

The keyword FRAGILITEpermet to determine the parameters \({A}_{m}\) and \(\beta\) (median and logarithmic standard deviation) of a fragility curve according to the log-normal model [U2.08.05]:

\({P}_{f\mid a}=\Phi (\frac{\mathrm{ln}(a/{A}_{m})}{\beta })\)

It is also possible to calculate the values of the curve for the values of the parameters Am and \(\beta\) obtained. Option FRACTILES (optional) also makes it possible to determine fractiles for the curve by a method of resampling the original sample that was entered in TABL_RESU.

3.1.1. Operand TABL_RESU#

♦ TAB_RESU = tables [table_saster]

We give the name of the table [table_sdaster] that we must have created before using CREA_TABLE [U4.33.02]. This table must have at least two columns with access keys (column label name): PARA_NOCI (this is the indicator fract = 0.5 ur characterizing the level of the excitation) and DEFA (the values in this column are 0 if no failure was observed or 1 if there was a failure or 1 if there was a failure) or DEMANDE (the values of the real variable of interest characterizing the failure or damage, also called seismic request in my literature).

3.1.2. Operand METHODE#

♦ METHODE =/ “EMV”

/'REGRESSION'

We choose between the two methods for calculating the lognormal fragility curve: EMVpour the maximum likelihood estimation or REGRESSIONpour the linear regression. More details on these two methods can be found in the documentation [U2,08.05]. If you choose REGRESSION, then the tableware must contain a column DEMANDErenseignant the seismic demand (variable of interest such as drift, maximum stress,…) and you must enter the failure threshold using the keyword SEUIL.

3.1.3. Operand SEUIL#

◊ SEUIL = threshold [R]


If table TAB_RESU contains a column DEMANDE, then it is necessary to enter the threshold of this variable at which the structure is considered to be faulty.

3.1.4. Operands LIST_PARA and VALE#

We can give a list of real values, values for which we evaluate the fragility curve.

This can be done in the form of a list containing the \(({a}_{\mathrm{1,}}{a}_{\mathrm{2,}}\mathrm{...},{a}_{n})\) calculation values:

◊ VALE =list [L_r]

or by giving the name of the listr8 type concept containing the list of values:

◊ LIST_PARA =listr8 list

3.1.5. Operands AM_INI and BETA_INI#

♦ AM_INI

◊ BETA_INI

If we have chosen METHODE =” EMV “, then it is imperative to give an initial value for the estimation of the parameter \({A}_{m}\) and it is advisable to give an initial estimate for \(\beta\) (starting points for the optimization algorithm).

3.1.6. Operands FRACTILES and NB_TIRAGE#

These operands must be filled in if it is desired to determine confidence intervals or, more precisely, fractionals for the fragility curve estimated by the maximum likelihood method ('EMV'). The resampling method (also called "bootstrap" in Anglo-Saxon literature) is used for this. The FRACTILES operand allows you to give the fractions you want to calculate.


◊ FRACTILES = fract listr8

By default, we draw as many "bootstrap" samples as we have data (this is the number :math:`N` of Monte Carlo simulation performed beforehand and whose results are stored in table TABL_RESU). However, the NB_TIRAGE command makes it possible to reduce the number of draws to be performed:


◊ NB_TIRAGE = NBT [I]

nbt must be less than or equal to the number of values in TABL_RESU (\(\mathrm{nbt}\le N\)). This feature makes it possible to reduce the calculation time but is not recommended in the general case because the results are unreliable.

3.2. Keyword INTERSPECTRE#

3.2.1. Operand INTE_SPEC#

♦ INTE_SPEC = inter

inter is the user name for the interspectral matrix.

The interspectral matrix can be obtained by various operators: read_inte_spec [U4.36.01], calc_inte_spec [U4.36.03], defi_inte_spec [U4.36.02], dyna_alea_modal [U4.53.22], DYNA_SPEC_MODAL [], [U4.53.23], or REST_SPEC_PHYS [U4.63.22].

Note:

Spectral moments are defined as integrals of power spectral density (DSP) :

\({\lambda }_{i}=\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}{\mid \omega \mid }^{i}{S}_{\text{XX}}(\omega )d\omega =2\underset{0}{\overset{\text{+}\infty }{\int }}{\omega }^{i}{S}_{\text{XX}}(\omega )d\omega\)

So, if DSP is given for positive frequencies only, POST_DYNA_ALEA multiplies by 2 the integrals of the DSP calculated for \(\omega >0\). In addition, DSP are defined according to the natural frequency \(f\mathrm{=}2\pi \omega\) ( \(\mathit{Hz}\) ) in POST_DYNA_ALEA. The following formulas are used [cf. R7.010.01]:

\(\begin{array}{}{S}_{\text{XX}}(f)=\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}{R}_{\text{XX}}(\tau ){e}^{-\mathrm{2i}\pi f\tau }d\tau ;\\ S{\text{'}}_{\text{XX}}(\omega )=\frac{1}{2\pi }{S}_{\text{XX}}(f)\end{array}\)

Readers are invited to consult the documentation for the DYNA_ALEA_MODAL [U4.53.22] command for more information on the meaning of the keyword parameters.

3.2.2. Operands NUME_ORDRE_I, NUME_ORDRE_J#

♦ NUME_ORDRE_J = number

These keywords make it possible to define the terms of the matrix whose functions will be processed.

When autospectra or interspectra are calculated on modes:

  • lnumi is the list of order numbers for the “i” modes. Example: (2,3.1).

  • lnumj is the list of order numbers for the “j” modes. Example: (2,1.4)

The indices are paired according to the same rank.

  • (2,2) corresponds to the autospectrum on mode 2,

  • (3.1) corresponds to the interspectrum between mode 3 and mode 1.

lnumi and lnumj must contain the same number of terms.

3.2.3. Operands NOEUD_I, NOEUD_J, NOM_CMP_I, NOM_CMP_J#

♦/♦ NOEUD_I = Monday

♦ NOEUD_J = long ♦ NOM_CMP_I = lcmpi ♦ NOM_CMP_J = lcmpj

When autospectra or interspectra are calculated on nodes in a given direction:

  • lnode is the list of nodes following « i »: \((\mathit{NO92},\mathit{NO95},\mathit{NO98})\)

  • lnoeudj is the list of nodes following « j »: \((\mathit{NO92},\mathit{NO92},\mathit{NO92})\)

  • lcmpi is the list of components following « i »: \((\mathit{DX},\mathit{DX},\mathit{DY})\)

  • lcmpj is the list of components following « j »: \((\mathit{DX},\mathit{DX},\mathit{DX})\)

The nodes and components are paired in the same order:

  • \((\mathit{NO92}\mathit{DX},\mathit{NO92}\mathit{DX})\) corresponds to the autospectrum at node \(\mathit{NO92}\) in the \(\mathit{DX}\) direction,

  • \((\mathit{NO98}\mathit{DY},\mathit{NO92}\mathit{DX})\) corresponds to the interspectrum between node \(\mathit{NO92}\) in direction \(\mathit{DX}\) and node \(\mathit{NO95}\) in direction \(\mathit{DY}\).

lnoeudi, lnoeudj, lcmpi, and lcpmj must contain the same number of terms.

3.2.4. Operand OPTION#

The calculations are carried out on all the interspectra of the matrix.

/OPTION = “DIAG”

The calculations are carried out on all the autospectra in the matrix and only for these.

Fract = 0.5

3.2.5. Key words DUREE and FRACTILE#

◊ DUREE = duration

If the keyword duration is entered, then the median maximum or any other fraction as well as the peak factor of the Gaussian stationary stochastic process is determined according to Vanmarcke formulas. duration then refers to the time interval considered to estimate these quantities. For example, in the context of a seismic analysis, duration may be taken equal to the duration of the strong phase of the seismic signal.

 ◊ FRACTILE = /frac [R]

3.2.6. Operand MOMENT#

◊ MOMENT = lmom

lmom is the list of the orders of spectral moments that will be calculated. By default, spectral moments of orders 0, 1, 1, 2, 3 and 4 are always calculated. It is therefore appropriate to mention in this list moments of order greater than 4. Example: (5,7,8).

3.3. Operand INFO#

◊ INFO =

1

printing the requested results.

2

like 1 but with more details.

3.4. Operand TITRE#

◊ TITRE = title

title is the title of the calculation. It will be printed at the top of the results. See [U4.03.01].