4. Rules for combining modal responses#
To evaluate a majority of the \(R\) response of the structure, we must now combine the modal responses \({R}_{\mathrm{imax}}^{k}\) defined previously. Several levels of combination are required:
combination of the selected natural modes \({\varphi }_{i}\), for \(i=\mathrm{1,}\mathrm{...},{N}_{r}\le N\),
static correction by pseudo-mode,
effect of different excitations applied to support groups,
combination according to the directions of earthquake excitation.
4.1. Earthquake direction and directional response#
Different considerations lead to the study of seismic behavior separately according to each direction of space:
for the study of a building on a ground, the accelerogram of the movement imposed vertically is different from that describing the horizontal movement, which is itself different in two orthogonal directions of space;
for the study of equipment, floor spectra differ significantly in the three directions of space, since they integrate the contributions of different building modes (floor bending, bending or twisting the frame, etc.).
This leads to the establishment of a directional modal response \({R}_{x}\) from different oscillator spectra and modal participation factors established in each direction \(X\) representing one of the directions of the coordinate system GLOBAL for defining the mesh (\(X,Y,Z\)) or a particular direction defined explicitly by the user.
4.2. Choice of specific modes to combine#
To correctly represent the deformation modes likely to be excited by the imposed movement, it would be necessary to know all the natural modes with a frequency lower than the cutoff frequency of the spectrum, beyond which there is no significant dynamic amplification. This condition can be difficult to meet for complex structures with a large number of specific modes.
The size of the required modal base (\({N}_{r}\) modes) must therefore be evaluated to ensure that no mode that has a significant contribution in internal forces and constraints has been omitted in each direction studied.
4.2.1. Expression of modal deformation energy#
The deformation energy associated with each natural mode \({U}_{i}=\frac{1}{2}{x}_{i\mathrm{max}}^{T}\mathrm{.}k\mathrm{.}{x}_{i\mathrm{max}}\) can be expressed for a particular direction:
\({U}_{\mathrm{iX}}=\frac{1}{2}{({p}_{i}\frac{{a}_{\mathrm{iX}}}{{\omega }_{i}^{2}})}^{2}{\varphi }_{i}^{T}\mathrm{.}k\mathrm{.}{\varphi }_{i}=\frac{1}{2}{({p}_{\mathrm{iX}}\frac{{a}_{\mathrm{iX}}}{{\omega }_{i}^{2}})}^{2}{\omega }_{i}^{2}{\mu }_{i}=\frac{1}{2}\frac{{a}_{\mathrm{iX}}^{2}}{{\omega }_{i}^{2}}{p}_{\mathrm{iX}}^{2}{\mu }_{i}\) eq 4.2.1-1
This expression corresponds to single-press excitation and can be extended to the case of multi-support.
The classification of modes with decreasing deformation energies makes it possible not to systematically select, for a global study of the structure, modes that do not produce significant deformations. On the other hand, to study the effect of stresses in a particular zone of the structure, it will be necessary to use « local » modes that can be detected by analyzing the distribution of deformation energy over groups of cells.
Note that an estimate of the total deformation energy is not available to quantify the error made by ignoring certain modes.
4.2.2. Expression of modal kinetic energy#
The kinetic energy associated with each eigenmode is written \({V}_{i}=\frac{1}{2}{\dot{x}}_{i\mathrm{max}}^{T}\mathrm{.}m\mathrm{.}{\dot{x}}_{i\mathrm{max}}\) which gives:
\({V}_{\mathrm{iX}}=\frac{1}{2}{({p}_{\mathrm{iX}}\frac{{a}_{\mathrm{iX}}}{{\omega }_{i}})}^{2}{\varphi }_{i}^{T}\mathrm{.}m\mathrm{.}{\varphi }_{i}=\frac{1}{2}\frac{{a}_{\mathrm{iX}}^{2}}{{\omega }_{i}^{2}}{p}_{\mathrm{iX}}^{2}{\mu }_{i}\) eq 4.2.2-1
The expression [éq 4.2.2-1] involves the effective modal mass \({p}_{\mathrm{iX}}^{2}{\mu }_{i}\) defined in [§2.2], which makes it possible to state the criterion for the accumulation of effective unit modal masses [éq 2.2.2-4].
Criterion for the accumulation of effective modal masss
The quality of a modal base, from the point of view of representing the inertial properties of the structure, is evaluated by combining, for this direction, the effective unit modal masses of the available modes. An eligibility threshold of 95% of the total mass is commonly accepted. The same criterion can be partially applied in the case of multi-press excitation with
modes by comparing \({\varphi }_{\mathrm{sj}}^{T}\mathrm{.}m\mathrm{.}{\varphi }_{\mathrm{si}}\) and \(\sum _{i=1}^{{N}_{r}}{p}_{\mathrm{ij}}^{2}{\mu }_{i}\) .
The sum of the effective modal masses is in fact worth the total mass that works on the chosen modal basis. In other words, this total working mass is worth the total mass minus the mass contributions that are carried by embedded degrees of freedom (which therefore do not work on a modal basis). So, for example, on a system with 1 degree of freedom mass-spring with one mass \(\mathit{M1}\) at the top and another mass \(\mathit{M2}\) at the base, then the working mass will be value \(\mathit{M1}\) and the total mass \(\mathit{M1}+\mathit{M2}\). Therefore, the effective unit modal mass for the system’s only mode will be value \(\mathit{M1}\mathrm{/}(\mathit{M1}+\mathit{M2})\). The total accumulation will therefore have the same value and, depending on the ratio in \(\mathit{M1}\) and \(\mathit{M2}\) , we will therefore not necessarily be able to reach 90% of the total mass \((\mathit{M1}+\mathit{M2})\) , even considering all the modes (we only have one mode in this example). In practice, the finer and more realistic the finite element model is, the smaller the difference between the working mass and the total mass will be.
Estimation of the error committed with an incomplete modal basis
The criterion for the accumulation of effective modal masses cannot always be satisfied. Indeed, we are generally limited to a modal base of \({N}_{r}\) specific modes with \({N}_{r}\ll N\) degrees of freedom. For rigid foundations, the spectrum of natural frequencies required commonly exceeds the cutoff frequency of the oscillator spectrum.
From the expression [éq 4.2.2-1], we can write the total kinetic energy in the form:
\({V}_{X}=\sum _{i=1}^{{N}_{r}}{V}_{\mathrm{iX}}+\sum _{i={N}_{r}+1}^{N}{V}_{\mathrm{iX}}\)
which allows you to express the absolute error from [éq 3.1-1]:
\(2\Delta {V}_{X}=\sum _{{N}_{r}+1}^{N}{V}_{\mathrm{iX}}=\sum _{{N}_{r}+1}^{N}\frac{{a}_{\mathrm{iX}}^{2}}{{\omega }_{i}^{2}}{p}_{\mathrm{iX}}^{2}{\mu }_{i}\le \frac{{a}_{({N}_{r}+1)X}^{2}}{{\omega }_{i}^{2}}\sum _{{N}_{r}+1}^{N}{p}_{\mathrm{iX}}^{2}{\mu }_{i}\)
by noting \({a}_{(n+1)X}=\mathrm{SRO}{\ddot{x}}_{X}(A,{\xi }_{\mathrm{min}},{\omega }_{{N}_{r}+1})\) the value read on the absolute pseudo-acceleration spectrum for \({\omega }_{n}\le {\omega }_{n+1}\) and the lowest modal damping \({\xi }_{\mathrm{min}}\) likely to give the increasing amplitude. If the maximum frequency of base \({f}_{n}\) exceeds the cutoff frequency, then \({a}_{(n+1)X}={a}_{\mathrm{nX}}=\mid A{(t)}_{\mathrm{max}}\mid\). This gives an increase of the absolute error:
\(\Delta {V}_{X}=\frac{1}{2}\frac{{a}_{(n+1)X}^{2}}{{\omega }_{i}^{2}}\sum _{{N}_{r}+1}^{N}{p}_{\mathrm{iX}}^{2}{\mu }_{i}=\frac{1}{2}\frac{{a}_{({N}_{r}+1)X}^{2}}{{\omega }_{i}^{2}}({m}_{T}-\sum _{1}^{{N}_{r}}{p}_{\mathrm{iX}}^{2}{\mu }_{i})\) eq 4.2.2-2
4.2.3. Conclusion#
The quantities allowing you to choose the modes necessary for each analysis are available in Code_Aster (operator POST_ELEM with the options MASS_INER, ENER_POT and ENER_CIN and modal parameters FACT_PARTICI_DX and MASS_EFFE_UN_DX in the mode_meca result concept).
No automatic eligibility criteria are currently programmed and the quantities \({\varphi }_{\mathrm{Sj}}^{T}\mathrm{.}m\mathrm{.}{\varphi }_{\mathrm{Sj}}\) and \(\sum _{1}^{{N}_{r}}{p}_{\mathrm{ij}}^{2}{\mu }_{i}\), necessary to verify the criterion for multi-support excitation, are not printed.
4.3. Pseudo-mode static correction#
4.3.1. Mono-support#
The evaluation of a major part of the response to seismic excitation requires, as the previous analysis suggests, a correction by a term representing the static contribution of the neglected eigenmodes.
If the structure is subjected to a uniform static acceleration in the \(X\) direction, the linear elastic response \({\varphi }_{\mathrm{aX}}\) is a solution of \(k\mathrm{.}{\varphi }_{\mathrm{aX}}=m\mathrm{.}{\delta }_{X}I\), without dynamic amplification. The elastic displacement field \({\varphi }_{\mathrm{aX}}\) of the nodes of the structure subjected to uniform acceleration in each direction is produced by the operator MODE_STATIQUE [U4.52.14] with the keyword PSEUDO_MODE.
By decomposing this deformation on the basis of natural modes, we obtain (cf. [§2.2.2]):
\(k\mathrm{.}{\varphi }_{\mathrm{aX}}=m\mathrm{.}\sum _{i}^{N}{p}_{\mathrm{IX}}{\varphi }_{i}\) from where \({\varphi }_{\mathrm{aX}}={k}^{\text{-1}}\mathrm{.}m\mathrm{.}\sum _{i=1}^{N}{p}_{\mathrm{iX}}{\varphi }_{i}=\sum _{i=1}^{N}\frac{{p}_{\mathrm{iX}}}{{\omega }_{i}^{2}}{\varphi }_{i}\)
This makes it possible to introduce a pseudo-mode \({\varphi }_{\mathrm{cX}}\), for each direction, by subtracting from the quasi-static mode \({\varphi }_{\mathrm{aX}}\) the static contributions of the \({N}_{r}\) modes used \({\varphi }_{i}\) in the low restraint:
\({\varphi }_{\mathrm{cX}}={\varphi }_{\mathrm{aX}}-\sum _{i=1}^{{N}_{r}}\frac{{p}_{\mathrm{iX}}}{{\omega }_{i}^{2}}{\varphi }_{i}\) eq 4.3.1-1
The expression [éq 4.3.1-1] is homologous to the term \(({m}_{T}-\sum _{i=1}^{{N}_{r}}{p}_{\mathrm{iX}}^{2}{\mu }_{i})\) in [eq 4.2.2-2] and the pseudo-mode makes it possible to introduce a correction of the static effects of the neglected modes.
The contribution of the static correction pseudo-mode is the value read from the absolute pseudo-acceleration spectrum for an infinite pulsation, called ZPA (zero period acceleration). In practice, however, as it is not always easy to respect the criterion of accumulating effective modal masses for the base of the \({N}_{r}\) selected natural modes, it is accepted to take the value: \({a}_{({N}_{r}+1)X}=\mathit{SRO}{\ddot{x}}_{X}(A,{\xi }_{\mathit{min}},{\omega }_{{N}_{r}})\) for the pulsation \({\omega }_{{N}_{r}+1}\ge {\omega }_{{N}_{r}}\) of the mode following the last of the selected mode base, and for the lowest modal damping \({\xi }_{\mathrm{min}}\). This is considered to increase this contribution in order to obtain a conservative result.
If the modal base has been correctly chosen, that is, if it includes all the modes up to the cutoff frequency of the seismic signal, the keyword FREQ_COUP of the COMB_SISM_MODAL operator offers the possibility of specifying the value of the frequency where the asymptote begins (value of ZPA). This ensures that the response is not increased unduly.
The correction to be made to the relative movements and the quantities that are deduced from them (generalized efforts, constraints, support reactions) in single-support excitation is then:
\({x}_{\mathrm{cX}}^{k}={a}_{({N}_{r}+1)X}\mathrm{.}{\varphi }_{\mathrm{cX}}^{k}\)
in accordance with the conditions for estimating the error cf. [§ 4.2.2].
For the evaluation of the absolute acceleration correction, we obtain:
\({\ddot{\mathrm{x}}}_{\mathit{cX}}^{k}={a}_{({N}_{r}+1)X}\mathrm{.}{\left({\mathrm{\delta }}_{\mathrm{X}}-\sum _{i=1}^{{N}_{r}}{p}_{\mathit{iX}}{\mathrm{\phi }}_{i}\right)}^{k}\)
This estimate is possible because we have the « theoretical » solution \({\mathrm{\delta }}_{\mathrm{X}}\).
4.3.2. Multi-support#
In multi-support excitation, the formulation of the pseudo-mode and its contribution take up the previous principle.
The elastic displacement field \({\phi }_{\mathit{ajX}}={\mathrm{k}}^{\text{-1}}\mathrm{.}\mathrm{m}\mathrm{.}{\phi }_{\mathit{SjX}}\) of the nodes of the structure subjected to a unit acceleration of the support \(j\) in the direction \(X\) is produced by the operator MODE_STATIQUE [U4.52.14] with the keyword PSEUDO_MODE.
In the same way as for mono-support, \({a}_{({N}_{\mathit{rj}}+1)\mathit{jX}}\) is defined as the « ZPA » to support \(j\) in the \(X\) direction. It is given, by default, by the value read on SRO for the last mode of the modal base. For a well-chosen base, it must correspond in practice to the asymptotic value of the acceleration. It can be adjusted, if necessary, using the FREQ_COUP keyword.
The correction to be made to the relative movements and the quantities that are derived from them can then be written, for support \(j\) in the \(X\) direction:
\({x}_{\mathit{cjX}}={\phi }_{\mathit{cjX}}{a}_{({N}_{\mathit{rj}}+1)\mathit{jX}}\) with \({\phi }_{\mathit{cjX}}={\phi }_{\mathit{ajX}}-\sum _{i=1}^{{N}_{r}}\frac{{P}_{\mathit{ijX}}{\phi }_{i}}{{\omega }_{i}^{2}}\)
For absolute acceleration, the correction is written as:
\({\ddot{x}}_{\mathit{cjX}}=\left({\mathrm{\phi }}_{\mathit{SjX}}-\sum _{i=1}^{{N}_{r}}{\mathrm{\phi }}_{i}{P}_{\mathit{ijX}}\right){a}_{({N}_{\mathit{rj}}+1)\mathit{jX}}\)
In practice, this estimate is not possible because we do not know how to estimate the vector \({\phi }_{\mathit{SjX}}\) which corresponds to the absolute acceleration values at all points when applying a unit acceleration to support \(j\).
4.4. Overview of combination rules#
The rules for combining or combining the various components, modal or directional, are multiple and more or less complex to implement.
The « natural » methods are presented from the point of view of their ability to provide a realistic majority of the stresses induced in a structure represented by a base of real eigenmodes from a linear elasticity model, an increase estimated without transient analysis for a component quantity \({G}^{k}\), which will be called \({G}_{\mathrm{max}}^{k}\). Hereinafter, the suffix \(\mathrm{max}\) designates the estimation of the maximum value reached during seismic excitation, ignoring the moment when it was reached, and the index \(r\) applies to natural modes, pseudo-modes, support directions…
Note:
Regardless of the combination method used, the value of a component obtained by combination cannot be used as data to calculate a new quantity: for example, the calculation of a plus of a differential displacement between two points must be calculated mode by mode, then combined.
4.4.1. Arithmetic combination#
\({G}_{\mathrm{max}}^{k}=\sum _{r}{G}_{r\text{max}}^{k}\)
It cannot be used for directional responses since the spectral method ignores the times when the maximum values are reached in two directions or for two different modes. No phase relationship, and therefore sign, exists between the contributions to be combined. It is therefore only available in the multi-support case, for the accumulation of modal directional support responses and for the accumulation of differential movements.
4.4.2. Absolute value combination#
\({G}_{\mathrm{max}}^{k}=\sum _{r}\mid {G}_{r\text{max}}^{k}\mid\)
Obviously, it can provide an upper bound, since it assumes that all contributions reach their maximum at the same time with the same sign. Too penalizing, it is available, but unusable industrially.
4.4.3. Simple quadratic combination#
This method is also known under the name SRSS (Square Root of Sum of Squares).
\({G}_{\mathrm{max}}^{k}=\sqrt{\sum _{r}{({G}_{r\text{max}}^{k})}^{2}}\)
Hypothesis:
The **hypothesis that justifies this combination method can be stated as follows:*
The probable maximum of the energy stored in the structure is the sum of the probable maximums of the energy stored on each of the modes and on each of the directional components of the earthquake, that is to say that, with respect to energy, the natural modes and the components of the earthquake are decoupled. It is analogous to the rule for the addition of Gaussian and zero-mean random variables.
The validity of this hypothesis, which will be discussed for each particular case of use of this combination method, is not established and various proposals have been presented to obtain a better approximation in cases where it is lacking cf. [§3.4.1.2].
In addition, we can refer to [bib3] for a criticism of this approach, in particular of its ability to estimate a probable maximum of deformations and constraints, but the alternative approach that it evokes has not been the subject of any development in Code_Aster.
4.5. Establishing the directional response in mono-support#
The directional answer, defined earlier, is obtained by simple quadratic combination of two terms that we are going to discuss:
\({R}_{X}=\sqrt{{R}_{m}^{2}+{({R}_{\mathit{qs}}+{R}_{c})}^{2}}\)
with:
\({R}_{m}\) dynamic combined response of the \({N}_{r}\) modal oscillators,
\({R}_{\mathit{qs}}\) combined quasistatic response of the modal oscillators (=0 except for the Gupta type combination),
\({R}_{c}\) contribution of the static correction of neglected modes (pseudo-mode).
The assumptions justifying the simple quadratic combination method, at this level, do not seem to need to be questioned [bib1].
To simplify the notations, we write \({R}_{m}\) instead of \({R}_{\mathrm{mX}}\),…
4.5.1. Combined response of modal oscillators#
The response of the structure \({R}_{m}\), in an earthquake direction, is obtained by one of the possible combinations of the contributions of each of the natural modes taken into consideration for this direction. The number of possible methods simply proves the difficulty of finding sufficient justification to guarantee a conservative and realistic estimate. If the simple quadratic combination (SRSS or CQS) is mentioned by everyone, we will remember from [bib1] that it is often faulted and the complete quadratic combination (CQC) will be preferred. The other methods are available for possible comparisons.
4.5.1.1. Sum of absolute values#
This combination corresponds to a hypothesis of complete dependence of the oscillators associated with each specific mode and leads to a systematic overvaluation of the response:
\({R}_{m}=\sum _{i=1}^{{N}_{r}}\mid {R}_{i}\mid\)
4.5.1.2. Simple quadratic combination (CQS)#
By considering that the contribution of each modal oscillator is an independent random variable, an estimate of the maximum response, for the displacement component \({x}_{\mathrm{max}}^{k}\), can be obtained by simple quadratic combination of the contributions of each mode, where, for a single-support excitation:
\({x}_{\mathrm{max}}^{k}=\sqrt{\sum _{i=1}^{{N}_{r}}{({x}_{i\mathrm{max}}^{k})}^{2}}=\sqrt{\sum _{i=1}^{{N}_{r}}{({\varphi }_{i}^{k}{p}_{i}{q}_{i\mathrm{max}})}^{2}}\) eq 4.5.1.2-1
In general, for any quantity \({R}_{i}\) associated with a modal oscillator \(({\omega }_{i},{\xi }_{i})\):
\({R}_{m}=\sqrt{\sum _{i=1}^{{N}_{r}}{R}_{i}^{2}}\)
It constitutes a good approximation of reality when the oscillator spectrum defining the earthquake is of a broad frequency band, and when the natural modes of the structure are well separated from each other and are located within or in the vicinity of this band. In particular, it is faulty in the case where natural modes are at similar frequencies or for modes far from the excitation peak [bib2]. The other methods of combining modal responses attempt to correct this point.
4.5.1.3. Full quadratic combination (CQC)#
The complete quadratic combination (established by DER KIUREGHIAN [bib5]) provides a correction to the previous rule by introducing correlation coefficients depending on damping and on the distances between neighboring eigenmodes:
\({R}_{m}=\sqrt{\sum _{{i}_{1}}\sum _{{i}_{2}}{\rho }_{{i}_{1}{i}_{2}}{R}_{{i}_{1}}{R}_{{i}_{2}}}\)
with the correlation coefficient:
\({\rho }_{\mathrm{ij}}=\frac{8\sqrt{{\xi }_{i}{\xi }_{j}{\omega }_{i}{\omega }_{j}}({\xi }_{i}{\omega }_{i}+{\xi }_{j}{\omega }_{j}){\omega }_{i}{\omega }_{j}}{{({\omega }_{i}^{2}-{\omega }_{j}^{2})}^{2}+4{\xi }_{i}{\xi }_{j}{\omega }_{i}{\omega }_{j}({\omega }_{i}^{2}+{\omega }_{j}^{2})+4({\xi }_{i}^{2}+{\xi }_{j}^{2}){\omega }_{i}^{2}{\omega }_{j}^{2}}\)
or by introducing the pulsation or frequency ratio between two \(\eta ={\omega }_{j}/{\omega }_{i}\) modes:
\({\rho }_{\mathrm{ij}}=\frac{8\eta \sqrt{{\xi }_{i}{\xi }_{j}\eta }({\xi }_{i}+{\xi }_{j}\eta )}{{(1-{\eta }^{2})}^{2}+4\eta {\xi }_{i}{\xi }_{j}(1+{\eta }^{2})+4{\eta }^{2}({\xi }_{i}^{2}+{\xi }_{j}^{2})}\)
and for \(\xi\) constant:
\({\rho }_{\mathrm{ij}}=\frac{8\eta {\xi }^{2}(1+\eta )\sqrt{\eta }}{{(1-{\eta }^{2})}^{2}+4\eta {\xi }^{2}(1+{\eta }^{2})+8{\eta }^{2}{\xi }^{2}}\)
4.5.1.4. Combination of ROSENBLUETH#
This rule (proposed by E. ROSENBLUETH and J. ELORDY [bib6]) introduces a correlation between modes, different from that of the CQC method. The responses of the oscillators are combined by double sum (Double Sum Combination):
\({R}_{m}=\sqrt{\sum _{{i}_{1}}\sum _{{i}_{2}}{\rho }_{{i}_{1}{i}_{2}}{R}_{{i}_{1}}{R}_{{i}_{2}}}\)
It requires additional data, the duration \(s\) of the « strong » phase of the earthquake. The correlation coefficient is then:
\({\rho }_{\mathrm{ij}}={(1+{(\frac{\omega {\text{'}}_{i}-\omega {\text{'}}_{j}}{\xi {\text{'}}_{i}{\omega }_{i}+\xi {\text{'}}_{j}{\omega }_{j}})}^{2})}^{\text{-1}}\) where \(\omega {\text{'}}_{i}={\omega }_{i}\sqrt{1-\xi {\text{'}}_{i}^{2}}\) and \(\xi {\text{'}}_{i}^{2}={\xi }_{i}+\frac{2}{s{\omega }_{i}}\)
4.5.1.5. Combination with the 10% rule#
Neighboring modes (whose frequencies differ by less than 10%) are first combined by summing the absolute values. The values resulting from this first combination are then combined quadratically (simple quadratic combination). This method was proposed by the American regulation U.S. Nuclear Regulatory Commission (Regulatory Guide 1.92 - February 1976) to mitigate the conservatism of the method of summing absolute values. It is still lacking for structures with a dense natural frequency spectrum and should no longer be used.
4.5.1.6. Combination according to Gupta#
GUPTA [NRC1.92], to take into account the correlations between modes due to the semi-static part of the response, introduces the rigid response factor, which varies from 0 to 1 the correlation between the modal responses of intermediate frequencies between \({f}_{1}\) and \({f}_{2}\), two frequencies to be determined (typically from \(\mathrm{2 }\mathrm{Hz}\) to \(20\mathrm{Hz}\)).
GUPTA breaks down each \({R}_{r}\) modal response into a dynamic part \({R}_{r}^{p}\) and a semi-static part \({R}_{r}^{\mathit{qs}}\): \({R}_{r}^{\mathit{qs}}\mathrm{=}{\alpha }_{r}{R}_{r}\) and \({R}_{r}^{p}\mathrm{=}\sqrt{1\mathrm{-}{\alpha }_{r}^{2}}{R}_{r}\)
Thus, for each \(r\) mode, we assign the rigid response factor \({\alpha }_{r}\) to the modal response \({R}_{r}\):
\({\alpha }_{r}\mathrm{=}0\) for \(f\mathrm{\le }{f}_{1}\) and \({\alpha }_{r}\mathrm{=}1\) for \(f\ge {f}_{2}\)
\({\alpha }_{r}\) is estimated for the \({f}_{r}\) frequency using the following formula:
\({\alpha }_{r}\mathrm{=}\frac{\text{ln}{f}_{r}\mathrm{/}{f}_{1}}{\text{ln}{f}_{2}\mathrm{/}{f}_{1}}\)
The dynamic combined response of the modal oscillators is carried out according to the combination “CQC”:
\({R}_{d}\mathrm{=}\sqrt{\mathrm{\sum }_{{r}_{1}}\mathrm{\sum }_{{r}_{2}}{\rho }_{{r}_{1}{r}_{2}}{R}_{{r}_{1}}^{p}{R}_{{r}_{2}}^{p}}\)
The combined quasistatic response of the modal oscillators is carried out according to an algebraic combination:
\({R}_{\mathrm{qs}}=\sum _{i=1}^{{N}_{r}}{R}_{r}^{\mathrm{qs}}\)
This combination according to GUPTA is only available in the single-support case.
4.5.2. Contribution of static correction of neglected modes#
The contribution of pseudo-fashion Cf. [§4.3.1] can be combined quadratically because independence with the contributions of vibration modes is not contested.
4.6. Establishing the directional response in multi-support#
4.6.1. Calculating the overall response#
The order of the combinations to be made differs depending on whether the excitations of the supports (or groups of supports) can be considered as correlated or uncorrelated with each other.
4.6.2. Separate calculation of the primary and secondary components of the response#
Each component is treated separately in a similar manner. This approach is adapted to the RCC -M post-treatments in force for the seismic analysis of pipes [§ 4.9]:
4.6.2.1. Primary component RIX (inertial response)#
The order of the combinations to be made differs depending on whether the excitations of the supports (or groups of supports) can be considered as correlated or uncorrelated with each other.
uncorrelated support groups:
for each imposed \(\ddot{{s}_{j}}\) movement, calculation of the directional support responses (accumulation over the modes):
\({R}_{\mathrm{IjX}}=\sqrt{{{R}_{j}^{m}}^{2}+{{R}_{j}^{c}}^{2}}\)
\({R}_{j}^{m}\) |
combined support response of the modal oscillators (accumulation over the modes) |
\({R}_{j}^{c}\) |
contribution of the static correction of neglected modes (pseudo-support mode) |
combination of answers \({R}_{\mathrm{IjX}}\) (accumulation of supports).
-correlated supports:
for each imposed movement \(\ddot{{s}_{j}}\), calculation of the modal directional responses (accumulation of supports):
\({R}_{\mathrm{IiX}}=\sqrt{{{R}_{i}^{m}}^{2}+{{R}^{c}}^{2}}\)
\({R}_{i}^{m}\) |
combined modal response of the modal oscillators (cumulative support) |
\({R}^{c}\) |
contribution of static correction of neglected modes (pseudo-mode) |
combination of answers \({R}_{\mathrm{IiX}}\) (accumulation of modes).
4.6.2.2. Secondary component RII (quasistatic response)#
combination of answers \({R}_{\mathrm{ej}}\)
4.6.3. Cumulation on modes#
The criterion for choosing the method for combining the contributions of the modes is the same as for single-support excitation and method CQC will preferentially be used.
4.6.4. Contribution of pseudo-fashion#
The term corrective by pseudo-mode Cf. [§4.3.2] can be combined quadratically.
4.6.5. Contribution of training movements#
Since the driving movement of the structure is not uniform, it is possible to add a term to the calculation of the directional response. This is not necessary if one chooses to consider this static contribution as a specific load case inducing secondary constraints. This term is defined on the basis of the maximum relative displacement which cannot be known from the absolute pseudo-acceleration spectra of the supports alone.
\({R}_{\mathrm{ej}}={\varphi }_{\mathrm{Sj}}{\delta }_{j\mathrm{max}}\)
\({\varphi }_{\mathrm{Sj}}\) |
static mode for \(j\) support |
\({\delta }_{j\mathrm{max}}\) |
maximum relative displacement of the \(j\) support in relation to a reference support (for which \({\delta }_{j\mathrm{max}}=0\)) |
4.6.6. Cumulation of supports#
This step is mandatory, but the choice of method for combining directional responses is still very open. In fact, the hypothesis of independence of \({\ddot{s}}_{j}\) depends strongly on the specific modes of the support structure of the equipment studied. An analysis of the system studied is necessary to possibly group the supports by groups: for example, for a pipe connecting two buildings, either all the supports are considered to be correlated with each other, or we can show groups that are uncorrelated with each other (the group of supports of building 1, that of building 2 and, finally, that of intermediate supports), whose supports within each group are correlated.
Intra-group cumulation
Since the excitations at the supports of the same group are assumed to be correlated with each other, the accumulation is carried out algebraically according to the linear combination defined by: \({R}_{X}=\sum {R}_{\mathrm{jX}}\)
Inter-group cumulation
Since the support groups are constituted in such a way that they are decorrelated to each other, inter-group accumulation is carried out by simple quadratic combination.
4.6.7. Equivalence: mono-support, multi-support#
A calculation relating to excitations with all identical supports can be carried out according to two different paths, theoretically equivalent:
single-support calculation;
multi-support calculation with excitations identical to the supports.
Attention is drawn to the fact that, although the results are strictly equal in the absence of static correction via the pseudo-mode, they may present slight differences when the static correction is taken into account, because the respective treatments of this static correction are then not strictly equivalent.
4.7. Combination of directional responses#
Two rules for combining directional responses are available.
4.7.1. Quadratic combination#
This combination corresponds to the hypothesis of strict independence of responses in each direction cf. [§ 3.3.3]. Recall that this combination rule has no geometric meaning, although the three directions of analysis are orthogonal.
\(R=\sqrt{{R}_{X}^{2}+{R}_{Y}^{2}+{R}_{Z}^{2}}\)
The assumptions justifying the simple quadratic combination method, at this level, do not seem to need to be questioned [bib3], but this method is not the most used.
4.7.2. Combination of NEWMARK#
This rule of empirical combination is the most commonly used and generally leads to estimates that are slightly stronger than the previous one. It assumes that when one of the directional responses is maximum, the others are at most equal to \(4/10\) of their respective maximum contributions. For each of the directions \(i(X,Y,Z)\), the 8 values are calculated:
\({R}_{l}=\pm {R}_{X}\pm \mathrm{0,4}{R}_{Y}\pm \mathrm{0,4}{R}_{Z}\)
This leads, by circular permutation, to 24 values and \(R=\text{max}({R}_{l})\)
4.8. Warning on combinations#
Several remarks are needed to warn the user about how to use combination methods and combined quantities in a study note.
Note 1:
If you want to use arithmetic combinations (direction) and quadratic combinations (modes), quadratic accumulations should always be done last.
Note 2:
Any quadratic combination only applies to quantities for which, in instantaneous values, the accumulation has the meaning of a sum: combination of displacement components, or generalized effort or constraint of each natural mode.
The modal or directional quadratic combination cannot therefore be applied to stress intensities (main, von Mises, Tresca stresses) .
Note 3:
The results of a combination, regardless of the accumulation rule, should not be used as data to calculate other quantities: for example, a differential displacement between two points (or a deformation) can only be calculated from the modal differential displacements that are then combined.
A fortiori, generalized forces and constraints can only be calculated mode by mode before any combination and not from inertia forces deduced from the acceleration fields obtained by combining modal accelerations.
4.9. Regulatory practices#
4.9.1. Partition of the primary and secondary components of the response#
The different supports of a pipe line can be animated by different movements. The same pipe section can be distributed over different buildings, levels or equipment. It therefore undergoes multiple excitation. This results in two types of loading [§ 2.1.2]:
an excitation whose frequency content varies from one medium to another and which constitutes a primary loading according to the RCC -M classification,
Differential Seismic Displacements (DDS) inducing a state of constraint by displacements imposed on the supports and classified as secondary.
The generalized moments resulting from these 2 loads intervene separately in the RCC -M sizing inequalities and at several levels.
For a thorough RCC -M post-processing, it is therefore necessary to have the primary and secondary components of the seismic response available.
More generally, the method for combining support responses may differ depending on whether the case of inertial or differential components is treated. Moreover, the number of supports concerned by these two summations may not be equal. It is often necessary to impose overall differential movements even for supports associated with different user spectra. On the other hand, DDS formulated in rotation are sometimes worth considering. They cannot be associated with inertial loading (limited to translations).
Code_Aster therefore offers two treatments:
Determination of the overall response:
The inertial and static training contributions are accumulated when calculating the directional support responses [§4.6].
Partition of the primary and secondary components of the global response:
The two previous contributions are no longer combined when calculating the directional responses and are subject to 2 independent treatments:
The inertial component is obtained by deleting the training term
in calculating the global response [§ 4.6].
The static component is determined by combining the training terms defined under the DEPL_MULT_APPUI keyword. The methods for combining these load cases DDS are specified in the COMB_DEPL_APPUI keyword.
4.9.2. Envelope spectrum method#
Even if the pipes are subjected to multiple seismic excitation, the current practice is to reduce themselves to the calculation of a mono-supported structure while maintaining load cases DDS. This simplified approach involves defining a single spectrum per direction for all pipe supports. For each direction, an « envelope » spectrum of the various spectra at the supports is then adopted. The spectra selected for the horizontal directions \(X\) and \(Y\) are identical.
In almost all cases, this method generates « sizing margin ».