What does superposition mean according to the CQC rule in the dynamic analysis?

Answer

The complete quadratic combination (CQC rule) must be applied if adjacent modal shapes whose periods differ by less than 10% are present when analyzing spatial models with mixed torsional / translational mode shapes. If this is not the case, the square root sum rule (SRSS rule) is applied. In all other cases, the CQC rule must be applied. The CQC rule is defined as follows:

${\mathrm E}_{\mathrm{CQC}}=\sqrt{\sum_{\mathrm i=1}^{\mathrm p}\sum_{\mathrm j=1}^{\mathrm p}{\mathrm E}_{\mathrm i}{\mathrm\varepsilon}_{\mathrm{ij}}{\mathrm E}_{\mathrm j}}$

with the correlation factor:

${\mathrm\varepsilon}_{\mathrm{ij}}=\frac{8\sqrt{{\mathrm D}_{\mathrm i}{\mathrm D}_{\mathrm j}}({\mathrm D}_{\mathrm i}+{\mathrm D}_{\mathrm j})\mathrm r^{\displaystyle\frac32}}{\left(1-\mathrm r^2\right)^2+4{\mathrm D}_{\mathrm i}{\mathrm D}_{\mathrm j}\mathrm r(1+\mathrm r^2)+4(\mathrm D_{\mathrm i}^2+\mathrm D_{\mathrm j}^2)\mathrm r^2}$

with:

$\mathrm r=\frac{{\mathrm\omega}_{\mathrm j}}{{\mathrm\omega}_{\mathrm i}}$

The correlation coefficient is simplified if the viscous damping value D is selected to be the same for all mode shapes:

${\mathrm\varepsilon}_{\mathrm{ij}}=\frac{8\mathrm D^2(1+\mathrm r)\mathrm r^{\displaystyle\frac32}}{\left(1-\mathrm r^2\right)^2+4\mathrm D^2\mathrm r(1+\mathrm r^2)}$

In analogy to the SRSS rule, the CQC rule can also be executed as an equivalent linear combination. The formula of the modified CQC rule is as follows:

${\mathrm E}_{\mathrm{CQC}}=\sum_{\mathrm i=1}^{\mathrm p}{\mathrm f}_{\mathrm i}{\mathrm E}_{\mathrm i}$

with:

${\mathrm f}_{\mathrm i}=\frac{{\displaystyle\sum_{\mathrm i=1}^{\mathrm p}}{\mathrm\varepsilon}_{\mathrm{ij}}{\mathrm E}_{\mathrm j}}{\sqrt{{\displaystyle\sum_{\mathrm i=1}^{\mathrm p}}{\displaystyle\sum_{\mathrm j=1}^{\mathrm p}}{\mathrm E}_{\mathrm i}{\mathrm\varepsilon}_{\mathrm{ij}}{\mathrm E}_{\mathrm j}}}$

Keywords

CQC

Reference

[1]   Meskouris, K. (1999). Baudynamik, Modelle, Methoden, Praxisbeispiele. Berlin: Ernst & Sohn.

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