# What is the meaning of the superposition according to the CQC rule in a dynamic analysis??

The complete quadratic combination (CQC rule) must be applied if there are the adjacent modal shapes, whose periods differ about less than 10%, when analyzing the spatial models with the combined torsional / translational mode shapes. If this is not the case, the square root of the sum of the squares (SRSS rule) applies. 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 coefficient:

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

where:

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

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

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

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

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

where:

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

#### Reference

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