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

Answer

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:

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

Keywords

CQC

Reference

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

Contact us

Did you find your question?
If not, contact us via our free e-mail, chat, or forum support, or send us your question via the online form.

+49 9673 9203 0

info@dlubal.com

RFEM Main Program
RFEM 5.xx

Main Program

Structural engineering software for finite element analysis (FEA) of planar and spatial structural systems consisting of plates, walls, shells, members (beams), solids and contact elements

Price of First License
3,540.00 USD
RSTAB Main Program
RSTAB 8.xx

Main Program

The structural engineering software for design of frame, beam and truss structures, performing linear and nonlinear calculations of internal forces, deformations, and support reactions

Price of First License
2,550.00 USD
RSTAB Dynamic Analysis
DYNAM Pro - Forced Vibrations 8.xx

Add-on Module

Dynamic and seismic analysis including time history analysis and multi-modal response spectrum analysis

Price of First License
1,120.00 USD
RSTAB Dynamic Analysis
DYNAM Pro - Equivalent Loads 8.xx

Add-on Module

Seismic and static load analysis using the multi-modal response spectrum analysis

Price of First License
580.00 USD
RFEM Dynamic Analysis
RF-DYNAM Pro - Equivalent Loads 5.xx

Add-on Module

Seismic and static load analysis using the multi-modal response spectrum analysis

Price of First License
760.00 USD
RFEM Dynamic Analysis
RF-DYNAM Pro - Forced Vibrations 5.xx

Add-on Module

Dynamic and seismic analysis including time history analysis and multi-modal response spectrum analysis

Price of First License
1,120.00 USD