FAQ 003414 EN

28 July 2021

003414

DYNAM Pro - Equivalent Loads

RF-DYNAM Pro - Equivalent Loads

RF-DYNAM Pro - Forced Vibrations

Steel Structures

Timber Structures

Dynamic and Seismic Analysis

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 when analyzing spatial models with combined torsional/translational eigenmodes that have adjacent modal shapes whose periods differ by less than 10%.

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:

Standard CQC Rule

$E_{C Q C} = \sqrt{\sum_{i = 1}^{p} \sum_{j = 1}^{p} E_{i} \cdot ε_{i j} \cdot E_{j}}$

ε_ij

Correlation coefficient

with the correlation coefficient:

Correlation Coefficient (CQC Rule)

$ε_{i j} = \frac{8 \cdot \sqrt{D_{i} \cdot D_{j}} \cdot (D_{i} + r \cdot D_{j}) \cdot r^{\frac{3}{2}}}{{(1 - r^{2})}^{2} + 4 \cdot D_{i} \cdot D_{j} \cdot r \cdot (1 + r^{2}) + 4 \cdot ({D_{i}}^{2} + {D_{j}}^{2}) \cdot r^{2}}$

D_i, D_j	Damping values
r	Quotient of the natural angular frequencies (ω_j/ ω_i)

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

Correlation Coefficient (CQC Rule, Simplified: D_i = D_j)

$ε_{i j} = \frac{8 \cdot D^{2} \cdot (1 + r) \cdot r^{\frac{3}{2}}}{{(1 - r^{2})}^{2} + 4 \cdot D^{2} \cdot r \cdot {(1 + r)}^{2}}$

D	Damping value
r	Quotient of the natural angular frequencies (ω_j/ ω_i)

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:

Modified CQC Rule - Equivalent Linear Combination

$E_{C Q C} = \sum_{i = 1}^{p} f_{i} \cdot E_{i} where f_{i} = \frac{\sum_{j = 1}^{p} ε_{ij} \cdot E_{j}}{\sqrt{\sum_{i = 1}^{p} \sum_{j = 1}^{p} E_{i} \cdot ε_{i j} \cdot E_{j}}}$

Keywords

CQC

Reference

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

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FAQ 003414 EN

Helpful Questions & Answers

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

Answer

Keywords

Reference