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Ανάλυση φάσματος απόκρισης: Μέθοδοι συνδυασμού τρόπων

When running a Ανάλυση φάσματος απόκρισης on a structure, one of the most important steps for getting the desired results is the definition of the modal combination method that will be used by the software to get a single result for displacement, αντιδράσεις, εσωτερικές δυνάμεις, και τα λοιπά… for each degree of freedom out of the raw results obtained for each vibration mode (modal responses). Those modal combined results are the ones used for designing the structure, that’s why properly defining the modal combination method is extremely important. Σε αυτό το άρθρο, we will dive into some modal combination methods for Response Spectrum Analysis.

It is important to mention that one of the most important steps during a Response Spectrum Analysis is the evaluation of the natural frequency for each vibration mode and its mass contribution, check our documentation on Dynamic Frequency Analysis.

Ανάλυση φάσματος απόκρισης, Μέθοδοι συνδυασμού τρόπων

Modal combination methods for Response Spectrum Analysis

Some of the most known and used methods for modal combination are:

  • ABS: Absolute Sum
  • SRSS: Square Root of Sum of the Squares
  • CQC: Complete Quadratic Combination

Γενικά, peak values of the total response (\(r_o\)) for displacements, εσωτερικές δυνάμεις, and reactions are of interest. Οπως προαναφέρθηκε, that peak value is obtained from properly combining the peak modal responses (\(r_{ν, ο}\))

Ανάλυση φάσματος απόκρισης: Μέθοδοι συνδυασμού τρόπων

Absolute Sum (ABS)

The Absolute Sum modal combination method takes the absolute value of the result (displacement or internal force) for each vibration mode and sums up all those absolute values. Physically, it assumes that all the peak modal responses occur at the same time. By doing that, it is the most conservative method, as it makes every modal contribution positive, και ως εκ τούτου, it is not popular in structural design applications. The formula for calculating the peak value of the total response is:

\(r_o=\sum_{n=1}^{Ν} |r_{ν,ο}|\)

Οπου,

\(n\) είναι το \(n^{ου}\) vibration mode considered in the analysis

\(N\) is the total number of vibration modes considered in the analysis

\(r_o\) is the peak total response

\(r_{ν,ο}\) is the peak modal response for the \(n^{ου}\) vibration mode

Square Root of Sum of the Squares (SRSS)

The Square Root of the Sum of the Squares modal combination method takes the square root of the sum of the squares of the result for each vibration mode, providing an estimate of the peak of the total response. This modal combination method provides excellent response estimates for structures with well-separated natural frequencies, for the cases in which the natural frequencies of the structure are not well-separated this method should not be used. Formally, the peak total response can be expressed as follows:

\(r_o=\left(\sum_{n=1}^{Ν} r^{2}_{ν,ο}\σωστά)^{1/2}\)

Οπου,

\(n\) είναι το \(n^{ου}\) vibration mode considered in the analysis

\(N\) is the total number of vibration modes considered in the analysis

\(r_o\) is the peak total response

\(r_{ν,ο}\) is the peak modal response for the \(n^{ου}\) vibration mode

Complete Quadratic Combination (CQC)

The Complete Quadratic Combination method overcomes the limitation mentioned for SRSS when combining modal responses in a structure with closely-spaced natural frequencies. Σε αυτή τη μέθοδο, the peak total response is obtained by applying the following formula:

\(r_o=\left(\sum_{= Μικτό εμβαδόν διατομής}^{Ν}\sum_{n=1}^{Ν} \Οι ροπές κάμψης υπολογίζονται σε κάθε κατεύθυνση σε τμήματα{σε} \cdot r_{Εγώ} \cdot r_{όχι} \σωστά)^{1/2}\)

Οπου,

\(ν, \; i\) είναι το \(n^{ου}, \; i^{ου}\) vibration mode considered in the analysis

\(N\) is the total number of vibration modes considered in the analysis

\(r_o\) is the peak total response

\(r_{ν,ο}, \; r_{Εγώ,ο}\) is the peak modal response for the \(n^{ου}, \; i^{ου}\) vibration mode

\(\Οι ροπές κάμψης υπολογίζονται σε κάθε κατεύθυνση σε τμήματα{σε}\) is the correlation coefficient for the two modes being combined at each summation step

The formula above can be rewritten as two separate groups of summations, the first being identical to the SRSS modal combination method. The second double summation includes all the cross (\(i \neq n\)) όροι, each of which may be positive or negative, this yields the fact that the estimate for the peak total response using CQC might be larger or smaller than the estimate provided by the SRSS :

\(r_o=\left( \sum_{n=1}^{Ν} r^{2}_{ν,ο} + \underbrace{\sum_{= Μικτό εμβαδόν διατομής}^{Ν}\sum_{n=1}^{Ν}}_{i \neq n} \Οι ροπές κάμψης υπολογίζονται σε κάθε κατεύθυνση σε τμήματα{σε} \cdot r_{Εγώ} \cdot r_{όχι} \σωστά)^{1/2}\)

Only one term in the equation for CQC has not been defined: the correlation coefficient. One of the most used equations for this coefficient is:

\(\Οι ροπές κάμψης υπολογίζονται σε κάθε κατεύθυνση σε τμήματα{σε} = frac{\xi^2(1+\βήτα_{σε})^ 2}{(1-\βήτα_{σε})^2+4\xi^2\beta_{σε}}\)

Οπου,

\(\βήτα_{σε}\) is the ratio between the natural frequencies of the \(i^{ου}\) και \(n^{ου}\) τρόπους λειτουργίας (\(\omega_i / \omega_n\))

\(\xi\) is the damping coefficient for the structure

βιβλιογραφικές αναφορές

Chopra, ΕΝΑ. (2015, Ιούνιος 4). Dynamics of Structures (4th ed.). Pearson Higher Education.

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