Seismic loading: Modal superposition
Modal superposition methods are used to calculate the response R of a seismic analysis. The term "response" (R) refers to the results obtained by a seismic analysis, i.e. displacements, velocities, accelerations, member forces or stresses.
Because the differential equations were uncoupled, a result will be obtained for each mode j (see previous chapter: Bases of modal analysis)
To obtain the global response Rtot of the structure, the individual modal responses R(j)have to be combined.
SRSS-method (Square Root of Sum of Squares)
This method is suitable when all modes can considered as decoupled, i.e. each mode has frequency that is far enough from its neighbours to consider, that those modes are not interfering with each other.
If it not the case (closely-spaced modes), different techniques may be used, such as CQC or SRSS with multiple eigenshapes (see below).
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(5.13) |
With:
: the response of mode j
CQC-method (Complete Quadratic Combination)
This method is suitable in the case of closely-spaces modes. It is identical to SRSS when the modes are fully decoupled. It is more general than SRSS, but also more computationally intensive.
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(5.14) |
With
: the response of mode i and j
: modal cross correlation coefficients.
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(5.15) |
: Circular frequency for mode i and j.
: Damping ratio for mode i and j.
This method is based on both modal frequency and modal damping. The CQC-method requires the input of a Damping Spectrum.
Let’s remark that the spectrum defined in the settings of the seismic action is overruled, in case the non-uniform damping method is used. See for details the chapter Non-uniform damping.
MAX-method
As for the SRSS-method, this method is suitable when all modes can considered as decoupled, i.e. each mode has frequency that is far enough from its neighbours to consider, that those modes are not interfering with each other.
If it not the case (closely-spaced modes), different techniques may be used, such as CQC or SRSS with multiple eigenshapes (see below).
|
(5.16) |
With:
: The response of mode j
: The maximum response of all modes
Multiple eigenshapes
This tool can be used in the seismic analysis in case the SRSS modal superposition method is selected.
Contiguous modes are taken together in case the precision condition is met
Standard SRSS
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SRSS with multiple eigenshapes
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If where
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then mode (i) and (j) are multiple. Then for example
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(5.17) |
where mode 2 and 3 are multiple.