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Modal analysis provides the entire response history for a given ground motion record. For design
purposes, its application requires a design ground motion record that is representative of the seismic
hazard at the site. For design purposes, we usually use the maximum value of a response parameter and
not the entire response history. Since every mode can be treated as an independent SDOF system, the
maximum response values of a mode can be easily obtained from the corresponding response
spectrum. If Sd(Tn, x), Sv(Tn, x), and Sa(Tn, x) denote the spectral displacement, velocity, and acceleration,
respectively, the maximum modal displacements are obtained from a response spectrum as
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The maximum displacement and the equivalent lateral force of the jth storey
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story
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It is used in the horizontal elastic design spectrum in the direction of a given earthquake and the maximum values of the response magnitudes in each vibration mode are calculated with the modal analysis method. The largest non-synchronous modal behavior magnitudes calculated for enough vibration modes are then combined statistically to obtain approximate values of the largest behavior magnitudes.
For each vibration mode considered, the largest modal behavior magnitudes namely displacements, relative floor displacements, internal forces and stresses are found. Located in the largest size modal behavior of Complete Quadratic Combination. It is combined using the (CQC) rule. In this analysis, it does not give information about when the said behavior magnitude occurred and its correlation with other loadings.
The Square Root of the Sum of Squares (SRSS) Rule
The most common rule for modal combination is the Square Root of Sum of Squares (SRSS) rule.
According to this rule, the peak response of every mode is squared and then the squares are summed.
The estimation of the maximum response quantity of interest is the square of the sum.
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The major limitation is that in order to produce satisfying estimates, the modes should be
well separated, i.e., the eigenfrequencies should not have close values. If this condition is not met,
the CQC method should be used instead. A criterion to determine if two modes are well separated is
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βnm = wm/wn =Tn /Tm ζn and ζm the damping ratio of modes n and m.
The Complete Quadratic Combination (CQC) Rule
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where ϵnm is a correlation coefficient that takes values in the 0,1 range and is equal to 1 when n=m.
βnm the correlation term is calculated as
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If the same modal damping is used for modes n and m (ζn = ζm = ζ), the equation reduces to
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