| Note |
|---|
Modal response spectrum analysis specified in 12.9.1 is done automatically. |
| Note |
|---|
In accordance with 12.9.1.1, the sufficient number of vibration modes to be taken into account is calculated automatically as 90%. |
...
In the modal response spectrum analysis method, the structure is decomposed into a number of single degree-of-freedomsystemsfreedom systems, each having its own mode shape and natural period of vibration. The number of modes available is equal to the number of mass degrees of freedom of the structure, so the number of modes can be reduced by eliminating mass degrees of freedom. For example, rigid diaphragm constraints may be used to reduce the number of mass degrees of freedom to one per story for planar models and to three per story (two translations and rotation about the vertical axis) for three-dimensional structures. However, where the vertical elements of the seismic force-resisting system have significant differences in lateral stiffness, rigid diaphragm models should be used with caution because relatively small in-plane diaphragm deformations can have a significant effect on the distribution of forces.
For a given direction of loading, the displacement in each mode is determined from the corresponding spectral acceleration, modal participation, and mode shape. Because the sign (positive or negative) and the time of occurrence of the maximum acceleration are lost in creating a response spectrum, there is no way to recombine modal responses exactly. However, statistical combination of modal responses produces reasonably accurate estimates of displacements and component forces. The loss of signs for computed quantities leads to problems in interpreting force results where seismic effects are combined with gravity effects, produce forces that are not in equilibrium, and make it impossible to plot deflected shapes of the structure.
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
...
The maximum displacement and the equivalent lateral force of the jth storey
...
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.
Modal Response Parameters
Modal response parameters are the magnitudes calculated according to the information obtained only in the direction of the earthquake considered and from the free vibration response of the structural system, regardless of the earthquake data. Modal response parameters are defined only for the (X) direction in this document. The same parameters are made for the (Y) direction. In the definition of modal response parameters, the degrees of freedom of the structural system are determined according to the defined masses. In case story floors are modeled as rigid diaphragm, the story mass is collected at the center of mass of the relevant floor. If story slabs are modeled as semi-rigid diaphragms, the masses are defined at the joints of Shell Finite Elements. In this case, instead of a total story mass, mThe masses of j (S) are taken into account.
Combined Response Parameters
...
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.
...
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
...
βnm = wm/wn =Tn /Tm ζn and ζm the damping ratio of modes n and m.
The Complete Quadratic Combination (CQC) Rule
...
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
...
If the same modal damping is used for modes n and m (ζn = ζm = ζ), the equation reduces to
...