# Modal Response Spectrum Analysis and Parameters per ASCE 7-16 with ideCAD

How does ideCAD perform modal response spectrum analysis, according to ASCE 7-16? • Individual directional results are automatically combined by CQC. And orthogonal loading effects are combined with SRSS. • 90% of mass participation ratio is automatically controlled and reported by the program. • Modal response spectrum analysis specified in 12.9.1 is done automatically. • In accordance with 12.9.1.1, the sufficient number of vibration modes to be taken into account is calculated automatically as 90%. • With the modal response analysis, seismic effects are combined automatically by CQC.

#### Symbols

mi   = total mass of the ith story
m   = mass moment of inertia of the ith story
mixn (X)  = (X) for the earthquake direction, the i'th story modal effective mass of the nth natural vibration mode of the building in the x-axis direction
miyn (X)  = (X) for the earthquake direction i'th story modal effective mass
miθn (X)  = (X) of the building's nth natural vibration around the z-axis for the earthquake direction i'th storey modal effective mass moment of inertia
mj (S) = Finite Element Analysis node j to effect individual masses
mtxn (X)   = (X) earthquake direction for building the x-axis direction of the nth vibration mode of base shear modal effective mass
mtyne (Y)  = (Y) earthquakes base shear in the building along the y axis for the direction of modal effective mass
rmax (X)  = (X) earthquake direction for any behavior variables (displacements and relative story drift, strain component) corresponding to the coupled typically to the maximum modal behavior of size
rn (X)  = Typical unit modal behavior magnitude corresponding to any action magnitude (displacement, relative floor displacement, internal force component) for the earthquake direction in the nth natural vibration mode (X),
rn,max (X)  = nth natural vibration mode ( X) Typical largest modal behavior magnitude corresponding to any action magnitude (displacement, relative floor displacement, internal force component) for the earthquake direction
SaR (Tn )  = reduced design spectral acceleration for the nth vibration mode
Tn  = nth mode natural vibration period
βmn  = ratio of mth and nth natural vibration periods
Φi (X) n  = nth natural vibration mode shape amplitude at i'th story (X) earthquake direction
Φixn  =nth natural vibration mode shape amplitude ati'th story in x-axis direction
Φiyn  = y-axis at i'th story nth natural vibration mode shape amplitude in the direction
θiθn  = nth natural vibration mode shape amplitude as rotation around the z-axis at the ith story
Γx (X)  = (X) for the earthquake direction, modal contribution of the nth vibration mode multiplier
ξn  = modal damping ratio of the nth vibration mode
ωn  = Natural vibration angular frequency of the nth vibration mode
ρmn  = Cross-correlation coefficient of the mth and nth natural vibration modes in the Complete Quadratic Combination Rule

Modal Response Analysis Method

In this method, maximum internal forces and displacements are determined by the statistical combination of maximum contributions obtained from each of the sufficient number of natural vibration modes considered. The number of modes available is equal to the number of mass degrees of freedom of the structure.

The displacement and internal forces in each mode are calculated from the corresponding spectral acceleration, modal participation, and mode shape. When the response spectrum curve is created, The sign (positive or negative) and the time of occurrence of the maximum acceleration are lost. Therefore, it is not possible to fully reassemble modal responses. However, displacements and component forces can be estimated closely by the statistical combination of modal responses produced. The loss of signs for computed quantities causes problems in interpreting force results where seismic effects are combined with gravity effects. Modal response analysis method produces 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 story  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. 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.

#### 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   JavaScript errors detected