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How does ideCAD perform modal response spectrum analysis, according to ASCE 7-16?

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Symbols

m_i   = total mass of the i'^th storey story
m_iθ   = mass moment of inertia of the i'^th storey story
m_ixn ^(X)  = (X) for the earthquake direction, the i'th storey story modal effective mass of the nth natural vibration mode of the building in the x-axis direction
m_iyn ^(X)  = (X) for the earthquake direction i'th storey story modal effective mass
m_iθn ^(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
m_j ^(S) = Finite Element Analysis node j to effect individual masses
m_txn ^(X)   = (X) earthquake direction for building the x-axis direction of the nth vibration mode of base shear modal effective mass
m_tyne ^(Y)  = (Y) earthquakes base shear in the building along the y axis for the direction of modal effective mass
r_max ^(X)  = (X) earthquake direction for any behavior variables (displacements and relative storey story drift, strain component) corresponding to the coupled typically  typically to the maximum modal behavior of size
r_n ^(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),
r_n,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
S_aR (T_n )  = reduced design spectral acceleration for the nth vibration mode
T_n  = nth mode natural vibration period
β_mn  = ratio of mth and nth natural vibration periods
Φ_{i (X) n}  = nth natural vibration mode shape amplitude at i'th storey story (X) earthquake direction
Φ_ixn  =nth natural vibration mode shape amplitude ati'th storey story in x-axis direction
Φ_iyn  = y-axis at i'th storey story nth natural vibration mode shape amplitude in the direction
θ_iθn  = nth natural vibration mode shape amplitude as rotation around the z-axis at the ith storeystory
Γ_x ^(X)  = (X) for the earthquake direction, modal contribution of the nth vibration mode multiplier
ξ_n  = modal damping ratio of the nth vibration mode
ω_n  = Natural vibration angular frequency of the nth vibration mode
ρ_mn  = Cross-correlation coefficient of the mth and nth natural vibration modes in the Complete Quadratic Combination Rule

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Modal Response Analysis Method

In the modal response spectrum analysis this method, the structure is decomposed into a number of single degree-of-freedomsystems, each having its own mode shape and natural period of vibrationmaximum 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, 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 .

The displacement and internal forces in each mode are calculated from the corresponding spectral acceleration, modal participation, and mode shape. Because the When the response spectrum curve is created, 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, . 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 produces reasonably accurate estimates of displacements and component forcesproduced. The loss of signs for computed quantities leads to causes problems in interpreting force results where seismic effects are combined with gravity effects, produce . 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 S_d(T_n, x), S_v(T_n, x), and S_a(T_n, 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 j^th 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.

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,  m_The  masses of _j ^(S) are taken into account.

Combined Response Parameters

SRSS and CQC methods are identical where applied to planar structures, or where zero damping is specified for the computation of the cross-modal coefficients in the CQC method. The modal damping specified in each mode for the CQC method should be equal to the damping level that was used in the development of the design response spectrum. For the spectrum in Section 11.4.6, the damping ratio is 0.05. The SRSS or CQC method is applied to loading in one direction at a time. Where Section 12.5 requires explicit consideration of orthogonal loading effects, the results from one direction of loading may be added to 30% of the results from loading in an orthogonal direction. Wilson (2000) suggests that a more accurate approach is to use the SRSS method to combine 100% of the results from each of two orthogonal directions where the individual directional results have been combined by SRSS or CQC, as appropriate.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

β_nm = w_m/w_n =T_n /T_m ζ_n and ζ_m the damping ratio of modes n and m.

The Complete Quadratic Combination (CQC) Rule

Mathinline
body --uriencoded--$$ \normalsize r_o \approx \sqrt%7B\displaystyle\sum_%7Bn=1%7D%5e%7BN%7D \displaystyle\sum_%7Bm=1%7D%5e%7BN%7D \epsilon_%7Bnm%7Dr_nr_m%7D $$

where ϵ_nm is a correlation coefficient that takes values in the 0,1 range and is equal to 1 when n=m. β_nm the correlation term is calculated as

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