Copy of Calculation of Maximum Behavior Magnitudes by Mode Combining Method (4.8.2.1)

Symbols

m _i   = total mass of the _i'th storey
m _iθ   = mass moment of inertia of the _i'th storey
m _ixn ^(X)  = (X) for the earthquake direction, the i'th storey 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 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 drift, strain component) corresponding to the coupled typically  to 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 (X) earthquake direction
Φ _ixn  =nth natural vibration mode shape amplitude ati'th storey in x-axis direction
Φ _iyn  = y-axis at i'th storey 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 storey
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

Modal Response Analysis Method

In the modal response spectrum analysis method, the structure is decomposed into a number of single degree-of-freedomsystems, 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.

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.