Calculation of Modal Response Parameters

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-freedom 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 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 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 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.

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

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

If the same modal damping is used for modes n and m (ζ_n = ζ_m = ζ), the equation reduces to