Considering orthogonality of the modes and assuming Rayleigh damping (Bellos, J., 1988, 1989a, 1989b), the problem of solving the system of coupled differential equations 1 ends up to solving N independent differential equations
equation (6.2.2) is transformed into
and the independent differential equations (6.2.3) are transformed into
Where the natural frequency of each mode is
the critical damping ratio is
And the modal participation factor for vibrations in a specific seismic direction may be expressed as
Another important quantity is the effective modal mass in a specific seismic direction
The participation of effective modal mass per mode and seismic direction, can be expressed as a percentage of the total mass of the structure
In order for an eigenvalue analysis to be sufficient, the sum of percentages of effective modal mass participation for all modes considered, should be significant enough for all primary seismic directions (usually greater than 90%), i.e.
Two eigenvalue analysis methods are available in the related software: the swift Lanczos method and the classical Subspace Iteration method.
Due to the maximum stress and strain resultants being sufficient for the dynamic analysis of buildings rather than the exact time responses themselves, the superposition is based on the ordinates of the seismic accelerations spectrum R_{d}(T) provided by Seismic Regulations corresponding to discrete values of the natural periods T_{i}, where
Provided that the spectral ordinates of displacements are obtained from the corresponding ordinates of accelerations in a proportion 1/ω _{i}^{2}, the terms of equation (6.2.5) yield that the maximum modal displacements per seismic action are given by the expression
The maximum modal values, of course, are usually asynchronous, thus they may not be directly superimposed to provide the total values of the requested stress and strain resultants. To this end, appropriate combination methods are used. According to the most well-known relevant methods, the individual maximum values V_{i,max} of the components of modal resultants are combined to give the maximum values V_{max}:
SRSS (Square Root of Sum of Squares)
and CQC (Complete Quadratic Combination)
where, for damping ratio x as defined by the EC8, the correlation factor of i and j modes is given by the expression
Using CQC method is recommended in the related software, as being more general and complete.