An Aqwa frequency domain dynamic analysis outputs the significant amplitudes of forces and responses

(12–28)

where in which is a force or response spectral density.

The significant amplitude given by Equation 12–28 can be used to estimate the probable maximum value over a given duration. If the force or response spectral density is assumed to be Rayleigh-distributed, and the mean zero crossing period of the force or response is assumed to be similar to the mean zero crossing period of the wave spectrum (as computed according to Irregular Waves), the number of independent maxima over the duration can be estimated as

(12–29)

Assuming that is large, the probable maximum value can be determined from

(12–30)

where the scaling factor is defined as

(12–31)

To similarly estimate a -fractile (or percentile) extreme value of the force or response this scaling factor can be modified as

(12–32)

where . Using the formulation, the median extreme value can be estimated by setting , while the expected extreme is estimated with 11.

Numerically, the density integration of spectral densities is performed over a finite frequency range. When the wave frequency responses only are concerned, this numerical integrating wave frequency range is given by (as defined in Equation 11–4 for a specified wave spectral group). When the total is required, consisting of both the wave frequency and drift frequency responses, the wave frequency range is given by .

Integration of the response spectral density is achieved using a 3-point Gaussian quadrature algorithm within program-selected frequency intervals. These intervals are chosen based on the natural frequencies of the equations of motion and the peak of the spectrum, as described below.

Let us assume that there are M natural frequencies falling into the frequency range of integration as defined above. Denoting as the j-th natural frequency of the structural system with a critical damping percentage , the frequency at the peak value of the receptance function of the j-th single degree of freedom system is

(12–33)

The effective critical damping percentage and effective natural frequency of each sub-directional wave spectrum are defined as

(12–34)

where is the peak frequency of the j-th sub-directional wave spectrum.

If the starting integration frequency is , then the frequency interval from this starting frequency is determined by

(12–35)

where

Three Gaussian evaluation points are chosen within the frequency interval of .

Subsequently, the starting frequency for the k-th 3-point Gaussian quadrature integration is found from

(12–36)

where , and the frequency interval is defined as

(12–37)

where

It should be noted that in order to limit the integration to a reasonable number of intervals, a minimum critical damping percentage of 0.5% assumed.