SYNTHESIS OF TIME HISTORIES FROM SPECTRA
159
model, the family of wave spectra can be derived. The following example illustrâtes these ideas.
Example Problem 6.1.
The hindcast studies for a particular site in the
Gulf of Mexico indicate that Hs for the 100-year storm is 12.2 m. Assume that
the wave spectral family can be described by the two-parameter Bretschneider
(1959) model, équation (6.24), and that the modal frequencies,
the corresponding periods Tm, the probabilities for ocurrence of Ha, for this family are
those listed is Table 6.3. The results in Table 6.3 were derived by Ochi (1978)
from selected wave data. Note that the sum of the probabilities for occurrence
(or weighting factors) of Hs is equal to unity.
The équation for the most probable spectrum is obtained by substituting
into équation (6.24) the frequency at the maximum of the spectrum: a)m = 0.41
rad/s from Table 6.3, for which the probability of occurrence is the maximum
of 0.25. With Hs = 12.2 m, the spectrum of équation (6.24) becomes
$„(«) =
e-° °3532^ m2 • s/rad
(6.26)
Equation (6.26), which is plotted in Figure 6.5, corresponds to the most probable
spectrum associated with a storm of a 100-year récurrence interval. As such,
this curve is not a design spectrum but a member of the family of spectra to be
considered in design. Other members of the family are also shown in Figure 6.5,
with selected frequencies u’m. and the corresponding probabilities of occurrence
as given in Table 6.3, together with their confidence limits.
The ideas illustrated in this example can be extended to more complex spectral formulations such as JONSWAP, which may be more appropriate for other
sites such as the North Sea. The reader is encouraged to consult the most recent references so that the most up-to-date data can be used to generate the
family of design spectra. Once these spectra are chosen, they can be employed
for preliminary dynamic analyses of alternative structural designs, as will be
illustrated in succeeding chapters.
6.5
SYNTHESIS OF TIME HISTORIES FROM SPECTRA
In the analysis of offshore structures, time domain solutions are frequent ly required. However, if the wave height excitation is expressed in the form of spectral density, it is necessary to transform this design spectra into an ensemble of
représentative time historiés. This may be accomplished by utilizing Borgman s
(1969) procedure for wave simulation. This is now presented with slight modifications.
The wave élévation T](t) can be represented as
rOO
______________
r](x,t) —
sïa(kx — urt + e)
dut
(6-27)
Jo
159
model, the family of wave spectra can be derived. The following example illustrâtes these ideas.
Example Problem 6.1.
The hindcast studies for a particular site in the
Gulf of Mexico indicate that Hs for the 100-year storm is 12.2 m. Assume that
the wave spectral family can be described by the two-parameter Bretschneider
(1959) model, équation (6.24), and that the modal frequencies,
the corresponding periods Tm, the probabilities for ocurrence of Ha, for this family are
those listed is Table 6.3. The results in Table 6.3 were derived by Ochi (1978)
from selected wave data. Note that the sum of the probabilities for occurrence
(or weighting factors) of Hs is equal to unity.
The équation for the most probable spectrum is obtained by substituting
into équation (6.24) the frequency at the maximum of the spectrum: a)m = 0.41
rad/s from Table 6.3, for which the probability of occurrence is the maximum
of 0.25. With Hs = 12.2 m, the spectrum of équation (6.24) becomes
$„(«) =
e-° °3532^ m2 • s/rad
(6.26)
Equation (6.26), which is plotted in Figure 6.5, corresponds to the most probable
spectrum associated with a storm of a 100-year récurrence interval. As such,
this curve is not a design spectrum but a member of the family of spectra to be
considered in design. Other members of the family are also shown in Figure 6.5,
with selected frequencies u’m. and the corresponding probabilities of occurrence
as given in Table 6.3, together with their confidence limits.
The ideas illustrated in this example can be extended to more complex spectral formulations such as JONSWAP, which may be more appropriate for other
sites such as the North Sea. The reader is encouraged to consult the most recent references so that the most up-to-date data can be used to generate the
family of design spectra. Once these spectra are chosen, they can be employed
for preliminary dynamic analyses of alternative structural designs, as will be
illustrated in succeeding chapters.
6.5
SYNTHESIS OF TIME HISTORIES FROM SPECTRA
In the analysis of offshore structures, time domain solutions are frequent ly required. However, if the wave height excitation is expressed in the form of spectral density, it is necessary to transform this design spectra into an ensemble of
représentative time historiés. This may be accomplished by utilizing Borgman s
(1969) procedure for wave simulation. This is now presented with slight modifications.
The wave élévation T](t) can be represented as
rOO
______________
r](x,t) —
sïa(kx — urt + e)
dut
(6-27)
Jo