182
STATISTICAL RESPONSES FOR LINEAR STRUCTURES
Also. as u>2 ~* oo, *-he argument of the natural logrithmic term approaches unity
and In (1) = 0. Thus, for idéal white noise, So is uniform for ail frequencies and
Example Problem 7.4- For the jackup rig described in Example Problème
5.4 and 7.3, compute the upper bound for the rms deck deflection, av, for each
of the three idealized loading spectra shown in Figure 7.6. In ail three cases,
choose
So = |G(w)|2Sn(w) = (4.76 x 108)(336.3) = 1.60 x 1011 lb2-sec/rad
Here the quantity 336.3 ft2-sec/rad is arbitrarily chosen as one-half the peak
value of the wave height spectrum shown in Figure 7.5, the value that corresponds to the frequency of a> = 0.324 rad/sec. The results are summarized.
(a) For the band-limited spectrum of Figure 7.6a, choose the same frequency
limits that were used in the direct intégration of équation (7.61), or oq = 0.16
rad/sec and u>2 = 1-6 rad/sec. Compute the variance by evaluating équations
(7.64)-(7.66), from which
a>2 = wc = 1.6 rad/sec. Compute the variance by evaluating équations (7.64)
and (7.65) using (7.67). The resuit is
av = 3.1079 ft
(c)
For idéal white noise, use équation (7.68) to give
the band width increases. However, these idealized approximations ail led to a
gross overestimate in av by about an order of magnitude, compared to the value
of 0.372 ft computed by direct intégration of équation (7.61). The conclusion is
that direct intégration gives the best answer, at least for this type of problem
in which the structure’s fundamental frequency cuq is at the very low end of the
wave height spectrum.
Extensions
There are several possible extensions to the classical statistical results obtained thus far in this chapter. These extensions involve the relaxation of certain
restrictive assumptions upon which the variance of the response given by équation (7.52) was based. Two of those assumptions were: there exists a single,
stationary excitation wave force of a known spectrum Spi(a>) and the wave
excitation force has a zéro mean value. This assumption of zéro mean was investigated by Tung (1974), who showed how a single, stationary wave excitation
STATISTICAL RESPONSES FOR LINEAR STRUCTURES
Also. as u>2 ~* oo, *-he argument of the natural logrithmic term approaches unity
and In (1) = 0. Thus, for idéal white noise, So is uniform for ail frequencies and
Example Problem 7.4- For the jackup rig described in Example Problème
5.4 and 7.3, compute the upper bound for the rms deck deflection, av, for each
of the three idealized loading spectra shown in Figure 7.6. In ail three cases,
choose
So = |G(w)|2Sn(w) = (4.76 x 108)(336.3) = 1.60 x 1011 lb2-sec/rad
Here the quantity 336.3 ft2-sec/rad is arbitrarily chosen as one-half the peak
value of the wave height spectrum shown in Figure 7.5, the value that corresponds to the frequency of a> = 0.324 rad/sec. The results are summarized.
(a) For the band-limited spectrum of Figure 7.6a, choose the same frequency
limits that were used in the direct intégration of équation (7.61), or oq = 0.16
rad/sec and u>2 = 1-6 rad/sec. Compute the variance by evaluating équations
(7.64)-(7.66), from which
a>2 = wc = 1.6 rad/sec. Compute the variance by evaluating équations (7.64)
and (7.65) using (7.67). The resuit is
av = 3.1079 ft
(c)
For idéal white noise, use équation (7.68) to give
the band width increases. However, these idealized approximations ail led to a
gross overestimate in av by about an order of magnitude, compared to the value
of 0.372 ft computed by direct intégration of équation (7.61). The conclusion is
that direct intégration gives the best answer, at least for this type of problem
in which the structure’s fundamental frequency cuq is at the very low end of the
wave height spectrum.
Extensions
There are several possible extensions to the classical statistical results obtained thus far in this chapter. These extensions involve the relaxation of certain
restrictive assumptions upon which the variance of the response given by équation (7.52) was based. Two of those assumptions were: there exists a single,
stationary excitation wave force of a known spectrum Spi(a>) and the wave
excitation force has a zéro mean value. This assumption of zéro mean was investigated by Tung (1974), who showed how a single, stationary wave excitation