180
STATISTICAL RESPONSES FOR LINEAR STRUCTURES
in this calculation requires that H3 be expressed in meters. The latter équation
thus becomes
S_(W) = ^e-00’38/-‘ft2.sec/rad
W5
(7.60)
The units displayed for équation (7.60) are appropriate only if uj is given in
rad/sec. A plot of équation (7.60), given in Figure 7.5, shows that for ail
practical purposes the wave energy is confined to frequency range of 0.16 to 1.6
rad/sec, the limits of intégration in the variance intégral. With équations (7.58)
and (7.60), the variance équation (7.52) is thus
2 / 3tt n2n \\ r1'6
8.40e-00138^
I —pgD'Cu
2 /
y 8
/
Jo.i6 w [(fci - mu’2) + c\u ]
(7.61)
where
pg
= 64.3 lb/ft3, water density
D
= 12 ft, single leg diameter
Cm = 2.0, assumed inertia coefficient
m
= 4.42 x 105 slugs (lb-sec2/ft), équivalent mass
ki
= 8.18 x 105 lb/ft, bending stiffness
cj
= 6.01 x 104 Ib-sec/ft, damping
When équation (7.61) was evaluated numerically, the variance was
= 0.526 ft for the rms deck deflection. For a Gaussian process,
the probability of exceeding the following dynamic responses is 0.26 percent:
deck displacement:
v = 3
fmax= k-pu/S = 4.30 x 105 1b
overturning moment:
Almax = £fmax = 1.14 x 108 ft-lb,
(leg height:
t — 265 ft )
Approximate Responses
In some applications, the spectral density of the loading Spi(u>) can be approximated as a constant So over a frequency band between the limits of wi
and u/2 and as zéro outside that frequency band. That is
•5Pi(to) = |G(co)|2Sn(u;) = So = const., uq < (üqui0) < co2
(7.62a)
Spi(ui) =0,
w < uq and w > w2
(7.62b)
Equations (7.62) define band-limited. white noise. The désignation white noise
originated with the description of white light for which the spectrum is nearly
uniform over the range of frequency for visible light. In the présent applications, it is noted that So is a one-sided spectrum since it is based on the
one-sided experimental wave spectrum Sr)(co). Further, it is assumed that the
undamped frequency u>0 of the single degree of freedom System is within the defined frequency band of équations (7.62). With these équations it follows that
the variance of the deflection response, équation (7.52), has the following form:
STATISTICAL RESPONSES FOR LINEAR STRUCTURES
in this calculation requires that H3 be expressed in meters. The latter équation
thus becomes
S_(W) = ^e-00’38/-‘ft2.sec/rad
W5
(7.60)
The units displayed for équation (7.60) are appropriate only if uj is given in
rad/sec. A plot of équation (7.60), given in Figure 7.5, shows that for ail
practical purposes the wave energy is confined to frequency range of 0.16 to 1.6
rad/sec, the limits of intégration in the variance intégral. With équations (7.58)
and (7.60), the variance équation (7.52) is thus
2 / 3tt n2n \\ r1'6
8.40e-00138^
I —pgD'Cu
2 /
y 8
/
Jo.i6 w [(fci - mu’2) + c\u ]
(7.61)
where
pg
= 64.3 lb/ft3, water density
D
= 12 ft, single leg diameter
Cm = 2.0, assumed inertia coefficient
m
= 4.42 x 105 slugs (lb-sec2/ft), équivalent mass
ki
= 8.18 x 105 lb/ft, bending stiffness
cj
= 6.01 x 104 Ib-sec/ft, damping
When équation (7.61) was evaluated numerically, the variance was
= 0.526 ft for the rms deck deflection. For a Gaussian process,
the probability of exceeding the following dynamic responses is 0.26 percent:
deck displacement:
v = 3
fmax= k-pu/S = 4.30 x 105 1b
overturning moment:
Almax = £fmax = 1.14 x 108 ft-lb,
(leg height:
t — 265 ft )
Approximate Responses
In some applications, the spectral density of the loading Spi(u>) can be approximated as a constant So over a frequency band between the limits of wi
and u/2 and as zéro outside that frequency band. That is
•5Pi(to) = |G(co)|2Sn(u;) = So = const., uq < (üqui0) < co2
(7.62a)
Spi(ui) =0,
w < uq and w > w2
(7.62b)
Equations (7.62) define band-limited. white noise. The désignation white noise
originated with the description of white light for which the spectrum is nearly
uniform over the range of frequency for visible light. In the présent applications, it is noted that So is a one-sided spectrum since it is based on the
one-sided experimental wave spectrum Sr)(co). Further, it is assumed that the
undamped frequency u>0 of the single degree of freedom System is within the defined frequency band of équations (7.62). With these équations it follows that
the variance of the deflection response, équation (7.52), has the following form: