242
APPLICATIONS OF MULTI-DEGREE OF FREEDOM ANALYSIS
S(yn,^) =
I' Rn(r)e~^TdT
(9.78)
J—oo
Substitute R(Çfe,r) of équation (9.76) in (9.77). Then interchange the order of
intégration and summation, assuming a well-behaved argument. Use équation
(9.78) in this resuit to obtain
TV
=
(9.79)
n=l
Now substitute équation (9.65) into (9.70), change the index from k to n, and
substitute this rewritten form of S(r/n,aj) into équation (9.79). The resuit is
- Sr,^) ■ g W _ ^2)2 + (2Q^3)2
( ■ )
10. Calculate the variance for each physical coordinate £k. Using the définition given by équation (7.22) and S(^fc,ur) of the last resuit, compute
/>OC
a2(efc)=2/ S^k,^)dw
(9.81)
Jo
With the wave height spectrum S^(w) in the Pierson-Moskowitz or JONSWAP form, and the limits of intégration in équation (9.81) replaced by 0.16
and 1.4 rad/s, the variance of each coordinate can be obtained by numerical intégration. Assuming that p(t) is Gaussian with zéro mean, then the responses will
also be Gaussian with zéro mean, since the System is linear. Thus, the rms value
of the fcth physical coordinate is given by the square root of équation (9.81).
For practical purposes, the extreme limits of
are ±3cr(£fc), fc = 1,2,... ,N.
For these extreme values, if the static stresses and deflections of the members
are within the allowable limits, then the structure is assumed safe, without considering other loadings and fatigue failure. For design purposes, it is generally
acceptable to superimpose the effects of the static or steady drag loadings due
to winds and currents.
9.4
A FIXED LEG PLATFORM: STATISTICAL RESPONSES
The statistical responses derived in the last section are now computed for the
two degree of freedom jacket template platform modeled in Figures 9.1 and 9.2.
The numerical parameters of this platform are summarized in Table 9-1- The
results of the first three steps of this calculation were obtained in Section 9.1
and are summarized as follows in terms of these numerical parameters:
1. The mathematical model is defined by the following two équations of
motion. The form of damping is defined later.
4.69 x 106
0
0
3.13 x 106
Cil
C21
C12
C22
G
. £2
APPLICATIONS OF MULTI-DEGREE OF FREEDOM ANALYSIS
S(yn,^) =
I' Rn(r)e~^TdT
(9.78)
J—oo
Substitute R(Çfe,r) of équation (9.76) in (9.77). Then interchange the order of
intégration and summation, assuming a well-behaved argument. Use équation
(9.78) in this resuit to obtain
TV
=
(9.79)
n=l
Now substitute équation (9.65) into (9.70), change the index from k to n, and
substitute this rewritten form of S(r/n,aj) into équation (9.79). The resuit is
- Sr,^) ■ g W _ ^2)2 + (2Q^3)2
( ■ )
10. Calculate the variance for each physical coordinate £k. Using the définition given by équation (7.22) and S(^fc,ur) of the last resuit, compute
/>OC
a2(efc)=2/ S^k,^)dw
(9.81)
Jo
With the wave height spectrum S^(w) in the Pierson-Moskowitz or JONSWAP form, and the limits of intégration in équation (9.81) replaced by 0.16
and 1.4 rad/s, the variance of each coordinate can be obtained by numerical intégration. Assuming that p(t) is Gaussian with zéro mean, then the responses will
also be Gaussian with zéro mean, since the System is linear. Thus, the rms value
of the fcth physical coordinate is given by the square root of équation (9.81).
For practical purposes, the extreme limits of
are ±3cr(£fc), fc = 1,2,... ,N.
For these extreme values, if the static stresses and deflections of the members
are within the allowable limits, then the structure is assumed safe, without considering other loadings and fatigue failure. For design purposes, it is generally
acceptable to superimpose the effects of the static or steady drag loadings due
to winds and currents.
9.4
A FIXED LEG PLATFORM: STATISTICAL RESPONSES
The statistical responses derived in the last section are now computed for the
two degree of freedom jacket template platform modeled in Figures 9.1 and 9.2.
The numerical parameters of this platform are summarized in Table 9-1- The
results of the first three steps of this calculation were obtained in Section 9.1
and are summarized as follows in terms of these numerical parameters:
1. The mathematical model is defined by the following two équations of
motion. The form of damping is defined later.
4.69 x 106
0
0
3.13 x 106
Cil
C21
C12
C22
G
. £2