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A methodology for assessing the reliability of taut and slack mooring systems against instability
Ocean Engineering 32 (2005) 1216–1234 www.elsevier.com/locate/oceaneng
A methodology for assessing the reliability of taut and slack mooring systems against instability A.A. Shah, A. Umar, N.A. Siddiqui* Department of Civil Engineering, Aligarh Muslim University, Aligarh 202 002, India Received 28 June 2004; accepted 3 November 2004 Available online 13 March 2005
Abstract In the present study, a methodology for reliability assessment of slack and taut mooring systems against instability has been presented. For this purpose, first, stability analysis of slack and taut mooring systems has been carried out and instable regions are obtained using procedure available in the literature. Having known the instable region(s), methodology for reliability assessment has been proposed which is based on Monte Carlo Simulation technique. After using the proposed methodology, probabilities of failure and reliability indices has been obtained for the above systems. Some parametric studies, such as, effect of lower and upper limits of instability and effect of frequency range of generations are also included to obtain the results of practical interest. q 2005 Elsevier Ltd. All rights reserved. Keywords: Mooring systems; Offshore vessels; Structural reliability; Instability
1. Introduction Moored offshore vessels and platform are becoming popular day by day in various offshore activities. This is so, because moored offshore vessels and platforms are economical, easy to install, and reusable offshore structures. The mooring of such offshore structures exhibits non-linear restoration, which may cause instability problem in the mooring system. In the event of instability, the excursions may become very large that may lead to the failure of moorings of the vessel. In the last two decades, good number of research papers appeared in various journals and conference proceedings on stability * Corresponding author. Tel.: C91 571 240 1074; fax: C91 571 270 0528. E-mail address: [email protected] (N.A. Siddiqui). 0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2004.11.002
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analysis of mooring systems (e.g. Umar and Datta, 2001, Gottlib and Yim, 1990, 1992, Oppenheim and Wilson, 1982, Nayfeh and Mook, 1979, Papoulias and Bernitsas, 1988) and reliability assessment of moorings of offshore platforms (e.g. Amanullah et al., 2002, Siddiqui and Ahmad, 2000, 2001, Umar and Siddiqui, 2002, Lotsberg, 1991). Gottlieb and Yim (1992) carried out the stability analysis of a taut mooring system. Umar and Datta (2003a,b) carried out a detailed dynamic analysis of slack mooring systems. Umar and Siddiqui (2002) carried out the reliability analysis of taut mooring system against yielding of taut moorings. They employed first-order reliability method (FORM) for this purpose. Siddiqui and Ahmad (2000, 2001) and Amanullah et al. (2002) performed the reliability analysis of taut mooring system of tension leg platforms (TLPs) against maximum tension, minimum tension, fatigue and fracture. From the detailed review, it has been observed that though considerable work has been done on the stability of moored vessels and reliability assessment of mooring systems against maximum tension, minimum tension, fatigue, fracture, etc. there is almost no work on reliability against instability. Keeping this point in view, an attempt has been made to propose a methodology for the reliability assessment of slack and taut mooring systems (Figs. 1 and 2) against instability. For this purpose, first, 15 m
Buoy
4m
m.s.l
1.75 m
150 m
Clump weight Seabed 30 m
Elevation
Mooring lines
Mooring lines Buoy
Plan Fig. 1. Slack mooring system (Umar and Datta, 2003a).
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b
Taut Mooring Lines
d vessel
d
PLAN
vessel
ELEVATION
Mooring Lines
Fig. 2. Symmetric multi-point taut mooring system (Gottlieb and Yim, 1992).
stability analysis of slack and taut mooring systems has been carried out and instable regions are obtained using procedure available in the literature. Having known the instable region(s), methodology for reliability assessment has been proposed which is based on Monte Carlo Simulation technique. After using the proposed methodology, probabilities of failure and reliability indices has been obtained for the above systems. Some parametric studies, such as, effect of lower and upper limits of instability and effect of frequency range of generations are also included to obtain the results of practical interest.
2. Mathematical formulation The reliability against instability requires the instable region(s), which can be obtained only after carrying out a non-linear dynamic and stability analyses of the moored vessels. Therefore, before discussing the formulation for reliability assessment, a brief formulation of dynamic analysis and stability analysis are also presented in the following section.
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2.1. Dynamic analysis Dynamic analysis is a prerequisite for the stability analysis and reliability assessment of moored vessels. In the present study, dynamic analyses of Taut and Slack mooring systems have been carried out assuming mooring system as a single degree of freedom non-linear oscillator and sea waves as regular sine waves. Under these assumptions, following governing equation of motion has been derived by considering the equilibrium of geometric restoring forces and small body motion under wave excitation (Umar and Datta, 2003a) M x€ C C x_ C RðxÞ Z F1 sin q
(1)
where MZmCm; m is the system mass; m is the added mass; C is the damping including structural damping and hydrodynamic damping, and R(x) is the displacement-dependent restoring force. Rearranging the equation of motion, Eq. (1) yields the following first-order autonomous equations x_ Z y;
y_ Z f1 sin q K rðxÞ K dy;
q_ Z u
(2)
where f1 Z
F1 ; M
rðxÞ Z
RðxÞ ; M
dZ
C ; M
and q Z ut C f
(3)
The restoring force r(x) is assumed to be represented by a fifth-order antisymmetric polynomial which closely describes the force displacement behavior of the mooring lines and is given by rðxÞ Z c1 x C c3 x3 C c5 x5
(4)
2.2. Stability analysis Let x0(t) be the approximate solution obtained by HBM (Appendix A). The stability of the approximate solution is determined by considering a perturbed solution given by xðtÞ Z x0 ðtÞ C 3ðtÞ
(5)
Differentiating with respect to time, we get _ Z x_0 ðtÞ C 3_ðtÞ xðtÞ
(6)
x€ ðtÞ Z x€ 0 ðtÞ C 3€ ðtÞ
(7)
_ y, we have Since xZ x€ 0 ðtÞ C 3€ ðtÞ Z f1 sin q K rðx0 C 3Þ K dx_0 ðtÞ K d_3ðtÞ
(8)
The above equation can be reduced into a standard form of Hill’s equation (Gottlieb and Yim, 1992) as 3€ C d_3 C aH½x0 ðtÞ3 Z 0
(9)
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where H½x0 ðtÞ Z
1 ½c C 3c3 x20 C 5c5 x40 a 1
(10)
leads to a general Hill’s variational Expanding H[x0(t)] in a Fourier series HðqÞ equation (Gottlieb and Yim, 1992) 3€ C d_3 C aH½qðtÞ3 Z0
(11)
where Z HðqÞ and 2 an Z p ( nZ
a0 X C an cos nq; 2 n ðp
Z ut C f C F qðtÞ
q; HðqÞcosðn qÞd
(12)
(13)
0
1; 2; 3; .; k;
½xðtÞ—unsymmetric
2; 4; 6; .; k;
½xðtÞ—symmetric
(14)
The solution of variational equation (11) gives the boundaries of the unstable region in sub and super harmonic regions as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 Z 2½aa0 K d2 G d4 K 2ad2 a0 C a2 a21 (15) and u2 Z
1 ½aða20 K a22 Þ K d2 a0 G 8a0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d4 a20 C 2ad2 ða22 K a20 Þ C a2 ða22 K 2a0 a4 Þ2
(16)
pffiffiffiffiffiffiffiffiffiffi For the undamped system Eq. (15) simplifies to u z 2aa0 and Eq. (16) simplifies to pffiffiffiffiffiffiffiffiffiffiffi u z aa0 =8.
3. Reliability assessment In the present study, for the reliability assessment of taut and slack mooring systems, first wave frequency bounds of instable region(s), as discussed above, have been determined in a deterministic fashion and then the frequency bounds are converted into random variables by assuming their certain probability distribution and essential statistics. The essential statistics for the present study are mean and standard deviation. Mean is obtained by assuming Mean/Nominal ratio and standard deviation by assuming COV (coefficient of variation). The nominal value is taken same as the estimated value. Having considered the randomness of instable regions, random wave frequencies are generated between minimum and maximum possible range of wave frequencies. For the generation of wave frequencies, its probability distribution is considered as uniform (Ahmad, 1996). The probability of failure has been obtained by drawing the samples of the basic variables
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Fig. 3. Strategy for reliability assessment of slack mooring system.
according to their probabilistic characteristics and then counting the failures. The failure is said to occur if wave frequency falls in the instable region as shown in Fig. 3 for slack mooring system and Fig. 4 for a taut mooring system. This process is nothing but the wellknown Monte Carlo Simulation Method (Madsen et al., 1986). Once the number of failure cycles is known for sufficiently large number of simulation cycles, one can estimate the probability of failure Pf by Pf Z
Nf N
(17)
where Nf is the number of simulation cycles in which failure occurs and N is the total number of simulation cycles. As N approaches infinity, Pf approaches the true probability of failure. The accuracy of Eq. (17) can be evaluated in terms of its variance, for a small probability of
Fig. 4. Strategy for reliability assessment of taut mooring system.
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failure and\or a small number of simulation cycles, the variance of Pf can be quite large. Consequently it may take a large number of simulation cycles to achieve a specified accuracy. The variance of the estimated probability of failure can be computed by assuming each simulation cycle to constitute a Bernoulli trial. Therefore, the number of failures in N trials can be considered to follow a binomial distribution. Then the variance of the estimated probability of failure can be computed approximately as (Siddiqui and Ahmad, 2001) VarðPf Þ Z
ð1 K Pf ÞPf Pf
(18)
It is recommended to measure the statistical accuracy of the estimated probability of failure by computing its coefficient of variation as qffiffiffiffiffiffiffiffiffiffiffiffiffiffi COVðPf Þ Z
ð1KPf ÞPf N
Pf
(19)
The smaller is the coefficient of variation, better is the accuracy of the estimated probability of failure. It is evident from Eqs. (18) and (19) that as N approaches infinity, Var(Pf) and COV(Pf) approaches zero. However, for all practical purposes, a smaller value of COV(Pf), e.g. less than 2% can be considered as a measure of convergence. Having known the value of Pf, the value of reliability index b, a measure of reliability, can be obtained using (Madsen et al., 1986) b Z KFK1 ðPf Þ
(20)
where F(x) is the cumulative distribution function of standard normal variant and is given by ðx 1 pffiffiffiffiffiffi expðKx2 Þdx FðxÞ Z (21) KN 2p
4. Numerical examples In order to demonstrate the application of methodology discussed above for mooring systems, two published mooring systems have been considered: one slack and another taut. The important data that are needed for their numerical study are given in Tables 3 and 4 and their geometrical configurations are shown in Figs. 1 and 2, respectively.
5. Discussion of results Using the data given in Tables 1 and 2 and employing the methodology discussed in Section 2 for dynamic analysis, we have obtained the response amplitude against the frequencies of excitation as shown in Fig. 5 for slack mooring system and Figs. 6 and 7 for taut mooring system. Fig. 5 shows that the region of instability for slack mooring system lies between 0.3 and 0.5 rad/s. However, Fig. 6 shows that there are two regions of
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Table 1 Specifications of slack mooring system (Umar and Datta, 2003a) Parameter
Slack mooring system
Structure type Mass of moored vessel (kg) Specific Gravity of concrete Number of anchored mooring lines Young’s Modulus of mooring lines (N/m2) Depth of sea (m)
Hollow cylindrical buoy 1.3!105 2.4 6 20.595!109 150
Table 2 Specifications of taut mooring system (Gottlieb and Yim, 1992) Parameter
Taut mooring system
Structure type Mass of moored vessel (kg) Specific Gravity of concrete Number of anchored mooring lines Young’s Modulus of mooring lines (N/m2) Depth of sea (m)
Vessel 1.2!105 2.4 4 20.595!109 150
instability in a taut mooring system, first region lies between 1.09 and 1.13 rad/s; and the other region lies between 2.3 and 2.7 rad/s. But due to scale effect, the first region of instability is not visible in Fig. 6. To show, therefore, this region, an enlarged view in a frequency range of 1–1.5 rad/s is shown in Fig. 7. These ranges of instable regions are considered as nominal values for obtaining the mean values of lower and upper limits of instability as shown in Tables 3 and 4. Other statistical data that are needed for reliability assessment against instability are also shown in these tables. It is to be noted here that COV and probability distributions are assumed for the reliability assessment due to unavailability of these in an approachable literature. 5.00
A (m)
4.00 3.00 2.00 1.00 0.00 0.0
1.0
2.0
3.0
4.0
5.0
6.0
Frequency (rad/sec) Fig. 5. Variation of maximum amplitude (A) with frequency of excitation (slack mooring).
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5.00
A (m)
4.00 3.00 2.00 1.00 0.00 0.00
1.00
2.00
3.00
4.00
5.00
6.00
Frequency (rad/sec) Fig. 6. Variation of maximum amplitude (A) with frequency of excitation (taut mooring).
5.00
A (m)
4.00 3.00 2.00 1.00 0.00 1.00
1.10
1.20
1.30
1.40
1.50
Frequency (rad/sec) Fig. 7. Enlarged view of Fig. 4a in a frequency range of 1–1.5 rad/s (taut mooring).
As it is obvious from Eq. (19), the number of simulation cycles for the application of Monte Carlo Simulation should be infinite. However, practically it is not feasible to have infinite number of simulation cycles (or zero COV). Therefore, we must have to find such a finite number of simulation cycles, which is practically feasible on present computing systems, without the sacrifice of accuracy. If COV(Pf) is less than 2% (i.e. 0.002 in Table 3 Statistical data for reliability assessment of slack mooring system Random variable
Mean/nominal
Mean (rad/s)
COV
Probability distribution
Others
u uls uus
1.0 1.0 1.0
– 0.3 0.5
0.15 0.15 0.15
Uniform Normal Normal
aZ0, b Z12 rad/s – –
u, exciting wave frequency; uls, lower frequency limit of instability; uus, upper frequency limit of instability; a, b, lower and upper limits for uniform probability distribution.
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Table 4 Statistical data for reliability assessment of taut mooring system Random variable
Mean/nominal
Mean (rad/s)
COV
Probability distribution
Others
u uls1 uls2 uus1 uus2
1.0 1.0 1.0 1.0 1.0
– 1.09 2.3 1.13 2.7
0.15 0.15 0.15 0.15 0.15
Uniform Normal Normal Normal Normal
aZ0, b Z12 rad/s – – – –
Subscript 1 and 2 indicates first and second regions of instability.
fraction), then the number of simulation cycles may be considered as sufficiently accurate (Siddiqui and Ahmad, 2001). Figs. 8 and 9 show that as the number of simulation cycles are increasing, COV(Pf) is decreasing correspondingly. At very high simulation cycles say above 2!105 simulations, 0.32 0.28
COV (Pf)
0.24 0.20 0.16 0.12 0.08 0.04 0.00 0.0E+00
1.0E+05
2.0E+05
3.0E+05
4.0E+05
5.0E+05
Simulation cycles (N) Fig. 8. Variation of COV(Pf) with simulation cycles in slack mooring system.
0.32 0.28
COV (Pf)
0.24 0.20 0.16 0.12 0.08 0.04 0.00 0.0E+00
1.0E+05
2.0E+05
3.0E+05
4.0E+05
5.0E+05
Simulation cycles (N) Fig. 9. Variation of COV(Pf) with simulation cycles in taut mooring system.
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Probability of failure
0.10 0.08 0.06 0.04 0.02 0.00 0.0E+00
1.0E+05
2.0E+05
3.0E+05
4.0E+05
5.0E+05
Simulation cycles (N) Fig. 10. Effect of simulation cycles on probability of failure in slack mooring system.
the COV(Pf) is less than 0.02 (i.e. 2%). With the further increase of simulation cycles, there is no significant change in COV(Pf). This shows the convergence. Figs. 10 and 11 show the variation of probability of failure of slack and taut mooring systems with simulation cycles. It can also be seen from these figures that at higher simulation cycles, e.g. above 2!105 simulations, there is almost no fluctuation in the value of Pf. These figures show that for present study, the simulation cycles above 2!105 would be appropriate. For the present study, therefore, we have taken 5!105 number of simulation cycles for the evaluation of probability of failure of moored vessels against instability. It is to be noted that on present computing machines, even 5!105 simulations require less than a minute to run. Table 5 shows that the slack mooring system has a considerable high probability of its failure against instability. This is so because in a structural reliability assessment any structure of importance having reliability index less than 3 or probability of failure above 10K4 is considered as unreliable or unsafe structure. Keeping this in view, one can say that the slack mooring system under study is unsafe against instability in the present sea environment.
Probability of failure
0.10 0.08 0.06 0.04 0.02 0.00 0.0E+00
1.0E+05
2.0E+05
3.0E+05
4.0E+05
5.0E+05
Simulation cycles (N) Fig. 11. Effect of simulation cycles on probability of failure in taut mooring system.
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Table 5 Results of reliability analysis for slack mooring system Mooring system
Pf
Slack mooring
1.62!10
K2
b
COV(Pf)
2.139
0.011
Table 6 Results of reliability analysis for taut mooring system Instable regions considered
Pf
b
COV(Pf)
Only I Only II Both I and II
3.27!10K3 3.28!10K2 3.61!10K2
2.719 1.841 1.798
0.020 0.008 0.007
Table 6 shows the Pf and b for taut mooring system when only first instable region is considered; only second instable region is considered; and when both the regions are considered simultaneously in the reliability assessment. The table shows that the Pf obtained against first instable region is quite small compared to Pf against second instable region. This is so because the first instable region (Figs. 6 and 7) is very small (1.09–1.13 rad/s) and obviously the chances of wave frequencies to fall in this region would be less. Moreover, we observe that the Pf value obtained after considering both the instable regions is quite close to Pf value obtained against second instable region alone. For approximate reliability analysis, therefore, this region can be omitted. Further, we again observe that the values are less than the desired range of safety (i.e. 3–4) that makes the taut mooring system under study also unsafe against instability in the present sea environment. 5.1. Parametric studies 5.1.1. Effect of lower and upper limit of instability Fig. 12 of slack mooring system and Figs. 13 and 14 of taut mooring system shows that the probability of failure decreases with the increase in the lower limit of instability.
Probability of failure
0.10 0.08 0.06 0.04 0.02 0.00 0.2
0.25
0.3
0.35
0.4
Lower limit of frequency (rad/sec) Fig. 12. Effect of lower limit of instability in slack mooring system.
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Probability of failure
0.10 0.08 0.06 0.04 0.02 0.00 0.75
0.875
1
1.125
1.25
Lower limit of frequency (rad/sec) Fig. 13. Effect of lower limit of instability in taut mooring system (region 1).
This is due to the fact that with the increase in lower limit of instability, the instable region is narrowing. Hence, the chances of wave frequency to fall in the instable region will decrease with the increase in the lower limit of instability. In other words, probability of failure will decrease and reliability index will increase. Similarly, with the increase in the upper limit of instability, the probability of failure is increasing due to the widening of the instable region as shown in Fig. 15 for slack and Figs. 16 and 17 for taut mooring systems. However, in taut mooring system, the effect of region I lower limit of instability or upper limit of instability is quite insignificant. This is so because the effect of this region on overall probability of failure itself is very small as shown in Table 4. 5.1.2. Effect of frequency range of generation Table 7 for slack mooring system and Table 8 for taut mooring system shows that the probability of failure is increasing with the decrease in the frequency range of
Probability of failure
0.10 0.08 0.06 0.04 0.02 0.00 1.5
1.75
2
2.25
2.5
2.75
3
Lower limit of frequency (rad/sec) Fig. 14. Effect of lower limit of instability in taut mooring system (region 2).
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Probability of failure
0.10 0.08 0.06 0.04 0.02 0.00 0.3
0.4
0.5
0.6
0.7
Upper limit of frequency (rad/sec) Fig. 15. Effect of upper limit of instability in slack mooring system.
Probability of failure
0.10
0.08
0.06
0.04
0.02 1
1.1
1.2
1.3
1.4
1.5
Upper limit of frequency (rad/sec) Fig. 16. Effect of upper limit of instability in taut mooring system (region 1).
Probability of failure
0.10 0.08 0.06 0.04 0.02 0.00 2
2.4
2.8
3.2
3.6
4
Upper limit of frequency (rad/sec) Fig. 17. Effect of upper limit of instability in taut mooring system (region 2).
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Table 7 Effect of frequency range of generation on reliability of slack mooring system a (rad/s)
b (rad/s)
Pf
0 0 0 0 0 0 0 0
12 11 10 09 08 07 06 05
1.62!10 1.77!10K2 1.93!10K2 2.13!10K2 2.43!10K2 2.84!10K2 3.36!10K2 4.08!10K2 K2
b
COV(Pf)
2.139 2.103 2.068 2.027 1.972 1.905 1.830 1.742
0.011 0.011 0.010 0.010 0.009 0.008 0.008 0.007
a, b, lower and upper limits of frequency.
generation. This is due to the fact that as the frequency range of generation is narrowing, it is becoming closer to the instable regions, which increases the chances of wave frequencies to fall in the instable region for the same number of generated frequencies. This trend is same for both slack and taut mooring systems. A random sea consists of range of wave frequencies and two different random seas may have different ranges of frequency. From this study, therefore, one can infer that a vessel, which is safe in one sea, may be quite unsafe in other sea against instability because of change in the range of wave frequencies. Through the above two parametric studies one reaches to this point that there can be two ways to control the reliability of vessel against instability: either by changing the region(s) of instability or by changing the wave frequency range. Changing the wave frequency range in a given sea, however, is beyond the human power. Therefore, the only alternative remains, is to change the region of instability of vessel. In a given sea environment (i.e. in a given range of wave frequency), by designing the vessel with changed material properties and geometric configurations region(s) of instability can be altered and the reliability of vessel against instability can be controlled.
Table 8 Effect of frequency range of generation on reliability of taut mooring system a (rad/s)
b (rad/s)
Pf
0 0 0 0 0 0 0 0
12 11 10 09 08 07 06 05
3.61!10 3.99!10K2 4.48!10K2 4.96!10K2 5.46!10K2 6.27!10K2 7.32!10K2 8.68!10K2
a, b, lower and upper limits of frequency.
K2
b
COV(Pf)
1.798 1.751 1.698 1.649 1.601 1.533 1.452 1.361
0.007 0.007 0.007 0.006 0.006 0.005 0.005 0.005
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6. Conclusions In the present study, a methodology for the reliability assessment of taut and slack mooring systems against instability has been presented. The proposed methodology was employed on two published mooring systems: one of slack type and other of taut type and reliabilities has been obtained. The reliability study showed that in the present sea environment, the slack and taut mooring systems of present study are not reliable, against instability, as desired. Through parametric studies it has been established that in a given sea environment (i.e. in a given range of wave frequency) desirable range of reliability for vessels can be achieved only after altering the region(s) of instability and for this purpose vessels have to be redesigned with changed material properties and geometric configurations.
Appendix A A.1. Solution of equation of motion by HBM In Harmonic Balance Method (HBM), first we assume the solution of equation of motion in terms of combination of sine and cosine functions with certain constant coefficients and phase angles. In the present study, we have assumed solution of equation of motion as xðtÞ Z A1 cosðut C fÞ C A3 cosð3ut C aÞ
(A1)
_ Z KuA1 sinðut C fÞ K 3uA3 sinð3ut C aÞ xðtÞ
(A2)
x€ ðtÞ Z Ku2 A1 cosðut C fÞ K 9u2 A3 cosð3ut C aÞ
(A3)
First we substitute Eq. (A1) in the expression of r(x), given by Eq. (4), and then we _ x€ ðtÞ and value of r(x) in the equation of motion given by Eq. (2). Now substitute x(t), xðtÞ, there are four unknowns A1, A3, a and f which can be obtained by equating cos ut terms, sin ut terms, cos 3ut terms and sin 3ut terms equal to zero. This provides us following four equations with four unknowns E1 C E 2 C E 3 C E 4 C E 5 C E 6 C E 7 C E 8 C E 9 Z 0
(A4)
E11 C E12 C E13 C E14 C E15 C E16 C E17 C E18 C E19 Z 0
(A5)
E21 C E22 C E23 C E24 C E25 C E26 Z 0
(A6)
E31 C E32 C E33 C E34 C E35 C E36 C E37 Z 0
(A7)
where E1 Z Ku2 A1 cos f C f1 sin f K duA1 sin f C c1 A1 cos f
(A8)
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3 E2 Z c3 A31 cos f 4
(A9)
3 E3 Z c3 A21 A3 ðcos a cos 2f C sin a sin 2fÞ 4
(A10)
3 E4 Z c3 A1 A23 cos f 2
(A11)
5 E5 Z c5 A41 fA1 cos f C 2A3 cos f cosða K 2fÞg 8
(A12)
E6 Z
15 c A3 A2 cos f 4 5 1 3
(A13)
E7 Z
5 c A4 A cosð4f K aÞ 16 5 1 3
(A14)
E8 Z
15 c A2 A3 ðcos a cos 2f K sin a sin 2fÞ 8 5 1 3
(A15)
E9 Z
15 c A A4 cos f 8 5 1 3
(A16)
E11 Z u2 A1 sin f C f1 cos f K duA1 cos f K c1 A1 sin f
(A17)
3 E12 Z K c3 A31 sin f 4
(A18)
3 E13 Z K c3 A21 A3 ðsin a cos 2f K cos a sin 2fÞ 4
(A19)
3 E14 Z K c3 A23 A1 sin f 2
(A20)
5 E15 Z K c5 A41 fA1 sin f C 2A3 sinða K 2fÞg 8
(A21)
E16 Z K
30 c A3 A2 sin f 8 5 1 3
(A22)
E17 Z K
5 c A4 A sinð4f K aÞ 16 5 1 3
(A23)
E18 Z K
15 c A2 A3 ðsin a cos 2f C cos a sin 2fÞ 8 5 1 3
(A24)
A.A. Shah et al. / Ocean Engineering 32 (2005) 1216–1234
E19 Z K
15 c A A4 cos f 8 5 1 3
E21 Z K9u2 A3 cos a K 3duA3 sin a C c1 A3 cos a C
(A25) c3 3 A cos 3f 4 1
3 E22 Z c3 A3 cos að2A21 C A23 Þ 4
E23 Z
5 c A4 ðA cos 3f C 2A3 cos aÞ 16 5 1 1
5 E24 Z c5 A31 A23 fcos 3fð2 C cos 2aÞ C sin 2a sin 3fg 8 E25 Z
1233
15 c A2 A3 cos a 4 5 1 3
(A26)
(A27)
(A28)
(A29)
(A30)
5 E26 Z c5 A53 cos a 8
(A31)
E31 Z 9u2 A3 sin a K 3duA3 cos a K c1 A3 sin a
(A32)
E32 Z K
c3 3 A sin 3f 4 1
3 E33 Z K c3 A3 sin að2A21 C A23 Þ 4 E34 Z K
5 c A4 ðA sin 3f C 6A3 sin aÞ 16 5 1 1
5 E35 Z K c5 A31 A23 fsin 3fð2 C cos 2aÞ C sin 2a cos 3fg 8 E36 Z K
15 c A2 A3 sin a 4 5 1 3
5 E37 Z K c5 A53 sin a 8
(A33)
(A34)
(A35)
(A36)
(A37)
(A38)
1234
A.A. Shah et al. / Ocean Engineering 32 (2005) 1216–1234
Eqs. (A4)–(A7) can be solved by iterative Newton Raphson method and the unknowns A1, A3, a and f are determined. Once these unknowns are known, the _ and x€ ðtÞ can be obtained from Eqs. (A1)–(A3). response x(t), xðtÞ
References Ahmad, S., 1996. Stochastic TLP response under long crested random sea. Computers and Structures 61 (6), 975–993. Amanullah, M., Siddiqui, N.A., Umar, A., Abbas, H., 2002. Fatigue reliability analysis of welded joints of a TLP tether system. Steel and Composite Structures: An International Journal 2 (5), 331–354. Gottlieb, O., Yim, S.C.S., 1990. The onset of chaos in a multipoint mooring system. Proceedings of First European Offshore Mechanics Symposium, International Society of Offshore and Polar Engineering (ISOPE) 1, 6–12. Gottlieb, O., Yim, S.C.S., 1992. Nonlinear oscillations, bifurcations and chaos in a multi-point mooring system with a geometric nonlinearity. Applied Ocean Research 14, 241–257. Lotsberg, I., 1991. Probabilistic design of the tethers of a tension leg platform. Journal of Offshore Mechanics and Arctic Engineering 113, 162–169. Madsen, H.O., Krenk, S., Lind, N.C., 1986. Methods of Structural Safety. Prentice-Hall, Englewood Cliffs, NJ. Nayfeh, A.H., Mook, D.T., 1979. Nonlinear Oscillations. Wiley, New York. Oppenheim, B.W., Wilson, P.A., 1982. Polynomial approximations to mooring forces in equations of lowfrequency vessel motions. Journal of Ship Research 26 (1), 16–24. Papoulias, F.A., Bernitsas, M.M., 1988. Autonomous oscillations, bifurcations and chaotic response of moored vessels. Journal of ship Research 32, 220–228. Siddiqui, N.A., Ahmad, S., 2000. Reliability analysis against progressive failure of TLP tethers in extreme tension. Reliability Engineering and System Safety 68, 195–205. Siddiqui, N.A., Ahmad, S., 2001. Fatigue and fracture reliability of TLP tethers under random loading. Marine Structures 14, 331–352. Umar, A., Datta, T.K., 2001. Stability analysis of a moored system, Proceedings of structural engineering convention, IIT Roorkee, India 2001 pp. 636–654. Umar, A., Datta, T.K., 2003a. Nonlinear response of a moored buoy. Ocean Engineering 30, 1625–1646. Umar, A., Datta, T.K., 2003b. Dynamic analysis of a moored vessel. Journal of Structural Engineering, SERC, India 29, 229–237. Umar, A., Siddiqui, N.A., 2002. Reliability analysis of a moored vessel. Journal of Structural Engineering, SERC, India 29 (3), 171–174.
A methodology for assessing the reliability of taut and slack mooring systems against instability A.A. Shah, A. Umar, N.A. Siddiqui* Department of Civil Engineering, Aligarh Muslim University, Aligarh 202 002, India Received 28 June 2004; accepted 3 November 2004 Available online 13 March 2005
Abstract In the present study, a methodology for reliability assessment of slack and taut mooring systems against instability has been presented. For this purpose, first, stability analysis of slack and taut mooring systems has been carried out and instable regions are obtained using procedure available in the literature. Having known the instable region(s), methodology for reliability assessment has been proposed which is based on Monte Carlo Simulation technique. After using the proposed methodology, probabilities of failure and reliability indices has been obtained for the above systems. Some parametric studies, such as, effect of lower and upper limits of instability and effect of frequency range of generations are also included to obtain the results of practical interest. q 2005 Elsevier Ltd. All rights reserved. Keywords: Mooring systems; Offshore vessels; Structural reliability; Instability
1. Introduction Moored offshore vessels and platform are becoming popular day by day in various offshore activities. This is so, because moored offshore vessels and platforms are economical, easy to install, and reusable offshore structures. The mooring of such offshore structures exhibits non-linear restoration, which may cause instability problem in the mooring system. In the event of instability, the excursions may become very large that may lead to the failure of moorings of the vessel. In the last two decades, good number of research papers appeared in various journals and conference proceedings on stability * Corresponding author. Tel.: C91 571 240 1074; fax: C91 571 270 0528. E-mail address: [email protected] (N.A. Siddiqui). 0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2004.11.002
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analysis of mooring systems (e.g. Umar and Datta, 2001, Gottlib and Yim, 1990, 1992, Oppenheim and Wilson, 1982, Nayfeh and Mook, 1979, Papoulias and Bernitsas, 1988) and reliability assessment of moorings of offshore platforms (e.g. Amanullah et al., 2002, Siddiqui and Ahmad, 2000, 2001, Umar and Siddiqui, 2002, Lotsberg, 1991). Gottlieb and Yim (1992) carried out the stability analysis of a taut mooring system. Umar and Datta (2003a,b) carried out a detailed dynamic analysis of slack mooring systems. Umar and Siddiqui (2002) carried out the reliability analysis of taut mooring system against yielding of taut moorings. They employed first-order reliability method (FORM) for this purpose. Siddiqui and Ahmad (2000, 2001) and Amanullah et al. (2002) performed the reliability analysis of taut mooring system of tension leg platforms (TLPs) against maximum tension, minimum tension, fatigue and fracture. From the detailed review, it has been observed that though considerable work has been done on the stability of moored vessels and reliability assessment of mooring systems against maximum tension, minimum tension, fatigue, fracture, etc. there is almost no work on reliability against instability. Keeping this point in view, an attempt has been made to propose a methodology for the reliability assessment of slack and taut mooring systems (Figs. 1 and 2) against instability. For this purpose, first, 15 m
Buoy
4m
m.s.l
1.75 m
150 m
Clump weight Seabed 30 m
Elevation
Mooring lines
Mooring lines Buoy
Plan Fig. 1. Slack mooring system (Umar and Datta, 2003a).
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A.A. Shah et al. / Ocean Engineering 32 (2005) 1216–1234 b
b
Taut Mooring Lines
d vessel
d
PLAN
vessel
ELEVATION
Mooring Lines
Fig. 2. Symmetric multi-point taut mooring system (Gottlieb and Yim, 1992).
stability analysis of slack and taut mooring systems has been carried out and instable regions are obtained using procedure available in the literature. Having known the instable region(s), methodology for reliability assessment has been proposed which is based on Monte Carlo Simulation technique. After using the proposed methodology, probabilities of failure and reliability indices has been obtained for the above systems. Some parametric studies, such as, effect of lower and upper limits of instability and effect of frequency range of generations are also included to obtain the results of practical interest.
2. Mathematical formulation The reliability against instability requires the instable region(s), which can be obtained only after carrying out a non-linear dynamic and stability analyses of the moored vessels. Therefore, before discussing the formulation for reliability assessment, a brief formulation of dynamic analysis and stability analysis are also presented in the following section.
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2.1. Dynamic analysis Dynamic analysis is a prerequisite for the stability analysis and reliability assessment of moored vessels. In the present study, dynamic analyses of Taut and Slack mooring systems have been carried out assuming mooring system as a single degree of freedom non-linear oscillator and sea waves as regular sine waves. Under these assumptions, following governing equation of motion has been derived by considering the equilibrium of geometric restoring forces and small body motion under wave excitation (Umar and Datta, 2003a) M x€ C C x_ C RðxÞ Z F1 sin q
(1)
where MZmCm; m is the system mass; m is the added mass; C is the damping including structural damping and hydrodynamic damping, and R(x) is the displacement-dependent restoring force. Rearranging the equation of motion, Eq. (1) yields the following first-order autonomous equations x_ Z y;
y_ Z f1 sin q K rðxÞ K dy;
q_ Z u
(2)
where f1 Z
F1 ; M
rðxÞ Z
RðxÞ ; M
dZ
C ; M
and q Z ut C f
(3)
The restoring force r(x) is assumed to be represented by a fifth-order antisymmetric polynomial which closely describes the force displacement behavior of the mooring lines and is given by rðxÞ Z c1 x C c3 x3 C c5 x5
(4)
2.2. Stability analysis Let x0(t) be the approximate solution obtained by HBM (Appendix A). The stability of the approximate solution is determined by considering a perturbed solution given by xðtÞ Z x0 ðtÞ C 3ðtÞ
(5)
Differentiating with respect to time, we get _ Z x_0 ðtÞ C 3_ðtÞ xðtÞ
(6)
x€ ðtÞ Z x€ 0 ðtÞ C 3€ ðtÞ
(7)
_ y, we have Since xZ x€ 0 ðtÞ C 3€ ðtÞ Z f1 sin q K rðx0 C 3Þ K dx_0 ðtÞ K d_3ðtÞ
(8)
The above equation can be reduced into a standard form of Hill’s equation (Gottlieb and Yim, 1992) as 3€ C d_3 C aH½x0 ðtÞ3 Z 0
(9)
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A.A. Shah et al. / Ocean Engineering 32 (2005) 1216–1234
where H½x0 ðtÞ Z
1 ½c C 3c3 x20 C 5c5 x40 a 1
(10)
leads to a general Hill’s variational Expanding H[x0(t)] in a Fourier series HðqÞ equation (Gottlieb and Yim, 1992) 3€ C d_3 C aH½qðtÞ3 Z0
(11)
where Z HðqÞ and 2 an Z p ( nZ
a0 X C an cos nq; 2 n ðp
Z ut C f C F qðtÞ
q; HðqÞcosðn qÞd
(12)
(13)
0
1; 2; 3; .; k;
½xðtÞ—unsymmetric
2; 4; 6; .; k;
½xðtÞ—symmetric
(14)
The solution of variational equation (11) gives the boundaries of the unstable region in sub and super harmonic regions as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 Z 2½aa0 K d2 G d4 K 2ad2 a0 C a2 a21 (15) and u2 Z
1 ½aða20 K a22 Þ K d2 a0 G 8a0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d4 a20 C 2ad2 ða22 K a20 Þ C a2 ða22 K 2a0 a4 Þ2
(16)
pffiffiffiffiffiffiffiffiffiffi For the undamped system Eq. (15) simplifies to u z 2aa0 and Eq. (16) simplifies to pffiffiffiffiffiffiffiffiffiffiffi u z aa0 =8.
3. Reliability assessment In the present study, for the reliability assessment of taut and slack mooring systems, first wave frequency bounds of instable region(s), as discussed above, have been determined in a deterministic fashion and then the frequency bounds are converted into random variables by assuming their certain probability distribution and essential statistics. The essential statistics for the present study are mean and standard deviation. Mean is obtained by assuming Mean/Nominal ratio and standard deviation by assuming COV (coefficient of variation). The nominal value is taken same as the estimated value. Having considered the randomness of instable regions, random wave frequencies are generated between minimum and maximum possible range of wave frequencies. For the generation of wave frequencies, its probability distribution is considered as uniform (Ahmad, 1996). The probability of failure has been obtained by drawing the samples of the basic variables
A.A. Shah et al. / Ocean Engineering 32 (2005) 1216–1234
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Fig. 3. Strategy for reliability assessment of slack mooring system.
according to their probabilistic characteristics and then counting the failures. The failure is said to occur if wave frequency falls in the instable region as shown in Fig. 3 for slack mooring system and Fig. 4 for a taut mooring system. This process is nothing but the wellknown Monte Carlo Simulation Method (Madsen et al., 1986). Once the number of failure cycles is known for sufficiently large number of simulation cycles, one can estimate the probability of failure Pf by Pf Z
Nf N
(17)
where Nf is the number of simulation cycles in which failure occurs and N is the total number of simulation cycles. As N approaches infinity, Pf approaches the true probability of failure. The accuracy of Eq. (17) can be evaluated in terms of its variance, for a small probability of
Fig. 4. Strategy for reliability assessment of taut mooring system.
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A.A. Shah et al. / Ocean Engineering 32 (2005) 1216–1234
failure and\or a small number of simulation cycles, the variance of Pf can be quite large. Consequently it may take a large number of simulation cycles to achieve a specified accuracy. The variance of the estimated probability of failure can be computed by assuming each simulation cycle to constitute a Bernoulli trial. Therefore, the number of failures in N trials can be considered to follow a binomial distribution. Then the variance of the estimated probability of failure can be computed approximately as (Siddiqui and Ahmad, 2001) VarðPf Þ Z
ð1 K Pf ÞPf Pf
(18)
It is recommended to measure the statistical accuracy of the estimated probability of failure by computing its coefficient of variation as qffiffiffiffiffiffiffiffiffiffiffiffiffiffi COVðPf Þ Z
ð1KPf ÞPf N
Pf
(19)
The smaller is the coefficient of variation, better is the accuracy of the estimated probability of failure. It is evident from Eqs. (18) and (19) that as N approaches infinity, Var(Pf) and COV(Pf) approaches zero. However, for all practical purposes, a smaller value of COV(Pf), e.g. less than 2% can be considered as a measure of convergence. Having known the value of Pf, the value of reliability index b, a measure of reliability, can be obtained using (Madsen et al., 1986) b Z KFK1 ðPf Þ
(20)
where F(x) is the cumulative distribution function of standard normal variant and is given by ðx 1 pffiffiffiffiffiffi expðKx2 Þdx FðxÞ Z (21) KN 2p
4. Numerical examples In order to demonstrate the application of methodology discussed above for mooring systems, two published mooring systems have been considered: one slack and another taut. The important data that are needed for their numerical study are given in Tables 3 and 4 and their geometrical configurations are shown in Figs. 1 and 2, respectively.
5. Discussion of results Using the data given in Tables 1 and 2 and employing the methodology discussed in Section 2 for dynamic analysis, we have obtained the response amplitude against the frequencies of excitation as shown in Fig. 5 for slack mooring system and Figs. 6 and 7 for taut mooring system. Fig. 5 shows that the region of instability for slack mooring system lies between 0.3 and 0.5 rad/s. However, Fig. 6 shows that there are two regions of
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Table 1 Specifications of slack mooring system (Umar and Datta, 2003a) Parameter
Slack mooring system
Structure type Mass of moored vessel (kg) Specific Gravity of concrete Number of anchored mooring lines Young’s Modulus of mooring lines (N/m2) Depth of sea (m)
Hollow cylindrical buoy 1.3!105 2.4 6 20.595!109 150
Table 2 Specifications of taut mooring system (Gottlieb and Yim, 1992) Parameter
Taut mooring system
Structure type Mass of moored vessel (kg) Specific Gravity of concrete Number of anchored mooring lines Young’s Modulus of mooring lines (N/m2) Depth of sea (m)
Vessel 1.2!105 2.4 4 20.595!109 150
instability in a taut mooring system, first region lies between 1.09 and 1.13 rad/s; and the other region lies between 2.3 and 2.7 rad/s. But due to scale effect, the first region of instability is not visible in Fig. 6. To show, therefore, this region, an enlarged view in a frequency range of 1–1.5 rad/s is shown in Fig. 7. These ranges of instable regions are considered as nominal values for obtaining the mean values of lower and upper limits of instability as shown in Tables 3 and 4. Other statistical data that are needed for reliability assessment against instability are also shown in these tables. It is to be noted here that COV and probability distributions are assumed for the reliability assessment due to unavailability of these in an approachable literature. 5.00
A (m)
4.00 3.00 2.00 1.00 0.00 0.0
1.0
2.0
3.0
4.0
5.0
6.0
Frequency (rad/sec) Fig. 5. Variation of maximum amplitude (A) with frequency of excitation (slack mooring).
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5.00
A (m)
4.00 3.00 2.00 1.00 0.00 0.00
1.00
2.00
3.00
4.00
5.00
6.00
Frequency (rad/sec) Fig. 6. Variation of maximum amplitude (A) with frequency of excitation (taut mooring).
5.00
A (m)
4.00 3.00 2.00 1.00 0.00 1.00
1.10
1.20
1.30
1.40
1.50
Frequency (rad/sec) Fig. 7. Enlarged view of Fig. 4a in a frequency range of 1–1.5 rad/s (taut mooring).
As it is obvious from Eq. (19), the number of simulation cycles for the application of Monte Carlo Simulation should be infinite. However, practically it is not feasible to have infinite number of simulation cycles (or zero COV). Therefore, we must have to find such a finite number of simulation cycles, which is practically feasible on present computing systems, without the sacrifice of accuracy. If COV(Pf) is less than 2% (i.e. 0.002 in Table 3 Statistical data for reliability assessment of slack mooring system Random variable
Mean/nominal
Mean (rad/s)
COV
Probability distribution
Others
u uls uus
1.0 1.0 1.0
– 0.3 0.5
0.15 0.15 0.15
Uniform Normal Normal
aZ0, b Z12 rad/s – –
u, exciting wave frequency; uls, lower frequency limit of instability; uus, upper frequency limit of instability; a, b, lower and upper limits for uniform probability distribution.
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Table 4 Statistical data for reliability assessment of taut mooring system Random variable
Mean/nominal
Mean (rad/s)
COV
Probability distribution
Others
u uls1 uls2 uus1 uus2
1.0 1.0 1.0 1.0 1.0
– 1.09 2.3 1.13 2.7
0.15 0.15 0.15 0.15 0.15
Uniform Normal Normal Normal Normal
aZ0, b Z12 rad/s – – – –
Subscript 1 and 2 indicates first and second regions of instability.
fraction), then the number of simulation cycles may be considered as sufficiently accurate (Siddiqui and Ahmad, 2001). Figs. 8 and 9 show that as the number of simulation cycles are increasing, COV(Pf) is decreasing correspondingly. At very high simulation cycles say above 2!105 simulations, 0.32 0.28
COV (Pf)
0.24 0.20 0.16 0.12 0.08 0.04 0.00 0.0E+00
1.0E+05
2.0E+05
3.0E+05
4.0E+05
5.0E+05
Simulation cycles (N) Fig. 8. Variation of COV(Pf) with simulation cycles in slack mooring system.
0.32 0.28
COV (Pf)
0.24 0.20 0.16 0.12 0.08 0.04 0.00 0.0E+00
1.0E+05
2.0E+05
3.0E+05
4.0E+05
5.0E+05
Simulation cycles (N) Fig. 9. Variation of COV(Pf) with simulation cycles in taut mooring system.
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Probability of failure
0.10 0.08 0.06 0.04 0.02 0.00 0.0E+00
1.0E+05
2.0E+05
3.0E+05
4.0E+05
5.0E+05
Simulation cycles (N) Fig. 10. Effect of simulation cycles on probability of failure in slack mooring system.
the COV(Pf) is less than 0.02 (i.e. 2%). With the further increase of simulation cycles, there is no significant change in COV(Pf). This shows the convergence. Figs. 10 and 11 show the variation of probability of failure of slack and taut mooring systems with simulation cycles. It can also be seen from these figures that at higher simulation cycles, e.g. above 2!105 simulations, there is almost no fluctuation in the value of Pf. These figures show that for present study, the simulation cycles above 2!105 would be appropriate. For the present study, therefore, we have taken 5!105 number of simulation cycles for the evaluation of probability of failure of moored vessels against instability. It is to be noted that on present computing machines, even 5!105 simulations require less than a minute to run. Table 5 shows that the slack mooring system has a considerable high probability of its failure against instability. This is so because in a structural reliability assessment any structure of importance having reliability index less than 3 or probability of failure above 10K4 is considered as unreliable or unsafe structure. Keeping this in view, one can say that the slack mooring system under study is unsafe against instability in the present sea environment.
Probability of failure
0.10 0.08 0.06 0.04 0.02 0.00 0.0E+00
1.0E+05
2.0E+05
3.0E+05
4.0E+05
5.0E+05
Simulation cycles (N) Fig. 11. Effect of simulation cycles on probability of failure in taut mooring system.
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Table 5 Results of reliability analysis for slack mooring system Mooring system
Pf
Slack mooring
1.62!10
K2
b
COV(Pf)
2.139
0.011
Table 6 Results of reliability analysis for taut mooring system Instable regions considered
Pf
b
COV(Pf)
Only I Only II Both I and II
3.27!10K3 3.28!10K2 3.61!10K2
2.719 1.841 1.798
0.020 0.008 0.007
Table 6 shows the Pf and b for taut mooring system when only first instable region is considered; only second instable region is considered; and when both the regions are considered simultaneously in the reliability assessment. The table shows that the Pf obtained against first instable region is quite small compared to Pf against second instable region. This is so because the first instable region (Figs. 6 and 7) is very small (1.09–1.13 rad/s) and obviously the chances of wave frequencies to fall in this region would be less. Moreover, we observe that the Pf value obtained after considering both the instable regions is quite close to Pf value obtained against second instable region alone. For approximate reliability analysis, therefore, this region can be omitted. Further, we again observe that the values are less than the desired range of safety (i.e. 3–4) that makes the taut mooring system under study also unsafe against instability in the present sea environment. 5.1. Parametric studies 5.1.1. Effect of lower and upper limit of instability Fig. 12 of slack mooring system and Figs. 13 and 14 of taut mooring system shows that the probability of failure decreases with the increase in the lower limit of instability.
Probability of failure
0.10 0.08 0.06 0.04 0.02 0.00 0.2
0.25
0.3
0.35
0.4
Lower limit of frequency (rad/sec) Fig. 12. Effect of lower limit of instability in slack mooring system.
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Probability of failure
0.10 0.08 0.06 0.04 0.02 0.00 0.75
0.875
1
1.125
1.25
Lower limit of frequency (rad/sec) Fig. 13. Effect of lower limit of instability in taut mooring system (region 1).
This is due to the fact that with the increase in lower limit of instability, the instable region is narrowing. Hence, the chances of wave frequency to fall in the instable region will decrease with the increase in the lower limit of instability. In other words, probability of failure will decrease and reliability index will increase. Similarly, with the increase in the upper limit of instability, the probability of failure is increasing due to the widening of the instable region as shown in Fig. 15 for slack and Figs. 16 and 17 for taut mooring systems. However, in taut mooring system, the effect of region I lower limit of instability or upper limit of instability is quite insignificant. This is so because the effect of this region on overall probability of failure itself is very small as shown in Table 4. 5.1.2. Effect of frequency range of generation Table 7 for slack mooring system and Table 8 for taut mooring system shows that the probability of failure is increasing with the decrease in the frequency range of
Probability of failure
0.10 0.08 0.06 0.04 0.02 0.00 1.5
1.75
2
2.25
2.5
2.75
3
Lower limit of frequency (rad/sec) Fig. 14. Effect of lower limit of instability in taut mooring system (region 2).
A.A. Shah et al. / Ocean Engineering 32 (2005) 1216–1234
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Probability of failure
0.10 0.08 0.06 0.04 0.02 0.00 0.3
0.4
0.5
0.6
0.7
Upper limit of frequency (rad/sec) Fig. 15. Effect of upper limit of instability in slack mooring system.
Probability of failure
0.10
0.08
0.06
0.04
0.02 1
1.1
1.2
1.3
1.4
1.5
Upper limit of frequency (rad/sec) Fig. 16. Effect of upper limit of instability in taut mooring system (region 1).
Probability of failure
0.10 0.08 0.06 0.04 0.02 0.00 2
2.4
2.8
3.2
3.6
4
Upper limit of frequency (rad/sec) Fig. 17. Effect of upper limit of instability in taut mooring system (region 2).
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Table 7 Effect of frequency range of generation on reliability of slack mooring system a (rad/s)
b (rad/s)
Pf
0 0 0 0 0 0 0 0
12 11 10 09 08 07 06 05
1.62!10 1.77!10K2 1.93!10K2 2.13!10K2 2.43!10K2 2.84!10K2 3.36!10K2 4.08!10K2 K2
b
COV(Pf)
2.139 2.103 2.068 2.027 1.972 1.905 1.830 1.742
0.011 0.011 0.010 0.010 0.009 0.008 0.008 0.007
a, b, lower and upper limits of frequency.
generation. This is due to the fact that as the frequency range of generation is narrowing, it is becoming closer to the instable regions, which increases the chances of wave frequencies to fall in the instable region for the same number of generated frequencies. This trend is same for both slack and taut mooring systems. A random sea consists of range of wave frequencies and two different random seas may have different ranges of frequency. From this study, therefore, one can infer that a vessel, which is safe in one sea, may be quite unsafe in other sea against instability because of change in the range of wave frequencies. Through the above two parametric studies one reaches to this point that there can be two ways to control the reliability of vessel against instability: either by changing the region(s) of instability or by changing the wave frequency range. Changing the wave frequency range in a given sea, however, is beyond the human power. Therefore, the only alternative remains, is to change the region of instability of vessel. In a given sea environment (i.e. in a given range of wave frequency), by designing the vessel with changed material properties and geometric configurations region(s) of instability can be altered and the reliability of vessel against instability can be controlled.
Table 8 Effect of frequency range of generation on reliability of taut mooring system a (rad/s)
b (rad/s)
Pf
0 0 0 0 0 0 0 0
12 11 10 09 08 07 06 05
3.61!10 3.99!10K2 4.48!10K2 4.96!10K2 5.46!10K2 6.27!10K2 7.32!10K2 8.68!10K2
a, b, lower and upper limits of frequency.
K2
b
COV(Pf)
1.798 1.751 1.698 1.649 1.601 1.533 1.452 1.361
0.007 0.007 0.007 0.006 0.006 0.005 0.005 0.005
A.A. Shah et al. / Ocean Engineering 32 (2005) 1216–1234
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6. Conclusions In the present study, a methodology for the reliability assessment of taut and slack mooring systems against instability has been presented. The proposed methodology was employed on two published mooring systems: one of slack type and other of taut type and reliabilities has been obtained. The reliability study showed that in the present sea environment, the slack and taut mooring systems of present study are not reliable, against instability, as desired. Through parametric studies it has been established that in a given sea environment (i.e. in a given range of wave frequency) desirable range of reliability for vessels can be achieved only after altering the region(s) of instability and for this purpose vessels have to be redesigned with changed material properties and geometric configurations.
Appendix A A.1. Solution of equation of motion by HBM In Harmonic Balance Method (HBM), first we assume the solution of equation of motion in terms of combination of sine and cosine functions with certain constant coefficients and phase angles. In the present study, we have assumed solution of equation of motion as xðtÞ Z A1 cosðut C fÞ C A3 cosð3ut C aÞ
(A1)
_ Z KuA1 sinðut C fÞ K 3uA3 sinð3ut C aÞ xðtÞ
(A2)
x€ ðtÞ Z Ku2 A1 cosðut C fÞ K 9u2 A3 cosð3ut C aÞ
(A3)
First we substitute Eq. (A1) in the expression of r(x), given by Eq. (4), and then we _ x€ ðtÞ and value of r(x) in the equation of motion given by Eq. (2). Now substitute x(t), xðtÞ, there are four unknowns A1, A3, a and f which can be obtained by equating cos ut terms, sin ut terms, cos 3ut terms and sin 3ut terms equal to zero. This provides us following four equations with four unknowns E1 C E 2 C E 3 C E 4 C E 5 C E 6 C E 7 C E 8 C E 9 Z 0
(A4)
E11 C E12 C E13 C E14 C E15 C E16 C E17 C E18 C E19 Z 0
(A5)
E21 C E22 C E23 C E24 C E25 C E26 Z 0
(A6)
E31 C E32 C E33 C E34 C E35 C E36 C E37 Z 0
(A7)
where E1 Z Ku2 A1 cos f C f1 sin f K duA1 sin f C c1 A1 cos f
(A8)
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3 E2 Z c3 A31 cos f 4
(A9)
3 E3 Z c3 A21 A3 ðcos a cos 2f C sin a sin 2fÞ 4
(A10)
3 E4 Z c3 A1 A23 cos f 2
(A11)
5 E5 Z c5 A41 fA1 cos f C 2A3 cos f cosða K 2fÞg 8
(A12)
E6 Z
15 c A3 A2 cos f 4 5 1 3
(A13)
E7 Z
5 c A4 A cosð4f K aÞ 16 5 1 3
(A14)
E8 Z
15 c A2 A3 ðcos a cos 2f K sin a sin 2fÞ 8 5 1 3
(A15)
E9 Z
15 c A A4 cos f 8 5 1 3
(A16)
E11 Z u2 A1 sin f C f1 cos f K duA1 cos f K c1 A1 sin f
(A17)
3 E12 Z K c3 A31 sin f 4
(A18)
3 E13 Z K c3 A21 A3 ðsin a cos 2f K cos a sin 2fÞ 4
(A19)
3 E14 Z K c3 A23 A1 sin f 2
(A20)
5 E15 Z K c5 A41 fA1 sin f C 2A3 sinða K 2fÞg 8
(A21)
E16 Z K
30 c A3 A2 sin f 8 5 1 3
(A22)
E17 Z K
5 c A4 A sinð4f K aÞ 16 5 1 3
(A23)
E18 Z K
15 c A2 A3 ðsin a cos 2f C cos a sin 2fÞ 8 5 1 3
(A24)
A.A. Shah et al. / Ocean Engineering 32 (2005) 1216–1234
E19 Z K
15 c A A4 cos f 8 5 1 3
E21 Z K9u2 A3 cos a K 3duA3 sin a C c1 A3 cos a C
(A25) c3 3 A cos 3f 4 1
3 E22 Z c3 A3 cos að2A21 C A23 Þ 4
E23 Z
5 c A4 ðA cos 3f C 2A3 cos aÞ 16 5 1 1
5 E24 Z c5 A31 A23 fcos 3fð2 C cos 2aÞ C sin 2a sin 3fg 8 E25 Z
1233
15 c A2 A3 cos a 4 5 1 3
(A26)
(A27)
(A28)
(A29)
(A30)
5 E26 Z c5 A53 cos a 8
(A31)
E31 Z 9u2 A3 sin a K 3duA3 cos a K c1 A3 sin a
(A32)
E32 Z K
c3 3 A sin 3f 4 1
3 E33 Z K c3 A3 sin að2A21 C A23 Þ 4 E34 Z K
5 c A4 ðA sin 3f C 6A3 sin aÞ 16 5 1 1
5 E35 Z K c5 A31 A23 fsin 3fð2 C cos 2aÞ C sin 2a cos 3fg 8 E36 Z K
15 c A2 A3 sin a 4 5 1 3
5 E37 Z K c5 A53 sin a 8
(A33)
(A34)
(A35)
(A36)
(A37)
(A38)
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A.A. Shah et al. / Ocean Engineering 32 (2005) 1216–1234
Eqs. (A4)–(A7) can be solved by iterative Newton Raphson method and the unknowns A1, A3, a and f are determined. Once these unknowns are known, the _ and x€ ðtÞ can be obtained from Eqs. (A1)–(A3). response x(t), xðtÞ
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