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On the estimation of mooring line characteristic resistance for reliability analysis
Applied Ocean Research 29 (2007) 239–241 www.elsevier.com/locate/apor
Short communication
On the estimation of mooring line characteristic resistance for reliability analysis Roberto Montes-Iturrizaga ∗ , Ernesto Heredia-Zavoni, Francisco Silva-Gonz´alez Instituto Mexicano del Petr´oleo, Programa de Explotaci´on de Campos en Aguas Profundas, Eje Central L´azaro C´ardenas 152, M´exico DF, 07730, Mexico Received 7 November 2007; accepted 27 December 2007 Available online 12 February 2008
Abstract The main aim of this work is to derive a correct formulation for the characteristic resistance of a mooring line segment with lognormally distributed component resistances and substitute an equation proposed by Vazquez-Hernandez et al. [Vazquez-Hernandez AO, Ellwanger GB, Sagrilo LVS. Reliability-based comparative study for mooring lines design criteria. Appl Ocean Res 2006; 28(6):398–406] in a paper published in this journal, which is not correct. The mooring line is considered as a series system and the resistances of individual components of a line segment are statistically independent and identically distributed; furthermore, the case of normally distributed component resistances is also discussed. A comparison with the corresponding equation proposed by DNV-OS-E301 is given. Results show that the formula proposed by Vazquez-Hernandez et al. [Vazquez-Hernandez AO, Ellwanger GB, Sagrilo LVS. Reliability-based comparative study for mooring lines design criteria. Appl Ocean Res 2006; 28(6):398–406] overestimates quite significantly the characteristic resistance of a mooring line segment. c 2008 Elsevier Ltd. All rights reserved.
Keywords: Structural reliability; Mooring system; Floating structures; Characteristic resistance
1. Introduction A paper by Vazquez-Hernandez et al. [1] presents the results of a reliability-based partial safety factor calibration study for mooring line design criteria. In that paper, the authors propose an expression for the most probable value of the global resistance of a segment of a mooring line. The mooring line is considered as a series system and the resistances of all individual components of a line segment are assumed to be statistically independent and identically distributed with the probability distribution FR (r ). If the number of components in the line segment, N , is very large then the global resistance distribution tends to an extreme distribution for minima. Under the assumption that FR (r ) is a lognormal distribution, VazquezHernandez et al. [1] derive the following expression for the characteristic resistance of the line segment, which is defined as the modal value or most probable value of the global resistance distribution:
Rk = q ×
µR 1 + δ 2R p
[1 − (ln(1 + δ 2R ))0.5 ]
2 ln(N ) −
ln(ln(N )) + ln(4π ) . √ 2 2 ln(N )
However, this equation is not correct. The correct expression is: 0.5 µR k R = q exp − ln(1 + δ 2R ) 1 + δ 2R p ln(ln(N )) + ln(4π ) × 2 ln(N ) − √ 2 2 ln(N ) which is derived in the following. By comparing the two expressions, it is quite likely that the original authors inadvertently made a small mistake in their derivation. In addition, the case where the resistance of components is normally distributed is also discussed here. 2. Distribution of minima
∗ Corresponding author. Tel.: +52 55 9175 8130; fax: +52 55 9175 8258.
E-mail address: [email protected] (R. Montes-Iturrizaga). c 2008 Elsevier Ltd. All rights reserved. 0141-1187/$ - see front matter doi:10.1016/j.apor.2007.12.003
Let X i (i = 1, 2, . . . N ) be a set of N independent and identically distributed random variables, and let Y be the
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R. Montes-Iturrizaga et al. / Applied Ocean Research 29 (2007) 239–241
minimum value of this set: Y = min(X 1 , X 2 , . . . , X N ).
(1)
If the probability distribution of each variable X i is FX (x), then the corresponding distribution for Y is FY (y) = 1 − [1 − FX (y)] N .
(2)
If N becomes very large, i.e. N → ∞, it can be shown that [2]: FY (y) = 1 − e−g(y) = 1 − e−N FX (y) .
(3)
Case I: Normal variables If variables X i are normally distributed with mean µ and standard deviation σ then using Eq. (3), the distribution for Y is given by [2]: i h 0 0 (4) FY (y) = 1 − exp −eα (y−u )
Fig. 1. Global characteristic resistance of line segment for normally distributed component resistance.
where u0 = µ + σ u α 0 = α/σ
(5a)
and p ln(ln(N )) + ln(4π ) u = − 2 ln(N ) + √ 2 2 ln(N ) p α = 2 ln(N ).
Fig. 2. Global characteristic resistance of line segment for lognormally distributed component resistance.
(5b)
Note that Eq. (4) has the form of a Type I extreme distribution for minima. In Eq. (5a) parameter u 0 is the modal value or characteristic smallest value of variable Y . Case II: Lognormal variables In this case Eq. (3) corresponds to a Type II extreme distribution for minima [2] " # v k FY (y) = 1 − exp − (6) y where: 0
v = eu k = −α 0
(7)
u0 = λX + ςX u α 0 = α/ς X
(8)
and µ λ X = ln √ 1 + δ2 p ς X = ln(1 + δ 2 ).
(9)
In Eq. (9), δ is the coefficient of variation of X i , and u and α are given in Eq. (5b). Substituting Eqs. (8) and (9) into Eq. (7) one obtains: h p i µ v=√ exp u ln(1 + δ 2 ) 1 + δ2 (10) α k = −p ln(1 + δ 2 )
where v is the modal value or characteristic smallest value of variable Y . 3. Global characteristic resistance Now let X i be the resistance of each component of a line segment, R, and let Y be the global resistance of the segment. If the resistance of each component is normally distributed with mean value µ R and coefficient of variation δ R , then it follows from Eqs. (5a) and (5b) that the characteristic resistance is: p ln(ln(N )) + ln(4π ) k R = µR 1 − δR 2 ln(N ) − . (11) √ 2 2 ln(N ) Similarly, if the resistance of each component is lognormally distributed, then from Eqs. (5b) and (10) q µR k R = q exp − ln(1 + δ 2R ) 2 1 + δR p ln(ln(N )) + ln(4π ) × 2 ln(N ) − . (12) √ 2 2 ln(N ) Eq. (12) should be used instead of that given in the paper by Vazquez-Hernandez et al. [1]. Comparison with DNV-OS-E301 DNV-OS-E301 [3] proposes the following equation for the mooring line global characteristic resistance: R k = µ R [1 − δ R (3 − 6δ R )]
δ R < 0.10.
(13)
Notice that Eq. (13) does not depend on N . Figs. 1 and 2 show the variation of normalized global resistance R k /µ R versus the resistance coefficient of variation δ R as a function of the
R. Montes-Iturrizaga et al. / Applied Ocean Research 29 (2007) 239–241
241
two expressions for the most probable value of global resistance of a line segment are quite significant. As seen in Fig. 3 the formulation by Vazquez-Hernandez et al. [1] overestimates the most probable resistance by more than 250%. Also notice that according to Eqs. (12) and (13), and as shown in Figs. 1 and 2, resistance R k cannot be greater than µ R . However the formulation given by Vazquez-Hernandez et al. [1] yields values of R k which are larger than µ R. 4. Conclusions Fig. 3. Comparison of correct equation for global characteristic resistance against that given by Vazquez-Hernandez et al. (2006); component resistance is lognormally distributed (N = 350).
number of components N . Fig. 1 shows the results for the case of normal resistance components and Fig. 2 for the lognormal case. The characteristic resistance decreases as the coefficient of variation gets larger and the ratio R k /µ R is always less than 1. The effect of the number of components N on the characteristic resistance becomes more noticeable for larger values of the coefficient of variation. It is seen that when component resistance is normally distributed, then for N = 150 the DNV formula and Eq. (11) yield practically the same results. In the case where the resistance of components are assumed to be lognormal then for N = 350 the DNV formula yields almost the same results as Eq. (12). Considering N = 350, Fig. 3 compares the results for normalized global resistance R k /µ R using Eq. (12) proposed in this communication and that proposed in the paper by VazquezHernandez et al. [1]. It can be seen that differences between the
In this work a correct formulation was derived for the characteristic resistance of a mooring line segment with lognormally distributed component resistances, in order to substitute the equation proposed by Vazquez-Hernandez et al. [1] which is shown not to be correct. Also, the case of normally distributed component resistances was discussed and a comparison with the corresponding equation proposed by DNV-OS-E301 was carried out. Results have shown that using the formulation by Vazquez-Hernandez et al. [1] overestimates quite significantly the characteristic resistance of a mooring line segment. References [1] Vazquez-Hernandez AO, Ellwanger GB, Sagrilo LVS. Reliability-based comparative study for mooring lines design criteria. Appl Ocean Res 2006; 28(6):398–406. [2] Ang AH-S, Tang WH. Probability concepts in engineering planning and design. John Wiley and Sons; 1984. [3] Det Norske Veritas [DNV]. Offshore standard DNV-OS-E301. Position mooring. 2004.
Short communication
On the estimation of mooring line characteristic resistance for reliability analysis Roberto Montes-Iturrizaga ∗ , Ernesto Heredia-Zavoni, Francisco Silva-Gonz´alez Instituto Mexicano del Petr´oleo, Programa de Explotaci´on de Campos en Aguas Profundas, Eje Central L´azaro C´ardenas 152, M´exico DF, 07730, Mexico Received 7 November 2007; accepted 27 December 2007 Available online 12 February 2008
Abstract The main aim of this work is to derive a correct formulation for the characteristic resistance of a mooring line segment with lognormally distributed component resistances and substitute an equation proposed by Vazquez-Hernandez et al. [Vazquez-Hernandez AO, Ellwanger GB, Sagrilo LVS. Reliability-based comparative study for mooring lines design criteria. Appl Ocean Res 2006; 28(6):398–406] in a paper published in this journal, which is not correct. The mooring line is considered as a series system and the resistances of individual components of a line segment are statistically independent and identically distributed; furthermore, the case of normally distributed component resistances is also discussed. A comparison with the corresponding equation proposed by DNV-OS-E301 is given. Results show that the formula proposed by Vazquez-Hernandez et al. [Vazquez-Hernandez AO, Ellwanger GB, Sagrilo LVS. Reliability-based comparative study for mooring lines design criteria. Appl Ocean Res 2006; 28(6):398–406] overestimates quite significantly the characteristic resistance of a mooring line segment. c 2008 Elsevier Ltd. All rights reserved.
Keywords: Structural reliability; Mooring system; Floating structures; Characteristic resistance
1. Introduction A paper by Vazquez-Hernandez et al. [1] presents the results of a reliability-based partial safety factor calibration study for mooring line design criteria. In that paper, the authors propose an expression for the most probable value of the global resistance of a segment of a mooring line. The mooring line is considered as a series system and the resistances of all individual components of a line segment are assumed to be statistically independent and identically distributed with the probability distribution FR (r ). If the number of components in the line segment, N , is very large then the global resistance distribution tends to an extreme distribution for minima. Under the assumption that FR (r ) is a lognormal distribution, VazquezHernandez et al. [1] derive the following expression for the characteristic resistance of the line segment, which is defined as the modal value or most probable value of the global resistance distribution:
Rk = q ×
µR 1 + δ 2R p
[1 − (ln(1 + δ 2R ))0.5 ]
2 ln(N ) −
ln(ln(N )) + ln(4π ) . √ 2 2 ln(N )
However, this equation is not correct. The correct expression is: 0.5 µR k R = q exp − ln(1 + δ 2R ) 1 + δ 2R p ln(ln(N )) + ln(4π ) × 2 ln(N ) − √ 2 2 ln(N ) which is derived in the following. By comparing the two expressions, it is quite likely that the original authors inadvertently made a small mistake in their derivation. In addition, the case where the resistance of components is normally distributed is also discussed here. 2. Distribution of minima
∗ Corresponding author. Tel.: +52 55 9175 8130; fax: +52 55 9175 8258.
E-mail address: [email protected] (R. Montes-Iturrizaga). c 2008 Elsevier Ltd. All rights reserved. 0141-1187/$ - see front matter doi:10.1016/j.apor.2007.12.003
Let X i (i = 1, 2, . . . N ) be a set of N independent and identically distributed random variables, and let Y be the
240
R. Montes-Iturrizaga et al. / Applied Ocean Research 29 (2007) 239–241
minimum value of this set: Y = min(X 1 , X 2 , . . . , X N ).
(1)
If the probability distribution of each variable X i is FX (x), then the corresponding distribution for Y is FY (y) = 1 − [1 − FX (y)] N .
(2)
If N becomes very large, i.e. N → ∞, it can be shown that [2]: FY (y) = 1 − e−g(y) = 1 − e−N FX (y) .
(3)
Case I: Normal variables If variables X i are normally distributed with mean µ and standard deviation σ then using Eq. (3), the distribution for Y is given by [2]: i h 0 0 (4) FY (y) = 1 − exp −eα (y−u )
Fig. 1. Global characteristic resistance of line segment for normally distributed component resistance.
where u0 = µ + σ u α 0 = α/σ
(5a)
and p ln(ln(N )) + ln(4π ) u = − 2 ln(N ) + √ 2 2 ln(N ) p α = 2 ln(N ).
Fig. 2. Global characteristic resistance of line segment for lognormally distributed component resistance.
(5b)
Note that Eq. (4) has the form of a Type I extreme distribution for minima. In Eq. (5a) parameter u 0 is the modal value or characteristic smallest value of variable Y . Case II: Lognormal variables In this case Eq. (3) corresponds to a Type II extreme distribution for minima [2] " # v k FY (y) = 1 − exp − (6) y where: 0
v = eu k = −α 0
(7)
u0 = λX + ςX u α 0 = α/ς X
(8)
and µ λ X = ln √ 1 + δ2 p ς X = ln(1 + δ 2 ).
(9)
In Eq. (9), δ is the coefficient of variation of X i , and u and α are given in Eq. (5b). Substituting Eqs. (8) and (9) into Eq. (7) one obtains: h p i µ v=√ exp u ln(1 + δ 2 ) 1 + δ2 (10) α k = −p ln(1 + δ 2 )
where v is the modal value or characteristic smallest value of variable Y . 3. Global characteristic resistance Now let X i be the resistance of each component of a line segment, R, and let Y be the global resistance of the segment. If the resistance of each component is normally distributed with mean value µ R and coefficient of variation δ R , then it follows from Eqs. (5a) and (5b) that the characteristic resistance is: p ln(ln(N )) + ln(4π ) k R = µR 1 − δR 2 ln(N ) − . (11) √ 2 2 ln(N ) Similarly, if the resistance of each component is lognormally distributed, then from Eqs. (5b) and (10) q µR k R = q exp − ln(1 + δ 2R ) 2 1 + δR p ln(ln(N )) + ln(4π ) × 2 ln(N ) − . (12) √ 2 2 ln(N ) Eq. (12) should be used instead of that given in the paper by Vazquez-Hernandez et al. [1]. Comparison with DNV-OS-E301 DNV-OS-E301 [3] proposes the following equation for the mooring line global characteristic resistance: R k = µ R [1 − δ R (3 − 6δ R )]
δ R < 0.10.
(13)
Notice that Eq. (13) does not depend on N . Figs. 1 and 2 show the variation of normalized global resistance R k /µ R versus the resistance coefficient of variation δ R as a function of the
R. Montes-Iturrizaga et al. / Applied Ocean Research 29 (2007) 239–241
241
two expressions for the most probable value of global resistance of a line segment are quite significant. As seen in Fig. 3 the formulation by Vazquez-Hernandez et al. [1] overestimates the most probable resistance by more than 250%. Also notice that according to Eqs. (12) and (13), and as shown in Figs. 1 and 2, resistance R k cannot be greater than µ R . However the formulation given by Vazquez-Hernandez et al. [1] yields values of R k which are larger than µ R. 4. Conclusions Fig. 3. Comparison of correct equation for global characteristic resistance against that given by Vazquez-Hernandez et al. (2006); component resistance is lognormally distributed (N = 350).
number of components N . Fig. 1 shows the results for the case of normal resistance components and Fig. 2 for the lognormal case. The characteristic resistance decreases as the coefficient of variation gets larger and the ratio R k /µ R is always less than 1. The effect of the number of components N on the characteristic resistance becomes more noticeable for larger values of the coefficient of variation. It is seen that when component resistance is normally distributed, then for N = 150 the DNV formula and Eq. (11) yield practically the same results. In the case where the resistance of components are assumed to be lognormal then for N = 350 the DNV formula yields almost the same results as Eq. (12). Considering N = 350, Fig. 3 compares the results for normalized global resistance R k /µ R using Eq. (12) proposed in this communication and that proposed in the paper by VazquezHernandez et al. [1]. It can be seen that differences between the
In this work a correct formulation was derived for the characteristic resistance of a mooring line segment with lognormally distributed component resistances, in order to substitute the equation proposed by Vazquez-Hernandez et al. [1] which is shown not to be correct. Also, the case of normally distributed component resistances was discussed and a comparison with the corresponding equation proposed by DNV-OS-E301 was carried out. Results have shown that using the formulation by Vazquez-Hernandez et al. [1] overestimates quite significantly the characteristic resistance of a mooring line segment. References [1] Vazquez-Hernandez AO, Ellwanger GB, Sagrilo LVS. Reliability-based comparative study for mooring lines design criteria. Appl Ocean Res 2006; 28(6):398–406. [2] Ang AH-S, Tang WH. Probability concepts in engineering planning and design. John Wiley and Sons; 1984. [3] Det Norske Veritas [DNV]. Offshore standard DNV-OS-E301. Position mooring. 2004.