9
Applied Ocean Research 33 (2011) 375–383 Contents lists available at ScienceDirect Applied Ocean Research journal homepage: www.elsevier.com/locate/apor Long-term response analysis of FPSO mooring systems A.O. Vázquez-Hernández a,, G.B. Ellwanger b , L.V.S. Sagrilo b a Instituto Mexicano del Petróleo, Deep Waters Explotation Department, Eje Central Lázaro Cárdenas Norte 152, Gustavo A. Madero, 07730, México, D.F., Mexico b COPPE Federal University of Rio de Janeiro, Civil Engineering Department, Cidade Universitária, Centro de Tecnologia, Bloco I-2000, Sala I-116, Ilha do Fundão 21945-970, Rio de Janeiro, Brazil a r t i c l e i n f o Article history: Received 3 May 2010 Received in revised form 5 April 2011 Accepted 21 May 2011 Available online 3 August 2011 Keywords: Mooring systems Long-term analysis Monte Carlo simulation a b s t r a c t The design of mooring systems for floating production units usually considers extreme environmental conditions as a primary design parameter. However, in the case of FPSO (Floating, Production, Storage and Offloading) units, the worst response for the mooring system may be associated with other sea state conditions due to the fact that its extreme response may be associated with a resonant period instead of an extreme wave height. The best way to deal with this problem is by performing long-term analysis in order to obtain extreme response estimates. This procedure is computationally very demanding, since many short-term environmental conditions, and their associated stochastic nonlinear time domain numerical simulations of the mooring lines, are required to obtain such estimates. A simplified approach for the long-term analysis is the environmental contour-line design approach. In this paper a Monte Carlo-based integration procedure combined with an interpolation scheme to obtain the parameters of the short- term response distribution is employed to hasten the long-term analysis. Numerical simulations are carried out for an FPSO at three different locations considering a North Sea joint probability distribution for the environmental parameters. The long-term analysis results are compared against those obtained using extreme environmental conditions and environmental contour-line methodology. These results represent the characteristic load effect for the design of mooring systems of floating units using the reliability analysis for mooring line. The results show that the long-term results are usually more critical than those obtained with the other approaches and even different mooring lines can be identified as the critical ones. © 2011 Elsevier Ltd. All rights reserved. 1. Introduction Mooring line responses are strongly dependent on the floater motions which are induced by the environmental actions due to waves, wind and current. These environmental actions in the long run are non-stationary processes with specific statistical characteristics for each location. However, for the purpose of dynamic analysis of marine structures, the environmental actions are approximately modeled as a sequence of piecewise short-term stationary processes, usually of a 3–6 h duration. For each short- term period wave, wind and current are represented by some deterministic parameters that can describe their random behav- ior, such as: significant wave height (H s ), peak wave period (T p ), mean wind velocity (V) and surface current velocity (C) and their associated directions. The ideal procedure for computing the mooring system response should be based on a long-term response analysis accounting for the contribution of every short-term condition. In Corresponding author. Tel.: +52 55 9175 7402; fax: +52 55 9175 8258. E-mail address: [email protected] (A.O. Vázquez-Hernández). other words, ideally, mooring lines should not be analyzed using only extreme response estimates associated with a given extreme short-term condition, once some important dynamic amplification can be found in other short-term conditions. However, in order to perform a long-term analysis, besides the availability of a joint dis- tribution of short-term wind, waves and current parameters for the location under consideration, it is necessary to solve a mul- tidimensional integral which accounts for the contribution of all short-term conditions to the long-term response. This is a very time-consuming approach, mainly when nonlinear time domain simulations are employed to analyze the mooring line response. How to solve this integral efficiently seems to be the most chal- lenging problem associated with long-term response analysis. An approximate and computationally cheaper way to solve this multidimensional integral is by means of using contour-line com- binations of environmental parameters [1,2] which are defined [3] using Inverse First Reliability Method (IFORM). However, as this approach does not take into account the randomness of the response itself, some correction (or an extreme response value associated with a higher fractile), based on some previous expe- riences, must be employed to get a better estimate of, for instance, a characteristic long-term extreme response. 0141-1187/$ see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.apor.2011.05.003

Long-term response analysis of FPSO mooring systems

Embed Size (px)

Citation preview

Page 1: Long-term response analysis of FPSO mooring systems

L

Aa

b

R

a

ARRAA

KMLM

1

mtlcdastdima

ra

0d

Applied Ocean Research 33 (2011) 375– 383

Contents lists available at ScienceDirect

Applied Ocean Research

journal homepage: www.elsevier.com/locate/apor

ong-term response analysis of FPSO mooring systems

.O. Vázquez-Hernándeza,∗, G.B. Ellwangerb, L.V.S. Sagrilob

Instituto Mexicano del Petróleo, Deep Waters Explotation Department, Eje Central Lázaro Cárdenas Norte 152, Gustavo A. Madero, 07730, México, D.F., MexicoCOPPE – Federal University of Rio de Janeiro, Civil Engineering Department, Cidade Universitária, Centro de Tecnologia, Bloco I-2000, Sala I-116, Ilha do Fundão 21945-970,io de Janeiro, Brazil

r t i c l e i n f o

rticle history:eceived 3 May 2010eceived in revised form 5 April 2011ccepted 21 May 2011vailable online 3 August 2011

eywords:ooring systems

ong-term analysisonte Carlo simulation

a b s t r a c t

The design of mooring systems for floating production units usually considers extreme environmentalconditions as a primary design parameter. However, in the case of FPSO (Floating, Production, Storageand Offloading) units, the worst response for the mooring system may be associated with other sea stateconditions due to the fact that its extreme response may be associated with a resonant period instead of anextreme wave height. The best way to deal with this problem is by performing long-term analysis in orderto obtain extreme response estimates. This procedure is computationally very demanding, since manyshort-term environmental conditions, and their associated stochastic nonlinear time domain numericalsimulations of the mooring lines, are required to obtain such estimates. A simplified approach for thelong-term analysis is the environmental contour-line design approach. In this paper a Monte Carlo-basedintegration procedure combined with an interpolation scheme to obtain the parameters of the short-term response distribution is employed to hasten the long-term analysis. Numerical simulations arecarried out for an FPSO at three different locations considering a North Sea joint probability distribution

for the environmental parameters. The long-term analysis results are compared against those obtainedusing extreme environmental conditions and environmental contour-line methodology. These resultsrepresent the characteristic load effect for the design of mooring systems of floating units using thereliability analysis for mooring line. The results show that the long-term results are usually more criticalthan those obtained with the other approaches and even different mooring lines can be identified as thecritical ones.

. Introduction

Mooring line responses are strongly dependent on the floaterotions which are induced by the environmental actions due

o waves, wind and current. These environmental actions in theong run are non-stationary processes with specific statisticalharacteristics for each location. However, for the purpose ofynamic analysis of marine structures, the environmental actionsre approximately modeled as a sequence of piecewise short-termtationary processes, usually of a 3–6 h duration. For each short-erm period wave, wind and current are represented by someeterministic parameters that can describe their random behav-

or, such as: significant wave height (Hs), peak wave period (Tp),ean wind velocity (V) and surface current velocity (C) and their

ssociated directions.

The ideal procedure for computing the mooring system

esponse should be based on a long-term response analysisccounting for the contribution of every short-term condition. In

∗ Corresponding author. Tel.: +52 55 9175 7402; fax: +52 55 9175 8258.E-mail address: [email protected] (A.O. Vázquez-Hernández).

141-1187/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.oi:10.1016/j.apor.2011.05.003

© 2011 Elsevier Ltd. All rights reserved.

other words, ideally, mooring lines should not be analyzed usingonly extreme response estimates associated with a given extremeshort-term condition, once some important dynamic amplificationcan be found in other short-term conditions. However, in order toperform a long-term analysis, besides the availability of a joint dis-tribution of short-term wind, waves and current parameters forthe location under consideration, it is necessary to solve a mul-tidimensional integral which accounts for the contribution of allshort-term conditions to the long-term response. This is a verytime-consuming approach, mainly when nonlinear time domainsimulations are employed to analyze the mooring line response.How to solve this integral efficiently seems to be the most chal-lenging problem associated with long-term response analysis.

An approximate and computationally cheaper way to solve thismultidimensional integral is by means of using contour-line com-binations of environmental parameters [1,2] which are defined[3] using Inverse First Reliability Method (IFORM). However, asthis approach does not take into account the randomness of the

response itself, some correction (or an extreme response valueassociated with a higher fractile), based on some previous expe-riences, must be employed to get a better estimate of, for instance,a characteristic long-term extreme response.
Page 2: Long-term response analysis of FPSO mooring systems

3 plied O

iaipnmcts1tpTdobt[

2

pmuaci(aflto

co[[de

oAttattmlsMnim

autottt[

76 A.O. Vázquez-Hernández et al. / Ap

In this paper the multidimensional integral is solved numer-cally by means of Monte Carlo Simulation (MCS). As it requires

lot of short-term analyses, an interpolation procedure is alsontroduced to hasten the numerical integration. This interpolationrocedure is based on linear interpolation functions found in con-ection with the isoparametric formulation of the finite elementethod [4]. Using a pre-defined set of short-term environmental

onditions, this interpolation procedure computes the parame-ers of the extreme short-term response distribution for any otherhort-term condition. This approach is employed to compute the00 yr extreme response of the intact mooring system of an FPSO athree different water depths. For both cases, a joint environmentalarameters distribution proposed for the North Sea [5] is employed.his distribution considers that wind, wave and current for a givenirection are collinear. For a given direction the joint distributionf the environmental parameters is formed by probability distri-utions conditioned on Hs. The results obtained are compared tohose obtained by the extreme environmental condition approach6] and by the environmental contour line methodology.

. Mooring line analysis

The dynamic analysis for deepwater floating systems is com-lex due to dynamic coupling between the platform and theoorings/risers system. Also, in the design of floating structures

ncertainties relating to structural-naval behavior are present andre caused by the multivariate environment of winds, waves andurrents. All of the latter make a significant contribution to moor-ng forces. Therefore several works have been done focused on:1) floating moored system analysis using methods that take intoccount the global behavior of platform, mooring and risers, (2)oating moored analysis using a combination of numerical simula-ion methods to obtain the short-term response and (3) other typef methods to determine the mooring long-term response.

In the first case, many researchers have been working in theouple effect topic such as Ormberg and Larsen [7] which devel-ped a tool for fully coupled time domain analysis. Nishimoto et al.8] develop a tool for FPSO time domain analysis. Low and Langley9,10] worked in a hybrid method considering a time/frequencyomain approach. Others authors have been working in severalfficient methods that have been applied by the offshore industry.

In the second case, researchers have been developing method-logies in which they apply numerical simulation methods such asrtificial Neuronal Network (ANN) [11,12] and Monte Carlo Simula-

ions [13–15] to obtained the extreme response of moored platformhrough the use of cheaper methods. Mazaheri and Downie [11]pplied an AAN to predict data series of platform’s excursion dueo long-term met-ocean data. Guarize et al. [12] also apply the ANNo obtain risers response that can be apply in a similar form to

ooring lines analysis. Spanos et al. [13] proposed a model for pre-iminary design regarding the combined Spar/riser mooring linesystem using the MCS tools. Karlsen and Naess [14] applied theCS on the modeling of nonlinear wave loads in addition to the

onlinear mooring characteristics. Later, Naess et al. [15] workedn a particular case of MCS application to obtain the horizontal surge

otions of a tension leg platform.Finally, in the third case we have different type of works such

s Morton and Bowers [16] that proposed a methodology to eval-ate the extreme response of a moored semi-submersible subjecto multivariate offshore environment conditions. This methodol-gy involves the construction of statistical models which describe

he inter-relationships of the extreme offshore environment. Inhis work the response is evaluated considering only the varia-ion of hs-wind parameters in the sea states. Grime and Langley17] present a frequency-domain response model coupled with a

cean Research 33 (2011) 375– 383

pseudo-asymptotic integration scheme to provide rapid estimatesof lifetime reliability.

According to the above, a common characteristic of the men-tion works is the use of different approaches and assumptions withgood results in the coupling effects involved. In this work the short-term mooring line tension was computed by means of an uncoupleddynamic analysis as shown in Fig. 1. Firstly, the floater motionstime-series were computed by means a dedicated computer codeDYNASIM [8] where the mooring lines were modeled simply ascatenary lines. Then, the nonlinear time-domain finite element-based computer program ANFLEX [18], using the floater motions asprescribed movements at the top of the line and 3-D truss elementsto represent the whole mooring line, was employed to compute theline tension. More details can be found in Vázquez-Hernández [19]and Vázquez-Hernández et al. [20].

Thus, in this work the vessel motions and line tensions arewell predicted by taking into account the wave and low frequencycomponents effects and their dynamic impact of geometric non-linearity applied to a FPSO moored in three different water depths(shallow water and ultra-deepwater depths). The work is also com-plemented with a Monte Carlo Simulation to obtain the long-termresponse.

3. Short-term mooring line tension

Line tension, mainly at the top of each line segment, is themost important response parameter to be observed in a mooringline analysis. In a given short-term condition, the line tension is astochastic process that can be represented by

SY( t| Y = y) = SD + S E|Y( t| Y = y) (1)

where SY( t| Y = y) is the total short-term tension, SD is the nomi-nal line pre-tension and S E|Y( t| Y = y) is the dynamic line tensionconditioned on the short-term condition Y = y. Hence, y is a set ofspecific values for the environmental parameters associated withwind, current and waves. Following the joint probability model ofthe environmental parameters used in this study [5], where theenvironmental parameters are considered to be collinear in a givendirection of incidence, the joint environmental parameters can thenbe represented by

Y =[Y′(�), �

](2)

where Y′ =[Hs, Tp, V, C

]and � is a set of discrete incidence angles.

A realization of SY( t| Y = y) is obtained, for instance, by means ofa nonlinear time-domain simulation using a finite element-basedcode for Y = y =

[y′(�i), �i

]. Dynamic tension is simply computed

by subtracting the pre-tension SD from the total tension. SinceS E|Y( t| Y = y) is usually a non-Gaussian process, an approximateprocedure must be employed to establish its peak distribution. Inthis study, a Weibull fitting model was employed. Then, genericallythe S E|Y( t| Y = y) peak distribution is given by

FpSE|Y( s| y) = 1 − exp

(−

(s

˛(y(�i), �i)

))ˇ(y(�i),�i)(3)

where ˛(y(�i), �i) and ˇ(y(�i), �i) are the Weibull distributionparameters, for the short-term condition y =

[y′(�i), �i

], obtained

by curve fitting according to Zurita [21]. According to the asymp-totic theory of extremes [22], the S E|Y( t| Y = y) short-term extremedistribution can be represented approximately by a Gumbel distri-bution given by

e

F SE|Y( s| y) = exp(− exp(−˛G(y(�i), �i)(s − u(y(�i), �i)))) (4)

where

u(y(�i), �i) = ˛(y(�i), �i)[ln(N)]1/ˇ(y(�i),�i)

Page 3: Long-term response analysis of FPSO mooring systems

A.O. Vázquez-Hernández et al. / Applied Ocean Research 33 (2011) 375– 383 377

n of t

˛

N

wtt

4

crtd

F

wmB

f

w(otfbB

Fig. 1. Schematic representatio

G(y(�i), �i) = ˇ(y(�i), �i)˛(y(�i), �i)

[ln(N)](ˇ(y(�i),�i)−1)/ˇ(y(�i),�i) (5)

= �p(y(�i), �i)TST

ith �p(y(�i), �i) as the frequency of the global peaks for the short-erm condition y =

[y′(�i), �i

]and TST is the duration of the short-

erm condition, assumed to be 3 h in the present study.

. Long-term mooring line tension

The long-term dynamic mooring line tension is a random pro-ess which results from the contribution of several short-termesponses. A characteristic value for long-term extreme dynamicension can approximately be obtained using the unconditionalistribution of its short-term extreme peak distribution given by

eSE

(s) =∫

FeSE|Y( s| y)fY(y) dy (6)

here fY(y) is the joint distribution of the short-term environ-ental parameters which, specifically in this study, is given by

itner-Gregersen and Haver [5].

Y(y) =N�∑i=1

f Y′|�( y′∣∣ �i)p�i(7)

here N� is the number of discrete directions considered, i.e., N� = 8N, NE, E, SE, S, SW, W, NW), and p�i

is the frequency of occurrencef short-term conditions in the incidence direction �i In this dis-

ribution wave, wind and current are considered to be collinearor a given direction of incidence and their joint distribution isased on probability functions conditioned on Hs, as described initner-Gregersen and Haver [5].

he structural analysis process.

Eq. (6) gives the extreme response distribution for a genericshort-term condition. However, using this distribution, forinstance, a characteristic extreme value of the long-term dynamictension can be associated with its 100 yr response Sk

E, given by:

FeSE

(SkE) = 1 − 1

100 × 2920(8)

where 2920 is the expected number of 3 h short-term conditionsin a given year.

5. Long-term response integration

To obtain the long-term distribution, it is necessary to evaluatethe multidimensional integral given by Eq. (6). It can be solved bymeans of Monte Carlo Simulation [23] as

FeSE

(s) =∫

FeSE|Y( s| y)fY(y)dy =

Ns∑j=1

FeSEY|( s| yj)

1Ns

(9)

where yj = (hjs, tj

p, vj, cj, �j) is a sample of the environmentalparameters randomly generated from fY(y) by standard randomnumber generator techniques [22,23] and Ns is the number of sim-ulations. It is well known that Eq. (9), especially for large levels ofs, gives accurate results only when there are many simulations. Inthis case, it means that a large number of nonlinear time-domaindynamic analysis of each mooring line in the mooring systemshould be performed, transforming the long-term analysis in a verytime-consuming task.

However, the evaluation of Eq. (9) for each mooring line can∣

be performed more efficiently if, instead of computing Fe

SE|Y( sp∣yi)

for each yj, its parameters could be obtained more quickly bymeans of an interpolation procedure. There are various interpola-tion schemes available in the field of long-term analysis of marine

Page 4: Long-term response analysis of FPSO mooring systems

3 plied O

sdtcsswtFwletTtm4p

ippnAi

6

icffw

F

ddmBae

u

wt

78 A.O. Vázquez-Hernández et al. / Ap

tructures, e.g. [24,25], however the vast majority of them are twoimensional-based schemes, i.e., they are restricted only to thewo wave environmental parameters, Hs and Tp. As in the presentase four environmental parameters are taken into account, a newtrategy was employed. The strategy adopted in this study was toelect a number of discrete points of wave parameters (Hs and Tp),ind velocity (V) and surface current velocity (C) for each direc-

ion of incidence in order to form a 4-D mesh of discrete points.or each one of these mesh points the parameters ˛G(·) and u(·)ere computed using a sampled time series obtained from a non-

inear time domain simulation of the mooring line [19]. Then, forvery integration sample yj generated by Monte Carlo Simulationhese two parameters were obtained by means of interpolation.he interpolation scheme implemented in this study is based onhe linear interpolation functions used in connection with isoperi-

etric formulation of the finite element method [4] expanded for a-D domain (Hs, Tp, V and C random variables). For each integrationoint yj the 24 = 16 nearest points on the discrete grid are selected

n order to obtain interpolated values ˛jG(·) and uj(·). As the nearest

oints are considered to obtain an approximate estimate for thesearameters and the interpolation functions are linear, in fact it doesot matter whether the result is being interpolated or extrapolated.

brief summary [19] of the whole integration scheme is presentedn Appendix A.

. Monte Carlo simulation process

In Monte Carlo simulation methodology it was consider thenverse transform method. To generate a series of samples it wasonsidered a basic variable ui for which the cumulative probabilityunction is FU(ui), as shown in Fig. 2, the technique of inverse trans-orm generates a variable uniformly distributed ri(0 ≤ ri ≤ 1) whichill be equals to FU(ui).

The latter can be mathematically expressed as:

U(ui) = ri or ui = FU−1(ui) (10)

However, there are specialized techniques for generating ran-om variables from some distributions as normal and lognormalistribution that are more computationally efficient than theethod of the inverse transform. One of them is the method of

ox–Muller [23], which produces a realization of a normal vari-ble with mean � and standard deviation � with the followingxpression:

i = [−2 ln(r1)]1/2sen(2 · � · r2) · � + � (11)

here r1 and r2 are independent random variables uniformly dis-ributed in the interval (0,1). In the case of random variables with

Fig. 2. Inverse transform method for th

cean Research 33 (2011) 375– 383

lognormal distribution Vi, these can be obtained from the aboveexpression as:

vi = exp(ui)

vi = [−2 ln(r1)]1/2sen(2 · � · r2) · � + � (12)

where and � are the parameters of the lognormal distribution.

7. Ultimate limit state design criteria

Most modern design codes are reliability-based and use the Loadand Resistance Factor Design (LRFD) format. They are based ondesign equations which compare loads and resistance parameters.These parameters have to be defined in such way that the line prob-ability of failure is equal or less than a certain target value. Loadsand resistances are subject to uncertainty and are introduced in thedesign equations by the design values. In the case of a mooring line[26] the ultimate limit state design condition can be expressed as:

(s · Ss + e · Ske ) ≤ Rk (13)

where Ss refers to the line tension component associated with func-tional loads (defined in this work as the nominal pre-tension of theline), s is the safety factor associated with Ss, Sk

e is the characteristicenvironmental line tension component associated with the envi-ronmental parameters of wave, wind and current, e is the safetyfactor associated with Se and Rk is the characteristic resistance ofthe line segment. The total line tension is usually obtained from afinite element-based nonlinear time domain dynamic analysis ofthe mooring line.

The probability of failure is evaluated considering the statisticaldescription of all parameters involved in the limit state equation.As the environmental tension acting on a mooring line is a dynamictime-dependent random variable a failure function associated withEq. (13) for structural reliability analysis can be defined as shownin Eq. (14).

G(t) = R(Z) − (Ss(Z) + Se(Z, Y, t)) (14)

where R is the line resistance and Ss is the line pre-tension, Se

is the line environmental tension component that includes low-frequency and wave-frequency components, Z is the vector oftime-independent random variables, as the model uncertaintiesand resistance parameters of the line, and Y are the short-termenvironmental parameters related to the wave, wind and currentgiven by

Y ={

Hs, Tp, �H, Vw, �w, Vc, �c

}(15)

where Hs is the significant wave height, Tp is the wave peakperiod, Vw is the mean hourly wind speed, Vc is the surface current

e generation of random variables.

Page 5: Long-term response analysis of FPSO mooring systems

A.O. Vázquez-Hernández et al. / Applied Ocean Research 33 (2011) 375– 383 379

tion for the FPSO in 200 m water depth.

vd

8

tmtdrtn(um

wptHdcmFwptd

TM

TM

Fig. 3. Mooring system configura

elocity and �H, �w and �c are the wave, wind and current incidenceirections, respectively [20].

. Numerical examples

A turret-moored 280 kDWT FPSO was analyzed consideringhe joint probability distribution for the environmental conditions

entioned above. In order to cover different dynamic behaviors forhe mooring system the FPSO was supposed to be installed in threeistinct locations having water depths of 200 m, 800 m and 3000 m,espectively. Risers were not included in the numerical model. Forhe lower water depth the mooring system is composed by cate-ary chain mooring lines. The other case considers a taut-leg systemchain-polyester rope-chain). These three mooring system config-rations are shown in Figs. 3 and 4, respectively. The details of eachooring system are presented in Tables 1 and 2.The traditional environmental extreme conditions associated

ith a 100 yr return period for the 8 directions of incidence areresented in Table 3 [19]. These sea state conditions were obtainedhrough the environmental model of the Bitner-Gregersen andaver [5]. These short-term conditions are employed in the stan-ard design methodology based only on extreme environmentalonditions [6]. The 100 yr contours [3] to be employed in the designethodology based on environmental contours [1] are shown in

ig. 5 for all incidence directions of the environmental loads. The

ind velocity and surface current velocity associated with eachair (Hs, Tp) on the environmental contour had been taken ashose most probable according to their conditional probabilityistributions.

able 1ooring line characteristics.

SWL (m) Material Length (m) Nominal diameter (m)

200 Top chain 650 0.095

Bottom chain 350 0.190

800 Top chain 150 0.100

Polyester 800 0.191

Bottom chain 150 0.100

3000 Top chain 150 0.102

Polyester 3500 0.191

Bottom chain 190 0.102

a EA – Quasi-Static axial stiffness (polyester).

able 2ooring line configurations.

Water depth (m) Configuration Pre-tension (kN) T

200 Catenary 2000 1800 Taut-leg 1400 1

3000 Taut-leg 3444 3

Fig. 4. Mooring system configuration for the FPSO in 800 m and 3000 m water depth.

Taking advantage of the mooring system array symmetry,only one direction of incidence needed to be considered for thelong-term response analysis. The discrete grid mapping for theintegration was composed by a mesh of 5 × 4 × 4 × 4 = 320 pointsassociated with Hs, Tp, V and C, respectively. These points are shownin Table 4. They were chosen in order to cover a significant part ofthe individual distributions of the environmental parameters. As aconsequence of this symmetry also only 4 mooring lines neededto be considered. Then, for the long-term analysis a total of 1280

3 h nonlinear time-domain dynamic mooring line analyses wereperformed. In order to get a good sample of peaks, the simulationlength was taken as 10,800 s. The long-term integral in Eq. (6) wasevaluated using Ns equal to 106.

MBL (kN) Weight (in sea water) (kN/m) Axial stiffness (EA) (kN)

9001 1.6822 710,26818,000 3.3640 1,420,536

9864 1.7069 766,0009810 0.0622 151,574a

9864 1.7069 766,000

10,217 2.0413 788,2079810 0.2364 151,574a

10,217 2.0413 788,207

otal length (m) Number of lines Angle between lines (◦)

000 8 45100 8 45840 8 45

Page 6: Long-term response analysis of FPSO mooring systems

380 A.O. Vázquez-Hernández et al. / Applied Ocean Research 33 (2011) 375– 383

Table 3Short-term extreme environmental conditions.

Case Wave (10 yr) Wind (10 yr) Current (100 yr)

Hs (m) Tp (s) Dir Velocity (m/s) Dir Velocity (m/s) Dir

1 11.93 15.65 N 38.99 N 0.825 N2 10.40 15.85 NE 44.15 NE 0.796 NE3 9.18 12.70 E 40.51 E 0.752 E4 6.36 9.70 SE 32.53 SE 0.639 SE5 6.47 10.60 S 32.97 S 0.641 S6 12.39 16.40 SW 39.97 SW 0.814 SW7 13.86 17.0 W 42.81 W 0.878 W8 11.70 16.45 NW 38.28 NW 0.821 NW

Case Wave (100 yr) Wind (100 yr) Current (10 yr)

9 13.43 16.70 N 42.35 N 0.763 N10 12.50 17.60 NE 49.68 NE 0.708 NE11 11.02 13.80 E 46.84 E 0.661 E12 7.22 10.10 SE 37.80 SE 0.572 SE13 7.24 10.90 S 37.89 S 0.578 S14 12.99 16.90 SW 41.85 SW 0.777 SW15 15.45 18.20 W 46.59 W 0.813 W16 13.32 17.55 NW 42.17 NW 0.752 NW

Table 4Environmental parameters mesh points.

Environmental parameter Mesh points

Hs (m) 1.5 3.1 5.8 8.5 11.2

l

TF

law

Tp (s) 7.0 11.0 15.0 20.0V (m/s) 2.5 7.5 12.5 17.5C (m/s) 0.0 0.20 0.40 0.55

The annual extreme distributions for the top tension of the mostoaded mooring lines, for each water depth, is shown in Figs. 6–8.

hese annual distributions are obtained by[Fe

SE(s)

]2920, where

e

SE

(s) is given by Eq. (6). It is observed that line L01 is the mostoaded mooring line for the FPSO in 200 m and 800 m water depthnd line L02 is the most loaded one when it is located in 3000 mater depth.

Fig. 5. 100 yr contour lines for each direction o

Fig. 6. Top tension annual extreme cumulative distribution for the 3 most loadedmooring lines (FPSO in 200 m water depth).

Table 5 presents the 100 yr characteristic top tensions and themost loaded mooring line for the three design criteria: (a) extreme100 yr short-term environmental condition, (b) 100 yr environ-mental contour and (c) extreme 100 yr long-term response. In case(a) the characteristic value was taken as the largest value amongall most probable short-term extreme values associated with eachone of the 16 extreme environmental conditions shown in Table 3.In case (b) the characteristic value was taken as the largest valueamong all most probable short-term extreme values found in 9points on the 100 yr contour line of each incidence direction. Forcase (c) the characteristic value was computed by means of Eq. (8).

It must be observed that, as a general trend, the other twomethodologies predict extreme values that are smaller than thosepredicted by the long-term response approach. It also must benoticed that lines L01 and L02 were identified as the most critical

f incidence of the environmental loads.

Page 7: Long-term response analysis of FPSO mooring systems

A.O. Vázquez-Hernández et al. / Applied Ocean Research 33 (2011) 375– 383 381

Table 5Most critical mooring system responses.

SWL (m) Pre-tension(kN)

100 yr Long-term response 100 yr extreme short-term environmentalcondition

100 yr environmental contour

Mooringline

Dyn. comp.(kN)

Total response(kN)

Mooringline

Dyn. comp.(kN)

Total response(kN)

Mooringline

Dyn. comp.(kN)

Total response(kN)

200 2000 L01 5140 7140 L02 4733 6733 L02 4998 6998800 1400 L01 5005 6405 L07 6754 8154 L01 5681 7081

3000 3444 L02 3497 6941 L03 3329 6773 L03 2868 6312

Fig. 7. Top tension annual extreme cumulative distribution for the 3 most loadedmooring lines (FPSO in 800 m water depth).

Fm

o8i

s1adHuo

Table 6Annual reliability index (ˇ1yr).

Water depth (m) Characteristic environmental tension criterion

100 yrresponse

100 yr Hs − Tp

contour100 yr environmentalcondition

200 3.707 3.695 3.691800 3.659 4.072 4.649

3000 3.736 3.676 3.760

ig. 8. Top tension annual extreme cumulative distribution for the 3 most loadedooring lines (FPSO in 3000 m water depth).

nes by the long-term response approach for water depths 200 m,00 m and 3000 m, respectively, while the lines L02 and L03 were

dentified by the other two approaches.Table 6 shows the reliability indexes of the three mooring line

ystem obtained for different approaches. As it can be seen the00 yr response criterion is the only one that leads to designs withlmost the same target safety level independently of the water

epth and environmental tension to nominal pre-tension ratio.owever, this methodology is computationally expensive to besed in every day design practice. Comparing the criterion basedn the environmental contour with that based on extreme environ-

mental condition it is observed that the reliability indexes are lessscattered for the former. Hence, the criterion based on the extremeenvironmental contour seems to be most effective for everydaymooring system design practice.

It is very important to notice that, in a general case, the totalnumber of nonlinear time-domain simulations for a single moor-ing line belonging to a given floater mooring system is roughly 16(2 × 8), 128 (16 × 8) and 2560 (320 × 8), respectively, for the 100 yrextreme condition, 100 yr contour-line and long-term responsemethodologies. If one decides to use a more refined interpolationmesh, the number of analyses for the latter methodology is evenhigher.

9. Conclusions

Extreme response prediction of the dynamic tension is a criticalstep for mooring line design of every moored offshore structure.One common way to evaluate this response parameter is by meansof using 100 yr short-term environmental conditions. The maindrawback of this approach is that it does not take into account thedynamic behavior of the whole system (floater and lines) itself. Theideal methodology to predict extreme response of mooring lines isthe one based on the long-term response statistics.

In this paper a Monte Carlo-based approach combined with aninterpolation scheme to cope with four environmental parameterswas developed to obtain the complete long-term response of anFPSO in three distinct locations but submitted to same environ-mental conditions. A simplified long-term analysis based on theextreme environmental contours was also employed.

Using the complete long-term analysis as a reference, the resultsobtained in this study indicated the extreme dynamic tensioncan be under-predicted by either the 100 yr environmental condi-tion approach or the 100 yr environmental contour methodology.However, even using some interpolation scheme, the number ofmooring line nonlinear time-domain simulations can become verylarge for the long-term response approach. Then, this approachshould be used either for code calibration purposes or as a finaldesign check for any innovative mooring system. Another possibil-ity, not pursued in this work, is to perform the long-term analysiscombining the proposed 4-D interpolation scheme with some other

simplified approach to compute the mooring tension instead ofusing time-demanding finite element-based numerical models.
Page 8: Long-term response analysis of FPSO mooring systems

3 plied O

A

tsletprTf

82 A.O. Vázquez-Hernández et al. / Ap

ppendix A.

This section shows the methodology used to obtain the long-erm response distribution through a multidimensional integrationcheme (MIS). With this process it is possible to take into account aarge number of sea states (combinations of environmental param-ters Hs, Tp, V, C and �). To run the MIS process it is necessaryo previously generate the short-term response (˛G(·) and u(·)arameters) database for a number of discrete points of the envi-onmental parameters as mentioned in Section 2 of this paper.hen, as shown in Fig. A.1, the steps of the MIS process are asollows:

(a) For each of the discrete incidence directions considered, i.e.,N� = 8 (N, NE, E, SE, S, SW, W, NW) NS samples of sea states aresimulated from the joint probability distribution of the envi-

ronmental parameters.

(b) For each nj sea state, by means of Monte Carlo simulationsfour independent random samples Uj = ( uj

1 uj2 uj

3 uj4

) uni-formly distributed between 0 and 1 are generated. These

Fig. A.1. Monte Carlo-based mooring l

cean Research 33 (2011) 375– 383

random variables will be used to obtain the environmentalparameters Hs, Tp, V and C through the inverse transformmethod [23] as shown in Fig. A.1, i.e.:

hs = F−1Hs

(u1)

tp = F−1Tp|Hs

(u2|hs)

v = F−1V |Hs

(u3|hs)

c = F−1C (u4)

(A.1)

(c) Once the environmental parameters set (hs, tp, v and c) isobtained for the ith sea state, the short-term extreme response(line top tension) parameters (˛G(·) and u(·)) are obtained basedon the initial discrete mesh of points and using the linear

interpolation/extrapolation functions found in connection withfinite element method [4];

(d) With the short-term extreme response parameters (˛G(·) andu(·)) it is possible to obtain the probability distribution of

ine long-term response analysis.

Page 9: Long-term response analysis of FPSO mooring systems

lied O

(

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[25] Videiro PM, Moan T. Efficient evaluation of long-term distributions. In: Pro-

A.O. Vázquez-Hernández et al. / App

response conditional on the ith incidence direction �i, i.e.,

FSE |�i(s) =

Ns∑j=1

FSE |�i,hsj,tpj,vj,cj(s|˛G(hsj, tpj, vj,

cj), u(hsj, tpj, vj, cj));

(e) When the process is finished for one incidence direction, it isnecessary to perform the same procedure for the other inci-dence directions.

(f) To determine the unconditional long-term extreme responsedistribution for the mooring line top tension it is necessaryto consider the frequency of occurrence (p�) of each incidence

direction, i.e., FSE(s) =

Ns∑j=1

F SE|�i(s)p�i

(g) Finally, the characteristic value SkE is derived according Eq. (8)

from the distribution obtained in the previous step;h) This process is repeated for each mooring line belonging to the

whole mooring system.

eferences

[1] DNV. Offshore Standard DNV-OS-E301, Position Mooring. Det Norske Veritas;2004.

[2] Haver S, Kleiven G. Environmental contour lines for design – why andwhen? Proceedings of the OMAE conference. Paper OMAE2004-51157. 2004.p. 337–45.

[3] Winterstein SR, Ude TC, Cornell CA, Bjerager P, Haver S. Environmental parame-ters for extreme response: inverse form with omission factors. In: Proceedingsof the International Conference on Structural Safety and Reliability (ICOSSAR).1993.

[4] Bathe KJ. Finite element procedures. Prentice Hall; 1995.[5] Bitner-Gregersen EM, Haver S. Joint environmental model for reliability cal-

culations. In: Proceedings of 1st International Offshore and Polar EngineeringConference (ISOPE), vol. I. 1991. p. 246–53.

[6] API. RP-2FPS: recommended practice for planning, designing, and constructingfloating production systems. American Petroleum Institute; 2001.

[7] Ormberg H, Larsen K. Coupled analysis of floater motion and moor-

ing dynamics for a turret-moored ship. Applied Ocean Research 1998;20:55–67.

[8] Nishimoto K, Fucatu CH, Masetti IQ. Dynasim – a time domain simulatorof anchored FPSO. Journal of Offshore Mechanics and Arctic Engineering2002;124:203–11.

[

cean Research 33 (2011) 375– 383 383

[9] Low YM, Langley RS. Time and frequency domain coupled analysis ofdeepwater floating production systems. Applied Ocean Research 2006;28:371–85.

10] Low YM, Langley RS. A hybrid time/frequency domain approach for effi-cient coupled analysis of vessel/mooring/riser dynamics. Ocean Engineering2008;35:433–46.

11] Mazaheri S, Downie MJ. Response-based method for determining theextreme behaviour of floating offshore platforms. Ocean Engineering 2005;32:363–93.

12] Guarize R, Matos NAF, Sagrilo LVS, Lima ECP. Neural networks in the dynamicresponse analysis of slender marine structures. Applied Ocean Research2007;29:191–8.

13] Spanos DP, Ghosh R, Finn DL. Coupled analysis of a spar structure: Monte Carloand statistical linearization solutions. Journal of Offshore Mechanic and ArcticEngineering 2005;127:11–6.

14] Karlsen HC, Naess A. Statistical response predictions for a nonlinearlymoored large volume structure in random seas. Applied Ocean Research2005;27:273–80.

15] Naess A, Gaidai O, Teigen PS. Extreme response prediction for nonlinear float-ing offshore structures by Monte Carlo simulation. Applied Ocean Research2007;29:221–30.

16] Morton D, Bowers J. Extreme value analysis in a multivariate offshore environ-ment. Applied Ocean Research 1996;18:303–17.

17] Grime AJ, Langley RS. Lifetime reliability based design of an offshore vesselmooring. Applied Ocean Research 2008;30:221–34.

18] Mourelle MM, Gonzalez EC, Jacob BP. ANFLEX computational system for flexibleand rigid riser analysis. In: Carneiro FLLB, et al., editors. Proceedings of 9th Inter-national Symposium Offshore Engineering. Chichester/New York: John Wiley& Sons; 1995. p. 441–58.

19] Vázquez-Hernández AO. Methodology for the calibration of partial safety fac-tors for mooring lines projects based on reliability. D.Sc. thesis. Rio de Janeiro,Brazil: Federal University of Rio de Janeiro, Civil Engineering Department,COPPE/UFRJ; 2004 [in Portuguese].

20] Vázquez-Hernández AO, Ellwanger GB, Sagrilo LVS. Reliability-based com-parative study for mooring lines design criteria. Applied Ocean Research2006;28:398–406.

21] Zurita BIG. Statistical analysis of extreme values of Gaussian and non-Gaussiantemporal series. M.Sc. thesis. Rio de Janeiro, Brazil: Civil Engineering Depart-ment, COPPE/UFRJ; 1999 [in Portuguese].

22] Ang AH, Tang WH. Probability concepts in engineering planning and design.John Wiley & Sons; 1984.

23] Melchers RE. Structural reliability analysis and prediction. 2nd ed. John Wiley& Sons; 2001.

24] Videiro PM. Reliability-based design of marine structures. Ph.D. thesis. Trond-heim, Norway: Department of Marine Structures, NTNU; 1998.

ceedings of the OMAE conference. Paper OMAE 99-6014. 1999.26] Horte T, Lie H, Mathisen J. Calibration of an ultimate limit state for moor-

ing lines. In: Proceedings of the OMAE conference. Paper OMAE98-1457.1998.