Long-term Response of Offshore Structures: Some Practical ......NATAF-BASED PDF MODEL •...

Long-term Response of Offshore

Structures: Some Practical Aspects

A. Papaleo, F.J.M. Sousa, E.C.P. Lima and

L.V.S. Sagrilo

COPPE

Federal University of Rio de Janeiro

Workshop: “Statistical models of the metocean environment for

engineering uses”

30/September/2013

Brest-France

INTRODUCTION

• Presentation Scope:

• Present two topics of a research work on response-based design of offshore structures;

Joint probabilty distribution model for metocean data;

Response-based methodology to define the design metocean conditions;

MOTIVATION

• Design of risers and mooring lines for turret- and spread-moored floating systems:

dynamic response depends on the intensity and directionality of the environmental actions

consistency of using the traditional 100-yr environmental conditions in the design ??

long-term response analysis is the best design methodology

RESPONSE-BASED ANALYSIS

• Long-term distribution of the response peaks

FR|S=s,d(r|s,di) – short-term distribution of the response peaks (r|s,d) – mean short-term frequency of response peaks p(di) – discrete probability distribution of the vessel draft along time (full,ballast,half-loaded, etc.) fS(s) – joint probability distribution function (pdf) of metocean data related to wave, wind and current

• Our reasearch:

obtain an fs(s) for a large number of metocean parameters solve and make practical use of FR(r)

dN

1i

iid,R

i

iR dpdf)d,r(F

d

d,rF sss

s

sSsS

METOCEAN JOINT PDF MODEL

• Availability of simultaneous data of waves, wind and current Conditional Modelling Approach (CMA)

difficult to be used for more than two metocean parameters

Model based on the Nataf’s Transformation

marginal distributions of metocean parameters and their correlation coefficients matrix

unlimited number of metocean parameters

ENVIRONMENTAL PARAMETERS

• The model developed deals with 10 metocean parameters:

Hsws = S1 wind sea significant wave height, Tpws = S2 wind sea wave spectral peak period ws = S3 wind sea direction Hsss = S4 swell significant wave height Tpss = S5 swell spectral peak period ws = S6 swell direction Vw = S7 mean wind velocity w = S8 wind direction Vc = S9 superficial current velocity and c = S10 superficial current direction

NATAF-BASED PDF MODEL

• Nataf-based joint pdf model (Der Kiureghian and Liu, 1986):

fSi(si) marginal PDF of the i

th variable FSi(si) marginal CPF of the i

th variable -1(.) inverse of the standard Gaussian CPF (.) PDF of the standard Gaussian distribution 10(.) joint PDF of ten correlated standard Gaussian variables “Nataf correlation coefficients” matrix

ρsS ,sF,sF

sF

sf

f 10S1

1S

1

1010

1i

iS

1

10

1i

iS

101

i

i

NATAF-BASED MODEL

• Nataf correlation coefficients

Ni,j Nataf correlation coefficient li,j linear correlation coefficient between Si and Sj Si,Sj mean values of Si and Sj Si,Sj standard deviations of Si and Sj

• Nataf model is a standard procedure for LINEAR variables !

• How to deal with the angular variables ???

for an angular variable 0 = 360 !!

21Nj,i212

S

S2

1

S

S

S1

1

Sl

j,i dydy,y,yyFyF

j

jj

i

ii

CIRCULAR STATISTICS

• Circular variable sample (Fisher,1993) :

sample mean:

sample standard deviation:

where C, S and R are given by

N21 ,, θ

R

Carccos

R

Sarcsin

5.0

N

Rlog2s

N

1iicosC

N

1iisinS 22 SCR

MARGINAL DISTRIBUTION OF A CIRCULAR VARIABLE

• Wrapped Normal distribution (unimodal)

• Equivalent Normal on the real line

1p

pc pcoss212

1f

2

deviation standard c

mean c

ircularss

ircular

2

l exp2

1f

2mod

deviation standardlinear slog2

meanlinear

alReCircle

MARGINAL DISTRIBUTION OF A CIRCULAR VARIABLE

• Wrapped Normal distribution (multimodal)

• Multimodal Normal on the real line

iNm

1i 1p

i

p

i pcoss212

1f

2

iNm

1i

2

i

iexp2

1f

modes of numberNm

0.1Nm

1i

i

CORRELATION INVOLVING CIRCULAR VARIABLES

• Sample circular correlation between = (1, 2, ... , N) and

= (1, 2,..., N) [Fisher and Lee, 1983]:

A to G: functions of i and j

• A “measure” of sample linear-circular correlation between

= (1, 2, ... , N) and X = (x1, x2,..., xN) [Mardia, 1976]:

r12, r13, r23: functions of i and xj

222222c

,

HGNFEN

CDAB4

11 c,

223

231312

2

13

2

122cl

X,r1

rrr2rr

• Correlation coefficients when circular variables are

represented on the real line

few theoretical solutions available numerical algorithms to solve the problem (Sagrilo et al.,

2011)

• Practical results: metocean database for a location in Campos Basin

offshore Brazil metocean database of 4000 measurements at each 3-h measurements made by PETROBRAS Research Center

CORRELATION INVOLVING CIRCULAR VARIABLES

FITTED MARGINAL DISTRIBUTIONS FOR LINEAR VARIABLES

0 2 4 6 8

Normalized Value - Hs/Hs

0

0.2

0.4

0.6

0.8

1

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n

Wind Sea - Sig. Wave Height

Data

Lognormal

2 4 6 8 10

Normalized value - Tp/Tp

0

0.1

0.2

0.3

0.4

0.5

Pro

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Wind Sea - Peak Period

Data

Weibull (3P)

Wind Sea

Significant wave height - Lognormal Spectral peak period – Weibull 3P


0 2 4 6 8

Normalized Value - Hs/Hs

0

0.2

0.4

0.6

Pro

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n

Swell Sea - Sig. Wave Height

Data

Weibull (2P)

2 4 6 8 10

Normalized value - Tp/Tp

0

0.04

0.08

0.12

0.16

0.2

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Swell Sea - Peak Period

Data

Weibull (3P)

Swell Sea

Significant wave height – Weibull 2P Spectral peak period – Weibull 3P


0 2 4 6 8

Normalized Value - Vc/Vc

0

0.4

0.8

1.2

1.6

2

Pro

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Current Velocity

Data

Weibull (2P)

0 2 4 6 8

Normalized Value - Vw/Vw

0

0.04

0.08

0.12

0.16

Pro

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ction

Wind Velocity

Data

Truncated Weibull (3P)

Wind velocity

– Truncated Weibull 3P

Superficial current velocity

– Weibull 2P

FITTED MARGINAL DISTRIBUTIONS FOR ANGULAR VARIABLES

Wind sea direction

Mixture of Wrapped Normals (3 modes)

Wind direction


-4 -2 0 2 4

Direction (rad)

0

0.2

0.4

0.6

Pro

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n

Wind Sea Direction

Data

3 Wrapped Normals

-4 -2 0 2 4

Direction (rad)

0

0.2

0.4

0.6

Pro

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Wind Direction

Data

3 Wrapped Normals

FITTED MARGINAL DISTRIBUTIONS FOR ANGULAR VARIABLES

Swell direction


Current direction


-4 -2 0 2 4

Direction (rad)

0

0.2

0.4

0.6

Pro

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Swell Sea Direction

Data

2 Wrapped Normals

-4 -2 0 2 4

Direction (rad)

0

0.2

0.4

0.6

0.8

1

Pro

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Current Direction

Data

2 Wrapped Normals

CORRELATIONS

Parameter Hsws Tpws Hsss Tpss Vw Vc

Hsws 1.00 0.68 -0.20 0.48 0.42 -0.05

Tpws 0.68 1.00 -0.24 0.69 -0.07 -0.04

Hsss -0.20 -0.24 1.00 -0.07 -0.14 -0.10

Tpss 0.48 0.69 -0.07 1.00 -0.07 -0.06

Vw 0.42 -0.07 -0.14 -0.07 1.00 0.10

Vc -0.05 -0.04 -0.10 -0.06 0.10 1.00

Data correlation coefficients - linear variables

CORRELATIONS

Data correlation coefficients – circular variables

Parameter ws ss w c

ws 1.00 0.12 0.43 0.09

ss 0.12 1.00 0.10 -0.08

w 0.43 0.10 1.00 0.11

c 0.09 -0.08 0.11 1.00

Parameter ws ss w c

ws 1.00 0.28 0.79 0.34

ss 0.28 1.00 0.38 -0.25

w 0.79 0.38 1.00 0.50

c 0.34 -0.25 0.50 1.00

Data on the circle

Data on the real line

CORRELATIONS

Data correlation coefficients – linear vs. circular variables

Angular data on the circle

All data on the real line


ws 0.14 0.17 -0.33 0.07 -0.35 0.09

ss 0.25 0.44 -0.27 0.64 -0.26 0.01

w 0.13 0.07 0.34 0.07 -0.43 -0.12

c 0.13 0.05 0.08 0.05 0.23 0.25


ws 0.18 0.22 0.45 0.10 -0.46 0.12

ss 0.27 0.47 -0.30 0.68 -0.29 0.01

w 0.24 0.12 0.60 0.13 -0.77 -0.25

c 0.16 0.06 0.12 0.06 0.29 0.32

SOME BI-DIMENSIONAL JOINT DISTRIBUTIOS

Hs x Tp wind sea

Hs wind sea x Wind velocity

Measured data Fitted joint pdf model

A PRACTICAL APPLICATION USING THE JOINT PROBABILITY MODEL

Long-term heading angle distribution of a half-loaded turret-moored FPSO (320 kDWT tanker)

0 60 120 180 240 300 360

Heading angle (degres)

0

0.2

0.4

0.6

0.8

1

Pro

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en

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un

ction

FPSO heading distribution

Original measured data ( 4000 sea states)

Data simulated JPM ( 50000 set points)

NATAF-BASED JOINT PDF MODEL SUMMARY

•Advantages

a simple model that can represent a large number of metocean parameters;

once the marginal distributions and correlation matrix are stablished, any low order joint pdf (nN) is easily obtained

“taylor-made” model for reliability analysis and Monte Carlo-based numerical simulations

• Cautions

an approximate model based on a weak measure of statistical dependency – the correlation coefficient

sometimes the correlation matrix may become numerically singular

• On going work: improve the treatment of correlations involving angular variables

LONG-TERM RESPONSE-BASED ANALYSIS

• Proposal: how to use response-based analysis to define short-term metocean data to be employed in the design of risers and mooring lines (storm wave design methodology)

define design metocean data associated to extreme responses of the structure instead of those related to extreme environmental events (e.g., 100-yr events)

LONG-TERM RESPONSE-BASED ANALYSIS

• Some remarks on long-term analysis:

full long-term response analyses of risers and mooring lines using FEM-based time-domain codes are very time-consuming ;

hardly used in the design practice;

an alternative is to perform long-term analyses using analytical and frequency-domain methods;

LONG-TERM RESPONSE EVALUATION

•Long-term integral evaluation

For more than two metocean data in S : Monte Carlo Simulation

Ns – number of Monte Carlo samples

sk – kth generated metocean data sample generated from fS(s)

Ns

1k

ikd,R

i

ik

id,R

i

i

dN

1i

i

Simulation Carlo Monteby Solved

id,R

i

iR

Ns

)d,r(Fd

d,

df)d,r(Fd

d,

dpdf)d,r(Fd

d,rF

ss

ssss

ssss

sS

sSsS

sSsS

LONG-TERM RESPONSE EVALUATION

• Response parameter: line top tension (Sousa et. all, OMAE 2012)

De-coupled analysis for each MCS generated set of metocean data (S = sk)

First step: Static analysis of the system by program APROA

Second step: short-term analysis by program FX_TENSION

aproximate frequency domain stochastic analysis

short-term distribution FR|S=s,d(r|s,di ): Hermite/Winterstein or Rayleigh

Long-term extreme response evaluation: program FX_LTERM

LONG-TERM RESPONSE ANALYSIS

•Turret-FPSO in 1300m water depth: 8” oil production flexible riser

60000 samples of metocean data for each Monte Carlo Simulation

5-6 distinct MC simulations (or 300000-360000 samples /3-4h PC)

100-yr response: mean + 2 x standard devations

EQUIVALENT DESIGN CONDITIONS

• Short-term design conditions (design metocean data set)

taken from MC simulations as the set of metocean parameters whose short-term extreme response (3-h return period) is numerically VERY CLOSE to the long-term N-yr response

this “equivalent” metocean condition S= si is used to perform more refined dynamic simulations for design check verifications

EQUIVALENT DESIGN CONDITIONS

• Short-term design wave approach

design metocean data : response-based instead extreme enviromental events (e.g.,100-yr metocean design conditions)

developed case by case, i.e., in the beggining of the design process a simplified long-term response analysis is performed to establish a “Response-based Metocean Technical Specification”

Plataform – RR 45 Design Metocean Data

Structure Limit State Wind sea Swell Current Wind

Flex Riser 8” Oil Top tension Hs, Tp, ws Hs, Tp, ss Vc, Vv,

Curvature Radius TDP

... ... ... ...

Flex Riser 6” Gas Top tension ... ... ... ...

Curvature Radius TDP

... ... ... ...

Line # 1 Top Tension ... ... ... ...

REMARKS ON THE DESIGN METOCEAN DATA BASED ON LONG-TERM ANALYSIS

• Advantages

response-based methodology;

less design conditions than those analysed today;

• Dificulties

metocean people and structural designers have to work togheter

it depends on the availability of simplified models for long-term analysis

floating system changes: update metocean data for design

• On going work

analytical/simplified time-domain models for representing bending moment behaviour along a riser – ther design limit states such as bending radius, section utilization factors for SCRs, etc.

Merci beaucoup! Thank you! Obrigado!

[email protected]

Conditonal Modelling Approach

Example - joint model for Hs and Tz

hs,hsfhstzfhsfhsf

hstzfhsftz,hsf

TzTzTzHsTz

HsHs

HsTzHsTz,Hs

data from obtained constants

d,c,b,a

hsdexpchs

hsbahs

Tz

Tz

0 2 4 6 8 10

Hs (m)

0

4

8

12

16

Tz (

s)

Closed-form solution for dynamic tension

• FX_TENSION: Short-term analytical line dynamic tension analysis Offset from static analysis

Motion RAOs are transfered to line top

Aproximate tension RAO using a closed-form solution for the line dynamic tension by Aranha et al. (2001) Short-term distribution: Rayleigh (neglecting LF second order effects) or Hermite (LF+WF)

Results benchmarked with ANFLEX

Static equilibrium analysis

APROA - Numerical Nonlinear Code

F – external forces on the hull and lines: • Sea and swell wave drift forces; • Static wind forces • Current loading

K(X)X - Restoring forces (analytical catenary equations)

• risers • mooring lines

X - Equilibrium position • vessel heading • static offset

FXXK )(

Long-term analysis summary

FX_LTERM : Long-term Monte Carlo Simulation

Long-term Response of Offshore Structures: Some Practical ......NATAF-BASED PDF MODEL •...

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