dynamic behavior of jack-up isope10

ISOPE 2010 Conference

1

Dynamic Response of Jackup Units Re-evaluation of

SNAME 5-5A Four Methods

Beijing, China24 June 2010

Xi Ying Zhang, Zhi Ping Cheng, Jer-Fang Wu and Chee Chow Kei

ABS

2

Main Contents

Background Multi-DOF method Four methods to predict MPME and DAF Random seed effect Concluding remarks

3

Background

DAF stands for dynamic amplification factor

The natural periods of jackup is 5-15s. It may be at or close to wave excitation period, hence the responses of jackup units may be amplified significantly.

Focus will be on DAF for base shear (BS) and overturning moment (OTM)

responseStaticresponseDynamicDAF =

4

1<Ω

ع -Damping ratio

n

w

ωω

=Ω

DAF vs Ω in SDOF System

For a SDOF system vibrating in sinusoidal waves, DAF can be obtained as follows

DAF = 1/(1-Ω2)2+(2 عΩ)2½

5

Dynamic Effect on Jackup

Dynamic effect needs to be considered (SNAME) when: 0.9 T w ≤ Tn ≤ 1.1 T w ;or

DAF > 1.05

Influence of dynamic effect: Magnify the hydrodynamic load

Lead to greater sway, then more P-∆ effect

SDOF model with deterministic excitation Simple but inaccurate (mainly for estimation of DAF)

SDOF model with random excitation Simple with non-Gaussian effects, not prevalent

MDOF model with deterministic excitation No non-Gaussian effect, widely use for jacket design

MDOF model with random excitation Most complicated one, widely use for jackup design

6

SNAME Dynamic Analysis Methods

7

Areas of Investigation

8

Main Contents


9

PHASE 1

Constructanalysis(equivalent) model withrespect tothe P-∆ effect

PHASE 2

Generate a random wave surface history and check the validity

PHASE 3

Carry out the non-linear dynamic analysis in time domain with the created random wave surface history

PHASE 4

Post process the simulation data to get the most probable maximum extreme (MPME) and DAF

Procedures to Obtain MPME

Leg stiffness Cross sectional area

Moment of inertia

Shear area

Torsional moment of inertia

P- effect-negative virtual spring Pg = weight of hull + leg above hull.

L = vertical distance from spudcan to hull CoG

Model the mass

Hydrodynamic loading

Damping

Calibrate the combined model with detailed model

10

Construct Equivalent Model

LPg /−∆

Build detailed leg model, fix it at 4 bay below lower guide

Apply unit load (6 DOF) on the spudcan end and obtained displacements

Compute the leg stiffness properties of detailed leg using unit load and corresponding displacements

11

The simplified leg can savecomputation time, whileloosing accuracy within areasonable range

Equivalent Leg Model

12

li = length of member iS = length of one bayDi = diameter of member iCDi = CD of member i

sDlDCC

e

iiDiiiiDe

5.1222 ]sincos[sin αββ ⋅+= slDD iie /)( 2∑=

∑⋅=⋅ MeieeMe CAAC

sAlACC

e

iiMiiiiMei ⋅−++= )]1)(sincos(sin1[ 222 αββ

slA

A iie∑=

Equivalent Leg Model

The hydrodynamic properties of the equivalent leg can be derived by empirical formula:

13

Natural Periods of Jackup Unit

Pierson-Moskowitz spectrum is used to generate the random wave surface profile

Check validity for sea state used Satisfy the Gaussianity of the sea surface

• Correct mean value • Standard deviation within Hs/4 plus minus 1%• -0.03 < skewness < 0.03• 2.9 < kurtosis < 3.1• Maximum crest elevation = (Hs/4)[2xln(N)]0.5 error within minus 5% to

plus 7.5%; N is number of cycle

Other miscellaneous requirement:• Number of wave components > 200• Component of division with equal energy, mean smaller pace at peak

frequency• First 100 second to be removed to get rid of transient effect• Time step < min Tz / 20 , Tn / 20

14

Random Sea States

15

Sample of sea surface history Wave spectrum type – PM

Wave height = 26.0 ft

Dom period =14.1s

Random Sea States

16

Main Contents


17

Prediction of MPME

Most probable maximum extreme (MPME) has 63% chance of being exceeded by the maximum of any three hour storm This level is reached by one in thousand peaks on average

Random seed is used to define the random phase angle of each wave components that are combined to create a simulated time history

There are 4 methods used for prediction of MPME D/I method: 60 minutes, 3 runs with different Cd, Cm, (study used

5 random seeds); (SNAME recommends one random seed)

Weibull method: 60 minutes, 5 random seeds

Gumbel method: 180 minutes, 10 random seeds

W/J method: 180 minutes, (study used 10 random seeds); (SNAME recommends one random seed)

18

With the obtained σRD, μRD ,σRS, μRS, σRI, μRI, DAF can be derived as below

μRS μRDμRIσRS σRD

DAF1= σRD / σRS

ρR= (σRD2 - σRS

2 - σRI2) / (2σRS σRI)CRI=[2ln(1000)]0.5 = 3.7 CRS to be determined

(MPMRD)2= (CRSσRS)2 + ( CRDσRD)2 +2* ρR(CRSσRS) ( CRDσRD) MPMRS= CRSσRS

DAF2= MPMRD / MPMRSCRD= MPMRD / σRD

MPMERS= MPMRS + μRSMPMERD= MPMRD + μRD

DAF3= MPMERD / MPMERS

Drag/Inertial Parameter Method

19

It is assumed that a standard process can be calculated by splitting it into two parts (static and inertial) with a correlation between the two

)()(2)()()( 222IneStaRIneStaDyn MPMMPMMPMMPMMPM ⋅++= ρ

MPMD

MPMSt

MPMI

n

θ

Perform quasi-static time history analysis to get RS(t)

Perform dynamic time history analysis to get RD(t)

Get inertial response from RI(t)= RD(t)- RS(t)

Get σRS, μRS by statistical analysis

Get σRI, μRI by statistical analysis

Get σRD, μRD by statistical analysis

The quasi-static analysis is achieved by simply set massand damping zero; while the dynamic one account them fully

Time domain analysis procedure

Drag/Inertial Parameter Method

Weibull distribution is fitted against the maxima values

F is the probability of non-exceedance

α = scaling; β = slope; γ = shift

Nonlinear data fitting, Levenber-Marquardt method, is to be used to produce the value of α, β and γ

MPM is the value of R when

MPME value is obtained by MPM + µ

Repeat above procedure for all response parameters

20

durationsimulationhourN

RF

311),,,(

max

−=γβα

])(exp[1),,,( β

αγγβα −

−−=RRF

Weibull Fitting Method

21

Curve Fitting

0.0000

0.2000

0.4000

0.6000

0.8000

1.0000

1.2000

0.0000 0.5000 1.0000 1.5000 2.0000 2.5000 3.0000 3.5000 4.0000 4.5000 5.0000

Original DataPredicted Data

Standardized Response

Cum

ulat

ive

Den

sity

Weibull Fitting

22

SNAME 5-5A suggests removing bottom 20% of the observed cycles in curve fitting. How about the top range?

The range of 20%-100% or 20%-98% generates more consistent DAFs with smaller standard deviation

OTM BS OTM BS OTM BS OTM BS OTM BS1 2.9582 1.9946 2.9598 1.9953 2.9323 1.9887 2.8528 1.9250 2.7890 1.8777

19 3.0354 2.0649 3.0415 2.0688 3.0603 2.0889 3.1177 2.1683 3.3065 2.333822 2.9796 2.0956 2.9867 2.1020 2.9787 2.1084 3.0130 2.1599 3.0051 2.281843 2.7440 1.7579 2.7402 1.7539 2.7270 1.7402 2.7043 1.7164 2.6190 1.641966 3.0055 2.0277 3.0080 2.0296 3.0221 2.0428 3.0587 2.0591 3.0611 2.041573 2.7331 1.8241 2.7337 1.8234 2.7204 1.8188 2.6830 1.7851 2.5946 1.730680 2.5810 1.9919 2.5920 2.0012 2.6146 2.0434 2.6569 2.1395 2.7169 2.223199 2.6872 1.8557 2.6839 1.8534 2.6443 1.8252 2.5853 1.7474 2.4472 1.6020

280 2.8175 1.7076 2.8274 1.7100 2.8635 1.7171 2.9892 1.7272 3.1290 1.6948320 3.3647 2.1996 3.3674 2.2010 3.3665 2.2013 3.3413 2.1905 3.3534 2.1925AVE 2.8906 1.9520 2.8940 1.9539 2.8930 1.9575 2.9002 1.9619 2.9022 1.9620SD 0.2262 0.1588 0.2263 0.1607 0.2298 0.1689 0.2435 0.2025 0.3133 0.2852

20%-85%Seed

20%-100% 20%-98% 20%-95% 20%-90%

Range of Data for Weibull Method

23

Extract maximum (and minimum) value for each of ten 3-hour response signal

A Gumbel distribution is fitted via 10 maxima/minima. Both maximum likelihood method or method of moment (preferable) can yield ψ and κ

F3h(MPME)= 1-0.63 = 0.37

Because the MPME in three hours will have probability of exceeding 0.63

The MPME then can be calculated by:

A similar procedure will generate the quasi-static MPME and so the DAF of overturning moment and base shear can be obtained

[ ] [ ] ψκψκψ ≈−−=−−= 37.0lnln)(lnln 3 MPMEFMPME h

)]exp(exp[)(3

κψ−

−−=xxF h

Gumbel Fitting Method

24

Moment fitting MLE Diff(%) Moment fitting MLE Diff(%)Dynamic OTM 526243.76 524183.16 0.39% 26336.50 35679.46 26.19%Static OTM 190671.82 191088.95 0.22% 12493.89 11162.27 11.93%Dynamic BS 5266.01 5259.65 0.12% 241.29 269.92 10.61%Static BS 2659.70 2658.73 0.04% 196.36 202.67 3.12%

DAF for OTM 2.760 2.743 0.61%DAF for BS 1.980 1.978 0.08%

Items ψ k

MPME is only related to , hence a moment fitting solution can be used for Gumbel fitting to replace the maximum likelihood method, which will simplify the calculation procedure

[ ] [ ] ψκψκψ ≈−−=−−= 37.0lnln)(lnln 3 MPMEFMPME h

ψ

Gumbel Fitting Method

25

It is assumed that a non-Gaussian process can be expressed as polynomial of zero mean, narrow band Gaussian process

The same relation exist between MPME of the 2 process. Since MPME of Gaussian process U is known, the MPME of R can be found if coefficient C0 ,C1 , C2 and C3 are determined.

The C1 , C2 and C3 can be obtained by equations below:

σ2 = C12 + 6C1C3 + 2C2

2 + 15C32

σ3α3 = C2(6C12 + 8C2

2 + 72C1C3 + 270C32)

σ4α4 = 60C24 + 3C1

4 + 10395C34 + 60C1

2C22 + 4500C2

2C32

+ 630C12C3

2 + 936C1C22C3 +3780C1C3

3 + 60C13C3

The following statistical quantities needed:

µ mean of the process σ standard deviationα3 skewness α4 kurtosis

R(U) = C0 + C1U + C2U2 +C3U3

Winterstein/Jensen Method

26

It is assumed that a non-Gaussian process can be expressed as polynomial of zero mean, narrow band Gaussian process

Newton-Raphson method could be utilized to solve the set of equations

The initial guess value can be:

c1 = σk(1-3h4)

c2 = σkh3

c3 = σkh4

The C0 , can be figured out by the MPME value is

RMPME = c0 + c1U1 + c2U2 + c3U3

R(U) = C0 + C1U + C2U2 +C3U3

2/124

23

44

433

]621[

18/]1)3(5.11[

])3(5.1124/[

−++=

−−+=

−++=

hhk

h

h

α

αα

30 hkc ⋅⋅−= σµ

Winterstein/Jensen Method

27

Main Contents


28

Item Rig 1 Rig 2

Length overall (ft) 210 240

Breadth overall (ft) 200 208

Water depth (ft) 300 350

Weight (kips) 25,000 27,000

Wave height (ft) 50 48

Wave period (s) 15 14

Current (knots) 1 1

Configuration of Two Rigs

Comparisons of Natural Periods

29

Mode

Rig 1 Rig 2

DetailModel

(s)

CombinedModel

(s)

Diff. (%)

DetailModel

(s)

CombinedModel

(s)

Diff. (%)

1 12.49 12.73 1.90 11.12 11.32 1.84

2 11.98 12.18 1.66 10.95 11.15 1.85

3 11.46 11.65 1.68 10.22 10.36 1.41

30

28%32%

Findings Both W/J and Weibull methods have significant variance in DAF

SNAME 5-5A recommends:

• For Weibull method, run number ≥ 5;

• For W/J method, run number = 1

• SNAME recommended run number may not be sufficient

OTM BS OTM BS43 0 2.6775 1.8827 2.7402 1.753999 0 2.8156 1.9379 2.6839 1.853480 0 2.3215 1.6221 2.5920 2.0012

320 0 3.1979 2.1559 3.3674 2.201073 0 2.5662 1.7856 2.7337 1.823466 0 2.6631 1.8345 3.0080 2.029622 0 2.8239 1.9093 2.9867 2.10201 0 2.9672 2.0371 2.9598 1.9953

19 0 2.8881 1.9426 3.0415 2.0688280 0 2.6590 1.8702 2.8274 1.7100

AVE 2.7580 1.8978 2.8940 1.9539SD 0.2391 0.1427 0.2263 0.1607

W/J METHOD WEIBULL METHODDEGREE

Statistical Properity

seed

Random Seed Effect: Rig 1

31

Findings Compared with W/J method, drag/inertia method is not

sensitive to the selection of random seeds and DAFs are pretty stable

Why?

2.0

2.5

3.0

3.5

1 2 3 4 5

Random Seed

DA

F fo

r Ove

rtur

ning

Mom

ent

DI 0 DegreeDI 30 DegreeDI 60 DegreeW/J 0 DegreeW/J 30 DegreeW/J 60 Degree

1.5

2.0

2.5

1 2 3 4 5

Random Seed

DA

F fo

r Bas

e S

hear


32

Because:

Drag/inertia method is only related to mean value and standard deviation (SD)

W/J method is related to mean value, standard deviation (SD), skewness and kurtosis. It can be seen that the skewness and kurtosis have not stabilized in the 3 hour run. Therefore a much longer duration would be required to obtain stable results for W/J method.

Dynamic overturning moment

0.00

1.00

2.00

3.00

4.00

5.00

6.00

1 2 3 4 5

Random Seed

Mean/10^4SD/10^5Skewness*100Kurtosis

Static Overturning Moment

0.001.002.003.004.005.006.007.008.00

1 2 3 4 5

Random Seed

Mean/10^4SD/10^5Skewness*100Kurtosis


33

Findings Five 1-hour runs (SNAME) may not yield comparable results to

10 3-hour runs

Among 10 3-hour runs, the difference between maximum and minimum DAF from 5 seeds is not negligible

0

0.5

1

1.5

2

2.5

3

3.5

DAF

s

3 hour (10 seeds)1 hour (5 seeds)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

DAF

s

10 seeds5 seeds to max DAF5 seeds to min DAF

OTM OTMBS BS

Weibull Method: Rig 1

34

Main Contents


35

Concluding Remarks

D/I Method

WeibullMethod

GumbelMethod

W/J Method

Running Period and

Number

60 minutes, 3 runs with

different Cd, Cm

60 minutes, runs

number ≥ 5

180 minutes, runs number

≥10

180 min, runs number

=1 (may not be sufficient)

Effect of Random Seed not sensitive sensitive sensitive

CharacteristicsWeak in

theory, but consistent

Sensitive to range of data

for fitting (20%-100% / 20%-98%),

DAFs scatter

Time consuming, but

reliable and stable,

moment fitting solution used

to replace MLM

Sensitive to random seed

selection, unstable

36

www.eagle.org

dynamic behavior of jack-up isope10

Documents