Dynamic Response of Jackup Units Re-evaluation of SNAME 5-5A Four Methods

Transcription

1 ISOPE 2010 Conference Beijing, China 24 June 2010 Dynamic Response of Jackup Units Re-evaluation of SNAME 5-5A Four Methods Xi Ying Zhang, Zhi Ping Cheng, Jer-Fang Wu and Chee Chow Kei ABS 1

2 Main Contents Background Multi-DOF method Four methods to predict MPME and DAF Random seed effect Concluding remarks 2

3 Background DAF = Dynamic response Static response 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) 3

4 DAF vs Ω in SDOF System -Damping ratio ع Ω = ω ω w n Ω <1 For a SDOF system vibrating in sinusoidal waves, DAF can be obtained as follows DAF = 1/{(1-Ω 2 ) 2 +(2 ( Ωع 2 } ½ 4

5 Dynamic Effect on Jackup Dynamic effect needs to be considered (SNAME) when: 0.9 T w T n 1.1 T w ;or DAF > 1.05 Influence of dynamic effect: Magnify the hydrodynamic load Lead to greater sway, then more P- effect 5

6 SNAME Dynamic Analysis Methods 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

7 Areas of Investigation 7

8 Main Contents Background Multi-DOF method Four methods to predict MPME and DAF Random seed effect Concluding remarks 8

9 Procedures to Obtain MPME PHASE 4 PHASE 1 Construct analysis (equivalent) model with respect to the 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 Post process the simulation data to get the most probable maximum extreme (MPME) and DAF 9

10 Construct Equivalent Model Leg stiffness Cross sectional area Moment of inertia Shear area Torsional moment of inertia P- effect-negative virtual spring P g = weight of hull + leg above hull. P g / L L = vertical distance from spudcan to hull CoG Model the mass Hydrodynamic loading Damping Calibrate the combined model with detailed model 10

11 Equivalent Leg Model Build detailed leg model, fix it at 4 bay below lower guide The simplified leg can save computation time, while loosing accuracy within a reasonable range 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

12 Equivalent Leg Model The hydrodynamic properties of the equivalent leg can be derived by empirical formula: C C C De Me Mei Dili 2 = [sin βi + cos βi sin αi ] C D D l s Di e = ( i i ) / D s A e = A e C Mei = [1 + (sin β + cos β sin α )( C 1)] i i i A e e = Mi Al s i i Ail i A s e l i S D i = length of member i = length of one bay = diameter of member i C Di = C D of member i 12

13 Natural Periods of Jackup Unit 13

14 Random Sea States 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 H s /4 plus minus 1% < skewness < < kurtosis < 3.1 Maximum crest elevation = (H s /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 { T z / 20, T n / 20 } 14

15 Random Sea States Wave height = 26.0 ft Dom period =14.1s Sample of sea surface history Wave spectrum type PM 15

16 Main Contents Background Multi-DOF method Four methods to predict MPME and DAF Random seed effect Concluding remarks 16

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) 17

18 Drag/Inertial Parameter Method With the obtained σ RD, μ RD,σ RS, μ RS, σ RI, μ RI, DAF can be derived as below μ RS σ RS μ RI σ RD μ RD DAF1= σ RD / σ RS C ρ R = (σ 2 RD - σ 2 RI =[2ln(1000)] 0.5 = 3.7 RS - σ RI2 ) / (2 σ RS σ RI ) C RS to be determined (MPM RD ) 2 = (C RS σ RS ) 2 + ( C RD σ RD ) 2 +2* ρ R (C RS σ RS ) ( C RD σ RD ) MPM RS = C RS σ RS C RD = MPM RD / σ RD DAF2= MPM RD / MPM RS MPME RD = MPM RD + μ RD MPME RS = MPM RS + μ RS DAF3= MPME RD / MPME RS 18

19 Drag/Inertial Parameter Method 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 ( MPM Dyn ) = ( MPM ) + ( MPM ) + 2ρ ( MPM Sta Ine R Sta ) ( MPM Ine ) MPM I n θ MPM Time domain analysis procedure Perform quasi-static time history analysis to get R S (t) St MPM D The quasi-static analysis is achieved by simply set mass and damping zero; while the dynamic one account them fully Perform dynamic time history analysis to get R D (t) Get inertial response from R I (t)= R D (t)- R S (t) Get σ RS, μ RS by statistical analysis Get σ RI, μ RI by statistical analysis Get σ RD, μ RD by statistical analysis 19

20 Weibull Fitting Method Weibull distribution is fitted against the maxima values R γ β F( R, α, β, γ ) = 1 exp[ ( ) ] α 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 F( R, α, β, γ ) = 1 N max 1 3hour simulation duration MPME value is obtained by MPM + µ Repeat above procedure for all response parameters 20

21 Weibull Fitting Curve Fitting Cumulative Density Original Data Predicted Data Standardized Response 21

22 Range of Data for Weibull Method SNAME 5-5A suggests removing bottom 20% of the observed cycles in curve fitting. How about the top range? Seed 20%-100% 20%-98% 20%-95% 20%-90% 20%-85% OTM BS OTM BS OTM BS OTM BS OTM BS AVE SD The range of 20%-100% or 20%-98% generates more consistent DAFs with smaller standard deviation 22

23 Gumbel Fitting Method 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 3 F h ( x) x ψ = exp[ exp( )] κ F 3h (MPME)= = 0.37 Because the MPME in three hours will have probability of exceeding 0.63 The MPME then can be calculated by: 3h { ln[ F ( MPME) ]} = ψ κ ln ln[ 0. ] { } ψ MPME = ψ κ ln 37 A similar procedure will generate the quasi-static MPME and so the DAF of overturning moment and base shear can be obtained 23

24 Gumbel Fitting Method Items ψ k Moment fitting MLE Diff(%) Moment fitting MLE Diff(%) Dynamic OTM % % Static OTM % % Dynamic BS % % Static BS % % DAF for OTM % DAF for BS % MPME 3h { ln[ F ( MPME) ]} = ψ κ ln ln[ 0. ] { } ψ = ψ κ ln 37 MPME is only related to ψ, hence a moment fitting solution can be used for Gumbel fitting to replace the maximum likelihood method, which will simplify the calculation procedure 24

25 Winterstein/Jensen Method It is assumed that a non-gaussian process can be expressed as polynomial of zero mean, narrow band Gaussian process R(U) = C 0 + C 1 U + C 2 U 2 +C 3 U 3 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 C 0,C 1, C 2 and C 3 are determined. The C 1, C 2 and C 3 can be obtained by equations below: σ 2 = C C 1 C 3 + 2C C 2 3 σ 3 α 3 = C 2 (6C C C 1 C C 32 ) σ 4 α 4 = 60C C C C 12 C C 22 C C 12 C C 1 C 22 C C 1 C C 13 C 3 The following statistical quantities needed: µ mean of the process σ standard deviation α 3 skewness α 4 kurtosis 25

26 Winterstein/Jensen Method 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: c 1 = σk(1-3h 4 ) c 2 = σkh 3 c 3 = σkh 4 R(U) = C 0 + C 1 U + C 2 U 2 +C 3 U 3 h h 3 4 = α /[4 + 2 = [ 3 k = [1 + 2h c { ( α 3)} 1]/ h { ( α 3)}] ] 1/ 2 = µ σ k 0 h 3 4 The C 0, can be figured out by the MPME value is R MPME = c 0 + c 1 U 1 + c 2 U 2 + c 3 U 3 26

27 Main Contents Background Multi-DOF method Four methods to predict MPME and DAF Random seed effect Concluding remarks 27

28 Configuration of Two Rigs Item Rig 1 Rig 2 Length overall (ft) Breadth overall (ft) Water depth (ft) Weight (kips) 25,000 27,000 Wave height (ft) Wave period (s) Current (knots)

29 Comparisons of Natural Periods Rig 1 Rig 2 Mode Detail Model (s) Combined Model (s) Diff. (%) Detail Model (s) Combined Model (s) Diff. (%)

30 Random Seed Effect: Rig 1 seed DEGREE W/J METHOD WEIBULL METHOD OTM BS OTM BS % % Statistical Properity AVE SD 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 30

31 Random Seed Effect: Rig 1 DAF for Overturning Moment Random Seed DI 0 Degree DI 30 Degree DI 60 Degree W/J 0 Degree W/J 30 Degree W/J 60 Degree DAF for Base Shear Random Seed Findings Compared with W/J method, drag/inertia method is not sensitive to the selection of random seeds and DAFs are pretty stable Why? 31

32 Random Seed Effect: Rig 1 Dynamic overturning moment 6.00 Mean/10^4 SD/10^ Skewness* Kurtosis Because: Random Seed Static Overturning Moment Mean/10^4 SD/10^5 Skewness*100 Kurtosis Random Seed 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. 32

33 Weibull Method: Rig hour (10 seeds) 1 hour (5 seeds) seeds 5 seeds to max DAF 5 seeds to min DAF DAFs DAFs OTM BS 0.0 OTM BS 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 33

34 Main Contents Background Multi-DOF method Four methods to predict MPME and DAF Random seed effect Concluding remarks 34

35 Concluding Remarks D/I Method Weibull Method Gumbel Method W/J Method Running Period and Number 60 minutes, 3 runs with different Cd, Cm 60 minutes, runs number minutes, runs number min, runs number =1 (may not be sufficient) Effect of Random Seed not sensitive sensitive sensitive Characteristics Weak 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 35

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