Download - Jackup Units Re-evaluation
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
Background Multi-DOF method Four methods to predict MPME and DAF Random seed effect Concluding remarks
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
Background Multi-DOF method Four methods to predict MPME and DAF Random seed effect Concluding remarks
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
Background Multi-DOF method Four methods to predict MPME and DAF Random seed effect Concluding remarks
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
Random Seed Effect: Rig 1
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
Random Seed Effect: Rig 1
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
Background Multi-DOF method Four methods to predict MPME and DAF Random seed effect Concluding remarks
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