Prediction of most probable extreme values for jackup dynamic analysis

Marine Structures 15 (2002) 15–34

Prediction of most probable extreme values for jackup dynamic analysis Yan Jenny Lu*, Youl-Nan Chen, Pao-Lin Tan, Yong Bai American Bureau of Shipping, 16855 Northchase Drive, Houston, TX 77060, USA Received 7 August 2000; received in revised form 1 March 2001; accepted 13 May 2001

Abstract Many important technical issues in the design of deep water jackup rigs stem from the prediction of extreme values of global design parameters such as platform sway, base shear, and overturning moment. Several analytical methods primarily based on time-domain dynamic simulation have been recommended in SNAME T&R Bulletin 5–5A (hereafter referred to as ‘‘the Bulletin’’) in the realm of the so-called most probable maximum extreme. This paper compares the four major dynamic, time-domain methods recommended in the Bulletin, investigates the random seed effect on each method, exams the convergence of the statistical properties within the recommended time simulation, and presents the impact on the dynamic response due to various parameters, e.g. leg-to-hull flexibility, P-delta effect and foundation fixity. Comparison among the four methods is presented in terms of the calculated extreme values and the corresponding dynamic amplification factors. The structural models employed in this investigation were constructed to reflect the behavior of two jackup rigs in service. These rigs were purposely selected to represent two of the most widely used jackup designs, which are of different leg types, different chord types, and designed for different water depth. r 2001 Published by Elsevier Science Ltd. Keywords: Time-domain simulations; Dynamic amplification factor; Most probable maximum extremes

1. Introduction The application of jackup drilling units have been extended into deeper water, where dynamic effect becomes a significant concern. The environmental loads to which a jackup is exposed are random in nature. Waves, wave loads and structural *Corresponding author. Tel.: +1-281-877-6732; fax: +1-281-877-5820. E-mail address: [email protected] (Y.J. Lu). 0951-8339/02/$ - see front matter r 2001 Published by Elsevier Science Ltd. PII: S 0 9 5 1 - 8 3 3 9 ( 0 1 ) 0 0 0 1 7 - X

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Y.J. Lu et al. / Marine Structures 15 (2002) 15–34

Nomenclature C0 ; C1 ; C2 ; C3 CD CM CR CRS CRi D DAF FR K; h3 ; h4 MPME N R Rd Rs TN TZ

coefficients of Jensen cubic model drag coefficient inertia coefficient standardized MPME standardized MPME of static response standardized MPME of inertia response outside diameter of a Morison member dynamic amplification factor non-exceedance probability coefficients of Winterstein Hermite model most probable maximum extreme number of zero-upcrossings of response time series for 3-h storm response variable dynamic response quasi-static response jackup natural period zero-upcrossing period of the wave spectrum

Greek letters a b e g k r m mR s sA sRd sRs sRi c

scale parameter of Weibull distribution, a > 0 slope parameter of Weibull distribution, b > 0 bandwidth threshold parameter of Weibull distribution, R
g > 0 scale parameter of a Gumbel distribution correlation coefficient mean value of a random variable mean value of response variable standard deviation of a random variable standard deviation of water particle acceleration standard deviation of dynamic response standard deviation of quasi-static response standard deviation of the so-called ‘‘inertia’’ response, which is the difference of dynamic response and quasi-static response location parameter of a Gumbel distribution

responses vary randomly with time and must be represented as stochastic variables. It is important to accurately predict the probability distributions of these random variables for assessing extreme responses of the structures. In this regard, it should be recognized that the structural response of a jackup is dominated by drag force, and in general is non-Gaussian.

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There are many different notions on how to measure the dynamic effect as well as many different approaches to account for dynamic effects in designs. One of the most common methods employed by the industry, as reflected in SNAME T&R Bulletin 5–5A [1], is the so-called inertia load set approach (see Fig. 1). In this method, a simplified structural model is generated, and the dynamic amplification factor (DAF) is calculated by comparing the resulting static and dynamic global responses of the structure. The total load acting on an offshore structure is then calculated as the combination of wave load multiplied by DAF and other static loads. The structural strength is evaluated against various failure modes, such as yielding, buckling, fatigue, etc. by detailed FE models using quasi–static analysis. While it is possible to frame the analysis in the frequency domain, as stated in the Bulletin, the present study focuses on random, time-domain simulation. Most probable maximum extreme (MPME) is defined as the mode value of the probability density function for extreme values of a random variable. In this paper, the dynamic load effect is estimated as the most probable maximum extreme corresponding to an exceedance probability of 1/1000. This level of exceedance typically reflects the situation of a 3-h storm. The term DAF refers to the ratio of the predicted extreme dynamic response to that of the associated extreme static response. The DAF is computed for the two basic global design parameters, i.e. base shear and overturning moment. Chen et al. [2] examined the effects of non-linearity and structural response upon various important parameters governing the behavior of the jackup structure in deep-water applications. The present paper compares the various methods of

Fig. 1. Inertia loadset approach to account for dynamics.

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predicting extreme responses based on simulated time series responding to the excitation of random wave action. These methods are recommended by the Bulletin. In this paper, deviations from what is specified in the Bulletin are suggested and their justifications are presented. A parametric study is performed to investigate the influence of various parameters on predicted extremes, which include structural stiffness, simulation time step, and random seeds. Transient response (from the first 100 s of the simulation) in the time series was removed before any statistical calculation was done. 2. Structural models Two structural models are generated for in-service jackups. Time-domain analysis is performed using the Extended Design Program, Version 4.2 by Digital Structures (EDP).1 Table 1 summarizes the major design parameters of the jackup rigs. The structural model consists of equivalent stick models of the legs, spudcans and representative beam-element hull framing. The equivalent legs are modeled using linear beam elements reflecting equivalent stiffness, mass, damping and hydrodynamic properties. Mass is distributed along the legs to represent the steel weight and added mass. The elevated weight is assumed to be uniformly distributed along the hull and is modeled as equally concentrated loads at each leg-to-hull connection joint. Structural responses were calculated by taking into account relative velocity and P-delta effect.

3. Comparison of structural stiffness Although it is possible to model the non-linear soil–structure interaction and legto-hull connection, a simplified model assuming pinned spudcan supports and fixed leg-to-hull connections is often used in dynamic analysis in the light of unknown soil properties (e.g. at design stage) and computational complexity. This study intends to investigate the variation of structural stiffness due to different ways of model simplification. Jackups are relatively flexible structures subject to high lateral displacement under environmental loading. These displacements result in an offset between the base of each leg and its point of fixity at the hull, and consequently lead to a reduced stiffness of legs under axial forces. The study also investigates the impact on the structural stiffness due to this so-called P-delta effect. Comparison of structural stiffness in terms of structural eigenperiod is presented in Table 2, with respect to various parameters as discussed above. It is generally accepted that a pinned spudcan support model is conservative, and a fixed leg-to-hull connection assumption is non-conservative. Between these two 1 The Extended Design Program (EDP), Version 4.2 developed by Digital Structures, Inc. is designed to perform structural analysis of general and pile supported structures. Static and dynamic analysis can be performed assuming either linear or non-linear member behaviors.

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Table 1 Design parameters Rig I Leg type Brace type

Triangular Inverse-K

Elevated weight (kips) Leg weight (incl.can) (kips)

Chord type Fixation system No. of pinions per chord Leg dist., long. (ft) Leg dist. trans. (ft) Leg length (ft) Bay height (ft) Chord distance (ft) Hull depth (ft)

Split-tube Rack chock 4 150 156 540 28 39 30

Water depth (incl. tide) (ft) Max. wave height (ft) Ass. wave period (s) Sig. wave height (ft) Peak wave period (s) Airgap (ft) Current at surface (knot) Current at mudline (knot)

Rig II Leg type Brace type

Square K

Elevated weight (kips) Leg weight (incl.can) (kips)

Chord type Fixation system No. of pinions per chord Leg dist., long. (ft) Leg dist. trans. (ft) Leg length (ft) Bay height (ft) Chord distance (ft) Hull depth (ft)

Triangular None 4 129 142 477 110
2 3/1600 30 26

Water depth (incl. Tide) (ft) Max. wave height (ft) Ass. wave period (s) Sig. wave height (ft) Peak wave period (s) Airgap (ft) Current at surface (knot) Current at mudline (knot)

28470 7945

350 55 15.5 29.6 15 42 1.5 0

15532 7354

258 40 15 21.5 14.4 58 0 0

simplifications, spudcan fixity is found to have far more influence than the leg-to-hull connection. Hence, it is important to consider the spudcan fixity when evaluating the dynamic response of jackups. For the purpose of a preliminary investigation, dynamic analysis may be conducted using fixed leg-to-hull connections and pinned spudcan supports in a simplified model. This contention, however, needs to be verified with more rig types and various sites. In general, the impact of the model simplification is site and rig specific.

4. Comparison of time-domain based dynamic analysis methods Jackups consist of slender cylindrical members as basic components. Wave and current induced loading is non-linear due to the non-linear drag force and free surface. Non-linearity in response is also induced by second order effects due to large structural motions and hydrodynamic damping caused by the relative velocity between the structure and water particles. Moreover, the leg-to-hull connection and

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Y.J. Lu et al. / Marine Structures 15 (2002) 15–34 Table 2 Comparison of stiffness in terms of eigenperiods (unit: second)FRig II Modeling

Eigenperiod

Bottom-pinned Leg-hull-fixed w/P-delta

10.24

Soil fixitya Leg-hull-flexible w/P-delta

7.9

Bottom-pinned Leg-hull-flexible w/P-delta

12.5

Bottom-pinned Leg-hull-flexible No P-delta

11.4

a

Spudcan fixity is calculated based on a 70 ft penetration at Block 338 in Gulf of Mexico. For pinned conditions, the pin end is applied at 10 ft below mudline.

soil–structure interaction induce structural non-linearity. As a result, although the random wave elevation can be considered as a Gaussian process, the response is nonlinear (e.g., with respect to wave height) and non-Gaussian. Hence, the prediction of the short-term extremes of the structural responses requires treatment different from that of floating platforms or even the more stocky seabed supported structures such as jacket drilling units operating in relatively shallow water. Basically, the prediction procedure is to select a proper class of probabilistic models for the simulation in question and then to fit the probabilistic models to the sample distributions. The T&R Bulletin 5–5A [1] recommends four methods to predict the MPMEs from time-domain simulations. The Bulletin recommends that random seas be generated from at least 200 components. The generated random wave elevation shall be Gaussian. The Bulletin also recommends that the simulation time step be shorter than the lesser of 5% of the incident wave zero crossing period and that of the structural fundamental period, i.e., DTominðTZ =20; Tn =20Þ:

ð1Þ

The structural response subjected to these random sea states can then be used to determine the MPME by applying one of the four methods as follows: 1. Drag/inertia parameter methodFcalculate MPMEs based on a so-called drag/ inertia parameter; an 1-h simulation is recommended. 2. Weibull fitting methodFuse the Weibull distribution to fit the individual peaks of the response series and calculate MPMEs based on the Weibull distribution

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function. It is recommended to run five 1-h simulations and take the result as an average of the five. 3. Gumbel fitting methodFuse the Gumbel distribution to fit the maximum/ minimum from each of ten 3-h simulations and calculate MPMEs based on the Gumbel distribution. 4. Winterstein/Jensen MethodFuse Winterstein’s Hermite polynomial model to predict MPMEs, with improvements by Jensen’s Cubic model if the Kurtosis is larger than 5. One 3-h simulation is recommended.

5. Random sea simulations and responses During the study, 10 random Gaussian sea states for Rig I and 12 for Rig II have been generated following the recommendation of the Bulletin. In a dynamic analysis, before extreme response can be estimated from the simulated time series, one should carefully check the statistical properties of the time series and ascertain the validity of the simulation scheme. Such examination includes the number of repeated simulations, the length (in time) of a simulation, and the length of the time step. The impact of such parameters upon the statistical stability and accuracy is estimated for the intended analysis. The first objective of this study is to gain the insight into the effect of these parameters on the convergence of the statistical properties (mean, standard deviation, skewness and kurtosis). For the cases studied, the maximum time step permitted by the Bulletin is 0.56 s on the premise given by Eq. (1). A study was performed with * * *

time step of 0.25 s, 2-h simulation, time step of 0.5 s, 4-h simulation, time step of 1 s, 6-h simulation.

As illustrated in Figs. 2A–F, this study indicates: (a) Skewness and kurtosis take a longer time to stabilize than does standard deviation. (b) For the cases considered here, it appears that difference is little in terms of statistical properties (including mean, standard deviations, skewness, kurtosis, predicted MPMEs and DAFs) among time simulations using * *

time step of 0.25 and 0.5 s (for the first 2 h), or 0.5 and 1.0 s (for the first 4 h).

The preceding observations require further confirmation covering a wider variety of test sample rigs. However, if these observations can be affirmed, it appears to be appropriate for designers to increase the time step to, e.g. 1.0 s (or the lesser of TZ =10 and TN =10).

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Fig. 2. (A) Calculated mean vs. various time step and simulation length. (B) Calculated standard deviation vs. various time step and simulation length. (C) Calculated skewness vs. various time step and simulation length. (D) Calculated kurtosis vs. various time step and simulation length. (E) Calculated MPME vs. various time step and simulation length. (F) Calculated DAFs vs. various time step and simulation length.

Y.J. Lu et al. / Marine Structures 15 (2002) 15–34

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Fig. 2. (Continued )

6. Prediction of short term extremes using simulated response series Parametric studies were performed to compare the predicted MPMEs using the four methods recommended by the Bulletin and to propose possible improvement. The structural model used in this part of the analysis employs pinned spudcan supports at 10 ft below the mudline and flexible leg-to-hull connections reflecting the actual stiffness of the rig. As a result of these simplified assumptions, the calculated structural responses are expected to be somewhat conservative. However, since the focus of this section is to examine the methodology of predicting MPMEs and DAFs using statistical calculation, the latter should not be significantly affected by the modeling conservatism. A time step of 0.5 s was used in the simulation generated for the following analysis. 6.1. Method IFdrag/inertia parameter method The drag/inertia parameter method is based on the assumption that the extreme value of a standardized process can be calculated by: splitting the process into two parts, evaluating the extreme values of each and the correlation coefficient between the two, then combining as ðmpmR Þ2 ¼ ðmpmR1 Þ2 þ ðmpmR2 Þ2 þ 2rR12 ðmpmR1 ÞðmpmR2 Þ:

ð2Þ

The extreme values of the dynamic response can therefore be estimated from extreme values of the quasi-static response and the so-called ‘‘inertia’’ response, which is in fact the difference between the dynamic response and the quasi-static response. The

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Y.J. Lu et al. / Marine Structures 15 (2002) 15–34

correlation coefficient of the quasi-static and ‘‘inertia’’ responses is calculated as rR ¼

s2Rd
s2Rs
s2Ri : 2sRs sRi

ð3Þ

The Bulletin recommends that the extreme value of the quasi-static response be calculated using one of the three approaches as follows: Approach 1: Static extreme can be estimated by combing the extreme of quasistatic response to the drag term of Morison’s equation and the extreme of quasistatic response to the inertia term of Morison’s equation, using Eq. (3) as above. Approach 2: Baar [3] suggested that static extreme may be estimated by using a non-Gaussian measure. The structural responses are non-linear and non-Gaussian. The degree of non-linearity and the deviation from a Gaussian process may be measured by the so-called drag–inertia parameter, K; which is a function of the member hydrodynamic properties and sea state. This parameter is defined as the ratio of the drag force to inertia force acting on a structural member of unit length. K ¼ ð2CD s2V Þ=ðpCM DsA Þ:

ð4Þ

As an engineering postulate, the probability density function of force per unit length may be used to predict other structural responses by obtaining an appropriate value of K from time-domain simulations. K can be estimated from the standard deviation of response due to drag force only and inertia force only. rffiffiffi p sR ðCM ¼ 0Þ : ð5Þ K¼ 8 sR ðCD ¼ 0Þ Approach 3: Alternatively K can be estimated from the kurtosis of structural response " #1=2 ðk
3Þ þ f26ðk
3Þ=3g1=2 K¼ : ð6Þ ð35
3kÞ It is observed from the study that: (a) The three approaches do not yield consistent results with each other. For the cases presented in Table 3, the first two approaches leads to similar results, but the last one produces quite different results (see Table 3). When comparing with other methods (see Table 8), the first two approaches yield similar results with Table 3 DAFs calculated by using the three approaches of Method 1 Approach

(1) (2) (3)

Rig I

Rig II

Base shear

OTM

Base shear

OTM

1.51 1.52 1.62

1.88 1.89 2.00

1.38 1.38 1.47

2.23 2.23 2.48

Y.J. Lu et al. / Marine Structures 15 (2002) 15–34

25

other methods for Rig I, while approach 3 yields similar results with other methods for Rig II. (b) The third approach may be unreliable because the estimation is based solely on kurtosis without the consideration of lower order moments. As explained by Hagemeijer [4], this approach ignores the effect of free-surface variation. The change in submerged area with time will produce a non-zero skewness in the probability density function of the structural response (say, base shear) which has not been accounted for in the equations for force on a sub-merged element of unit length. Hagemeijer also pointed out that the skewness and kurtosis estimated (as is the parameter K) from short simulations (say 1–2 h) are unreliable, which is also verified in this study as mentioned in Section 4. (c) The correlation coefficient rR of quasi-static and ‘‘inertia’’ responses is not necessarily positive as indicated in the Bulletin. Especially when the load frequency is higher than the structural eigenfrequency, the dynamic response lags behind quasi-static response for a phase angle of 90–1801. 6.2. Method IIFWeibull fitting Weibull fitting is based on the assumption that for drag dominated platforms, structural response can be fitted to a Weibull class of distribution "   # R
g b FR ¼ 1
exp
: ð7Þ a The extreme value for a specified exceedance probability (say 1=N) can therefore be calculated as R ¼ g þ a½
lnð1
FR Þ 1=b : Using a uniform level of exceedance probability of 1=N; Eq. (8) leads to  1=b : RMPME ¼ g þ a
lnð1=NÞ

ð8Þ

ð9Þ

The key for using this method is therefore to calculate the parameters a; b and g; which can be estimated by regression analysis, maximum likelihood estimation, or static moment fitting. For a 3-h storm simulation, N is approximately 1000. The time series record is first standardized ðRn ¼ R
m=sÞ; and all positive peaks are then sorted in ascending order. As recommended in the Bulletin, only a small fraction (e.g., the top 20%) of the observed cycles is to be used in the curve fitting and least squares regression analysis is to be used for estimating the Weibull parameters. This study indicates that the predicted MPMEs and DAFs are sensitive to the percentage of data extracted for regression analysis. It is true that for predicting extreme values in order statistics, the upper tail data is far more important than lower tail data. What percentage of the top ranked data should be extracted for regression analysis is, however, very hard to establish. Scattered data are found at the very top tail of the ranked data (see Fig. 3). For the two rigs being studied, it seems that better results have been achieved by using the

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Y.J. Lu et al. / Marine Structures 15 (2002) 15–34

Fig. 3. Weibull fitting.

Fig. 4. Comparison of Weibull fitting results with simulation data.

upper 10% of the ranked data and excluding the top 2% to avoid scattered data. As indicated by Fig. 4, using 10% of the top ranked data results in a better fit at the top tail, which is the most significant in order statistics. A proper fraction of the data to be extracted for regression analysis should be further investigated. The random seed effect can also be significant for this method, which complicates the situation (Table 4). 6.3. Method IIIFGumbel fitting Gumbel fitting method is based on the assumption that for a wide class of parent distributions whose tail is of the form FðXÞ ¼ 1
expð
gðxÞÞ;

ð10Þ

where gðxÞ is a monotonically increasing function of x, the distribution of extreme values is Gumbel (or Type I, maximum) with the form  

1 Fðxextreme pXMPME Þ ¼ exp
exp
ðXMPME
cÞ : ð11Þ k

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Y.J. Lu et al. / Marine Structures 15 (2002) 15–34 Table 4 Calculated DAFs using Weibull fitting method Simulation

1

Rig I Upper 20% Shear 1.59 OTM 2.00 Upper 10%, excluding top 2% Shear 1.58 OTM 2.01 Rig II Upper 20% Shear 1.58 OTM 2.70 Upper 10%, excluding top 2% Shear 1.47 OTM 2.66

2

3

4

5

Mean

1.49 1.92

1.63 2.03

1.57 1.99

1.71 2.20

1.60 2.03

1.42 1.76

1.51 1.81

1.59 1.98

1.54 2.03

1.53 1.92

1.59 2.64

1.47 2.58

1.64 2.74

1.48 2.55

1.55 2.64

1.45 2.30

1.33 2.32

1.72 2.74

1.52 2.50

1.50 2.50

The most probable maximum extreme discussed in this paper corresponds to an exceedance probability of 1/1000 in a distribution function of individual peaks or to 0.63 in an extreme distribution function. The MPME of the response can therefore be calculated as XMPME ¼ c
k lnð
lnðFðXMPME ÞÞÞ ¼ c
k lnð
lnð0:37ÞÞÞEc:

ð12Þ

Now the key is to estimate the parameters c and k based on the response signal records obtained from time-domain simulations. The Bulletin recommends to extract the maximum simulated value for each of the ten 3-h response signal records, and to compute the parameters by maximum likelihood estimation. Similar calculations are also to be performed using the ten 3-h minimum values. Although it is always possible to apply the maximum likelihood fit numerically, the method of moments (as explained below) may be preferred by designers for computing the Gumbel parameters in light of the analytical difficulty involving the type-I distribution in connection with the maximum likelihood procedure. For the type-I distribution, the mean and variance are given by mean: m ¼ c þ gk; g=Euler constant (0.5772) Variance : s2 ¼ p2 k2 =6 by which means the parameters c and k can be directly obtained using the moment fitting method pffiffiffi 6s k¼ ; c ¼ m
0:57722k: ð13Þ p

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Table 5 MPMEs and DAFs calculated based on Gumbel fitting Various methods to estimate parameters

Base shear Static (kips)

Dynamic (kips)

DAF

Rig I Closed-form MLEa Least squares

664 652 685

1023 1010 1041

1.54 1.55 1.52

Rig II Closed-form MLEa Least squares

563 564 565

862 828 873

1.53 1.47 1.54

a ‘‘MLE’’ stands for maximum likelihood estimation, ‘‘Static’’ stands for MPME of quasi-static response, ‘‘Dynamic’’ stands for MPME of dynamic response.

Table 5 compares the MPMEs and DAFs calculated by the moment fitting method (closed-form solution), maximum likelihood estimation, and least squares regression, respectively. It illustrates that the moment fitting method yields reasonably accurate results. As maximum likelihood estimation obviously takes much more effort dealing with complicated numerical iterations, a closed-form solution (moment fitting) is believed to be a more favorable method in practical designs. 6.4. Method IVFWinterstein/Jensen method The basic premise of Winterstein/Jensen method is in accordance with Winterstein [5] and Jensen [6], that a non-Gaussian process can be expressed as a polynomial (e.g., a power series or an orthogonal polynomial) of a zero mean, narrow-banded Gaussian process (represented here by the symbol U). In particular, the orthogonal polynomial employed by Winterstein is the Hermite polynomial. In both cases, the series is truncated after the cubic terms as follows: Winterstein:   RðUÞ ¼ mR þ sR K U þ h3 ðU 2
1Þ þ h4 ðU 3
3UÞ ; ð14Þ Jensen: RðUÞ ¼ C0 þ C1 U þ C2 U 2 þ C3 U 3 :

ð15Þ

Within this framework, the solution is essentially separated into two phases. First, the coefficients of the expansions, i.e., K; h3 ; and h4 in Winterstein’s formulation and C0 to C3 in Jensen’s formulation are obtained.2 Subsequently, upon substituting the most probable extreme value of U in Eq. (14) or (15), the MPME of the responses 2

Extracts from the commentary to the Bulletin [1] are appended in an Annex to demonstrate the solution method for the Wintenstein formulation.

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29

will be determined. While the procedure of Method IV appears perfectly simple, there are a number of technical issues that should be raised as discussed below. (a) In accordance with the Bulletin, MPME of U is to be calculated as follows: ( )1=2 2 1oge ðv0 TÞ pffiffiffiffiffiffiffiffi U¼ ; ð16Þ 1
2fð
U ðpzÞÞ pffiffiffiffiffiffiffiffi where v0 T ¼ N is the number of zero-up crossings. The term ‘‘1
2fð
U ðpzÞÞ’’ is to account for finite bandwidth effect as stated in the Bulletin. In either Wintersten’s or Jensen’s formulation, the variable U; is considered to be a zero-mean, narrow-banded Gaussian process. Ochi [7] presented the expression for the most probable value of a random process that satisfies the generalized Rayleigh distribution (i.e. the wide-banded Rayleigh). The bandwidth e; of this random variable is determined from the zeroth, 2nd and 4th spectral moments. For e less than 0.9, the short-term most probable extreme value of U is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffiffiffiffiffiffiffiffiffiffiffi u u 1
e2 t pffiffiffiffiffiffiffiffiffiffiffiffiffi N : U ¼ 2 ln ð17Þ 1 þ 1
e2 For a narrow-banded process, e approaches zero and the preceding reduces to the more well-known expression pffiffiffiffiffiffiffiffiffiffiffiffiffi U ¼ 2 ln N ð18Þ Comparison of Eqs. (17) and (18) clearly indicates that consideration of the bandwidth effect for a Gaussian process, U; results in a reduction of the most probable value (see Fig. 5). As illustrated by the above graph, the inclusion of bandwidth effect will reduce the predicted value of U and consequently reduce the responses. Ignoring the bandwidth effect is therefore slightly conservative, and results in very little difference in the predicted MPME when the bandwidth e is small. On the other hand, however, the inclusion of the band width effect in the Bulletin leads to a very significant increase in the most probable extreme value of U; and in turn, the MPME of the response. This behavior is in contrast with the discussion given above. In a benchmarking study supported by the IADC [8], Method IV was carried out using the EDP software that incorporated the bandwidth effect precisely according to the Bulletin. In so doing, it results in a discrepancy in MPME by an order of almost 100% compared with Methods II and III. Had bandwidth effect not been applied, the EDP analysis would have come much closer to those obtained by applying the other methods and it would have been reasonable. In the present study, similar striking effectsFover predicted MPMEsFwere observed as shown in Table 6. Moreover, it is observed that the Bulletin approach leads to unstable DAFs with respect to random simulations. It should be noted that only one simulation is recommended for Method 4.

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Y.J. Lu et al. / Marine Structures 15 (2002) 15–34

Fig. 5. Bandwidth effect.

(b) It is indicated in the Bulletin that Winterstein’s Hermite Polynomial model is to be used unless the kurtosis of the time series is greater than 5. In such cases, Jensen’s analysis should be used. This statement may contain a subtle but fundamental flaw in that the two cited approaches are theoretically identical. Furthermore, while the value 5 for kurtosis as a threshold appears to be reasonable, it only covers the cases where the skewness is positive. In cases where the skewness is negative, the need to apply Jensen’s formulation over the Winterstein approximation would have to be extended to a much smaller value of kurtosis. For this reason, it appears wise to apply Jensen’s formulation uniformly, in light of the fact that most of the computational effort is spent on the time-domain simulation. The numerical work stemming from either formulation is negligible comparing with producing the time series from simulation. Alternatively, a more precise statement may be ‘‘yIf the skewness is positive and the kurtosis is greater than 5, the Winterstein closed form results can be applied. Otherwise, Jensen’s exact analysis should be used’’. Winterstein [9] has further developed his Hermite model to improve the accuracy of the calculated results, which may be investigated and employed later. 6.5. Random seed effect Except for Method III (Gumbel fitting), all other methods can predict MPMEs and DAFs from one simulation. Whether the results from one simulation are reliable and robust depends on the effect of the starting random seed. The following tables present the predicted results from five simulations for Rig I. It is observed that the MPMEs resulting from any of the methods vary upon the selection of the starting random seed. This observation, as shown in Table 7, implies that multiple simulations may always be required in order to achieve reliable MPMEs. On the other hand, if only DAFs are concerned, random seed effects seem small when using Method I (drag/inertia parameter) and Method IV(Winterstein/ Jensen), but more significant when using Method II (Weibull fitting). The

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Y.J. Lu et al. / Marine Structures 15 (2002) 15–34 Table 6 MPMEs and DAFs calculated based on Method IV Simulations

Static (kips)

Dynamic (kips)

DAF

Rig I Revised Method IV: ignore bandwidth effect, apply Jensen’s method uniformly 1 644.49 992.75 2 611.82 913.77 3 699.2 1073.1 4 721.1 1078.2 5 797.48 1224.6

1.54 1.49 1.53 1.50 1.54

Mean SDa

1.06E+03 1.16E+02

1.52 0.02

Method IVFSNAME T&R: exactly follow SNAME T&R Bulletin 1 1.01E+03 1.68E+03 2 9.56E+02 1.45E+03 3 1.13E+03 1.90E+03 4 1.18E+03 1.90E+03 5 1.36E+03 2.45E+03

1.67 1.51 1.68 1.61 1.80

Mean SD

1.65 0.11

6.95E+02 7.19E+01

1.13E+03 1.57E+02

1.88E+03 3.72E+02

Rig II Revised Method IV: ignore bandwidth effect, apply Jensen’s method uniformly 1 631 949 2 592 901 3 575 862 4 609 931 5 578 856

1.50 1.52 1.50 1.53 1.48

Mean SD

1.51 0.02

596.87 23.22

899.90 40.80

Method IVFSNAME T&R: exactly follow SNAME T&R Bulletin 1 1212 1741 2 1109 1623 3 1048 1461 4 1119 1726 5 1036 1443

1.44 1.46 1.39 1.54 1.39

Mean SD

1.45 0.06

a

1104.85 70.21

1598.78 141.58

SD=standard deviation.

recommended five simulations for Method II and one simulation for Methods I and IV as per the Bulletin are appropriate. It is, however, always a good practice to confirm the results with multiple simulations. In this study, one among the 12 simulations generated for Rig II yielded a DAF 10% higher than the others.

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Y.J. Lu et al. / Marine Structures 15 (2002) 15–34

Table 7 Random seed effect, Rig I Simulations

Static (kips* in)

Dynamic (kips* in)

DAF

987 918 950 1058 941

1.51 1.50 1.53 1.48 1.49

646 40

971 55

1.50 0.02

Method IIFWeibull fitting 1 651 2 658 3 709 4 697 5 643 Mean 672 SD 29

1029 933 1072 1107 989 1026 68

1.58 1.42 1.51 1.59 1.54 1.53 0.07

Method IIIFGumbel fitting 1 644 2 612 3 699 4 721 5 797

993 914 1073 1078 1225

1.54 1.49 1.53 1.50 1.54

Mean SD

1056 116

1.52 0.02

Method IFdrag/inertia parameter 1 651 2 612 3 623 4 713 5 630 Mean SD

695 72

6.6. Comparison of four methods With the observations made above for the four time-domain based dynamic analysis methods, MPMEs and DAFs are calculated in accordance with the Bulletin approach but with deviations as follows (results are presented in Table 8): *

*

*

For Method II, Weibull fitting used 10% of the top ranked data excluding the top 2%. For Method III, results obtained from a closed-form solution as suggested in this paper are presented. For Method IV, band width effect is ignored and Jensen’s cubic model is applied uniformly.

Three of the four methods are found to provide reasonably close predictions of DAF for the two rigs studied. Method I does not yield consistent results. However, if MPME values are of concern, the results from the four methods can be rather different.

33

Y.J. Lu et al. / Marine Structures 15 (2002) 15–34 Table 8 Comparison of predicted MPMEs and DAFs

Rig I Base shear

OTM

Rig II Base shear

OTM

MPME (kips) DAF MPME (kips* in) DAF

MPME (kips) DAF MPME (kips* in) DAF

Static Dynamic Static Dynamic

Static Dynamic Static Dynamic

Method I

Method II

Method III

Method IV

646 971 1.50 2.84E+06 5.30E+06 1.87

695 1056 1.53 3.18E+06 6.08E+06 1.92

672 1026 1.52 3.05E+06 5.84E+06 1.91

664 1023 1.54 3.06E+06 5.82E+06 1.90

675 938 1.39 1.99E+06 4.48E+06 2.25

605 903 1.49 1.85E+06 4.62E+06 2.50

563 862 1.53 1.76E+06 4.56E+06 2.59

597 900 1.51 1.88E+06 4.76E+06 2.53

Based on the above observations, in light of the required simulation length, number of simulations and robustness of calculation results, method IV is considered preferable from the design viewpoint. Method III is theoretically the most accurate, if enough simulations are generated. Ten simulations are minimum in which may, however, not be sufficient in some cases. Methods I and II require further investigation.

7. Conclusions and recommendations Further investigation and study covering a wider variety of test sample rigs should be performed to verify the observations and results obtained so far. However, the above studies appear to provide strong support for the contention as follows in predicting dynamic effects in jackup designs: *

*

*

*

Among the four methods, the Winterstein/Jensen method is believed to be the most efficient. Guidance provided by the Bulletin to apply the Winterstein/Jensen method seems to yield over-predicted MPMEs and unstable DAFs; and, therefore, it ought to be revised. It is believed that the bandwidth effect can be ignored and Jensen’s method can be applied uniformly. A closed-form (static moment fitting) solution can be used for Gumbel fitting to replace the maximum likelihood method, which will simplify the calculation procedure. Method I (drag/inertia parameter) and Method II (Weibull fitting) require further investigation in order to provide more robust results.

34 *

*

Y.J. Lu et al. / Marine Structures 15 (2002) 15–34

Criteria with regards to simulation time step and simulation length for timedomain dynamic analysis should be further investigated. It may be appropriate to double the time step. Modeling of soil–structure interaction, hull-to-leg connection and P-delta effect have significant impact on the structural stiffness and, consequently, on the predicted dynamic effect. Spudcan fixity has much more influence than other factors and therefore needs to be given more careful consideration.

The readers are recommended to read Ref. [10–16] for more discussions on dynamic response of jackups.

Acknowledgements The authors extend their appreciation to the management of ABS for their permission to publish this paper. This includes Donald Liu, Jack Spencer, Malcolm Sharples and many others.

References [1] SNAME Technical & Research Bulletin 5–5A (1994), Guideline for site specific assessment of mobile jack-up units, Recommended practice for site specific assessment of mobile jack-up units, Commentaries to recommended practice for site specific assessment of mobile jack-up units’’. [2] Chen YN, Chen YK, James PC. Extreme dynamic response and fatigue damage assessment for selfelevating drilling units in deep water. SNAME Trans 1990;98:143–68. [3] Baar JJM. Extreme values of morison-type processes. Appl Ocean Res 1992;14:65–8. [4] Hagemeijer PM. Estimation of drag/inertia parameters using time-domain simulations and the prediction of the extreme response. Appl Ocean Res 1990;12:134–40. [5] Winterstein SR. Non-linear vibration models for extremes and fatigue. J Eng Mech 1988;114:10. [6] Jensen JJ. Dynamic amplification of offshore steel platform response due to non-gaussian wave loads. Mar Struct 1994;7:91–105. [7] Ochi MK. On prediction of extreme values. J Ship Res 1973;17:1. [8] IADC Committee. IADC jack-up study for Letourneau 116C Hull for Gulf of Mexico prepared by Zentech; SNAME TR5-5A Site Specific Assessment LeTourneau 116C prepared by Digital Structures, 1998. [9] Winterstein SR, Kashef T. Moment-based load and response models with wind engineering applications. Proceedings of the 1999 ASME Wind Energy Symposium, 1999. [10] Borgman LE. Ocean wave simulation for engineering design, Civil engineering in the oceans. Proceedings of the ASCE Conference, San Francisco, September 1967. [11] Hattori Y, et al. Full-scale measurement of natural frequency and damping ratio of jackup rigs and some theoretical considerations. OTC 4287, 1982. [12] Brouwers JJH, Verbeek PHJ. Expected fatigue damage and expected extreme response for Morisontype wave loading. Appl Ocean Res 1983;5(3):129–33. [13] Grenda KG. Wave dynamics of jackup rigs. OTC 5304, 1986. [14] P. Liu et al. Response of jackup models to irregular waves. OTC 6591, 1991. [15] Karunakaran D. Nonlinear dynamic response and reliability analysis of drag-dominated offshore platforms, Doctoral dissertation. Norwegian University of Science and Technology, 1993. [16] Greeves EJ, Jukui BH, Sliggers PGF. Evaluating jack-up dynamic response using frequency domain methods and the inertial load set technique. J Mar Struct 1996;9:101–28.