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Ocean Engineering 35 (2008) 484–493

www.elsevier.com/locate/oceaneng

Optimum design of ice-resistant offshore jacket platforms in the BohaiGulf in consideration of fatigue life of tubular joints

Gang Lia,�, Xiang Liua, Yuan Liub, Qianjin Yuea

aState Key Laboratory of Structural Analysis of Industrial Equipment, Department of Engineering Mechanics,

Dalian University of Technology, Dalian 116024, ChinabChina Classification Society Tianjin Branch, Tianjin, China

Received 25 April 2007; accepted 19 December 2007

Available online 31 December 2007

Abstract

Economy and ice-resistance are two factors that must be considered in the design of offshore oil platforms in the Bohai Gulf. This

paper focuses on the optimal design of ice-resistant offshore jacket platforms in the Bohai Gulf in consideration of fatigue life of tubular

joints, with the minimum initial weight and satisfying dynamic properties. An efficient fatigue analysis procedure based on spectral

analysis is proposed, in which a hybrid finite element model is adopted to simulate the jacket platform and the tubular joints, and the

pseudo-excitation method is used to calculate the power spectral density of the hot spot stress. Finally, a practical jacket platform with an

ice-breaking cone in the Bohai Gulf is optimized, and the results demonstrate that the fatigue life of tubular joints could be improved and

the weight of jackets decreased simultaneously.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Jacket platform; Ice-resistant design; Optimum design; Fatigue analysis; Pseudo-excitation method

1. Introduction

The Bohai Gulf is located in the northeast of China andfreezes during the winter due to its middle latitude andmonsoon climate (Yang, 2000). The force caused by theinteraction of the ice sheet and the structure is thedominant environmental load on offshore structures inthe Bohai Gulf. There are two recorded cases whereplatforms collapsed due to ice-floe loading in 1969 and1979, respectively. Since then, much attention has beenpaid to ice loading for platform design in the Bohai Gulf.In the early 1980s ice-resistant design techniques that weredeveloped in other countries were applied in the BohaiGulf, which unfortunately, led to overly conservativedesigns, requiring large mounts of construction material.With the progress in the research of sea ice conditions inthe Bohai Gulf, the relatively economical ice-resistant

front matter r 2007 Elsevier Ltd. All rights reserved.

eaneng.2007.12.005

ng author. Tel.: +86411 8470 7267;

70 7267.

ss: ligang@dlut.edu.cn (G. Li).

platforms were designed by determining the extreme iceforce and ice parameters properly. However, these plat-forms were designed based on the extreme static ice forceand did not account for the dynamic effects of ice loading.From the full-scale field tests and observations, Yue and Bi(2000) and Yue and Liu (2003) found that these platformsunderwent severe ice-induced vibrations during winters,which had the potential to cause fatigue problems.Research on the fatigue of the offshore structures has

attracted much attention during recent decades. Nolte andHansford (1976) developed closed-form mathematicalexpressions for determining the fatigue damage of struc-tures due to ocean waves. These expressions incorporatedrelationships between the wave height and the stress range,considering the stress range versus the number of cycles tofailure and the probability distribution for the occurrenceof wave heights. Almar-Næss (1985) discussed the mostimportant subjects related to fatigue of the offshore steelstructures, such as calculation of fatigue stresses andfatigue lives. Dover and Madhava Rao (1996a, b) reviewedand summarized the knowledge in the area of SCF (stress

ARTICLE IN PRESSG. Li et al. / Ocean Engineering 35 (2008) 484–493 485

concentration factor), fatigue and fracture mechanics ofthe tubular joints and damage assessment and reliabilityanalysis of joints. Ramachandra Murthy et al. (1994) andGandhi et al. (2000a, b) performed several studies aboutfatigue of stiffened steel tubular joints, such as corrosionfatigue and fatigue behaviors. Vugts (2005) discussedfatigue damage assessments and the influence of wavedirectionality, investigated for a rotationally symmetricstructure (a vertical circular cylinder) and presentedquantitative results.

Most of the research on the fatigue analysis of offshorestructures mentioned above focuses on the wave-inducedvibration. There was very little research on fatigue analysiscaused by ice-induced vibration. Fang et al. (1997, 2000,2001, 2003) developed a method to handle fatigue lifecalculation of tubular joints of offshore platforms sub-jected to ice force, and conducted research on reliabilityanalysis of ice-induced fatigue and fatigue life prediction ofwhole system for offshore platforms in view of wave andsea ice. With the development of studies of dynamic iceforce, Yue et al. (2007a) developed a method of ice-inducedfatigue analysis using the spectral approach for offshorejacket platforms.

In this paper, the authors propose an optimum designmethod of the ice-resistant offshore jacket platformconsidering fatigue life of tubular joints for the purposeof both ice-resistant capacity and lower initial weight in themarginal fields of Bohai Gulf, because some designedplatforms seem conservative and uneconomical by requir-ing a large amount of material. Considering that structuraloptimization is an auto-redesign process with a lot of time-consuming iterations, the authors propose an efficientapproach to calculate the fatigue life of irregular andcomplex tubular joints by spectral approach. In theproposed method, a hybrid finite element model isemployed to simulate the global structure and the localtubular joints, and the pseudo-excitation method (PEM)(Lin and Zhang, 2004, 2005) is also used to calculate powerspectral density (PSD) of the hot spot stress. Finally anexample of a real platform in the Bohai Gulf demonstratesthe feasibility and applicability of our method.

2. Optimum design formulation for ice-resistant

jacket platform

The oil and natural gas resources of the Bohai Gulf aremainly in marginal oil fields, and the ice force is thedominant environmental force on the offshore structures.Therefore, to design the offshore platforms with bothice-resistant capacity and lower cost is the main task ofthe designers. However, the conventional design is actuallydriven by a ‘‘trial-and-error’’ process, in which designerspropose an initial design first, then manually update thedesign based on the structural responses while complyingwith various design performance criteria. This ‘‘trial-and-error’’ process is generally time-consuming and it can onlyprovide a design satisfying the regulations of codes rather

than a better or optimal design. Furthermore, the finaldesign depends a lot on the designers’ experience andeducation. Structural optimization technique is an improve-ment of the design method, which has become an importantassistant tool for the structural design and has manyapplications in practical engineering. Structural optimiza-tion is actually an automated synthesis technique thatoptimizes structures for the purpose of achieving theoptimum design while satisfying the specified design criteria,using computers to implement auto-redesign effectively.The key issue in structural optimization is to formulate theoptimal problem, which consists of design variables, theobjective function and constraints. Fig. 1 shows the flowchart of conventional design and structural optimization. Inthis paper, the authors incorporate structural optimizationinto ice-resistant jacket platform design in order to improvecurrent design methods and to provide a more rational andeconomical design, especially considering the ice-induceddynamic effects, such as fatigue of tubular joints.

2.1. General optimization formulation

In general, the optimization formulation of the offshoreplatform can be stated as

Find : X

Min : F ¼ F ðX Þ

s:t: : Extreme Static Performance

Dynamic Performance

(1)

where X is the vector of design variables; F is the objectivefunction. The constraints contain extreme static perfor-mance and dynamic performance. The former consists ofthe strength and stability requirements of the structure andstructural elements along with the extreme deformation ofthe structure. The latter consists of dynamic effects such asfatigue of tubular joints and other requirements related tothe ice-induced vibration of the structure or sub-structure.To design an ice-resistant jacket platform according

to the current codes, the design ice condition is oftendetermined based on a given return period that is usuallyseveral times the life of the structure, for example, the icethickness of about 45 cm for the return period of 50 years isused in JZ20-2 region of the Bohai Gulf. The offshoreplatforms in service in the Bohai Gulf usually have enoughcapacity to resist the extreme static ice force. However,because of the lack of understanding of the dynamic iceforce caused by the ice-structure interaction, the presentdesign codes do not account for the dynamic performancesspecifically. Thus, the ice-induced vibration is the chiefproblem for the design of ice-resistant offshore jacketplatforms (Yue and Liu, 2003). In recent years, somedevelopments on the dynamic ice force have been made inthe Bohai Gulf (Yue and Bi, 2000; Qu et al., 2006; Yueet al., 2007b) and the characteristic of marginal fields in thisregion requires rational and economical ice-resistant jacketplatforms, which may cause dynamic problems. Thus, it is

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Structural modeling

Structural analysis—FEM

Check design behavior

Modify design based on experience

Does design meet criteria? Stop

No

Yes

Conventional design

Structural modeling

Structural analysis—FEM

Check design behavior

Modify design based on optimization method

Does design meet

optimum criteria?Stop

No

Yes

Development of optimal problem

Computer-Based Structural Optimization

Fig. 1. Conventional design and computer-based structural optimization.

G. Li et al. / Ocean Engineering 35 (2008) 484–493486

urgent to account for the dynamic effect in design of ice-resistant platforms in the Bohai Gulf directly. The authorsclassify the performances of the offshore platforms intostatic and dynamic performances and handle themseparately. First carry out the static optimum design ofthe offshore platforms based on the requirements under theextreme static ice force and other loads according to thedesign codes, which is a preliminary step and can provide afeasible region of design variables. Then perform thedynamic optimum design considering the requirements ofice-induced vibration under the dynamic ice force. Thispaper focuses on the dynamic optimum design.

2.2. Dynamic optimum design formulation

The dynamic optimum design formulation of offshorejacket platforms in consideration of fatigue life can bewritten as

Find : X ¼ fxi; i ¼ 1; 2; . . . ; ng

Min : Weight

s:t: : f 1 pf 1pf 1

DpD

xi pxipxi

(2)

where X is the vector of design variables as the dimensionsof components (diameter and thickness of pipes); f1 is thestructural fundamental frequency, f 1 and f 1 are the lowerand upper bounds; D is the fatigue damage index oftubular joints, D is the threshold value; xi and xi are thelower and upper bounds of the design variable xi,respectively.In Eq. (2), the weight of construction material is treated

as the optimization objective function in this study becauseit affects the cost of an offshore platform to a great extent.It should be noted that the actual ‘‘fabricated’’ cost is madeup of much more than just the cost of the raw material, forexample, the fabrication requirement of standard size forthe components to be designed is also a major factor thataffects the total cost.The constraints are the structural fundamental fre-

quency, fatigue damage of tubular joints and size limita-tions. As discussed above, the static performances,including strength, stability and deformation of thestructure and components based on design codes, areimplied in size limitations of design variables. It should bepointed out that in reality, the physical size of membersand joints in an offshore structure is often dictated bymany different requirements and some additional con-straints often come from fabrication, transportation,

ARTICLE IN PRESSG. Li et al. / Ocean Engineering 35 (2008) 484–493 487

installation, stiffness requirements, etc. These requirementsshould be considered further in the future research.

In the Bohai Gulf, the frequency of the ice force is closeto the fundamental frequency of the jacket platforms,which may lead to resonance of the structure. Thus, thefundamental frequency of the ice-resistant jacket platformis treated as a constraint.

In this paper, the ice-induced fatigue damage of tubularjoints is considered explicitly in the optimum design.Generally speaking, fatigue damage in an offshorestructure can be induced by other causes as well, such aswave-induced vibrations, and sometimes transportation(specially for long tows) and/or pile driving. In all thesecases, the increased flexibility that appears to reduce theice-induced fatigue damage might lead to increases in thedamage from other sources. However, for the jacketplatforms in the Bohai Gulf, the ice-induced fatiguedamage is the major cause. Furthermore, API RP 2A(API, 2000) has given a detailed simplified approach totreat the wave-induced fatigue, which is suitable to jacketplatforms with a fundamental period less than 3 s, installedin less than 122m (400 ft)-deep water. The designed ice-resistant jacket platforms in the Bohai Gulf satisfy theseconditions, with a fundamental period around 1 s, installedin 10–25m-deep water. In this sense the wave-inducedfatigue analysis for the jacket platforms in the Bohai Gulfcan be treated approximately using the method providedby API RP 2A if needed. Therefore, the ice-induced fatiguedamage is focused on in the dynamic optimum designformulation of offshore jacket platforms herein, which hasnot yet been specified in current codes explicitly.

3. Estimation of fatigue life of tubular joint in optimization

Because both fatigue analysis and optimization are time-consuming, the authors have chosen to incorporate thePEM (Lin and Zhang, 2004, 2005) into the spectralapproach (Yue et al., 2007a) to estimate the fatigue lifeof tubular joints efficiently in optimization.

3.1. Ice fatigue environment

The ice fatigue environments of JZ20-2 field of the BohaiGulf consist of the ice thickness, ice velocity, ice period andso on.

3.1.1. Ice force and parameters

The ice force spectrum on the conical structure (Yueet al., 2007b) is

sðf Þ ¼10F

2

0T�2:5

f 3:5exp �

5:47

T0:64

f 0:64

!, (3)

where F 0 is the ice force amplitude of the conical structure,which can be obtained by Hirayama–Obara formulation:

F ¼ Bsfh2 D

Lc

� �0:34

, (4)

where F is the ice force; B is the constant (here the value of3.7 is taken); sf is the ice bending strength, 0.7MPa; h isthe ice thickness; D is the diameter of cone; Lc is thecharacteristic length of ice plate, Lc ¼ ðEh3=12grwÞ

0:25,where E is the ice elastic modulus, 0.5GPa, and g is thegravity acceleration and rw is the water density. T is the iceforce period of the conical structure, which can bedetermined by T ¼ Lb=V (Lb is the ice breaking length,Lb ¼ kh, h is the ice thickness; V is the ice velocity). Thenumerous observations in the Bohai Gulf showed thatthe ratio of breaking length and ice thickness, k, wasdistributed in the range of 4–10.Here ice thickness h and ice velocity V are random

variables (Yue et al., 2007a). The ice thickness h (cm)follows the lognormal distribution with the probabilitydensity function (PDF) as

f ðhÞ ¼1

0:5503hffiffiffiffiffiffi2pp exp �

1

2

ln h� 1:8671

0:5503

� �2" #

. (5)

The ice velocity V (cm/s) follows Rayleigh distributionwith the PDF as

f ðV Þ ¼V

826:5512exp �

V 2

1653:1024

� �. (6)

3.1.2. Cyclic number

The average effectual ice period is defined as follows:

Teff ðdayÞ ¼X

TiPi, (7)

where Pi is the probability of the ith grade of ice conditionand Ti is the corresponding effectual ice period. The gradeof ice condition and the effectual ice period in JZ20-2region in the Bohai Gulf has been recorded, based onwhich the average effective ice can be obtained as 42 days(Yue et al., 2007a) The cyclic number of the stress underthe kth fatigue case is

nk ¼3600� 24� T eff � pk

Tk

, (8)

where Tk is the ice force period of the conical structure forthe kth fatigue case, which is equal to the ratio of the icebreaking length to the ice velocity, as defined above; pk isthe occurrence probability of the kth fatigue case; Teff is theaverage effectual ice period.

3.2. Analysis of random hot spot stress

Because of the random ice action, the hot spot stress isalso random. The stress amplitude of hot spot (the peakvalue) is assumed to follow Rayleigh distribution (Lu et al.,1992).

pðsÞ ¼s

m0exp �

s2

2m0

� �, (9)

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DT

1=

Calculate hot spot stress spectrum under unit pseudo excitation, Shot,0 (f)

K fatigue cases

Calculate nk by Eq.(8)

Calculate kth ice force spectrum Sk (f) by Eq.(3)

Calculate real hot spot stress spectrum by Eq.(15)

Shot,k (f)dfm0,k ∫=

Determine cyclic number of nki

Determine cyclic number of Nki

D =nki

Nkiik

k=k+1 i=i+1∑∑

Fig. 2. Fatigue analysis using pseudo-excitation method.

G. Li et al. / Ocean Engineering 35 (2008) 484–493488

where m0 is the zero-order moment of the hot spot stressspectrum.

Thus, the distribution of the stress range can bedescribed by Rayleigh distribution too.

pðsÞ ¼s

4m0exp �

s2

8m0

� �, (10)

where s is the stress range of the hot spot; m0 is the same asin Eq. (9).

In the present paper, the hot spot stress spectrum is calcu-lated by the PEM that can improve the efficiency of analysisof random vibration considerably (Fig. 2). The following isa brief introduction of PEM (Lin and Zhang, 2005).

Consider a linear system subjected to a zero-meanstationary random excitation and with a given PSD Sxx,the response is y and its auto-PSD is Syy ¼ jHj

2Sxx

(H is the frequency-response function). If the inputis a sinusoidal excitation exp(io), the harmonic responsewould be H(o) exp(io). Thus, if the actual input PSDSxx is replaced by a sinusoidal pseudo-excitation~x ¼

ffiffiffiffiffiffiffiSxx

peiot, the response would be ~y ¼

ffiffiffiffiffiffiffiSxx

pH eiot.

Consider two arbitrary responses, ~y1 ¼ffiffiffiffiffiffiffiSxx

pH1 e

iot and

~y2 ¼ffiffiffiffiffiffiffiSxx

pH2 e

iot (H1 and H2 are the frequency-responsefunction, respectively), PEM provides the followingformulae:

Syy ¼ ~y� ~y ¼ j ~yj2 ¼ jHj2Sxx, (11)

Sy1y2 ¼ ~y�1 ~y2 ¼ H�1SxxH2, (12)

Sy2y1 ¼ ~y�2 ~y1 ¼ H�2SxxH1, (13)

where ~y� is the conjugated value of ~y.PEM means that the auto- and cross-PSD functions of

two arbitrarily selected random responses can be computedusing the corresponding pseudo-harmonic responses due tothe PEM. Hence, PEM can be employed to obtain theauto-PSD of the hot spot stress. Firstly a unit pseudo-excitation is imposed on the structure to get the corre-sponding hot spot stress Fhot,0, then the auto-PSD ofthe hot spot stress under unit pseudo-excitation can beobtained

Shot;0ðf Þ ¼ Fhot;0F�hot;0 ¼ jFhot;0j2, (14)

where F�hot;0 is the conjugated value of Fhot,0.

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For a linear system under the kth fatigue case with theice force spectrum Sk(f) obtained by Eq. (3), the real hotspot stress spectrum is

Shot;kðf Þ ¼ Skðf ÞShot;0ðf Þ. (15)

Then the zero-order moment of the hot spot stressspectrum corresponding to the kth fatigue case, m0,k, is

m0;k ¼

Z 1�1

Shot;kðf Þdf . (16)

By now the PDF of the stress range of the hot spotcorresponding to the kth fatigue case pk(s) can bedetermined by Eq. (10).

3.3. Flowchart of estimation of ice-induced fatigue life

For the PDF of the stress range of the hot spotcorresponding to the kth fatigue case pk(s), the stress rangeis divided into N intervals Dsi ¼ si�si�1, where i ¼ 1,2, y,N, and s0 ¼ 0.

The occurrence number of Dsi is

nki ¼ nkPðDsiÞ ¼ nk

Z si

si�1

pkðsÞds

ðk ¼ 1; 2; . . . ;K ; i ¼ 1; 2; . . . ;NÞ (17)

where nk is the cyclic number of the stress under the kthfatigue case, defined as in Eq. (8). Based on Miner’s Rulethe total fatigue damage index is

D ¼XK

k¼1

XN

i¼1

nki

Nki

, (18)

where Nki ¼ Nki(sri) is the representative value sri of thestress range Dsi, determined according to S–N curve.

The fatigue life is

T ¼1

D. (19)

4. Numerical example

Here the JZ20-2 NW platform is used as a numericalexample to demonstrate the dynamic optimal design of ice-resistant offshore jacket platforms in consideration offatigue life of tubular joints. The JZ20-2 NW platform,whose design life is 20 years, is installed in 13.5m-deepwater with 16.0m height of deck and 250 t mass on deck.A 6m-diameter ice-breaking cone is installed at sea level.

In order to consider the hot spot accurately andefficiently for complex tubular joints (see Fig. 3), weemploy a hybrid finite element model, which uses shellelements in the region of tubular joints, beam elements inother parts of the pipes and the constraint equations forconnection between different types of elements in view ofSaint-Venant’s principle. Here the finite element modelconsists of the lumped mass elements, beam elements andshell elements, which are Mass21, Shell93 and Pipe16 in

ANSYS. Mass21 is used to model the top mass on thedeck, Pipe16 the jacket structure and Shell93 the tubularjoints. The mesh density of the tubular joints is based onthe guidelines in DNV code (DNV, 2001). Foundationsare modeled as the equivalent piles, fixed under the mudlevel at a depth of six times the diameters of the piles.Connections between Pipe16 and Shell93 are linked byconstraint equations in ANSYS.The ice force is imposed on the structure at the sea level.

Because JZ20-2 NW platform has one-leg, the ice-driftingdirection has little influence. Thus we consider the icethickness and ice velocity and assume they are mutuallyindependent, and divide them into 10 groups, respectively,listed in Table 1. There are 100 ice fatigue cases in total,whose occurrence probabilities are listed in Table 2.In the optimum design formulation, the design variables

include the diameter and thickness of the piles, the columnsupporting the deck, the pipes under water, the braces andreinforced joints, as shown in Fig. 3(d). Constraints consistof the structural fundamental frequency and the fatigue lifeof the representative tubular joint. Fatigue life of thetubular joint must meet the requirement of API RP 2A(API, 2000) that the design fatigue life of each joint andelement should be at least twice of the design life of thestructure. The design life of the JZ20-2 NW platform is 20years, so the fatigue life of the tubular joint should belonger than 40 years using the X/ S–N curve of API RP 2A.Because the configuration of the structure and the loadingmode are not changed during the optimization process, weassume that the location of the hot spot will not change.The hot spot is located at the intersection of the columnand the braces, as shown in Fig. 4(a). The size constraintscan account for the static performance requirements, suchas strength, stiffness and stability in some sense. Moreover,due to the meshing and computational consideration ofdifferent types of elements in the hybrid finite elementmodel, some geometric constraints are added. For exam-ple, the diameter and thickness of the chord are larger thanthose of braces; and also point 1 and point 2 in Fig. 4(b)and (c) must not be superimposed in order to make theBoolean operation perform correctly during optimization.They are listed in Table 3 as Geo. 1, Geo. 2. Geo. 3 andGeo. 4, meaning that the joints should be reinforced. Thesymbol ‘‘–’’ in Table 3 means null.In the optimization process, the constraints and objective

function are normalized for numerical stability and thetolerance of convergence is taken as 0.001 for theconstraints and 0.01 for the objective function. Duringthe use of the design optimization module in ANSYS, thefirst-order method based on the gradient information isemployed to perform the optimization process. Theoptimization process converged after seven iterations,which provides seven feasible design sets. Fig. 5 showsthe optimization history normalized by the initial values.The optimal results demonstrate that the optimizationcan minimize the weight of the jacket structure andimprove the fatigue life of tubular joints simultaneously.

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Table 1

Groups of ice parameters

Ice

thickness,

h (m)

Label Ph Ice

velocity, v

(m/s)

Label Pv

0–0.04 h1 0.191137 0–0.1 v1 0.058699

0.4–0.08 h2 0.459065 0.1–0.2 v2 0.156221

0.08–0.12 h3 0.21901 0.2–0.3 v3 0.204908

0.12–0.16 h4 0.080849 0.3–0.4 v4 0.200283

0.16–0.20 h5 0.029803 0.4–0.5 v5 0.159488

0.20–0.24 h6 0.011533 0.5–0.6 v6 0.107102

0.24–0.28 h7 0.004724 0.6–0.7 v7 0.061693

0.28–0.32 h8 0.002043 0.7–0.8 v8 0.030779

0.32–0.36 h9 0.000929 0.8–0.9 v9 0.013379

0.36–0.40 h10 0.000907 0.9–1.0 v10 0.007447

Note: The upper bound of each group is used to generate the ice force

spectrum.

Fig. 3. JZ20-2 NW platform: (a) picture of JZ20-2 NW; (b) hybrid finite element model; (c) mesh of tubular joints; (d) statement of design variables.

G. Li et al. / Ocean Engineering 35 (2008) 484–493490

The optimal weight of the jacket is reduced to 64.37% ofthe initial design, and the structure becomes more flexiblewith the fundamental frequency reduced to 74.26% of the

initial design. Also the fatigue life of tubular joints wasimproved by 16.34%, from 62.8720 to 73.1440 years.Moreover, Table 3 lists optimization information, whichmay be useful for decision-making with several availableoptions.To discuss the optimal results further, let us consider the

ratio of the hot spot stress distribution parameter, rk ¼

mopt:0;k =mini:

0;k (mopt:0;k and mini:

0;k are the distribution parameters in

Eq. (9) for the optimal and initial design), and the ratio ofthe ice force period of conical structure, rT ;k ¼ T

ini:k =T

opt:k

(Topt:k and T

ini:k are the effective ice periods in Eq. (8) for the

optimal and initial design).Assuming S–N curves as (Nolte and Hansford, 1976)

smN ¼ A, (20)

where s is the real stress range; A and m are the parametersof the given S–N curve.For the stationary narrow-band Gaussian process of the

stress range of the hot spot, the closed form of fatigue

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Table 2

Occurrence probability of fatigue case

h1 (%) h2 (%) h3 (%) h4 (%) h5 (%) h6 (%) h7 (%) h8 (%) h9 (%) h10 (%)

v1 1.122 2.695 1.286 0.475 0.175 0.068 0.028 0.012 0.005 0.005

v2 2.986 7.172 3.421 1.263 0.466 0.180 0.074 0.032 0.015 0.014

v3 3.917 9.407 4.488 1.657 0.611 0.236 0.097 0.042 0.019 0.019

v4 3.828 9.194 4.386 1.619 0.597 0.231 0.095 0.041 0.019 0.018

v5 3.048 7.322 3.493 1.289 0.475 0.184 0.075 0.033 0.015 0.014

v6 2.047 4.917 2.346 0.866 0.319 0.124 0.051 0.022 0.010 0.010

v7 1.179 2.832 1.351 0.499 0.184 0.071 0.029 0.013 0.006 0.006

v8 0.588 1.413 0.674 0.249 0.092 0.035 0.015 0.006 0.003 0.003

v9 0.256 0.614 0.293 0.108 0.040 0.015 0.006 0.003 0.001 0.001

v10 0.142 0.342 0.163 0.060 0.022 0.009 0.004 0.002 0.001 0.001

Fig. 4. Sketch map of tubular joint: (a) location of hot spot; (b) elevation; (c) platform.

Table 3

Optimization information of JZ20-2 NW platform

Initial set Optimal set Lower bound Upper bound Comment

X1 (m) 3.5000 3.1480 2.000 4.000 Diameter of column

X2 (m) 0.0380 0.0213 0.020 0.070 Thickness of column

X3 (m) 0.0600 0.0518 0.030 0.080 Thickness of chief column at joints

X4 (m) 1.3000 1.0000 1.000 2.500 Diameter of piles

X5 (m) 0.0500 0.0388 0.030 0.080 Thickness of piles

X6 (m) 0.7620 0.5012 0.500 1.500 Diameter of braces

X7 (m) 0.0250 0.0200 0.020 0.050 Thickness of braces

X8 (m) 0.6100 0.5213 0.400 1.000 Diameter of braces

X9 (m) 0.0250 0.0200 0.020 0.050 Thickness of braces

X10 (m) 1.3000 1.2200 0.500 1.500 Diameter of braces at joints

X11 (m) 0.0500 0.0429 0.020 0.080 Thickness of braces at joints

f1 (Hz) 1.0050 0.7463 0.630 2.000 Structural fundamental frequency

T (Year) 62.8720 73.1440 40 – Fatigue life of tubular joint

Geo. 1 (1) 43.608 45.602 – 901 Geometric configuration

Geo. 2 (m) 0.230 0.268 6.0 (cm) –

Geo. 3 1.706 2.434 1.00 –

Geo. 4 2.000 2.147 1.00 –

Weight (t) 319.6690 205.7860 – – Weight of jacket

G. Li et al. / Ocean Engineering 35 (2008) 484–493 491

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Fig. 5. Optimization history.

G. Li et al. / Ocean Engineering 35 (2008) 484–493492

damage index can be derived from Eqs. (10) and (20) as

Dk ¼nk

A2ffiffiffiffiffiffiffiffiffiffiffi2m0;k

p� �m

Gm

2� 1

� �, (21)

where G( � ) is the gamma function.Because the ice fatigue environments, Teff and pk, do not

change during the optimization process, the followingresult is obtained according to Eqs. (8) and (21)

Dopt:k

Dini:k

¼nopt:k

nini:k

mopt:0;k

mini:0;k

!m=2

¼T

ini:k

Topt:k

mopt:0;k

mini:0;k

!m=2

¼ rT ;k � ðrkÞm=2.

(22)

From Eq. (22), it is found that the ratio of the fatiguedamage of the optimal design to the initial design for thekth fatigue case is determined by both the ratio of the hotspot stress distribution parameter rk and the ratio of the iceforce period of the conical structure, rT,k. Thus, the fatiguedamage Dk may increase for some fatigue cases anddecrease for other cases after optimization. According toMiner’s Rule, the total damage index D is summation of Dk

for all fatigue cases, which means the occurrence prob-ability of the ice fatigue case should be considered at thesame time. Finally, the total effects of the hot spot stressdistribution, the ice force period of conical structure andthe occurrence probability of each fatigue case resultedin the improvement of the structural fatigue life afteroptimization design.

As shown here, the optimal results for JZ20-2 NWshow that the optimal structure becomes more flexible(the structural fundamental frequency has reduced from1.0050 to 0.7463Hz, see Table 3), which leads to a slightincrease of the hot spot stress, and a dramatic decrease ofthe cyclic number of stress. The final effect improves thefatigue life of the tubular joint. This demonstrates that thecyclic number of stress plays an important role in fatiguedamage if the stress range is relatively acceptable, and

flexible ice-resistant jacket platforms can significantlyreduce the cyclic number of stress.

5. Conclusion

The purpose of this paper is to deal with the optimumdesign of the ice-resistant offshore jacket platforms consider-ing dynamic performance requirements, especially the ice-induced fatigue of tubular joints, in arctic marginal oil fieldssuch as the Bohai Gulf. The applications of the PEM and thehybrid finite element model can improve the computationalefficiency considerably. The numerical example of JZ20-2NW platform in Bohai Gulf demonstrated the effectivenessand feasibility of the proposed optimum design approach, bywhich the economical and ice-resistant offshore jacketplatforms with the less weight and improved fatigue lifecan be obtained. It should be pointed out that the flexibleice-resistant jacket platform can significantly reduce thenumber of stress cycles, which is helpful in improving thefatigue life of the tubular joints; and the structural optimumdesign can achieve the automated redesign with manyfeasible options, which is helpful to decision-making.It should be pointed out that the optimum design of the

ice-resistant offshore jacket platforms in this study doesnot include the constraints of fabrication, transportation,installation and stiffness requirements, and the optimumdesign is limited to the structural members itself, notincluding the shape of the ice-breaking cone that is closelyrelated to the dynamic force. All these need furtherresearch in the future.

Acknowledgments

The support of the National High Technology Researchand Development Program of China (No. 2001AA602015),the National Natural Science Foundation of China (No.50578028 and No. 10421002), the National Basic ResearchProgram of China (No. 2006CB705403) and the Programfor New Century Excellent Talents in University are veryappreciated. The authors gratefully acknowledge the threeanonymous reviewers for their kind suggestions andcomments. The authors also thank Daniel R. Currit(Magnusson Klemencic Associates, Seattle) and RobertOberlies (University of Florida) for their work in improv-ing English presentation of the manuscript, who hadworked together with the authors in the summer of 2004and 2007, respectively, as part of the US National ScienceFoundation Research Experience for Undergraduates inMarine Science and Engineering in China.

References

Almar-Næss, A., 1985. Fatigue Handbook: Offshore Steel Structures.

Tapir.

API RP 2A, 2000. Recommended Practice for Planning, Designing

and Constructing Fixed Offshore Platforms—Working Stress Design.

American Petroleum Institute.

ARTICLE IN PRESSG. Li et al. / Ocean Engineering 35 (2008) 484–493 493

DNV, 2001. Recommended Practice DNV-RP-C203 Fatigue Strength

Analysis of Offshore Steel Structures. October 2001.

Dover, W.D., Madhava Rao, A.G., 1996a. Fatigue in Offshore Structures,

vol. 1. A.A. Balkema, Rotterdam/Brookfield.

Dover, W.D., Madhava Rao, A.G., 1996b. Fatigue in Offshore Structures,

vol. 2. A.A. Balkema, Rotterdam/Brookfield.

Fang, H., Xu, F., Chen, G., 1997. A new method for fatigue life

calculation of tubular joint of offshore platform in ice region. China

Offshore Platform 12 (6), 259–264 (in Chinese).

Fang, H., Duan, M., Xu, F., Shen, Z., Liu, Y., 2000. Reliability analysis of

ice-induced fatigue and damage in offshore engineering structures.

China Ocean Engineering 14 (1), 15–24.

Fang, H., Duan, M., Wu, Y., Fang, X., Xu, F., 2001. An integrated

approach to fatigue life prediction of whole system for offshore

platforms. China Ocean Engineering 15 (2), 177–184.

Fang, H., Duan, M., Jia, X., Xie, B., 2003. Fuzzy fatigue reliability

analysis of offshore platforms in ice-infested waters. China Ocean

Engineering 17 (3), 307–314.

Gandhi, P., Raghava, G., Ramachandra Murthy, D.S., 2000a. Fatigue

behavior of internally ring-stiffened welded steel tubular joints.

Journal of Structural Engineering 126 (7), 809–815.

Gandhi, P., Ramachandra Murthy, D.S., Raghava, G., Madhava Rao,

A.G., 2000b. Fatigue crack growth in stiffened steel tubular joints in

seawater environment. Engineering Structures 22 (10), 1390–1401.

Lin, J.H., Zhang, Y.H., 2004. Pseudo Excitation Method of Random

Vibration. Science Press, Beijing (in Chinese).

Lin, J.H., Zhang, Y.H., 2005. In: de Sila, W. (Ed.), Vibration and Shock

Handbook. CRC Press, Clarence.

Lu, W.F., Li, L.P., Gao, M.D., 1992. Offshore Jacket Platform. Ocean

Press, Beijing (in Chinese).

Nolte, K.G., Hansford, J.E., 1976. Closed-form expressions for deter-

mining the fatigue damage of structures due to ocean waves.

In: Proceedings of the Eighth Annual Offshore Technology

Conference, Offshore Technology Conference, Dallas, TX, vol. 2,

pp. 861–872.

Qu, Y., Yue, Q.J., Bi, X.J., Karna, T., 2006. A random ice force model for

narrow conical structures. Cold Regions Science and Technology 45

(3), 148–157.

Ramachandra Murthy, D.S., Madhava Rao, A.G., Santhakumar, A.R.,

1994. Corrosion fatigue of stiffened offshore steel tubular joints.

Journal of Structural Engineering 120 (7), 1991–2010.

Vugts, J.H., 2005. Fatigue damage assessments and the influence of wave

directionality. Applied Ocean Research 27 (3), 173–185.

Yang, G.J., 2000. Bohai sea ice condition. Journal of Cold Regions

Engineering 14 (2), 54–67.

Yue, Q.J., Bi, X.J., 2000. Ice-induced jacket structure vibrations in Bohai

Sea. Journal of Cold Regions Engineering 14 (2), 81–92.

Yue, Q.J., Liu, L., 2003. Ice problems in Bohai Sea oil exploitation. In:

Proceedings of the 17th International Conference on Port and Ocean

Engineering Under Arctic Conditions, vol. 1, pp. 151–163.

Yue, Q.J., Liu, Y., Zhang, R., Qu, Y., Wang, R., 2007a. Ice-induced

fatigue analysis by spectral approach for offshore jacket platforms

with ice-breaking cones. China Ocean Engineering 21 (1), 1–10.

Yue, Q.J., Qu, Y., Bi, X.J., Karna, T., 2007b. Ice force spectrum on

narrow conical structures. Cold Regions Science and Technology 49

(2), 161–169.