An Analytical Method for Jack-Up Riser’s Fatigue Life...

Research ArticleAn Analytical Method for Jack-Up RiserrsquosFatigue Life Estimation

FengdeWang 1 Wensheng Xiao1 Qi Liu1 Lei Wu 1 and Zhanbin Meng2

1College of Mechanical and Electronic Engineering China University of Petroleum (East China) Qingdao Shandong 266580 China2Drilling Technology Research Institute of Shengli Oilfield Dongying Shandong 257061 China

Correspondence should be addressed to Fengde Wang b14040125supceducn

Received 30 November 2017 Accepted 29 March 2018 Published 10 May 2018

Academic Editor Francesco Pellicano

Copyright copy 2018 Fengde Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to determine whether a special sea area and its sea state are available for the jack-up riser with surface blowout preventersan analytical method is presented to estimate the jack-up riserrsquos wave loading fatigue life in this study In addition an approximateformula is derived to compute the random wave force spectrum of the small-scale structures The results show that the responseof jack-up riser is a narrow band random vibration The infinite water depth dispersion relation between wavenumber and wavefrequency can be used to calculate the wave force spectrum of small-scale structures The riserrsquos response mainly consists of theadditional displacement response The fatigue life obtained by the formula proposed by Steinberg is less than that of the Bendatmethod

1 Introduction

The jack-up platform often uses surface blowout preventers(BOP) to drill exploratory well in shallow sea (under 90mwater depth as shown in Figure 1) This kind of drillingprocess is widely used due to its low cost However as themainstream jack-up platform can be operated in 122m waterdepth the lateral rigidity of the platform decreases withthe increase of water depth which makes the platform in arandom vibration status The platformrsquos vibration can reducethe drilling riserrsquos service life Therefore it is necessary tostudy the random response of the jack-up riser

In this study we choose the jack-up riser as the researchobject The riser is modelled as a Bernoulli-Euler beam thejoint between the riser and the platform can be regarded asa hinge and the riserrsquos lower end is clamped on the seabedThe platformrsquos vibration could be treated as the riserrsquos time-dependent boundary condition and the riser bears the axialcompressive load resulting from surface BOPThus the jack-up riser is modelled as an axial loaded Bernoulli-Euler beamwith time-dependent boundary condition

Many researches have been done on the fatigue analysis ofmarine riser Trim et al [1] introduced a series of experimentson riser models over a range of scales and current conditions

their research results suggest that a key consideration invortex-induced vibration (VIV) fatigue design is the lengthof suppression coverage as well as the nature of the flowto which the bare section of the riser is exposed Basedon the energy equilibrium theory and the forced vibrationexperimental data of a rigid cylinder Xue et al [2] pre-sented a prediction model of VIV fatigue damage forriser accounting for both cross-flow and in-line vibrationsConsidering parametric excitations Zhang and Tang [3]investigatedVIV fatigue analysis of deep-water risers in shearflow Low and Srinil [4]made aVIV fatigue reliability analysisof marine risers with uncertainties in the wake oscillatormodel Xu et al [5] proposed a quick and accurate VIVmodel to predict the fatigue life of marine drilling risers Low[6] extended a timefrequency domain hybrid method forthe fatigue analysis of risers and the linearized frequencydomain approach is proven to be adequate in mild seastates Li and Low [7] studied the influence of low-frequencyvessel motions on the fatigue response of steel catenaryrisers at the touchdown point Elosta et al [8] studied thefatigue performancersquos sensitivity to geotechnical parameterswith a parametric study Low [9] proposed a variancereduction approach for the long-term fatigue analysis ofoffshore structures by Monte Carlo simulation As for beamrsquos

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 5843525 9 pageshttpsdoiorg10115520185843525

2 Mathematical Problems in Engineering

riser

platform

BOP

Figure 1 Realwork conditions of the jack-up riserwith surface BOP

dynamic analysis the vibratory response of a beamwith time-dependent boundary conditions can be obtained by Laplacetransform [10 11] and the Mindlin-Goodman method [12ndash16] In theMindlin-Goodmanmethod the nonhomogeneousboundary conditions are transformed into homogeneousones Therefore the method of separation of variables can beused to solve the beamrsquos response [17] Besides the influenceof axial load on the dynamics of structures has attractedmuch attention as a result of its wide applications [18ndash20]However only few researches on the jack-up riserrsquos fatigue lifeestimation can be found in literature and it is necessary todetermine the riserrsquos random response before drilling an oilwell in the deep-shallow sea The riserrsquos fatigue life can beapplied as a quantitative criterion to judge whether a jack-up riser with surface BOP can be used in a special seaarea Therefore we propose an analytical method to estimatethe jack-up riserrsquos wave loading fatigue life by the Mindlin-Goodman method and the Steinberg method in frequencydomain

The current paper is organized as follows Firstly themathematical model is established in Section 2 And then theanalytical procedure to solve the problem is presented inSection 3 Subsequently a case study is carried out inSection 4 Finally several conclusions are summarized inSection 5

2 Mathematical Model

Since the jack-up riser has characteristics of small inclinationangle and little deformation its lateral motion is governed bythe following equation [3 5]

1198641198681205974119910 (119909 119905)1205971199094 + 119873 (119909) 1205972119910 (119909 119905)1205971199092 + 1198981205972119910 (119909 119905)1205971199052+ 119888120597119910 (119909 119905)120597119905 = 0 (1)

In (1) 119905 is the time 119909 is the coordinatemeasured along theaxis of riser 119910(119909 119905) is the transverse deflection of the beamaxis 119864 is the modulus of elasticity 119868 is the area moment ofinertia119873(119909) is the beam with an axial compressive force 119898is the mass per unit length and 119888 is the damping coefficientIn this study to simplify the computational process the axial

force at the riserrsquos midpoint is used as the average axial force119873 [5]Equation (2) presents the riserrsquos time-dependent bound-

ary conditions at 119909 = 0 and 119909 = 119897120597119910 (0 119905)120597119909 = 0 119910 (0 119905) = 01205971199102 (119897 119905)1205971199092 = 0 119910 (119897 119905) = 119906 (119905) (2)

where 119897 is the length between the riserrsquos clamped end andhinged end and 119906(119905) is the random vibration response of theplatform The wave is a stationary Gaussian random processand the jack-up platform can be modelled as a linear systemBy the Morison equation and the linearization methodproposed by Borgman [21] 119906(119905) can be treated as a stationaryGaussian random process

The beamrsquos transverse deflection 119910(119909 119905) is decomposedinto two parts according to the Mindlin-Goodman methodone is quasi-static displacement 119910119904(119909 119905) resulting from theplatform vibration and the other one is additional displace-ment 119910119889(119909 119905) due to the dynamic inertial force [22]119910 (119909 119905) = 119910119904 (119909 119905) + 119910119889 (119909 119905) (3)

119910119904 (119909 119905) = 119903sum119894=1

119892119894 (119909) 119906119894 (119905) (4)

119910119889 (119909 119905) = infinsum119899=1

120593119899 (119909) 119902119899 (119905) (5)

where 119892119894(119909) is static influence function 120593119899(119909) is the shapefunction of the riser and 119902119899(119905) is the modal coordinate of 119894thmode The boundary conditions of 119910119904(119909 119905) and 119910119889(119909 119905) areformulated by the following equations120597119910119904 (0 119905)120597119909 = 0 119910119904 (0 119905) = 0

1205971199102119904 (119897 119905)1205971199092 = 0 119910119904 (119897 119905) = 119906 (119905) (6)

120597119910119889 (0 119905)120597119909 = 0 119910119889 (0 119905) = 01205971199102119889 (119897 119905)1205971199092 = 0 119910119889 (119897 119905) = 0

(7)

The index 119903 is determined by the boundary conditions forthe jack-up riser 119903 = 1 and 119892119894(119909) can be obtained by initialparametric method [22]

119892 (119909) = 3119909221198972 minus 119909321198973 (8)

Substituting (3) into (1)

11986411986812059741199101198891205971199094 + 11987312059721199101198891205971199092 + 11989812059721199101198891205971199052 + 119888120597119910119889120597119905 = 119865eq (9)

119865eq = minus11986411986812059741199101199041205971199094 minus 11987312059721199101199041205971199092 minus 11989812059721199101199041205971199052 minus 119888120597119910119904120597119905 (10)

where 119865eq is the equivalent load

Mathematical Problems in Engineering 3

As a result of (8) the first itemrsquos value in the right side of(10) is zero therefore (10) can be rewritten as

119865eq = minus11987312059721199101199041205971199092 minus 11989812059721199101199041205971199052 minus 119888120597119910119904120597119905 (11)

The frequency equation of the riserrsquos lateral vibration isderived as

radic1205732 + 120578 sdot tanh (120573119897) = 120573 tan(119897radic1205732 + 120578) (12)

120573 = radic(1205824 + 12057824 )12 minus 1205782 (13)

In (13) 120573 represents solutions of (12) 120582 is frequency coeffi-cient and 120578 is axial force impact factor

1205824 = 1198981205962119864119868 120578 = 119873119864119868

(14)

where 120596 is circular frequencyMode functions of the riserrsquos lateral vibration are formu-

lated by the following equation

120593119899 (119909) = sin 120582119899119909 minus sinh 120582119899119909 + sin 120582119899119897 + sinh 120582119899119897cos 120582119899119897 + cosh 120582119899119897

times (cosh 120582119899119909 minus cos 120582119899119909) (119899 = 1 2 infin)

(15)

3 Analysis

31 Random Vibration Analysis of the Jack-Up Platform Asthe lateral vibration of the riser is mainly induced by theplatformrsquos horizontal vibration the response of the platformhas to be calculated firstly

According to the rules of the China Classification Society(CCS) we choose the single degree of freedom model todescribe jack-up platform Therefore the motion of theplatform can be formulated by the following equation

119898119890 (119905) + 119888 (119905) + 119896119890119906 (119905) = 119901 (119905) (16)

where 119898119890 is equivalent mass of the platform 119896119890 is equivalentbending stiffness and 119901(119905) is random wave load

The frequency response function of the platform isformulated as

119879 (120596) = 1minus1198981198901205962 + 119895119888120596 + 119896119890 (17)

In this study Pierson-Moskowitz spectrum is used tomodel the wave

119878 (120596) = 0781205965 exp(minus 31112059641198672119904 ) (18)

In (18) 119878(120596) is Pierson-Moskowitz spectrum and 119867119904 is thesignificant wave height

According to the Morison equation and Borgmanrsquos lin-earization method the wave force spectrum of one leg atarbitrary height can be obtained by the following equation

119878119891 (120596) = [12119862119863120588119863119900120590radic 8120587120596cosh (119896ℎ)sinh (119896119889) ]2 119878 (120596)

+ [11986211987212058812058711986324 1205962 cosh (119896ℎ)sinh (119896119889) ]

2 119878 (120596) (19)

where 119862119863 is drag force coefficient 119862119872 is inertia forcecoefficient 119896 is wavenumber 120588 is the density of seawater119892 is the gravitational acceleration ℎ is the coordinate alongthe leg 119863119900 is legrsquos equivalent outer diameter and 119889 is waterrsquosdepth

Based on (19) the wave force spectrum of one leg 119878119865(120596)can be obtained by the integral along the direction of thewater depth

119878119865 (120596) = [119862119863120588119863119900120596sinh (119896119889)radic 2120587 int119889

0120590 cosh (119896ℎ) 119889ℎ]2 119878 (120596)

+ [11986211987212058811989212058711986324 tanh (119896119889)]2 119878 (120596) (20)

In (20)

1205902 = intinfin0(120596cosh 119896ℎ

sinh 119896119889)2 119878 (120596) 119889120596 (21)

Under a certain water depth 1205902 is the variance of waterparticlersquos horizontal speed Considering the fact that the lin-earization of the drag force is an approximate treatment theintegral in (21) is calculated by Pierson-Moskowitz spectrumonly and the variance can be obtained as

120590 asymp 025119867119904120596cosh (119896ℎ)sinh (119896119889) (22)

Substituting (22) into (20) the approximate expression ofone legrsquos total wave force spectrum can be obtained as follows

119878119865 (120596)= (119862119863120588119863119900119892119867119904)232120587 [ sinh (2119896119889) + 2119896119889

sinh (2119896119889) ]2119878 (120596)+ [141198621198721205881198921205871198632119900 tanh (119896119889)]

2 119878 (120596) (23)

According to the definition of power spectral densityfunction and the relationship between autocorrelation func-tion and power spectral density function the total wave


force spectrum of the platform 119878119901(120596) can be derived by thefollowing equation

119878119901 (120596)= 9 (119862119863120588119863119900119892119867119904)232120587 [ sinh (2119896119889) + 2119896119889

sinh (2119896119889) ]2119878 (120596)+ [341198621198721205881198921205871198632119900 tanh (119896119889)]

2 119878 (120596) (24)

Based on (17) and (24) the random response spectrum ofthe platform can be obtained as

119878119906 (120596) = |119879 (120596)|2 119878119901 (120596) (25)

32 Random Vibration Analysis of the Jack-Up Riser Substi-tuting (4) and (5) into (9) generates

119864119868 infinsum119899=1

11988941205931198991198891199094 119902119899 (119905) + 119873infinsum119899=1

11988921205931198991198891199092 119902119899 (119905)+ 119898 infinsum119899=1

120593119899 (119909) 119902119899 (119905) + 119888 infinsum119899=1

120593119899 (119909) 119902119899 (119905)= minus119898119892 (119909) (119905) minus 119888119892 (119909) (119905) minus 1198731198892119892 (119909)1198891199092 119906 (119905)

(26)

It can be deduced from (7) that the additional displace-ment has the homogeneous boundary condition thereforeits mode shape functions conform to the following relation-ship

1198641198681198894120593119899 (119909)1198891199094 + 1198731198892120593119899 (119909)1198891199092 minus 1198981205962119899120593119899 (119909) = 0 (27)

Equation (28) can be obtained by substituting (27) into(26)

119898 infinsum119899=1

120593119899 (119909) 119902119899 (119905) + 119888 infinsum119899=1

120593119899 (119909) 119902119899 (119905)+ 1198981205962119899 infinsum

119899=1


(28)

Then (28) can be decoupled to (29) by the orthogonalityconditions119898 119902119899 (119905) + 119888 119902119899 (119905) + 1198981205962119899119902119899 (119905)= minus119898120575119899 (119905) minus 119888120575119899 (119905) minus 119873120576119899119906 (119905) (29)

120575119899 = int1198970119892 (119909) 120593119899 (119909) 119889119909int11989701205932119899 (119909) 119889119909

120576119899 = int119897011989210158401015840 (119909) 120593119899 (119909) 119889119909int11989701205932119899 (119909) 119889119909

(30)

where 120575119899 and 120576119899 are weight coefficients of equivalent load

According to the linear theory the frequency responsefunction of the 119899th-order modal coordinates of the riserrsquosadditional displacement versus 119906(119905) can be obtained in thefollowing equation

119867119899 (120596) = 1198981205751198991205962 minus 119895119888120575119899120596 minus 1198731205761198991198981205962119899 minus 1198981205962 + 119895119888120596 (31)

And then the frequency response function of additionaldisplacement versus 119906(119905) can be obtained by (5) and (31)

119867(119909 120596) = infinsum119899=1

120593119899 (119909)119867119899 (120596) (119899 = 1 2 infin) (32)

Let 119878119889(119909 120596) denote the power spectral density functionof 119910119889(119909 119905) and let 119878119889119906(119909 120596) denote the cross power spectraldensity function between 119910119889(119909 119905) and 119906(119905) Based on (25) and(32) 119878119889(119909 120596) and 119878119889119906(119909 120596) can be obtained as follows

119878119889 (119909 120596) = 119878119906 (120596) |119867 (119909 120596)|2 (33)

119878119889119906 (119909 120596) = 119878119906 (120596)119867 (119909 120596) (34)

The relationship between displacement and stress can beexpressed as

119884 (119909 119905) = 119864119868119882 1205971199102 (119909 119905)1205971199092 (35)

where 119884(119909 119905) denotes stress and 119882 is the bending modulusof the riser

On the basis of (32) and (35) we can get the frequencyresponse function of the riserrsquos stress resulting from 119910119889(119909 119905)119867119884 (119909 120596) = 119864119868119882

infinsum119899=1

119867119899 (120596) 11988921198891199092120593119899 (119909)(119899 = 1 2 infin)

(36)

Then the power spectral density function of stress result-ing from 119910119889(119909 119905) can be obtained by (33) and (36)

119878119889119884 (119909 120596)= 119878119906 (120596) (119864119868119882)2 infinsum

119899=1

1003816100381610038161003816119867119899 (120596)10038161003816100381610038162 [ 11988921198891199092120593119899 (119909)]2

(119899 = 1 2 infin) (37)

And the cross power spectral density function of the stresscan be obtained in the following equation

119878119889119906119884 (119909 120596) = 119878119906 (120596) 119864119868119882infinsum119899=1

119867119899 (120596) [ 11988921198891199092120593119899 (119909)](119899 = 1 2 infin)

(38)

As the definition of autocorrelation function and theWiener-Khintchine principle the power spectral density


Table 1 Parameters of nontubular jointsrsquo S-N curves in sea water(CCS)

Grade 119870 119901B 337 times 1014 40C 141 times 1013 35D 507 times 1011 30E 347 times 1011 30F 210 times 1011 30F2 143 times 1011 30G 833 times 1010 30W 533 times 1010 30

functions of the stress resulting from119910119904(119909 119905) can be generatedin the following equation

119878119906119884 (119909 120596) = 119878119906 (120596) (119864119868119882)2 [ 11988921198891199092119892 (119909)]2

(119899 = 1 2 infin) (39)

According to the spectrumrsquos summation formula thepower spectral density functions of the riserrsquos bending stresscan be obtained by combining (37) (38) and Eq (39)

119878119884 (119909 120596) = 119878119889119884 (119909 120596) + 2Re [119878119889119906119884 (119909 120596)]+ 119878119906119884 (119909 120596) (40)

In Eq (40) Re[119878119889119906119884(119909 120596)] denotes the real part of119878119889119906119884(119909 120596)33 Fatigue Life Estimation Based on the above analysis thefatigue life estimation is carried out subsequently

Because the jack-up riser is operated in corrosion envi-ronment the S-N curve of nontubular joints in sea water isapplied in this studyThe curve is formulated in the followingequation

log (119873) = log (119870) minus 119901 log (119878) (41)

where 119878 is stress range119873 is cycle number119870 is constant and119901 is slope the values of119870 and119898 can be selected from Table 1As for narrow band random vibration Bendat [23]

proposed a formula to estimate structuresrsquo fatigue life basedon the power spectral density function and the formula iswritten as follows

119879119861 = 119870119864 [0] (radic2120590)119898 Γ (1 + 1198982) (42)

In (42) Γ(119911) is Gamma function which is formulated in thefollowing equation

Γ (119911) = intinfin0119905119911minus1119890minus119905 119889119905 (43)

where 119911 is independent variableSteinberg [24] also proposed an empirical equation for

the fatigue life estimation based on the assumption that the

PSD of wave load under infinite water depth conditionPSD of wave load under finite water depth condition

times1012

0

2

4

6

8

10

12

14

S p(

)(

2middots

rad)

02 04 06 08 1 12 14 160 (rads)

Figure 2 The power spectral density (PSD) function of waveforce under finite water depth condition and infinite water depthcondition (119867119904 = 10m)

maximum stress response would not exceed the six timesmean square deviation The equation is presented as

119879119878 = 119870119864 [0] sdot [0683120577119898 + 0271 (2120577)119898 + 0043 (3120577)119898] (44)

4 Case Study

Taking a jack-up platform as the example [25] the quanti-tative research of the riserrsquos fatigue life estimation is carriedout by the method proposed in this study The platformrsquosparameters are listed as follows 119898119890 = 648 times 106 kg 119896119890 =471 times 106Nm 119888 = 877 times 105Nsdotms 119863119900 = 362m 119864119868 =824 times 108Nsdotm2119898 = 461 kgm 119897 = 110m 120578 = 7 times 10minus4mminus2119867119904 = 10m 119862119863 = 20 119862119872 = 20 119889 = 100m 120588 = 1025 kgm3and 119892 = 98ms2

When we compute the spectrum of the random waveforce there are two independent variables of wavenumber119896 and circular frequency 120596 in (24) Under the condition ofthe finite water depth these two variables have the dispersionrelation as

1205962 = 119892119896 tanh (119896119889) (45)

With the increase of water depth (45) is simplified intothe following equation

1205962 = 119892119896 (46)

Under the finite water depth condition three transcen-dental equations need to be solved in the calculation ofthe platformrsquos power spectral density function In this studythe power spectral density function of the platform wascalculated by (45) and (46) respectively The data are listedin Table 2 and plotted in Figure 2 the results show that thereis a minor error between the finite water depth conditionand the infinite water depth condition Therefore (46) isaccurate enough to calculate platformrsquos wave force spectrumBy calculating the mean square deviation of the random


Table 2 The data of the wave force spectrum calculation (119867119904 = 10m)

Sequencenumber

Wavefrequency Wavenumber [tanh (119896119889)]2 [ sinh (2119896119889) + 2119896119889

sinh (2119896119889) ]2 Wave force spectrumunder finite water

depth

Wave force spectrumunder infinite water

depth1 005 00016 00252 39328 00000 000002 010 00032 00958 37437 00000 000003 015 00050 02136 34259 00000 000004 020 00069 03576 30320 00000 000005 025 00089 05061 26177 123 times 1011 101 times 1011

6 030 00113 06578 21811 348 times 1012 311 times 1012

7 035 00141 07876 17886 104 times 1013 987 times 1012

8 040 00174 08840 14753 134 times 1013 132 times 1013

9 045 00213 09451 12552 122 times 1013 121 times 1013

10 050 00258 09773 11220 949 times 1012 946 times 1012

11 055 00310 09919 10510 695 times 1012 695 times 1012

12 060 00368 09975 10188 498 times 1012 498 times 1012

13 065 00431 09993 10062 357 times 1012 357 times 1012

14 070 00500 09998 10018 257 times 1012 257 times 1012

15 075 00574 10000 10005 188 times 1012 188 times 1012

16 080 00653 10000 10001 139 times 1012 139 times 1012

17 085 00737 10000 10000 105 times 1012 105 times 1012

18 090 00827 10000 10000 796 times 1011 796 times 1011

19 095 00921 10000 10000 613 times 1011 613 times 1011

20 100 01020 10000 10000 477 times 1011 477 times 1011

21 105 01125 10000 10000 376 times 1011 376 times 1011

22 110 01235 10000 10000 299 times 1011 299 times 1011

23 115 01349 10000 10000 241 times 1011 241 times 1011

24 120 01469 10000 10000 195 times 1011 195 times 1011

25 125 01594 10000 10000 159 times 1011 159 times 1011

26 130 01724 10000 10000 131 times 1011 131 times 1011

27 135 01860 10000 10000 109 times 1011 109 times 1011

28 140 02000 10000 10000 909 times 1010 908 times 1010

29 145 02145 10000 10000 763 times 1010 763 times 1010

30 150 02296 10000 10000 645 times 1010 645 times 1010

31 155 02452 10000 10000 547 times 1010 548 times 1010

32 160 02612 10000 10000 467 times 1010 468 times 1010

stress response an approximate calculation formula for theplatformrsquos wave force spectrum can be obtained as follows

119878119901 (120596)=

9 (119862119863120588119863119900119892119867119904)232120587 [ sinh (21198891205962119892) + 21198891205962119892sinh (21198891205962119892) ]2

sdot 119878 (120596) + [341198621198721205881198921205871198632119900 tanh (1198891205962119892)]2 119878 (120596)

(47)

The power spectral density function of the platformrsquoslateral displacement can be obtained by (25) and (47) and isplotted in Figure 3 The results show that the power spectraldensity function of the platform has two peak frequencies

which are wave force spectrumrsquos dominant frequency and theplatformrsquos natural frequency and the frequency band of theplatform is mainly between these two peak frequencies

In order to get the response of the riser we have to calcu-late the riserrsquos natural frequencies and weight coefficients ofequivalent load firstly In (37) there is an item of (11988921205931198991198891199092)2which can reduce the seriesrsquo convergence rate Therefore it isnecessary to consider the influence of high-order modesThefirst ten orders data are calculated and listed in Tables 3 and4

In this study we take the randomwave in one-year returnperiod (119867119904 = 204m) as the excitation Based on (47) thepower spectral density function of the stress resulting fromadditional displacement was calculated by (37) and plotted inFigure 4 and the power spectral density function of the stress


02 04 06 08 1 12 14 16 18 20 (rads)

0

02

04

06

08

1

12

14

S u(

)(Ｇ

2middots

rad)

Figure 3The power spectral density function of the platform (119867119904 =10m)

Table 3 Natural frequencies of the riserrsquos lateral vibration

119899 120582119899 120596119899(rads)1 00312 130432 00618 510043 00910 1107184 01200 1926765 01490 2966966 01776 4216137 02062 5687058 02349 7379739 02636 92871510 02922 1141480

Table 4 Weight coefficients of the riserrsquos equivalent load

119899 120575119899 1205761198991 03478 599 times 10minus5

2 minus02250 180 times 10minus3

3 01558 416 times 10minus5


5 00934 278 times 10minus5


7 00673 207 times 10minus5


9 00518 165 times 10minus5


originated from 119910119904(119909 119905) was computed by (39) and presentedin Figure 5 Next the power spectral density function of thetotal stress response was obtained by (40) and plotted inFigure 6

The power spectral density function of the riser hasthe maximum value when its vibration frequency is equalto the platformrsquos natural frequency From the comparativeanalysis between Figures 4 and 6 it can be deduced thatthe jack-up riserrsquos response mainly comprises the additionaldisplacement response

Subsequently the mean square deviation of the riserrsquosrandom stress response was calculated and depicted in

(rads)

times1016

x (m)0 20 40 60 80 100

004081216202468

1012

S dY(x

)

(０2

sra

d)

Figure 4 PSD of the riserrsquos stress response resulting from additionaldisplacement (119867119904 = 204m)

(rads)

times1013

x (m)0 20 40 60 80 1000040812162

02468

1012

S uY(x

)

(０2

sra

d)Figure 5 PSD of the riserrsquos stress response resulting from quasi-static displacement (119867119904 = 204m)

times1016

(rads) x (m)0 20 40 60 80 1000040812162

02468

1012

S Y(x

)

(０2

sra

d)

Figure 6 PSD of the riserrsquos stress response (119867119904 = 204m)

times108

395

4

405

41

415

42

425

(P

a)

50 4025 50 853520 4515 55 60 65 70 75 8030 90 9510 110

105

100

x (m)

Figure 7 The mean square deviation of the riserrsquos random stressresponse (119867119904 = 204m)

Figure 7 The data shows that the riser has the maximumstress value at its clamped end In order to determine the typeof the riserrsquos random response the spectral width factor of theriserrsquos stress response was computed by (48)The result showsthat the spectral width factor at 119909 = 0m is approximately


Table 5 Fatigue life estimation of the riser

Significant wave height (m) Fatigue life (d)Bendat method Steinberg method119867119904 = 05 508 times 106 478 times 106119867119904 = 06 188 times 105 177 times 105119867119904 = 07 183 times 104 172 times 104119867119904 = 08 265 times 103 250 times 103119867119904 = 09 49588 46660119867119904 = 10 11495 10817119867119904 = 11 3268 3075119867119904 = 12 1120 1054

equal to 016 (less than 03) therefore the riserrsquos response is anarrow band random vibration

120585 = radic1 minus ( 119864 [0]119864 [119875])2 (48)

where 120585 is the spectral width factor 119864[119875] is the peakfrequency

119864 [119875] = 12120587radicintinfin

minusinfin1205964119878119884 (119909 120596) 119889120596intinfin

minusinfin1205962119878119884 (119909 120596) 119889120596 (49)

Figure 7 shows that the riserrsquos clamped end (119909 = 0m)is the most dangerous part Therefore the fatigue life of thewhole riser system is decided by the fatigue life of the riserrsquosclamped end In this study we chose grade B in Table 1 toestimate the riserrsquos fatigue life The fatigue life of the riserwas computed by (42) and (44) respectively and the data arelisted in Table 5 The data in Table 5 shows that the Steinbergmethod generates the shorter fatigue life for the sake ofsafety the Steinberg method is recommended to estimate thestructuresrsquo fatigue life It also can be deduced that the riser inthis study cannot be used above level 3 sea condition

5 Conclusions

Based on the Mindlin-Goodman method and the formulaproposed by Steinberg an analytical procedure for the jack-up riserrsquos fatigue life estimation is proposed During theresearch we also derived an approximate formula to solve thewave force spectrum

It is found that the wave loading vibration of jack-up riser is a narrow band random vibration The infinitewater depth dispersion relation between wavenumber 119896 andcircular frequency 120596 is accurate enough to compute the waveforce spectrum The additional displacement response is themain component of the riserrsquos response and the fatigue lifegenerated by the Steinberg method is less than that of theBendat method

The method proposed in this paper can be used todetermine whether a special sea area and its sea state areavailable for a jack-up riser with surface BOP Besides themethod can be extended to estimate the fatigue life of theBernoulli-Euler beam with other boundary conditions underrandom support excitation

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

Theauthors would like to acknowledge theMinistry of Indus-try and Information Technology of China for supporting thisstudy through the project ldquoJack-Up Platform Brand Project(II)rdquo (Grant no 10200001-15-ZC0607-0018)

References

[1] A D Trim H Braaten H Lie and M A Tognarelli ldquoExperi-mental investigation of vortex-induced vibration of longmarinerisersrdquo Journal of Fluids and Structures vol 21 no 3 pp 335ndash361 2005

[2] H Xue W Tang and X Qu ldquoPrediction and analysis of fatiguedamage due to cross-flow and in-line VIV for marine risers innon-uniform currentrdquoOcean Engineering vol 83 no 2 pp 52ndash62 2014

[3] J Zhang and Y Tang ldquoFatigue analysis of deep-water risersunder vortex-induced vibration considering parametric excita-tionsrdquo Journal of Coastal Research vol 73 pp 652ndash659 2015

[4] Y M Low and N Srinil ldquoVIV fatigue reliability analysis ofmarine risers with uncertainties in the wake oscillator modelrdquoEngineering Structures vol 106 pp 96ndash108 2016

[5] J Xu DWang HHuangMDuan J Gu andC An ldquoA vortex-induced vibration model for the fatigue analysis of a marinedrilling riserrdquo Ships and Offshore Structures vol 12 supplement1 pp S280ndashS287 2017

[6] YM Low ldquoExtending a timefrequency domain hybridmethodfor riser fatigue analysisrdquo Applied Ocean Research vol 33 no 2pp 79ndash87 2011

[7] F Z Li and Y M Low ldquoInfluence of low-frequency vesselmotions on the fatigue response of steel catenary risers at thetouchdown pointrdquo Ships and Offshore Structures vol 9 no 2pp 134ndash148 2014

[8] H Elosta S Huang and A Incecik ldquoWave loading fatigue reli-ability and uncertainty analyses for geotechnical pipeline mod-elsrdquo Ships and Offshore Structures vol 9 no 4 pp 450ndash4632014

[9] Y M Low ldquoA variance reduction technique for long-termfatigue analysis of offshore structures using Monte Carlo sim-ulationrdquo Engineering Structures vol 128 pp 283ndash295 2016

[10] G A Nothmann ldquoVibration of a cantilever beam with pre-scribed end motionrdquo Transactions of ASME Journal of AppliedMechanics vol 15 pp 327ndash334 1948

[11] T C Yen and S Kao ldquoVibration of a beam-mass system withtime-dependent boundary conditionsrdquo Transactions of ASMEJournal of Applied Mechanics vol 26 pp 353ndash356 1959

[12] S Y Lee and SM Lin ldquoDynamic analysis of nonuniformbeamswith time-dependent elastic boundary conditionsrdquo Transac-tions of ASME Journal of Applied Mechanics vol 63 no 2 pp474ndash478 1996

[13] S Y Lee and S M Lin ldquoNon-uniform timoshenko beams withtime-dependent elastic boundary conditionsrdquo Journal of Soundand Vibration vol 217 no 2 pp 223ndash238 1998

[14] S M Lin and S Y Lee ldquoThe forced vibration and boundarycontrol of pretwisted timoshenko beams with general time


dependent elastic boundary conditionsrdquo Journal of Sound andVibration vol 254 no 1 pp 69ndash90 2002

[15] M Li ldquoAnalytical study on the dynamic response of a beamwithaxial force subjected to generalized support excitationsrdquo Journalof Sound and Vibration vol 338 pp 199ndash216 2015

[16] Y-W Kim ldquoDynamic analysis of Timoshenko beam subjectedto support motionsrdquo Journal of Mechanical Science and Technol-ogy vol 30 no 9 pp 4167ndash4176 2016

[17] R DMindlin and L E Goodman ldquoBeam vibrations with time-dependent boundary conditionsrdquo Journal of Applied Mechanicsvol 17 no 4 pp 377ndash380 1950

[18] S Caddemi and I Calio ldquoThe influence of the axial force on thevibration of the Euler-Bernoulli beamwith an arbitrary numberof cracksrdquo Archive of Applied Mechanics vol 82 no 6 pp 827ndash839 2012

[19] Y Yesilce ldquoFree and forced vibrations of an axially-loadedTimoshenko multi-span beam carrying a number of variousconcentrated elementsrdquo Shock and Vibration vol 19 no 4 pp735ndash752 2012

[20] J Murin M Aminbaghai V Kutis and J Hrabovsky ldquoModalanalysis of the FGM beams with effect of axial force underlongitudinal variable elastic Winkler foundationrdquo EngineeringStructures vol 49 pp 234ndash247 2013

[21] L E Borgman ldquoSpectral analysis of ocean wave forces on pil-ingrdquo Journal of the Waterways Harbor and Coastal EngineeringDivision ASCE vol 83 no 2 pp 129ndash156 1967

[22] R W Clough and J Penzien Eds Dynamics of StructuresComputers amp Structures University Ave 3rd edition 1995

[23] J S Bendat ldquoProbability functions for random responsesrdquoNASA report on Contact NASA-5-4590 1964

[24] D S Steinberg Vibration Analysis for Electronic EquipmentJohn Wiley amp Sons New York NY USA 2nd edition 1988

[25] Y G Tang G G Shen and L Q LiuHydrodynamics of MarineStructures Tianjin University Press Tianjin China 2008

Hindawiwwwhindawicom Volume 2018

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Volume 2018

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Submit your manuscripts atwwwhindawicom


riser

platform

BOP

Figure 1 Realwork conditions of the jack-up riserwith surface BOP

dynamic analysis the vibratory response of a beamwith time-dependent boundary conditions can be obtained by Laplacetransform [10 11] and the Mindlin-Goodman method [12ndash16] In theMindlin-Goodmanmethod the nonhomogeneousboundary conditions are transformed into homogeneousones Therefore the method of separation of variables can beused to solve the beamrsquos response [17] Besides the influenceof axial load on the dynamics of structures has attractedmuch attention as a result of its wide applications [18ndash20]However only few researches on the jack-up riserrsquos fatigue lifeestimation can be found in literature and it is necessary todetermine the riserrsquos random response before drilling an oilwell in the deep-shallow sea The riserrsquos fatigue life can beapplied as a quantitative criterion to judge whether a jack-up riser with surface BOP can be used in a special seaarea Therefore we propose an analytical method to estimatethe jack-up riserrsquos wave loading fatigue life by the Mindlin-Goodman method and the Steinberg method in frequencydomain

The current paper is organized as follows Firstly themathematical model is established in Section 2 And then theanalytical procedure to solve the problem is presented inSection 3 Subsequently a case study is carried out inSection 4 Finally several conclusions are summarized inSection 5

2 Mathematical Model

Since the jack-up riser has characteristics of small inclinationangle and little deformation its lateral motion is governed bythe following equation [3 5]

1198641198681205974119910 (119909 119905)1205971199094 + 119873 (119909) 1205972119910 (119909 119905)1205971199092 + 1198981205972119910 (119909 119905)1205971199052+ 119888120597119910 (119909 119905)120597119905 = 0 (1)

In (1) 119905 is the time 119909 is the coordinatemeasured along theaxis of riser 119910(119909 119905) is the transverse deflection of the beamaxis 119864 is the modulus of elasticity 119868 is the area moment ofinertia119873(119909) is the beam with an axial compressive force 119898is the mass per unit length and 119888 is the damping coefficientIn this study to simplify the computational process the axial

force at the riserrsquos midpoint is used as the average axial force119873 [5]Equation (2) presents the riserrsquos time-dependent bound-

ary conditions at 119909 = 0 and 119909 = 119897120597119910 (0 119905)120597119909 = 0 119910 (0 119905) = 01205971199102 (119897 119905)1205971199092 = 0 119910 (119897 119905) = 119906 (119905) (2)

where 119897 is the length between the riserrsquos clamped end andhinged end and 119906(119905) is the random vibration response of theplatform The wave is a stationary Gaussian random processand the jack-up platform can be modelled as a linear systemBy the Morison equation and the linearization methodproposed by Borgman [21] 119906(119905) can be treated as a stationaryGaussian random process

The beamrsquos transverse deflection 119910(119909 119905) is decomposedinto two parts according to the Mindlin-Goodman methodone is quasi-static displacement 119910119904(119909 119905) resulting from theplatform vibration and the other one is additional displace-ment 119910119889(119909 119905) due to the dynamic inertial force [22]119910 (119909 119905) = 119910119904 (119909 119905) + 119910119889 (119909 119905) (3)

119910119904 (119909 119905) = 119903sum119894=1

119892119894 (119909) 119906119894 (119905) (4)

119910119889 (119909 119905) = infinsum119899=1

120593119899 (119909) 119902119899 (119905) (5)

where 119892119894(119909) is static influence function 120593119899(119909) is the shapefunction of the riser and 119902119899(119905) is the modal coordinate of 119894thmode The boundary conditions of 119910119904(119909 119905) and 119910119889(119909 119905) areformulated by the following equations120597119910119904 (0 119905)120597119909 = 0 119910119904 (0 119905) = 0

1205971199102119904 (119897 119905)1205971199092 = 0 119910119904 (119897 119905) = 119906 (119905) (6)

120597119910119889 (0 119905)120597119909 = 0 119910119889 (0 119905) = 01205971199102119889 (119897 119905)1205971199092 = 0 119910119889 (119897 119905) = 0

(7)

The index 119903 is determined by the boundary conditions forthe jack-up riser 119903 = 1 and 119892119894(119909) can be obtained by initialparametric method [22]

119892 (119909) = 3119909221198972 minus 119909321198973 (8)

Substituting (3) into (1)

11986411986812059741199101198891205971199094 + 11987312059721199101198891205971199092 + 11989812059721199101198891205971199052 + 119888120597119910119889120597119905 = 119865eq (9)

119865eq = minus11986411986812059741199101199041205971199094 minus 11987312059721199101199041205971199092 minus 11989812059721199101199041205971199052 minus 119888120597119910119904120597119905 (10)

where 119865eq is the equivalent load






120573 = radic(1205824 + 12057824 )12 minus 1205782 (13)


1205824 = 1198981205962119864119868 120578 = 119873119864119868

(14)





(15)

3 Analysis



119898119890 (119905) + 119888 (119905) + 119896119890119906 (119905) = 119901 (119905) (16)



119879 (120596) = 1minus1198981198901205962 + 119895119888120596 + 119896119890 (17)


119878 (120596) = 0781205965 exp(minus 31112059641198672119904 ) (18)



119878119891 (120596) = [12119862119863120588119863119900120590radic 8120587120596cosh (119896ℎ)sinh (119896119889) ]2 119878 (120596)

+ [11986211987212058812058711986324 1205962 cosh (119896ℎ)sinh (119896119889) ]

2 119878 (120596) (19)



119878119865 (120596) = [119862119863120588119863119900120596sinh (119896119889)radic 2120587 int119889

0120590 cosh (119896ℎ) 119889ℎ]2 119878 (120596)

+ [11986211987212058811989212058711986324 tanh (119896119889)]2 119878 (120596) (20)

In (20)

1205902 = intinfin0(120596cosh 119896ℎ

sinh 119896119889)2 119878 (120596) 119889120596 (21)


120590 asymp 025119867119904120596cosh (119896ℎ)sinh (119896119889) (22)


119878119865 (120596)= (119862119863120588119863119900119892119867119904)232120587 [ sinh (2119896119889) + 2119896119889

sinh (2119896119889) ]2119878 (120596)+ [141198621198721205881198921205871198632119900 tanh (119896119889)]

2 119878 (120596) (23)




119878119901 (120596)= 9 (119862119863120588119863119900119892119867119904)232120587 [ sinh (2119896119889) + 2119896119889

sinh (2119896119889) ]2119878 (120596)+ [341198621198721205881198921205871198632119900 tanh (119896119889)]

2 119878 (120596) (24)


119878119906 (120596) = |119879 (120596)|2 119878119901 (120596) (25)


119864119868 infinsum119899=1

11988941205931198991198891199094 119902119899 (119905) + 119873infinsum119899=1

11988921205931198991198891199092 119902119899 (119905)+ 119898 infinsum119899=1

120593119899 (119909) 119902119899 (119905) + 119888 infinsum119899=1


(26)


1198641198681198894120593119899 (119909)1198891199094 + 1198731198892120593119899 (119909)1198891199092 minus 1198981205962119899120593119899 (119909) = 0 (27)


119898 infinsum119899=1

120593119899 (119909) 119902119899 (119905) + 119888 infinsum119899=1

120593119899 (119909) 119902119899 (119905)+ 1198981205962119899 infinsum

119899=1


(28)


120575119899 = int1198970119892 (119909) 120593119899 (119909) 119889119909int11989701205932119899 (119909) 119889119909

120576119899 = int119897011989210158401015840 (119909) 120593119899 (119909) 119889119909int11989701205932119899 (119909) 119889119909

(30)





119867(119909 120596) = infinsum119899=1

120593119899 (119909)119867119899 (120596) (119899 = 1 2 infin) (32)


119878119889 (119909 120596) = 119878119906 (120596) |119867 (119909 120596)|2 (33)

119878119889119906 (119909 120596) = 119878119906 (120596)119867 (119909 120596) (34)


119884 (119909 119905) = 119864119868119882 1205971199102 (119909 119905)1205971199092 (35)



infinsum119899=1

119867119899 (120596) 11988921198891199092120593119899 (119909)(119899 = 1 2 infin)

(36)


119878119889119884 (119909 120596)= 119878119906 (120596) (119864119868119882)2 infinsum

119899=1

1003816100381610038161003816119867119899 (120596)10038161003816100381610038162 [ 11988921198891199092120593119899 (119909)]2

(119899 = 1 2 infin) (37)


119878119889119906119884 (119909 120596) = 119878119906 (120596) 119864119868119882infinsum119899=1

119867119899 (120596) [ 11988921198891199092120593119899 (119909)](119899 = 1 2 infin)

(38)






119878119906119884 (119909 120596) = 119878119906 (120596) (119864119868119882)2 [ 11988921198891199092119892 (119909)]2

(119899 = 1 2 infin) (39)


119878119884 (119909 120596) = 119878119889119884 (119909 120596) + 2Re [119878119889119906119884 (119909 120596)]+ 119878119906119884 (119909 120596) (40)



log (119873) = log (119870) minus 119901 log (119878) (41)



119879119861 = 119870119864 [0] (radic2120590)119898 Γ (1 + 1198982) (42)






times1012

0

2

4

6

8

10

12

14

S p(

)(

2middots

rad)

02 04 06 08 1 12 14 160 (rads)



119879119878 = 119870119864 [0] sdot [0683120577119898 + 0271 (2120577)119898 + 0043 (3120577)119898] (44)

4 Case Study



1205962 = 119892119896 tanh (119896119889) (45)


1205962 = 119892119896 (46)




Sequencenumber



depth



6 030 00113 06578 21811 348 times 1012 311 times 1012

7 035 00141 07876 17886 104 times 1013 987 times 1012

8 040 00174 08840 14753 134 times 1013 132 times 1013

9 045 00213 09451 12552 122 times 1013 121 times 1013

10 050 00258 09773 11220 949 times 1012 946 times 1012

11 055 00310 09919 10510 695 times 1012 695 times 1012

12 060 00368 09975 10188 498 times 1012 498 times 1012

13 065 00431 09993 10062 357 times 1012 357 times 1012

14 070 00500 09998 10018 257 times 1012 257 times 1012

15 075 00574 10000 10005 188 times 1012 188 times 1012

16 080 00653 10000 10001 139 times 1012 139 times 1012

17 085 00737 10000 10000 105 times 1012 105 times 1012

18 090 00827 10000 10000 796 times 1011 796 times 1011

19 095 00921 10000 10000 613 times 1011 613 times 1011

20 100 01020 10000 10000 477 times 1011 477 times 1011

21 105 01125 10000 10000 376 times 1011 376 times 1011

22 110 01235 10000 10000 299 times 1011 299 times 1011

23 115 01349 10000 10000 241 times 1011 241 times 1011

24 120 01469 10000 10000 195 times 1011 195 times 1011

25 125 01594 10000 10000 159 times 1011 159 times 1011

26 130 01724 10000 10000 131 times 1011 131 times 1011

27 135 01860 10000 10000 109 times 1011 109 times 1011

28 140 02000 10000 10000 909 times 1010 908 times 1010

29 145 02145 10000 10000 763 times 1010 763 times 1010

30 150 02296 10000 10000 645 times 1010 645 times 1010

31 155 02452 10000 10000 547 times 1010 548 times 1010

32 160 02612 10000 10000 467 times 1010 468 times 1010


119878119901 (120596)=

9 (119862119863120588119863119900119892119867119904)232120587 [ sinh (21198891205962119892) + 21198891205962119892sinh (21198891205962119892) ]2

sdot 119878 (120596) + [341198621198721205881198921205871198632119900 tanh (1198891205962119892)]2 119878 (120596)

(47)






02 04 06 08 1 12 14 16 18 20 (rads)

0

02

04

06

08

1

12

14

S u(

)(Ｇ

2middots

rad)



119899 120582119899 120596119899(rads)1 00312 130432 00618 510043 00910 1107184 01200 1926765 01490 2966966 01776 4216137 02062 5687058 02349 7379739 02636 92871510 02922 1141480


119899 120575119899 1205761198991 03478 599 times 10minus5


3 01558 416 times 10minus5


5 00934 278 times 10minus5


7 00673 207 times 10minus5


9 00518 165 times 10minus5





(rads)

times1016

x (m)0 20 40 60 80 100

004081216202468

1012

S dY(x

)

(０2

sra

d)


(rads)

times1013

x (m)0 20 40 60 80 1000040812162

02468

1012

S uY(x

)

(０2

sra


times1016

(rads) x (m)0 20 40 60 80 1000040812162

02468

1012

S Y(x

)

(０2

sra

d)


times108

395

4

405

41

415

42

425

(P

a)

50 4025 50 853520 4515 55 60 65 70 75 8030 90 9510 110

105

100

x (m)







120585 = radic1 minus ( 119864 [0]119864 [119875])2 (48)


119864 [119875] = 12120587radicintinfin


minusinfin1205962119878119884 (119909 120596) 119889120596 (49)


5 Conclusions






Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018













120573 = radic(1205824 + 12057824 )12 minus 1205782 (13)


1205824 = 1198981205962119864119868 120578 = 119873119864119868

(14)





(15)

3 Analysis



119898119890 (119905) + 119888 (119905) + 119896119890119906 (119905) = 119901 (119905) (16)



119879 (120596) = 1minus1198981198901205962 + 119895119888120596 + 119896119890 (17)


119878 (120596) = 0781205965 exp(minus 31112059641198672119904 ) (18)



119878119891 (120596) = [12119862119863120588119863119900120590radic 8120587120596cosh (119896ℎ)sinh (119896119889) ]2 119878 (120596)

+ [11986211987212058812058711986324 1205962 cosh (119896ℎ)sinh (119896119889) ]

2 119878 (120596) (19)



119878119865 (120596) = [119862119863120588119863119900120596sinh (119896119889)radic 2120587 int119889

0120590 cosh (119896ℎ) 119889ℎ]2 119878 (120596)

+ [11986211987212058811989212058711986324 tanh (119896119889)]2 119878 (120596) (20)

In (20)

1205902 = intinfin0(120596cosh 119896ℎ

sinh 119896119889)2 119878 (120596) 119889120596 (21)


120590 asymp 025119867119904120596cosh (119896ℎ)sinh (119896119889) (22)


119878119865 (120596)= (119862119863120588119863119900119892119867119904)232120587 [ sinh (2119896119889) + 2119896119889

sinh (2119896119889) ]2119878 (120596)+ [141198621198721205881198921205871198632119900 tanh (119896119889)]

2 119878 (120596) (23)




119878119901 (120596)= 9 (119862119863120588119863119900119892119867119904)232120587 [ sinh (2119896119889) + 2119896119889

sinh (2119896119889) ]2119878 (120596)+ [341198621198721205881198921205871198632119900 tanh (119896119889)]

2 119878 (120596) (24)


119878119906 (120596) = |119879 (120596)|2 119878119901 (120596) (25)


119864119868 infinsum119899=1

11988941205931198991198891199094 119902119899 (119905) + 119873infinsum119899=1

11988921205931198991198891199092 119902119899 (119905)+ 119898 infinsum119899=1

120593119899 (119909) 119902119899 (119905) + 119888 infinsum119899=1


(26)


1198641198681198894120593119899 (119909)1198891199094 + 1198731198892120593119899 (119909)1198891199092 minus 1198981205962119899120593119899 (119909) = 0 (27)


119898 infinsum119899=1

120593119899 (119909) 119902119899 (119905) + 119888 infinsum119899=1

120593119899 (119909) 119902119899 (119905)+ 1198981205962119899 infinsum

119899=1


(28)


120575119899 = int1198970119892 (119909) 120593119899 (119909) 119889119909int11989701205932119899 (119909) 119889119909

120576119899 = int119897011989210158401015840 (119909) 120593119899 (119909) 119889119909int11989701205932119899 (119909) 119889119909

(30)





119867(119909 120596) = infinsum119899=1

120593119899 (119909)119867119899 (120596) (119899 = 1 2 infin) (32)


119878119889 (119909 120596) = 119878119906 (120596) |119867 (119909 120596)|2 (33)

119878119889119906 (119909 120596) = 119878119906 (120596)119867 (119909 120596) (34)


119884 (119909 119905) = 119864119868119882 1205971199102 (119909 119905)1205971199092 (35)



infinsum119899=1

119867119899 (120596) 11988921198891199092120593119899 (119909)(119899 = 1 2 infin)

(36)


119878119889119884 (119909 120596)= 119878119906 (120596) (119864119868119882)2 infinsum

119899=1

1003816100381610038161003816119867119899 (120596)10038161003816100381610038162 [ 11988921198891199092120593119899 (119909)]2

(119899 = 1 2 infin) (37)


119878119889119906119884 (119909 120596) = 119878119906 (120596) 119864119868119882infinsum119899=1

119867119899 (120596) [ 11988921198891199092120593119899 (119909)](119899 = 1 2 infin)

(38)






119878119906119884 (119909 120596) = 119878119906 (120596) (119864119868119882)2 [ 11988921198891199092119892 (119909)]2

(119899 = 1 2 infin) (39)


119878119884 (119909 120596) = 119878119889119884 (119909 120596) + 2Re [119878119889119906119884 (119909 120596)]+ 119878119906119884 (119909 120596) (40)



log (119873) = log (119870) minus 119901 log (119878) (41)



119879119861 = 119870119864 [0] (radic2120590)119898 Γ (1 + 1198982) (42)






times1012

0

2

4

6

8

10

12

14

S p(

)(

2middots

rad)

02 04 06 08 1 12 14 160 (rads)



119879119878 = 119870119864 [0] sdot [0683120577119898 + 0271 (2120577)119898 + 0043 (3120577)119898] (44)

4 Case Study



1205962 = 119892119896 tanh (119896119889) (45)


1205962 = 119892119896 (46)




Sequencenumber



depth



6 030 00113 06578 21811 348 times 1012 311 times 1012

7 035 00141 07876 17886 104 times 1013 987 times 1012

8 040 00174 08840 14753 134 times 1013 132 times 1013

9 045 00213 09451 12552 122 times 1013 121 times 1013

10 050 00258 09773 11220 949 times 1012 946 times 1012

11 055 00310 09919 10510 695 times 1012 695 times 1012

12 060 00368 09975 10188 498 times 1012 498 times 1012

13 065 00431 09993 10062 357 times 1012 357 times 1012

14 070 00500 09998 10018 257 times 1012 257 times 1012

15 075 00574 10000 10005 188 times 1012 188 times 1012

16 080 00653 10000 10001 139 times 1012 139 times 1012

17 085 00737 10000 10000 105 times 1012 105 times 1012

18 090 00827 10000 10000 796 times 1011 796 times 1011

19 095 00921 10000 10000 613 times 1011 613 times 1011

20 100 01020 10000 10000 477 times 1011 477 times 1011

21 105 01125 10000 10000 376 times 1011 376 times 1011

22 110 01235 10000 10000 299 times 1011 299 times 1011

23 115 01349 10000 10000 241 times 1011 241 times 1011

24 120 01469 10000 10000 195 times 1011 195 times 1011

25 125 01594 10000 10000 159 times 1011 159 times 1011

26 130 01724 10000 10000 131 times 1011 131 times 1011

27 135 01860 10000 10000 109 times 1011 109 times 1011

28 140 02000 10000 10000 909 times 1010 908 times 1010

29 145 02145 10000 10000 763 times 1010 763 times 1010

30 150 02296 10000 10000 645 times 1010 645 times 1010

31 155 02452 10000 10000 547 times 1010 548 times 1010

32 160 02612 10000 10000 467 times 1010 468 times 1010


119878119901 (120596)=

9 (119862119863120588119863119900119892119867119904)232120587 [ sinh (21198891205962119892) + 21198891205962119892sinh (21198891205962119892) ]2

sdot 119878 (120596) + [341198621198721205881198921205871198632119900 tanh (1198891205962119892)]2 119878 (120596)

(47)






02 04 06 08 1 12 14 16 18 20 (rads)

0

02

04

06

08

1

12

14

S u(

)(Ｇ

2middots

rad)



119899 120582119899 120596119899(rads)1 00312 130432 00618 510043 00910 1107184 01200 1926765 01490 2966966 01776 4216137 02062 5687058 02349 7379739 02636 92871510 02922 1141480


119899 120575119899 1205761198991 03478 599 times 10minus5


3 01558 416 times 10minus5


5 00934 278 times 10minus5


7 00673 207 times 10minus5


9 00518 165 times 10minus5





(rads)

times1016

x (m)0 20 40 60 80 100

004081216202468

1012

S dY(x

)

(０2

sra

d)


(rads)

times1013

x (m)0 20 40 60 80 1000040812162

02468

1012

S uY(x

)

(０2

sra


times1016

(rads) x (m)0 20 40 60 80 1000040812162

02468

1012

S Y(x

)

(０2

sra

d)


times108

395

4

405

41

415

42

425

(P

a)

50 4025 50 853520 4515 55 60 65 70 75 8030 90 9510 110

105

100

x (m)







120585 = radic1 minus ( 119864 [0]119864 [119875])2 (48)


119864 [119875] = 12120587radicintinfin


minusinfin1205962119878119884 (119909 120596) 119889120596 (49)


5 Conclusions






Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018










119878119901 (120596)= 9 (119862119863120588119863119900119892119867119904)232120587 [ sinh (2119896119889) + 2119896119889

sinh (2119896119889) ]2119878 (120596)+ [341198621198721205881198921205871198632119900 tanh (119896119889)]

2 119878 (120596) (24)


119878119906 (120596) = |119879 (120596)|2 119878119901 (120596) (25)


119864119868 infinsum119899=1

11988941205931198991198891199094 119902119899 (119905) + 119873infinsum119899=1

11988921205931198991198891199092 119902119899 (119905)+ 119898 infinsum119899=1

120593119899 (119909) 119902119899 (119905) + 119888 infinsum119899=1


(26)


1198641198681198894120593119899 (119909)1198891199094 + 1198731198892120593119899 (119909)1198891199092 minus 1198981205962119899120593119899 (119909) = 0 (27)


119898 infinsum119899=1

120593119899 (119909) 119902119899 (119905) + 119888 infinsum119899=1

120593119899 (119909) 119902119899 (119905)+ 1198981205962119899 infinsum

119899=1


(28)


120575119899 = int1198970119892 (119909) 120593119899 (119909) 119889119909int11989701205932119899 (119909) 119889119909

120576119899 = int119897011989210158401015840 (119909) 120593119899 (119909) 119889119909int11989701205932119899 (119909) 119889119909

(30)





119867(119909 120596) = infinsum119899=1

120593119899 (119909)119867119899 (120596) (119899 = 1 2 infin) (32)


119878119889 (119909 120596) = 119878119906 (120596) |119867 (119909 120596)|2 (33)

119878119889119906 (119909 120596) = 119878119906 (120596)119867 (119909 120596) (34)


119884 (119909 119905) = 119864119868119882 1205971199102 (119909 119905)1205971199092 (35)



infinsum119899=1

119867119899 (120596) 11988921198891199092120593119899 (119909)(119899 = 1 2 infin)

(36)


119878119889119884 (119909 120596)= 119878119906 (120596) (119864119868119882)2 infinsum

119899=1

1003816100381610038161003816119867119899 (120596)10038161003816100381610038162 [ 11988921198891199092120593119899 (119909)]2

(119899 = 1 2 infin) (37)


119878119889119906119884 (119909 120596) = 119878119906 (120596) 119864119868119882infinsum119899=1

119867119899 (120596) [ 11988921198891199092120593119899 (119909)](119899 = 1 2 infin)

(38)






119878119906119884 (119909 120596) = 119878119906 (120596) (119864119868119882)2 [ 11988921198891199092119892 (119909)]2

(119899 = 1 2 infin) (39)


119878119884 (119909 120596) = 119878119889119884 (119909 120596) + 2Re [119878119889119906119884 (119909 120596)]+ 119878119906119884 (119909 120596) (40)



log (119873) = log (119870) minus 119901 log (119878) (41)



119879119861 = 119870119864 [0] (radic2120590)119898 Γ (1 + 1198982) (42)






times1012

0

2

4

6

8

10

12

14

S p(

)(

2middots

rad)

02 04 06 08 1 12 14 160 (rads)



119879119878 = 119870119864 [0] sdot [0683120577119898 + 0271 (2120577)119898 + 0043 (3120577)119898] (44)

4 Case Study



1205962 = 119892119896 tanh (119896119889) (45)


1205962 = 119892119896 (46)




Sequencenumber



depth



6 030 00113 06578 21811 348 times 1012 311 times 1012

7 035 00141 07876 17886 104 times 1013 987 times 1012

8 040 00174 08840 14753 134 times 1013 132 times 1013

9 045 00213 09451 12552 122 times 1013 121 times 1013

10 050 00258 09773 11220 949 times 1012 946 times 1012

11 055 00310 09919 10510 695 times 1012 695 times 1012

12 060 00368 09975 10188 498 times 1012 498 times 1012

13 065 00431 09993 10062 357 times 1012 357 times 1012

14 070 00500 09998 10018 257 times 1012 257 times 1012

15 075 00574 10000 10005 188 times 1012 188 times 1012

16 080 00653 10000 10001 139 times 1012 139 times 1012

17 085 00737 10000 10000 105 times 1012 105 times 1012

18 090 00827 10000 10000 796 times 1011 796 times 1011

19 095 00921 10000 10000 613 times 1011 613 times 1011

20 100 01020 10000 10000 477 times 1011 477 times 1011

21 105 01125 10000 10000 376 times 1011 376 times 1011

22 110 01235 10000 10000 299 times 1011 299 times 1011

23 115 01349 10000 10000 241 times 1011 241 times 1011

24 120 01469 10000 10000 195 times 1011 195 times 1011

25 125 01594 10000 10000 159 times 1011 159 times 1011

26 130 01724 10000 10000 131 times 1011 131 times 1011

27 135 01860 10000 10000 109 times 1011 109 times 1011

28 140 02000 10000 10000 909 times 1010 908 times 1010

29 145 02145 10000 10000 763 times 1010 763 times 1010

30 150 02296 10000 10000 645 times 1010 645 times 1010

31 155 02452 10000 10000 547 times 1010 548 times 1010

32 160 02612 10000 10000 467 times 1010 468 times 1010


119878119901 (120596)=

9 (119862119863120588119863119900119892119867119904)232120587 [ sinh (21198891205962119892) + 21198891205962119892sinh (21198891205962119892) ]2

sdot 119878 (120596) + [341198621198721205881198921205871198632119900 tanh (1198891205962119892)]2 119878 (120596)

(47)






02 04 06 08 1 12 14 16 18 20 (rads)

0

02

04

06

08

1

12

14

S u(

)(Ｇ

2middots

rad)



119899 120582119899 120596119899(rads)1 00312 130432 00618 510043 00910 1107184 01200 1926765 01490 2966966 01776 4216137 02062 5687058 02349 7379739 02636 92871510 02922 1141480


119899 120575119899 1205761198991 03478 599 times 10minus5


3 01558 416 times 10minus5


5 00934 278 times 10minus5


7 00673 207 times 10minus5


9 00518 165 times 10minus5





(rads)

times1016

x (m)0 20 40 60 80 100

004081216202468

1012

S dY(x

)

(０2

sra

d)


(rads)

times1013

x (m)0 20 40 60 80 1000040812162

02468

1012

S uY(x

)

(０2

sra


times1016

(rads) x (m)0 20 40 60 80 1000040812162

02468

1012

S Y(x

)

(０2

sra

d)


times108

395

4

405

41

415

42

425

(P

a)

50 4025 50 853520 4515 55 60 65 70 75 8030 90 9510 110

105

100

x (m)







120585 = radic1 minus ( 119864 [0]119864 [119875])2 (48)


119864 [119875] = 12120587radicintinfin


minusinfin1205962119878119884 (119909 120596) 119889120596 (49)


5 Conclusions






Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018












119878119906119884 (119909 120596) = 119878119906 (120596) (119864119868119882)2 [ 11988921198891199092119892 (119909)]2

(119899 = 1 2 infin) (39)


119878119884 (119909 120596) = 119878119889119884 (119909 120596) + 2Re [119878119889119906119884 (119909 120596)]+ 119878119906119884 (119909 120596) (40)



log (119873) = log (119870) minus 119901 log (119878) (41)



119879119861 = 119870119864 [0] (radic2120590)119898 Γ (1 + 1198982) (42)






times1012

0

2

4

6

8

10

12

14

S p(

)(

2middots

rad)

02 04 06 08 1 12 14 160 (rads)



119879119878 = 119870119864 [0] sdot [0683120577119898 + 0271 (2120577)119898 + 0043 (3120577)119898] (44)

4 Case Study



1205962 = 119892119896 tanh (119896119889) (45)


1205962 = 119892119896 (46)




Sequencenumber



depth



6 030 00113 06578 21811 348 times 1012 311 times 1012

7 035 00141 07876 17886 104 times 1013 987 times 1012

8 040 00174 08840 14753 134 times 1013 132 times 1013

9 045 00213 09451 12552 122 times 1013 121 times 1013

10 050 00258 09773 11220 949 times 1012 946 times 1012

11 055 00310 09919 10510 695 times 1012 695 times 1012

12 060 00368 09975 10188 498 times 1012 498 times 1012

13 065 00431 09993 10062 357 times 1012 357 times 1012

14 070 00500 09998 10018 257 times 1012 257 times 1012

15 075 00574 10000 10005 188 times 1012 188 times 1012

16 080 00653 10000 10001 139 times 1012 139 times 1012

17 085 00737 10000 10000 105 times 1012 105 times 1012

18 090 00827 10000 10000 796 times 1011 796 times 1011

19 095 00921 10000 10000 613 times 1011 613 times 1011

20 100 01020 10000 10000 477 times 1011 477 times 1011

21 105 01125 10000 10000 376 times 1011 376 times 1011

22 110 01235 10000 10000 299 times 1011 299 times 1011

23 115 01349 10000 10000 241 times 1011 241 times 1011

24 120 01469 10000 10000 195 times 1011 195 times 1011

25 125 01594 10000 10000 159 times 1011 159 times 1011

26 130 01724 10000 10000 131 times 1011 131 times 1011

27 135 01860 10000 10000 109 times 1011 109 times 1011

28 140 02000 10000 10000 909 times 1010 908 times 1010

29 145 02145 10000 10000 763 times 1010 763 times 1010

30 150 02296 10000 10000 645 times 1010 645 times 1010

31 155 02452 10000 10000 547 times 1010 548 times 1010

32 160 02612 10000 10000 467 times 1010 468 times 1010


119878119901 (120596)=

9 (119862119863120588119863119900119892119867119904)232120587 [ sinh (21198891205962119892) + 21198891205962119892sinh (21198891205962119892) ]2

sdot 119878 (120596) + [341198621198721205881198921205871198632119900 tanh (1198891205962119892)]2 119878 (120596)

(47)






02 04 06 08 1 12 14 16 18 20 (rads)

0

02

04

06

08

1

12

14

S u(

)(Ｇ

2middots

rad)



119899 120582119899 120596119899(rads)1 00312 130432 00618 510043 00910 1107184 01200 1926765 01490 2966966 01776 4216137 02062 5687058 02349 7379739 02636 92871510 02922 1141480


119899 120575119899 1205761198991 03478 599 times 10minus5


3 01558 416 times 10minus5


5 00934 278 times 10minus5


7 00673 207 times 10minus5


9 00518 165 times 10minus5





(rads)

times1016

x (m)0 20 40 60 80 100

004081216202468

1012

S dY(x

)

(０2

sra

d)


(rads)

times1013

x (m)0 20 40 60 80 1000040812162

02468

1012

S uY(x

)

(０2

sra


times1016

(rads) x (m)0 20 40 60 80 1000040812162

02468

1012

S Y(x

)

(０2

sra

d)


times108

395

4

405

41

415

42

425

(P

a)

50 4025 50 853520 4515 55 60 65 70 75 8030 90 9510 110

105

100

x (m)







120585 = radic1 minus ( 119864 [0]119864 [119875])2 (48)


119864 [119875] = 12120587radicintinfin


minusinfin1205962119878119884 (119909 120596) 119889120596 (49)


5 Conclusions






Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018










Sequencenumber



depth



6 030 00113 06578 21811 348 times 1012 311 times 1012

7 035 00141 07876 17886 104 times 1013 987 times 1012

8 040 00174 08840 14753 134 times 1013 132 times 1013

9 045 00213 09451 12552 122 times 1013 121 times 1013

10 050 00258 09773 11220 949 times 1012 946 times 1012

11 055 00310 09919 10510 695 times 1012 695 times 1012

12 060 00368 09975 10188 498 times 1012 498 times 1012

13 065 00431 09993 10062 357 times 1012 357 times 1012

14 070 00500 09998 10018 257 times 1012 257 times 1012

15 075 00574 10000 10005 188 times 1012 188 times 1012

16 080 00653 10000 10001 139 times 1012 139 times 1012

17 085 00737 10000 10000 105 times 1012 105 times 1012

18 090 00827 10000 10000 796 times 1011 796 times 1011

19 095 00921 10000 10000 613 times 1011 613 times 1011

20 100 01020 10000 10000 477 times 1011 477 times 1011

21 105 01125 10000 10000 376 times 1011 376 times 1011

22 110 01235 10000 10000 299 times 1011 299 times 1011

23 115 01349 10000 10000 241 times 1011 241 times 1011

24 120 01469 10000 10000 195 times 1011 195 times 1011

25 125 01594 10000 10000 159 times 1011 159 times 1011

26 130 01724 10000 10000 131 times 1011 131 times 1011

27 135 01860 10000 10000 109 times 1011 109 times 1011

28 140 02000 10000 10000 909 times 1010 908 times 1010

29 145 02145 10000 10000 763 times 1010 763 times 1010

30 150 02296 10000 10000 645 times 1010 645 times 1010

31 155 02452 10000 10000 547 times 1010 548 times 1010

32 160 02612 10000 10000 467 times 1010 468 times 1010


119878119901 (120596)=

9 (119862119863120588119863119900119892119867119904)232120587 [ sinh (21198891205962119892) + 21198891205962119892sinh (21198891205962119892) ]2

sdot 119878 (120596) + [341198621198721205881198921205871198632119900 tanh (1198891205962119892)]2 119878 (120596)

(47)






02 04 06 08 1 12 14 16 18 20 (rads)

0

02

04

06

08

1

12

14

S u(

)(Ｇ

2middots

rad)



119899 120582119899 120596119899(rads)1 00312 130432 00618 510043 00910 1107184 01200 1926765 01490 2966966 01776 4216137 02062 5687058 02349 7379739 02636 92871510 02922 1141480


119899 120575119899 1205761198991 03478 599 times 10minus5


3 01558 416 times 10minus5


5 00934 278 times 10minus5


7 00673 207 times 10minus5


9 00518 165 times 10minus5





(rads)

times1016

x (m)0 20 40 60 80 100

004081216202468

1012

S dY(x

)

(０2

sra

d)


(rads)

times1013

x (m)0 20 40 60 80 1000040812162

02468

1012

S uY(x

)

(０2

sra


times1016

(rads) x (m)0 20 40 60 80 1000040812162

02468

1012

S Y(x

)

(０2

sra

d)


times108

395

4

405

41

415

42

425

(P

a)

50 4025 50 853520 4515 55 60 65 70 75 8030 90 9510 110

105

100

x (m)







120585 = radic1 minus ( 119864 [0]119864 [119875])2 (48)


119864 [119875] = 12120587radicintinfin


minusinfin1205962119878119884 (119909 120596) 119889120596 (49)


5 Conclusions






Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018









02 04 06 08 1 12 14 16 18 20 (rads)

0

02

04

06

08

1

12

14

S u(

)(Ｇ

2middots

rad)



119899 120582119899 120596119899(rads)1 00312 130432 00618 510043 00910 1107184 01200 1926765 01490 2966966 01776 4216137 02062 5687058 02349 7379739 02636 92871510 02922 1141480


119899 120575119899 1205761198991 03478 599 times 10minus5


3 01558 416 times 10minus5


5 00934 278 times 10minus5


7 00673 207 times 10minus5


9 00518 165 times 10minus5





(rads)

times1016

x (m)0 20 40 60 80 100

004081216202468

1012

S dY(x

)

(０2

sra

d)


(rads)

times1013

x (m)0 20 40 60 80 1000040812162

02468

1012

S uY(x

)

(０2

sra


times1016

(rads) x (m)0 20 40 60 80 1000040812162

02468

1012

S Y(x

)

(０2

sra

d)


times108

395

4

405

41

415

42

425

(P

a)

50 4025 50 853520 4515 55 60 65 70 75 8030 90 9510 110

105

100

x (m)







120585 = radic1 minus ( 119864 [0]119864 [119875])2 (48)


119864 [119875] = 12120587radicintinfin


minusinfin1205962119878119884 (119909 120596) 119889120596 (49)


5 Conclusions






Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018












120585 = radic1 minus ( 119864 [0]119864 [119875])2 (48)


119864 [119875] = 12120587radicintinfin


minusinfin1205962119878119884 (119909 120596) 119889120596 (49)


5 Conclusions






Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018




























Journal of












Journal of







Volume 2018






Volume 2018








An Analytical Method for Jack-Up Riser’s Fatigue Life...

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Transcript of An Analytical Method for Jack-Up Riser’s Fatigue Life...