reliability analysis of universal joint of a compliant platform 2010 fatigue and fracture of...

8/3/2019 Reliability Analysis of Universal Joint of a Compliant Platform 2010 Fatigue and Fracture of Engineering Materials an

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doi: 10.1111/j.1460-2695.2010.01453.x

Reliability analysis of universal joint of a compliant platform

M . M . Z A H E E R a n d N . I S L A M

Department of Civil Engineering, Jamia Millia Islamia, New Delhi 110025, India

Received in final form 29 December 2009

A B S T R A C T The paper describesa methodology forcomputation of fatigue reliability of universal jointin an articulated offshore tower. Failure criteria were formulated using the conventionalPalmgren-Miner rule (S-N curve approach) and the fracture mechanics (F-M) principle. The dynamic analysis of double hinged articulated tower under wind and waves is carriedout in time domain. The response histories of hinge shear stresses are employed for thereliability analysis. Advanced first-order reliability method and Monte Carlo simulationmethod were used to estimate the reliability. Various parametric studies were carried out, which yield important information for the reliability based design. The S-N curve ap-proach yieldsa significantly conservative estimate of probability of failure when compared

to the F-M approach.Keywords articulatedjoint; fatigue life;FORM;fracture mechanics;offshore platforms;sensitivity.

N O M E N C L AT U R E A= fatigue strength coefficienta =crack size

a = initial crack sizeacr =critical crack size

B =stress modelling errorC =Paris coefficient

[C] =damping matrixC , m =crack propagation parameters

C d =drag coefficientD =accumulated damageda / d N =crack growth rate

E [ .] =expectation f i =zero crossing frequency

f x( X ) =probability density function of the random variable X g ( X ) = limit state function

H s =significant wave height[I] =mass matrix K = intercept of the S-N curve at R equals to one

[K] =stiffness matrix L = load effect

{ M

} =forcing function

m =slope of the S-N curvem = fatigue exponentmo =nth moment of the stress spectrum N =number of cycles to fatigue failure

Correspondence: M. M. Zaheer. E-mail: [email protected]

408 c 2010 Blackwell Publishing Ltd.Fatigue Fract Engng Mater Struct 33 , 408419

Fatigue & Fracture ofEngineering Materials & Structures


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R E L I AB I L I TY A NA LY SI S O F U N I VE R S AL J O IN T OF A CO M P LI A N T P L ATF O R M 409

P f =probability of failure R =structural resistanceT F = time to failureT L = lifetime of the structureT z =zero crossing period

u( z) = wind velocity Y (a) = function of crack geometry y j = value of the variable at the design point in the standard normal space

j =sensitivity of a random variable F = MinerPalmgren damage index at failure

K = range of stress intensity factor=Gamma function

i = fraction of time spent in the i th sea state = Wirschings wide band correction factor=stress parameter

=cumulative probability function of a standard normal distribution( ) = inverse of the standardized normal distribution function

R =constant amplitude stress range R

=nominal stress range

i = r.m.s. value of the stress process in the i th sea state{ },{}and {} =structural acceleration, velocity and displacement vectors

I N T R O D U C T I O N

An articulated tower platform as shown in Fig. 1 is acompliant offshore structure which is provided with largebuoyancy chambers near the water surface. Being com-pliant in nature, they are free to move with wind and waves. The time variant loadings due to wind and waves

cause fluctuating shear stresses in the articulated joint of the tower. In fact, the platform is attached to the sea bedthrough it that allows controlled articulation under envi-ronmental loadings, and its failure will lead severe con-sequences. Therefore, its fatigue and fracture reliability assessment is of great importance.Low-frequency response is an inherent characteristic

of compliant platforms and in general is caused by dy-namic wind and wave loadings. Fatigue life assessmentsof the compliant articulated tower require treatment of the low-frequency responses which can aggravate the fa-tigue damage in thearticulationpoints. The impacton thefatigue damage due to the low-frequency responses, how-ever, cannotbe fully captured by the traditional frequency domain analysis. In order to offset and complement thefrequencydomain approach,a time domaindirectintegra-tion approach has to be adopted for fatigue life assessmentof the articulated tower.

L I T E R AT U R E S U RV E Y

Fatigue cracking of structuraldetails in offshorestructuresdue to cyclic loading has gained considerable attention inthe past few years. Numerous research studies have been

conducted in this field on both the theoretical and practi-cal aspects. Consequently, a great number of papers havebeen published resulting in various topics relating to fa-tigue assessment. Reliability analysis of structures under wave loading (Moseset al.,1 Wirshing, 2 and Committeeon Structural Reliability of Offshore Structures 3) formsa distinct class of problems in the literature called relia-

bility and safety of offshore structures. Sedillot et al.4

dis-cussed the design and fatigue analysis of a laminated rub-ber articulated joint for the deep-water gravity articulatedtower.

Fatigue and wear studies were carried out on Baldpatecompliant tower by Chen and Will. 5 They concludedthat the impact of fatigue damage due wind loads aresignificant and cannot be ignored. Siddiqui and Ahmad 6

studied fatigue and fracture reliability of tension leg plat-form tethers under wind and wave loading. Dong andHong 7 presented a master S-N curve approach using themesh-insensitive structural stress parameter and its di-rect linkage to fracture mechanics (F-M) principle. Withthe master S-N curve method, tubular joints in offshorestructures and pipe joints for riser applications can becollapsed into a single curve, referred to as the masterS-N curve. In their study, its applications were illustratedby using various offshore/marine examples. Yamashitaand Sekita8 carried out fatigue damage analysis on off-shore wind turbines subjected to wind and wave loads. Jinet al.9 proposed a methodology of system reliability-basedassessment for the single point mooring jacket platforms.It was concluded that system reliability-based assessmentmethod could provide important and reliable referential

c 2010 Blackwell Publishing Ltd.Fatigue Fract Engng Mater Struct 33 , 408419


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410 M. M. ZAHEER AND N. ISLAM

Fig. 1 Articulated tower.

suggestions for the design, maintenance and repair of single point mooring jackets. Pillai andVeena 10 describe amethodology forcomputation of reliability of members of fixed offshore platforms with respect to fatigue using F-Mprinciple. Uraga and Moan 11 investigated alternate S-Nand F-M formulations of fatigue. They include a crack growth formulation based on bi-linear crack growth law,considering both segments of the crack growth law ascorrelated and non-correlated in the fracture probability calculation.Despite continuing research in the area of reliability

analysis of offshore structures against dynamic forces,there is still scope of work in this area because it is as-sociated with large number of uncertainties and a highdegree of complexity. In addition, very little literatureexists on the reliability analysis of articulated joint. Thispaper deals with a comprehensive study on the fatigue re-liability analysis of articulated tower under wind and waveenvironment.

T H E A RT I C U L AT I O N S Y S T E M

It is composed of two different devices as shown in Figs 2and 3. The rubber articulated joint, located at the upperpart of the fixed base and on the vertical axis of the struc-ture, supports the vertical and horizontal reactions. Thetorsional moment is resisted by a wide torsional frame asshown in Fig. 3. This frame is provided in the horizontaldiametral plane of the joint, having two opposite anglesconnected to the base structure and the two other onesto the steel tower. These devices collectively allowed animposed rotation due to the tilting of the tower by envi-ronmental forces.

Fig. 2 Articulated joint [Courtesy: Sedillot et al.4).

Fig. 3 The articulation system: torsional frame (Courtesy: Sedillotet al.4).

Description of the ball joint

The ball joint for the articulated tower has been takenfrom Deep Water Gravity Tower, 4 which consists of a series of laminated rubber pads inserted between twohemispherical steel shells, as shown in Fig. 2. A centralcircular pad, about 1.20 m in diameter, is surrounded by two rows of eight trapezoidal pads. Each pad is composedof a sandwich of layers of rubber, 10 to 15 mm thick,and of curved metallic plates, about 6 mm thick. The twoextreme plates are 40 mm thick. The pads are firmly fastened to the inner shell which

is connected to the central vertical member of the steeltower, and to the outer shell which is located in thebase central member. When a rotation is imposed to the



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articulation point, the inner shell rotates and opposes atangential distortion to the pads, which develop only neg-ligible resisting forces, mainly due to their low shear stiff-ness. Further, these elastomeric pads have a very highstiffness in the axial direction, so that variation of verticalreaction, due to live loads on deck or to effects of environ-

mental loads. Results in very little vertical displacement.

S I M U L AT I O N O F S E A S TAT E

The description of sea states have been given by threeparameter Weibull distribution, as given by Karadenizet al.12 The long-term distribution of the significant waveheight based on these parameters is given as under

f ( H s ) =C B

H s AB

C 1e(

H s AB )C

(1)

where A = lower limit of H s ; B = scale parameter andC =shape parameter. The parameters A, B and C have been obtained from ascatter diagram of North Sea location. These constantsare found to be as A = 0.594; B = 2.290 and C = 1.385(Mathinsen and Bitner-Gregersen 13)Corresponding to a known significant wave height H s ,

zero crossing period T z and mean wind velocity u( z) may be obtained assuming the same probability of occurrencefor T z and u( z) as H s . Sarpakaya and Isaacson14 gave thefollowing empirical relations for significant wave height H s and zero crossing period T z. In this formulation, windand waves are considered to be in correlated fashion.

T z = 32 H s g (2)

u( z) =g H s

0.283. (3)

In the present study, sea states have been simulated by using Eqs (1)(3). Twelve significant wave heights areselected such that f ( H s ) H s 1; i.e. the whole areaunder the ( f ( H s ) versus H s ) curve have been divided into12 rectangular strips of width H s . The area of each stripprovides the magnitude of the corresponding probabil-ity of occurrence of sea state. These results have beenpresented in Table 2. The synthetic waves corresponding to a particular sea

state, represented by a seasurface elevation, are generatedby using wave superposition technique. The linearizedsmall amplitude wave theory allows the summation of velocity potential, wave elevation and water particle kine-matics of individual waves to form a random wave madeup of a number of components. The generated syntheticrandom wave is considered to be adequately representedby the summation of K number of sinusoids (harmonics)

in random phase. The series representation of sea surfaceelevation (t ) is given by the equation

(t ) = K

i =1 Ai cos(ki x i t +i ), (4)

where

Ai = 2[ S ( i ) i ] (5)in which, Ai is the amplitude of the i th component wave,ki is the wave number of the i th component wave, i isthe wave frequency of the i th component wave, i is thephase angle of the i th component wave randomly chosenbetween 0 and 2 , and following the normal distribution, K is the number of wave harmonics considered in thesimulation, x is the structural displacement, S () is thespectral density value of one sided sea surface elevationspectrum at the frequency i . The selection of frequency i is done such that these

frequencies are uncorrelated so that they do not consti-tute harmonics with each other. First, the range of thefrequencies, from the lowest frequency, min to the high-est frequency, max is divided into (k 1) sub-ranges withthe dividing frequencies constituting a power series of:

1 =min +max min

k 1(6)

2 =1C k (7)

i = i 1C k (8)

k1 =1C (k2)k , (9)

where

C k =max1

1k2

. (10)

Then, the secondary dividing frequencies 1 , 2 ,3 , . . . , k1 are chosen, at random, in respective sub-ranges. The initial frequency 0 is set equal to min andthe last one is k =max. The selection is done with theaid of a random-number-generation process program on

the computer. Finally the component frequency, i andits bandwidth, i is calculated as:

i =0.5 i 1 +i (11)

i = i i 1 i =1, 2, 3, . . . , k. (12) The above process of random selection of component

frequency is repeated for each run of each sea surface el-evation spectrum. The random phase angle i must bechosen such that the resultant function (t ) follows the



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Gaussian distribution. This has been done with the gen-eration of random numbers normally distributed between0 and 2 .Basedon theperformedstudies, theasymptoticapproach

to the Gaussian distribution is found time-consuming forthe number of component waves above 50 and hence the

simulation is carried out with 50 component waves.

M AT H E M AT I C A L F O R M U L AT I O N

An articulated tower as shown in Fig. 1 has been consid-ered for mathematical formulation in the present study. A universal joint is provided near the sea bed. The re-liability assessment of universal joint against fatigue andfracture is the objective of the present study. For this pur-pose, dynamic analysis of the tower was carried out intime domain and then using its response, reliability analysis is performed. The formulation for these two analyses

is presented in the following sections.

Dynamic analysis

The dynamic analysis of the platform hasbeen carried outunder the following assumptions:

The flexural deformations of the platform are assumed tobe small as compared to its displacement as a rigid body.

The topsides wind loading estimation is based on theSimiu wind spectrum assuming an overall drag coefficient, C d

= 2.0. An assumption is made that the wind direction isperpendicular to the largest projected area.

The tower has uniform properties over the segments of uniform diameters. The equations of equilibrium at each time step modify

elements of the stiffness matrix to account for fluctuatingbuoyancy.

Equation of motion

The governing equation of motion is given as:

[I] +[C] +[K]{}= { M}, (13) where [I] is the mass matrix consisting of structural massand added mass moment of inertia. [C], the damping ma-trix and [K], the stiffness matrix. { }, and {}are thestructural acceleration, velocity and displacement vectors.

{ M}is the forcing function at any instant of time due to wind and waves.

Having determined the various matrices and force vec-tors, the next step is to integrate the equation of motion,i.e. Eq. (13). In the present study, equation of motion issolved in time domain using Newmarks beta integrationmethod. Responses thus obtained, are employed for thesubsequent reliability analysis.

R E L I A B I L I T Y A N A LY S I S

The reliability of a structure is concerned with the cal-culation and prediction of the probability of limit state violation at any stage during its entire life. A limit statefunction or failure equation is a mathematical represen-tation of a particular limit state of failure. This failure

equation is mainly governed by the failure criteria andrandom variables. In the present study, reliability assess-ment of articulated joint has been carried out against fa-tigue and fracture limit state using probabilistic analysissolver NESSUS.

Fatigue reliability formulation

Assumptions

Following assumptions are made for the reliability analy-sis:

Platform failure is defined as the failure of the articulated joint.

There is no correlation among the two articulated joints. There is no inspection or repair program of articulated

joints before failure.

Limit state function

The reliability of a structure can be determined based ona failure equation in terms of basic variables for structuralresistance and loads. Mathematically, the failure equation g ( X ) can be expressed as

g ( X ) = R L, (14) where R =structural resistance and L = load effect Thesurface g ( X ) =0 which separates thesafe andunsafe

regions is termed as the limit surface. The probability of failure is thus expressed as

P f =Prob[ g ( X ) 0] = g ( X )< 0 f x ( X ) d x, (15) where f x( X ) is the probability density function of therandom variable X . The failure probability in terms of areliability index can be expressed as:

P f = ( ) , (16) where =cumulative probability function of a standardnormal distribution. The complement, 1 p f , is accord-ingly referred to as the reliability. Equation (16) may alsobe written in terms of the reliability index as follows:

= 1 P f , (17) where () is the inverse of the standardized normal dis-tribution function.



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FAT I G U E A N A LY S I S A N D D E S I G NA P P R O A C H E S

Two general approaches are widely reported in the liter-ature for fatigue and fracture limit state formulations: (i)S-N curve approach (Kjerengtroen and Wirsching 15) and(ii) F-M approach (Torng and Wirsching 16).

The S-N approach is based on experimental measure-ment of fatigue life in terms of cycles to failure for differ-ent stress ranges. On the other hand, the F-M approachis based on the existence of an initial crack in a stress freestructure. Several investigators have demonstrated thatthe PalmgrenMiner hypothesis cannot be used to accu-rately predict material behaviour. However, metal fatigueis an extremely complicated process involving two dis-tinct phases (crack initiation and propagation) and subjectto influence by many factors such as mean stress, load-ing rate, surface conditions, etc. It is not surprising thata very simple model would fail to provide an accurate

description of a complex phenomenon. Nevertheless, thePierson-Moskowitz (P-M rule) remains, for general de-sign purposes, as the most viable fatigue failure criterionin the case of variable amplitude loading.

The characteristic S-N approach

The characteristic S-N approach is based on the assump-tion that fatigue damage accumulation is a linear phe-nomenon (Miners rule). According to miners rule, thefatigue life of a structure under different stress ranges isthe weighted sum of the individual lives at constant stress( ) as given by the S-N curves, with each being weightedaccording to fractional exposure to that level of stressrange. The S-N model is generally used for high cycle fa-tigue and the basic equation that represents the S-N curveis given by:

N = K m R

(18)

where N =number of cycles to fatigue failure, K =the in-tercept of the S-N curve at R equals to one, R =constantamplitude stress range at N and m =slope of the S-Ncurve. Equation (6) can also be written as:

log N

=log K

m log R (19)

The ALP is subjected to combined action of wind and waves which are random in nature. Consequently, thestress process is stochastic in nature and therefore, eachstress is a random variable. In this approach, it has beenassumed that thedamage on thestructural componentperload cycle D j is constant at a given stress range Rj and isgiven by:

D j =1

N ( Rj )(20)

where N ( Rj ) is the number of cycles to cause failure atstressrange Rj . Thetotal damageaccumulated in lifetimeof the structure T L is given by:

D = N (T L)

j =1

1 N Rj

(21)

in which N (T L) is the total number of stress cycles tofailure at stress range R j in time T L.

Using the S-N curve, accumulated damage D is writtenas:

D = N (T L)

j =1

m Rj K

. (22)

If N (T L) is relatively large, then the associated uncertainty with the sum is to be very small and the sum can bereplaced by its expected value. Therefore,

E N (T L)

j =1 m Rj = E [ N (T L)] E m Rj . (23)

Stress ranges are Rayleigh distributed for a narrow bandGaussian process, and the mean value of the stress rangefollows directly as:

E m Ri = 0 (2 p)m p p exp 12 p p2

d p (24)

= (2 2)m m p 1 +m2

. (25)

Hence, the accumulated damage D is given as:

D =1

K E [ N (T L)] E m R . (26)

The total damage can be obtained by summing up theaccumulated damage over all the sea states which yields:

D =T L K

(27)

where, is a stress parameter given as:

= 2 2m

1 +m2

n

i

=1

f i i mi (28)

in which is a Gamma function, f i = 12 m2mo , is zerocrossing frequency of the stress process in the i th seastate, mo = 0

n S ()d is the nth moment of the stressspectrum, i is the fraction of time spent in the i th seastate to account for long-term sea state effect. i = mois the r.m.s. value of the stress process in the i th sea state.

Failure occurs if D F , where F is the value of the MinerPalmgren damage index at failure. Failure is as-sumed to occur when the damage measured D =1. This



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formulation has the advantage of simplicity, but the dam-age measure D is not related to a direct physical quantity such as crack length, and it ignores sequence effects.Letting D = F , the basic damage expression can be

expressed in terms of time to failure T F of the articulated joint as:

T F = F K . (29)

The time of failure T F for the joint, considering theassociated factors as random variables may be given as:

T F = F K i

Bmi . (30)

Define the intended service life of the structure as T L. Then the probability of failure of the joint is

P f = P (T F T L). (31)

The fatigue failure occurs when the random variable T F is less than T L. Thus, the limit state function is:

g ( X ) = F K i

Bmi T L. (32)

If x1 = F , x2 = K i , x3 = Bi then limit state function inEq. (32) can be written as:

g ( X ) = g ( x1, x2, x3) = x1 x2 xm3 T L. (33)

The probability of failure P f is given as:

P f =

Prob( T F

T L)

=P [ g ( X )

0]. (34)

The reliability is thus obtained by the expression:

=1( P f ), (35) where ( ) is the inverse of standardized normal distri-bution function.

The fracture mechanics approach

F-M approach is concerned with the study of the be-haviour of structures containing flaws or cracks and isbased on crack growth data. However, this model will notmodel the crack initiation phase. This approach is moredetailed andit involves examiningcrackgrowthanddeter-mining the number of loadcycles that are neededfor smallinitial defects to grow into cracks large enough to causefracture. The growth rate is proportional to the stressrange. It is expressed in terms of a stress intensity factor

K , which accounts for the magnitude of the stress, cur-rent crack size, and weld and joint details. Linear-elasticF-M relate the growth of a crack of size a to the numberof fatigue cycles N . The most common relationship is the

Paris fatigue crack growth law (Paris17)

dad N =C ( K )

m K > 0 (36)

in which dad N = crack growth rate, k = range of stressintensity factor, a = crack size, N = number of fatiguecycles and C

and m are crack propagation parameters.

The range of the stress intensity factor is given by:

K =Y (a) R a , (37) where R is the nominal stress range and Y (a) is a func-tion of crack geometry. When crack size a, reaches crit-ical crack size acr , failure is assumed to have occurred.Combining the above two equations and accounting for variable amplitude loading, we have:

acr

a

d xY (a)m x m =C N T E

m R , (38)

where N T is the total number of stress cycles over theservice lifeT L (years); E [ .] is expectation; a is initial crack size and ac r is the crack size at failure.Equation (38) is modified to account for stress process

from all sea states as:

acr

a

d xY (a)m( x)m =C T L , (39)

where is a stress parameter given by Eq. (28). The probabilistic model for the time to failure T F is

defined byEquation (40),given below;taking into accountthe uncertainties involved in the F-M approach

T F =

1

C Bmi mi acr

a

d x

i Y (a)m x m(40)

in which Bi and i are introduced to model errors inthe estimation of the stress range R and the geometry function Y (a).For fatigue failure, T F is smaller than T L. The limit state

function takes the form as:

g ( X ) =1

C Bmi

mi

acr

a

d x i Y (a)m x m T L (41)

If x1 =C , x2 = Bi , x3 =a, x4 = i then g ( X ) = g ( x1, x2, x3, x4) =

1

x1 xm

2 xm

4 m/ 2

acr

z3

d xY (a)m xm/ 2 T L. (42)

The probability of failure and safety index is given by Eqs. (34) and (35).

S T R E S S R A N G E C A L C U L AT I O N

The fatigue stress process over the service life of an off-shore structure is non-stationary. However, it can be



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modelled by a sequence of several discrete stationary sea states, each being characterized by significant waveheight, H s and wave period,T z. The long-term descriptionof the sea, which in turn represents stress, is complete if the probability of occurrence of each sea state is known. The long-term statistical properties of stress history are

obtained by evaluating the short-term statistics for eachsea state separately and weighing them according to thelong-term sea state probabilities. The probability of oc-currence of each sea state is usually available as sea scatterdiagram.For the model of a simple bottom founded structure, it

is established that predicted fatigue life, using the timesimulation and rainflow counting methods was approx-imately 30% higher than the estimated life employingthe commonly used narrow band assumption. The fa-tigue stress in an offshore environment is generally a wideband stochastic process. In this study, Wirschings wideband correction factor is applied to account for the possi-ble error in the narrow band assumption. The expressionfor stress parameter after applying wide band correctionfactor is given by

= 2 2m

1 +m2

n

i =1 f i i mi i (43)

or

= 2 2m

1 +m2

D f , (44)

where

D f

=f i i mi i , (45)

where i isWirschingswide band correction factorfor i thseastate.Theconstant D f is a measure of theaccumulateddamage for all sea states. After the structure is analysedfor all the sea states, the value of D f will be evaluatedfor the joint. Estimates of i is obtained by the followingempirical expressions given by:

i ( i , m) =a (m) +[1 a (m)] (1 i )b(m) , (46)

where a(m) =0.9260.033 mb (m) =1.587 m 2.323

and i is the spectral width parameter for i th sea state.For a typical ocean structure problem, if i > 0.5, then i =0.79 for m =4.38 and i =0.86 for m =3

M E A S U R E S O F S E N S I T I V I T Y

First-order reliability method (FORM) analysis providesa measure of the sensitivity of the reliability index to therandom variables modelled in the analysis. The sensitivity of a random variable is computed as:

Table 1 Properties of double pendulum articulated towerplatform

Features Value

Height of bottom tower 240 mHeight of top tower 160 m

Structural mass of top and bottom tower 2.0 E 5 N/mStructural mass of ballast 448 400 N/mDeck mass 2.5 E 07 N Time periods (lower and upper shaft) 37.03 s, 32.26 sService life 20 yearsEffective diameters (tower shafts)For drag, buoyancy, inertia and added mass 17.0, 7.50 and 4.5 mEffective diameter (buoyancy chamber)For drag, buoyancy, inertia and added mass 20, 19.5 and 7.5 m

j

=

g 1 y j

n

j =1

g 1 y j

2

1/ 2(47)

where g 1 is the failure surface in the normalized coordi-nate; y j is the value of this variable at the design point inthe standard normal space. A study of the sensitivity analysis helps to identify the

variables of the model that most significantly influencethe reliability of the structure. The lower the magnitudeof j , the lesser is the influence of the j th random variableon the reliability (Val et al.18).

N U M E R I C A L S T U D Y

The geometrical and mechanical characteristics of doublehinged articulated tower used in the reliability study aregiven in Table 1. The ball joint of the tower has beentaken from Deep Water Gravity Tower 4 whose detailsare shown in Fig. 2. For 12 simulated sea states (Table 2),a detailed dynamic analysis has been carried out for waveonly, and combined action of wind and waves. The re-spective time histories so obtained are statistically anal- ysed and response statistics are obtained (Tables 3 and 4)for the two loading environments. Hydrodynamic load-ing for the structure were evaluated using modified Mori-sons equation. Dynamic wind loads based on the Simiu wind spectrum were accounted for in the time domainfatigue analysis. Critical structural damping ratios of 3% were used in the fatigue analysis. The random wind and wave loads are treated as two independent processes, andare derived using a Monte Carlo simulation technique. The simulated time histories are long enough (3600 s)in order to produce stable tower response statistics andaccurately predict the platform low-frequency responses.



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Table 2 Simulated sea states

Wind Significant Fraction of Sea velocity wave Dominant time instate, i u (m/s) height, H s (m) period, T z (s) each sea state, i

1 24.38 17.15 13.26 0.00000037

2 23.29 15.65 12.66 0.000002383 22.15 14.15 12.04 0.000014374 20.94 12.65 11.39 0.000079805 19.66 11.15 10.69 0.000405726 18.29 9.65 9.94 0.001871297 16.81 8.15 9.14 0.007738248 15.18 6.65 8.26 0.028221229 13.36 5.15 7.26 0.08851105

10 11.25 3.65 6.12 0.2283116211 8.63 2.15 4.69 0.4354235812 4.75 0.65 2.58 0.20942036

Table 3 Statistics of hinge shear stresses under long crestedrandom wave only

Sea Fraction of time in Zero crossing R.m.s. stressstate, i each sea state, i rate f i mi (MPa)

1 0.00000037 0.042 40.522 0.00000238 0.153 35.663 0.00001437 0.337 31.364 0.00007980 0.508 26.695 0.00040572 0.401 18.756 0.00187129 0.332 28.497 0.00773824 0.381 14.438 0.02822122 0.395 13.589 0.08851105 0.404 10.86

10 0.22831162 0.613 6.1911 0.43542358 0.674 3.3312 0.20942036 0.711 1.46

Table 4 Statistics of hinge shear stresses under long crestedrandom waves + wind

Sea Fraction of time in Zero crossing R.m.s. stressstate, i each sea state, i rate f i mi (MPa)

1 0.00000037 0.149 38.632 0.00000238 0.187 36.153 0.00001437 0.313 43.714 0.00007980 0.640 27.265 0.00040572 0.401 18.756 0.00187129 0.325 16.077 0.00773824 0.373 14.098 0.02822122 0.392 12.889 0.08851105 0.417 9.50

10 0.22831162 0.581 5.4811 0.43542358 0.602 2.7812 0.20942036 0.631 1.07

Table 5 Data for fatigue reliability investigation (S-N model)

Random variable Mean/median COV Distribution

P-M damage index 1.0 0.30 Log normalat failure, F

Fatigue strength 5.27 E +12 MPa 0.63 Log normalcoefficient, AStress modelling 1.0 0.20 Log normalerror, B

Fatigue exponent, m 3.0 - Constant

Table 6 Data for fatigue reliability investigation (F-M model)

Random variable Mean/median COV Distribution

Paris coefficient, C 1.8 E -12 MPa 0.63 Log normalStress modelling error, B 1.0 0.20 Log normal Modelling error in Y(a), i 1.0 0.10 Log normalInitial crack length, 0.005 - Exponential

a (mm)Critical crack length, 8.0 - Constant

ac (mm)Paris exponent, m 3.0 - Constant

Table 7 Probability of failure and reliability index using S-Ncurve approach

FORM Monte CarloReliability method ()Sea environment () P f P f

Wave only 0.142E -4 4.186 0.260E -4 4.046 Wave + wind 0.236E -5 4.577 0.333E -5 4.504

The description of random variables considered in thereliability analysis based on the MinerPalmgren damagemodel and F-M model has been given in Tables 5 and6. Subsequently, reliability analyses have been carried outusing S-N curve and F-M approaches.

D I S C U S S I O N O F R E S U LT S Wave alone environment

To study the joint reliability under wave alone, responsestatistics of hinge shear stresses have been obtained for12 simulated sea states as shown in Table 3. The prob-ability of failure and reliability indices of the joint ob-tained from S-N and F-M model are shown in Tables 7and 8. The reliability indices values from the twoapproaches obtained for wave only are 4.186 and 4.381,



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Table 8 Probability of failure and reliability index using F-Mapproach

FORM Monte CarloReliability method ()Sea environment () P f P f Wave only 0.593E -5 4.381 0.306E -4 4.007 Wave + wind 0.133E -5 4.694 0.113E -4 4.237

respectively (using advanced FORM simulation). Theseresults show that the S-N curve approach gives somewhatconservative results as compared to F-M approach. Thisis because in S-N approach, through thickness crack isnormally defined as failure criteria. On the other hand inF-M approach, failure occurs when an initial crack size(original imperfections) reaches to final crack size (unsta-ble length). Furthermore, Tables 7 and 8 also shows thatadvanced FORM formulation gave results in close prox-imity with Monte Carlo method. Therefore, advancedFORM formulation provides equally viable solution forfatigue reliability as Monte Carlo simulation method.

Combined wind and wave environment

Analysisunder wave alone is nota realisticproposition be-cause wind is the major source of sea wave generation. Inthe present study, correlated wind and waves are consid-ered (Table 2). It makes a more realistic model of oceanenvironment. Response statistics of hinge shear stressesunder combined action of wind and waves are given in

Table 4. On comparing Tables 3 and 4, it is observedthat zero crossing rate increases when wind is included. This may be due to the fact that by the incorporationof wind loading, effective wave height exponent reducesprior to wave breaking. Again from Tables 7 and 8, it isseen that wind causes a reduction of probability of failure. This reduction is of the order of 8.5 and 6.6% for S-N andF-M approaches using FORM formulation. It is mainly due to the attenuating effect of the wind 19 which actson the exposed superstructure of the articulated tower.Due to this attenuation effect, reliability index has beenimproved from 4.186 to 4.577 for S-N model and from4.381 to 4.694 for F-M model using advanced FORMformulation.

Sensitivity analysis

Sensitivity analysis gives quantitative dependence of reliability on the value of sensitivity factor for ran-dom variables. Table 9 shows the sensitivity factor val-ues of random variables in the limit state functionbased on S-N curve approach. The sensitivity factorsfor MinerPalmgren damage index ( F ), and fatigue

Table 9 Sensitivity factors using S-N curve approach

Sensitivity factor ()Sea environment () 1 2 3 Wave only 0.3374 0.6645 0.6668 Wave + wind 0.3372 0.6641 0.6673 1 =P-M damage index at failure F , 2 =Fatigue strength coef-ficient A, 3 =Stress modelling error B.

Table 10 Sensitivity factors using F-M approach

Sensitivity factor ()Sea environment () 1 2 3 4 Wave only 0.55718 0.57252 0.52784 0 .28835 Wave + wind 0.56313 0 .57866 0 .51296 0 .29144 1 =Paris coefficient C , 2 =Stress modelling error B, 3 =Initialcrack length a and 4 = Modelling error in Y(a), i .

strength coefficient ( A) are negative, hence, they are re-sistance variables. Whereas, sensitivity factor for stressmodelling error ( Bi ) is positive, hence, it will contributeto load part of the limit state function. It is also seen thatout of the two resistance variables, reliability of the jointis more sensitive to ( A) than ( F ). It is also seen that outof the two resistance variables, the reliability of the jointis influenced more by variation in the fatigue strengthcoefficient ( A) than P-M damage index at failure F forboth the loading environments. It is therefore essentialto accurately model the statistical information of fatiguestrength coefficient ( A). Table 10 shows results of sensitivity analysis for F-M

based model. It is seen that all the random variables ap-pearing in thelimit state functionarepositive whichshowsthat these variables will contribute to the load part only. This is due to the fact that in F-M approach, criticalcrack size (a) is the resistance parameter which has beenassumed as deterministic. For a given uncertainty, reli-ability of articulated joint will decrease for any increasein magnitude of random variables as all of them are load variables.

Design point or most probable point

A point on the limit state surface that corresponds to theshortest distance from the origin in the reduced coordi-nate system is defined as the design point or most proba-ble point. Tables 11 and 12 show the values of the designpoint on failure surface for the two approaches. Designpoint values are essential for reliability based probabilisticdesign of articulated joint. In such designs partial safety factors for load and resistance variables are determined



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Table 11 Design point or most probable point using S-N curveapproach

Random variables ()Sea environment () z1 z2 z3 Wave only 0.642856 0.949E+12 1.6686 Wave + wind 0.618272 0.816E+12 1.7589 z1 =P-M damage index at failure F , z2 =Fatigue strength coef-ficient A and z3 =Stress modelling error B.

Table 12 Design point or most probable point using F-Mapproach

Random variables ()Sea environment () z1 z2 z3 z4 Wave only 6.9 E -10 1.561 0.0194 1.093 Wave + wind 7.7 E -10 1.681 0.0213 1.114

z1 =Paris coefficient C , z2 =Stress modelling error B, z3 =Initialcrack length a and z4 = Modelling error in Y(a), i .

Table 13 Effect of probability distribution in random variables onreliability index (S-N model)

FORMReliability method Sea environment A b C d Wave only 4.18 3.26 1.54 1.54Wave +wind 4.57 3.27 1.56 1.56 a = All random variables are log normally distributed.b = F and Bi (normal); Ai (log normal).c = F and Bi (log normal); Ai (normal).d = All random variables are normally distributed.

for achieving the target reliability (i.e. target reliability index).

Effect of probability distribution

To study the effect of probability distributions of ran-dom variables on the reliability of articulated joint, fourcases have been considered as shown in Tables 13 and14. The results show that for S-N model, the reliability is under estimated for case d when all the random vari-ables are normally distributed. Same results have beenobtained for case c also. The reliability index for case b isimproved significantly indicating that exact distributionof variable Ai governs the probability of failure. For F-Mmodel, reliability is overestimated if all the variables areconsidered as normally distributed. Thus it may be con-cluded that for fatigue analysis an accurate assumption of random variable distribution is highly significant.

Table 14 Effect of probability distribution in random variables onreliability index (F-M model)

FORMReliability method Sea environment a b c d Wave only 4.38 8.29 10.27 8.07 Wave +wind 4.78 7.34 10.01 9.67 a = All variables have original distribution except a(exp).b=C o (normal); Bi and i (log normal); a =Exponential.c =C o (log normal);Bi and i (normal); a =Exponential.d = All variables are normally distributed except a(exp).

Table 15 Effect of design life on probability of failure using S-Ncurve approach

Wave only Wind+ waveService lifeT L (years) P f P f

10 0.353 106 4.960 0.433 107 5.35315 0.326 105 4.508 0.491 106 4.89520 0.142 104 4.186 0.236 105 4.57725 0.421 104 3.932 0.756 105 4.326

Table 16 Effect of design life on probability of failure using F-Mapproach

Wave only Wind+ waveService lifeT L (years) P f P f

10 0.853 106 4.785 0.171 106 5.09815 0.269 105 4.549 0.581 106 4.86220 0.593 105 4.381 0.133 105 4.69425 0.106 104 4.250 0.249 105 4.565

Effect of design life

Design life directly affects the probability of failure of ar-ticulated joint. Tables 15 and 16 shows an expected trend. As the design life requirement increases, correspondingprobability of failure also increases.

Effect of reliability method

Probabilities of failure and reliability indices of three dif-ferent reliability methods have been compared in Table17 for wave only and wave plus wind. These reliability methods are: (i) FORM, (ii) Advanced First Order Reli-ability Method (AFORM) and (iii) Monte Carlo method. Table 17 shows that reliability indices are in close agree-ment; therefore, FORM is also equally good method of reliability estimation.



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Table 17 Effect of reliability method (S-M model)

FORM AFORM Monte Carlo

Sea idealization P f P f P f

Wave only 0.284104 4.025 0.142 104 4.186 0.260 104 4.046 Wave

+ wind 0.489

105 4.218 0.236

105 4.577 0.333

105 4.504

C O N C L U S I O N S

Following conclusions may be drawn from the study:

S-N curve approach yields a significantly conservative es-timate of probability of failure as compared to the F-Mapproach.

Probabilities of failure and reliability indices obtainedfrom advanced FORM and Monte Carlo methods areobserved to be in close proximity. Therefore, advancedFORM is computationally efficient, economical and accu-rate method for the reliability estimation.

The inclusion of wind with waves in the analysis causes areduction of probability of failure of the universal joint.

In S-N curve approach, the reliability is more sensitiveto fatigue strength coefficient ( A) while in F-M approach,reliabilityis most sensitiveto stressmodelling error ( B)andleast sensitive to modelling error in geometry function ( ).

Forfatigueanalysisan accurateassumption of random vari-able distribution is highly significant.

Fatigue reliability of universal joint is inversely propor-tional to its service life.

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