random fatigue analysis of a steel domain ......in order to perform the fatigue verification,...

RANDOM FATIGUE ANALYSIS OF A STEEL

CATENARY RISER IN FREQUENCY AND TIME

DOMAIN

Ana Lucia Fernandas Lima TorresMarcio Martins MourellePETROBRAS/CENPES/DIPREX

Marcos Queija de SiqueiraGilberto Bruno EllwangerCOPPE/UFRJ

Abstract - The structural fatigue verification of a steel catenary riser model isperformed by means of two random procedures of analysis. One of them is anonlinear time-domain approach, based on simulation technique. The otherone is a linearized frequency-domain approach, that considers that thestructure presents a nonlinear static response and the dynamic response isalmost linear around the static deformed configuration. Fatigue damage iscalculated based on S-N curves and the Palmgren-Miner's rule.PETROBRAS in-house software were used for the analyses. Results arecompared and commented.

INTRODUCTION

The steel catenary riser (SCR) was adopted by PETROBRAS S.A. as analternative for the oil and gas exploitation on fields located at deep waters,where flexible risers with large diameter present technical and economiclimitations.

The semi-submersible platform P-18^, in the Marlim Field, Campos Basin,will have a SCR installed that will be monitored for almost a year. A SCR forthe taut-leg moored semi-submersible platform P-19 was designed^, thatproved to be a technically feasible alternative. The analysis of these two SCRshowed that fatigue damage due to platform motions is significant anddetermines the riser final configuration.

Transactions on the Built Environment vol 29, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509

362 Offshore Engineering

The SCR structures are subjected to several types of loads during their lifethat may be static or time-varying ones. When installed, the action ofenvironmental phenomena like wind, current and sea waves on the floatingunit, generates movements that will be transferred to the riser.

In order to perform the fatigue verification, deterministic or randomapproaches may be used. The random approach is considered to be moreadequate for the representation of loading and structural response, and it wasused in the SCR analyses. The fatigue analysis model used in this work isbased on the Palmgren-Miner rule for the assessment of the cumulativedamage, and the S-N curve to access the number of cycles allowed.

Two random approaches to perform the structural analysis and the fatigueverification were used in this work. One is the frequency-domain approachand the other the time-domain approach. The analysis were performed bymeans of PETROBRAS's in-house computer software's, developed andimplemented as part of projects from "CENPES-The Research andDevelopment Center of PETROBRAS" with "COPPE/UFRJ-TheEngineering Post-Graduating Coordination of the Federal University of Riode Janeiro".

The time-domain random analysis is considered to be the more appropriate tobe used due to the possibility of representing the existing nonlinearities ofthe model. The sea-state spectra were treated by a time-simulation method,so fluid load nonlinearities and fluid-structure interaction were represented.The dynamic analysis was carried out in time-domain by means of a directintegration method. Structural nonlinearities, drag forces, fluid-structurerelative velocity, or sea surface level variations were taken into account. Asthe fatigue damage calculation depends on the stresses variations during alllife of the structure, the set of loads used in the analysis should be completeenough to represent all possible situations, that leads to a long computer timenecessary for the analysis in time-domain.

For the frequency-domain approach, a linearized method was used forloading treatment. Dynamic response was obtained through a direct methodof integration in the frequency-domain. This approach is more attractive dueto the lower computer time, but it demands the previous linearization of theexisting nonlinearities, both in the applied loading and in the structuralmodel. This restriction limits the range of application of this analysis, anddemands care in its use.

In this work both methods were used in order to verify the fitness of thelinearized frequency-domain approach, when compared to a time-domainapproach.


Offshore Engineering 363

LOAD DEFINITION

The environmental loads considered were due to the action of current andwave. The wave scatter diagram was separated into a set of sea-states, eachof them characterized by its significant height and average zero crossingperiod. A set of current velocity profiles was used, associated to the sea-states.

The modified Pierson-Moskowitz spectrum was adopted for all sea-states:

(1)

where B = —| - | and A = —H*

Tz is the mean zero crossing period (s)

H, is the significant wave height (m).

co is the frequency (rad/s)

The motions of the platform were defined by prescribed movementsrepresented as RAO's, that were combined with the sea-state spectra in orderto determine the movements spectra:

(2)

where S (co) is the movement spectrum and S(co) is the sea-state spectrum.

Besides, dead weight and buoyancy were considered, too.

RANDOM TIME-DOMAIN APPROACH

For each sea-state associated to a percentage of occurrence, a time-domainstructural analysis was performed. The random response of the structure wasobtained by a random nonlinear time-domain analysis, that generated time-histories of member end forces for the fatigue program. The PETROBRASin-house software "ANFLEX - Nonlinear Dynamic Analysis of Lines" ^was used to perform this analysis.

In the simulation process, the sea elevations time-history was represented bythe summation of a finite number of harmonic waves associated to randomphases, obtained from the spectrum discretization into intervals :



N

(3)

where G3^ is a frequency and G^ has a sufficient high value such that

S(o>) =

Let GSn = (co,, - C0n_i ) / 2 and the quantity "a, " is given by:

a. =j2S(Q.)Aw,, (4)

in which Ao) = CD,, - co,,'n-l

The phase angles (()„ are independent random variables uniformly distributedover the interval (0,2 n), and kn is the wave number.

The time-histories of horizontal and vertical components of water particlesvelocities and accelerations at some elevation z above the sea bottom wererelated to the sea surface elevation through the linear Airy theory. The usualMorison's equation was applied with the time-histories of velocities andaccelerations.

The technique of integration employed for the time-domain analysis was theHHT method or which is also called a-method , where some constantswere determined in the beginning of the analysis for both the prescribeddegrees of freedom and the free degrees of freedom that were applied to thecalculation of the effective stiffness matrix and effective loading vector. Thealgorithm employed is called Modified Newton-Raphson, where the effectivestiffness matrix is calculated in the beginning of each step and is keptconstant during the internal iterations. The member end forces results weregenerated as time histories.

RANDOM FREQUENCY-DOMAIN APPROACH

For each sea-state associated to a percentage of occurrence, a linearizedfrequency-domain structural analysis was performed in order to determinespectra of member end forces.

The random response of the structure was obtained by a random linearfrequency-domain analysis, that generated spectra of member end forces forthe fatigue program. The PETROBRAS in-house software "ALFREQ -Risers Frequency Domain Random Analysis"* was used to perform thisanalysis. It is a linearized frequency random analysis system. The structural



nonlinearities were approximately considered by means of a previousnonlinear static analysis, and the linear dynamic analysis was performedusing the deformed configuration. The loading nonlinearities, as drag forces,were obtained using statistical approximations.

Loading spectra were obtained as linear transformation of velocities andaccelerations spectra. The fluid-structure relative velocity was treatedthrough an iterative procedure. The non-linear drag term in Morison'sequation was treated by means of a statistical linearization technique, basedon the Krolikowsky and Gay^ procedure. The wave and current loadingwere expressed in the linearized way as:

P(w) = pC —D^A(w) + -pC,DB, I V(w) - X(w) 1 (5)4 2 I J

where:

2 d 2 c

B,H4*PF|- +2u I2PI-

B, = 2aPF p- +u. 1+ -^ 2PI:[ v o) \_ \yj JL v«j jj

(7)

PF(V,= ' '-"'

PI(v) = —j= J* exp du = erf function

where a is the relative velocity standard deviation, Uc is the current velocity,BI and B% are the dynamic and static linearization coefficients, respectively.

The relative fluid-structure velocity is treated by an iterative procedure,where the structure velocity spectrum is obtained from the one calculated atthe previous iteration.



The hydrodynamic damping matrix is converted to the loading parcel inorder to optimize the convergence:

U(G>) = [- coM + ico (CpD+Q + K]-' P(CD) (8)

where:

Cm=jpC,DB, (9)

In this method, the harmonic components and standard deviations werecalculated from the fluid velocities and accelerations spectra. The linearizedforce spectrum Sp(co) was obtained directly from the velocities andaccelerations harmonics.

The structural response density function was obtained from the basicrelation:

Su(co) = H(co) Sp(w) H(co) (10)

where H(co) is the structure's frequency response; H(co) is the conjugate

complex matrix of the

The loading spectral density is :

Sp(CO) = P(CO)P(CO) (11)

where P(co) is a vector that is frequency dependent.

Finally, for the response spectral density:

Su(co) = U((o)U(CG) (12)

where U(co) = H(co) P(co) is obtained from:

[- co M + ico C + K] U(co) = P(co) ( 1 3)

where C is a coupled structural damping matrix.



FATIGUE DAMAGE EVALUATION

The random fatigue analysis was performed by means of PETROBRAS's in-house software, the "POSFAL-Random Fatigue Analysis"* *'*\ that aims atcalculating fatigue damage and lifetime of welded steel tubular joints.

The load conditions were defined related to their percentage of occurrence.For each sea-state, fatigue damage was calculated at 8 points around thejoint's section.

The S-N curves model was used for the calculation of the expected fatiguelife of tubular structural joints. The fatigue behavior of a material describedby an S-N curve is assumed to be of the form:

NS"=K (14)

where S is the stress range, N is the number of cycles to failure, m and K arematerial constants obtained from experimental tests.

In order to calculate the total fatigue damage and the fatigue life, the linearcumulative fatigue damage law known as Miner's rule was assumed. Thetotal damage caused by the stress process was calculated by the relation: :

(15)

where the index i stands for each stress cycle in the time-varying stressprocess; S is the stress range, N is the number of cycles, m and K arematerial constants for S-N curve.

From the long term description, the percentage of occurrence of each loadcondition was used. Total damage was obtained from the summation of eachdamage related to a sea-state, with the associated probability of occurrence,7, resulting:

(16)

where S-- is the i-th stress cycle associated to the J-th load condition. The

expected fatigue life is assumed to be the inverse of the fatigue damage.



Fatigue Damage - Time-domain ApproachThe hot-spot stresses time-histories, S(t), were generated based on the resultsof the nonlinear time-domain analysis, whose resultant member end forceswere increased by means of the stress concentration factors:

(17)

where i=l,..,8 stands for the cross section points; F% (t) ,M,(t) and M,(t)

are the time-histories of member axial force and local bending moments; A isthe cross section area, ly and Iz are the inertia moments related to sectionaxes y and z; SCFx, SCFy and SCFz are the stress concentration factors; andyi, zj are the point distances to section axes y and z.

The rainflow algorithm* was used in order to identify and count eachstress cycle. So, for each sea-state, damage was calculated by the equation(15).

Fatigue Damage - Frequency-domain Approach

In this case, frequency components of stresses were calculated using the

following expression:

(18)

where sign * is used to identify the complex form, i=l,..,8 stands for the

cross section points; j=l,..(number of frequency components) stands for the

frequency discretization of spectral density function; Fj ,M* and M* are

the frequency components for member axial force, in-plane bending and out-

of-plane bending moments, respectively.

The stress spectral density function was obtained from the stress frequency

components as:



where the bar stands for the conjugate complex and Aw, is the frequency

interval associated to the frequency ®..

Assuming that the stress time-history constitutes a narrow-banded Gaussianprocess S(t), the fatigue damage for a particular load condition can becalculated as*°:

(20)

where n = f*T is the expected number of stress cycles within the time

duration T of the load condition being considered, p(a) is the Rayleighprobability distribution and F(.) is the Gamma function. The total fatiguedamage can be rewritten as:

D? =(2V2rr( + i)Yfo,y,Jm (21)& ^ i=l

where the index /=7,2,...,M stands for each particular load condition used torepresent the long-term process, and 7, is the corresponding probability of

occurrence.

In the case of a wide-banded process, aiming at maintaining the simplicity ofpredicting the fatigue damage for a narrow-band Gaussian process, theWirshing and Light empirical formula may be used, relating the actualfatigue damage caused by a wide-band process to the damage obtainedassuming an equivalent narrow-band Gaussian process with the samevariance and zero upcrossing frequency, that is:

where e is the spectral width parameter, m is the S-N curve parameter, D^

is the damage assuming narrow-band hypothesis and X(&, m) is the

correction factor for a wide-banded process given by

X(E, m) = a(m) + (1 - a(m))(l - e)*™> (23)

where a(m) = 0.926 - 0.033m and b(m) = 1.587m - 2.323 .



CASE STUDY

A steel catenary riser model presented on figure 1 was analyzed with bothtime-domain and frequency domain approaches. PETROBRAS' in-housesoftwares ANFLEX, ALFREQ and POSFAL were used in order to performthe analyses.

The objective of this application was to compare fatigue damage resultscalculated through both methods, considering that the time-domain approachfurnishes more confident results.

The fatigue verification was performed at the top section and touchdownpoint, that are the critical regions of the riser. The finite element mesh wascomposed of 493 joints, 491 nonlinear space frame elements, for a waterdepth of 770 m. Structural and added hydrodynamic masses were calculatedautomatically by the software's, and applied at model joints. The flexjointwas represented by a rotational spring at the top of the SCR.

Figure 1 - SCR Model


Offshore Engineering 3 71

For the time-domain analysis, each sea-state spectrum was divided into 45frequency intervals associated to constant areas, plus 15 intervals in order torefine extremities.

For the frequency-domain analysis, each sea-state spectrum was divided intoconstant frequency intervals of 0.05 rad/s, from 0.2 rad/s until 3 rad/s.

The Near, Far and Cross loading situations were analyzed, consideringalmost 10 sea-states for these situations, combined to collinear currents andcross current, leading to a total of 48 loading conditions. This selection wasbased on percentage of occurrence and maximum static offsets. Thecorresponding semi-submersible movements were applied at the top of theriser.

The S-N curve API X'"* was used, in order to take into account the type ofwelding technique adopted.

In the case for the time-domain approach, the technical bibliography in thearea of offshore structural random analysis recommends, for a convenientstatistical representation of fatigue damage, the use of a set of stress time-histories obtained from different simulations, or realizations, of each sea-state, instead of only one. Nevertheless, in common practice of design, this isdifficult to perform due to the computer time required, so only onesimulation for each sea-state was adopted. A correction factor wasdetermined based on the study of several simulations of a unique sea-state, inorder to obtain the maximum difference in fatigue life results based on thesimulation variation. Another study was carried out in order to determine thecorrection for using a shorter time-history. These studies led to a correctionfactor that was applied at final fatigue life results.

For the frequency-domain approach no correction factor due to timesimulation was applied.

At Table 1 results from time-domain and frequency-domain analyses arepresented, for top an touchdown spots associated to the worst lifetime. Theresults are presented in terms of the relation between time-domain fatiguelife divided by frequency-domain fatigue life. It can be seen that, in this case,the frequency-domain approach furnished more conservative results. Thecorrected time domain results took into account the correction factor due tonumber of simulations and signal time duration.

Table 1 - LIFETIME RESULTS (years)•/ r, ..,.%. K ; :h: v\ -ANALYSIS %' ;z 4 ! ' -^ = • V , V

TIME DOMAIN/ FREQUENCY DOMAINCORRECTED TIME DOMAIN / FREQUENCY DOMAIN

TOP4,51,8

TDP2,71,08



The differences are due to the linearization of hydrodynamic loading, mainlyat the top of the riser, and the linear characteristic of the dynamic analysis.The variation of the touchdown point region was represented at the staticanalysis, but in the frequency-domain analysis it was not represented.

At frequency domain touch down point variation isn't accounted for, whichleads to the more conservative results at that region when compared to timedomain. At the model employed, the same soil stiffness has been used forboth techniques. One possibility is to calibrate frequency domain soilstiffness for a certain range of touchdown point variation.

In table 2, the CPU time spent for the dynamic analysis of one sea-state infrequency and time domains are presented. Differences are significant and interms of a project the frequency domain approach is more attractive.

Table 2 - CPU TIME

CONCLUSIONS

The use of frequency-domain approach is attractive in order to attainconfident structural response associated to feasible time of analysis. The steelcatenary riser is a type of structure whose behaviour may be represented by anonlinear static analysis associated to a linear dynamic analysis of thedeformed configuration. As the fatigue analysis demands a large number ofloading cases to be considered, the use of a linearized method appears as anoption to perform such a design.

In the design work, it will be necessary to perform time domain analysis for afew loading cases, in order to find out if it's necessary to change somethingat the FD model or to run TD for the most severe conditions. PETROBRASis at the moment working at the front-end of risers for the Barracuda Field atCampos Basin. ALFREQ system is being used for the fatigue analyses.

The linearized frequency-domain method presented in this work may be auseful tool if nonlinearities are not significant.



REFERENCES

1. "ALFREQ - Input Data Manual", doc. 1.0, PETROBRAS/CENPES/DIPREX/ SEDEM, April, 1996 (in Portuguese).

2. "ANFLEX - User's Manual", doc. 3.0, PETROBRAS/CENPES/DIPREX/SEDEM (in Portuguese).

3. API (American Petroleum Institute) "Recommended Practice for Planningand Constructing Fixed Offshore Platforms", API RP2A, 1989.

4. BorgmanJLE., "Ocean Wave Simulation for Engineering Design",Journal of the Waterways and Harbors Division, Proceedings of theAmerican Society of Civil Engineers, vol. 55, no WWS, 1969.

5. Chakrabarti,S.K., "Hydrodynamics of Offshore Structures", Springer-Verlag, Berlin, 1987.

6. Franciss R., Torres, A.L.L., Mourelle, M.M., Pinto, F.J.C.P., Souza,L.F.A., "Steel Catenary Riser for a Taut-Leg Moored Semi-SubmersiblePlatform", OTC 8515, Offshore Technology Conference, 1997.

7. Lima, E.C.P., Ellwanger, G.B, Siqueira, M.Q. "ALFREQ - TheoreticalManual", doc. 1.0, COPPE/UFRJ e PETROBRAS/CENPES/DIPREX/SEDEM, April, 1996 (internal report).

8. Mourelle,M.M., Gonzalez,E.C, (1991). "ANFLEX Program - UtilizationCourse" (in Portuguese).

9. Mourelle,M.M., Gonzalez,E.C., Jacob, B.P. - "ANFLEX - ComputationalSystem for Flexible and Rigid Riser Analysis", Proceedings of the 9thInternational Symposium on Offshore Engineering, Brazil Offshore 95,Rio de Janeiro, September 1995.

lO.Newland, D.E., "An Introduction to Random Vibrations and SpectralAnalysis", London, Longman Group Limited, 1975.

ll."POSFAL: Random Fatigue Analysis - User's Manual", PETROBRAS/ CENPES / DIPREX/SEDEM (in Portuguese), 1995.

12.Serta, O.B., Mourelle, M.M., Grealish,F.W., Harbert, S.J., Souza, L.F.A.,"Steel Catenary Riser for the Marlim Field FPS P-XVUT, OTC 8069,Offshore Technology Conference, 1996.



1 S.Torres, A.L.F.L., Sagrilo, L.V.S., Siqueira, M.Q., Lima, E.C.P - "AProcedure for Random Fatigue Analysis of Offshore Structures",Proceedings of the 9th International Symposium on Offshore Engineering,Brazil Offshore 95, Rio de Janeiro, September 1995.

14.Wirshing,P.H. and Light,M.C, "Fatigue Under Wide-Banded RandomStresses", Journal of Structural Division, 106, ST7:1593-1607,1980.

15.Wirshing,P.H. and Shehata, A.M., "Fatigue Under Wide-banded RandomStresses Using the Rainflow Method", Journal of Engineering Materialsand Technology, 99, 3:205-211, 1977.

16.Hilber,H.M., Hughes,T.J., Taylor,R.L.,"Improved Numerical Dissipationfor Time Integration Algorithms in Structural Dynamics", EarthquakeEngineering and Structural Dynamics, vol.5,1977.

17.Krolikowsky,L.P. and Gay, T.P. - An Improved Linearization Techniquefor Frequency Domain Riser Analysis - Proceedings of the 12th AnnualOffshore Technology Conference - Houston - pp 341-353 - 1980


random fatigue analysis of a steel domain ......in order to perform the fatigue verification,...

Documents