RANDOM FATIGUE ANALYSIS OF A STEEL CATENARY RISER IN FREQUENCY AND TIME DOMAIN

Transcription

1 RANDOM FATIGUE ANALYSIS OF A STEEL CATENARY RISER IN FREQUENCY AND TIME DOMAIN Ana Lucia Fernandas Lima Torres Marcio Martins Mourelle PETROBRAS/CENPES/DIPREX Marcos Queija de Siqueira Gilberto Bruno Ellwanger COPPE/UFRJ Abstract - The structural fatigue verification of a steel catenary riser model is performed by means of two random procedures of analysis. One of them is a nonlinear time-domain approach, based on simulation technique. The other one is a linearized frequency-domain approach, that considers that the structure presents a nonlinear static response and the dynamic response is almost linear around the static deformed configuration. Fatigue damage is calculated based on S-N curves and the Palmgren-Miner's rule. PETROBRAS in-house software were used for the analyses. Results are compared and commented. INTRODUCTION The steel catenary riser (SCR) was adopted by PETROBRAS S.A. as an alternative for the oil and gas exploitation on fields located at deep waters, where flexible risers with large diameter present technical and economic limitations. 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 for the taut-leg moored semi-submersible platform P-19 was designed^, that proved to be a technically feasible alternative. The analysis of these two SCR showed that fatigue damage due to platform motions is significant and determines theriserfinalconfiguration.

2 362 Offshore Engineering The SCR structures are subjected to several types of loads during their life that may be static or time-varying ones. When installed, the action of environmental phenomena like wind, current and sea waves on the floating unit, generates movements that will be transferred to the riser. In order to perform the fatigue verification, deterministic or random approaches may be used. The random approach is considered to be more adequate for the representation of loading and structural response, and it was used in the SCR analyses. The fatigue analysis model used in this work is based on the Palmgren-Miner rule for the assessment of the cumulative damage, and the S-N curve to access the number of cycles allowed. Two random approaches to perform the structural analysis and the fatigue verification were used in this work. One is the frequency-domain approach and the other the time-domain approach. The analysis were performed by means of PETROBRAS's in-house computer software's, developed and implemented as part of projects from "CENPES-The Research and Development Center of PETROBRAS" with "COPPE/UFRJ-The Engineering Post-Graduating Coordination of the Federal University of Rio de Janeiro". The time-domain random analysis is considered to be the more appropriate to be used due to the possibility of representing the existing nonlinearities of the 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 direct integration method. Structural nonlinearities, drag forces, fluid-structure relative velocity, or sea surface level variations were taken into account. As the fatigue damage calculation depends on the stresses variations during all life of the structure, the set of loads used in the analysis should be complete enough to represent all possible situations, that leads to a long computer time necessary for the analysis in time-domain. For the frequency-domain approach, a linearized method was used for loading treatment. Dynamic response was obtained through a direct method of integration in the frequency-domain. This approach is more attractive due to the lower computer time, but it demands the previous linearization of the existing nonlinearities, both in the applied loading and in the structural model. This restriction limits the range of application of this analysis, and demands care in its use. In this work both methods were used in order to verify the fitness of the linearized frequency-domain approach, when compared to a time-domain approach.

3 Offshore Engineering 363 LOAD DEFINITION The environmental loads considered were due to the action of current and wave. The wave scatter diagram was separated into a set of sea-states, each of them characterized by its significant height and average zero crossing period. A set of current velocity profiles was used, associated to the seastates. 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 movements represented as RAO's, that were combined with the sea-state spectra in order to determine the movements spectra: where S^(co) is the movement spectrum and S(co) is the sea-state spectrum. Besides, dead weight and buoyancy were considered, too. (2) RANDOM TIME-DOMAIN APPROACH For each sea-state associated to a percentage of occurrence, a time-domain structural analysis was performed. The random response of the structure was obtained by a random nonlinear time-domain analysis, that generated timehistories of member end forces for the fatigue program. The PETROBRAS in-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 by the summation of a finite number of harmonic waves associated to random phases, obtained from the spectrum discretization into intervals^:

4 364 Offshore Engineering 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 distributed over the interval (0,2 n), and kn is the wave number. The time-histories of horizontal and vertical components of water particles velocities and accelerations at some elevation z above the sea bottom were related to the sea surface elevation through the linear Airy theory. The usual Morison's equation was applied with the time-histories of velocities and accelerations. The technique of integration employed for the time-domain analysis was the HHT method or which is also called a-method ^, where some constants were determined in the beginning of the analysis for both the prescribed degrees of freedom and the free degrees of freedom that were applied to the calculation of the effective stiffness matrix and effective loading vector. The algorithm employed is called Modified Newton-Raphson, where the effective stiffness matrix is calculated in the beginning of each step and is kept constant during the internal iterations. The member end forces results were generated as time histories. RANDOM FREQUENCY-DOMAIN APPROACH For each sea-state associated to a percentage of occurrence, a linearized frequency-domain structural analysis was performed in order to determine spectra of member end forces. The random response of the structure was obtained by a random linear frequency-domain analysis, that generated spectra of member end forces for the fatigue program. The PETROBRAS in-house software "ALFREQ - Risers Frequency Domain Random Analysis"*^ was used to perform this analysis. It is a linearized frequency random analysis system. The structural

5 Offshore Engineering 365 nonlinearities were approximately considered by means of a previous nonlinear static analysis, and the linear dynamic analysis was performed using the deformed configuration. The loading nonlinearities, as drag forces, were obtained using statistical approximations. Loading spectra were obtained as linear transformation of velocities and accelerations spectra. The fluid-structure relative velocity was treated through an iterative procedure. The non-linear drag term in Morison's equation was treated by means of a statistical linearization technique, based on the Krolikowsky and Gay^ procedure. The wave and current loading were 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 at the previous iteration.

6 366 Offshore Engineering The hydrodynamic damping matrix is converted to the loading parcel in order 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 were calculated from the fluid velocities and accelerations spectra. The linearized force spectrum Sp(co) was obtained directly from the velocities and accelerations harmonics. The structural response density function was obtained from the basic relation: 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.

7 FATIGUE DAMAGE EVALUATION Offshore Engineering 367 The random fatigue analysis was performed by means of PETROBRAS's inhouse software, the "POSFAL-Random Fatigue Analysis"* *'*\ that aims at calculating 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 the joint's section. The S-N curves model was used for the calculation of the expected fatigue life of tubular structural joints. The fatigue behavior of a material described by 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 are material constants obtained from experimental tests. In order to calculate the total fatigue damage and the fatigue life, the linear cumulative fatigue damage law known as Miner's rule was assumed. The total 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 stress process; S is the stress range, N is the number of cycles, m and K are material constants for S-N curve. From the long term description, the percentage of occurrence of each load condition was used. Total damage was obtained from the summation of each damage 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. expected fatigue life is assumed to be the inverse of the fatigue damage. The

8 368 Offshore Engineering Fatigue Damage - Time-domain Approach The hot-spot stresses time-histories, S(t), were generated based on the results of the nonlinear time-domain analysis, whose resultant member end forces were increased by means of the stress concentration factors: (17) where i=l,..,8 stands for the cross section points; F% (t),m,(t) and are the time-histories of member axial force and local bending moments; A is the cross section area, ly and Iz are the inertia moments related to section axes y and z; SCFx, SCFy and SCFz are the stress concentration factors; and yi, zj are the point distances to section axes y and z. The rainflow algorithm*^ was used in order to identify and count each stress cycle. So, for each sea-state, damage was calculated by the equation (15). M,(t) 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 outof-plane bending moments, respectively. The stress spectral density function was obtained from the stress frequency components as:

9 Offshore Engineering 369 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 Gaussian process S(t), the fatigue damage for a particular load condition can be calculated 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 Rayleigh probability distribution andf(.) is the Gamma function. The total fatigue damage 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 to represent 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 of predicting the fatigue damage for a narrow-band Gaussian process, the Wirshing and Light empirical formula may be used, relating the actual fatigue damage caused by a wide-band process to the damage obtained assuming an equivalent narrow-band Gaussian process with the same variance 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) = m and b(m) = 1.587m

10 370 Offshore Engineering CASE STUDY A steel catenary riser model presented on figure 1 was analyzed with both time-domain and frequency domain approaches. PETROBRAS' in-house softwares ANFLEX, ALFREQ and POSFAL were used in order to perform the analyses. The objective of this application was to compare fatigue damage results calculated through both methods, considering that the time-domain approach furnishes more confident results. The fatigue verification was performed at the top section and touchdown point, that are the critical regions of the riser. The finite element mesh was composed of 493 joints, 491 nonlinear space frame elements, for a water depth of 770 m. Structural and added hydrodynamic masses were calculated automatically by the software's, and applied at model joints. The flexjoint was represented by a rotational spring at the top of the SCR. Figure 1 - SCR Model

11 Offshore Engineering 3 71 For the time-domain analysis, each sea-state spectrum was divided into 45 frequency intervals associated to constant areas, plus 15 intervals in order to refine extremities. For the frequency-domain analysis, each sea-state spectrum was divided into constant 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, considering almost 10 sea-states for these situations, combined to collinear currents and cross current, leading to a total of 48 loading conditions. This selection was based on percentage of occurrence and maximum static offsets. The corresponding semi-submersible movements were applied at the top of the riser. The S-N curve API X'"* was used, in order to take into account the type of welding technique adopted. In the case for the time-domain approach, the technical bibliography in the area of offshore structural random analysis recommends, for a convenient statistical representation of fatigue damage, the use of a set of stress timehistories obtained from different simulations, or realizations, of each seastate, instead of only one. Nevertheless, in common practice of design, this is difficult to perform due to the computer time required, so only one simulation for each sea-state was adopted. A correction factor was determined based on the study of several simulations of a unique sea-state, in order to obtain the maximum difference in fatigue life results based on the simulation variation. Another study was carried out in order to determine the correction for using a shorter time-history. These studies led to a correction factor that was applied at final fatigue life results. For the frequency-domain approach simulation was applied. no correction factor due to time At Table 1 results from time-domain and frequency-domain analyses are presented, for top an touchdown spots associated to the worst lifetime. The results are presented in terms of the relation between time-domain fatigue life divided by frequency-domain fatigue life. It can be seen that, in this case, the frequency-domain approach furnished more conservative results. The corrected time domain results took into account the correction factor due to number of simulations and signal time duration. Table 1 - LIFETIME RESULTS (years) / r,..,.%. K ; :h: v\^-analysis %' ;z 4! '-^ ^ = V, V TOP TIME DOMAIN/ FREQUENCY DOMAIN 4,5 CORRECTED TIME DOMAIN / FREQUENCY DOMAIN 1,8 TDP 2,7 1,08

12 372 Offshore Engineering The differences are due to the linearization of hydrodynamic loading, mainly at the top of the riser, and the linear characteristic of the dynamic analysis. The variation of the touchdown point region was represented at the static analysis, but in the frequency-domain analysis it was not represented. At frequency domain touch down point variation isn't accounted for, which leads to the more conservative results at that region when compared to time domain. At the model employed, the same soil stiffness has been used for both techniques. One possibility is to calibrate frequency domain soil stiffness for a certain range of touchdown point variation. In table 2, the CPU time spent for the dynamic analysis of one sea-state in frequency and time domains are presented. Differences are significant and in terms 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 attain confident structural response associated to feasible time of analysis. The steel catenary riser is a type of structure whose behaviour may be represented by a nonlinear static analysis associated to a linear dynamic analysis of the deformed configuration. As the fatigue analysis demands a large number of loading cases to be considered, the use of a linearized method appears as an option to perform such a design. In the design work, it will be necessary to perform time domain analysis for a few loading cases, in order to find out if it's necessary to change something at the FD model or to run TD for the most severe conditions. PETROBRAS is at the moment working at the front-end of risers for the Barracuda Field at Campos Basin. ALFREQ system is being used for the fatigue analyses. The linearized frequency-domain method presented in this work may be a useful tool if nonlinearities are not significant.

13 REFERENCES Offshore Engineering "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 Planning and Constructing Fixed Offshore Platforms", API RP2A, BorgmanJLE., "Ocean Wave Simulation for Engineering Design", Journal of the Waterways and Harbors Division, Proceedings of the American Society of Civil Engineers, vol. 55, no WWS, Chakrabarti,S.K., "Hydrodynamics of Offshore Structures", Springer- Verlag, Berlin, 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-Submersible Platform", OTC 8515, Offshore Technology Conference, Lima, E.C.P., Ellwanger, G.B, Siqueira, M.Q. "ALFREQ - Theoretical Manual", doc. 1.0, COPPE/UFRJ e PETROBRAS/CENPES/DIPREX/ SEDEM, April, 1996 (internal report). 8. Mourelle,M.M., Gonzalez,E.C, (1991). "ANFLEX Program - Utilization Course" (in Portuguese). 9. Mourelle,M.M., Gonzalez,E.C., Jacob, B.P. - "ANFLEX - Computational System for Flexible and Rigid Riser Analysis", Proceedings of the 9th International Symposium on Offshore Engineering, Brazil Offshore 95, Rio de Janeiro, September lo.newland, D.E., "An Introduction to Random Vibrations and Spectral Analysis", London, Longman Group Limited, ll."posfal: Random Fatigue Analysis - User's Manual", PETROBRAS / CENPES / DIPREX/SEDEM (in Portuguese), 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.

14 374 Offshore Engineering 1 S.Torres, A.L.F.L., Sagrilo, L.V.S., Siqueira, M.Q., Lima, E.C.P - "A Procedure for Random Fatigue Analysis of Offshore Structures", Proceedings of the 9th International Symposium on Offshore Engineering, Brazil Offshore 95, Rio de Janeiro, September Wirshing,P.H. and Light,M.C, "Fatigue Under Wide-Banded Random Stresses", Journal of Structural Division, 106, ST7: , Wirshing,P.H. and Shehata, A.M., "Fatigue Under Wide-banded Random Stresses Using the Rainflow Method", Journal of Engineering Materials and Technology, 99, 3: , Hilber,H.M., Hughes,T.J., Taylor,R.L.,"Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics", Earthquake Engineering and Structural Dynamics, vol.5, Krolikowsky,L.P. and Gay, T.P. - An Improved Linearization Technique for Frequency Domain Riser Analysis - Proceedings of the 12th Annual Offshore Technology Conference - Houston - pp