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I'¢IuTTERWO~THEngineeringStructures,Vol. ~ E I N E M A N N 0141--0296(95)00027-5 17, No. 4, pp. 293-304, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0141-0296/95 $10.00 + 0.00 Review of flexible riser modelling and analysis techniques M. H. Patel Department of Mechanical Engineering, University College London, Torrington Place, London WC1E 7JE, UK F. B. S e y e d Farman & Associates Ltd, Latymer Court, Hammersmith Road, London W6 7JE, UK The past decade has seen significant developments in the use of flexible risers for floating production duties throughout the world. The technological developments in the construction of these pipes have, in parallel, been followed by advances in the analysis of their hydrodynamic and mechanical behaviour. At present, the hydrodynamic analysis is well established with several commercial analysis packages at the disposal of designers. The analysis of flexible pipe mechanical behaviour has produced new analytical and numerical models for the prediction of pipe internal wear and fatigue behaviour. These models are currently being exploited in the optimization of pipe construction and design. New issues of reliability and risk assessment are also entering the scene and are beginning to influence the development of design codes for flexible risers. This paper presents an historical overview of the development of hydrodynamic analysis techniques for flexible risers. It highlights key issues addressed during these developments including the effects of internal and external hydrostatic pressures and of internal flow. The paper also presents an overview of the current status of analysis techniques for flexible riser design and engineering and identifies those uncertainties that remain. Keywords: flexible riser analysis, catenary risers Static analysis methods The initial stage of any analysis of suspended pipe (such as in Figure 1) is the computation of its profile under a set of static forces. The classical catenary equations provide a good first approximation for this but in their original form, only consider loads due to self-weight and assume a pipe of zero bending stiffness. These equations have been presented and discussed ~-9 However, an adequate static analysis of flexible risers requires a solution for large deflection nonlinear behaviour with the effects of bending stiffness included. There have been many different approaches to obtaining this solution with most contributors using a combination of numerical techniques and use of tile classical catenary equation. Two commonly used schemes, based on incremental and hybrid techniques are illustrated in Figures 2 and 3. These figures illustrate two different computational approaches that are utilized to arrive at a solution of flexible riser nonlinear statics. The incremental method is an example of one of the earliest techniques relying on solutions at multiple load increments to achieve accuracy. However, the hybrid technique, relying as it does on a combination of incremented loading and iteration, offers a more modem alternative. Cowan and Andris 1° used an incremental shifting solution for static analysis of pipeline-stinger-vessel systems. This involves starting with a horizontal pipeline profile and incrementally lowering the pipe until it comes into contact with the seabed. Since the pipe laying system undergoes large deformations during loading, albeit incrementally, significant errors are likely to accumulate. These are usually rectified through intermediate iterations for equilibrium although there is a sensitive area near the seabed that requires a larger concentration of finite elements. Sparks 11 conducted a series of parametric studies and provided results for rigid risers which show that the effects of bending rigidity are confined to the pipe extremes. These results were expanded upon by Bratu and NarzuP 2. 293 X. Steep-S 7 o __Xl X2 exact Displacement Figure 3 Hybrid method schematic (key as in Figure 2) Steep-wave Lazy-S Lazy-wave Figure 1 Rigid and flexible riser geometries Force F 7 _x. force increment. Pate/and F. Finally. Seyed Force F i! --! / i ~fJfJffJ ~ffff. displacement array at Step (i). K. The method involves an iterative solution of a series of nonlinear catenary equations. Tikhonov and Fisher t4 employed a mathematical approach to determine the behaviour of an ocean mining pipe carrying an attached mass at its lower end whilst being towed through water at constant velocity. The formulation is limited to cables of zero bending rigidity and cannot be modified to account for bending effects but is useful in generating an initial starting profile for a more accurate numerical analysis. large differences exist at the extremities. 5 for guyed towers' mooring lines composed of a lead cable. The program incorporates the linear effects of axial elongation and uniform temperature change with the analysis formulation terms of force components at the supports. One of the few methods employing a finite difference scheme for suspended catenary pipes has been reported by Langer ~3. H. / . F. Ractliffe7 has investigated the validity of approximate formulae in dynamic analysis of flexible pipes and cables through a discussion of some available exact and approximate formulae. the dynamic variation of tensile forces due to vessel motions is studied using a series of time domain simulations and it is shown that the results are very close to quasistatic approximations.exact displacement vector solution Peyrot and Goulois 2 provided a reliable computer program for the calculation of the profile of a catenary suspended between two arbitrary points. B. The formulation is based on asymptotic theory of differential equations using Tupchiev's method for the solution of singularly perturbed boundary-value problems. The work shows that although the bending moment distribution along the pipe is accurately predicted by both methods. The method includes the effects of flexural stiffness. A different approach to the problem has been presented by Orgill e t al. each representing one segment of the cable. The method can be readily extended to model flexible risers composed of multiple pipe sections of different weights or buoyancies. remains very specific to the problem addressed.294 Review of flexible riser modelling and analysis techniques: M. The solution. however. Displacement Figure2 Incremental method schematic. Huang and Chucheepsaku115 and Huang and Rivero 16 have proposed a Lagrangian formulation where the total energy of a riser pipe with a sliding top connection has . Xo. •F. It considers a rigid pipe and includes hydrostatic forces arising during transport without taking account of hydrodynamic loading due to waves and vessel motions. a trailing cable and a connecting heavy segment. total force vector. An investigation of the effects of longitudinal stress waves on tension variations along the pipe is presented and some cancellation features at high frequencies are illustrated. The relative lack of interest in these calculation schemes amongst flexible pipe iterative techniques such as the Gauss-Seidel or Jacobi iteration is due to the possibility of nonconvergence of the solution and the complexity of defining arbitrary boundary conditions. Simple Catenary / . )(exact. tangent stiffness matrix at Step (i)./ f//~//////////. a nonlinear bending-curvature relationship and allows pipe properties to vary along its length. The algorithm provides an analysis of general cable systems in three dimensions. A comparison is then made between a stiffened catenary solution and the proposed technique. starting nodal geometry array. The resulting equations are rewritten in terms of the base horizontal tension and solved through iteration for the global equilibrium of the forces. . : ....295 -66... :.... / / .:t'b"' • ~ > 70.. ignoring axial extensibility and torsional effects together with a finite-difference time integration procedure... In considering this method for implementation." ..0 Horizontal Projection (m) Figure4 Incremental shifting for calculation of static profile due to Mathisen and Bergen 22 and Engseth et el? 9 small for risers of low flexural rigidity 13 little advantage is gained with this method and the simple catenary equations provide a sufficiently accurate solution for the same regions of the pipe.." ":=::: .-".3 Z -21)0. little difference is noted whilst for typical riser applications discrepancies of the order of 10% appear near the pipe extremities. The finite element method was then used to model the pipe and obtain the equilibrium configuration iteratively using the Newton-Raphson method.:~ii~:::~::... The linear techniques are therefo~e deemed conservative in this respect..... a measure against numerical problems....0 ' 420...3 -400. Further... /'/:~. It is hence concluded that the least expensive method is one which starts with a shape which is the closest to the final expected profile... ~:. 2° have continued to include comparisons with some analytical solutions which serve to validate the technique. Seyed been derived and minimized using a variational approach to yield the equilibrium relationships and their associated boundary conditions...... extra complexity is introduced by considering the large displacement relationships and the implementation of the boundary equations is made more difficult.~:. In the first two cases an initial prestressing of about 1% strain was applied to avoid numerical problems. lead to overestimation of displacements and forces. Comparisons with small deflection approaches are not presented which makes it difficull: to decide whether the increased accuracy warrants the extra computational efforts. again. The work of Nordgren TMpreceded that of Bernitsas et al.: :." . y . nonlinearities resulting from pipe extensibility and coupling between translational and rotational degrees-of-fre..0 ' 2gO.. vertical and slanting geometries. ~//// '~"ii--~~:).'" . 350. Bernitsas et al. albeit slightly.. tensile forces were noted to be underestimated...0 "~. Three starting profiles have been studied which include horizontal.'"i.. However." ... The first general conclusion is that ignoring deformation depende. Analytical solutions for pipes of significant flexural stiffness are difficult to obtain and the limited number available tend not to be general. However.. its advantages must be weighted against extended implementation and run times arising from the complexity of its equations and the extended computer storage requirements resulting from the nonsymmetric nature of the matrices involved....."...210..7 -133. "01 2so.. As a result.="... historically. however....0 . Difficulties do arise with this technique from the coupling of difference equations for displacements and time......4.. These effects can also be noted from Nordgren 18.. element weight and buoyancy loads have been applied in a number of steps which is...o / / / / i i/ .. ! i / i i i i/"i.. i 1 •. Mathisen and Bergan22 have addressed the problem of computation of the static profile of flexible risers of steepwave geometry. The formulation uses exact expressions for pipe curvature and hence obtains a higher-order representation.... It has been stipulated that the conventional techniques overestimate bending distributions by a maximum of 25%...0 ....19 who used the same approach... Pate/and F....0 ~.. The formulation is comprehensive as it uses the general vectorial equilibrium equations in three dimensions.: . whilst equivalent stresses are only affected by about 9%." .. Bernitsas et al...~ 210..zdom are preserved...:. / / ! / / -~.. . The method is considered quite accural:e although the implementation is likely to be lengthy and modifications complicated...Review of flexible ri. deformation independent and deformation dependent forms of the formulation.. For simple cases..~er modelling and analysis techniques: M..: :::-. .".0 2~0 Horizontal Projection (m) 140.ncy of loading will.. .0 7.~~:~::!!::::~..... since bending effects are restricted to 1:he extremities of the pipe and are 0.. typically......... 0 210. 17 have proposed a three-dimensional incremental large deflection finite element method for static analysis of risers.:i..'".. Figures 4 and 5 illustrate the development of these incremental solutions using horizontal and slanting starting profiles."" .O ' 350. .. 00 7d0 1..:... An incremental solution technique has been used to shift the riser pipe from a given starting position to its final equilibrium... Secondly. Test cases are presented to illustrate the discrepancies between linear....11 "~ -266.:i. and appears to have paved the way for this technique..7 -333..." .0 Figure 5 Comparison of in-plane profiles with those of O'Brien and McNamara 2e . One such method was presented by Owen and Qin 2~ and used an asymptotic expansion to obtain an approximation to the exact solution of a stiffened catenary. This stress is. The resulting equations are solved using an incremental finite element method employing a predictor-corrector scheme at each increment... B.. The method provides a numerical approximation and the derived series solution is considered convergent for risers of low flexural rigidity.. by linear methods and those assuming deformation independent loads...0 ~ 0. :::. H..... The slanting profile is found to use the least computation time.. It also gives an interesting illustration of the use of the method where the free fall of a pipe from a stinger duLring laying is simulated. relieved incrementally.:~: ..0 ~::"' • 0. 296 Review of flexible riser modelling and analysis techniques: M. Patel and F.e. attempts have also been made at considering the linear effects of axial elongation. Such techniques are centred on the idea that every random process. A general purpose computer program was employed in conjunction with complementary software developed by the investigators. Whilst the inextensible catenary equations have been by far the most widely used. However. The paper includes two case studies and contains a detailed listing of input parameters and output results which may be used for comparison. The shifting procedure is based on the formulation of Mathisen and Bergan 22 and Hansen and Bergan 3°. Loads are imposed incrementally during the shifting procedure. B. tangentially or temporally linear Detailed discussions of the method have been given by T h o m s o n 31. A convected co-ordinate system is then used to shift the pipe from an initially straight or slightly sagging configuration to its final equilibrium. various techniques are usually employed which estimate the response to random excitation using a frequency domain analysis. a sufficiently accurate profile may be obtained by dividing the riser into a series of catenaries where the buoyancy modules are described as inverted catenaries and solving the resulting set of simultaneous nonlinear equations. Standard widely available computer algorithms or the method of Peyrot and Goulois 2 could be employed for this purpose. The case study of these authors is duplicated and the static and dynamic results compared with good agreement. The first uses the simple catenary equations to obtain the equilibrium profile and is a fast way of obtaining the geometry. Engseth et al. (1) The loading on the structure may be decomposed into a series of periodic components at different frequencies (2) The total response can be calculated from the summation of responses over all frequencies. reduced computer storage requirements and shorter run times. These are essentially based on a discretization of the pipe into connecting catenary segments and solving the resulting simultaneous nonlinear equations using a computer algorithm of one form or other. Other static loads are then imposed incrementally or through a direct iteration scheme to obtain Frequency domain dynamic analysis Dynamic analysis in the frequency domain derives its popularity from its ease of implementation. the computational method is quite time consuming and except for cases where particularly difficult buoyancy arrangements are used. The works of McNamara and O'Brien provide a structured approach to the computation of the flexible riser static profile. A new set of equations are thus introduced which exactly negate the axial extensions of the pipe. The shifting of the pipe to its equilibrium position is carried out using the HilberHughes-Taylor implicit time integration operator. Owen and Qin 28 described the formulation of the governing differential equations for a flexible riser and carried out a series of model tests to provide a check against computer simulations. The validity of such a decomposition depends on the statistics of the given process and . given its statistics. i. The catenary solutions are simple in concept and are extremely fast and reliable. The use of the method is recommended for its controllable timestep size. Static analysis techniques for flexible risers started with the study of catenary equations for pipe-laying and mooring applications. O'Brien et a l Y have described their formulation with special consideration to the detailed discussion of the mathematical basis of their work. 29 have reported on the development and verification of a computer program for analysis of general flexible riser systems. The buoyancy arch is. Dynamic analysis is then carried out as a perturbation about the mean static profile. This paper contains a validation of their method using a vertical cantilever example and the study of an offshore loading tower in three dimensions. His work shows that the incremental shifting technique is only recommended for unusual riser geometries where a simple solution would not be adequate. Oppenheimer and W i l s o n 27 offer another approach to the problem. and compared in detail with the results of the program employed by the above authors. Engseth demonstrated that the use of a simple catenary formulation for computation of the initial configuration is quite accurate and compares very well with the more elaborate shifting procedures. With a rising interest in flexible riser applications and the advent of new geometries such as 'S' and 'wave' configurations. however. Dynamic analyses were also carried out and compared against model tests. Frequency domain analysis techniques do not yield the response in random seas directly. McNamara et al. not modelled and a discontinuity exists at the transition from the upper to lower catenary. The second method uses a procedure where the pipe is shifted to its final equilibrium starting from an initial unstressed state. However. may be decomposed into a series of harmonic processes. the indeterminacy of the shape prompted investigators to consider shifting procedures. output at different frequencies can be superimposed (3) The structural response to each excitation frequency is confined to that frequency (4) Structural response is absolutely. The authors are of the opinion that the use of a controllable-step time integration method is essential in order to avoid problems with start-up transient oscillations. The practical application of this formulation to flexible risers is found in O'Brien and McNamara 26 which presents studies of catenary and steep-wave profiles. Discrepancies were noted in the results which were attributed to differences in the levels of structural damping in the model and computer simulation. A lazy-wave riser was studied whose static profile was generated using two adjacent catenaries. Seyed the final static equilibrium. The basic method is constructed on the following assumptions. In this way buoyancy modules can be modelled as inverted catenaries with negative self-weight connecting the adjacent positive weight catenary segments. this trend has reversed towards the use of catenary solutions with shifting techniques reserved for the more elaborate cases. All papers provide detailed sets of input and output data for comparison. The program first carries out a static analysis of the pipe to determine the initial profile under self-weight and buoyancy loads using one of two approaches. The method is formulated using a Lagrangian constraint which assumes the pipe to be axially inextensible. 23 have presented a method for static and dynamic analysis of flexible pipes and risers using a hybrid element formulation which avoids the problems of ill-conditioning reported by others 24. H. Having discovered the computational cost of these procedures. Their approach is attractive in enforcing the condition of zero axial strain mathematically. linearized methods have had to be developed. Further.38. the cross-correlation of forces along the pipe are calculated. conservative results are obtained using linear spectral methods. The same method is then used for the case of a regular wave with current. Langley's method requires an iterative procedure for determining the linearized drag coefficients. The findings indicate that linearized techniques overpredict the higher frequency response whilst underestimating lower frequency components which contain a greater proportion of the total wave energy and are often of greater interest to the designer. Eatock Taylor and Rajagopalan 34 have examined loading on rigid slender offshore structures subjected to the combined action of currents and waves. Patel and F. have . These issues are. incorrect use of such methods can lead to en:oneous results which may differ significantly from the correct solution. Krolikowski and Gay's linearization methods for regular waves have been widely used in the offshore industry. not directly relevant to this work and their further examination will be avoided. B. although some workers have used a lower number of frequency bands. A major shortcoming of frequency domain methods is their inability to model ~aonlinearities directly. The results of the improw~d linearized frequency domain method are shown to be in very good agreement with time domain solutions whilst the quasistatic approach is found to give quite different results. Krolikowski and Gay 33 have followed on to propose a linearization procedure for frequency domain analysis in irregular seas with and without current. Most conventional techniques are frame variant which implies that the drag force calculated is a function of the orientation of the co-ordinate system orthogonal to the pipe axis. For a 15-band discretization some 25% error may be expected whilst the use of 100 bands could reduce this error to around 10%. Since wave loading o n slender bodies is drag dominated. where the resonant response may be underpredicted. Terms of the linearized coefficient matrices are computed through fairly time consuming numerical integrations in two dimensions. Kirk and Etok 3 have presented a typical application of the frequency domain method for analysis of a pipeline subjected to random oscillations. Although the paper does not include an application of the method to structural analysis of pipes. It should be noted. averaging of the spectra over a number of simulations with different starting phases should be attempted. The use of frequency domain techniques for the analysis of flexible pipes in irregular sea states avoids the cost of time domain simulations. It also illustrates that terms up to third-order must be retained if a realistic simulation is to be achieved. Krolikowski and Gay 33 have given an overview of the techniques available for the linearization of drag forces. In the first technique discussed. The paper is a good illustration of the use of modal analysis and its combination with spectral techniques. This is achieved by minimizing the expected RMS error involved in representing the nonlinear drag force by a'linear expression. A large number of methods have been proposed with different levels of sophistication. However. Hamilton 37. This often provides an adequate resolution. Langley 36 has proposed an attractive method for linearization of the drag force in irregular unidirectional or multidirectional seas. This method assumes that the steady current drag can be superimposed onto the dynamic component and that the effect of the higher harmonics can be ignored as these would be filtered out by the riser dynamics. Trials showed that run times increased significantly with increasing grid size under the probability density function and indeed the largest proportion of computer effort was expended in carrying out these integrations. that omitting the higher-order terms from the expansion of the nonlinear drag force is not necessarily valid. One possible source of inaccuracy in a computer implementation of Langley's method is the two-dimensional integration for the calculation of linearized coefficients. the nonlinear nature of the drag force requires a linearization technique for a frequency domain simulation. the forcing spectrum derived and the response spectrum computed using the frequency response function. Rodenbush et al. be found in Ochi 32 which provides a thorough treatment. the conventional zero-current method is derived which relies on a Fourier expansion of the nonlinear drag term where the coefficients 01! the series are calculated and harmonics above the fundamental ignored. having identified the difficulty of carrying out the double integrations involved in Langley's method.. A starting point on the topic may. The situation is reversed for compliant structures such as flexible risers and guyed towers. Discussion in the paper concentrates on the influence of boundary conditions and derivation of extreme values of bending stresses from the RMS values obtained from the calculated spectra.ter modelling and analysis techniques: M. An approximately inverse square-root relationship exists between the number of frequency bands and the expected accuracy of calculated forces. i1: is shown that the second method reduces to the first for the case of zero current. Given the wave height spectrum. however. a mixed Lagrangian and Rayleigh-Ritz method is used to evaluate the natural frequencies of the pipeline and its complex frequency response function.Review of flexible ri. Simulations of random sea states using linearized spectral analysis techniques are known to offer substantial computational advantages. Smoothing using the average of 100 realizations was used and it was discovered that the RMS values stabilized after about 20-30 iterations. The general conclusion of the paper is that for stiff systems with natural periods of 5 s or less. Seyed many other consideratio:as. e. The wave spectrum was divided into 100 bands and this was adhered to throughout the work. in passing. The program implementation is also likely to be of greater complexity compared with a time domain technique. The linearized techniques appear to be very suitable for two-dimensional analyses and indeed offer great time savings over expensive time domain simulations. Reasonably large 'dynamic' computer storage is required to store the calculated properties for each frequency band but the final output is far less than that produced from typical time domain simulations. Firstly. An example is given 297 by Kao 35 where an investigation into the shortcomings of linear spectral analysis methods has been carded out. however. Their work provides evidence that omitting higher-order terms may yield nonconservative drag loads. It is inferred that for realistic computation of the response. The method relies on a statistical linearization of the drag force across the frequency spectrum. H. a Monte Carlo simulation is presented which illustrates the large improvement over conventional techniques. A major advantage of this technique is its frame invariance property. However. The results show that conventional technique can give rise to large errors and highlight the significant improvement achieved using the second combined technique (Figure 6 illustrates some results).g. especially n~3t with structures whose velocities of motion are large compared with those induced by waves. Pate/and F.0 Horizontal Projection (m) Figure 7 Combined wave.0 I 200. The formulation is of mixed Lagrangian and Rayleigh-Ritz type where displacements are estimated using the combination of a linear function representing the vessel-induced motions and a Fourier series for other dynamic displacements.0 80. The outline of the method and its results are provided but details of its derivation have been left to a later publication.0 320. (--) riser profile developed a novel method for reducing these to single integrals using a series of successive transformations.0 I 300.0 0. B. The paper considers a vertical riser connected at its top extremity to a tension-leg platform which is undergoing motions in regular seas. It should be noted that Rodenbush's reductions must be slightly modified for the special case of zero current. 25 s. H. Wave: 12 m. I I I k=iQ NONLINEARDRAG FORCE DCo EA! -kJ I I I I I I I I I TIME I Ol~d3 UN~TI3N 4 kuc I1¢ 0 Z k~' k. It . The method offers time savings over numerical integration techniques in two dimensions such as the trapezoidal or Simpson's rule and its use in conjunction with Langley's formulation is recommended.0 0. 15 s.SUPE~OSmON MOOK ~ kuc~ U¢ 2 kuc OK bc k = -~ 00Co I I I t | I TLU£ Figure 6 Linearization results from Krolikowski and Gay z3 400. 4~ A modal analysis is then carried out using a linearized form of the drag force to yield the approximate response.0 ~ 160. The combination is believed to be the most accurate method for irregular unidirectional and multidirectional seas in three dimensions including current.~ OCo 3 be 2 ku¢j .0 / I -I00.298 Review of flexible riser modelling and analysis techniques: M. J.. The assumed displacements are then used to compute the eigenvalues using a Rayleigh-Ritz approach.0 ~ • 240. Slug: 230 m.0 400.0 / I 100. Kirk et aL 39 have complemented Kirk4° by including the effects of surface vessel motions. vessel and slug induced dynamics in 15 s regular sea state. Seyed 2 kA~ I I I I I I . The resulting linearized coefficients are thus functions of the standard deviation of the relative velocity. It is shown that a linearization based on independent linearization of the orthogonal components of the relative velocity vector fails all frame invariance tests whilst the minimum mean-square error approach suggested by Langley 36 lacks the second and third properties. simplifying calculations considerably. The validity of the assumption concerning the ratio of the natural frequencies is. The stiffness and mass properties of the system are allowed small perturbations and the corresponding multipliers are adjusted until a minimum least squares solution is obtained. open to question. calculating the responses at each frequency and summing to obtain the overall response. calculating the force in the direction of this resultant and transforming to the required normal axes'.Review of flexible riser modelling and analysis techniques: M. The results of the works of Borgman 49. For example. apparent weight. since the force is calculated in a given direction. Another internationally compiled comparison of analysis results for flexible risers has been prepared by Committee V7 of ISSC 45. its projections onto any set of orthogonal axes normal to pipe axes will maintain the same resultant. the method includes a convolution term from the Fourier expansion of the drag term and shows that the resulting bending moments calculated in this way compare very well with the nonlinear and linear time domain simulations and offer an advantage over other linearizations. However. Seyed appears that only hydrodynamic damping has been considered and structural damping has been ignored. often using the maximum wave approach. Leira 4s has continued the work presented in his joint 1985 paper 43 to identify and compare the merits of different methods of linearization of the nonlinear drag force. B. Mclver and Lunn 51. The paper also includes a linearized spectral approach where the drag force linearization due to Wu and McDermott 42 is employed. The paper discusses two time domain and two frequency domain solut!ions. top tension and so forth are represented as multipliers. This method is based on division of the wave spectrum into frequency bands. This allowed the modal analysis techniques to be used. however. A modified form of Langley's technique is then presented which enforces the colinearity condition to obtain a diagonal (2 × 2) linearization matrix with equal terms of magnitude (80"(v)/~')i where o(v) represents the standard deviation of the relative velocity. Further. The nature of the formulation is such that the use of the method is confined to vertical risers. A thorough mathematical description of the frame variance property is given. This linearization is based on the assumptions that the relative velocity is a Gaussian process and that the input-output cross-correlation function is proportional to the cross-correlation of the input relative velocity. Additionally. The calculated force is therefore unique. The result is physically sound and intuitively very simple and may be restated in the following way: 'The two dimensional drag force should be calculated by computing the resultant relative velocity magnitude and direction normal to the pipe. To quantify the errors involved. Nevertheless. The adopted approach illustrates the preliminary stage in the use of the frequency domain technique.' These provide three fundamental conditions for frame invariance. i. Leira and Remseth 43 and Leira 48 have made a major contribution to the development of linearized frequency domain techniques. a series of frequency domain approximations have so far been suggested. Further. the resultant relative velocity direction may always be rederived from the resultant forces. The required set of natural frequencies are then specified as perturbations about the existing values and changes in system parameters such as inner and outer diameters. The linearised vector should ideally be colinear with the velocity vector. Larsen and Bech 46 have used an approach which is based on the assumption that the natural frequencies of a riser in in-line vibration under a uniform current are twice those in transverse oscillations. This result should not come as a surprise since it is equivalent to calculating the magnitude and direction of the resultant relative velocity vector normal to the pipe and linearizing in this direction. The same applies to the force direction as measured relative to a fixed reference direction. Linearized spectral methods have either been based on a summation of independently calculated . The versatility of the method stems from its ability to eliminate the trial and error stage from design optimization. An interesting application of the frequency domain method has been given by Bernitsas et aL 2° where a nonlinear inverse perturbation technique is used for the redesign of risers in situations where the computed natural frequencies of the riser are found to be undesirable for the sea state under consideration. for a distributed parameter system. The frequency domain solutions employ the irregtdar sea linearization discussed by Krolikowski and Gay 33. K i m 47 has investigated the use of a distributed parameter frequency domain method for dynamic analysis of rigid risers with different support conditions. As a result of its complexity. to quote: 'The force magnitude as measured by the Euclidian norm must be invariant under a rotation of the reference co-ordinate system. H. the eigenvalue analysis. The paper includes a reformulation of Langley's (1984)36 work. This survey indicates that frequency domain techniques offer large reductions in computer time in comparison with their time domain equivalents. a derivation of which was first provided by Gelb and Vander Velde 44. some time domain simulations would then be carried out. Gumestad and C o n n o : °. This work is one of few attempts at riser design optimization that the authors have encountered in the literature and possibilities exist for its development to other applications. Langley 36. In the past. Patel and F. It is stated that for the calculated force to be frame invariant.e. A comparison of four linear and nonlinear methods for dynamic analysis of risers has been provided by Leira and Remseth 43. there have always been questions as to the validity of the linearization methods employed and the extent of their inaccuracy. Consequently. The paper concentrates on the calculation of natural frequencies of vertical risers and presents com- 299 parisons to illustrate the accuracy of the results. The frame invariance is implicit as there can only be one direction in which the resultant can act at any one time. Kao 35. most analyses were carded out in the frequency domain. analysis of vortex shedding is usually carried out in the time domain. The solution uses asymptotic series and shows that in the limit these approach an exact solution. the method is essentially based on a reduction to the one-dimensional form suggested by Borgman 49. The technique starts with calculation of the natural frequencies and mode shapes of the riser and uses a modal normalization to decouple the equations of motion. Krolikowski and Gay 33. The magnitude of the linearised vector should be independent of the velocity vector direction. Wu and McDermott 42. Gardner and Kotch 55 have used a finite element analysis in conjunction with the Newmark-/3 method to provide a time domain technique for the analysis of vertical risers and caissons in waves. test cases and with a standard computer bureau program. The results are shown to lie within the envelopes described by these programs.300 Review of flexible riser modelling and analysis techniques: M. This is followed by an eigenvalue analysis using a diagonalized representation of the mass matrix. A discussion on the generation of elemental forces. Houbolt. Comparison with the API test cases shows reasonably good agreement with slight discrepancies in some cases. Patel e t al. The dynamic equations of motion are then integrated in the time domain using responses at different frequencies or have employed a more elaborate stochastic linearization method to minimize the error involved in linearization over the entire frequency range. The vessel. Equations of dynamic equilibrium in time have been developed using a stiffness proportional damping matrix. the stinger and the pipe are considered and detailed derivation of the equations of motion for each system is described with additional notes on the computer implementation of the equations. The selection of the method has been made on the basis of its ability to introduce artificial damping of the higher modes of vibration. they had a very slow start in hydrodynamics-this being primarily due to the computer storage and costs associated with their use. Bratu and Narzu112 have employed cable elements for analysis of flexible risers. Wilson-0 and Newmark-13 methods. H. Typical methods include the finite-difference. Although they are invariably more expensive in computer time. The frequency domain solution employs a linearized drag force which is calculated by equating the work done by the nonlinear drag force within one wave period to that done by the equivalent linearized form. Patel and Jesudasen 57 have addressed the analysis of vertical free hanging risers subjected to vessel motions and wave loading using the Newmark-/3 time domain technique. good accuracy and ease of implementation52. There follows a time domain analysis which considers the dynamic variations of the system parameters as perturbations about their mean static values. they should potentially provide greater accuracy than their frequency domain equivalents. Clough and Penzien41. additionally. The nondiagonal terms of the equivalent drag-damping matrix are ignored and a diagonalized mass matrix is used. The time domain solution employs the Newmark-/3 scheme. Rayleigh structural damping is assumed with damping in the first two modes of vibration. zero period elongation. Flow charts are provided but details of the time integration technique are not disclosed. a static analysis is carried out using a modified Newton's method to solve the equations of static equilibrium. Recent developments in computer hardware have led to large increases in computer memory and have. The results were validated using model tests at 1:28 scale. Integrations in time are carried out using the Wilson0 method which updates all the system matrices if the structural rotations become larger than 8 ° . low amplitude decay. Example results are given for a case study in deep water and it is concluded that the dynamic range of stresses is narrowed with increasing top tension. the simulation of random seas and imposition of vessel motions is presented and followed with a detailed decription of the Wilson-0 method and the additional improvements incorporated into its formulation. made available large computer disk storage capacities with reduced access times. The formulation of cable geometry and stiffness matrices closely follows the work of Peyrot and Goulois 2. A higher-order stiffness matrix is used which incorporates the nonlinear effects of both tensile and shear forces. The results of the work are compared against the results of other programs collected and published by the American Petroleum Institute 59. The resulting equations are then solved in complex form using an iteration scheme for structural velocities. Details of the formulation are presented and element matrices and their associated equivalent force vectors provided for a selection of boundary conditions and loadings. Further iterations are then carried out at each step to equilibrate the equations of motion. Furthermore. Examples of both approaches may be found in the references already quoted in this section. Patel and F. As a consequence. The first step in the analysis involves an incremental finite element static analysis using a modified form of Newton's method. B. application and limitations of several commonly used techniques. This is the only method which has considered the difference in behaviour of the geometric stiffness matrix in tension and compression and has included the exact deflection shape functions into the equations. The Newmark-/3 method has also been used by Cowan and Andris 1° who employed this for a time domain analysis of a pipe laying system. time domain simulations have rapidly grown in popularity. The next stage of the evolution of dynamic analysis methods was started with the use of time domain techniques. Hashemi-Safai 58 has outlined a general three-dimensional method for analysis of rigid risers. It has been stipulated that the effect of structural damping in comparison with fluid damping is small but its inclusion has the important effect of stabilizing the problem numerically. Time domain dynamic analysis Time domain techniques are well suited to modelling nonlinearities in structural geometry. The problem of the nonlinear dynamics of deep water risers is addressed using a finite element formulation. Results of the time domain program show good agreement with those of the bureau program. Initially. Natvig 56 has chosen the Wilson-0 method for time inte- . The results of both analyses were compared with the standard American Petroleum Institute 59. The material stiffness is assumed temporally constant at the instant of loading. Although these techniques had been in use in many branches of engineering and science. Seyed gration. An initial incremental static analysis is performed with the stiffness matrix updated at each stage. loading and material behaviour. Such damping was considered to have the effect of filtering out these higher modes and allowing larger time steps to be used for integration. The popularity of this method arises from its unconditional stability. General treatments of time domain integration techniques are found in Bathe 52. The treatment of a frictional ball joint capable of developing a bending moment at this point is also explained. A large majority of solutions used in offshore applications utilize the Newmark-13 method to integrate the governing equations in time. The later joint work by Vogel and Natvig 24 on the dynamics of flexible hoses provides further details of the methods. Warburton 53 and Zienkiewicz54 with the first providing a detailed account of the derivation. 6° have presented frequency and time domain analysis procedures for vertical risers. Archimedes' principle is only applicable to bodies fully immersed in a fluid which is not the case with continuous pipes where the ends of a given section of the pipe are clearly not subjected to fluid pressures. Bratu and Narzu112 have attempted to nondimensionalize the equations of motion and in this respect the work is of particular interest. The Houbolt method is. Despite its advantages. recognizing that the pressure forces may be computed directly. Furthermore. B. O'Brien and McNamara 26 have emphasized that the Hilber-Hughes-Taylor variable step method has the ability to adaptively control the progress of calculations through the initial transients until an optimal step size is obtained. Young's derivation suffers from a mathematical error in the calculation of the vertical component of the pressure force. The frequency domain methods were discussed in the previous section. Experience gained through the 301 use of the software has led the authors to the conclusion that for most flexible riser applications. Later works 25.26 have described the extension of the method to three dimensions and a change of the time integration scheme to the Hilber-Hughes-Taylor. it is not clear whether the use of a Newmark-/3 algorithm with a time step equal to the stabilized time step. The static analysis preceding the dynamic analysis is also carried out using the time domain technique. 4z. The problem of dynamic time domain analysis of flexible risers in two dimensions has been examined by McNamara et a l .Review of flexible riser modelling and analysis techniques: M. Dynamic analyses all follow an incremental static analysis with inLtermediate Newton-Raphson equilibrium iterations. It is clear that most authors have been reluctant to become involved in the detailed mathematics of the more advanced methods even in cases where these have been shown to lead to savings in computer effort and storage. more so than other methods such as the Wilson-0 and the Newmark-/3 methods 52. it is of much greater complexity than the Newmark-/3 method which is the favourite with the great majority of investigators. papers by these authors have not provided details on the computer implementation. The integration is carried out on an initially vertical segment of pipe curved into the arc of a circle. These results have been studied and compared later in this paper. Additionally. Traditionally. For the steep-wave case. risers and cables considering the linear effects of axial elongation and torsion. Young and Fowler65. a double steep-wave configuration has been studied which comprises two steepwave risers in parallel planes. Newton's method is used for time integration. Detailed input and output data are provided allowing other investigators to compare results. a buoyancy analogy has been used which involves adding and subtracting the missing end pressure discussed above. The results have been compared against a series of full-scale tests carried out on a 14 in OD pipe of 2 in WT in the Norwegian sector of the North Sea (Frigg Field) in the 1982-83 period. Seyed Adam's multistepping algorithm to yield time histories of the structural response. The criterion used in choosing this method is stated as its suitability !for slow periods of motion experienced by flexible risers.cts and example results have been obtained using 30 frequency bands and 5 wave directions. A study of the time domain techniques so far illustrates their historical development for riser applications. Three static analysis methods and two time domain techniques are presented. amplitude decay properties and the period elongation characteristics of the method. nevertheless.5 s. This requirement arises as a result of the continuity of the pipe which prevents the use of traditional buoyancy calculations. Internal and external pressure effects Remarkably little work has been done to identify and quantify the effects of internal and external pressures on a continuous curved cylinder. take longer to eliminate the transients or compare closely. is likely to give incorrect results. Although the choice of the method affects the ease of programming. have carried out what is perhaps the earliest work on the analytical integration of pressure forces over the curved surface of a pipe. It is therefore difficult to compare this integration method with other algorithms. a maximum time step of 50 s for a total of 2500-4500 s has been used for the static cases whilst the corresponding maximum time step for dynamic runs was 0. Of the four. the accuracy of the results for a given time step and the run time may always be improved by reducing the time step. namely. as obtained from the Hilber-Hughes-Taylor algorithm. The combined dynamics of the stinger and the pipeline subjected to a regular wave are then examined and the range of dynamic stresses computed. expressions are presented for the total pressure resultant on the pipe. Four different methods for dynamic analysis of flexible risers have been compared by Leira and Remseth. the analyses have included wave directionality effe. Although the paper continues to discuss the apparent discrepancy with other methods and presents modified equa- . two are frequency domain techniques and two time domain solutions. The static analysis teclhniques were mentioned earlier in this paper. An example is the work of Felippa and Park 64 who developed a technique for reduction of the equations of motion to first-order with special emphasis on structural dynamic problems. 29 The resulting package provides a facility for the analysis of a multitude of riser geometries. H. The time domain techniques offer linear and nonlinear analysis options. the adaptive control of the time step. Additionally. 23 A mixed finite element formulation was used where the axial force is interpolated independently. However. Back-Pederson61 and Sodah162 also present reviews and some further developments in analysis for flexible risers. Malahy 63 has reported a three-dimensional method for the analysis of pipelines. Patel and F. Assuming a small change of slope between the ends of the element. The development and verification of flexible riser analysis software has been described by Engseth et al. known to suffer from amplitude decay and period elongation properties. These assume that the fluid pressures vary linearly between the base and the top of the curved element under consideration. Some exceptions such as the joint works of McNamara and O'Brien using the Hilber-Hughes-Taylor algorithm are worthy of special consideration where the choice of the method has had a direct purpose. use of the linear techniques suffices and that nonlinear methods are only required for selected cases of material or geometric nonlinearity. The time domain techniques are the linear and nonlinear methods which only differ in that the nonlinear form includes regular updating of the stiffness matrix. This assumption may lead to inaccuracies for larger element sizes in areas of rapidly changing curvatures but is nevertheless consistent with the small angle assumption. Results are presented for single catenary and steep-wave riser configurations. The resulting equations are then solved using the Houbolt method. the flow transfers a tangential force to the pipe through friction against the walls of the pipe. Seyed a straight continuous pipe which provides its exact buoyancy. The method is exact for circular arcs and is applicable to arbitrarily curved elements in two dimensions. In the later sections the authors derive the catenary equations using the pressure integration results and including internal flow terms. The flow of a liquid through a curved pipe will result in forces whose magnitude and direction will depend on pipe curvature. Secondly. The paper considers the governing equations used by many investigators and concludes that although the dynamic analysis methods have undergone major improvements. these may be evaluated using pressure integration techniques. The validity of the three-dimensional method has been verified against the results of two-dimensional cases. A steady flow has different effects on the pipe. The complexity of the problem is amplified by considerations regarding the transfer along the pipe of a mass composed of a mixture of liquids and gases whose mixing and interactions are governed by a multitude of factors ranging from their solubilities. Unfortunately. Chakrabarti and Frampton 68 have published a review of riser analysis techniques with the aim of tracing the historical development of rigid and flexible riser analysis techniques. Thirdly. Details of all derivations have been presented in the paper. Notes have also been included on calculation of stresses and design implications of the work. the resulting forces may be calculated from a momentum balance between the inlet and outlet of the pipe. They have also presented the corrected form of Young and Fowler's equations for small curvature pipes. but for curved pipes depends on the pressures at the ends of the pipe element considered. Sparks addressed a large spectrum of riser problems including the effects of inner and outer hydrostatic pressure forces. rigid risers were primarily used in marine applications. The pressure force calculation method provides a physical and mathematical explanation of the buoyancy analogy to justify the use of effective tension in the governing equations of the riser. the flow momentum steers the flow around the curvature of the pipe. The formulation of Chakrabarti and Frampton 68 has then been corrected and validated against the former. This feature may be exploited in numerical modelling of flexible pipes where the element sizes involved may be increased. A new method for the calculation of pressure forces in three dimensions is then given which is general and may be used for pipes of arbitrary curvature and orientation in space. thereby reducing the computer storage requirements and improving the calculation speed for no loss of accuracy. the original error has propagated through and invalidated the results. The mode of interaction of these forces is of great complexity and as such difficult to compute with any accuracy. Since the same equations are used to derive the governing equations of riser equilibrium. This paper remains a major source of reference in rigid riser technology and encompasses the body of knowledge on rigid risers at the time of its writing. the influence of riser flexural rigidity on bending stresses along its length and at its connections. For simpler cases. several controversial issues involving drag linearization. Patel and Seyed 7° have presented a method for the analysis of flexible risers subjected to a time varying internal flow. For simpler flows. arbitrarily inclined columns. Mclver and O 1 s o n 66 have also described the concept of effective tension as well as providing a thorough treatment of the concept of buoyancy on straight. the role of boundary conditions and stiffness of the connections at the ends of the riser pipe and finally proposes methods for calculation of the natural frequencies for a riser pipe. The complexity of multiphase flows and the empirical nature of the information available on their behaviour. which for a straight pipe is independent of the fluid pressure. not exact and remains valid within the small angle assumption highlighted above. Young and Fowler's work was succeeded by that of Sparks 1~ who addressed a variety of considerations pertaining to the design of marine risers in a very comprehensive paper. Seyed and Patel 9 have provided exact integration results for Internal flow effects The mechanism of internal flow in pipes has long been the subject of concern in the onshore as well as the offshore industry. B. Although this work is not as comprehensive as that of Sparks. The formulation is. it draws attention to many common sources of confusion and error in interpretation of the concept of effective tension and provides a complementary paper to that of Sparks. Results of the analysis have been compared against in-house model tests with reasonable agreement. these are invalidated by the error. Seyed and Patel 9 have presented a detailed treatment of the equations governing internal flow and illustrated that internal flow contributes a term to the effective tension expression which is linearly proportional to the fluid density and cross-sectional area of the pipe whilst being quad- . Patel and F. pressures and temperatures to their compressibilities and viscosities.302 Review of flexible riser modelling and analysis techniques: M. effective tension and nonlinearities resulting from large deformations have remained unresolved. When Sparks' work was published. Firstly. Fyllings and B e c h 69 a r e an example of such work. There is also an increasing trend to use the results of the internal structural behaviour of flexible pipes to compute global riser response. inclination. it imparts a lateral force onto the pipe inner surface. The advantage of this work over that of Young is that the expressions derived are exact and are hence applicable to all element sizes and curvatures. H. tions for rigid risers. The method assumes a flow with a sinusoidal density variation about a mean and has been implemented in the frequency domain. the mathematics of the derivation are in error with the result that the expressions provided for pressure forces are incorrect. and flexible riser technology was in its infancy. has been a major deterrent to their theoretical simulation. The paper includes a detailed derivation of pressure forces over a circular segment of pipe. It is worth noting a subsequent work by S p a r k s 67 which also forms an explanatory paper to the earlier work and discusses pressure forces in much greater detail. The magnitude of these forces can be calculated empirically for the given pipe surface roughness and flow Reynold's number and may often be approximated by constant functions of pipe length. nevertheless. The paper also includes a derivation of the traditional buoyancy approach and an illustration of its equivalence with pressure integration methods. Young and Fowler's work has been investigated in detail by Seyed and Patel 9 where a correction to the error has been presented and the equivalence of the effective tension approach to that of Young and Fowler demonstrated mathematically. roughness and flow velocity. Houston. T. 10. U. BOSS 1985. S. ASME. 'Three-dimensional analysis for submarine cables'. 805-813 3 Kirk. J. K. Appl. I (I). J. Other issues of lesser importance are the effects of vortex shedding and out of plane oscillations of midwater buoys. Houston. C. The programs included several commercial packages as well as in-house programs developed by engineering companies and research institutions. 51-60 4 Meriam. Houston. However. Larsen TM has provided a comparison between the results of 10 flexible riser analysis programs. References 1 De Zoysa. Amsterdam. 'Wave induced random oscillations of pipelines during laying'. Winter Annual Meeting of the American Society of Mechanical Engineers. E. 104. 1985. M. 'An efficient procedure of static analysis of long ocean mining pipe. For several years there has been a need for similar example cases for flexible risers. 7. Tokyo. The work includes a detailed treatment of pressure effects on the pipe and shows an overlap in the modes of action of the pressure forces and internal flow forces. Ocean Res. Seyed and Pate173 have reported on additional results from model tests as well as providing detailed results from a series of numerical verification cases. 1982. OMAE 1986. Hoff. 'Mathematics of flexible risers including pressure and internal flow effects'. TX 11 Sparks. OMAE 1986. OMAE 1985. Struct. E. Energy Resour. R. F. C. 209-223 2 Peyrot. G. Japan 15 Huang. BOSS 1985. model testing and some limited field measurements. L. J. Seyed ratically proportional to the flow velocity. A. J. P. and Chucheepsakul. R. and Rivero. Technol. S. However. BOSS 1985. B. Patel and Seyed 7° have reported on a series of steady-state model tests with the pipe subjected to internal pressure and flow as well as including a series of comparisons with results from independent model tests with the riser subjected to wave loading. 'Mechanical behaviour of marine risers mode of influence of principal parameters'. In 1977 The American Petroleum Institute 59 published a series of example cases for use in verification of programs for dynamic analysis of rigid risers. matters such as drag loading on buoyancy attachments. Details of input parameters have been provided for all test cases which cover many of the cases used by other investigators. 1985. Unless such a vigorous link with reality is maintained. Proc. Validation of numeriical analysis It has taken flexible risers about a decade to become established. It is further illustrated that the resulting term is unaffected by flow direction. G. 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