f t

214

Unit

er,

e w

an

ling

volv

lly a

vib

ull-

nts

s for oell-heap-watetheir e

minimer, itn intera a buwave,

current related problem of vortex-induced vibrations (VIV) have

ulting

Contents lists available at ScienceDirect

lse

Ocean Eng

Ocean Engineering 38 (2011) 13971402results. A large tank is used for a small-scale study of the riser,E-mail address: l.virgin@duke.edu (L.N. Virgin).been conducted (Cheng et al., 2009; Niedzwecki and Moe, 2008). For one conguration, the lower boundary condition involves atangential lift-off point (also encountered in upheaval buckling ofpipelines, for example), and this requires careful modeling.

The riser is also modeled experimentally to verify the numerical

0029-8018/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.oceaneng.2011.06.009

Corresponding author: Tel.: 1 919 660 5342; fax: 1 919 660 8963.(1973) and Bylsma et al. (1988). Vikestad et al. (2000) studied theeffect of ow past a circular section, and other studies on the

boundary conditions, location and magnitude of the buoyant force.An extensive parametric study of risers, including dynamicsfor the steep-wave conguration, was performed by Seyed andPatel (1991), and for the lazy wave by Liu (2000). Some relatedvibration studies for hanging cables were conducted by Irvine andCaughey (1974), Ahmadi-Kashani (1989), Smith and Thompson

states are computed for varying parameter values. The rescongurations depend on a number of factors including

weight, water depth,steep-S, according to whether the riser rests along the sea bed andthe means by which the buoyancy is achieved (Bai and Bai, 2005).Furthermore, it is reasonable to assume that the riser possessessome bending stiffness. These factors make the analyses of riserssomewhat challenging (Matulea et al. (2008)).

appropriate boundary conditions) (Santillan et al., 2006; Virgin, 2007;Plaut, 2006; Timoshenko and Gere, 1961). Two typical congurationsare compared (Bai and Bai, 2005) with a common feature involvingan intermediate buoyant section designed to relieve some of the axialforces acting on the riser, especially at the end points. Equilibriummodeling, a freely hanging riser withbe considered as a catenary. Howevincorporate a buoyant portion on ariser or an upward force at a point viin the standard congurations of lazysusceptibility to buckling and vibration problems. In terms of simple behavior either ignore bending stiffness or rely on numerical techni-The use of highly exible riserwidespread. The riser connects the wvessel. These systems are used in deepresent design challenges due toil drilling purposes isd with a oating controlr drilling situations andxtreme exibility and

al bending stiffness canis common practice tomediate section of theoy. This addition resultssteep-wave, lazy-S, and

A static analysis is made here as a foundation for a more thoroughdynamic analysis.

Many different solutionmethods have been used to model exiblerisers, including nite element models and analytical methods(Hosseini Kordkheili and Bahai, 2008; Larsen, 1992; Rodrigues et al.,2005). Traditional analytic approaches to modeling slender structural

ques such as the nite element method. However, both of theseapproaches have their limitations, and this research introduces a newtechnique that is able to model the behavior of highly deformedequilibria. Here the governing equations are derived using an elasticaformulation and solved using a nite difference approach (given1. IntroductionNumerical and experimental analysis odeformed risers

S.T. Santillan a, L.N. Virgin b,

a Department of Mechanical Engineering, United States Naval Academy, Annapolis, MDb Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708,

a r t i c l e i n f o

Article history:

Received 24 August 2010

Accepted 26 June 2011

Editor-in-Chief: A.I. IncecikAvailable online 23 July 2011

Keywords:

Geometric nonlinearity

Elastica

Finite differences

Flexible riser

a b s t r a c t

This paper models a slend

industry that connects th

water drilling applications

and susceptibility to buck

with a common feature in

acting on the riser, especia

results of small-amplitude

that closely resemble the f

considered, and experime

journal homepage: www.e02, United States

ed States

exible structure used as a drill string or riser in the offshore oil and gas

ell-head with a oating control vessel. These systems are used in deep-

d present considerable design challenges due to their extreme exibility

and vibration. Two typical congurations are used (Bai and Bai, 2005),

ing the attachment of a buoy designed to relieve some of the axial forces

t the attachment points. Previous work by the authors studied numerical

rations and two other equilibrium congurations using parameter values

scale application (Santillan et al., 2008). Here, two new congurations are

are designed and conducted to verify these equilibrium results.

& 2011 Elsevier Ltd. All rights reserved.he static behavior of highlyvier.com/locate/oceaneng

ineering

modeled using exible uoropolymer tubing. The experimentallows for accurate deection and end point load measurements.A buoyant force can be applied at an intermediate point on thetubing, and both its location along the tube length and its verticalcoordinate can be modied. To compare the numerical andexperimental results, the following parameters are varied system-atically: location along the riser length of the attached buoy;vertical location of the buoy relative to the water surface; totallength of the riser relative to water depth; riser congurationtype. Deections and force measurements are compared, andthere is good agreement between these two models.

A number of simplications are used in this work. The analysis isplanar and thus no twisting behavior is considered (Chai andVaryani, 2006), the riser is assumed to be inextensible, the connec-tion to the vessel is not subject to oscillations, and the study doesnot include internal ow. Water pressure caused by depth is alsoneglected. All of these factors can be incorporated in the analysis ifnecessary, and in general this approach is somewhat more efcientthan a nite element analysis.

2. Analytical formulation

The nonlinear boundary value problem is modeled using non-dimensional equations that are derived using an elastica formulation.Fig. 1 includes a schematic of a marine riser that attaches a oatingvessel to the sea bed. A buoyant force is sometimes incorporated toreduce stresses in the system, especially in deeper water (Bai and Bai,2005). The attachment point at the sea bed can be especiallychallenging from a modeling point of view, for example, a commonconguration is shown in part (b) in which the system rests for acertain length on the sea bed (this is sometimes called a lazy

conguration). Although this gure is a schematic, the congurationsshown are in fact computed solving the governing elasticity equa-tions using a nite difference approach. An important innovation isthat the governing equations are developed in terms of an elasticaformulation (Santillan et al., 2006), and hence no restrictions areplaced on the magnitude of the deections or curvature. A keyelement in this modeling is taking into account the weight of astructure. Normally this effect is negligible for a conventionalstructure, but with such slender, highly deformed structures (in agravitational/buouyant environment) the effect of weight on struc-tural loading becomes an important parameter. A series of para-metric studies was conducted by the authors in which twoequilibrium congurations of the system (steep-wave and lazy wave)were computed as a function of various factors including cablelength, velocity of a cross ow or current, degree of buoyancy, etc.(Santillan et al., 2010).

Here, two new congurations are studied: lazy-S and steep-S.Unlike the wave congurations, the buoyant force here is appliedat a single point. This load, in practice, is applied with a buoy. Theparameters of the riser are depicted in parts (c) and (d) of Fig. 1.The steep-S conguration is shown in part (a), and the lazy-S isshown in (b). The riser is assumed to be inextensible and to have arelatively small, nonzero bending stiffness. The weight and effectof buoyancy are included in the analysis. Points on the riser havecoordinates XS,T and YS,T, and rotation yS,T with respect tothe X-axis, where S is the arc length and T is the time. The totallength of the riser is L L1L2, and the sea depth is H. Thelocation measured along the riser arc length from the seaoorattachment to the buoy connection point is L1, and the height ofthe riser at this point measured from the seaoor is H1. In the caseof the lazy-S conguration, the length L1 includes the risersegment that is resting on the seaoor and the suspended length

-S r

S.T. Santillan, L.N. Virgin / Ocean Engineering 38 (2011) 139714021398E,I,WL1

L2

SY

X

D

H1

Fig. 1. Riser schematics. (a) Steep-S conguration, (b) lazy-S conguration, (c) steepcurrent velocity, V

g

sea water density,

buoyedsectionlazy wave conguration, L1 includes the segment resting on the seaoor.H

Lift-off point

L1

X

Y

Lift-off point

D

iser schematic, (d) lazy-S riser near the seaoor attachment point. Note that for the

tow tank, allowing for accurate deection and load measure-

The end at the bottom of the tank was pinned to a plate resting onthe tank oor so that the tubing could freely rotate, while the topend is similarly pinned at the water surface and mounted on a scaleto measure vertical forces. A light cord could be attached at anypoint along the length of the tubing with the other end tied to aoating ring. The cord length could be adjusted to x specic h1values. The experiment was assembled so that the upward force atthe top end of the riser could be measured with a small scale. Toaccurately measure the riser position, a grid was constructed andplaced against the back wall of the tank. The riser deection atintermediate points along the length could then be measured andcompared with the numerical data.

To verify the numerical data with a baseline case, the tubing wascongured with no intermediate upward force, and the experi-mental results for this simple conguration were compared withthe numerical results (Fig. 4). These results agree well, and thevertical force magnitude at the water surface was also measured.The experimental nondimensional force was ql 4:97, agreeing

S.T. Santillan, L.N. Virgin / Ocean Engineering 38 (2011) 13971402 1399below the buoy attachment. The horizontal distance between theriser ends is D. The internal forces in the strip are denoted PS,Tand Q S,T parallel to the X- and Y-axes, respectively, and thebending moment isMS,T. For both congurations, the upper endof the riser is pinned (ML,T 0. The lower end of the steep-wave riser is also pinned (M0,T 0.

The governing equations, based on geometry, momentcurva-ture relation, and dynamic equilibrium, are

XS cos y,

YS sin y,

yS M=EI,

MS Q cos yP sin y,

PS 2BW=gXTT2mfUXSTmf U2XSSF sin y,

QS 2BW=gYTT2mfUYSTmf U2YSSWB, 1where the subscripts S and T denote partial derivatives. The riser hasbending stiffness EI. The buoyancy coefcient is B 1rgA= W,and the magnitude of a cross-current force, if present, is F rrCdV2.The weight per unit length of the riser in air is given by W. Thesteady current velocity is V, r denotes the sea water density, and rand A are the riser radius and the cross-sectional area, respectively.The drag coefcient is Cd. The inner uid velocity and mass per unitlength are given by U and mf , respectively.

In this study, only static congurations are considered, and thefollowing nondimensional quantities are dened:

x X=H, y Y=H, s S=H, dD=H,

l L=H, l1 L1=H, l2 L2=H, uUHmf =EI

q,

mMH=EI, p PH2=EI, qQH2=EI,

f FH3=EI, wWH3=EI, h1 H1=H: 2In nondimensional terms, the governing equilibrium equations

become

xs cos y,

ys sin y,

ys m,

ms q cos yp sin y,

ps u2xss f sin y,

qs u2yssBw: 3To obtain numerical results for the equilibrium shapes of the

riser, a second-order nite difference method is used. The equa-tions are used to create difference equations for internal nodes ofthe riser. The boundary conditions for the steep-S conguration ats0 (x0 y0 m0 0 and at s l (xl d, yl 1, ml 0are applied at the end nodes. For the lazy-S conguration, theboundary conditions are given by x0 y0 y0 0, xl d,yl 1, and ml 0. The equations are written to accommodatean intermediate point load; at the location s l1, the differenceequation corresponding to the vertical internal force q is replacedwith the condition that yl1 h1. Shooting and continuationmethods are used to approximate solutions to the boundary valueproblem for varying parameter values.

There is nothing in the steep-S analysis restricting the riserfrom lying beneath the seaoor; however, with reasonable para-

meter values, the uplift caused by the buoy load prevents seaoorments. The riser is modeled with uoropolymer tubing and alarge tank at the US Naval Academy. These experiments aredesigned to validate the numerical results for varying congura-tions and conditions.

Experimental images are shown in Fig. 3. The tank being usedis 36.6 m long and approximately 1.5 m deep. The water depth inthe tank uctuates, and this depth was measured regularly. Clearuoropolymer tubing with lengths varying from 4.57 to 6.18 mwas lled with dye through the unsealed ends. The tubing ishollow with a circular cross-section. The outer diameter of thetubing is 14 mm with a wall thickness of 0.7 mm. The measuredmodulus of elasticity, E, is 434 MPa, and the density is 2150 kg/m3.contact. The lazy-S conguration, however, requires that a nitelength rests on the seaoor. To prevent the analysis from predict-ing a riser segment lying beneath the seaoor, an alteration of thenumerical analysis is required. To model this condition numerically,after each numerical iteration is performed, an additional normalforce is included in the vertical force q(s) at any nodes where yo0.The location where the riser makes contact with the seaoor iscalled the lift-off point. Numerical results for varying h1 values areshown in Fig. 2. For a buoyant weight of Bw4.47, nondimensionallength l4.0, and buoy attachment point l11.5, the riser transi-tions from lazy-S to steep-S at h1 0:26.

3. Experiments

Experiments have also been designed and conducted in a large

0 0.5 1 1.5 2 2.5 3 3.50.5

0

0.5

1

1.5

Fig. 2. Numerical riser congurations, l4.0, Bw4.47, and xed heights atl11.5 of h1 0, 0.1, and 0.4.well with the numerical value, ql 5:00. The dimensional baseline

First, deected equilibrium shapes were compared by recordinghorizontal and vertical positions along the length of the tubing.These values were also found numerically for corresponding valuesof total length l, l1, and horizontal distance d. Two resulting steep-Scongurations are shown in Fig. 5. Continuous curves give numericalresults, and solid data points represent experimental riser coordi-nates. The open circle gives the location of the buoy attachment

S.T. Santillan, L.N. Virgin / Ocean Engineering 38 (2011) 139714021400case values and corresponding nondimensional values appear inTable 1, along with numerical boundary conditions for each of thetwo congurations.

4. Results

The experiment was congured with varying parameter values,and deections and forces were compared with numerical results.

point at s l1. Experimental nondimensional upward force values ats l are ql 3:22 and 3.46 for congurations (a) and (b), respec-tively; numerical values for each conguration are 3.07 and 4.09.

A similar experiment was created for the lazy-S conguration.Results are shown in Fig. 6. The buoy is attached closer to thebottom attachment point in part (b), requiring that the water-level attachment carry more of the tubing weight. In congura-tion (a) the experimental and numerical values of q(l) are 3.46 and3.77, respectively, and as expected these values increase signi-cantly for conguration (b). For that case the values increase to an

Fig. 3. Experimental riser and upper attachment. A grid is placed behind the riserto measure displacements, and the upper attachment allows rotation of the riser

end and is placed on a scale to measure vertical forces on the tubing.

0 0.5 1 1.5 2 2.5 0.2

0

0.2

0.4

0.6

0.8

1

Fig. 4. Experimental riser (data points), and numerical results (solid line): l2.99,d2.50, and Bw 4:47.

Table 1Dimensional and nondimensional values and numerical boundary conditions used

in the present study.

Dimensional Nondimensional

Baseline values

EI0.2680 kN m2 w8.357H1.53 m, D3.82 m d2.50r 2150 kg=m3 l2.99ro7 mm, ri6.3 mm, L4.57 m B0.5349

Boundary conditions

x(0)0 xl dy(0)0 yl 1m(0)0 (steep-S), y0 0 (lazy-S) ml 0

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

y

y

x

x

1

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

Fig. 5. Steep-S riser congurations, Bw 4:47, l3.02, d2.7. Data points:experimental; solid line and open circle: numerical. (a) l11.26, h10.45.(b) l11.01, h10.61.

y

0.810 0.5 1 1.5 2 2.5 3 3.50

0.5

y

x

x

1

0 0.5 1 1.5 2 2.5 3 3.50

0.20.40.6

Fig. 6. Lazy-S riser congurations, Bw 4:47, l3.94. Data points: experimental;solid line and open circle: numerical. (a) d3.67, l12.03, h1 0:48. (b) d3.55,

l11.67, h1 0:37.

experimental value of 5.18 and a numerical value of 5.13. Thisforce value varies signicantly with the system parameters andconguration types. The dependence of q(l) on the xed height atl1 was studied experimentally by xing d and the buoy attach-ment location and varying only h1. Congurations resulting fromthree values were found numerically and are shown in Fig. 7(a).Calculated force values for the three specied values of h1 0:01,0.29, and 0.57 are ql 6:34, 4.41, and 3.86, respectively. Theupward force as a function of h1 is shown numerically withexperimental values in Fig. 7(b). The results agree well for thislazy-S conguration.

A similar study was then performed for a set of parametervalues that cause a transition from the lazy-S to steep-S cong-uration with increasing h1. Again, some intermediate numericalequilibrium shapes are shown in Fig. 8(a). For the xed valuesshown (h1 0, 0.34, and 0.68), numerical nondimensional forcevalues at the upper end are 4.93, 4.49, and 4.02. These numericaland experimental forces are shown in Fig. 8(b). For this specicset of parameter values, there is a small range of h1 values thatresult in a segment in contact with the ground along l1oso l(i.e., between the buoy and the water-level attachment point).This segment loses contact at h1 0:05. As h1 increases beyondthis value, the buoy carries a larger segment of the tubing, causinga decrease in the required upward force at the upper end. Theconguration transitions from lazy-S to steep-S at h1 0:31, butthe force value at the s l is unaffected by this transition becausethe buoy, not the upper attachment, carries the weight of theuplifted segment at the lower end.

Finally, q(l) is measured for varying buoy attachment location, l1with xed h1. For this case, the horizontal distance between thetubing ends was not xed; the upper end was allowed to move and

for each l1 value, d was recorded. Numerical values of q(l) werecalculated with these varying d values. The congurations shown inFig. 9(a) correspond to l1 1:55, 1.94, 2.33, and horizontal lengthvalues d3.54, 3.57, and 3.65. Calculated upper end force magni-tudes are 5.72, 4.16, and 2.82. Results for several values are comparedand plotted in Fig. 9(b); numerical and experimental values agreewell for this lazy-S conguration.

5. Conclusions

Steep-S and lazy-S risers have been considered for the planarequilibrium case. The risers were modeled using the elastica,where the bending stiffness is included and the bending momentis assumed to be proportional to the curvature. The upper end ofeach riser was pinned, and an upward buoyant force was appliedat an intermediate point along the length. Numerical results wereobtained using a nite difference method; unlike with niteelement packages, this approach is easy to implement andefcient for all cases, including that of a very low bendingstiffness. Results of a parametric study were then compared withexperimental results that were conducted using a large tank anduoropolymer tubing.

The effects of the conguration type and the following para-meters on the equilibrium shape and upper end force wereinvestigated: total length, buoy attachment location, and thexed height of the attachment point. There is strong agreementbetween the experimental and numerical results of the study;

0 0.1 0.2 0.3 0.4 0.5 0.63.5

h1

Fig. 8. Riser with varying h1. Bw 4:47, l3.94, d3.42, l11.74. (a) Numericalcongurations for h1 0, 0.34, and 0.68. (b) Data points: experimental; solid line:numerical.

S.T. Santillan, L.N. Virgin / Ocean Engineering 38 (2011) 13971402 14010 0.1 0.2 0.3 0.4 0.53.5

4

4.5

5

5.5

6

6.5

h1

q(l)

0 0.5 1 1.5 2 2.5 3 3.50

0.20.40.60.8

1

x

y

Fig. 7. Lazy-S riser with varying h1. Bw 4:47, l3.94, d3.5, l11.84. (a) Numericalcongurations for h1 0:01, 0.29, and 0.57. (b) Data points: experimental; solid line:

numerical.4

4.55

5.5

6q(l)

0 0.5 1 1.5 2 2.5 30

0.20.40.60.8

1

x

yfuture verication of the numerical results can be conducted for

References

Ahmadi-Kashani, K., 1989. Vibration of hanging cables. Comput. Struct. 31,

699715.Bai, Y., Bai, Q., 2005. Subsea Pipelines and Risers. Elsevier, Amsterdam.Bylsma, R., Nguyen, A., Van Baak, D.A., 1988. Oscillations of a suspended chain. Am.

J. Phys. 56, 10241032.Chai, Y.T., Varyani, K.S., 2006. An absolute coordinate formulation for three-

dimensional exible pipe analysis. Ocean Eng. 33, 2358.Cheng, Y.M., Xu, L.X., Lambrakos, K.F., Muriby, K., 2009. Investigation of VIV fatigue

prediction for a top tensioned riser. In: 28th International Conference on

Ocean Offshore and Arctic Engineering.Hosseini Kordkheili, S.A., Bahai, H., 2008. Non-linear nite element analysis of

exible risers in presence of buoyancy force and seabed interaction boundary

condition. Arch. Appl. Mech. 78, 765774.Irvine, H.M., Caughey, T.K., 1974. The linear theory of free vibrations of a

suspended cable. Proc. R. Soc. A 341, 299315.Larsen, C.M., 1992. Flexible riser analysis-comparison of results from computer

programs. Marine Struct. 5, 107119.Liu, Y., Bergdahl, L., 2000. Frequency-domain dynamic analysis of cables. Eng.

Struct. 19, 499506.Matulea, I.C., Nastase, A., Talmaciu, N., Slamnoiu, G., Goncalves-Coelho, A.M., 2008.

On the equilibrium conguration of mooring and towing cables. Appl. Ocean

catenary riser model. In: Oceans 2008 Conference.Plaut, R.H., 2006. Postbuckling and vibration of end-supported elastica pipes

conveying uid and columns under follower loads. J. Sound Vib. 289, 264277.Rodrigues, G.J.O., Cardoso, D.C.T., de Lima, B.S.L.P., Jacob, B.P., Fernandes, A.C.,

2005. An analytical/numerical procedure for structural analysis of hybrid riser

5.5

6

6.5q(l)

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

x

y

S.T. Santillan, L.N. Virgin / Ocean Engineering 38 (2011) 1397140214021.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.42.5

3

l13.5

4more complex cases. The numerical model allows for the applica-tion of an oscillating upper attachment point that would modelwave motion and for a horizontal current. The model can beextended to allow for three-dimensional congurations.

Acknowledgments

This work has been supported by Dr. Kelly Cooper under ONRGrant 000141-0W-X2-1-287.

Fig. 9. Lazy-S riser with varying l1. Bw 4:47, l3.94, h1 0:31. (a) Numericalcongurations for l1 1:55, 1.94, 2.33. (b) Data points: numerical; open circles:experimental.systems. Proceedings of the 24th International Conference on Offshore

Mechanics and Arctic Engineering, vol. 1; 2005, pp. 549555.Santillan, S.T., Virgin, L.N., Plaut, R.H., 2006. Post-buckling and vibration of heavy

beam on horizontal or inclined rigid foundation. J. Appl. Mech. 73, 664671.Santillan, S.T., Virgin, L.N., Plaut, R.H., 2008. Nonlinear two-dimensional elastica

analysis of subsea risers. In: Proceedings of EURODYN 2008, 7th EuropeanConference on Structural Dynamics Paper E122.

Santillan, S.T., Virgin, L.N., Plaut, R.H., 2010. Static and dynamic behavior of highlydeformed risers and pipelines. J. Offshore Mech. Arctic Eng. 132.

Seyed, F.B., Patel, M.H., 1991. Parametric studies of exible risers. In: Proceedings

of the First International Offshore and Polar Engineering Conference, vol. 2,pp. 147156.

Smith, C.E., Thompson, R.S., 1973. The small oscillations of a suspended exibleline. J. Appl. Mech. 40, 624626.

Timoshenko, S.P., Gere, J.M., 1961. Theory of Elastic Stability. McGraw-Hill, New York.Vikestad, K., Vandiver, J.K., Larsen, C.M., 2000. Added mass and oscillation

frequency for a circular cylinder subjected to vortex-induced vibrations andexternal disturbance. J. Fluids Struct. 14, 10711088.

Virgin, L.N., 2007. Vibration of Axially Loaded Structures. Cambridge UniversityPress, New York.Res. 30, 8191.Niedzwecki, J.M., Moe, G., 2008. An investigation of VIV response of a steel4.55

Numerical and experimental analysis of the static behavior of highly deformed risersIntroductionAnalytical formulationExperimentsResultsConclusionsAcknowledgmentsReferences