Journal of Ship Research, Vol. 26, No. 1, March 1982, pp. 59-64

A Three-Dimensional Nonlinear Large-Deflection Model Behavior of Risers, Pipelines, and CaNes

Michael M. Bernitsas 1

for Dynamic

A comprehensive nonlinear model for the dynamic behavior of marine risers, cables, and pipelines has been developed. Large three-dimensional lateral oscillations are modeled in the local principal, osculating and rectifying planes. Longitudinal extensional oscillations in the local tangential direction are also taken into account. The derived model shows the significance of three-dimensional bending effects and the con- tribution of the nonlinear terms. Lateral and longitudinal oscillations are coupled due to the curvature and geometric torsion of the centerline of the structure. The model also shows the effects of the external hy- drostatic and the internal mud static pressure forces--which are integrated exactly along the structure--on the structure's tensile and bending rigidity.

In t roduct ion

TIlE MATHEMATICAL MODEL developed ill this paper, in its most general form, describes the dynamic behavior of marine risers. Simplified versions of it model the dynamic response of cables and pipelines used in underwater operations.

Marine risers (see Fig. 1) are long tubular beams connecting the drill ship with the wellhead at the seabed and are composed of rigid pipes with average length of 15 m (49 ft) and outer diameter be- tween 0.40 and 1.0 m (1.8 and 8.2 ft) [1]. 2 The riser pipes are connected by the riser connectors. A ball joint is used at the lower end of the riser and a combination of a ball and a slip joint at the upper end. The ball joints alleviate high bending stresses and the slip joint compensates for the heave motion of the drill ship. A tensioning system is installed on the drill ship and applies a tension at the top of the riser (TTII). This tension provides part of the support required to keep the riser tight and prevent buckling or collapse. Additional supporting force is provided by buoyancy modules properly distributed along the riser. The kill and choke lines (K&C) are high-pressure small-diameter pipes which run along the riser. K&C lines are mounted directly on the connectors through which they exert concentrated forces and moments of the riser.

Inside the riser is the drill string, which runs from the drill ship to the well. Its outer diameter is 10 to 16 cm (4 to 6.4 in.). In the drill string and between the riser and the drill string circulates the drilling mud. It exerts on the riser static pressure force, Coriolis and centrifugal forces due to the riser's local rotation [21, and vertieal and torsional frictional forces.

The basic sources of external forces exerted on the riser are the ocean currents, the surface and internal waves [8], the drill ship and the riser end systems above the upper joint and below the lower ball joint.

Due to the drifting of the drill ship, the action of the dynamic control system, which tends to reset the ship at its original position above the wellhead, and due to the variation of the direction of the current with depth, the centerline of the riser is in general de- formed to a three-dimensional curve with large deflections with respect to its original equilibrium position [1].

The purpose of this work is to develop a comprehensive math- ematical model for the dynamic behavior of risers which will take

1 Assistant professor, Department of Naval Architecture and Marine Engineering, The University of Michigan, Ann Arbor, Michigan.

Numbers in brackets designate References at end of paper. Manuscript received at SNAME headquarters April 22, 1980; revised

manuscript received November 28, 1980.

Z DHILL SH IP

TENSIONING SYSTEM

SL IP JO INT

JO INT

X ILL AND CHOKE L INES

MODULE

R ISER CONNECORS

BALL JO INT

BLOWOUT PREVENTER Y SEABED

IqELL HEAD

Fig. 1 Marine riser system

into account all major forces and effects. The developed model is a set of consistent equations [4] for large three-dimensional de- flections of tubular beams under internal and external pressure. The extensional oscillations of the beam are also included in this formulation. Finally, the initial position of the riser may be any three-dimensional curve as long as the riser material remains linear elastic and the fundamental assumptions of beam bending theory are not violated [5].

MARCH 1982 0022-4502/82/2601-0059500.37/0 59

Simpl i f icat ions, assumpt ions , and features of the model

Simplifications The riser system can be modeled as described in the introduction

and depicted in Fig. 1. However, the inclusion of some terms which complicate the analysis considerably and which if omitted would introduce only minor errors is not justified. Thus, the model derived in the following section is based on the following simpli- fications.

1. The riser can be modeled as a beam rather than a shell be- cause its diameter-to-length ratio is small. If local detailed analysis is necessary, the riser should locally be modeled as a shell. How- ever, our knowledge of the distribution of hydrodynamic loads around the cylinder is very limited and such analysis could only be approximate and possibly not more accurate than the one de- veloped in this paper.

2. The drill string occasionally comes in contact with the riser through the tool joints and exerts concentrated forces and moments on the riser. The presence of the drill string can be neglected [6]. This assumption underestimates the weight of the riser and its contents. An alternative approach [2] is to assume that the drill string is in rigid contact with the riser through the collars, which results in overestimation of the riser's bending rigidity. In this work, the former approach is adopted. Should the latter be used instead, a simple modification of the equations of motion would be required.

8. The translational and rotational veloeity of the drilling mud is small [7]. Consequently, the centrifugal force exerted on the mud and by reaction on the riser, and the Coriolis force exerted on the riser, can be neglected. For the same reason, the frictional force exerted on the riser due to the mud's viscosity is not taken into account.

4. The variation of temperature of water and drilling mud with depth, which may induce stresses, is negleeted.

Assumptions

In addition to the foregoing simplifications of the riser system, the following assumptions are made:

1. The riser material is isotropic, homogeneous, elastic, and linear.

2. The developed model is a Bernoulli-Euler type of beam model and not a Timoshenko type [8]; that is, planes remain plane after bending and normal to the neutral axis. Bernoulli-Euler models are satisfactory for modeling low-frequency vibrations of beams.

3. Torsional deformation of the riser is neglected. This should not be confused with the torsion of the centerline of the riser which is taken into account.

Assumptions 2 and 8 imply that the differential element ds of a riser has 3 translational degrees of freedom but no rotational ones.

Features of the model Based on these assumptions we can derive the model for the

dynamic behavior of risers. The basic features of the model de- rived in the following sections are:

1. It models the bending of tubular beams in 3 dimensions. 2. It is a set of consistent equations for large lateral deflections

of beams [4]. 3. It takes into account the extensional oscillations of the

riser. 4. The hydrostatic pressure force exerted on the riser is inte-

grated exactly over the external surface of the riser. 5. The mud static pressure force exerted on the riser is inte-

grated exactly over the internal riser surface. 6. The validity of the model is not limited by the boundary

conditions, which may change during drilling operations due to drifting of the drill ship.

7. Initial conditions do not limit the applicability of the model. The initial configuration of the riser may be any three-dimensional curve as long as none of the previously stated assumptions is vio- lated [9].

Incorporation of all the preceding features was made possible by modeling the lateral oscillations of the riser in the local oscu- lating and rectifying planes and the extensional oscillations in the local tangential direction.

The mathemat ica l model

The centerline of the riser is, in general, a three-dimensional curve and can be described by three equations in terms of one parameter [10]. If these parametric equations give the coordinates of the eurve in a cartesian system of coordinates and the parameter is the length s of the curve, then the various properties of the centerline of the riser can be expressed in simple forms.

Thus if x,y,z are the coordinates of a point s along the riser at time t, then

x = X[s(t),t] = X[s(to),to] + u[s(t),t] (1)

y = Y[s(t),t] = Y[s(to),to] + v[s(t),t] (2)

z = Z[s(t),t] = Z[s(to),to] + w[s(t),t] (3)

where X[s(to),to], Y[s(to),to], and Z[s(to),to] are the coordinates of point s at the initial time to and u,v,w are the displacements of point s from its original position.

For the sake of simplicity, time will occasionally be omitted from the notation.

The equations of motion of the riser are derived in the local principal coordinate system defined by the unit vectors in the tangental, normal and binormal directions [10], where the tan- gential unit vector is

~bs] + ~s ] + ~'s j~ = (~l,c~2,c~s) (4)

the principal normal unit vector h(s) is

and the unit vector is the binormal direction/~(s) is

fi(8) = ("/1,Y2,YS) where

1 w(s) =

1

1 v~/s/= ~-~

by bez bz b2y] bs bs ~ bs bsZJ bz bex bx bZz] as bs2 bs Ts 2j bx b2y by b%] bs bs e bs bs 2]

(5)

(6)

(6a)

(6b)

(6c)

and K(s) is the local curvature

(b2x/2 [b2y/2 [b2z/2 K2(s) = ~S~] + ~bs z] + ~SJs~l (7)

For the sake of clarity, the forces exerted on the differential element ds are shown in four different figures (see Figs. 2 through 5).

Figure 2 shows the local principal directions defined by equa- tions (4), (5), and (6) and the hydrodynamic forces exerted on the riser. The structural restoring forces exerted on the element ds are shown in Fig. 8.

The force due to the hydrostatic pressure can be computed by integrating the pressure force on the wetted sides of the element ds, taking into account the curvature of the element due to bending. This method has been applied to the case of two-di- mensional bending of beams [2] and cables [11,12] and yields re- sults correct to first order.

60 JOURNAL OF SHIP RESEARCH

Z

O(o,o,o)

J Fig. 2

Z

ft(s)ds

J fb(~)d s

CtS~~S) ~(S~ Qn(S_ds)

f (M Ob(S-2 ds) n

R

Free-body diagram for differential element ds: local principal directions and hydrodynamic forces

0(o,o,o)

J An alternative method, which yields the exact force, Hw, is

shown in Fig. 4. If the element ds were disconnected from the riser, and were fully submerged in water, the vector sum of/lw and the hydrostatic pressure forces exerted on the bases of the cylindrical element ds would be equal to the buoyancy. There- fore

R(s*lds) i

T(s+ds)

-~T" n( (s- d s)

~" f R(s-~d s)

Fig. 3 Free-body diagram for differential element ds: structural restoring forces and moments

The integral of the mud static pressure,/lm' is computed in a similar way (Fig. 5). So

+'4'( ' sl

Figures 8, 4, and 5 and equations (8) and (9) show that the ten- sion T(s), the hydrostatic pressure term D(s), and the mud static pressure term Din(s) have similar effects on the riser and can be

Z

R 0(o,o,o).

J

B(s)ds

D(s*l 2 ds)

f / WR(s)d s

D(s-~ds)

; y

Fig. 4 Free-body diagram for differential element ds: external hydrostatic pressure and riser's weight

Z

R 0(o,o,o)

J Fig. 5

D(S+dS)

Wm(S)d s

Dm(S-i ds)

i Y

Free-body diagram for differential element ds: mud static pressure

MARCH 1982 61

combined to another force Pc(s), called effective tension:

Pe(s) = T(s) + owgTr [hw - Z(s)] - tim ~--g[hm - Z(s)]

(10)

Likewise, the buoyancy B(s), the riser we ight WR(s), and the mud weight W., ( s ) - -per uni t l ength - -can be combined to yield the ef fect ive weight , We(S), of the riser and its contents in water:

W~(s) = -B(s) + WR(s) + WIn(S) (11)

Equat ions o f equ i l ib r ium

The equ i l ib r ium of forces exerted on the d i f ferent ia l e lement ds yields:

Ve(s+~ds) t (8+lds ) -Pe(s -~ds) t (8 -~ dS)

+ ft(s)ds [(s) + fn(s)ds fi(s) + fb(s)ds [ffs) b2

-- We(s)ds ~ - [me(s)]ds "~ [u(s,t)i + v(s,t)]

+ w(s,t);:] = 0 (12) 3

where [me(s)] is the mass of the riser plus the mass of the riser contents ent ra ined in the d i rect ion of mot ion of the e lement ds, that is, the d irect ion of f(s,t), where

~(s,t) = u(s,t)~ + v(~,t)] + w(s,t)~ (is)

In general , the computat ion of [me~(S)] is diff icult . However , in the local pr inc ipa l d irect ions (t, h, [~), this task is much easier. Since the viscosity of the dr i l l ing mud is small , the per-

8 X, Y, Z, Pc, T, Qb, R, M,~, Mb, [, fit, and ~ are functions of s and t. For the sake of simplicity, however, t is occasionally omitted from the nota- tion

eentage of its mass ent ra ined by the cy l inder in the tangent ia l di- rect ion is very smal l and can be neglected. So

[n~e(S)] t = ~ WR(S ) ds (14)

On the contrary, the whole mass of the mud is ent ra ined in the pr inc ipa l normal d i rect ion fi and the b inormal one ~. Thus

1 [me(s)]~ = g [Wa(s) + Wm(s)] ds (15)

and

[me(S)]b = 1__ [Wrt(s) + Win(s)] ds (16) g

The equ i l ib r ium of the moments exerted on the d i f ferent ia l e lement ds, about its center of gravity, yields

+ + +

- Ot [{Jc(s)} &(s,t)] ds = 0 (17)

,Nomenclature

NOTE: For the sake of simplicity, t is occa- sionally omitted from the notation throughout the text and for all time-dependent quantities.

~(s),[J(s,t) = unit vector in binormal direction B(s) = buoyancy of riser per unit length Db =outer diameter of buoyaney

modules D~ = riser inner diameter Do = riser outer diameter

D(s,t) = force due to hydrostatic pressure Dm(s,t) =foree due to mud static pressure

E(s) = Young's modulus fb,ft ,fn = dimensionless force per unit length

in direction indicated by sub- script

hm = z-coordinate of mud free surface hw = z-coordinate of water free sur-

face Ibb,In,, = second moment of area about axis

indicated by subscript Jc = mass inertia tensor

Jbb,Jnn,Jtt = mass inertia about principal axis indicated by subscript

K(s,t),K(s) = local curvature Ime(S)]l = mass of riser and entrained mass of

contents in Direction 1 indicat- ed by subscript

Mb,M,, = bending moment h(s,t),fi(s) = unit vector in local principal nor-

mal direction Pe(S,t),Pe(S) = e~feetive tension

Qb(s,t), Qb(S),

Q,~(s,t), Q.Cs) ~(s,t)

a(s,t),a(s) t

?(s,t),i(s)

r(s,t),T(s) TTB

u(s,t),~(s) v(s,t),v(s)

w(s,t),w(s)

= shear force = direction of motion of differential

element ds = torsional moment = time = unit veetor in tangential direc-

tion = actual tension = tension at top of riser = displacement in x-direction = displacement in y-direction = displacement in z-direction

We(S) = effective riser weight per unit length

W,~(s) =drilling mud weight per unit length

Wa(s) = riser weight per unit length X,x(s,t),x(s) = x-coordinate of point s along

riser Y,y(s,t),y(s) = Y-coordinate of point s along the

riser Z,z(s,t),z(s) = z-coordinate of point s along

riser

Greek symbols

0q,o%o~3 = directional cosines of local tan- gential unit vector

fll,flz,fl3 = directional cosines of local princi- pal normal unit vector

3q,'Ye,'Ya = directional cosines of local binor- mal unit vector

e(s,t),e(s) = strain pR = density of riser material

pw = water density r(s,t),'r(s) = torsion or second curvature

62 JOURNAL OF SHIP RESEARCH

where {J~(s)} is the inertia tensor and &(s,t) is the local angular velocity.

If we consider the element ds as a small right circular cylinder, then t, r~, and ~ are the principal directions and the inertia tensor becomes diagonal. Thus

:1 {J~l = Jbb (18) 0 Jtt where

Wa D~ Wm D~ z J,,,, = Jbb - - - + - - (19) 8g 8g

w~ D~ Jtt - - - (20)

4g

D-D + D, (21) 2

Equations (12) and (17) can be simplified by using the Frenet formulas (22), (23), and (24) [10]:

~i(s) _ K(s ) ~(s) 5s

~,~(s) _ K(s) i (s) - r (s ) ~(s)

bs ' ~(s)

- r (s) r~(s) bs

where ~-(s) is the local torsion of the riser's centerline:

(22)

(2~)

(24)

.~(~) - K~(s)

~x/~s ~v/~s ~z/os ]

~2x/~s2 ~/~s ~ ~z /~s ~] (25)

~ax/~)s~ b~/~s a ~z /~s ~j

Substitution of (22), (23), and (24) in (12) and (17) yields the following equations of motion:

Equilibrium of forces in the tangential direction:

i)z bPe(S)~s K(s)Q~(s) + f,(s) - W~(s) -~s

g [5t 2 ~)s + ~ s + 5t e 5s] = 0 (26)

Equilibrium of forces in the principal normal direction:

bQ,,(s) 1 ~2z - - + K(s)Pe(s) + T(S)Qb(S) + fn(S) -- We(s ) - -

~S K(s) ~s 2

_ ! [WR(s) + Wm(S)] 1 ]~h, F

52x K-~ tbS~ b-Ts ~

~2 v62y + =0 (27) + bt---g 5s--~ 5t 2 5s z]

Equilibrium of forces in the binormal direction:

i:)Ob(S) 7-(S)Qn(S) -t- fb(S) -- We(S) 3/8(8) ~S

--g~-- [WR(8) + Wm(8)] lot2 Vl(S) bzv , , b2w ,.]

+ Si572~s) + -S~w(s)] = 0 (28)

Equilibrium of moments in the tangential direction:

~B(s) K(s)M~'(s)- ?) [ ] 5s ~ IJc(s)}&(s,t) [(s)= 0 (29)

Equilibrium of moments in the principal normal direction:

~Mn(s) ~ + K(s)R(s) + r(s)Mb(S) - Qb(S) - ~ [{Jc(s)}&(s,t)]

,~(s) = o (30)

Equilibrium of moments in the binormal direction:

c)Mb(s) ~t 5s + Qn(s) - r(s)Mn(s) - [{Jc(s)}&(s,t)]~(s) = 0 (31)

Constitutive relations To complete the model of the dynamic response of risers, we

should derive the constitutive relations. In order to have a set of consistent equations valid for large deflections, all nonlinear terms should be included. The constitutive relation of bending in the osculating plane (t,fi) is [8]

Mb(S) = E(s) Ibb(S) K(S) (32)

where K(s) is the local curvature of the centerline of the riser and is given by equation (7). K(s) is the rate of change of the principal normal direction in the osculating plane [10]. The local torsion of the centerline of the structure, r(s), called second curvature, is the rate of change of the binormal direction in the rectifying plane. Consequently, the constitutive relation of bending in this plane (L/;) is [51

M~(s) = E(s) I,,~(s) r(s) (33) where

D3o Ibb(S) = Inn(S) = rr ~- ~ (34)

5 - Do - Di (35) 2

In the tangential direction t(s), the constitutive relation of ten- sion is

T(s,t) = e(s,t)E(s) 4 [O~(s) - Of(s)] + T(s,to) (36)

where e is the strain of the riser element ds(to)

E(s,t ) - ds(t ) - ds(to) ds(to)

and to is the time of initial equilibrium position. Equations (26) to (33) and equation (36) constitute a set of

consistent equations which describe the dynamic behavior of risers for large three-dimensional lateral oscillations and extensional oscillations.

Pipel ines and cables In general, cables and pipelines during laying operations are

subject to large lateral deflections and extensions. The principles behind the dynamic behavior of pipelines and cables are the same as those behind the model developed in the previous section.

The boundary and the initial conditions for the pipelines, cables, and risers are different. However, none of the equations of the riser model is restricted by these conditions. The basic differences of these two structures from the riser can be handled as trivial cases of the riser model.

Pipelines do not have internal pressure or buoyancy modules. In this case the drilling mud density in the riser model should be set equal to zero and the buoyancy module diameter D~ should be equal to Do. Other modifications of the model may be re- quired depending on the particular configuration of the pipeline [13].

Similarly, the model of dynamic behavior of cables is a trivial case of the riser model. Cables do not have bending rigidity and buoyancy modules and are not hollow. Consequently, in the riser model the inner diameter D~ = 0, the buoyancy module diameter Db = Do, and Elbb = Elnn = 0 [14].

MARCH 1982 63

Conc lud ing remarks

A mathematical model has been developed for the dynamic behavior of risers, pipelines, and cables. It models lateral, large, three-dimensional bending of the aforementioned structures in the local osculating and rectifying planes of the centerline of the structure. Longitudinal extensional oscillations are modeled in the local tangential direction.

The external hydrostatic pressure force is integrated exactly over the wetted riser surface. The internal mud static pressure force is integrated exactly over the internal riser surface. The results of this formulation show that the hydrostatic force increases and the internal mud static force decreases the actual tension in the riser. The resulting tension, called effective tension, defines the buckling loads and the required tension at the top of the riser [5].

Three-dimensional bending effects may be significant for large values of the curvature, K(s), and torsion r(s), of the centerline of the structure. Lateral and longitudinal oscillations of the structure are coupled by K(s) and 7-(s) and in general cannot be decoupled [15].

Boundary conditions and initial conditions do not limit the ap- plicability of the model. Consequently, the initial position of the structure may be any three-dimensional curve as long as the basic beam bending assumptions are not violated.

Acknowledgments The author would like to thank Professors C. Chryssostomidis

and K. Vandiver for their advice and help during this work. The work was partially supported by the Rackham School of Graduate Studies of the University of Michigan, under Grant No. 387565. This support is gratefully acknowledged.

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183-2A, National Engineering Science Co., Jan. 1965. 2 Botke, J. C., "'An Analysis of the Dynamics of Marine Risers," Delco

Electronics, Aug. 1975. 3 Tritton, D. J., Physical Fluid Dynamics, Van Nostrand Reinhold,

New York, 1977. 4 Jones, N., "Consistent Equations for the Large Deflections of

Structures,'" Bulletin on Mechanical Engineering Education, Vol. 10, 1971, pp. 9-20.

5 Bernitsas, M. M., "Contributions Towards the Solution of Marine Riser Design Problem," Ph.D. Dissertation, M.I.T., Cambridge, Mass., 1979.

6 "Comparison of Marine Drilling Riser Analyses,'" American Pe- troleum Institute Bulletin 2J, Jan. 1977.

7 Morgan, G. W., "Marine Riser Systems," Offshore Drilling and Production Technology, 1976, pp. 44-48.

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9 Bernitsas, M. M., "Problems in Marine Riser Design, "Marine Technology, Vol. 19, No. 1, Jan. 1982, pp. 73-82.

10 Eisenhart, L. P., "'An Introduction to Differential Geometry," Princeton University Press, Princeton, N.J., 1947.

11 Breslin, J. P., "Dynamic Forces Exerted by Oscillating Cables,'" Journal of Hydronautics, Vol. 8, No. l, Jan. 1974 pp. 19-31.

12 Goodman, T. R. and Breslin, J. P., "Si.atics'and Dynamics of An- choring Cables," Journal of Hydronautics, Vol. 10, No. 4~ Oct. 1976, pp. 113-120.

13 Lamb, M. J., "'Designing, Laying, and'Maintaining Underwater Pipelines," Offshore Drilling and Production Technology, Petroleum Engineer Publishing Co., 1976.

14 Reid, R. O., "'Dynamies of Deep Sea Mooring Lines," Texas A&M University, Project No. 204 Reference No. 61-11F, July 1968.

15 Azar, J. J., "A Comprehensive Study of Marine Drilling Risers," American Society of Mechanical Engineers Petroleum Division, PET-61, 1978.

64 JOURNAL OF SHIP RESEARCH