Equilibrium of offshore cables and pipelines during laying

THE DANISH CENTER FOR

APPLIED MATHEMATICS AND MECHANICS

Søren Christiansen Frank A. Engelund Svend Gravesen Erik Hansen ¡var G. Jonsson P. Scheel Larsen Hugo Møltmann

Frithiol L Niordson

Pauli Pedersen Kristian Refslund

Scientific Council

Laboratory of Applied Mathematical Physics

Institute of Hydrodynamics and Hydraulic Engineering

Structural Research Laboratory

Laboratory of Applied Mathematical Physics

Institute of Hydrodynamics and Hydraulic Engineering

Department of Fluid Mechanics

Structural Research Laboratory

Department of Solid Mechanics

Department of Solid Mechanics

Department of Fluid Mechanics

Secretary

Frithiof

_{I. Niordson, Professor, Ph. D.}

Department of Solid Mechanics, Building 404

Technical University of Denmark

2800 Lyngby, Denmark

boratorlu

Sc

.-

..'

Archief

e

2, 2628

' DG!ft

O1 .y35

EQUILIBRIUM OF OFFSHORE CABLES

AND PIPELINES DURING LAYING

by

P. Terndrup Pedersen

DEPART ME NT

OF

OCEAN ENGINEERING

THETECHNICAL UNIVERSITY OFDENP.IARK

A Cn Ct D EI F f n' n f,,g.

J)

H Hb.hb H1 ,h. M N,n px r T Tb V Vb, Vb V

_i'

V.

_i

Wb wt

Distance between barge deck and water surface Normal drag coefficient

Tangential drag coefficient Outer diameter of pipe or cable Bending stiffness of pipe

Normal drag force per unit length Tangential drag force per unit length Auxiliary functions

Water depth

Horizontal force component at the ocean floor Horizontal force component at upper end of suspended length

Suspended length Bending moment Axial tension

Components of load per unit length Arc length

Shear force

Horizontal anchor force Current velocity

Vertical force component at ocean floor Vertical force component at upper end of suspended length

Buoyancy per unit length Weight per unit length X,Y,x,y Rectangular coordinates

3 _{Tangent angle}

Mass density of water

P. Terndrup Pedersen Department of Ocean Engineering

The Technical University of Denmark, Lyngby, Denmark

ABSTRACT - An efficient solutin technique for

determination of the static

equilibrium form of a cable or a pipeline suspended between the ocean floor

and a laying barge or a stinger is presented.

Variations in bending

stiff-ness, weight and buoyancy and forces due to ocean current are taken

into

account. The governing non-linear boundary value problem

is

transformed into a non-dimensional form such that the a priori unknown suspended length

of the pipeline or cable acts as a scaling parameter.

The numerical

solu-tion is then based

on

successive integrations.

The

ajìication

of

the

method is illustrated

by the analysis of equilibrium curves

and stresses

in pieeiines laid with or without the use of

stingers and during abandon

and recovery operations.

-1. INTRODUCTION

The object of this paper is to present a technique that will provide an efficient and direct solution to the static

equilibrium form of a cable or a pipeline suspended between the ocean floor and a laying barge in a steady ocean current.

With increasing water depth the length of the suspended pipeline between the pipe laying barge or the lift-off point from the stinger and the ocean floor becomes greater. This may cause the stresses in the pipe to become so high that the pipe buckles or the stresses in the stinger to reach a level where the stinger breaks. In calm seas these considerations establish the limits for water depths in which pipes can be layed.

The problem of predicting the suspended geometry and there-by the stresses of marine pipelines during laying in the ocean is one of large deflection beam theory, where the length of the

suspended beam is nota priori known. There has been some

Stiffened catenary solutions have been developed [i), _[2) for laying of pipes with relatively small bending stiffness in extremely deep water. The stiffened catenary solution is

ob-tainod by applying the method of matched asymptotic expansions

irto _{is based on the assumption that the pipeline takes a shape}

which approximates a natural catenary over most of its length ,nd that the influence of the boundary conditions is confined to small "boundary layers" near the pipe-laying barge and the ocean floor. Unfortunately, the method is difficult to apply when the pipe is laid with the use of a stinger, but in those cases where all the basic assumptions of the theory are

satis-fied the method is attractive because of relatively small

demands onl calculations.

For the analysis of pipes laid in shallower water or of pipes with larger bending stiffness much more involved finite element methods [3), finite difference methods [41 - [51, and methods involving transformation of the non-linear,two-point boundary value problem into initial-value problems have been proposed. Common to these methods is that they are based on successive iterations and that each main iteration step in-volves a second set of successive iterations for the calcula-tion of the suspended length, thereby making these methods relatively expensive as regards computer size and time. Furthermore, these methods are usually not flexible enough to handle all the existing types of pipe-laying procedures without non-trivial modifications.

In this paper a method is presented which provides a

relatively direct solution to cable and pipe-laying problems. The governing non-linear, two-point boundary value problem is

derived and transformed into a non-dimensional form such that the a priori unknown suspended length of the pipeline or cable acts as a scaling parameter. _{The method of solution is then based} on successive integrations. _{The requirements to computer storage} and time are thereby limited to the same order of magnitude as for the stiffened catenary calculations. _{In order to demonstrate} the flexibility of the method, examples are given on the ana-lysis of the pipe-laying operations, where the pipe is laid without the use of a stinger or with an articulated _stinger,

and with the use of a rigid stinger. Finally pipe abandon! recovery operations are treated. In the last section of the paper we demonstrate how cable-laying problems and one-leg mooring problems can be analysed by the proposed method.

2. LOADING ON THE SUSPENDED PIPE.

We consider a deformed inextensible pipe as shown in Fig. 2.1. As independent variable in our formulatiqn we shall employ the arc length s from the point where the pipe touches the ocean floor. This point will also serve as origin for the rectangular coordinate system X-Y shown in Fig. 2.1. The tangent angle to the pipe is denoted O(s). and we designate the depth of the ocean by H.

1he load on an element of unit length of the suspended pipe is composed of: the weight wt(s), the buoyancy w0(s)

and,due to a steady ocean current with velocity V(Y) , also a normal drag force Fe(s) and a tangential drag force _Fe(s)

The mass density of the water is given by

'

the

gravity by g, and the cross sectional area of the pipe by A.

Y

The buoyancy load on the pipe due to the water pressure is determined as follows. Corlsider the segment of length ds shown in Fig. 2.2. The total buoyancy of the segment with

7open ends

equals

JgAds

and acts in the Y-direction. This load has to be corrected for the lack of pressure at the ends of the segment. From Fig. 2.2 it follows that the resulting buoyancy load w0ds acts in the direction normal to the centreline of the pipe segment and with a magnitude given by

Fig. 2.2

Buoyancy on pipe element.

dO

W0 =

WiCOSO

+ (H-YY--j

The loading due to the current takes the form

F = p CV2A

2v

c

where C is the drag coefficient, V the flow velocity, and

Ac a characteristic area. The axial and tangential load per Unit length can be obtained from (2.2) as

F =

pCVVIDsin2O

Ft = 1TJ _{C VIVIDcOS2O}

Vt

(2.1) = FcosO - (wü_Fn)sinO (2.4) p(s) = FsinO + (w0_Fn)coSO -respectively.

3. GOVERNING EQUATIONS FOR PIPES

In this section we shall set up the governing equations for the plane, one-dimensional, finite-strain beam theory which will be used to model the pipe.

As constitutive law for the pipe we will assume a linear relation between the bending moment M and the curvature

dO/ds. Thus

where EI is the bending stiffness of the pipe.

The moment equilibrium condition for segments of the pipe

(2.2) gives the shear force T(s) as

dO

dM

-T(s) = _{ã ds}

The shear force T at any section of the pipe can be found from Fig. 2.1 by equilibrium considerations. We find

d dO

(2.3) T(s)

_{- (EI) = _HbsinO(s) + vbcosO(s)}

(3.2)

where

Wb = PgA

for O < Y < H and Wb = O for Y > H. M(s) = EIP (3.1)

where D _{is the diameter of the pipe, and C, C are drag}

-

Cos8(S)J(5l)dsi

₊

_{sine(s)J(si)dsi}

(3.3)

coefficients.

where 11.0, V, are the horizontal and vertical force components,

Thus, the resulting horizontal and vertical load inten- respectively, at the support point at the ocean floor.

form

+ coso(s)J P(S1)dS1 - sinO(s)1 p (s )ds

_lxi

O JO and N(s) = Hbcose(s) + vbsinO(s) whe re = Ftcoso(s) + F sinO(s) -

siñO(s)J(si)dsi

_{COSO(5)jx(Si)dS1} p(s) = Fsin0(s) - F cosO(s) _{- (WL_wb)} 11b + HwbcosOb _{Vb + HwbsinOb} and N = N + {H_Y(s)}wb

Equation (3.5) shows that the effect of the buoyancy on the equilibrium curve of the pipe can be accounted for by

intro-ducing the submerged weight of the pipe. However, it will be seen from Eq. _{(3.6) that taking care of the buoyancy simply by} introducing the submerged weight results in an apparent axial

for N _{which equals the real axial force N plus the}

hydro-static force _{wb(H_y) . Here, we may note that for the} eva-luation of the buckling strength of a pipe it is the real axial

force N _{that is of importance, whereas for the determination}

of a reference stress for a solid cable or mooring line we will

(3.6)

-2 d de

A _{-(y-) = hbsinø - VbCOSO +}

+A{cosOJPdi

-

sinOJPdi}

and Eg.

(3.6)

takes the form

(3.7)

n() = hbcosO + vbsinO - A{sinOfPdi + coser

d1}

(3.8)

0 '0

Neglecting the axial extension of the pipe, the relation between the dimensionless natural coordinates (O,E) and the dimension-less rectangular coordinates (x,y) are

dy = sinOd and dx = cosOd (3.9)

The boundary conditions at the ocean floor are

y(0) = x(0) = O (3.10)

O(0) = (3.11)

(de\

(3. 12)

The boundary conditions at the upper end of the suspended pipe depend on the method of operation (for example the type of

stinger used) . But we note, for future use, that the dimen-= s/L {x, y) = {X, Y)/L A = L/B

The equations (3.3) and (3.4) can also be written in the

EI

Y = ----

_x'

Pyf

₌

d dO - wtH

= HbsinO(s) - VbcosO(s) and

(s

Is

hb Vb, n,t} = (Hb, Vb, j:, T}/(wH)

(3.5)

0.

where w is a characteristic value of the weight per unit

length w of the pipe. Then Eg. (3.5) takes the form be concerned with the adjusted axial force which here is suspended pipe as

denoted N

N(s) = HbcosO(s) + VbsinO(s) In order to isolate the unknown suspended length L of

the pipe let us then introduce the following dimensionless

_sinOsJys1dsl

-

cOsOsJxsidsi

(3.4)

o

and that the vertical component of the tension is given by

i

vi v - A

The non-linear boundary value problem will be solved nu-meri.cally by successive iterations. The physical background of of the numerical method is as follows: first the loading on

the pipe (p,p) associated with an arbitrary deflection _curve

and is determined; based on this loading a new deflection curve is found (8. ,x. ,y , and À. } which is

3+1 j+1 j+l j+1

then used In the next iteration step for the calculation of the loading on the pipe. This continues until a certain re-quired degree of equilibrium is obtained. _{Mathematically,} the method Is based on a series of formal integrations of

the thfferential equations, together with a determination of the dimensionless suspended length A , which acts as a

scaling parameter.

In all the examples presented in the following the iteration procedure is stopped when the following inequa-lity is fulfilled

(1

{(x -x )2 +

(Y31-Y)21d1

2

j_O j+i i

(3.14)

where

t

is a small number (lOs).

(4.1)

o

Consequently, tiere the non-dimensional forces h.,v. are related to the

to the equilibrium app)ied forces H.,V. by gration we can h1 = (H1 _{+ (H-Yj)wbcosOj}/(wH)} where and f1(.1) =

y.

_i

= (V.

_i

+ (H-Yjw sinO.}/(wH)

_i

_b

_i

in

The differential equations (3.7) and (3.9) and the bound- f3R)

let us assume that curve (8..x..Y}.

determine the functions

Ir,

;

f.(F,)

= I

I y 1 I

-

pd1

i

fU)

=

we have an approximation Then by numerical

inte-f.();

i = i ...6,

X

Y 1

sionless applied horizontal tension can be found from the following equilibrium equation

h. _{= hb}

-pxl

(3.13)

ary conditions of the problem constitute the non-linear bound- O O

(4.2)

ary value problem to be solved.

f5()

= (f3sinø. + f4cosO.}d1

4. METHOD OF SOLUTION FOR PIPE-LAYING PROBLEMS, and

f6;)

=

f5

O

Following the outlined method of solution we get from Eq. (3.7) ,wlth the notation (4.2)

dO -2 d

A.1

-(ï d) -

hbslnO. -

vbcosO.

(4.3)

+ A . (f sinO f cosO 3 j 4 j

and from Eq. (3.8) we get

n1(E)

_{= hbcosO.}

+ vbsin8.

(4.4)

Fin1ly,the equations (3.13) and (3.14) result in

h1 = hb + À.1f3(l)

y. = y - A. f (1)

i b

j+l

4

respectively.

A first step in the determination of the new approximation to the equilibrium curve is a formal integration of Eq. (4.3),

which yields

do,

A21(y

lf1)

-

h y - y x + A f + C

b j b j+1 S

The static boundary condition (3.12) expressing the fact that the bending moment is zero for = O imposes the value zero for the integration constant C. Now the tangent angle

can be obtained from (4.7) by a second formal integra-tien resulting in

+ A2

_0b

A2

0. (1) _{hbfl - vbf2 + A.1 f6}

j+l

1+1 3+1

when the kinematic boundary condition (3.11) is introduced. In order to proceed we have to introduce the boun-dary conditions at the upper end of the suspended pipe. We

shall here consider three important cases.

a. Pipe-laying without stinger or with an articulated stinger.

Fig. 4.1

Pipe-laying without stinger.

First we shall consider a case where the pipe is layed without the use of a stinger. We shall assume that at the

upper end of the suspended pipe we have the kinematic

bound-(4.5) _{ary conditions} (4.6) (4.7) where (4.8)

_{g () = O}

_{+ (0.-0}

f2)

i b i b f2(l) f(ì) = hi(f1() - f2(1) f2()} jo

The as yet unk'own suspended length A1 of the pipe

can now be determine. by solving the transcendental equation obtained from the bourìary condition (4.10) and Eq. (4.15)

0(L) = 0.

1

Y(L) = H + A

where A _{is the distance between the pipe support ori the}

barge and the ocean surface. We shall also assume that the applied horizontal tension H. at the support is known.

Introducing the kinematic boundary condition (4.9) in (4.8) gives

= 1(0 -0.)AT2 _{+ hbfl(l) + f6(1)A1)/f2(1)}

b i. j+1

Introducing (4.11) and (4.5) In (4.8) yields

2 3

= g1() + A.1 g2() + A1 g3()

g3(E) =f6(F) +(f3(l)f1(1) f6(1)}f(1) f3(i)f1U) (4.11) (4.12) (4. 13)

From (3.9) and (3.10) we get

X.

_j+1

_{= jo}

cosO

d1

i+1 (4.14) = _sinO (4.15) (4.9) (4.10)

1

sin(g + A2 g + A3 _{g3dE1 = 1 + a}

i j+l 2 j+i,

o

where a = A/H.

Thus, starting with an arbitrary integrable approximation to the equilibrium curve the functions _{g1, g2, and g3 can} be determined from (4.13) and a new approximation to the sus-pended length of the pipe _{can be found from (4.16). The} improved approximation to the equilibrium functions _are here-after found from (4.12), (4.14), and (4.15).

The sequence of successive iterations may be started with an arbitrary regular function satisfying the kinematic boundary conditions. _{We can, for example, use the deflection curve} corresponding to a natural catenary or a solution of the linearized Bernoulli-Euler beam equation that satisfies all

the boundary conditions at the ocean floor and the kinematic boundary conditions at the upper end of the suspended pipe.

These methods of obtaining the first approximation also supply us with a first estimate of the suspended length.

The effect of having part of the equilibrium curve above the surface of the ocean (A > O), or support buoys along the pipe, or, an articulated stinger, is easily taken care of in the present formulation by introducing _{a variation in the} distributed buoyancy and/or weight of the pipe.

As an application of the foregoing, Fig. 4.2 shows _the results of the numerical analysis of a pipe-laying procedure where the pipe is laid without _{the use of a stinger. The water} depth H _{is 50 m, the pipe leaves the pipe-laying barge 2 m}

above the water surface at an angle equal to 200. The hori-zontal tension H. applied at the barge is 2.200l05N.

The uniform bending stiffness EI of the pipe is 2.256'109Nm2, the buoyancy per unit length in water _{Wb is} l.6l4l04N/m and

the weight per unit length w is l.843l04N/m.

Starting with a deflection curve corresponding to the solution of the linearized beam equation, where the effect of the applied horizontal tension _{h1 is neglected, the} so-lution presented in Fig. 4.2 is obtained in 3 iteratIon steps.

(4.16) A.2 so Lt 2 fl$ t0 'b.' -, 'to' 41. a t &t4 to' t41. 4, L200 tO'N o

Fig. 4.2 Results of numerical analysis of pipe-laying procedure without the use of a stinger.

b. Pipe-laying using a rigid stinger.

We will now consider pipelaying with the use of a rigid

stinger with a fixed curvature i/R as shown in Fig. 4.3. Let us

assume that the applied horizontal tension is H. at the upper end of the suspended pipe (the lift-off point from the stinger) The tangent angle of the stinger at the point where the stinger

is hinged to the barge is denoted O. _{The angle e will}

nor-mally be a non-linear func-tion of the posifunc-tion of the lift-off point given

by Y and and the magnitude of the concen-trated force T1

per-pendicular to the stinger axis at the lift-off point. Due to the constant curva-ture of the stinger the force T1 equals the shear force in the pipe Just be-low the lift-off point. The function = 0(Y.,T.) can be determined when the geometry and the weight distribution of the stinger are known. See Fig.4.4

Besides the static boundary condition expressing the fact that the horizontal force is H at the lift-off point the bending moment M1 is also given

= -EI/R (4.18)

Finally, kinematic considerations give us the relationship

I-1-Y1 +A

cosO. = cose

-i u R

Let us now construct the iteration algorithm using these boundary conditions. _{Applying the boundary condition (4.18) to} the equation (4.7) results in

Fig.

4.4

_{The stinger supported part} of the pipe. X- (1) _____ - I hb _{Vb y(l)} r (l) A141 (l) f5(l) (4.19) (4.20)

where r R/H. _{The equations (4.20) and (4.5) yield}

y.(l) f5(l)

f (1)) + ) Vb

_x1(l)

1h1 + A11{(1)

3 1+1 r y(l) (4.21)

substituting (4.21) and (4.5) into (4.8) gives

8. () = O + A.

g () +

1g2() + b j+l i J+i where Y = r x.(1)

-g () = h.(f ()

f 2 i 1 x.(1) and 1f5(1) y.(l)

g3() =f6() -f3(l)f1()

_2jx.(l)

_f3(l)3(1)}

J i

The unknown dimensionsless suspended length A141 is determined from the following transcendental equation obtained from the boundary condition (4.19) together with (4.15)

cose. (1) = cosO - À(l

+ a - À1y1(l)}

3+1 u r

Thus, starting with an arbitrary integrable approximation to the equilibrium curve one step in the iteration procedure is to determine the functions g1, g2, and g3 from (4.23) and

from (4.24). The improved approximation to the equi-librium functions are hereafter given by (4.22), (4.14) and

(4.15). The iteration scheme is stopped when the convergence criterion (4.1) is fulfilled.

The method outlined is,of course,only valid when the stinger is so long that the calculated lift-off point is on the stinger. If this is not the case,a slightly different itera-tion scheme is called for.

An example of the numerical analysis of a pipe-laying procedure using a stinger with fixed curvature Is shown In Fig. 4.5. The stinger radius is assumed to be 300 m and, in this example, the stinger is assumed to be rigidly connected to the pipe-laying barge. The water depth and the pipe data are assumed to be the sanie as in the previous example.

(4.24) (4.22)

El 226 i0 N,..2

i soi io'

8 101o 10' 1W...

ii, 2.200 lO N

-I

Fig. 4.5 Results of numerical analysis of pipe-laying procedure with the use of a rigid stinger.

c. Pipe abandon/recovery operations.

Pipe abandon and recovery operations may be modelled as

Fig- 4.6 pipe abandon/recoverY operation

shown in Fig. 4.6. These operations can be performed in a number of different ways. As an example we will assume that the wire passes over the stinger rollers and that the horizontal anchor force transmitted to the pipe _{Tb and}

the V-coordinate of the pipe end Vi are known, whereas the wire tension _T is considered as a dependent

vari-able.

Taking into account tohe water pressure on the lid which is normally welded onto the pipe end during these operations,the boundary conditions for the upper end of the pipe take the form

Y(L) = Y.

i

M(L) = O

FI. = T - w (H-Y.)cosh(L)

i b b i

Let us again construct one step in the iteration algo-rithm using these boundary conditions. _{From F.gs. (4.5) and}

(4.7) and the boundary conditions (4.25) - (4.26) we get

y,(l) f5(l) i

j+13

)J

+ Vb

{hk.

f (1) x.(l) 3+1 X (1) J j

Substituting into Eg. (4.8) results in

e.

()

e

j+l = b

1g2() + À1g3()

where

g2()

and g3(r,) are given by (4.23). The dimen-sionless suspended length _{is determined from the} boundary conditions (4.25).

Fig. 4.7 shows the results of a numerical analysis of a pipe abandon or recovery operation. _{The same pipe data as} in the two previous numerical examples have beell assumed. The position of the upper end of the suspended length is

specified as 25 m above the ocean floor and the horizontal anchor force as 105N. The necessary wire tension is found

(4.25) (4.26) (4.27) 100 200 i,. 300 X (4 .28) (4.29)

to be 2.148105N.

dN

= wsinO - Ft

Fig. 4.7 ResultS of numerical analysis of abandon/recovery operation

Cables are characterized by a negligible bending stiff-ness (EI O) . The iteration method presented in section 4

for determination of equilibrium curves for pipes fails if we let the bending stiffness approach zero. Therefore, in this section we shall present a similar iteration method for the analysis of equilibrium curves and elastic restoring forces of cables as that used, for example, for mooring systems, taking into account drag forces on the cable and variations in cable weight per unit length.

and that equilibrium in the direction normal to the neutral

axis of the cable is obtained when

F _{+ (w_wb)cosO - wb(H_Y)}

ds n

Introducing the adjusted normal force

= N + wb(H'SY)

the governing equations (5.j), (5.2) take the simpler form

A dN (wt_wb) sinO * Ft

=j"

ds ç (wt_wb)cosø + F)

The boundary conditions for these equations express the fact that, at the upper end of the Suspended cable,

Y(L) H+A

5. STATIC EQUILIBRIUM OF CABLES.

and = H./cosO(L)

where H. _{is the applied tension, while at the ocean floor}

we have

X(0) O and Y(0) = O (5.6a-b)

If we consider a problem where part of the cable is lying on the ocean floor we have the additional boundary condition

e(o) -

(5.7)

In non-dimensional forni,the governing equations (5.3)

-(5.4) take the form

(5.1) _{pt() = {(wt_wb)sino -} and n(F;) _{= N/wtH}

AO

(5.2)

(5.3)

(5.4)

(5. Sa-b)

Governing equations. dn

Ap()

dO A (5 .8a-b)

It follows from Fig. 2.1 that force equilibrium in the _where tangential direction for an element of the cable is expressed

= f

(w-w)cos8 +

by

V

El 223S *0PI.

I I3 Io' *4 Witt I.V.n

05 I SII. IO' *V4..

o u. I0'N *4 . I0N

05..

.o-Numerical solution.

An iteration procedure similar to that outlined in section 4 will now be applied for the numerical solution of the non-linear boundary value problem given by (5.5) - (5.8). Assume that an initial approximation to equilibrium curve

is given so that the iteration algorithm can be started by a formal integration of Eq. (5.8e)

nU)

= 1j+l ptdl + C1

Jo

It follows from the boundary condition (5.5b) that the inte-gration constant is determined by

i h. J-

-À

C1 = cosO j+l'

ti

i JO

Formal integration of Eg. (5.8b) results in

=

À11[

d

+

n 1

o

Long Cables. If we are dealing with a mooring cable problem where the length of the cable is so large that part of the cable lies on the ocean floor,then the integration constant in (5.10) is given by the inclination of the ocean floor. Fig.5.l.

Fig. 5.1 Mooring-cable geometry, long cable.

(5.9)

(5. 10)

Using the boundary conditions (5.5a) and (5.6b) it is seen that in this case the unknown,unsupported length of the cable can be determined from the transcendental equation

Il

A sinO

d1

= i + a

j+l _j+l

Thus, the static equilibrium of the cable can be determined by successive iterations, where each iteration step involves the

calculation of n and

A41

using Eqs. (5.9) - (5.11). The sequence of iterations can be stopped when the convergence criterion (4.1) is fulfilled.

Shorter Cables. When dealing with a mooring system where _the cable is of shorter length so that the point of initial contact is the anchor point, the inclination of the _{cable at the anchor} point

0b is usually unknown. However, in this case we will as-sume that the length L of the cable is known. This leaves _us with sufficient information to construct one step in the iteration procedure. _{See Fig. 5.2.}

(5.11)

X 7,

Dropping the subscripts on the known dimensionless length A Eq. (5.10) can be written as

e.

_j+1

()

= n.

() + O

+i

b where n

() =

A'

j+1

J0 n

From (5.13) the unknown angle _{8b can be determined as}

8 = Arcsin( l+a

)

Arctan

A/C+C

()

h

Thus, in order to determine the static equilibrium position of a short mooring cable,each iteration

step must consist of a calculation of n (5.9) and 8.4k (5.10), where the angle

is determined from (5.14).

(5.12)

(5.14)

6. CONCLUSION

The method of successive integrations presented for the determination of equilibrium forms and stresses for submarine pipelines during laying possesses several advantages over other available methods. The principal advantage is the ex-tremely modest requirements to computer storage and computer

time. Since, in principle, the method only involves integration of known functions,the me'thod is well suited for programming on shipboard computers for control of the actual pipe-laying procedure. The numerical calculations for the three pipe-laying examples presented in this paper were done as one job on an IHM 370/165 computer. The total computer time is 0.88 sec. and the storage requirement,including object code and array area for this general program,is 32 K8. Another

ad-vantage of the method is its flexibility. For example the effect of variations of pipeline bending stiffness due to variations in the coating, current variations with depth, and auxiliary support buoys can easily be accounted for.

The primary limitation of the present method for the analysis of equilibrium forms of pipes is failure to converge

for some extreme input values such as very deep water or small bending stiffness. However, since the present method and the stiffened catenary method has a common convergence region it is possible to shift to stiffened catenary solutions if the pipe data are such that the method of successive integrations fails to converge. If it is necessary to take ocean current and variations in pipe data into account the stiffened cate-nary solution can be based on the numerical method presented for the analysis of cables.

Introducing

in

where C1 =

(5.12) into the boundary condition

CicosOb + C2slflOb =

Il

sinn. dF1 and C2 I 3+1 (5.4a)

Il

cosri i

j+1

û results (5. 13)

REF ERENC ES

il) PLUNKETT, R.: Static bending stresses in catenaries and drill strings, Trans. ASME, Journal of Engineering for Industry, Vol. 89, No. 1, pp 31-36, 1967

(2) DIXON, D.A. and RUTLEDGE, D.R.: Stiffened catenary cal-culations In pipeline laying problems, Trans. ASME, Journal of Engineering for Industry, Vol. 90, No. 1,

pp 153-160, 1968

[3] POWERS, J.T. and FINN. L.D.: Stress analysis of off-shore pipelines during installation, Offoff-shore Technology Conference, Houston, Texas, Paper No. OTC 1071, May, 1969

[4) PALMER, A.C., HUTCHINSON, G. and ELL.S, J.W.: Configuration of submarine pipelines during laying operations, ASME, Paper No. 73-WA/OCT-4, 1973

[5] DAREING, D.W., NEATHERY, R.F.: Marine pipeline analysis based on Newtons's method with an arctic application, Trans. ASME, Journal of Engineering for Industry, Vol. 92, No. 4, pp 827-833, 1970

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Institute

0f Hydrodynamics and

Hydisulic Engineering, Department of Solid

Mechanics and Structural Research Laboratory at the Technical University of

Denmark.

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