D . A. D I X O N Senior Research Eng ineer .

D . R. R U T L E D G E Junior Research Engineer .

Pan Amer i can Petroleum Corp . Research Cen te r

Tulsa , O k l a -

Stiffened Catenary Calculations in Pipeline Laying Problem According to a granted U. S. Patent, it is contemplated that deep water pipelines be laid on the sea floor from an inclined derrick on a lay barge without the aid of a stinger. Means are presented in this paper for calculation of required tension and inclination of pipeline at the lay barge. Charts giving necessary tension and inclination, taking pipe-line stiffness into consideration, are presented.

A s

Introduction Is THE search for natural resources beneath the sea

floor will shift to deep waters, conventional techniques which have been developed for marine production will become obsolete. For example, laying pipeline on the sea floor by means of a stinger will be uneconomical due to the size of stinger required for deep water operation.

According to a granted U. S. Patent,1 a deep water pipeline could be made up in an angular position on the lay barge and maintained under a predetermined high tensile force, while lower-ing it to the bottom, as shown in Fig. 1(a). The axial force is chosen large enough to avoid overstressing and kinking of the pipe-line near its point of tangency with the sea floor. In addition, the upper end of the pipeline must be securely clamped to the lay barge at the work level during fabrication of each successive section of pipe. If the angle at the upper end of the pipeline were arbitrary, there would be a large bending moment imposed at that point. To avoid damaging the pipe laying apparatus and over-stressing the pipeline, this angle should be predetermined to give zero bending moment.

Lacking sufficiently advanced mathematical theory, it had been at first necessary to assume the unsupported pipeline would take the form of a natural catenary.1 Although this assumption neglected the bending stiffness of the pipe, the authors believed that making the upper end conditions compatible with those required by natural catenary calculations would lead to safe operations when placing pipe in deep water. However, neither calculations nor measurements were available to justify this belief.

In a recent publication, R. Plunkett presented a formal asymptotic expansion which is valid to describe the deflection of an unsupported pipeline considering stiffness if, over most of the length, the tension has more influence on the deflection than does the bending stiffness.2 Therefore, it is now possible to obtain a better estimate of the shape of an unsupported pipeline and, con-sequently, to more accurately determine the end conditions which are required at the lay barge to avoid damaging the pipeline.

In the following paper, equations for determining required end conditions are developed, first, assuming the pipeline to take the form of a natural catenary and, second, assuming it takes the shape of a stiffened catenary. To facilitate application of the re-sults, dimensionless curves prepared from these equations are included. The curves demonstrate that the minimum tension at

1 Postlewaite, W. R., and Ludwig, M., "Method for Laying Sub-marine Pipe Lines," Patent No. 3,266,256, United States Patent Office, August 16, 1966.

2 Plunkett, R., "Static Bending Stresses in Catenaries and Drill Strings," JOURNAL OF ENGINEERING FOR INDUSTRY, TRANS. ASME, Series B, Vol. 89, No. 1, Feb. 1967, pp. 31-36.

Contributed by the Petroleum Division and presented at the Petroleum Mechanical Engineering Conference, Philadelphia, Pa., Sept. 1 7 - 2 0 , 1967, of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Paper received at ASME Headquarters, June 5, 1967. P a p e r N o . 6 7 — P e t - 6 .

the la}' barge is calculated with sufficient accuracy for any water depth by neglecting the bending stiffness, i.e., using the natural catenary method. On the other hand, for calculation of the angle at the lay barge, bending stiffness may be neglected in deep waters only. It is further shown by examples that the stiffened catenary calculations for angle are necessary in water depths less than 400 ft for 8-in. pipe and S00 ft for 30-in. pipe.

Discussion Though pipe stiffness may have significant importance in the

analysis of short pipeline sections, long sections of unsupported pipeline are analogous to a string or cable; i.e., shear is negligible and curvature exists without any appreciable moment. The pipeline takes a shape which approximates a natural catenary over most of its length. Near either end the shape diverts from that of a natural catenary due to the bending stiffness and boundary conditions which are not compatible with those of the natural catenary.

Utilization of a lay barge equipped with an adjustable angle ramp and a means to provide large axial support to the pipeline allows control of the critical pipeline curvature. If the pipeline is assumed to take the shape of a natural catenary over its entire length, the maximum curvature occurs at its point of contact with the sea floor assuming the sea floor is horizontal. The shape of the pipeline and the required minimum tension at lay barge are selected to minimize the curvature at the sea floor and, therefore, to control the maximum stress due to bending.

The shape of the natural catenary is described by the classic equation

V =

H_ IF M?H ( i )

which is presented in a slightly different form than by Korkut and Hebert,3 where H is the horizontal component of axial force (lb), or

«6 (2)

IF is the buoyant unit weight of the pipeline ( lb/ ft) , and x and y are the Cartesian coordinates shown in Fig. 1(&).

The equations for minimum line tension and corresponding ramp angle which were developed by A. Lubinski for the Mohole Project are

6 = cot - 1

- • K i ] (3)

(4)

3 Korkut, M. D., and Hebert, E. J., "Presenting in Condensed Usable Form Equations to Find Anchor Chain Curve," Ocean In-dustry, Aug. 1966, p. 14A.

Journal of Engineer ing for Industry F E B R U A R Y 1.9 6 8 / 1 5 3

Copyright © 1968 by ASME

Table 1

Pipe Dimensions and

Allowable Bending Stress

Wra

Thickness (in)

pper Weight in Air

(Xb/ft3)

Pipe Line Bouyant Weight

( l b / f t )

Stiffness Factor

Water Depth at which

- 5 - = i - 0

( f t )

8.625" P x .59V 0. = 26.25 ksi 0

1 140 44.90 63.72

(A) (W)

.00814

.00574 (A) (W) 410

8.625" 0 x .59k" ab = 31-5 ksi 1 140 44.90

63.72 (A) (W)

.01406

.00991 (A) (W) 342

10.75" <j> x .844" ob = 31.5 ksi 1 i4o 72.85

100.80 (A) (W)

.01195

.00863 (A) (W) 427

12.75" 0 x .844" a.. = 31-5 ksi D

1 140 •78.22 119.9

(A) (W)

.01156

.00754 (A) (W) 506

18.0" 0 x 1.156" ob = 31-5 ksi 1 130 130

213 (A) (W)

.00962

.00584 (A) (W) 714

22.0" ff x 1.375" ob = 31.5 ksi 1 130 177

303 (A) (W)

.00841

.00491 (A) (w) 873

30.0" 0 x .625" a. = 46.8 ksi

D 3 210 183

A.71 (A) (W)

.01377

.00535 (A) (W) 800

(A) is for air f i l l ed pipe

(W) is for vater f i l l ed pipe

Fig. 1(a) Method proposed by Postlewaite and Ludwig1 for laying deep water pipelines

V T

Fig. 1(b) In deep water the pipeline configuration between water surface and bottom approximates a catenary

where D is the water depth ( ft ) ; eb is the maximum allowable beading strain (in/in.); c is the external radius of pipeline cross section ( ft ) ; T is the line tension (lb); and 9 is the ramp angle measured with respect to the vertical. These equations have been presented in a different form by Postlewaite and Ludwig1

and are rederived in the Appendix for completeness. One aspect of the natural catenary which makes this a very

attractive method for laying pipeline is that the horizontal com-ponent of force at the lay barge H is independent of water depth D, as demonstrated by equation (2). Therefore, the capacity re-quired of the mooring system or propulsion system which pro-duces this component of force does not increase with water depth.

The maximum allowable bending strain e6 should be chosen such that the corresponding bending stress <rb = ebE is a fraction of the yielding stress. While axial stress also exists at the point of maximum curvature, this stress is negligible in most cases, but should be calculated in case the total stress does exceed its per-missible range.

As water depths increase, the tension T increases, H remaining constant, until the axial stress at the lay barge end of the pipe-line becomes equal to the allowable bending stress, or

f - f [ > + ; ] = ^ The maximum water depth for which the most exacting stress is at or near the sea floor and not at the lay barge is, therefore,

- ^ ( f ) - ( { ) Use of (6) on numerical examples has shown that D is of such magnitude that for all practical considerations, the tensile stress at the barge may be disregarded.

It should be noted that the natural catenary method of analysis, as well as the stiffened catenary method which follows, neglects the dynamic stresses in the pipeline due to motions of the lay barge and due to current and wave forces acting on the pipe-line. Thus, until such an investigation is made, a margin of safety for the total stress must allow for these possible dynamic variations.

In assuming the pipeline takes the form of a natural catenary, the end effects due to the pipeline bending stiffness have been neglected. If, on the other hand, one considers a stiffened

1 5 4 / F E B R U A R Y 1 9 6 8 Transactions of the A S M E

catenary, then the pipeline must be considered moment-free at its point of tangency with the bottom. Therefore, the point of maximum bending is at some distance above the sea bottom and, it can be shown, the maximum bending stress is less than that predicted by the natural catenary calculations. Furthermore, since the curvature of the natural catenary is never zero, the moment-free end condition at the lay barge cannot be satisfied by the natural catenary.

While the exact solution of an unsupported line is not known, the asymptotic expansion by Plunkett2 provides a better ap-proximation of the shape of an unsupported pipeline and a means for measuring the effect of bending stiffness. The shape of the pipeline is best described by its angle with respect to the vertical,

6{z) = tan"1 { ^ j - ^ e x p [ - c r ( z - zo)gi(z)]

+ ah

+ exp t _ a ( 1 - z ) q M ] (7)

where a is a nondimensional quantity which measures the in-fluence of bending stiffness; z and h are the nondimensionalized vertical and horizontal components of axial force, respectively; and qi and g-2 are defined in the Appendix where (7) is derived.

The equations for required lay-barge end conditions are con-siderably more complicated than those for the natural catenary. For minimum tension and ramp angle the equations are

T = IF£(fc2 + l ) ' / s

= tan _ 1 ( '0 + ah

(fc2 + l )"7 .

(8a)

(86)

6 bL h C ~ (fc2 + z2)

(fc2 + 22)'A

exp [ - ( j ( 0 - z0)(fc2 + eo)'7"]

(9a)

and

0 = 2fcz

(fc2 + s2)2

(fc2 + z 2 ) ' A

fc3/2 2 fc3/'(fc2 + z2)3/,

X exp [ —a(s - z„)(fc2 + z0 2) 'A] (96)

The water depth corresponding to values of a, fc, and L is

= L j(fc2 + 1 ?/* - (fc2 + z„2)

+ [fc1/2(fc2 + Zo2)V< _ (fc2 + l ) 2 .

D

M U W \<

which will be called the "stiffness factor." The stiffness

factor measures the relative influence of bending stiffness on de-flection in comparison to the influence of axial tension and has the same order of magnitude as a2/fc3, whose significance is dis-cussed in the Appendix, or

0 < a* h*

< EI IF

a1

fc3" < 2.

1.010

where L is the length of unsupported pipe, and h is chosen such that fc and z simultaneously satisfy

Values of the stiffness factor for several standard pipeline sizes, each with adequate corrosion protection and concrete wrapper, are tabulated in Table 1. For the sizes of pipe listed, the stiffness factor is less than 0.015 for air or gas filled pipeline and, when the pipeline is flooded, the stiffness factor is less than 0.010.

In Fig. 2 the influence of bending stiffness on T/WD is shown

to be very small and, in fact, the curve for — ( ^ j = 0.1

could not be distinguished from that of the natural catenary

— ( - J = 0.00 when the curves were plotted. Thus, for the W \c/ range of pipeline listed in Table 1, calculation of T by the natural catenary method gives sufficient accuracy. Furthermore, while not demonstrated in this text, it has been found that H/WD is influenced even less by the bending stiffness than is T/WD. Consequently, it is safe to make the additional conclusion that H is calculated with more than sufficient accuracy by the natural catenary method (2).

However, the family of curves for the angle at the lay barge 6, Fig. 3, show a greater dependence on the stiffness factor. For

example, if the water is shallow enough to give = 0.1 and the

pipeline properties are such that the stiffness factor is 0.010, then the angle at lay barge should be 69.5 deg instead of 65.5 deg

as would be indicated by the natural catenary — = 0.0,

or a difference of approximately 6 percent. This difference be-tween the two methods of calculation vanishes with increasing depth until, for the same pipeline, the influence of bending stiff-

10.00

5 . 0 0

(10)

1.00

0 . 5 0

In the case of the stiffened catenary, fc and L cannot be elimi-nated from (8a) and (86) and, consequently, T and 6 cannot be expressed as direct functions of the water depth D and the pipe properties EI, W, eb and c as was done for the natural catenary. Therefore, curves instead of formulas must be employed to dis-play the significance of the stiffened catenary calculations in com-parison to the natural catenary calculations.

Nondimensional Curves In the absence of closed-form expressions for T and d in terms

of EI, IF, eb, c, and D, nondimensional curves are the most con-venient means of representing the results for the stiffened catenary. These curves have been prepared from equations (8), (9), and (10) by use of an I B M 704 computer.

Fig. 2 presents curves for the nondimensionalized minimum tension at lay barge T/WD as a function of ebD/c and a parameter

Journal of Engineer ing for Industry

6b D c

0.10

0 . 0 5

0,01

-v \

v \ y> W ) \ W ) 0

- E l <•

\

" W

F i f f b V u I r - \l i f f b V u I r - u . u : :

\ \

\

N A T U R A L C A T E N A R Y — \ \

\ \ V

1.0 50.0 100.0 5.0 10.0 T

WD Fig. 2 Nondimensional curves for minimum tension at lay barge

F E B R U A R Y 1.9 6 8 / 1 5 5

10 2 0 3 0 4 0 5 0 6 0 7 0

ANGLE WITH RESPECT TO VERTICAL AT LAY BARGE (DEGREES)

Fig. 3 Nondimensional curves for required angle at lay barge

ness is negligible at depths for which — = 1.0. Table 1 gives

Liitj uepuus itu » uiuii luu pi j JUii i H: buiuneaa la mjgii

lation of angle at lay barge ^i.e., = 1.0^

and

eb = 0.000875 (in/in.)

eb/c = 0.002435 ( f t ) "

TENSION IN P IPE AT LAY BARGE (K IPS)

20 40 60 80

the depths at which the pipeline stiffness is negligible in the calcu-ejD

c pipe sizes.

for a selection of

100

1000

Examples Two examples are presented with corresponding curves to

demonstrate the range in which the stiffened catenary calculations take precedence for specific size of pipeline.

8-in. Schedule 100. For the first example, the pipeline is made of 8.625-in-dia by 0.594-in. wall-thickness pipe weighing 50.93 lb / f t with a 3-lb/ft corrosion coat and a 1-in. concrete wrapper of density 140 lb/ft3 in air. The buoyant weight of the pipeline is 44.90 lb / f t if air or gas filled and 63.72 lb / f t if water filled. As-suming the maximum allowable bending stress to be 75 percent of the yielding stress, 35 ksi, gives

Fig. 4(a)

0 10 20 30 40 50 60 70 80 90 100 ANGLE WITH RESPECT TO VERT ICAL

AT LAY BARGE (DEGREES ) Required end conditions for air-filled 8-in. schedule 100 pipeline

TENSION IN P IPE AT LAY BARGE (KIPS)

20 40 60 80

The stiffness factor for a moment of inertia of 121.49 in.4 is 0.00814 for air or gas-filled and 0.00573 for water-filled pipeline.

Using the digital computer, Fig. 4(a) and Fig. 4(6) have been prepared for air-filled and water-filled pipeline, respectively. The solid curves give minimum tension and corresponding angle at lay barge obtained from stiffened catenary equations. The broken curve gives angle at lay barge obtained from the natural catenary equation (4) while the tension at lay barge from the natural catenary equation (3) coincides with the stiffened catenary calculations.

It is apparent from both figures that the stiffened catenary equations are necessary for the calculation of 6 in water shallower than 400 ft while the natural catenary equation (4) is accurate in more than 400 ft of water.

30-in. X-52. Fig. 5(a) and Fig. 5(6) give similar curves for 30-in-dia by 0.625-in. wall-thickness pipe with negligible corro-sion coat but a 3-in. concrete wrapper weighing 433 lb / f t in air. In this case the buoyant weight is 183 lb / f t if the pipeline is air filled or 471 lb / f t if water filled; maximum bending stress is 90 percent of the yielding stress, 52 ksi, which gives

— = 0.00124S(ft)"' c

and the stiffness factor is 0.01377 for air-filled and 0.00535 for water-filled pipeline.

The dependence of angle at lay barge upon the pipeline stiffness is more significant in this example than in the preceding one.

100

^ 8 SCHEDULE 100 WATER F ILLED

BOUYANT WEIGHT = 63.72 l b 4 t

ALLOWABLE S T R E S S

= 26.25 KSI

TENSION STIFFENED CATENARY

METHOD

- NATURAL CATENARY

1000 0 10 20 30 40 50 60 70 80 90 100

ANGLE WITH RESPECT TO VERTICAL AT LAY BARGE (DEGREES )

Fig. 4(b) Required end conditions for water-filled 8-in. schedule 100 pipeline

Furthermore, the bending stiffness does not become negligible until the water is deeper than 800 ft.

Conclusions Equations and curves are presented giving required end con-

ditions at the lay barge for laying pipeline in deep waters without the aid of a stinger. The problem is analyzed by two means. First, the pipeline is assumed to take the form of a natural catenary, neglecting bending stiffness, and the end conditions

1 5 6 / F E B R U A R Y 1 9 6 8 Transact ions of the A S M E

TENSION IN PIPE AT LAY BARGE (KIPS)

200 400 600 800 1000

t V ^

/30-INCH X-52 K AIR FILLED

BOUYANT WEIGHT = 183 l b / f t

ALLOWABLE STRESS = 46.8 KSI

STIFFENED CATENARY

METHOD

NATURAL CATENARY

METHOD

1 0 0 0

Fig. 5(a)

0 10 20 30 40 50 60 70 80 90 100 ANGLE WITH RESPECT TO VERTICAL

AT LAY BARGE (DEGREES) Required end conditions for air-filled 30-in. X-52 pipeline

TENSION IN PIPE AT LAY BARGE (KIPS)

200 400 600 800 1000

1000 0 10 20 30 40 50 60 70 80 90 100

ANGLE WITH RESPECT TO VERTICAL AT LAY BARGE (DEGREES)

Fig. 5(b) Required end conditions for water-filled 30-in. X-52 pipeline

are chosen to make the pipeline compatible with the catenary. The second method is to describe the form of the pipeline by an asymptotic expansion which takes into consideration the bending stiffness of the pipe. The former technique is applicable if the influence of the stiffness on deflection is negligible in comparison to the influence of the axial force.

Comparison of curves prepared from both methods show that the minimum tension at lay barge is not significantly influenced by bending stiffness. Thus, the values of tension resulting from the natural catenary calculations are accurate for most sizes of pipe. However, the corresponding angle at the lay barge has greater dependence on stiffness. Therefore, the stiffened catenary calculations for angle shoidd be used in preference to the natural catenary calculations when laying pipeline at water depths which are normally encountered in present practice.

In both of these approaches the sea floor is assumed to be horizontal. In addition, dynamic stresses due to motions of the lay barge and due to current and wave forces on the pipeline are neglected. Thus, the margin of safety in choosing the allowable range of bending stress should allow for variation from these assumptions.

The mathematical limits which must be imposed on the stiff-ened catenary equations prevent application of these results for pipeline placement in very shallow water. From the example for 30-in. X-52 pipe neither tension nor angle can be evaluated in less than 75 ft of water. In very shallow waters the problem could probably be solved using the theory of beams under tension. Such a study is, however, outside of the scope of this paper.

A P P E N D I X Natural Catenary. The equilibrium equations for the element of

rod shown in Fig. 6, which is subjected to horizontal and vertical forces H and V, bending moment M, and unit bouyant weight W, are

EI- b H cos d - V sin 6 = 0 f7.<J2

dH ds

= 0 or H = const

H+dH

Fig. 6 Coordinates and sign conventions

dV — = W or V = Ws ds

where s is measured from the point at which V — 0,

dy ds

and from beam theory

dx

M = -EI dd_ ds '

(13)

(14)

(15)

(11 )

(12)

Using the same nondimensional notation employed by Plunkett2

z = s/L = V/WL, 20 < z < 1

h = H/WL

a2 = EI/WL3

and introducing a nondimensional moment equal to the negative nondimensional curvature

m = ML /EI = -dd/dz,

equation (11) becomes

Journal of Engineer ing for Industry F E B R U A R Y 1.9 6 8 / 1 5 7

d2d + h cos 6 — z sin 6 = 0 (16)

where L is the length of pipe whose total bouyant weight equals the maximum value of V, or

= WL.

If the bending stiffness is neglected, or a 2 = 0, as in the case of a natural catenary, the solution of (16) is simply

0(z) = tan - 1 = cot - 1 (j^j. (17)

Substitution of (17) into (14a) and (146) and assuming the boundary conditions shown in Fig. 1(6), or

lead to

2/|«-o = x\*-o = 0

(i) x/L = h s inh - 1 I —

V/L = h

where 0 <. z < 1, or upon elimination of z

V = Lh cosh

V ^ i T - ] limination c

( § ) - 0 - ( f )

(18o)

(186)

cosh ( ^ H (19)

D/L = (y/L)|,=1 = (ft2 + l ) 1 ' 2 - h (20)

0(1) = cot.-1 a >

G> i

(r) c

where c is the outer radius of the pipe cross section. For simplest application of these results, T and 9 at the lay

barge should be direct functions of the pipe properties IF, c, and e6, and the water depth D. Solution of (20) and (23) for h and L

h =

L =

and substitution into.(21) and (22) lead to

9(1) = cot"1

and

(24a)

(246)

Stiffened Catenary. In the case of a stiffened catenary, a 2 0, the asymptotic expansion which Plunkett chose to satisfy the pipeline problem is

9(z) = ftz, of) + pi(z, a) exp [~<j(z - z0)?i(z)]

+ \pi(z, a) exp [ —tr(l - zMz)} (25)

which is the classic solution for a natural catenary.3

Since this solution depends on IF being constant, it is necessary to assume the upper end of the pipeline is at or below the water surface. If the barge supports the pipeline just at the water sur-face, the water depth follows directly from (186), or

where a = 1 /a . The first term of the solution is a series expan-sion which satisfies the differential equation (16) but not the boundary conditions, or

ftz, a) = £ « 2 r f r (z ) r = 0

where the first three coefficients are

ft = tan" (!) and, from (17), the angle of the pipe at the upper end, with re-spect to the vertical, is

(21)

The axial tension at any point in the pipeline is

T(z) = (V2 + H2)'^

= TFL(z2 + ft2)1/*

and at the lay barge, z = 1, this becomes

T(l) = WL(h2 + 1)1/2. (22)

The nondimensional moment at any point is

dd h m { z ) = ~lh = ( ^ T w

and the maximum moment, which is at the point of tangency with the sea floor, z = 0, is

= m( 0) = 7". h

Consequently, the maximum strain due to bending becomes

_ Mc 1 x I E max

ft = 2zh(h2 + 22)-i/!

ft = 10z/i(4z2 - 3h2)(h2 + z2)-5 .

The first coefficient ft is recognized as the solution for the natural catenary (17). This series is absolutely convergent if 0 < a 2 < 1 and a 2 < h3.

The remaining terms in (25) are exponentially decaying func-tions where tpi and i/'i will be chosen so that 6(z) satisfies the boundary conditions at z = z0 and z = 1. By requiring the co-efficients of ipi and ipi to each independently satisfy (16), equa-tions have resulted for qi(z) and qi(z), or

2i(z) = (z — zo)

?z(z)

P ( f 2 + J zo

= ̂ f1 (r + vy/w d - z ) J*

(z0 = Z = 1)

(26a)

(266)

The quantities (pi, \f/lt qi, and qi are each well behaved functions of z and, in addition, (fli and ij/i are the first members of rapidly decaying series expansions in a.

If the range of applicability is restricted to pipe sizes and water depths such that a2 is much less than 1.0, then 9(z) is represented with sufficient accuracy by neglecting all but the first member of each of the expansions in a, or

9(z) = tan - 1 ( — ) + (pi(z, a) exp [-a(z - zo)ei(z)]

+ \f/i(z, a) exp [ —cr(l - z)5a(z)].

At z = zo the pipeline is tangent to the horizontal sea floor

(23) 9(z0) = tan - a ) + </>i(zo, a ) = (27a)

1 5 8 / F E B R U A R Y 1 9 6 8 Transactions of the A S M E

and moment free The maximum strain due to bending follows directly from (35), or

f <*>> = + 1 <*.«) = ( | ) - U - ( | ) { - cr^zo, a)(/i2 + *•)'/« = o, (276) (fe2 + z 2 ) v ' . . . , 1 . „ „ ,

- ^ exp [-a(z - zo)ffi(z)] j . (36)

mu

<r(/i2 + z2)1 / '

where the coefficient of i^i(zo, a ) has been assumed negligible at z = zo- Therefore, Since this is the point of maximum curvature, z must satisfy

* , < « , « ) = o o t - i ( A ) ( 2 8 ) = gftg _ r \«o/ dz \ dz /2=g (/i2 + z2)2 L

- 2 , V , ( ; ; + - P [ - * « - zo)9l(,)l = o (37)

' ' + t _ l ( | 0 (A2 + Zo2)^' \ zo / where for (z — z0) « 1, a Taylor series expansion of (z — zo)gi(z)

about Zo leads to A \ A \ Expansion of c o t - 1 I — I into a Taylor series in 1 — 1 and making (g _ Zo)qi(,z) = (z — z0)(/i2 + zo2)'^1.

the assumption that < 1 lead to

" h 1 / 2

and

= ~t t t „ (29) / . l

Assuming the upper end of the pipeline to be at the water sur-face, the water depth D corresponding to values of a, h, and L is found by substituting (33) into (14), or

D/L = cos ddz 'zo

<pA.z, a) (30)

The pipeline is assumed to be moment free at the lay barge end, = j { — — - — - j t cos (U — V)

f «/ zo

= J* cos j ^ t a u - 1 (j^J - [/ + y j dz

(/i2 + z 2 ) I / !

f W) = - ( F T d + f a «) + «x» + D'A = 0 + sin (c/ - y ) ) (31) where

where the coefficient of <£,(1, a ) is negligible at z = 1. Therefore, a H*. a) must be U = e x P ~

*< '• a ) = ( F + 1 ) V . <32> F - exp [ - . ( 1 - z)32(z)].

and the equation for 6{z) becomes

0(z) = tan"1 ^ j j - exp [ —cr[(z - 20)31(2)]

Since U decays rapidly near z = z0 and V decays rapidly at z = 1, D is approximated by

ah ~ I (h2 4- 22V/ j „ ... 1 zdz d / l r

1 r 1

+ / . , • 2i'A ^ c o s - 1] + A sin V)dz (38) (/i2 + zo2) ' 2

+ 4! 1}»/, j ] ' i Icos F - 1] - si» U)dz-The first integral in (38) becomes

Thus, the angle of the pipeline at the lay barge measured with respect to vertical is

0(1) - t a n - M + ^ ^ v , , (34)

while the minimum tension is given by the same expression as for the natural catenary (22), or

1X1) = WL(h2 + l )1 / ' . TT , . T , Since ?7 = 0 at 2 = 1, the second integral is approximated as However, the values of L and li are different from the stiffened follows' catenary equations.

In this case the location of maximum moment is not at the 1 point of tangency with the sea floor, but it occurs at 2 = z which 2 z02)l/,i

is slightly greater than z0. If z Ls close enough to z0 to neglect the coefficient of ^ ( z , a) , the maximum nondimensional moment is —a

C1 zdz J2o + = W + - W + ^ W

f f2o[cos U - 1] + h sin U}dz = J zo

(/i2 + Z„2)V<

dz (/i2 + z2) r° (z0[Cos U - 1] + h sin U) d u

J U=c/h'/' I + Z2) 'A r . . . , 3 - , Jv-a/kV* exp [ - c r ( z - zo)ji(z)]. l-d0 ' '

U

Journal of Engineering for Industry F e b r u a r y i ? 6 s / 1 5 9

(h2 +

r t/2

(40) (Conl.)

(h2 + Z„2)V '

+ A I

a T a 2

d(7 =

fi/L = (h2 + 1)1/2 - (h2 + z0 2 ) 'A

+ a 2 /i2

_A1/!(/i2 + z02)3/4 (/i2 + l )2 . • (42)

For ease in applying the stiffened catenary equations it would be best to express 2"(1) and 0(1) directly in terms of IF, c, eb, EI, and D as was done for the natural catenary. However, this re-quires solving simultaneously the complex set of equations (36), (37), and (42) for z0, h, and L, which cannot be done in their existing form.

/iI/s(7i2 + z0 2)V i

where higher order terms in a are neglected for a small. Simi-larly, since V = 0 at z = z0, the third integral is approximated by

(k* + ~TJyA J ' { I c o s V - l] - h sin V}dz = a2h2

(h2 + l)2' (41)

Substitution of (39), (40), and (41) into (38) leads to the expres-sion for water depth

A c k n o w l e d g m e n t The authors wish to thank Pan American Petroleum Corpora-

tion for the authorization of releasing the results of this study. They are grateful to the following Pan American personnel for

valuable help and constructive criticism: Mr. Arthur Lubinski, Dr. F. H. Hsu, Mr. W. D. Greenfield, and Dr. K. A. Blenkarn.

They also wish to express their appreciation to Mr. L. E. Minor, Vice President, Brown and Root, Houston, Texas, and his per-sonnel for bringing to our attention the potentialities of laying pipelines by the method investigated in this paper.

1 6 0 / F E B R U A R Y 1 9 6 8 Transact ions of the A S M E