Asymmetric Effects of Prop Imperfections on the Upheaval Bucling of Pipelines

Thin- Walled Structures 13 (1992) 355-373

Asymmetric Effects of Prop Imperfections on the Upheaval Buckling of Pipelines

J. P. Bal le t

l~cole Nationale des Ponts et Chaussdes. Paris, France

&

R. E. H o b b s *

Professor of Structural Engineering, Imperial College of Science, Technology and Medicine, London UK, SW7 2BU

(Received 25 July 1990; accepted 28 March 1991)

ABSTRACT

Submarine pipelines often carry products which are much hotter than the surrounding seawater. The potential thermal expansion is restrained by friction between the pipeline and the seabed, causing the development of large compressive axial forces in the line, which can lead to buckling of the pipedne.

This paper takes a fresh look at the vertical buckling of a pipeline encountering a point irregularity on an otherwise perfectly fiat seabed, the so-called 'prop case'. Some approximations and assumptions in earlier work in this area are reexamined and their effects are calculated. Most importantly, the assumption that buckling is symmetric about the prop is tested. Asymmetric results are found, at a lower critical temperature than the symmetric mode, a

fact which may have significant implications for design.

NOTATION

A Cross-sectional area a Distance between prop and nearest separation point

*To whom correspondence should be addressed.

355 Thin-Wailed Structures 0263-8231/92/$05.00© 1992 Elsevier Science Publishers Ltd, England. Printed in Great Britain

356 J. P Ballet, R. E. Hobbs

E H I

J L L~ M P /'o S, SI,

T V w

X

Y

Young's modulus Prop height Second moment of area see equation 17 Length of buckle Slipping length Bending moment Axial force in buckle Prebuckling axial force s2 Slips Temperature increment Prop reaction Submerged weight of pipeline per unit length Coordinate along pipe axis Deflection perpendicular to pipe axis

a Coefficient of linear thermal expansion 6 Maximum deflection A see equation 5 ~'3, ~'4, ~vIR, ~2 Beam-column functions, see eqn (16) g Axial friction coefficient e Average axial strain

INTRODUCTION

A submarine pipeline laid at ambient temperature and subsequently pressurised and used to carry hot oil away from a wellhead will experience significant axial compressive forces as a consequence of the restraint of axial expansion by seabed friction. The compressive forces are frequently large enough to induce lateral buckling ('snaking') of untrenched lines, or vertical buckling ('upheaval') of trenched lines. The critical temperature for upheaval is raised by back-filling the trench or by covering an untrenched line with rock or gravel. This whole question of thermally induced buckling is of considerable economic importance, and a substantial literature has built up in the past l0 years. The papers presented at a special session of the 1990 Offshore Technology Confer- ence 1 are perhaps the best introduction to the current state of the art and to prior work in the field.

It is clear that the buckling behaviour of the pipeline will be influenced by the initial imperfections in the line as laid. Concentrating on upheaval

Asymmetric effects of prop imperfections 357

here, trenching and the natural contours of the seabed can create two main classes of imperfections. In the first, contact is maintained between the line and a smooth mound in the foundation. Reference I will provide access to work in this area. In the second, the pipe is in contact with an isolated high spot or 'prop' (such as a substantial rock in the trench bottom) and forms free spans either side of the prop before regaining contact with the seabed some distance away. Upheaval from the propped condition has been studied previously 2.3 in some detail. This paper examines some approximations in these earlier studies, and presents what is thought to be an improved upheaval history for the standard symmetric mode of prop upheaval. The paper also examines the possibility that prop upheaval may become asymmetric, a possibility suggested by the well known asymmetry present in the buckling of shallow arches. 4

ANALYSIS m 'PERFECT' CASE

The 'perfect" case is the case of a semi-infinite line under uniform frictional restraint lying on a perfectly fiat and rigid seabed (Fig. 1). This case has been thoroughly analysed by Martinet 5 in connection with the cognate problem in continuous railway tracks and (perhaps more accessibly) by Hobbs 6 for pipelines. Assuming the pipe to be a long slender beam and the deflections, slopes and curvatures to be small, the bending moment becomes

- ~ - x 2 + P y (1)

where P is the force within the buckle.

Since

d 2 y _ M dx 2 E1 (2)

W

I.Zw ~, ~ f I.Zw

I- - r -I- -1

Fig. 1. U p h e a v a l - - perfect case.

358 J. P. Ballet, R. E. Hobbs

and using the boundary conditions: y ' ( L / 2 ) = y ( L / 2 ) = 0 (rigid seabed at x = L/2), the following results for buckle force and slope are found after integration:

E1 P = 8 0 . 7 6 ~ (3)

w[ sin (Zx) ] y ' ( x ) = fi [Acos (AL/2) - x (4)

where

p ]m A = LE-IJ (5)

Part of the line has lifted, and because the arc length around the buckle, L~, exceeds the initial length L, P is smaller than P0, the compressive force before buckling, which was equal to a E A T . This force reduction causes the pipe to extend slightly. For small buckle lengths and high enough friction coefficient, g, .the length increment matches the arc length increase around the buckle, and no slipping is observed in regions adjacent to the buckle. The extension is given by

Z l - - L = L(Po - P ) / E A (6)

while the arc length change is

L/2 L I - L = [ y'2(X) dx (7)

aO

Carrying out the integration and combining these results, the following relation is found:

P0 = P + 1.597 × 10 -5 w2AEL6/ (EI ) 2 (8) When the temperature T, force P0 and buckle length increase, a point is

reached at which the difference between P0 and P exceeds the friction force available (= 1awL/2) at the two lift-off points, x = + L/2 . Two slipping lengths, Ls, then form adjacent to the buckle and slide in towards the buckle to maintain equilibrium. Equation (7) is still valid, but now:

L~ - L = 2s + L (Po - P ) / E A (9)

where s is the inward movement at the separation points, given by


(Po - P - p w L / 2 ) 2 s = (10)

2 E A g w

Finally, we obtain

Po = P + [ w L / E I ] [1.597 X 10 -5 E A p w L s - 0.25 (/./E/)2] I/2 (11)

Equation (11) is valid for

(,uE/)2 -]02 (12) L > [6"388 X 1--OZrEAIuwJ

and eqn (8) should be used for smaller values of L. From eqns (8) and (11) it is possible to plot the temperature change, T,

against L for several values of the parameter/.t. As an example, in Fig. 2 the following values of the parameters appropriate for a pipe of 406-4 mm diameter and 6.35 mm wall thickness have been used:

A = 7980 m m 2

I = 159.65 × 106 mm 4 w = 0.6414 kN/m E = 207 kN/mm 2 a = 11 × 10-6/°C

ANALYSES - - PROP CASE

The 'prop' case (Fig. 3) is the case of a pipeline supported by a rigid point object of height H above an otherwise flat seabed. It is noted that the

2t, O

16Q

G o.. 80

0

/ <_

Y 120

J

0

~,0 80 160 200 Buckle length L [rn]

Fig. 2. Upheaval: influence of friction coefficient p in the perfect case.

360 J. P. Ballet, R. E. Hobbs

w • Y H

P ~ " l ~ / / / / ~ . ~ 2 / / / / / / / / / / / / / / / / / ~ - / / ' / / L ~l -] / /

Fig. 3. The prop imperfection.

p

boundary conditions are the same as in the perfect case if it is assumed that the prop reaction Vis applied in the middle of the buckle (symmetric buckling).

Symmetric buckling

Determination of P, 6 and V Expressions linking the buckle force P, the prop force V and the prop height H have been obtained by Richards (Boer et al.2). It was assumed that the buckling is symmetric, and the similarity between the prop case and the well-known standard case of a simply supported beam, 7 loaded by a uniform lateral load w, an axial force P and a central vertical force V (Fig. 4), was used to establish the relations between the loads and the boundary conditions. This leads to the pre-upheaval relations where the pipe is still in contact with the prop:

,U)~,4~ : C2(/)v,3(/)J

v - w4(J) [ ~3U) ],/4 ~3(J) Ig4(/') IgllU) - I//3(/) I//12(.]) [HEIw3]I/4 (14)

P= [U/ll(]) ff'/4(J) -- iPrl2q) lPf3(]')]l/2 [~---] ) (15)

M1 M2 el e2

l_ L ~1

Fig. 4. Simply supported beam-co lumn.


where

1 1] ~t3(]) = 2 [cos 1 (]/2)

~4(J) = t a n ( j / 2 ) - j / 2 1

I¢~t0") = ~ (tan (/'/2) - j /Z )

1~12(]) cos

(16)

and

j = L [P/(EI)] I/2 (17)

At this stage, of course,6 - H. The pre-upheaval stage ends when V falls to zero, and the pipe lifts off the prop. This occurs when j = j * (say) -- 8.9868.

From the relations given above it is easy to plot the evolution of L, V and P against j during the pre-upheaval stage, and only a few trivial calculations are required to be able to plot the evolution of V/(HEIw3) ~25 and P/P* against L/L*, where P* = P(]*) and L* -- L(]*) (Fig. 5). It is supposed that the compressive force in the line is initially zero, so that the initial values of L and Vare those of the well-known zero compression case described by Hobbs, 8 say:

L0 = 2(72 EIH/w) TM ~- 1"29L* (18)

V0 = (4w/3) (72 EIH/w) TM ~- 3"88 (HEIw3) °'25 (19)

The subsequent post-upheaval behaviour is characterised by the relations:

p = j , 2 [ ~ q V = Oand6 - Wl2(J*)[ -wL4] ' J*' L El_] (20)

The decrease of P/P* and the increase of 6 /H are plotted against L/L* in Fig. 6.

Upheaval history for the prop case The main objective is still to determine the evolution of the temperature increment against L, and thus the upheaval and 'safe' temperatures. Unfortunately, the results for the perfect case are no longer valid. This is obvious during the pre-upheaval stage when Vis still large, but it is (more

362 J. P. Ballet. R. E. Hobbs

1.0

O.B

0.6

.K, a. O.t, 0 . .

• ~ 0.0

%

eovol

1.10 Buckle length L/L*

1.20

/ /

j ~ f upheaval

Buckle length L/L*

Fig. 5.

1.30

1.10 1.20 1.30

Pre-upheava l stage in p rop case.

surprisingly) true, as will be demonstrated below, during the post- upheaval stage when the perfect and prop cases seem at first sight to be totally similar. Therefore, a new method has to be implemented.

The requirement for displacement compatibility used in the perfect case can also be expressed as a requirement for axial strain compatibility at any point in the buckle. Neglecting the pressure term, 2 strain compatibility requires:

CI + /~2 "~- •3 "¥ /~4 = c o n s t a n t (21)

where

a x i a l w a l l s t r a i n - - p

A E

e2 = thermal strain = a T

Asymmetric effects of prop imperfections

1.0

0.8

O.G

O.L

0",

Q..

e~

Buckle length L / L* 300

- . - - . . _ . _ ~

20{3

z

10(

0

B~kle length

Fig. 6.

I

LIL"

Pos t -upheava l stage in p rop case.

363

83 = average curvature or arclength 'strain'

s e4 = average boundary displacement 'strain' -- 2 ~

Assuming again that there is initially no compression on the line, the constant is equal to the curvature strain in the zero compression case, which can be found in the usual way:

e 3 = w2L6/(483840E212)

At this stage it can be noted that the original work by Richards 2 appears to include two approximations.

Firstly, it neglects the effect of the concentrated force at the separation points, taking the slip as simply:

(P0 - p)2 (22) s - 2EAlaW

364 J. P Ballet, K E. Hobbs

so that

(Po - e)2 e 4 - E A I ~ w L (23)

Secondly, instead of integrating the general expression for the curvature strain:

1 fL/2 = y'2(x) dx (24) ~3 ~---'J0

where y ' is the real slope in the buckle, an approximation to the true curvature strain is used:

c3 = - ~- (25)

which is obtained by considering the deflected shape to be sinusoidal. This step is forced by the use of the beam-column formulae (eqns (13- (17)) which do not include a description of the displacement pattern. (In passing, it is noted that Ref. 2 also omits the power of 2 on the term in Jr, but this is assumed to be a misprint.)

Removing the first approximation is easy, rectifying the second, rather more demanding.

As suggested by Martinet's analysis, 5 the accurate expression for the boundary displacement sWain involves the concentrated force w L / 2 - V /2

at the separation points. This gives, by analogy with eqn (10):

(Po - P - ! u ( w L / 2 - V/2)) 2 e4 = EAII w L (26)

The general expression for the slope in the buckle is needed to find the accurate value of the curvature strain. It requires a complete review of the case. Using Hobbs's nonzero tension case study 8 and making the changes appropriate for the nonzero compression case, it is easy to show that

M ( x ) = 0.5 w x 2 - 0.5 ( w L - l O x + P y for0 < x < L / 2 (27)

(with the origin for x at the lift-off point.) Hence, the integration yields for 0 < x < L / 2 :

w sin (zlx) + w L - V(1 w x y ' ( x ) = ~ 2P - cos (A.x)) - -if- (28)

and e3 follows by numerical integration.


Every term in eqn (21) is now known accurately and can be calculated easily. Equation (21) can be considered as a quadratic equation for T, which can be solved in the usual way for any given value of L by substituting the corresponding values of P, V and 6.

Results Figures 7, 8 and 9 show the complete upheaval history of the prop case for three different values of the prop height. In each figure, the perfect case curve has been drawn, and the prop case results given by Richards' approximations and the present analysis have been plotted. Ti and T2 represent the upheaval and minimum temperatures according to the first analysis, and T[ and T~ those according to the second.

The following numerical values, appropriate for a buried pipeline of 304-8 mm diameter and lff3 mm wall thickness were used throughout the study of the prop case:

21,0

k S/. .."

160 / .- '~ ~ l e c t cose

"'""..~-~ ,~ . . l ...'" ' .......... " i 120 ~ Improved Approximete

prop enalysis prop onolyiis

8(~ H =O.2m w =5 k N / m ' T~ = 196.6 ° T2 = 126.30

TI= 19/,.7 ° T 2 = 120.50

' - I,O

20 2[, 28 32 36 {,0 [d,

Buckle length L {m)

Fig. 7. U p h e a v a l h i s to ry : H = 0-2 m.

366 J. P BalM, R. E. Hobbs

2/,0

200

160

120

80

/,[l a a

Perfect case •

' ' . . . . . . . . o . , ' ' " ' "

~ lmpro d plop annlylis

T~ = 13/,.9"

T 1 = 131.7'

0 20 25 30 35 /,0 Buckle length L (m)

Fig. 8. U p h e a v a l history: H = 0.5 m.

/ "/

T,~ = 122.8" T 2 = 117./, 0

{,5 50

E = 207 kN/mm 2

a = 11 x 10-6/°C

A = 48.538 × 10 -4 m 2

I = 5449.22 X 10 -8 m 4

/ / = 0 - 5

Three main observations can be made:

1. The classical main feature of the prop case history is confirmed: T rises to the upheaval value and then falls to a post-upheaval minimum, before rising again. Thus, for small enough values of H, at the upheaval temperature the pipeline experiences a possibly catastrophic jump to much greater deformation. The post-upheaval minimum temperature is a possible design limit of safe temperatures although it has also been argued that the upheaval temperature TI is a design criterion if a design bound on H can be specified.


180

160

170

o...

Perfect cose J

~ -"°°°

~ ' f o.. °" ..

k...Jmproved prop analysis

prop

w = 5kN/m- -

T11 = 112.0'

25 30 Buckle length L (m)

1,0 /,8

Fig. 9. U p h e a v a l h i s tory : H = 1 m.

.

.

It now becomes plain that the post-upheaval curves and the perfect case curve must not be confused. The post-upheaval curves depend on H, while the perfect case curve, of course, does not. The post- upheaval curves are, however, asymptotic to the perfect case curve at large values of L. The two approximations used by Richards turn out to cause a significant inaccuracy, as proved by the values of T~, T2, T~ and T~. This is above all true when H is large and V = 0 (post-upheaval stage). It is found that the real upheaval and minimum temperatures are above Richards' ones. The corrected post-upheaval curve tends much more quickly towards the perfect case curve than does Richards' curve.

The effect of the submerged weight on the temperature history for the prop case is illustrated in Fig. 10, obtained with the corrected values of the curvature and boundary displacement strains. T~, T2 and T3 represent

368 J. P Ballet. R E. Hobbs

3so ~: 1 H =0.2m

250 I ~w--15 % I kN/m

i Tc =10 kN / m

~ T b ~

w =SkN/m To

100

I1 =196.6 o Ta 126.3 o

13:31.lO* T c =229.5'

t = ab ,- O 15 20 25 30 35 /.O i,5

Buckle Length L (m)

F i g . 10 . Upheaval history: influence of submerged weight.

the upheaval temperatures forw = 5, 10 and 15 kN/m, and Ta, Tb and Tc for the corresponding minimum temperatures.

The main feature of the upheaval history related to the weight is confirmed: both upheaval and minimum temperatures increase with weight, which justifies gravel covering operations. It was again found that the minimum and upheaval temperatures were underestimated by Richards' analysis.

Nonsymmetrie buckling

In all the calculations above, the buckle has been assumed to be completely symmetric with respect to the prop. This section investigates the possibility of nonsymmetric buckling in the prop case, a possibility suggested by the remarkable sharpness of the cusp in the symmetric characteristic at upheaval, and the well-known asymmetry found in the


buckling of shallow arches under a central point load. Denoting the length between the application point of Vand the closest lift-offpoint on the line by a, the bending moment is:

( M(x) - ~ + P y - - V 1 - x f o r O < x < a

M(x) - ~ + P y - - V 1 - x - V ( x - a ) f o rx> /a

(29)

with y" (x) = - M(x)/(EI) . Using the boundary condit ionsy (0) = y ' (O) = O, the complete solution

of eqn (29), for 0 < x < a, is

w y(x) = ~ ( 1 - c o s ( X x ) ) - - - wX22p + [wL/2 - V ( 1 -

Ix sin(Zx)] (30)

Equating the deflections and slopes left and right of the prop position (x = a), the solution of eqn (29) for x > a is

sin ( A , x ) [ V ( 1 - a / L - c o s ( A a ) ) - wL/2] +cos (A,x) y(x) -

A, L P J A

Vsin (A,a) w -] w/ / l 2 - Va + Vax/L + w x ( L - x ) / 2 P FXJ + P (31)

The other boundary conditions are: y(a) = H, y(L) = y ' (L) = O. Using these three equations leads to a nonlinear system of three

equations in the variables P, V and a:

L(1 + cos (AL))/2 - sin (A,L)/A V = w a(1 - cos (AL))/L + cos (AL) - cos (A(L - a)) (32)

w [ H = ~ ( 1 - c o s ( A a ) ) + a 1

wa 2 2P (33)

370 J. P Ballet. R. E. Hobbs

w 0 = ~ + (Vsin (Aa) - w / A ) cos (AL) + (II(1 - a / L

- cos (Aa)) - w L / 2 ) sin (AL) (34)

It is now possible to determine the equilibrium values of P, V and a during the pre-upheaval stage.

Such a system can be solved by using the NAG library routine C05NBF 9 (which uses a convex combination of the Newton and scaled gradients iterative method). Asymmetric values of P, Vand a are indeed found.

Hence, asymmetric pre-upheaval values of the temperature can be found by substituting the values obtained above in a strain compatibility equation. This equation is now slightly different from eqn (21), since the two concentrated forces at the separation points are no longer equal. It becomes.

s l + 8 2 8 4 - L

where

S 1 -----

and

(35)

(Po - P - l a ( w L / 2 - II(1 - a / L ) ) ) 2

2 E A t J w (36)

(Po - P - I J ( w L / 2 - V a / L ) ) 2 (37) s2 = 2 E A I J w

The curvature strain is now

- ± r y , 2 e3 - 2LJ0 (x) dx (38)

where y ' is given by differentiating the equations for the deflection obtained above.

Resu l t s

The results for H = 0.2 m and w = 5 kN/m are shown in Fig. 11. Dots represent the asymmetric combinations of a, L and the temperature T found by the NAG routine. 9 The standard symmetric upheaval history is also shown for reference. It has to be admitted that a specific asymmetric upheaval equilibrium path has yet to be determined. It can only be noted that:


20~

1/E

150

~lOO

, ° . , .

• . " . , '

20 22 2~ 26 Buckle length L Im)

Fig. l l . Asymmetric upheaval buckling.

B =0.2m w

S ymmetric Asymmetric

=SkN/m

28

- - A path above the symmetric post-upheaval curve seems likely. This path, associated with large values of lL/2 - a [ (very small or large values of a) and large prop forces V, is equivalent to the standard upheaval with a firm support close to one end, giving a higher equilibrium temperature. This 'path' seems likely to be of academic rather than of practical interest.

- - Most of the asymmetric data points form a broad scatter band under the symmetric post-upheaval curve. Their existence casts considerable doubt on the stability of the symmetric curve.

- - Asymmetric combinations of the temperature and a and L are indeed found close to the pre-upheaval curve.

372 J. P. Ballet. R. E. Hobbs

If enough data points were accumulated it would be possible to draw a lower bound to the scatter band. This lower bound curve would intersect the primary pre-upheaval path well below T1 and also pass below T 2, the minimum of the symmetric post-upheaval curve.

It would be expected that this lower bound would approximate to the true asymmetric equilibrium path, although more direct calculations of such a path are clearly desirable.

The apparent erosion of TI and T2 has very clear design implications, undermining the safety margin of existing designs whether they are based o n T2, or on T~ and a design prop height.

.

,

3.

CONCLUSIONS

The approximations in the previous symmetric prop upheaval analysis 2 cause a significant underestimate of the upheaval and minimum temperatures T~ and T2. It now appears clear that the perfect case curve and the prop post- upheaval curve do not coincide, and must not be confused. Even if the nonsymmetric analysis has not so far permitted the establishment of complete numerical results and requires further investigation, it proves that an asymmetric mode of buckling is possible. Thus, the symmetric pre-upheaval equilibrium path might branch at a temperature below T~ into a new asymmetric equilibrium path which would reach the post-upheaval curve after passing through a minimum ('safe') temperature below T2. This means that T~ and T2 may not be that relevant to describe the buckling for the prop case, and the pipe may experience a potentially catastrophic jump to greater deformation before the temperature reaches T~. Even though T1 and 7"2 are underestimated by the methods of Ref. 2, it is concluded that some erosion of the expected safety margins is implied by the hitherto unsuspected presence of an asymmetric buckling mode.

REFERENCES

1. Proc. 22nd Annual Offshore Technology Conf., Houston, TX, Offshore Technology Conference (Inc). 1990. Papers 6332-5, Vol. 2, pp 519-60; Papers 6486-8, Vol. 4, pp. 563-600.

2. Boer, S., Hulsbergen, C. H., Richards, D. M., Klok, A. & Biaggi, J. P., Buckling considerations in the design of the gravel cover for a high temperature oil


line. Proc. 18th Annual Offshore Technology Conf., Houston, TX. Paper OTC 5294, 4 1986, 9-26.

3. Richards, D. M. & Andronicou, A., Seabed irregularity effects on the buckling of heated submarine pipelines. WEMT Symp. 'Advances in Offshore Technology', Amsterdam, 1986, pp. 5-12.

4. Thompson, J. M. T. & Hunt, G. W.,A General Theory of Elastic Stability. John Wiley, New York, 1973, pp. 57-9.

5. Martinet, A., Flambement des voies sans joints sur ballast et rails de grande longueur. Rev. G~n. Chemins defer, 55(2) (1936) 212-30.

6. Hobbs, R. E., In-service buckling of heated pipelines. ASCE J. Transport. Eng., 110 (1984) 175-89.

7. Timoshenko, S. P. & Gere, J. M., Theory of Elastic Stability. 2nd edn, McGraw-Hill, New York, 1961, pp. 3-15.

8. Hobbs, R. E., The lifting of pipelines for repair or modification. Proc. Inst. Civil Eng., Part 2, 67 (1979) 1003-13.

9. The NAG Fortran Workstation Library Handbook, Vol. 1, 1st edn, Numerical Algorithms Group, Oct. 1986.

Documents

Asymmetric Effects of Prop Imperfections on the Upheaval Bucling of Pipelines