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Pipeline buckling caused by axial loads
Pipeline Buckling Caused by Axial Loads Dr R. E. Hobbs LECTURER, DEPARTMENT OF CIVIL ENGINEERING, IMPERIAL COLLEGE OF SCIENCE AND TECHNOLOGY, LONDON
SYNOPSIS Compressive loads are commonly induced in pipelines by the frictional restraint of axial extensions due to temperature changes or internal pressure. It is shown that these forces can cause buckling in the presence of the initial imperfections which are certainly present in pipelines as laid, particularly in submarine lines. Two buckling modes which have occurred in practice (and their interaction) are considered. The first mode involves an upward movement from the sea bed while the second involves snaking lateral movements on the sea bed itself. Both modes have been reported in the literature on railway tracks. These results are reviewed, and extended in the case of the analysis of the lateral mode. For normal coefficients of friction, the lateral mode occurs at a lower axial load than the vertical mode and is therefore dominant in pipeline work unless the line is laid in a trench. In this case the lateral restraint makes a vertical buckle up and out of the trench possible, which may be followed by an interactive lateral movement and/or rolling movement of the elevated part of the p(peline. The theoretical solutions are illustrated by numerical calculations for a typical pipeline and some design implications are reviewed.
NOTATION A E / k 1, k 2, k 3, k 4 L Ls m M n p P P0
cross-sectional area Young's modulus second moment of area of cross-section constants - see Table 1 length of buckle length of slipping pipe adjacent to buckle w/E1 bending moment (P/EI) 1/2 pressure axial force in buckled pipe prebuckling axial force
r
t T w x y tt v
radius wall thickness of pipe temperature increment submerged weight of pipeline (including weight coat) per unit length coordinate along pipe axis coordinate perpendicular to pipe axis coefficient of linear thermal expansion strain Poisson's ratio coefficient of friction
Superscripts
"
peak, maximum value first derivative, slope second derivative, curvature value associated with the 'safe' temperature increment
INTRODUCTION The economic importance of submarine pipelines has increased greatly in recent years with the development of offshore oil and gas fields in many parts of the world. The cost of a failure in such a line is so high (not only for repairs but also for lost production) that considerable interest has been focussed on the stresses caused during the laying (1'2) and modification or repair (3'4) of submarine pipelines. Similar arguments on a somewhat lower scale of expense apply to pipelines on dry land. However, comparatively little attention has been paid to problems occurring in lines during routine operation. This paper addresses one such problem, the assessment of compressive axial forces in the pipeline and their consequences. The size of the axial load in the pipe depends on many factors. As well as the mechanical properties of the line and its weight coat, the axial force is a function of the initial tension at the sea bed just after laying, the pressure difference across the pipe wall and tem-
Pipeline buckling caused by axial loads perature variations due to hot oil passing through the lhae. These factors interact with sea bed geometry and frictional and/or trench backfill effects, as well as the influence of end restraints in shorter lines. Finally, the previous loading history and time-dependent changes due to scour, currents and tides are also relevant. Thus it is extremely difficult to say with any certainty what axial force exists at any point in a given pipeline at a given time. Nonetheless, two major causes of compressive forces can be identified, arising from the restraint of the strains associated with thermal and internal pressure loadings. With oil temperatures up to 100°C above water temperature and operating pressures over 10 N/mm 2, these effects can produce very significant forces indeed in a long line where the necessary frictional force can develop between pipe and sea bed, or in shorter fines with effective end restraints. The force created by full restraint of thermal expansion is, simply,
Po= EAaT
(1)
The free axial strain, t, due to a positive pressure difference p across the wall of the pipe is given in terms of the well-known thin wall axial and hoop stresses in the pipe by e=
v
E
(2)
where v is Poisson's ratio. Then, if t is completely restrained, the axial compressive force generated and available to participate in buckling is
Po = EAt =
apr t
m
(4)
where m = w/E1 and n 2 = P/E1, with the solution
(o)
iiiii1~11111111111
sea bed
Elevotion
b)
Plan view
Figure 1
Analysis - vertical mode
This mode of buckling has attracted a great deal of attention over the years from railway engineers concerned with the buckling of rail track, particularly continuously welded track. Kerr (~) has presented a review paper with nearly fifty references to this problem, while Marek and Daniels (~) have described an apparently independent analysis of the vertical buckling of crane rails. Their work (as corrected in discussion by Granstr6m (8)) agrees in all respects with the theory first developed in 1936 by Martinet (9). Accordingly, it seems appropriate merely to summarise the method and results for the vertical mode at this point. The first step is to solve the linear differential equation for the deflected shape of the buckled part of the pipeline, which is treated as a beam column under uniform lateral load. It is assumed that the bending moment at the rift-off point is zero. This assumption has been examined elsewhere (1°), and it should also be noted that the use of the linear equation reties on the usual column buckling assumption of small slopes, a point to be borne in mind later when assessing the results. In the notation of Figure 2(a),
(3)
Having assessed the sources of compressive force, there are two possible buckling modes in response to the force. These are both overall buckling modes without gross distortions of the cross-section which may be contrasted with the propagating buckle triggered by excessive bending and external pressurisation during laying operations. (5) The first mode (Figure l(a)) involves part of the line lifting itself vertically from the sea bed while the second (Figure 1(b)) results in various snaking lateral movements in the
J~.11111111111~1~11
horizontal pIane against frictional resistance. Both of these modes have been demonstrated in a very simple model employing rubber strips resting on a horizontal machined surface and loaded by a small screw jack, and both are known to occur in real pipelines under suitable conditions.
y" + n2y + --~ (4x 2 - L 2) = 0 (0.5 -- v).
3
Vertical a n d lateral b u c k l i n g m o d e s
m ( Y=-n4
cosnx cos (nL/2)
n2x 2
n2L 2
- - + ~ +
2
8
) 1 . (5)
The unknown length of buckle L is then determined from the condition that the slope at the ends of the buckle should be zero. This yields
nL
nL
tan - - -- - 2 2
(6)
nL = 8.9868...
(7)
or as lowest root
The next step is to compare the axial load P in the buckle with the axial load P0 well away from the buckle (Figure 2(b)). P is clearly less than P0 because of the extra length round the buckle compared to L itself. The drop in force is not as large as would be expected at first glance because two adjacent lengths of pipe L s sfide in towards the buckle - giving the axial force distribution shown in Figure 2(b). The discontinuities in this distribution at each lift-off point are associated with the concentrated vertical reactions of 0.5wL which occur there.
4
Journal of Constructional Steel Research: Vol. 1, No. 2: January 1981 (o)
w
t /
/
/
/
,-
/ / ~ - -
qr-
k l-
~
~
¢
, i r - ~
Ls
z
J_ I
r
L
"
.-r
I"
wL,2 i
Dimensions ---p -.p
-~
--p
~
/
/
/
z
z
z
z
/
'~I
,s
o,w I
(b) Axiol force distribution
Figure 2
Details of vertical buckle
Setting up a compatibility equation (7), the following results are obtained
P = 80.76EI/L 2
Equations 15 and 16 are useful when solving equation 9 for particular cases as described later.
(8) Analysis - lateral mode
wL
P0 = P + - - [
El
1.597 x !0 -5 EA~wL 5 - 0.25(~EI)2] ~ (9)
The maximum amplitude of the buckle wL 4
fi = 2.408 x 10 -3 - -
(10)
E1
and the maximum bending moment, at x = 0, is 37/= 0.06938wL 2
(11)
while the maximum slope is f ' = 8.657 × 10 -3 wL3/EI.
(12)
This last result is useful for checking the validity of the small slope assumption in particular numerical cases, i.e. conventionally )~' x<0.1 for 'small' slopes. A further result of practical interest is the size of the slipping length adjacent to the buckle, P~-- P L s - - -
0.5L.
(13)
Thus the minimum theoretical distance between the centres of two adjacent but independent vertical buckles is L + 2L s -
2(P o - P)
Mode 2
Mode I
I-
(14)
Equation 9 is awkward and may be compared with the result for a very large coefficient of friction (i.e. L s = O)
EI
The lateral modes observed in a small-scale experiment (e.g. Figure l(b)) all resemble sine curves decaying either side of a central peak amplitude. As a first tentative step, a lateral mode of the same shape as the vertical mode was considered, mode 1 in Figure 3. With the vertical analysis to hand, it was simple to determine the relationship between buckle length L and axial force P0 for a lateral coefficient of friction assumed equal to the friction coefficient for axial movements of the slipping lengths L s. Unfortunately, for equilibrium, this mode requires concentrated lateral forces of ~wL/2 at the ends of the buckle analogous to the concentrated vertical forces at the lift-off points in the vertical mode. On a rigid sea bed concentrated vertical forces are possible (and practical cases closely approach the rigid ideal (1°)) but it is not possible to generate a truly concentrated lateral force by friction. References 9, 11 and 12 which came to hand rather later deal with this point at length, but in the investigation described here the next step was to repeat the analysis using a different assumption which avoids the difficulty.
Mode3
1
L
j
Model,
w2AEL 6
P0 = 80.76 ~-S + 1.597 × 10-5 (EI)---------~
(15)
It is easy to show that equation 15 has a minimum at
[ = ( 1 . 6 8 5 6 x 106(EI)3) °'12' w2AE
M o d e c~o
(16)
Figure 3
-I ---_i
Lateral buckling modes
Pipeline buckling caused by axial loads Looking at the experimentally observed lateral modes once more, it was assumed that the family of decaying trigonometric curves were all initial imperfection generated variants of one fundamental constantamplitude periodic curve. In other words, it was assumed that an initially perfect pipe would buckle into an indefinite series of half waves as shown in Figure 3, mode oo. This assumption has the computational advantages that the nodes of the half wave pattern do not slide parallel to the axis of the pipe and, more importantly, can be made to satisfy lateral equilibrium. Looking at the consequences of this new assumption in detail, the linear differential equation governing the deflected shape is unchanged, equation 4, except that m is now qJw/EI, i.e. it is assumed that the lateral frictional force is fully mobilised everywhere. The displacement boundary conditions are unchanged but for the zero slope condition at x = +L/2 which is replaced by a shear force condition at the same location. Thus equation 5 still holds with the new value of m, while after careful consideration of signs the shear condition yields the length of the buckle from
nL
tan - - = 0 2
(17)
or as the lowest non-trivial root
nL = 2zr.
(18)
(20)
Po=P + 1.4545 x 10-SAE \( ~Wt2L6. EI]
Then
(19)
that is the reduction in axial force in the buckle equals the product of the axial stiffness and the extension round the curve. This leads to the following results for the infinite lateral buckling mode
p = 4ztEEI/L 2
linear differential equation is probably sufficiently accurate. References 9, 11 and 12 were obtained after completing this analysis. Martinet (9), as long ago as 1936, considered mode 1 as well as the vertical mode, observed the lack of lateral equilibrium at the ends of the buckle and predicted that mode 3 (with a smaller lack of equilibrium at its ends) was likely to develop as a result (Figure 3). He then analysed this higher mode, confirming that buckling would be initiated at a lower axial force for a given out-of-straightness than necessary for mode 1. Kerr (m (and in a condensed form (m) confirmed Martinet's work for modes 1 and 3 using a variational formulation and presented results for two antisymmetric modes, modes 2 and 4 (Figure 3). These are initiated at very similar axial force/initial imperfection combinations which are rather smaller than those needed to trigger mode 3. Martinet and Kerr's results are discussed later, but it is worth summarising their formulae for modes 1-4 in the notation of the present paper at this point. Taking the half wavelength of the most significant part of the buckle as L in each case (Figure 3), and using the constants of Table 1, the reduced axial force within the buckle is given by P = k 1EI/L2 (26)
AE~wL 5~ 1/2 ] --1.0]. Po=P+k3,wL[(1.O+k2-- ~S
Compatibility then requires that
Po--P= AE f~_/2 -1 '2dx L -L/2 2 y
(27) The maximum amplitude of the buckle relative to the original axis is fi = k, ~ w L4" Table 1
Constants Mode
~ 2 ~-E-
(22)
The maximum amplitude of the buckle fi = 4.4495 x 10-3 # w
E1
L4
Constants for lateral buckling modes
(21)
0.125
(28)
E1
For computational use, it is noted that this equation has a minimum at ~ , = ( 9 . 0 4 7 4 x 105 (EI)3 t ]
5
kt (Eqn. 26)
1 2 3 4
80.76 47t2 34.06 28.20
oo
47t~
k2 (Eqn. 27) 6.391 1.743 1.668 2.144
x x x x
10-5 10 -4 10 -4 10 -4
),.4545 x 10 -5
k3 (Eqn. 27) 0.5 1.0 1.294 1.608
k4 (Eqn. 28) 2.407 5.532 1.032 1.047
x x x x
10-3 10 -3 10 -2 10 -2
(Eqn. 21) 4.4495 x 10-3
(23)
RESULTS
and the maximum bending moment, at x -- 0, is )1~ = 0.05066#wL 2
(24)
while the maximum slope is
fi' = O.O1267fJwL3/EI.
(25)
Again, if y' <~ 0.1 in a particular case, the small slope
Figures 4-6 summarise the results of the vertical and lateral mode calculations for a typical pipeline for a variety of coefficients of friction, including the practical range 0.3 ~<~ ~<0.7. The pipe considered has an outside diameter of 650 mm, and a wall thickness of 15 mm giving a cross-sectional area of 299.2 cm 2 and a second
6
Journal of Constructional Steel Research: Vol. 1, No. 2: January 1981
moment of area 150900 cm 4. Its submerged weight (including concrete coating) has been taken as 3.8 kN/ m. The results have been presented in terms of the temperature rise necessary to generate the axial force in equilibrium with the buckle lengths and amplitudes shown, and the coefficient of linear thermal expansion, ct, has been taken as 11 x 10-~/°C for this purpose. The results were obtained from a small computer program using the following procedure which recognises that it is much easier to find the force (or temperature change) corresponding to a given buckle length L than vice versa.
Po using equation 21, T using equation 1 and the buckle amplitude fi using equation 23. (d) Lateral modes - modes 1-4 Repeat steps (c) (i) and (ii) using equations 27 and 28 for P0 and f. To establish the minimum for mode 1, values of L up to 2.0L may have to be used.
DISCUSSION It is apparent from Figures 4 and 5 that the lateral modes become possible at a smaller temperature change than the vertical mode. Thus unless lateral restraint is provided, for example by burying the line, the lateral modes will be dominant. This feature is confirmed by small-scale experiments and practical experience. An unburied line will snake laterally, while a buried line may burst out of the sea bed (or desert!). Once a buried line has lifted, an interaction with the lateral mode may occur, as the buckle itself now has no lateral restraint (4 = 0). Alternatively the buckle may roll or twist laterally. Figures 4 and 5 are otherwise very similar and it is useful to discuss the common features in terms of simplified load or temperature against length of buckle and amplitude curves, Figure 7. The first point is that in theory an ideal perfectly straight pipeline would not buckle. The pipe is in equilibrium for all values of axial load in an undeformed configuration and while the
(a) Vertical mode - high friction coefficient (i) Compute i, using equation 16. (ii) For a range of 20 values of L between 0.1L and 3L, compute P0 using equation 15, T using equation 1, and the buckle amplitude fi using equation 10. (b) Vertical mode - real friction coefficients For a range of friction coefficients 4, and 20 values of L between 0.1L and 3L, compute P0 using equation 9, T using equation 1, and the buckle amplitudefi using equation 10. (c) Lateral modes - infinite mode (i) For each of a range of friction coefficients 4, compute [, using equation 22. (ii) For each friction coefficient 4, and a range of 20 values of L between 0.ST, and 1.5T, compute 320
!
2.o
/
/
~06o
large
0"4
//
240 Temperature rise T ('c) 200
0'3
/
160
O,
120
8oFI
0'01
r J
40
¢=0 I
t
80
20 0.05 0.1 i I
0"2 i
0.5 i
1
100 120 1}.0 Buckle length L (rn) 2
5
10
20
I
i
I
i
Buckle
Figure 4 Resultsfor vertical buckling
160
omptitude
y (m)
180
200 5O I
Pipeline buckling caused by axial loads
7
140
120
100 Temperature rise T ('c) 80
60
40
20
I
I
4O
20
I
60
0.05 0.1
02
i
I
80
0'5
i
I
I
100 120 140 B u c k l e l e n g t h L (m)
i
2
5
i
i
10
I
I
160
180
200
2O i
0
Buckte a m p l i t u d e
Figure 5
m)
¢
:
0.5
)
Results.for lateral buckling - oo mode
150
. . . . .
130 TemperotUrerise T ('c)
(for
,
--
=~~t
110 .
.
.
.
.
.
.
i .
.
.
.
.
.
. I
Mode
oo
30 /(
0.2
I 04
I 0.6
I 08
-7-
J i i i 10 12 1-4 16 B u c k l e o m p l i t u d e } (m)
_,,
l
I
I
1.8
2.0
2.2
2.4
Figure 6 Comparisonof the lateral buckling modes
curve for the deformed shape (equations 9, 15 or 21) approaches the vertical axis asymptotically at high temperatures and small wavelengths, it never quite meets the axis. As Kerr u2) has remarked, this is a consequence of the assumption of fully mobilised friction even for vanishingly small displacements, and the situation may be contrasted with the conventional
pin-ended Euler column where the post-buckling equilibrium path actually crosses the initial equilibrium path on the axis. Supposing, next, something marginally less than initial perfection in the pipeline, it is clear that at some temperature T the imperfection will be enough to bridge the gap between the two equilibrium paths, bringing it
8
Journal of Constructional Steel Research: Vol. 1, No. 2: January 1981
c
"\ '
I~
I'
P,
X
',\
L
L
E
I
I
/
I
/
! I I
/ . . . . . . . ~'"/ ~ ~4" z
I /
I
I
!
J
I II I 11 I
/
/
I
i I
/ /
I
III
//
It
I I ~11
/
I
J
/
II -- ~ - : - - - ' f ' z " z Ill ~-'." z /
o I I
/ / /
/
Buckle
length
L
Buckle
amplitude
~,
Figure 7 Equilibrium paths and imperfection sensitivity-schematic
to the equilibrium position A. The part of the secondary equilibrium path from A to B is clearly unstable, and a dynamic snap will occur from A to C (chain dotted line) at a constant load P0 in the pipeline well away from the buckle, although the buckle itself lengthens and partially unloads. Because, on this argument, no snap can occur below the temperature T at B, T has been called the 'safe' temperature in the railway literature. These ideas can be presented with slightly more rigour by considering the temperature/buckle length curves for initially imperfect systems sketched in Figure 7 (dashed lines). For very small imperfections a large snap is indeed seen experimentally. As the initial out-of-straightness is increased the snap occurs at lower forces and less dramatically. Eventually, for large enough imperfections the snap is eliminated, to be replaced by a single-valued magnification of the initial bow. For any initial imperfection, the behaviour is ultimately asymptotic to the stable curve BC for the perfect system. In practical terms it may even be desirable to deliberately create initial out-of-straightness, to eliminate the possibility of a snap. This is simple enough for land-based lines which often include dog legs and expansion loops, but it is not so easy for submarine lines. Working from a lay barge, the pipe lengths are welded together with a 1 : 480 straightness tolerance and sent over the stinger. Deliberate kinks could overstress the line during laying. As the weather worsens, involuntary large radius curves may be induced at the sea bed by lay barge motion but it is not practical to lay the line to specified radii. Nor is it worth trying; Martinet (9) and small-scale experiments both show that
a large radius is ineffective in eliminating the snap or in changing its wavelength. On unloading the pipeline differences appear between the vertical and horizontal modes. A vertical buckle will trace the secondary path back down to B, and if the load is further reduced, drop back to the straight configuration although a small residual compression will arise at the site of the buckle because of frictional effects. A chain of horizontal buckles will initially try to trace the path down to B but the changes in wavelength coupled with the reversal of the lateral friction force will leave it at zero temperature with some residual out-of-straightness and a small tension in the line. On reloading the pipeline these imperfections must predispose the line to buckle laterally at a lower load than before by magnification of the imperfections. In the vertical mode, with small residual axial forces, the line is likely to buckle at the same location as before but not necessarily at a significantly reduced temperature as the straightness of the pipe is unimpaired. As might have been predicted, it is clear from Figures 4 and 5 that the value of the friction coefficient ~ only become particularly relevant to the results in the post-buckling regime. However, these results do not show any extreme sensitivity to the value of 4; since the value of ~ for a given stretch of sea bed is only known approximately, this is just as well. Turning now to a comparison of the various lateral modes, Figure 6 presents temperature/amplitude results for the five possibilities considered here for a single friction value, ~b= 0.5. These curves correspond to the unstable part of the equilibrium path, AB, in Figure 7, and if extended for large amplitudes would all rise, corresponding to BC in Figure 7. Thus each point
Pipeline buckling caused by axial loads shown represents a temperature/amplitude combination sure to initiate buckling in the mode shown for a suitable initial waveform. It is clear, as Kerr tH) has shown for rail tracks, that the antisymmetric modes 2 and 4 occur at very similar temperature/amplitude combinations, slightly smaller than those needed for mode 3. Mode I, occurring at higher temperatures, seems unlikely to be of practical interest. Mode oo, on the other hand lies below modes 2 and 4 for amplitudes less than 1.4 metres, and has a 'safe' temperature some 3°C lower. Indeed, there are grounds for suspecting that mode ~ lies below modes 2 and 4 at even larger amplitudes: it is recognised tH) that the analysis of modes 3 and 4, and to a lesser extent mode 2 contains an approximation which will lead to an overestimate of the amplitude. In these analyses it has been assumed that the axial force P is constant throughout the entire buckle, in spite of the inward movement from the adjacent unbuckled pipeline. On the contrary, the axial force in the inner half waves of modes 3 and 4 must be significantly less than in the outer, smaller, half waves because of axial friction and the true amplitude of the inner waves will be smaller as a result. If these arguments were to be extended to modes with 5, 6, 7 . . . . half waves, a situation would rapidly be reached where the inner, most significant, buckles were not receiving any energy at all from the inward sliding of unbuckled line, in other words the inner waves would approach the conditions assumed in the analysis of mode ~. These arguments lend weight to the idea that practical buckles are imperfection generated variants of mode ~ . These considerations suggest that mode oo might be a very useful lower bound for design purposes. It is certain that most pipelines would buckle in modes approximating to modes 2, 3, 4. It is not certain that higher modes closer to oo will never arise if bad luck produces the wrong initial imperfection pattern. Thus the 'safe' temperature of mode ~ , and the associated amplitude/wavelength combinations for temperatures and axial forces above the 'safe' level are proposed as providing a conservative estimate of permissible imperfection sizes and wavelengths for design use with appropriate safety factors. Finally, the following limitations of the analyses presented in this paper should be borne in mind. (a) Only perfect systems have been rigorously examined. In the vertical mode no account has been taken of the initial out-of-straightness. It is not easy to see how to include this factor, short of an approximate numerical (e.g. finite element) approach. The classical column (Fourier) analysis of initial lack of straightness becomes difficult here because the buckled length changes progressively as the load increases. It is some comfort to know that the magnitude and nature of the initial imper fections in pipelines as laid are also unknown, and
9
extremely difficult to assess, so that an improved analysis would not of itself be very helpful. (b) Perfect elasticity and small slopes have been assumed. The analyses presented assume that the full elastic modulus of the pipe is available to resist bending. Once plasticity occurs the analysis loses its validity: yielding will obviously leave its mark on the pipe on unloading and act as an imperfection on subsequent reloading. Of course, most linepipe is competent to develop a plastic hinge without effect on its serviceability. The analyses also use the linear differential equation for bending which is strictly true only for small slopes, conventionally less than 0.1 radian. However, more exact elliptic integral solutions have been obtained t13,14} for cantilevers and pin-ended columns. In the cantilever, for an axial load 1.5 per cent greater than that predicted by the linear equation, the end slope is 0.35 radian. Similarly, in the pin-ended column, for an axial load 1.2 per cent greater than the Euler load, the greatest slope is 0.31 radian. These results suggest that the 0.1 radian limit on 'small' slopes is rather conservative; cut-off lines corresponding to this limit have been marked on Figures 4 and 5 as a guide, and true loads beyond this limit may be rather higher than shown.
CONCLUSIONS Two potential buckling mechanisms in pipelines subjected to axial compression have been identified and analysed. It is found that horizontal snaking modes occur at a lower axial load than the vertical mode, and a horizontal mode is therefore dominant unless lateral restraint is provided by trenching when an interactive buckling mode becomes possible. Full account has been taken of friction between the pipe and the ground in the vertical mode, relying on two independent but formally identical references on the buckling of rail tracks and crane rails. An analysis of a lateral mode involving an infinite sequence of half waves with fully developed lateral friction has been presented and compared with earlier work on isolated buckles over a shorter length which involve longitudinal as well as lateral sliding. It is concluded that the infinite mode gives a lower bound to permissible imperfection amplitude/wavelength combinations which may be useful for design purposes. The effects of imperfections, unloading and reloading are discussed. The theoretical solutions are illustrated by numerical results for a typical pipeline.
A CKNO WLEDGEMENTS The author wishes to record his gratitude to Jack Ells, Chris Lawlor and David Walker of the Engineering Department of BP Trading Lid, and Chris Burgoyne of the Civil Engineering Department, lmperial College, for the stimulating discussions which led to this paper.
10
Journal of Constructional Steel Research: Vol. 1, No. 2: January 1981
However, the opinions expressed (and any mistakes made) are the author's alone.
7
8
ings of the American Society of Civil Engineers, Journal of the Structural Division, January 1972, ST l, 360o61.
REFERENCES 9 1
WILHOIT, J. C., JR. and MERWIN, J. E. 'Pipe stresses induced in laying offshore pipelines.' Transactions of the American
Society of Mechanical Engineers, Journal of Engineering for Industry, 89, 1967, 37-43. PALMER, A. C., HUTCHINSON, G. and ELLS, J. W. 'Configuration of submarine pipelines during laying operations.' Transactions of the American Society of Mechanical Engineers, Journal of Engineering for Industry, 96, 1974, 1112-18. 3 HOBBS, R. E. 'A beam bending problem with a free boundary.' Computers and Structures, 10, December 1979, 915-20. 4 HOBBS, R. E. 'The lifting of pipelines for repair or modification.' Proceedings of the Institution of Civil Engineers, Part 2, 67, December 1979, 1003-13. 5 PALMER, A. C. and MARTIN, J. H. 'Buckle propagation in submarine pipelines.' Nature, 254, March 1975, 46-8. 6 KERR, A. D. 'On the stability of the railroad track in the vertical plane.' RailInternational, 5, February 1974, 132-42.
MAREK, P. J. and DANIELS, J. H. 'Behaviour of continuous crane rails.' Proceedings of the American Society of Civil Engineers, Journal of the Structural Division, April 1971, ST4, 1081-95. GRANSTR(}M, A. Discussion on Reference 7 above. Proceed-
10
2
ll 12
13
14
MARTINET, A. 'Flambement des voies sans joints sur ballast et rails de grande longueur.' (Buckling of tracks without joints on ballast and long rails, in French.) Revue Gdn~rale des Chemins deFer~ 55/2, 1936, 212-30. HOBBS, R. E. 'The effect of soil modulus on pipeline stresses.' (Awaiting publication.) KERR, A. D. 'Analysis of thermal track buckling in the lateral plane.'Acta Mechanica, 30, 1978, 17-50. KERR, A. D. 'On thermal buckling of straight railroad tracks and the effect of track length on the track response.' Rail International, 9, September 1979, 759-68. TIMOSHENKO, S. P. and GERE, J. M. Theory of Elastic Stability, 2nd edition. McGraw-Hill Kogakusha, New York, 1961, 76-81. HORNE, M. R. and MERCHANT, W. The Stability of Frames. Pergamon, Oxford, 1965, 10-12.
Contributions discussing this paper should be received by the Editor before 1 May 1981.
SYNOPSIS Compressive loads are commonly induced in pipelines by the frictional restraint of axial extensions due to temperature changes or internal pressure. It is shown that these forces can cause buckling in the presence of the initial imperfections which are certainly present in pipelines as laid, particularly in submarine lines. Two buckling modes which have occurred in practice (and their interaction) are considered. The first mode involves an upward movement from the sea bed while the second involves snaking lateral movements on the sea bed itself. Both modes have been reported in the literature on railway tracks. These results are reviewed, and extended in the case of the analysis of the lateral mode. For normal coefficients of friction, the lateral mode occurs at a lower axial load than the vertical mode and is therefore dominant in pipeline work unless the line is laid in a trench. In this case the lateral restraint makes a vertical buckle up and out of the trench possible, which may be followed by an interactive lateral movement and/or rolling movement of the elevated part of the p(peline. The theoretical solutions are illustrated by numerical calculations for a typical pipeline and some design implications are reviewed.
NOTATION A E / k 1, k 2, k 3, k 4 L Ls m M n p P P0
cross-sectional area Young's modulus second moment of area of cross-section constants - see Table 1 length of buckle length of slipping pipe adjacent to buckle w/E1 bending moment (P/EI) 1/2 pressure axial force in buckled pipe prebuckling axial force
r
t T w x y tt v
radius wall thickness of pipe temperature increment submerged weight of pipeline (including weight coat) per unit length coordinate along pipe axis coordinate perpendicular to pipe axis coefficient of linear thermal expansion strain Poisson's ratio coefficient of friction
Superscripts
"
peak, maximum value first derivative, slope second derivative, curvature value associated with the 'safe' temperature increment
INTRODUCTION The economic importance of submarine pipelines has increased greatly in recent years with the development of offshore oil and gas fields in many parts of the world. The cost of a failure in such a line is so high (not only for repairs but also for lost production) that considerable interest has been focussed on the stresses caused during the laying (1'2) and modification or repair (3'4) of submarine pipelines. Similar arguments on a somewhat lower scale of expense apply to pipelines on dry land. However, comparatively little attention has been paid to problems occurring in lines during routine operation. This paper addresses one such problem, the assessment of compressive axial forces in the pipeline and their consequences. The size of the axial load in the pipe depends on many factors. As well as the mechanical properties of the line and its weight coat, the axial force is a function of the initial tension at the sea bed just after laying, the pressure difference across the pipe wall and tem-
Pipeline buckling caused by axial loads perature variations due to hot oil passing through the lhae. These factors interact with sea bed geometry and frictional and/or trench backfill effects, as well as the influence of end restraints in shorter lines. Finally, the previous loading history and time-dependent changes due to scour, currents and tides are also relevant. Thus it is extremely difficult to say with any certainty what axial force exists at any point in a given pipeline at a given time. Nonetheless, two major causes of compressive forces can be identified, arising from the restraint of the strains associated with thermal and internal pressure loadings. With oil temperatures up to 100°C above water temperature and operating pressures over 10 N/mm 2, these effects can produce very significant forces indeed in a long line where the necessary frictional force can develop between pipe and sea bed, or in shorter fines with effective end restraints. The force created by full restraint of thermal expansion is, simply,
Po= EAaT
(1)
The free axial strain, t, due to a positive pressure difference p across the wall of the pipe is given in terms of the well-known thin wall axial and hoop stresses in the pipe by e=
v
E
(2)
where v is Poisson's ratio. Then, if t is completely restrained, the axial compressive force generated and available to participate in buckling is
Po = EAt =
apr t
m
(4)
where m = w/E1 and n 2 = P/E1, with the solution
(o)
iiiii1~11111111111
sea bed
Elevotion
b)
Plan view
Figure 1
Analysis - vertical mode
This mode of buckling has attracted a great deal of attention over the years from railway engineers concerned with the buckling of rail track, particularly continuously welded track. Kerr (~) has presented a review paper with nearly fifty references to this problem, while Marek and Daniels (~) have described an apparently independent analysis of the vertical buckling of crane rails. Their work (as corrected in discussion by Granstr6m (8)) agrees in all respects with the theory first developed in 1936 by Martinet (9). Accordingly, it seems appropriate merely to summarise the method and results for the vertical mode at this point. The first step is to solve the linear differential equation for the deflected shape of the buckled part of the pipeline, which is treated as a beam column under uniform lateral load. It is assumed that the bending moment at the rift-off point is zero. This assumption has been examined elsewhere (1°), and it should also be noted that the use of the linear equation reties on the usual column buckling assumption of small slopes, a point to be borne in mind later when assessing the results. In the notation of Figure 2(a),
(3)
Having assessed the sources of compressive force, there are two possible buckling modes in response to the force. These are both overall buckling modes without gross distortions of the cross-section which may be contrasted with the propagating buckle triggered by excessive bending and external pressurisation during laying operations. (5) The first mode (Figure l(a)) involves part of the line lifting itself vertically from the sea bed while the second (Figure 1(b)) results in various snaking lateral movements in the
J~.11111111111~1~11
horizontal pIane against frictional resistance. Both of these modes have been demonstrated in a very simple model employing rubber strips resting on a horizontal machined surface and loaded by a small screw jack, and both are known to occur in real pipelines under suitable conditions.
y" + n2y + --~ (4x 2 - L 2) = 0 (0.5 -- v).
3
Vertical a n d lateral b u c k l i n g m o d e s
m ( Y=-n4
cosnx cos (nL/2)
n2x 2
n2L 2
- - + ~ +
2
8
) 1 . (5)
The unknown length of buckle L is then determined from the condition that the slope at the ends of the buckle should be zero. This yields
nL
nL
tan - - -- - 2 2
(6)
nL = 8.9868...
(7)
or as lowest root
The next step is to compare the axial load P in the buckle with the axial load P0 well away from the buckle (Figure 2(b)). P is clearly less than P0 because of the extra length round the buckle compared to L itself. The drop in force is not as large as would be expected at first glance because two adjacent lengths of pipe L s sfide in towards the buckle - giving the axial force distribution shown in Figure 2(b). The discontinuities in this distribution at each lift-off point are associated with the concentrated vertical reactions of 0.5wL which occur there.
4
Journal of Constructional Steel Research: Vol. 1, No. 2: January 1981 (o)
w
t /
/
/
/
,-
/ / ~ - -
qr-
k l-
~
~
¢
, i r - ~
Ls
z
J_ I
r
L
"
.-r
I"
wL,2 i
Dimensions ---p -.p
-~
--p
~
/
/
/
z
z
z
z
/
'~I
,s
o,w I
(b) Axiol force distribution
Figure 2
Details of vertical buckle
Setting up a compatibility equation (7), the following results are obtained
P = 80.76EI/L 2
Equations 15 and 16 are useful when solving equation 9 for particular cases as described later.
(8) Analysis - lateral mode
wL
P0 = P + - - [
El
1.597 x !0 -5 EA~wL 5 - 0.25(~EI)2] ~ (9)
The maximum amplitude of the buckle wL 4
fi = 2.408 x 10 -3 - -
(10)
E1
and the maximum bending moment, at x = 0, is 37/= 0.06938wL 2
(11)
while the maximum slope is f ' = 8.657 × 10 -3 wL3/EI.
(12)
This last result is useful for checking the validity of the small slope assumption in particular numerical cases, i.e. conventionally )~' x<0.1 for 'small' slopes. A further result of practical interest is the size of the slipping length adjacent to the buckle, P~-- P L s - - -
0.5L.
(13)
Thus the minimum theoretical distance between the centres of two adjacent but independent vertical buckles is L + 2L s -
2(P o - P)
Mode 2
Mode I
I-
(14)
Equation 9 is awkward and may be compared with the result for a very large coefficient of friction (i.e. L s = O)
EI
The lateral modes observed in a small-scale experiment (e.g. Figure l(b)) all resemble sine curves decaying either side of a central peak amplitude. As a first tentative step, a lateral mode of the same shape as the vertical mode was considered, mode 1 in Figure 3. With the vertical analysis to hand, it was simple to determine the relationship between buckle length L and axial force P0 for a lateral coefficient of friction assumed equal to the friction coefficient for axial movements of the slipping lengths L s. Unfortunately, for equilibrium, this mode requires concentrated lateral forces of ~wL/2 at the ends of the buckle analogous to the concentrated vertical forces at the lift-off points in the vertical mode. On a rigid sea bed concentrated vertical forces are possible (and practical cases closely approach the rigid ideal (1°)) but it is not possible to generate a truly concentrated lateral force by friction. References 9, 11 and 12 which came to hand rather later deal with this point at length, but in the investigation described here the next step was to repeat the analysis using a different assumption which avoids the difficulty.
Mode3
1
L
j
Model,
w2AEL 6
P0 = 80.76 ~-S + 1.597 × 10-5 (EI)---------~
(15)
It is easy to show that equation 15 has a minimum at
[ = ( 1 . 6 8 5 6 x 106(EI)3) °'12' w2AE
M o d e c~o
(16)
Figure 3
-I ---_i
Lateral buckling modes
Pipeline buckling caused by axial loads Looking at the experimentally observed lateral modes once more, it was assumed that the family of decaying trigonometric curves were all initial imperfection generated variants of one fundamental constantamplitude periodic curve. In other words, it was assumed that an initially perfect pipe would buckle into an indefinite series of half waves as shown in Figure 3, mode oo. This assumption has the computational advantages that the nodes of the half wave pattern do not slide parallel to the axis of the pipe and, more importantly, can be made to satisfy lateral equilibrium. Looking at the consequences of this new assumption in detail, the linear differential equation governing the deflected shape is unchanged, equation 4, except that m is now qJw/EI, i.e. it is assumed that the lateral frictional force is fully mobilised everywhere. The displacement boundary conditions are unchanged but for the zero slope condition at x = +L/2 which is replaced by a shear force condition at the same location. Thus equation 5 still holds with the new value of m, while after careful consideration of signs the shear condition yields the length of the buckle from
nL
tan - - = 0 2
(17)
or as the lowest non-trivial root
nL = 2zr.
(18)
(20)
Po=P + 1.4545 x 10-SAE \( ~Wt2L6. EI]
Then
(19)
that is the reduction in axial force in the buckle equals the product of the axial stiffness and the extension round the curve. This leads to the following results for the infinite lateral buckling mode
p = 4ztEEI/L 2
linear differential equation is probably sufficiently accurate. References 9, 11 and 12 were obtained after completing this analysis. Martinet (9), as long ago as 1936, considered mode 1 as well as the vertical mode, observed the lack of lateral equilibrium at the ends of the buckle and predicted that mode 3 (with a smaller lack of equilibrium at its ends) was likely to develop as a result (Figure 3). He then analysed this higher mode, confirming that buckling would be initiated at a lower axial force for a given out-of-straightness than necessary for mode 1. Kerr (m (and in a condensed form (m) confirmed Martinet's work for modes 1 and 3 using a variational formulation and presented results for two antisymmetric modes, modes 2 and 4 (Figure 3). These are initiated at very similar axial force/initial imperfection combinations which are rather smaller than those needed to trigger mode 3. Martinet and Kerr's results are discussed later, but it is worth summarising their formulae for modes 1-4 in the notation of the present paper at this point. Taking the half wavelength of the most significant part of the buckle as L in each case (Figure 3), and using the constants of Table 1, the reduced axial force within the buckle is given by P = k 1EI/L2 (26)
AE~wL 5~ 1/2 ] --1.0]. Po=P+k3,wL[(1.O+k2-- ~S
Compatibility then requires that
Po--P= AE f~_/2 -1 '2dx L -L/2 2 y
(27) The maximum amplitude of the buckle relative to the original axis is fi = k, ~ w L4" Table 1
Constants Mode
~ 2 ~-E-
(22)
The maximum amplitude of the buckle fi = 4.4495 x 10-3 # w
E1
L4
Constants for lateral buckling modes
(21)
0.125
(28)
E1
For computational use, it is noted that this equation has a minimum at ~ , = ( 9 . 0 4 7 4 x 105 (EI)3 t ]
5
kt (Eqn. 26)
1 2 3 4
80.76 47t2 34.06 28.20
oo
47t~
k2 (Eqn. 27) 6.391 1.743 1.668 2.144
x x x x
10-5 10 -4 10 -4 10 -4
),.4545 x 10 -5
k3 (Eqn. 27) 0.5 1.0 1.294 1.608
k4 (Eqn. 28) 2.407 5.532 1.032 1.047
x x x x
10-3 10 -3 10 -2 10 -2
(Eqn. 21) 4.4495 x 10-3
(23)
RESULTS
and the maximum bending moment, at x -- 0, is )1~ = 0.05066#wL 2
(24)
while the maximum slope is
fi' = O.O1267fJwL3/EI.
(25)
Again, if y' <~ 0.1 in a particular case, the small slope
Figures 4-6 summarise the results of the vertical and lateral mode calculations for a typical pipeline for a variety of coefficients of friction, including the practical range 0.3 ~<~ ~<0.7. The pipe considered has an outside diameter of 650 mm, and a wall thickness of 15 mm giving a cross-sectional area of 299.2 cm 2 and a second
6
Journal of Constructional Steel Research: Vol. 1, No. 2: January 1981
moment of area 150900 cm 4. Its submerged weight (including concrete coating) has been taken as 3.8 kN/ m. The results have been presented in terms of the temperature rise necessary to generate the axial force in equilibrium with the buckle lengths and amplitudes shown, and the coefficient of linear thermal expansion, ct, has been taken as 11 x 10-~/°C for this purpose. The results were obtained from a small computer program using the following procedure which recognises that it is much easier to find the force (or temperature change) corresponding to a given buckle length L than vice versa.
Po using equation 21, T using equation 1 and the buckle amplitude fi using equation 23. (d) Lateral modes - modes 1-4 Repeat steps (c) (i) and (ii) using equations 27 and 28 for P0 and f. To establish the minimum for mode 1, values of L up to 2.0L may have to be used.
DISCUSSION It is apparent from Figures 4 and 5 that the lateral modes become possible at a smaller temperature change than the vertical mode. Thus unless lateral restraint is provided, for example by burying the line, the lateral modes will be dominant. This feature is confirmed by small-scale experiments and practical experience. An unburied line will snake laterally, while a buried line may burst out of the sea bed (or desert!). Once a buried line has lifted, an interaction with the lateral mode may occur, as the buckle itself now has no lateral restraint (4 = 0). Alternatively the buckle may roll or twist laterally. Figures 4 and 5 are otherwise very similar and it is useful to discuss the common features in terms of simplified load or temperature against length of buckle and amplitude curves, Figure 7. The first point is that in theory an ideal perfectly straight pipeline would not buckle. The pipe is in equilibrium for all values of axial load in an undeformed configuration and while the
(a) Vertical mode - high friction coefficient (i) Compute i, using equation 16. (ii) For a range of 20 values of L between 0.1L and 3L, compute P0 using equation 15, T using equation 1, and the buckle amplitude fi using equation 10. (b) Vertical mode - real friction coefficients For a range of friction coefficients 4, and 20 values of L between 0.1L and 3L, compute P0 using equation 9, T using equation 1, and the buckle amplitudefi using equation 10. (c) Lateral modes - infinite mode (i) For each of a range of friction coefficients 4, compute [, using equation 22. (ii) For each friction coefficient 4, and a range of 20 values of L between 0.ST, and 1.5T, compute 320
!
2.o
/
/
~06o
large
0"4
//
240 Temperature rise T ('c) 200
0'3
/
160
O,
120
8oFI
0'01
r J
40
¢=0 I
t
80
20 0.05 0.1 i I
0"2 i
0.5 i
1
100 120 1}.0 Buckle length L (rn) 2
5
10
20
I
i
I
i
Buckle
Figure 4 Resultsfor vertical buckling
160
omptitude
y (m)
180
200 5O I
Pipeline buckling caused by axial loads
7
140
120
100 Temperature rise T ('c) 80
60
40
20
I
I
4O
20
I
60
0.05 0.1
02
i
I
80
0'5
i
I
I
100 120 140 B u c k l e l e n g t h L (m)
i
2
5
i
i
10
I
I
160
180
200
2O i
0
Buckte a m p l i t u d e
Figure 5
m)
¢
:
0.5
)
Results.for lateral buckling - oo mode
150
. . . . .
130 TemperotUrerise T ('c)
(for
,
--
=~~t
110 .
.
.
.
.
.
.
i .
.
.
.
.
.
. I
Mode
oo
30 /(
0.2
I 04
I 0.6
I 08
-7-
J i i i 10 12 1-4 16 B u c k l e o m p l i t u d e } (m)
_,,
l
I
I
1.8
2.0
2.2
2.4
Figure 6 Comparisonof the lateral buckling modes
curve for the deformed shape (equations 9, 15 or 21) approaches the vertical axis asymptotically at high temperatures and small wavelengths, it never quite meets the axis. As Kerr u2) has remarked, this is a consequence of the assumption of fully mobilised friction even for vanishingly small displacements, and the situation may be contrasted with the conventional
pin-ended Euler column where the post-buckling equilibrium path actually crosses the initial equilibrium path on the axis. Supposing, next, something marginally less than initial perfection in the pipeline, it is clear that at some temperature T the imperfection will be enough to bridge the gap between the two equilibrium paths, bringing it
8
Journal of Constructional Steel Research: Vol. 1, No. 2: January 1981
c
"\ '
I~
I'
P,
X
',\
L
L
E
I
I
/
I
/
! I I
/ . . . . . . . ~'"/ ~ ~4" z
I /
I
I
!
J
I II I 11 I
/
/
I
i I
/ /
I
III
//
It
I I ~11
/
I
J
/
II -- ~ - : - - - ' f ' z " z Ill ~-'." z /
o I I
/ / /
/
Buckle
length
L
Buckle
amplitude
~,
Figure 7 Equilibrium paths and imperfection sensitivity-schematic
to the equilibrium position A. The part of the secondary equilibrium path from A to B is clearly unstable, and a dynamic snap will occur from A to C (chain dotted line) at a constant load P0 in the pipeline well away from the buckle, although the buckle itself lengthens and partially unloads. Because, on this argument, no snap can occur below the temperature T at B, T has been called the 'safe' temperature in the railway literature. These ideas can be presented with slightly more rigour by considering the temperature/buckle length curves for initially imperfect systems sketched in Figure 7 (dashed lines). For very small imperfections a large snap is indeed seen experimentally. As the initial out-of-straightness is increased the snap occurs at lower forces and less dramatically. Eventually, for large enough imperfections the snap is eliminated, to be replaced by a single-valued magnification of the initial bow. For any initial imperfection, the behaviour is ultimately asymptotic to the stable curve BC for the perfect system. In practical terms it may even be desirable to deliberately create initial out-of-straightness, to eliminate the possibility of a snap. This is simple enough for land-based lines which often include dog legs and expansion loops, but it is not so easy for submarine lines. Working from a lay barge, the pipe lengths are welded together with a 1 : 480 straightness tolerance and sent over the stinger. Deliberate kinks could overstress the line during laying. As the weather worsens, involuntary large radius curves may be induced at the sea bed by lay barge motion but it is not practical to lay the line to specified radii. Nor is it worth trying; Martinet (9) and small-scale experiments both show that
a large radius is ineffective in eliminating the snap or in changing its wavelength. On unloading the pipeline differences appear between the vertical and horizontal modes. A vertical buckle will trace the secondary path back down to B, and if the load is further reduced, drop back to the straight configuration although a small residual compression will arise at the site of the buckle because of frictional effects. A chain of horizontal buckles will initially try to trace the path down to B but the changes in wavelength coupled with the reversal of the lateral friction force will leave it at zero temperature with some residual out-of-straightness and a small tension in the line. On reloading the pipeline these imperfections must predispose the line to buckle laterally at a lower load than before by magnification of the imperfections. In the vertical mode, with small residual axial forces, the line is likely to buckle at the same location as before but not necessarily at a significantly reduced temperature as the straightness of the pipe is unimpaired. As might have been predicted, it is clear from Figures 4 and 5 that the value of the friction coefficient ~ only become particularly relevant to the results in the post-buckling regime. However, these results do not show any extreme sensitivity to the value of 4; since the value of ~ for a given stretch of sea bed is only known approximately, this is just as well. Turning now to a comparison of the various lateral modes, Figure 6 presents temperature/amplitude results for the five possibilities considered here for a single friction value, ~b= 0.5. These curves correspond to the unstable part of the equilibrium path, AB, in Figure 7, and if extended for large amplitudes would all rise, corresponding to BC in Figure 7. Thus each point
Pipeline buckling caused by axial loads shown represents a temperature/amplitude combination sure to initiate buckling in the mode shown for a suitable initial waveform. It is clear, as Kerr tH) has shown for rail tracks, that the antisymmetric modes 2 and 4 occur at very similar temperature/amplitude combinations, slightly smaller than those needed for mode 3. Mode I, occurring at higher temperatures, seems unlikely to be of practical interest. Mode oo, on the other hand lies below modes 2 and 4 for amplitudes less than 1.4 metres, and has a 'safe' temperature some 3°C lower. Indeed, there are grounds for suspecting that mode ~ lies below modes 2 and 4 at even larger amplitudes: it is recognised tH) that the analysis of modes 3 and 4, and to a lesser extent mode 2 contains an approximation which will lead to an overestimate of the amplitude. In these analyses it has been assumed that the axial force P is constant throughout the entire buckle, in spite of the inward movement from the adjacent unbuckled pipeline. On the contrary, the axial force in the inner half waves of modes 3 and 4 must be significantly less than in the outer, smaller, half waves because of axial friction and the true amplitude of the inner waves will be smaller as a result. If these arguments were to be extended to modes with 5, 6, 7 . . . . half waves, a situation would rapidly be reached where the inner, most significant, buckles were not receiving any energy at all from the inward sliding of unbuckled line, in other words the inner waves would approach the conditions assumed in the analysis of mode ~. These arguments lend weight to the idea that practical buckles are imperfection generated variants of mode ~ . These considerations suggest that mode oo might be a very useful lower bound for design purposes. It is certain that most pipelines would buckle in modes approximating to modes 2, 3, 4. It is not certain that higher modes closer to oo will never arise if bad luck produces the wrong initial imperfection pattern. Thus the 'safe' temperature of mode ~ , and the associated amplitude/wavelength combinations for temperatures and axial forces above the 'safe' level are proposed as providing a conservative estimate of permissible imperfection sizes and wavelengths for design use with appropriate safety factors. Finally, the following limitations of the analyses presented in this paper should be borne in mind. (a) Only perfect systems have been rigorously examined. In the vertical mode no account has been taken of the initial out-of-straightness. It is not easy to see how to include this factor, short of an approximate numerical (e.g. finite element) approach. The classical column (Fourier) analysis of initial lack of straightness becomes difficult here because the buckled length changes progressively as the load increases. It is some comfort to know that the magnitude and nature of the initial imper fections in pipelines as laid are also unknown, and
9
extremely difficult to assess, so that an improved analysis would not of itself be very helpful. (b) Perfect elasticity and small slopes have been assumed. The analyses presented assume that the full elastic modulus of the pipe is available to resist bending. Once plasticity occurs the analysis loses its validity: yielding will obviously leave its mark on the pipe on unloading and act as an imperfection on subsequent reloading. Of course, most linepipe is competent to develop a plastic hinge without effect on its serviceability. The analyses also use the linear differential equation for bending which is strictly true only for small slopes, conventionally less than 0.1 radian. However, more exact elliptic integral solutions have been obtained t13,14} for cantilevers and pin-ended columns. In the cantilever, for an axial load 1.5 per cent greater than that predicted by the linear equation, the end slope is 0.35 radian. Similarly, in the pin-ended column, for an axial load 1.2 per cent greater than the Euler load, the greatest slope is 0.31 radian. These results suggest that the 0.1 radian limit on 'small' slopes is rather conservative; cut-off lines corresponding to this limit have been marked on Figures 4 and 5 as a guide, and true loads beyond this limit may be rather higher than shown.
CONCLUSIONS Two potential buckling mechanisms in pipelines subjected to axial compression have been identified and analysed. It is found that horizontal snaking modes occur at a lower axial load than the vertical mode, and a horizontal mode is therefore dominant unless lateral restraint is provided by trenching when an interactive buckling mode becomes possible. Full account has been taken of friction between the pipe and the ground in the vertical mode, relying on two independent but formally identical references on the buckling of rail tracks and crane rails. An analysis of a lateral mode involving an infinite sequence of half waves with fully developed lateral friction has been presented and compared with earlier work on isolated buckles over a shorter length which involve longitudinal as well as lateral sliding. It is concluded that the infinite mode gives a lower bound to permissible imperfection amplitude/wavelength combinations which may be useful for design purposes. The effects of imperfections, unloading and reloading are discussed. The theoretical solutions are illustrated by numerical results for a typical pipeline.
A CKNO WLEDGEMENTS The author wishes to record his gratitude to Jack Ells, Chris Lawlor and David Walker of the Engineering Department of BP Trading Lid, and Chris Burgoyne of the Civil Engineering Department, lmperial College, for the stimulating discussions which led to this paper.
10
Journal of Constructional Steel Research: Vol. 1, No. 2: January 1981
However, the opinions expressed (and any mistakes made) are the author's alone.
7
8
ings of the American Society of Civil Engineers, Journal of the Structural Division, January 1972, ST l, 360o61.
REFERENCES 9 1
WILHOIT, J. C., JR. and MERWIN, J. E. 'Pipe stresses induced in laying offshore pipelines.' Transactions of the American
Society of Mechanical Engineers, Journal of Engineering for Industry, 89, 1967, 37-43. PALMER, A. C., HUTCHINSON, G. and ELLS, J. W. 'Configuration of submarine pipelines during laying operations.' Transactions of the American Society of Mechanical Engineers, Journal of Engineering for Industry, 96, 1974, 1112-18. 3 HOBBS, R. E. 'A beam bending problem with a free boundary.' Computers and Structures, 10, December 1979, 915-20. 4 HOBBS, R. E. 'The lifting of pipelines for repair or modification.' Proceedings of the Institution of Civil Engineers, Part 2, 67, December 1979, 1003-13. 5 PALMER, A. C. and MARTIN, J. H. 'Buckle propagation in submarine pipelines.' Nature, 254, March 1975, 46-8. 6 KERR, A. D. 'On the stability of the railroad track in the vertical plane.' RailInternational, 5, February 1974, 132-42.
MAREK, P. J. and DANIELS, J. H. 'Behaviour of continuous crane rails.' Proceedings of the American Society of Civil Engineers, Journal of the Structural Division, April 1971, ST4, 1081-95. GRANSTR(}M, A. Discussion on Reference 7 above. Proceed-
10
2
ll 12
13
14
MARTINET, A. 'Flambement des voies sans joints sur ballast et rails de grande longueur.' (Buckling of tracks without joints on ballast and long rails, in French.) Revue Gdn~rale des Chemins deFer~ 55/2, 1936, 212-30. HOBBS, R. E. 'The effect of soil modulus on pipeline stresses.' (Awaiting publication.) KERR, A. D. 'Analysis of thermal track buckling in the lateral plane.'Acta Mechanica, 30, 1978, 17-50. KERR, A. D. 'On thermal buckling of straight railroad tracks and the effect of track length on the track response.' Rail International, 9, September 1979, 759-68. TIMOSHENKO, S. P. and GERE, J. M. Theory of Elastic Stability, 2nd edition. McGraw-Hill Kogakusha, New York, 1961, 76-81. HORNE, M. R. and MERCHANT, W. The Stability of Frames. Pergamon, Oxford, 1965, 10-12.
Contributions discussing this paper should be received by the Editor before 1 May 1981.