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Submarine pipeline buckling—imperfection studies
Thin-Walled Structures 4 (1986) 295-323
Submarine Pipeline Buckling Imperfection Studies
Neil T a y l o r a n d A i k B e n G a n Department of Civil Engineering, SheffieldCity Polytechnic,Pond Street, SheffieldS1 1WB, UK
ABSTRACT In-service buckling of submarine pipelines can occur due to the institution of axial compressive forces caused by the constrained expansions set up by thermal and internal pressure actions. Previous attempts at modelling the appropriate behaviour have been based on idealised or perfect pipelines; further, such analyses have also employed fully mobilised friction forces. Herein presented is a set of analyses which incorporate structural imperfections and deformation-dependent axial friction resistance. Not only do these features enable a more rational interpretation of submarine pipeline buckling behaviour to be established, but, in addition, an inherent limitation existing in the previous mathematical modelling of the vertical buckling mode is elucidated and o vercome.
NOTATION fA q r t u Ua uf us
Axial friction parameter. Submerged weight of pipeline per unit length. External radius of pipe. Wall thickness of pipe. Axial deformation of the pipe. Tensile extension of the buckle. Compressive flexural end shortening of the buckle. Resultant longitudinal movement at buckle/slip length interface.
295 Thin-Walled Structures 0263-8231/86/$03.50 © Elsevier Applied Science Publishers Ltd, England, 1986. Printed in Great Britain
296
v
Vm Vo Votll w wm
Wo 141om X
A E I L, LI Lo
L~ M Mm
P Pa Po T Tm Train
Tp Ty V o~ O'm O'y
Neil Taylor, Aik Ben Gan
Fully mobilised axial displacement. Vertical deformation of the pipe. Maximum vertical amplitude of the buckled pipe. Vertical deformation of the imperfection topology. Maximum vertical amplitude of the imperfection topology. Lateral deformation of the pipe. Maximum lateral amplitude of the buckled pipe. Lateral deformation of the imperfection topology. Maximum lateral amplitude of the imperfection topology. Spatial co-ordinate Cross-sectional area. Young's modulus. Second moment of area of cross-section. Buckle lengths. Buckle length of the imperfection topology. Slip length. Bending moment. Maximum bending moment. Buckling force. Axial force component. Prebuckling axial force. Temperature rise. Maximum temperature rise. Minimum safe temperature rise. Permissible temperature rise. Temperature rise at first yield. Total potential energy. Coefficient of linear thermal expansion. Maximum compressive stress. Yield stress. Axial friction coefficient. Lateral friction coefficient.
1 INTRODUCTION The circumstances concerning the means by which the in-service buckling of submarine pipelines can occur have been discussed at length elsewhere. ;-4 Analyses have been primarily oriented about thermal action
Submarine pipeline buckling--imperfection studies
297
with oil and gas temperatures potentially ranging up to 100°C above that of the water environment. That is, a uniform temperature increase, T, in a perfectly straight submarine pipeline will create an axial compression force due to constrained thermal expansion. Within the elastic range of the pipeline response, this force can be represented by Po = A E a T
(1)
where A E is the axial rigidity of the pipe and ot is the respective coefficient of linear thermal expansion. Should buckling occur, part of the constrained thermal expansion is released in a buckled region which, taken together with the friction resistance of the sea bed/pipeline interface, results in a reduction in the axial compression to some buckling force P. Studies to date have been based on idealised buckling phenomena with the simplifying non-conservative assumption that the friction forces are fully mobilised throughout. The five established lateral buckling modes which relate to snaking movements across the sea bed are illustrated in Fig. 1. t'3 Axial and lateral deformations are denoted by u and w respectively. The appropriate buckling lengths are denoted by L and L 1, whilst Ls represents the slip length. Axial friction resistance is generated through the slip length whilst lateral friction resistance occurs through the buckling lengths. The vertical buckling mode which involves part of the pipeline buckling out of the sea bed can also be represented by Fig. l(a) except that w is now replaced by v, the appropriate vertical deformation. It can be safely assumed that submarine pipelines suffer structural imperfections under field conditions. Initial lack-of-straightness will occur during laying operations and sea bed conditions will also generally preclude a perfect lie. The present study concentrates attention on the vertical mode and lateral modes 1 and 2. With regard to the five lateral modes depicted in Fig. 1, it is considered that modes 1 and 2 are more susceptible than the remaining lateral modes to the most basic initially deformed pipeline topologies, these imperfections taking the form of fundamental symmetric and skew-symmetric modal deviations from the idealised lie. Further support for this choice of modes comes from established idealised analyses' stress trends. 2 That is, values of the minimum safe temperature rise, Tmi,, for all six identified submarine pipeline buckling modes, employing the typical parametric values 2 given in Table 1, are given in Table 2 together with the corresponding buckling amplitudes, Vmor Win,and maximum compressive stresses, or,,. It can be
Neil Taylor, Aik Ben Gan
298
l_ibU
Ls
L/2
L/2
Ls
,',o) Mode 1
Ls (b)
<
Ls (c)
<
Ls
L
L
Ls
Mode 2
~
L1
-'~<" L/2
~
L/2
~
L1
~'
L
~
L1
Ls
;
Mode 3
~
L1
~
L
~
~
Ls
>
(d) M o d e 4
Fig. 1. Lateral buckling mode topologies.
seen that on the basis of Train, lateral modes 3 and 4 are more critical than lateral modes 1 and 2. This ordering is indicated parenthetically adjacent to the respective Tmi. values. However, the corresponding stress levels are consistent with an underlying and alternative interpretation with the vertical m o d e and lateral mode 1 generating the highest stress levels. That is, accepting that imperfections will be present in practice, lateral mode 1
Submarine pipeline buckling---imperfection studies
299
TABLE 1
Pipeline Parameters Parameter
Symbol
External radius Wall thickness Sectional area Young's modulus Second moment Self-weight Yield stress Thermal coefficient Axial coefficient Lateral coefficient
r t A E I q O-y a ~bAIU6 CbL
Value 325 mm 15 mm 29 920 mm 2 206 kN/mm 2 1.509 × 109 mm 4 3.8 kN/m 448 N/ram 2 11 × 10-6/°C 0.7 at 5 mm 1.0
TABLE 2
Idealised/Fully Mobilised Analyses Mode Vertical Lateral 1 Lateral 2 Lateral 3 Lateral 4 Lateral oo
Train(°C)
vm or wm I Train (m)
crm I Tmin (N/mm 2)
65.3 (5) 63.6 (4) 61.6 (3) 60-6 (2) 60.5 (1) 77.5 (6)
2.3 2-4 1.4 1.9 1.3 0.5
597 606 495 547 479 261
is n o t a b l e for g e n e r a t i n g the highest stress trend o f all the lateral m o d e s . T a k e n in c o n j u n c t i o n with the basic nature o f the physical i m p e r f e c t i o n c o n c e r n e d , it is c o n s i d e r e d that, in practical terms, lateral m o d e 1 is the m o s t significant s y m m e t r i c lateral m o d e with respect to i m p e r f e c t i o n c o n s i d e r a t i o n s . Similar reasoning suggests that lateral m o d e 2 is the m o s t significant s k e w - s y m m e t r i c m o d e , lateral m o d e s 3 and 4 being, in e n g i n e e r i n g terms, s u b o r d i n a t e forms o f lateral m o d e s 1 and 2, respectively. All f o u r m o d e s can be c o n s i d e r e d to be localised versions o f lateral m o d e ~ , this f e a t u r e being r e i n f o r c e d by the p r e s e n c e o f a particular i m p e r f e c t i o n in a particular locality. In all t h r e e cases studied, variable or d e f o r m a t i o n - d e p e n d e n t axial
300
Neil Taylor, Aik Ben Gan
resistance forces are included in the respective analyses; this feature is of particular importance with respect to the vertical mode study as the established modelling L2 is shown to be invalid at small values of vertical deformation. The refined friction force modelling uses a recently established axial friction response lOCUS. 4 The analyses presented should provide for a more logical interpretation of the submarine pipeline buckling problem and a design chart, based upon the onset of first yield, is provided for a typical pipeline parameter set. The studies employ small deformation theory and material behaviour is taken to be elastic; gross sectional distortion as associated with laying operations ~ is not a factor here.
2 G E O T E C H N I C A L FACTORS Recent geotechnical experimentation 4 with respect to North Sea conditions 6 has provided information relating to the nature of the deformation-dependent friction force behaviour with regard to typical submarine pipeline parameters. Based upon these findings, a refined axial
.,/~Futly Mobilised Locus 1.0 ExperimentQL Locus & 0.8 ¸ fA
-
Full Scale Design Curve
041
0.6-
fA/~A = 1 - e - 25u/u¢
0.2
fAq=
.- = u
q = Submerged serf-weight per unit length of pipe Conventicn
0
0.'i 0.2 013 01~ o'.s o'-G 0'.7 0~8 019 u/u¢
Fig. 2. Generalised axial friction characteristics.
~.0
Submarine pipeline buckling--imperfection studies
301
friction force-deformation locus is depicted in Fig. 2. The appropriate fully mobilised axial friction coefficient is given by qbA; the movement corresponding to the attainment of full mobilisation is denoted by u~. The friction force parameter fA, 0--
/
u,~ = 5 mm
J
(2)
Consequently, the design curve given in Fig. 2 employs jk/~bA = 1 - e-ZSu/"~
(3)
noting the appropriate convention given in the Figure. This expression gives good agreement with the designated non-dimensionalised experimental curve and, in addition, accords with the appropriate fully mobilised asymptote with fA/CbA ~ 1 at u/u, = 1. It is to be noted that eqn (3) offers superior modelling characteristics to its precursor given elsewhere. 4 Finally, accepting that lateral resistance is taken to be fully mobilised, ~-3 the value for ~bL,the fully mobilised lateral friction coefficient, is taken to b e 1.0. 4
3 VERTICAL MODE ANALYSIS The essential features of the vertical mode are shown in Figs 3(a) and (b). The former figure details the topology whilst the latter depicts the axial force distribution within the pipe. The submerged self-weight of the pipe per unit length is denoted by q and Vmrepresents the maximum amplitude of the buckled pipe. Regarding the imperfection parameters, Voand Yore represent the respective vertical deformation and maximum amplitude whilst the appropriate imperfection buckling length parameter is denoted by Lo. It is to be noted that Vom/Lo forms the essential imperfection parameter. This parameter is unique in as much as Yore and Lo are
302
Neil Taylor, Aik Ben Gan
q &
I
&
&
&
&
&
&
&
&
I
vm -Yore
7,.z~,Tr~bA1q~1:/e22~u;)-Tf " ....
~""-- ~"'-- -
I~---- Ls-°°
~
#
~
/
~
Lo/2
LI2
'
/
I
#
.
-
,
~x LO/2
~
LI2
~
~Aq(1-e2~ .
14
u~7.#,. ~.
..--..~ ..-.--.-~. . . .
~l~
Ls--°° _ _ _ _ ~
(o) Topology
/ #
--~.Pa
PO =~'Aq(1.e25usILk~)L12
IZai
(b) Axial Force Distribution
Fig. 3. Details of vertical mode. dependent. The linearised vertical deflection equation, appertaining to idealised pipelines, 1.3is given by q( v = ~
1+
n2 L 2 8
n~x,2
cosnx ) - L/2 <-x <- L/2 cosnL/2
(4)
where n L = 8.986 8 and E1 is the flexural rigidity of the pipe. The m a x i m u m amplitude of the buckle, at x -- 0, is Klq
aL4
Vrn - E-7~n - 2.407x 10 ~'-I/z
(5)
where L is the unprescribed buckling length and Kt = (1 + n 2L:/8 - 1 / cos n L / 2 ) = 15.698 465, such that substituting eqn (5) into eqn (4) yields the transverse d e f e c t i o n expression Vm( n2L = n=x2 v = ~ 1+ T 2
cosnx ) cosnL/2
(6)
From the foregoing, the imperfection deflection expression can be assumed to be
Vo
= yore{ n2oL~-o nox 2 Ki \ 1 + ~ 2
cosnox ) _ L o / 2 < _ x < L o / 2 cosnoL,>/2
(7)
Submarine pipeline buckling---imperfection studies
303
where noLo = 8.986 8 and Yore, the maximum imperfection amplitude, noting eqn (5), takes the form Vom = 2"407 X 10-3 qL4 E1
(8)
This equation displays the previously denoted dependence of Yoreand Lo. The deflection expressions eqns (7) and (8) relate, in practical terms, to a vertical deviation from the intended or idealised lie; this will act as a potential 'pop-up' trigger. Employing symmetry, the total potential energy relating to the deformed state is given by V ~
i* Lo/2 El~2 ( d 2 v / d x 2 - d 2 v o / d x 2 ) 2 d x || ,,Jo
q'- f L/2 El~2 (d 2v/dx 2 Lo/2 + f L°/2 ",10 -- f L°/2 "sO -- ; L/2
q(v-Vo)+ f
d 2
vo/dx2)2dx
L/2 Lo/2
q(v-vo)dx
P/2[(dv/dx) 2 - (dvo/dx) 2]dx PI2[(dv/dx) 2 - (dvo/dx)2]dx = 0
Lo/2
(9)
the corresponding equilibrium state being given by d V / d v m = 0. Noting that for 0-< x <_ Lo/2, the derivatives of initial curvature, slope and deflection with respect to Vm are null, as are the actual values of initial curvature, slope and deflection for Lo/2 <-x <-L/2, then applying the statics criterion, 45.36260Eln3vm/K]
-
2 2 RlElvomnno/K1
+ 30.241 74q/(nKl) - 75"604 3 4 P n v m / K ]
=
0
(1o)
where F Rt = 4.603 14 ]sin(4.4934Lo/L)
+ 2.301 57
sin4.493 4(1 + Lo/L) (L/Lo + 1)
sin 4.493 4(l - Lo/L)
(it)
304
Neil Taylor, Aik Ben Gan
Simplification of eqn (10), noting eqns (5) and (8), yields the buckling force
= + 7+ +,
]
(12)
Bending m o m e n t is afforded by M = El(d2v/dx 2
-
d2vo/dx 2)
(13)
which gives the maximum moment, at x = 0, upon substitution from eqns (6) and (7), to be Mm = - 0.06938q(L 2- L~)
(14)
the negative sign correctly indicating flexural compression to be acting on the lower part of the pipeline section. The maximum compressive stress O-minduced in the pipe is thereby obtained from
p/A +IM~r/I I
O" m :
(15)
where r is the external radius of the pipe. Having established the relationships between v, L, Mm, or,, and the buckling force P, it is now necessary to determine the dependence of P u p o n T, the temperature rise. This is achieved by considering the slip length characteristics. It is proposed to employ the deformationd e p e n d e n t axial friction force modelling discussed previously and detailed in Figs 2 and 3. As established elsewhere, v the slip length field equation is given by d2u
AE-~
= --]Aq
L/2~x<-L/2+L,
(16)
T h e associated slip length boundary conditions take the form lim [u, du/dx] = 0
X~m
(17)
and
dx
I
-L/2
(Po - P) - Pa AE
(18)
Submarine pipeline buckling--imperfection studies
305
where force Pa, detailed in Fig. 3, represents the axial force c o m p o n e n t frictionally induced in the pipe by the vertical shear reaction q L / 2 at the buckle length/slip length interface. Equation (17) can be considered to be a regularity condition which assists in precluding the prejudicial requirem e n t of assuming some function u = f ( x ) . C o m b i n i n g eqns (3) and (16) affords the non-linear slip length field equation d2 A E ~ = - ~bAq(1 -- e 25u/"~) dx
(19)
noting that the change of sign of the exponent with respect to that given in eqn (3) is due to u and fA being co-oriented within the slip length regionwrecall that Figs 1 and 3 employ a different convention from Fig. 2. Employing the identity 2d 2u / d x : = d [ ( d u / d x ) 2 ] / d u in conjunction with eqns (17) and (19) gives dx
[ AE
5
u
/]
(20)
Equating this expression with the boundary condition given in eqn (18), and with u] L/2 = us yields (Po - P) =
24)AqAE
5
- us
L
+ 4~aq ~- ( 1 - e 25"S/"*)
(21)
It is necessary to set up a matching compatibility expression at the interface of the slip and buckling lengths, x = L / 2 . This can be expressed in the form us = ua + uf where Ua and uf denote the tensile extension and compressive flexural end shortening of the buckle length L / 2 . Incorporating the presence of the imperfection v o / L o gives
us -
2AE
2
dx -
which yields, given eqns (6) and (7), 2
(Po- P)L us 2AE
7.9883 x l0 -6 ~q
(Z 7 -
Lo7)
(23)
306
Neil Taylor, Aik Ben Gan 140-
Idealised
-
Fulty Mobihsed
120
v 100 F-
Snap
2 BO D
E E
6O
i
z,O o
°1o! ,,05 5
20
~P'r
~
i
. . . . . 4
~ 5
,
, 7
Buckle Amphtude Vm(m)
C' '
d0
75
90
100 '
18 5
1~0 '
115 '
120 '
,2's
Buckle Le,ngth L ( m )
Fig. 4. Thermalaction characteristics--vertical mode,
Solutions for (Po - P) and us are thereby obtained in terms of discrete values of L employing a computerised non-linear iterative algorithm involving eqns (21) and (23). The numerical evaluations have been carried out, noting eqn (8), for a series of eleven imperfection ratios Vom/Lo from 0.003 through to 0.010, the remaining pipeline parameters 2 employed being as denoted in Table 1. Results for (Po - P) are then substituted into eqns (1), (5), (11) and (12) to produce the loci depicted in Figs 4 and 5(a). Only those loci for Vom/Lo of 0.003, 0.007 and 0.010 are shown for reasons of clarity. The loci in Fig. 5(b) are obtained by substituting the respective values of P into eqn (15), given eqn (14).
Submarine pipeline buckling--imperfection studies
307
ldeolised / Fully Mobilised
~6 v t3_
,~ Snap
c~ r-5
2~2
sFl~
;.~.
;I,I ° . . . . 1
I, 2
~lopeo.~ ,
,
,
,
,
l
3 4 5 Buckle Amplitude vm or w m (m)
i ~--7
(o) Buckling Force
~-'~E 800 l ~ I d e Q l i s e d / F u l l y k4obilised
f(/,
. . . . 1
2
. . . . . 3
4
5
6
; 7
Buckle Amplitude vm or wm (m) (b) Ma×imum Compressive Stress
Fig. 5. Dependent parameter characteristics--vertical mode and lateral mode 1.
308
Neil Taylor, Aik Ben Gan
It can be shown that the vertical mode modelling in the established submarine pipeline t'2 and related crane rail u'~2 studies is invalid at small values of vertical deformation. Using the notation of the present paper, eqn (19) of Ref. 11 can be rewritten "" ~4~Aq~AEL7 - P~, ( P o - p)Z = 1.5977 × lu - ( E l f
(24)
this being the fully mobilised equivalent to the expression obtained if eqn (23) were to be substituted into eqn (21). Clearly, noting Fig. 3(b), both approaches require (Po - P) -> P~
(25)
For the fully mobilised analysis (26)
P~ = eflAqL/2
w h e r e u p o n , substituting eqns (25) and (26) into eqn (24), there is the requirement L5->3.1296×10
(27)
That is, for the pipeline parameters given previously L -> 39.014 m, such that, given eqn (5), Vm-----68" 17 ram. These restrictions on the established fully mobilised studies t,2 have not been previously reported. In the present study there is no such restriction upon eqns (2 I) and (23) due to the variable, deformation-dependent axial friction force modelling; that is, eqn (25) is valid for all values of Po, P and P,.
4 LATERAL MODE 1 ANALYSIS T h e essential features of mode 1 are shown in Fig. 6(a) and (b) which depict the respective topology and axial force distribution. The m a x i m u m amplitude of the buckled pipe is denoted by Win. With regard to the imperfection parameters, Wo and Worn represent the respective lateral deformation and maximum amplitude. The analysis for the buckling
Submarine pipeline buckling--imperfection studies
I
~
•
~
~
{
wm- Worn ~ q ( 1- e 25u/u¢ ) ,~ . . . .
.,e-...- ~
~'
i,
-
" ~.-tot?
~ L/2 (a)
{
I
-.
- ~ ~
Ls-e°
¢
309
J
j,x _ _~o/~
"r-
L/2
~ q ( 1 - e 25 u/u¢) - " - " P '--'P . . . .
"
LS-~
Topology
<__ i!oi (b) Axial Force Distribution
Fig. 6. Detailsof lateralmode 1. region is identical to that of the vertical mode. Equations (4) through (15) remain valid except that v, together with its related components, and q are now replaced by w and cbLq respectively, noting that +Lq is the appropriate fully mobilised lateral resistance force per unit length acting against the lateral buckling mechanism. The physical imperfection corresponding to eqns (7) and (8) therefore takes the form of a basic symmetric wave. Regarding the slip length region, eqns (16) through (20) remain valid except for eqn (18) where the respective boundary condition is replaced by dUldxL/2 _ __(P°AEP)
(28)
This results in eqn (21) being replaced by ( P o - P) =
[
2c~AqAE
(e2--,..l 5
(29)
us
The matching compatibility condition at the ends of the buckle remains as previously, subject to the appropriate replacement of v. That is, eqns (6), (7) and (22), so modified, generate us -
(Po- P) L _ 7.988 3 x 10-6 - 2AE
(L 7- L~)
(30)
Neil Taylor, Aik Ben Gan
310
140,
=
Idealised / Fully Mobilised
120
~" 100
~ so
_
_
S~p
_
Z0
20 ¸
I
1
2
I
3
I
I
z,
I
I
5
I
1
6
I
7
Buckle Amplitude w m (m)
6 6~ 7~
9'o
~6o t&
~io
~is
~o
~
Buckle Length L (m)
Fig. 7. Thermal action characteristics--lateral mode 1.
this expression being analogous to that given in eqn (23) with q replaced
by q~Lq. The solution procedure for (Po - P) and us is as previously, eqns (29) and (30) being solved iteratively in terms of discrete values of L. It is to be noted that since in the present study ~bL = 1-0, 4 the revised equations are actually the same as eqns (4) through (15) apart from v being replaced by w. Numerical evaluations have been undertaken, recalling eqn (8) and Table 1, for the previously denoted range of 11 imperfection ratios (herein Wom/Lo). Results for (Po - P) are then substituted into eqns (1), (5), (11) and (12) to produce the w-loci depicted in Fig. 7. The loci in Figs
Submarine pipeline buckling---imperfectionstudies
311
5(a) and (b), noting eqns (14) and (15), are also valid for mode 1 since 4~L =
1"0.
5 LATERAL MODE 2 ANALYSIS The topology and axial force distribution depicted in Figs 8(a) and (b) respectively represent the essential features of mode 2. The linearised lateral deflection equation, obtained from related idealised rail track studies, 13is given by
(~-~-)
qbhqL4
(-~ + 2LX( (31)
O<-x<-L
The maximum amplitude of the buckle, at x = 0.346 4L, is
Kzd~eqL4 Wm
=
x
= 5-532
16";7"4 E1
chLqL4
10 -3
(32)
E1
where L is the unprescribed buckling length and K2 : 8.621 149 6. Substituting eqn (32) into eqn (31) yields the lateral deflection expression w :
w'( --T-m [l-c°s 1 - L('~)) +'rrsin ] K (-~f-) 2 + 2L~xI
1'
t
t
t
I
Wm .
.
.
.
_~q<~-e2Su'u¢/
_
M L (a)
_ - - /
,,I,
"i'
~ ~
Lo
,I,
I
~.
" 9 ' o m ~
Lo Ls--C°
$
, ~w-',~o
(33)
" -..
j L "1
'~qIl-e25u~u~z
.
.
.
.
Ls-C°
Topotogy
(b) Axial Force Distribution
Fig. 8. Details of lateral mode 2.
~""'~-
_
312
Neil Taylor, Aik Ben Gan
F r o m the foregoing, the imperfection deflection expression can be assumed to be w,, = K-~ ()<--x < - Lo
(34)
where Wom represents the maximum imperfection amplitude which, noting eqn (32), takes the form & aL 4.
Wom= 5"532 x 10 3~t.-,-~-
(35)
El
The physical imperfection corresponding to eqns (34) and (35) therefore takes the form of a basic skew-symmetric wave. Employing skewsymmetry, the energy formulation is similar to that of the vertical m o d e analysis except that the regions under consideration for the deformed pipeline and imperfection topology are 0<-x-< L and 0<-x<- Lo respectively. Accordingly, eqn (9) can again be employed with Lo/2 and L / 2 being replaced by Lo and L respectively whilst 0bcq and w appropriately supersede q and v. Applying the statics criterion d V / d w m = 0, then eqn (9), modified as d e n o t e d above, affords (24",'7"4+ 87"r6) Elwm K~L 3
+
87r3ElwomR2 K~L~,L
~q(77"2) PWrn{ . 1+-~- - ~ 2 L k l O w 2 +
10"B'4x~
3 ] =0
(3~)
where R2 = sin (2rrLo/L)
(L/Lo)(rr2 + L/Lo) ] 1 - (L/Lo) 2 + 2rrLo/L
+ rc[ l -cos(2zrLo/ L ) ]
(L)
(37)
Submarine pipeline buckling---imperfection studies
313
Simplification of eqn (36), given eqns (32) and (35), yields the buckling force 4rr 2 E 1
_ 3 [
R2
(38)
Bending moment is given by M = El(d2w/dx
2-
d2wo/dx2)
(39)
which generates the maximum moment from eqns (33) and (44), then Mm = - 0.108 8 4 t b k q ( L
Mm
at x = 0.299L; substituting (40)
2 - L~)
The maximum compressive stress induced in the pipe is thereby obtained employing eqn (15) subject to the substitution of the appropriate parameters. Having established the essential relationships appertaining to the buckling region, those concerning the slip length are now determined. The formulation is similar to that given in the vertical mode analysis except that the slip length region is now restricted by L -< x -< L + Ls. Therefore, eqns (16) through (20) can be employed with L / 2 in eqn (16) being replaced by L. Equations (28) and (29) remain valid subject to the buckle length/slip length interface for lateral mode 2 being specified at L with respect to the former equation. The matching compatibility condition at the end of the buckle length takes the form U
=
t/s
L
--
AE
2
o
(41) which yields, noting equations (33) and (34) us -
AE
8"715 ×
10 -5
( q)2 - -
(L 7 - L 7)
(42)
The solution procedure for (Po - P) and Usis as previously; eqns (29) and (42) are solved iteratively in terms of discrete values of L. Numerical
314
Neil Taylor, Aik Ben Gan
~4i9
----
Ideolised / Fud~ Mobiiised
120
~ ~oo rY
SnGp
& E
6O
40
2°-I I"o I°1 .& 0.5
10
1.5
20
25
30
35
Buckle Amplitude wm (m)
Buckle Length L (m)
Fig. 9. Thermal action characteristics--lateralmode 2. evaluations have been carried out, noting eqn (35) and Table 1, for the previously denoted range of 11 imperfection ratios (Wo~,/Lo). Results for (Po - P) are then substituted into eqns (1), (32), (37) and (38) to produce the loci depicted in Figs 9 and 10(a). The loci in Fig. 10(b) are obtained by substituting the respective values of P into eqn (t5), noting eqn (40).
6 DISCUSSION The imperfection ratios explicitly considered are denoted in Tables 3 and 4 together with the appropriate permissible temperature rise values,
Submarine pipeline buckling---4mperfection studies
315
8J
~~Fullyldeolised / Fully MobilisShOp ed~ v
G.
~4 ~2
~:o E I
z 600/~ ,,
~ Max. Slope 0.1 I
0,5
l
I
1.0
I
I
I
I
I
I
I
1.5 2.0 2.5 Buckle Amplitude wm (m) (O) Buckling Force
Ideotised/Fu[[yMobilised
0.5
l.O
I
3,0
~
3.5
.-..0.003
.
1.5 2.0 2.5 Buckte Amplitude wm (m) (b) Maximum CompressiveStress
.
.
.
3-0
.
3.5
Fig. 10. Dependent parametercharacteristics--lateralmode 2.
based upon the onset of first yield, and the corresponding buckling amplitudes. It is to be noted that the squash load for the pipeline parameters considered, denoted in Table 1, is 13.4 MN, which corresponds to a temperature rise of 197.7°C. Perusal of Tables 3 and 4 therefore shows that buckling action will occur in practice. Prior to
316
Neil Taylor, A i k Ben Gan
TABLE 3 Maximum Temperature Rise, T,-. Mode
Vertical
Lateral 1
Lateral 2
Tm (°C) Vmor wm (m) vmorwm(m)
79.6 0.30 5"95
+h
78.9 0.30 6"21
+
69-7 (I-25 2"86
Tm (°C) 0.{~3 5 Vmor wm (m) vmorwm(m)
72.9 0"38 4.79
~1
72.1 0"37 5.03
$
64.1 (1"32 2.21
T,, (°C) Vmor Wm(m) Vmor Wr. (m)
67-8 0.48 3-83
~1
66-9 0-47 4.05
~1
%0-0 "0-42 "1.64
Tm (°C) 0"004 5 vm or W m (m) vmor wm (m)
63"9 0-60 2-99
~1
62"8 0.59 3.20
~
"57" 1 "(/.60 "1.02
Tm (°C) vm or Wm(m) vm or wm (m)
%0-9 ~().77 "2.18
~1
"59-8 '~0.74 ~2-41
~1
NA NA NA
~
NA NA NA
0.003
.3 :e
.3 0.004 O
© o O
E
0.005
Tm (°C) 0"005 5 Vmor Wm (m) VmOrWm(m)
NA NA NA
~57"5 ~l.06 "1'43
"Non-crucial values followed by yield Tm 4= T~,--refer to Table 4. h $ Indicates snap.
further consideration of Tables 3 and 4, and the ensuing implications, it is pertinent to assess the behaviour associated with the three imperfection ratios identified in detail previously--that is, with Vom/Lo and Wom/Lo taking the values 0.003, 0.007 and 0.010. With regard to the respective temperature rise/buckling amplitude loci given in Figs 4, 7 and 9, it can be seen that only the relatively small imperfection ratio case (0.003) displays a maximum temperature rise, Tm, together with the associated snap buckling phenomenon. The remaining two cases (0-007 and 0.010) generate stable post-buckling paths. In all three modes, the respective loci are of converging form as is to be anticipated; indeed, the modes afford characteristics which are in general agreement with those reported in the related field of rail track buckling, 14-16 although in the present study it is to be recalled that the buckling length
317
Submarine pipeline buckling---imperfection studies TABLE 4 Temperature Rise at First Yield, Ty
Mode
Vertical
Lateral 1
Lateral 2
Ty (°C) vm or wm (m)
NA NA
NA NA
a64"2 °2"26
0.0045 TY(°C) Vmor Wm(m)
NA NA
NA NA
364"9 O2.37
Ty (°C) Vmor Wm(m)
362.3 32.67
a60"5 O2.67
65.6 2-49
0.005 5 Ty (°C) Vmor Wm(m)
62.3 2"81
a60"5 O2"81
66"3 2"60
0"006
Ty (°C) Vmor w~ (m)
62.4 2.95
60.6 2-95
67.0 2.71
0.007
Ty (°C) Vmor Wm(m)
62.8 3-22
60-9 3.22
68.6 2.93
0.008
Ty (°C) Vmor Wm(m)
63.3 3.48
61.4 3.48
70.1 3.14
0-009
Ty (°C) Vmor Wm(m)
64-0 3"74
62"0 3.74
71"8 3"35
0-010
Ty (°C) Vmor wm (m)
64.7 4.00
62.7 4.00
73-4 3-55
0.004
0.005
O
o
a
Snap followed by yield--refer to footnote a of Table 3.
L is unprescribed. It is to be noted that the results obtained relate to small deformation studies, the onset of slopes in excess of 0.1 radian being indicated in the three figures concerned. The loci are therefore conservative in this larger deformation range. 1.~7 The general characteristics for the respective buckling force/buckling amplitude and maximum compressive stress/buckling amplitude loci for all three modal studies, given in Figs 5 and 10, are again of common form. Figure 5 is directly applicable to both the vertical mode and lateral mode 1 (q~L = 1"0) studies. As illustrated in Figs 5(a) and 10(a), all imperfection ratio cases generate maximum buckling force states; it should be noted that in the low imperfection ratio case, this state does not exactly coincide with the corresponding maximum temperature rise state. This feature is
318
Neil Taylor, Aik Ben Gan
again in agreement with studies in the related field of rail track buckling. ~6 Figures 5(b) and 10(b), read in conjunction with Figs 4, 7 and 9, suggest that for the stable configurations, the temperature rise required for the onset of first yield increases with increasing imperfection ratio. This feature will be discussed further when consideration is given to all the imperfection ratios studied. Care must be taken with the small imperfection ratio snap buckling cases, however, as the first yield state is incurred during snap. This implies that yield is first incurred with the onset of the respective maximum temperature rise, this feature being indicated by use of the dashed locus. The corresponding idealised buckling loci are included in Figs 4, 5, 7, 9 and 10. The inclusion of imperfections can be seen to result in a significant reassessment of 'safe temperature rise', ,~-3identified in idealised terms in Figs 4, 7 and 9 by Tm~,;note also Table 2. Credibility with regard to the idealised loci given in Figs 4, 5, 7, 9 and 10 is also to be restricted given the possible snap phenomenon associated with such studies. Formal definition of this snap requires identification of the idealised critical state, as yet only quantitatively evaluated in lateral pipeline buckling studies. 7 Noting this, it is d e e m e d appropriate to employ "permissible' rather than 'safe" with respect to the design interpretation of limiting temperature rise. Considering the whole set of imperfection ratios studied, then Tables 3 and 4 present data relating to the establishment of permissible temperature rises Tp. The cases given in Table 3 are those involving a maximum temperature rise state (Tin)--recall the 0.003 imperfection ratio loci given in Figs 4, 7 and 9---whilst those cases in which the first yield state is incurred statically, at temperature rise Ty with or without a prior snap, are listed in Table 4. There is some overlap between the tables due to the fact that there are essentially three possible configurations d e p e n d e n t upon the magnitude of the delineated imperfection ratio. First, there are those cases involving a Tm state in which the ensuing snap buckling results in achievement of a stable state which involves stresses in excess of first yield. These cases are associated with relatively low imperfection ratios and are given in Table 3 with T r = Tin. Second, there are those cases involving a Tm state in which snap buckling occurs but the post-snap stable state is sub-yield; these cases, involving middle-order imperfection ratios, are denoted in Tables 3 and 4 by superscript a, with Tp = Ty > Tm. Third, there are the cases relating to higher imperfection ratios which involve a fully stable path such that Tp = Ty; these cases are
Submarine pipeline buckling---imperfection studies
d
Key :
1\ \
(D
\
,-~ 75-
319
V - Vertical Mode
~ 70-
~ 6sE a_ 60-
55
_
i
0-002
_
_
,__
xx
_
i
_
_\1,~" !
i
i
i
i
0.004 0006 0.008 Imperfection RQtio Vom/Lo or Wom/L o
!
~
0.010
Fig. 11. Permissible temperature rise/imperfection ratio graph.
given in Table 4. The pairs of deformation values (Vm or Wm) given in conjunction with each Tm value in Table 3 correspond to the amplitudes associated with the temperature rise Tm pre- and post-snap. The singlevalued amplitudes given in Table 4 simply denote the first yield state. It can therefore be suggested that there are three imperfection range classifications: low, medium and high. Low imperfection ratios are associated with permissible temperature rises restricted, subject to safety factors, by maximum temperature rises associated with snap buckling. Medium ratios will involve careful consideration of sub-yield snap buckling whilst high ratios will afford the most stable and predictable basis for yield stress based permissible temperature rises. The imperfection ratio ranges associated with these classifications vary between the different modes and will be subject to individual pipeline parameters; note Table 1. The above considerations are illustrated in Fig. 11 and shown qualitatively in Fig. 12. A possible fourth classification, wherein snap and yield occur simultaneously, is also denoted in Fig. 12. Figure 11 is, subject to the imposition of safety factors, effectively a design chart for thermal submarine pipeline buckling with respect to the particular pipeline parameters employed. The general trends are that for
320
Neil Taylor, Aik Ben Gan
1
,lP y l --Ty
---
~j
~p
vm or w m
vm OF Wm
fm>!y
T m < ly
predominant
Low rQho E
I S
Ty only Gtobte p o s t buckling path
$nO,p involved ~-
r~
Vm Or Wm
Medium ratio
High rotio
L
I N
\ \ \
Vnq or wn/ *F', z [
t
"1
Imperfection Ratio Vorr,,'L o or Wom/L o
Fig. 12. Qualitative imperfection ratio relationships.
low imperfection ratios, the permissible temperature rise decreases with increasing imperfection ratio, whilst for high ratios, the permissible temperature rise increases with increasing imperfection ratio. For medium range imperfection ratios, the closely related vertical mode and lateral mode 1 loci possess a minimum turning point whilst lateral mode 2 generates a locus wherein the permissible temperature rise steadily increases with increasing imperfection ratio. Finally, noting the previous argument relating to the role of the respective idealised modal maximum compressive stress levels in determining the relative importance of the various lateral modes, it is suggested that for high imperfection ratios, the (stable) curves appertaining to lateral modes 3 and 4 appropriate to Fig. 11 will lie between those given for lateral modes 1 and 2 and above that for lateral
Submarine pipeline buckling--imperfection studies
321
mode 2 respectively. The corresponding infinity mode locus will then lie above that corresponding to lateral mode 4.
7 CONCLUSIONS An archetypal design chart for use in thermal submarine pipeline buckling has been produced. Clearly, such a chart could be prepared for alternative pipeline parameter and imperfection ratio data. The study has illustrated the effects of basic physical imperfections appropriate to the problem involved. The modes investigated have been considered upon the basis of their being the most sympathetic and susceptible to such localising physical 'triggers'. A yield stress based permissible temperature rise criterion, suited to both the design of new pipelines and the assessment of existing pipelines, has been suggested. This is considered to be superior to the somewhat vague safe temperature rise concept previously delineated despite the general absence of quantitative knowledge of the probable idealised critical state snap. That is, the idealised criterion offered little help in determining the appropriate post-buckling characteristics, making identification of the location of the pipeline, should buckling occur, impossible. The proposed approach affords formal post-buckling displacement characteristics, including information on the onset, or otherwise, of plasticity during buckling. Whilst it is not suggested that the present study totally overcomes such problems, difficulty lying with identifying any given physical imperfection, data trends can now be determined. Of additional interest is the possibility that recovery upon subsequent cooling would not be total, due to the non-conservative nature of the frictional forces involved. This leads to the realisation that, upon recovery, the imperfection ratio would have increased with a consequent change in post-buckling characteristics should thermal buckling recur. Such post-buckling considerations are perhaps particularly important with regard to trenched pipelines undergoing vertical buckling--'popups'--in view of the possibility of raised pipelines fouling, for example, anchor cables. Finally, it must be recalled that the effects of residual stresses remain to be considered, although their effects could perhaps be interpreted in terms of equivalent initial lack-of-straightness. Further, the eccentricity
322
Neil Taylor, Aik Ben Gan
of the frictional forces with respect to the pipe centreline has been neglected. Both features require further study.
ACKNOWLEDGEMENT The authors wish to thank Professor Alastair Walker and Dr Charles Ellinas of J. P. Kenny and Partners, London, for discussions held with regard to the contents of this article. However, the opinions expressed, and any errors incurred, are the authors' alone. REFERENCES 1. Hobbs, R. E., Pipeline buckling caused by axial loads, Journal of Constructional Steel Research, 1(2) (January 198 l) 2-10. 2. Hobbs, R. E., In-service buckling of heated pipelines, Journal of the Transportation Engineering Division, ASCE, 110(2) (March 1984) 175-89. 3. Taylor, N. and Gan, A. B., Regarding the buckling of pipelines subject to axial loading, Journal of Constructional Steel Research, 4(l) (1984) 45-50. 4. Taylor, N., Richardson, D. and Gan, A. B., On submarine pipeline frictional characteristics in the presence of buckling, Proceedings of the 4th International Symposium on Offshore Mechanics and Arctic Engineering, ASME, Dallas, Texas, 17-21 February 1985, 5(18-15. 5. Palmer, A. C. and Martin, J. H., Buckle propagation in submarine pipelines, Nature, 254 (March 1975) 46-8. 6. Bjerrum, L., Geotechnical problems involved in foundations of structures in the North Sea, Geotechnique, 23(1) (1973) 319-58. 7. Taylor, N. and Gan, A. B., Refined modelling for the lateral buckling of submarine pipelines, Journal of Constructional Steel Research, 6(2) 143--62. 8. Lyons, C. G., Soil resistance to lateral sliding of marine pipelines, 5th Offshore Technology Conference, OTC 1876, 2 (1973) 479-84. 9. Gulhati, S. K., Venkatapparao, G. and Varadarajan, A., Positional stability of submarine pipelines, IGS Conference on Geotechnical Engineering, 1 (1978) 430-4. 10. Anand, S. and Agarwal, S. L., Field and laboratory studies for evaluating submarine pipeline frictional resistance, Transactions of the ASME, Journal of Energy Resources Technology, 103 (September 198l) 250-4. l l. Marek, P. J. and Daniels, J. H., Behaviour of continuous crane rails, Journal of the Structural Division, ASCE, ST4 (April 1971) 1081-95. 12. Granstrom, A., Behaviour of continuous crane rails, Journal of the Structural Division, ASCE, ST1 (January 1972) 360-1. 13. Kerr, A, D., Analysis of thermal track buckling in the lateral plane, Acta Mechanica, 30 (1978) 17-50.
Submarine pipeline buckling---imperfection studies
323
14. Kerr, A. D., A model study for vertical track buckling, High Speed Ground Transportation Journal, 7 (1973) 351-68. 15. Kerr, A. D., The effect of lateral resistance on track buckling analyses, Rail International, 1 (1976) 30-8. 16. Tvergaard, V. and Needleman, A., On localized thermal track buckling, International Journal of Mechanical Sciences, 23(10) (1981) 577-87. 17. Timoshenko, S. P. and Gere, J. M., Theory of Elastic Stability, 2nd edition, McGraw-Hill--Kogakusha Ltd, New York, 1961, 76-81.
Submarine Pipeline Buckling Imperfection Studies
Neil T a y l o r a n d A i k B e n G a n Department of Civil Engineering, SheffieldCity Polytechnic,Pond Street, SheffieldS1 1WB, UK
ABSTRACT In-service buckling of submarine pipelines can occur due to the institution of axial compressive forces caused by the constrained expansions set up by thermal and internal pressure actions. Previous attempts at modelling the appropriate behaviour have been based on idealised or perfect pipelines; further, such analyses have also employed fully mobilised friction forces. Herein presented is a set of analyses which incorporate structural imperfections and deformation-dependent axial friction resistance. Not only do these features enable a more rational interpretation of submarine pipeline buckling behaviour to be established, but, in addition, an inherent limitation existing in the previous mathematical modelling of the vertical buckling mode is elucidated and o vercome.
NOTATION fA q r t u Ua uf us
Axial friction parameter. Submerged weight of pipeline per unit length. External radius of pipe. Wall thickness of pipe. Axial deformation of the pipe. Tensile extension of the buckle. Compressive flexural end shortening of the buckle. Resultant longitudinal movement at buckle/slip length interface.
295 Thin-Walled Structures 0263-8231/86/$03.50 © Elsevier Applied Science Publishers Ltd, England, 1986. Printed in Great Britain
296
v
Vm Vo Votll w wm
Wo 141om X
A E I L, LI Lo
L~ M Mm
P Pa Po T Tm Train
Tp Ty V o~ O'm O'y
Neil Taylor, Aik Ben Gan
Fully mobilised axial displacement. Vertical deformation of the pipe. Maximum vertical amplitude of the buckled pipe. Vertical deformation of the imperfection topology. Maximum vertical amplitude of the imperfection topology. Lateral deformation of the pipe. Maximum lateral amplitude of the buckled pipe. Lateral deformation of the imperfection topology. Maximum lateral amplitude of the imperfection topology. Spatial co-ordinate Cross-sectional area. Young's modulus. Second moment of area of cross-section. Buckle lengths. Buckle length of the imperfection topology. Slip length. Bending moment. Maximum bending moment. Buckling force. Axial force component. Prebuckling axial force. Temperature rise. Maximum temperature rise. Minimum safe temperature rise. Permissible temperature rise. Temperature rise at first yield. Total potential energy. Coefficient of linear thermal expansion. Maximum compressive stress. Yield stress. Axial friction coefficient. Lateral friction coefficient.
1 INTRODUCTION The circumstances concerning the means by which the in-service buckling of submarine pipelines can occur have been discussed at length elsewhere. ;-4 Analyses have been primarily oriented about thermal action
Submarine pipeline buckling--imperfection studies
297
with oil and gas temperatures potentially ranging up to 100°C above that of the water environment. That is, a uniform temperature increase, T, in a perfectly straight submarine pipeline will create an axial compression force due to constrained thermal expansion. Within the elastic range of the pipeline response, this force can be represented by Po = A E a T
(1)
where A E is the axial rigidity of the pipe and ot is the respective coefficient of linear thermal expansion. Should buckling occur, part of the constrained thermal expansion is released in a buckled region which, taken together with the friction resistance of the sea bed/pipeline interface, results in a reduction in the axial compression to some buckling force P. Studies to date have been based on idealised buckling phenomena with the simplifying non-conservative assumption that the friction forces are fully mobilised throughout. The five established lateral buckling modes which relate to snaking movements across the sea bed are illustrated in Fig. 1. t'3 Axial and lateral deformations are denoted by u and w respectively. The appropriate buckling lengths are denoted by L and L 1, whilst Ls represents the slip length. Axial friction resistance is generated through the slip length whilst lateral friction resistance occurs through the buckling lengths. The vertical buckling mode which involves part of the pipeline buckling out of the sea bed can also be represented by Fig. l(a) except that w is now replaced by v, the appropriate vertical deformation. It can be safely assumed that submarine pipelines suffer structural imperfections under field conditions. Initial lack-of-straightness will occur during laying operations and sea bed conditions will also generally preclude a perfect lie. The present study concentrates attention on the vertical mode and lateral modes 1 and 2. With regard to the five lateral modes depicted in Fig. 1, it is considered that modes 1 and 2 are more susceptible than the remaining lateral modes to the most basic initially deformed pipeline topologies, these imperfections taking the form of fundamental symmetric and skew-symmetric modal deviations from the idealised lie. Further support for this choice of modes comes from established idealised analyses' stress trends. 2 That is, values of the minimum safe temperature rise, Tmi,, for all six identified submarine pipeline buckling modes, employing the typical parametric values 2 given in Table 1, are given in Table 2 together with the corresponding buckling amplitudes, Vmor Win,and maximum compressive stresses, or,,. It can be
Neil Taylor, Aik Ben Gan
298
l_ibU
Ls
L/2
L/2
Ls
,',o) Mode 1
Ls (b)
<
Ls (c)
<
Ls
L
L
Ls
Mode 2
~
L1
-'~<" L/2
~
L/2
~
L1
~'
L
~
L1
Ls
;
Mode 3
~
L1
~
L
~
~
Ls
>
(d) M o d e 4
Fig. 1. Lateral buckling mode topologies.
seen that on the basis of Train, lateral modes 3 and 4 are more critical than lateral modes 1 and 2. This ordering is indicated parenthetically adjacent to the respective Tmi. values. However, the corresponding stress levels are consistent with an underlying and alternative interpretation with the vertical m o d e and lateral mode 1 generating the highest stress levels. That is, accepting that imperfections will be present in practice, lateral mode 1
Submarine pipeline buckling---imperfection studies
299
TABLE 1
Pipeline Parameters Parameter
Symbol
External radius Wall thickness Sectional area Young's modulus Second moment Self-weight Yield stress Thermal coefficient Axial coefficient Lateral coefficient
r t A E I q O-y a ~bAIU6 CbL
Value 325 mm 15 mm 29 920 mm 2 206 kN/mm 2 1.509 × 109 mm 4 3.8 kN/m 448 N/ram 2 11 × 10-6/°C 0.7 at 5 mm 1.0
TABLE 2
Idealised/Fully Mobilised Analyses Mode Vertical Lateral 1 Lateral 2 Lateral 3 Lateral 4 Lateral oo
Train(°C)
vm or wm I Train (m)
crm I Tmin (N/mm 2)
65.3 (5) 63.6 (4) 61.6 (3) 60-6 (2) 60.5 (1) 77.5 (6)
2.3 2-4 1.4 1.9 1.3 0.5
597 606 495 547 479 261
is n o t a b l e for g e n e r a t i n g the highest stress trend o f all the lateral m o d e s . T a k e n in c o n j u n c t i o n with the basic nature o f the physical i m p e r f e c t i o n c o n c e r n e d , it is c o n s i d e r e d that, in practical terms, lateral m o d e 1 is the m o s t significant s y m m e t r i c lateral m o d e with respect to i m p e r f e c t i o n c o n s i d e r a t i o n s . Similar reasoning suggests that lateral m o d e 2 is the m o s t significant s k e w - s y m m e t r i c m o d e , lateral m o d e s 3 and 4 being, in e n g i n e e r i n g terms, s u b o r d i n a t e forms o f lateral m o d e s 1 and 2, respectively. All f o u r m o d e s can be c o n s i d e r e d to be localised versions o f lateral m o d e ~ , this f e a t u r e being r e i n f o r c e d by the p r e s e n c e o f a particular i m p e r f e c t i o n in a particular locality. In all t h r e e cases studied, variable or d e f o r m a t i o n - d e p e n d e n t axial
300
Neil Taylor, Aik Ben Gan
resistance forces are included in the respective analyses; this feature is of particular importance with respect to the vertical mode study as the established modelling L2 is shown to be invalid at small values of vertical deformation. The refined friction force modelling uses a recently established axial friction response lOCUS. 4 The analyses presented should provide for a more logical interpretation of the submarine pipeline buckling problem and a design chart, based upon the onset of first yield, is provided for a typical pipeline parameter set. The studies employ small deformation theory and material behaviour is taken to be elastic; gross sectional distortion as associated with laying operations ~ is not a factor here.
2 G E O T E C H N I C A L FACTORS Recent geotechnical experimentation 4 with respect to North Sea conditions 6 has provided information relating to the nature of the deformation-dependent friction force behaviour with regard to typical submarine pipeline parameters. Based upon these findings, a refined axial
.,/~Futly Mobilised Locus 1.0 ExperimentQL Locus & 0.8 ¸ fA
-
Full Scale Design Curve
041
0.6-
fA/~A = 1 - e - 25u/u¢
0.2
fAq=
.- = u
q = Submerged serf-weight per unit length of pipe Conventicn
0
0.'i 0.2 013 01~ o'.s o'-G 0'.7 0~8 019 u/u¢
Fig. 2. Generalised axial friction characteristics.
~.0
Submarine pipeline buckling--imperfection studies
301
friction force-deformation locus is depicted in Fig. 2. The appropriate fully mobilised axial friction coefficient is given by qbA; the movement corresponding to the attainment of full mobilisation is denoted by u~. The friction force parameter fA, 0--
/
u,~ = 5 mm
J
(2)
Consequently, the design curve given in Fig. 2 employs jk/~bA = 1 - e-ZSu/"~
(3)
noting the appropriate convention given in the Figure. This expression gives good agreement with the designated non-dimensionalised experimental curve and, in addition, accords with the appropriate fully mobilised asymptote with fA/CbA ~ 1 at u/u, = 1. It is to be noted that eqn (3) offers superior modelling characteristics to its precursor given elsewhere. 4 Finally, accepting that lateral resistance is taken to be fully mobilised, ~-3 the value for ~bL,the fully mobilised lateral friction coefficient, is taken to b e 1.0. 4
3 VERTICAL MODE ANALYSIS The essential features of the vertical mode are shown in Figs 3(a) and (b). The former figure details the topology whilst the latter depicts the axial force distribution within the pipe. The submerged self-weight of the pipe per unit length is denoted by q and Vmrepresents the maximum amplitude of the buckled pipe. Regarding the imperfection parameters, Voand Yore represent the respective vertical deformation and maximum amplitude whilst the appropriate imperfection buckling length parameter is denoted by Lo. It is to be noted that Vom/Lo forms the essential imperfection parameter. This parameter is unique in as much as Yore and Lo are
302
Neil Taylor, Aik Ben Gan
q &
I
&
&
&
&
&
&
&
&
I
vm -Yore
7,.z~,Tr~bA1q~1:/e22~u;)-Tf " ....
~""-- ~"'-- -
I~---- Ls-°°
~
#
~
/
~
Lo/2
LI2
'
/
I
#
.
-
,
~x LO/2
~
LI2
~
~Aq(1-e2~ .
14
u~7.#,. ~.
..--..~ ..-.--.-~. . . .
~l~
Ls--°° _ _ _ _ ~
(o) Topology
/ #
--~.Pa
PO =~'Aq(1.e25usILk~)L12
IZai
(b) Axial Force Distribution
Fig. 3. Details of vertical mode. dependent. The linearised vertical deflection equation, appertaining to idealised pipelines, 1.3is given by q( v = ~
1+
n2 L 2 8
n~x,2
cosnx ) - L/2 <-x <- L/2 cosnL/2
(4)
where n L = 8.986 8 and E1 is the flexural rigidity of the pipe. The m a x i m u m amplitude of the buckle, at x -- 0, is Klq
aL4
Vrn - E-7~n - 2.407x 10 ~'-I/z
(5)
where L is the unprescribed buckling length and Kt = (1 + n 2L:/8 - 1 / cos n L / 2 ) = 15.698 465, such that substituting eqn (5) into eqn (4) yields the transverse d e f e c t i o n expression Vm( n2L = n=x2 v = ~ 1+ T 2
cosnx ) cosnL/2
(6)
From the foregoing, the imperfection deflection expression can be assumed to be
Vo
= yore{ n2oL~-o nox 2 Ki \ 1 + ~ 2
cosnox ) _ L o / 2 < _ x < L o / 2 cosnoL,>/2
(7)
Submarine pipeline buckling---imperfection studies
303
where noLo = 8.986 8 and Yore, the maximum imperfection amplitude, noting eqn (5), takes the form Vom = 2"407 X 10-3 qL4 E1
(8)
This equation displays the previously denoted dependence of Yoreand Lo. The deflection expressions eqns (7) and (8) relate, in practical terms, to a vertical deviation from the intended or idealised lie; this will act as a potential 'pop-up' trigger. Employing symmetry, the total potential energy relating to the deformed state is given by V ~
i* Lo/2 El~2 ( d 2 v / d x 2 - d 2 v o / d x 2 ) 2 d x || ,,Jo
q'- f L/2 El~2 (d 2v/dx 2 Lo/2 + f L°/2 ",10 -- f L°/2 "sO -- ; L/2
q(v-Vo)+ f
d 2
vo/dx2)2dx
L/2 Lo/2
q(v-vo)dx
P/2[(dv/dx) 2 - (dvo/dx) 2]dx PI2[(dv/dx) 2 - (dvo/dx)2]dx = 0
Lo/2
(9)
the corresponding equilibrium state being given by d V / d v m = 0. Noting that for 0-< x <_ Lo/2, the derivatives of initial curvature, slope and deflection with respect to Vm are null, as are the actual values of initial curvature, slope and deflection for Lo/2 <-x <-L/2, then applying the statics criterion, 45.36260Eln3vm/K]
-
2 2 RlElvomnno/K1
+ 30.241 74q/(nKl) - 75"604 3 4 P n v m / K ]
=
0
(1o)
where F Rt = 4.603 14 ]sin(4.4934Lo/L)
+ 2.301 57
sin4.493 4(1 + Lo/L) (L/Lo + 1)
sin 4.493 4(l - Lo/L)
(it)
304
Neil Taylor, Aik Ben Gan
Simplification of eqn (10), noting eqns (5) and (8), yields the buckling force
= + 7+ +,
]
(12)
Bending m o m e n t is afforded by M = El(d2v/dx 2
-
d2vo/dx 2)
(13)
which gives the maximum moment, at x = 0, upon substitution from eqns (6) and (7), to be Mm = - 0.06938q(L 2- L~)
(14)
the negative sign correctly indicating flexural compression to be acting on the lower part of the pipeline section. The maximum compressive stress O-minduced in the pipe is thereby obtained from
p/A +IM~r/I I
O" m :
(15)
where r is the external radius of the pipe. Having established the relationships between v, L, Mm, or,, and the buckling force P, it is now necessary to determine the dependence of P u p o n T, the temperature rise. This is achieved by considering the slip length characteristics. It is proposed to employ the deformationd e p e n d e n t axial friction force modelling discussed previously and detailed in Figs 2 and 3. As established elsewhere, v the slip length field equation is given by d2u
AE-~
= --]Aq
L/2~x<-L/2+L,
(16)
T h e associated slip length boundary conditions take the form lim [u, du/dx] = 0
X~m
(17)
and
dx
I
-L/2
(Po - P) - Pa AE
(18)
Submarine pipeline buckling--imperfection studies
305
where force Pa, detailed in Fig. 3, represents the axial force c o m p o n e n t frictionally induced in the pipe by the vertical shear reaction q L / 2 at the buckle length/slip length interface. Equation (17) can be considered to be a regularity condition which assists in precluding the prejudicial requirem e n t of assuming some function u = f ( x ) . C o m b i n i n g eqns (3) and (16) affords the non-linear slip length field equation d2 A E ~ = - ~bAq(1 -- e 25u/"~) dx
(19)
noting that the change of sign of the exponent with respect to that given in eqn (3) is due to u and fA being co-oriented within the slip length regionwrecall that Figs 1 and 3 employ a different convention from Fig. 2. Employing the identity 2d 2u / d x : = d [ ( d u / d x ) 2 ] / d u in conjunction with eqns (17) and (19) gives dx
[ AE
5
u
/]
(20)
Equating this expression with the boundary condition given in eqn (18), and with u] L/2 = us yields (Po - P) =
24)AqAE
5
- us
L
+ 4~aq ~- ( 1 - e 25"S/"*)
(21)
It is necessary to set up a matching compatibility expression at the interface of the slip and buckling lengths, x = L / 2 . This can be expressed in the form us = ua + uf where Ua and uf denote the tensile extension and compressive flexural end shortening of the buckle length L / 2 . Incorporating the presence of the imperfection v o / L o gives
us -
2AE
2
dx -
which yields, given eqns (6) and (7), 2
(Po- P)L us 2AE
7.9883 x l0 -6 ~q
(Z 7 -
Lo7)
(23)
306
Neil Taylor, Aik Ben Gan 140-
Idealised
-
Fulty Mobihsed
120
v 100 F-
Snap
2 BO D
E E
6O
i
z,O o
°1o! ,,05 5
20
~P'r
~
i
. . . . . 4
~ 5
,
, 7
Buckle Amphtude Vm(m)
C' '
d0
75
90
100 '
18 5
1~0 '
115 '
120 '
,2's
Buckle Le,ngth L ( m )
Fig. 4. Thermalaction characteristics--vertical mode,
Solutions for (Po - P) and us are thereby obtained in terms of discrete values of L employing a computerised non-linear iterative algorithm involving eqns (21) and (23). The numerical evaluations have been carried out, noting eqn (8), for a series of eleven imperfection ratios Vom/Lo from 0.003 through to 0.010, the remaining pipeline parameters 2 employed being as denoted in Table 1. Results for (Po - P) are then substituted into eqns (1), (5), (11) and (12) to produce the loci depicted in Figs 4 and 5(a). Only those loci for Vom/Lo of 0.003, 0.007 and 0.010 are shown for reasons of clarity. The loci in Fig. 5(b) are obtained by substituting the respective values of P into eqn (15), given eqn (14).
Submarine pipeline buckling--imperfection studies
307
ldeolised / Fully Mobilised
~6 v t3_
,~ Snap
c~ r-5
2~2
sFl~
;.~.
;I,I ° . . . . 1
I, 2
~lopeo.~ ,
,
,
,
,
l
3 4 5 Buckle Amplitude vm or w m (m)
i ~--7
(o) Buckling Force
~-'~E 800 l ~ I d e Q l i s e d / F u l l y k4obilised
f(/,
. . . . 1
2
. . . . . 3
4
5
6
; 7
Buckle Amplitude vm or wm (m) (b) Ma×imum Compressive Stress
Fig. 5. Dependent parameter characteristics--vertical mode and lateral mode 1.
308
Neil Taylor, Aik Ben Gan
It can be shown that the vertical mode modelling in the established submarine pipeline t'2 and related crane rail u'~2 studies is invalid at small values of vertical deformation. Using the notation of the present paper, eqn (19) of Ref. 11 can be rewritten "" ~4~Aq~AEL7 - P~, ( P o - p)Z = 1.5977 × lu - ( E l f
(24)
this being the fully mobilised equivalent to the expression obtained if eqn (23) were to be substituted into eqn (21). Clearly, noting Fig. 3(b), both approaches require (Po - P) -> P~
(25)
For the fully mobilised analysis (26)
P~ = eflAqL/2
w h e r e u p o n , substituting eqns (25) and (26) into eqn (24), there is the requirement L5->3.1296×10
(27)
That is, for the pipeline parameters given previously L -> 39.014 m, such that, given eqn (5), Vm-----68" 17 ram. These restrictions on the established fully mobilised studies t,2 have not been previously reported. In the present study there is no such restriction upon eqns (2 I) and (23) due to the variable, deformation-dependent axial friction force modelling; that is, eqn (25) is valid for all values of Po, P and P,.
4 LATERAL MODE 1 ANALYSIS T h e essential features of mode 1 are shown in Fig. 6(a) and (b) which depict the respective topology and axial force distribution. The m a x i m u m amplitude of the buckled pipe is denoted by Win. With regard to the imperfection parameters, Wo and Worn represent the respective lateral deformation and maximum amplitude. The analysis for the buckling
Submarine pipeline buckling--imperfection studies
I
~
•
~
~
{
wm- Worn ~ q ( 1- e 25u/u¢ ) ,~ . . . .
.,e-...- ~
~'
i,
-
" ~.-tot?
~ L/2 (a)
{
I
-.
- ~ ~
Ls-e°
¢
309
J
j,x _ _~o/~
"r-
L/2
~ q ( 1 - e 25 u/u¢) - " - " P '--'P . . . .
"
LS-~
Topology
<__ i!oi (b) Axial Force Distribution
Fig. 6. Detailsof lateralmode 1. region is identical to that of the vertical mode. Equations (4) through (15) remain valid except that v, together with its related components, and q are now replaced by w and cbLq respectively, noting that +Lq is the appropriate fully mobilised lateral resistance force per unit length acting against the lateral buckling mechanism. The physical imperfection corresponding to eqns (7) and (8) therefore takes the form of a basic symmetric wave. Regarding the slip length region, eqns (16) through (20) remain valid except for eqn (18) where the respective boundary condition is replaced by dUldxL/2 _ __(P°AEP)
(28)
This results in eqn (21) being replaced by ( P o - P) =
[
2c~AqAE
(e2--,..l 5
(29)
us
The matching compatibility condition at the ends of the buckle remains as previously, subject to the appropriate replacement of v. That is, eqns (6), (7) and (22), so modified, generate us -
(Po- P) L _ 7.988 3 x 10-6 - 2AE
(L 7- L~)
(30)
Neil Taylor, Aik Ben Gan
310
140,
=
Idealised / Fully Mobilised
120
~" 100
~ so
_
_
S~p
_
Z0
20 ¸
I
1
2
I
3
I
I
z,
I
I
5
I
1
6
I
7
Buckle Amplitude w m (m)
6 6~ 7~
9'o
~6o t&
~io
~is
~o
~
Buckle Length L (m)
Fig. 7. Thermal action characteristics--lateral mode 1.
this expression being analogous to that given in eqn (23) with q replaced
by q~Lq. The solution procedure for (Po - P) and us is as previously, eqns (29) and (30) being solved iteratively in terms of discrete values of L. It is to be noted that since in the present study ~bL = 1-0, 4 the revised equations are actually the same as eqns (4) through (15) apart from v being replaced by w. Numerical evaluations have been undertaken, recalling eqn (8) and Table 1, for the previously denoted range of 11 imperfection ratios (herein Wom/Lo). Results for (Po - P) are then substituted into eqns (1), (5), (11) and (12) to produce the w-loci depicted in Fig. 7. The loci in Figs
Submarine pipeline buckling---imperfectionstudies
311
5(a) and (b), noting eqns (14) and (15), are also valid for mode 1 since 4~L =
1"0.
5 LATERAL MODE 2 ANALYSIS The topology and axial force distribution depicted in Figs 8(a) and (b) respectively represent the essential features of mode 2. The linearised lateral deflection equation, obtained from related idealised rail track studies, 13is given by
(~-~-)
qbhqL4
(-~ + 2LX( (31)
O<-x<-L
The maximum amplitude of the buckle, at x = 0.346 4L, is
Kzd~eqL4 Wm
=
x
= 5-532
16";7"4 E1
chLqL4
10 -3
(32)
E1
where L is the unprescribed buckling length and K2 : 8.621 149 6. Substituting eqn (32) into eqn (31) yields the lateral deflection expression w :
w'( --T-m [l-c°s 1 - L('~)) +'rrsin ] K (-~f-) 2 + 2L~xI
1'
t
t
t
I
Wm .
.
.
.
_~q<~-e2Su'u¢/
_
M L (a)
_ - - /
,,I,
"i'
~ ~
Lo
,I,
I
~.
" 9 ' o m ~
Lo Ls--C°
$
, ~w-',~o
(33)
" -..
j L "1
'~qIl-e25u~u~z
.
.
.
.
Ls-C°
Topotogy
(b) Axial Force Distribution
Fig. 8. Details of lateral mode 2.
~""'~-
_
312
Neil Taylor, Aik Ben Gan
F r o m the foregoing, the imperfection deflection expression can be assumed to be w,, = K-~ ()<--x < - Lo
(34)
where Wom represents the maximum imperfection amplitude which, noting eqn (32), takes the form & aL 4.
Wom= 5"532 x 10 3~t.-,-~-
(35)
El
The physical imperfection corresponding to eqns (34) and (35) therefore takes the form of a basic skew-symmetric wave. Employing skewsymmetry, the energy formulation is similar to that of the vertical m o d e analysis except that the regions under consideration for the deformed pipeline and imperfection topology are 0<-x-< L and 0<-x<- Lo respectively. Accordingly, eqn (9) can again be employed with Lo/2 and L / 2 being replaced by Lo and L respectively whilst 0bcq and w appropriately supersede q and v. Applying the statics criterion d V / d w m = 0, then eqn (9), modified as d e n o t e d above, affords (24",'7"4+ 87"r6) Elwm K~L 3
+
87r3ElwomR2 K~L~,L
~q(77"2) PWrn{ . 1+-~- - ~ 2 L k l O w 2 +
10"B'4x~
3 ] =0
(3~)
where R2 = sin (2rrLo/L)
(L/Lo)(rr2 + L/Lo) ] 1 - (L/Lo) 2 + 2rrLo/L
+ rc[ l -cos(2zrLo/ L ) ]
(L)
(37)
Submarine pipeline buckling---imperfection studies
313
Simplification of eqn (36), given eqns (32) and (35), yields the buckling force 4rr 2 E 1
_ 3 [
R2
(38)
Bending moment is given by M = El(d2w/dx
2-
d2wo/dx2)
(39)
which generates the maximum moment from eqns (33) and (44), then Mm = - 0.108 8 4 t b k q ( L
Mm
at x = 0.299L; substituting (40)
2 - L~)
The maximum compressive stress induced in the pipe is thereby obtained employing eqn (15) subject to the substitution of the appropriate parameters. Having established the essential relationships appertaining to the buckling region, those concerning the slip length are now determined. The formulation is similar to that given in the vertical mode analysis except that the slip length region is now restricted by L -< x -< L + Ls. Therefore, eqns (16) through (20) can be employed with L / 2 in eqn (16) being replaced by L. Equations (28) and (29) remain valid subject to the buckle length/slip length interface for lateral mode 2 being specified at L with respect to the former equation. The matching compatibility condition at the end of the buckle length takes the form U
=
t/s
L
--
AE
2
o
(41) which yields, noting equations (33) and (34) us -
AE
8"715 ×
10 -5
( q)2 - -
(L 7 - L 7)
(42)
The solution procedure for (Po - P) and Usis as previously; eqns (29) and (42) are solved iteratively in terms of discrete values of L. Numerical
314
Neil Taylor, Aik Ben Gan
~4i9
----
Ideolised / Fud~ Mobiiised
120
~ ~oo rY
SnGp
& E
6O
40
2°-I I"o I°1 .& 0.5
10
1.5
20
25
30
35
Buckle Amplitude wm (m)
Buckle Length L (m)
Fig. 9. Thermal action characteristics--lateralmode 2. evaluations have been carried out, noting eqn (35) and Table 1, for the previously denoted range of 11 imperfection ratios (Wo~,/Lo). Results for (Po - P) are then substituted into eqns (1), (32), (37) and (38) to produce the loci depicted in Figs 9 and 10(a). The loci in Fig. 10(b) are obtained by substituting the respective values of P into eqn (t5), noting eqn (40).
6 DISCUSSION The imperfection ratios explicitly considered are denoted in Tables 3 and 4 together with the appropriate permissible temperature rise values,
Submarine pipeline buckling---4mperfection studies
315
8J
~~Fullyldeolised / Fully MobilisShOp ed~ v
G.
~4 ~2
~:o E I
z 600/~ ,,
~ Max. Slope 0.1 I
0,5
l
I
1.0
I
I
I
I
I
I
I
1.5 2.0 2.5 Buckle Amplitude wm (m) (O) Buckling Force
Ideotised/Fu[[yMobilised
0.5
l.O
I
3,0
~
3.5
.-..0.003
.
1.5 2.0 2.5 Buckte Amplitude wm (m) (b) Maximum CompressiveStress
.
.
.
3-0
.
3.5
Fig. 10. Dependent parametercharacteristics--lateralmode 2.
based upon the onset of first yield, and the corresponding buckling amplitudes. It is to be noted that the squash load for the pipeline parameters considered, denoted in Table 1, is 13.4 MN, which corresponds to a temperature rise of 197.7°C. Perusal of Tables 3 and 4 therefore shows that buckling action will occur in practice. Prior to
316
Neil Taylor, A i k Ben Gan
TABLE 3 Maximum Temperature Rise, T,-. Mode
Vertical
Lateral 1
Lateral 2
Tm (°C) Vmor wm (m) vmorwm(m)
79.6 0.30 5"95
+h
78.9 0.30 6"21
+
69-7 (I-25 2"86
Tm (°C) 0.{~3 5 Vmor wm (m) vmorwm(m)
72.9 0"38 4.79
~1
72.1 0"37 5.03
$
64.1 (1"32 2.21
T,, (°C) Vmor Wm(m) Vmor Wr. (m)
67-8 0.48 3-83
~1
66-9 0-47 4.05
~1
%0-0 "0-42 "1.64
Tm (°C) 0"004 5 vm or W m (m) vmor wm (m)
63"9 0-60 2-99
~1
62"8 0.59 3.20
~
"57" 1 "(/.60 "1.02
Tm (°C) vm or Wm(m) vm or wm (m)
%0-9 ~().77 "2.18
~1
"59-8 '~0.74 ~2-41
~1
NA NA NA
~
NA NA NA
0.003
.3 :e
.3 0.004 O
© o O
E
0.005
Tm (°C) 0"005 5 Vmor Wm (m) VmOrWm(m)
NA NA NA
~57"5 ~l.06 "1'43
"Non-crucial values followed by yield Tm 4= T~,--refer to Table 4. h $ Indicates snap.
further consideration of Tables 3 and 4, and the ensuing implications, it is pertinent to assess the behaviour associated with the three imperfection ratios identified in detail previously--that is, with Vom/Lo and Wom/Lo taking the values 0.003, 0.007 and 0.010. With regard to the respective temperature rise/buckling amplitude loci given in Figs 4, 7 and 9, it can be seen that only the relatively small imperfection ratio case (0.003) displays a maximum temperature rise, Tm, together with the associated snap buckling phenomenon. The remaining two cases (0-007 and 0.010) generate stable post-buckling paths. In all three modes, the respective loci are of converging form as is to be anticipated; indeed, the modes afford characteristics which are in general agreement with those reported in the related field of rail track buckling, 14-16 although in the present study it is to be recalled that the buckling length
317
Submarine pipeline buckling---imperfection studies TABLE 4 Temperature Rise at First Yield, Ty
Mode
Vertical
Lateral 1
Lateral 2
Ty (°C) vm or wm (m)
NA NA
NA NA
a64"2 °2"26
0.0045 TY(°C) Vmor Wm(m)
NA NA
NA NA
364"9 O2.37
Ty (°C) Vmor Wm(m)
362.3 32.67
a60"5 O2.67
65.6 2-49
0.005 5 Ty (°C) Vmor Wm(m)
62.3 2"81
a60"5 O2"81
66"3 2"60
0"006
Ty (°C) Vmor w~ (m)
62.4 2.95
60.6 2-95
67.0 2.71
0.007
Ty (°C) Vmor Wm(m)
62.8 3-22
60-9 3.22
68.6 2.93
0.008
Ty (°C) Vmor Wm(m)
63.3 3.48
61.4 3.48
70.1 3.14
0-009
Ty (°C) Vmor Wm(m)
64-0 3"74
62"0 3.74
71"8 3"35
0-010
Ty (°C) Vmor wm (m)
64.7 4.00
62.7 4.00
73-4 3-55
0.004
0.005
O
o
a
Snap followed by yield--refer to footnote a of Table 3.
L is unprescribed. It is to be noted that the results obtained relate to small deformation studies, the onset of slopes in excess of 0.1 radian being indicated in the three figures concerned. The loci are therefore conservative in this larger deformation range. 1.~7 The general characteristics for the respective buckling force/buckling amplitude and maximum compressive stress/buckling amplitude loci for all three modal studies, given in Figs 5 and 10, are again of common form. Figure 5 is directly applicable to both the vertical mode and lateral mode 1 (q~L = 1"0) studies. As illustrated in Figs 5(a) and 10(a), all imperfection ratio cases generate maximum buckling force states; it should be noted that in the low imperfection ratio case, this state does not exactly coincide with the corresponding maximum temperature rise state. This feature is
318
Neil Taylor, Aik Ben Gan
again in agreement with studies in the related field of rail track buckling. ~6 Figures 5(b) and 10(b), read in conjunction with Figs 4, 7 and 9, suggest that for the stable configurations, the temperature rise required for the onset of first yield increases with increasing imperfection ratio. This feature will be discussed further when consideration is given to all the imperfection ratios studied. Care must be taken with the small imperfection ratio snap buckling cases, however, as the first yield state is incurred during snap. This implies that yield is first incurred with the onset of the respective maximum temperature rise, this feature being indicated by use of the dashed locus. The corresponding idealised buckling loci are included in Figs 4, 5, 7, 9 and 10. The inclusion of imperfections can be seen to result in a significant reassessment of 'safe temperature rise', ,~-3identified in idealised terms in Figs 4, 7 and 9 by Tm~,;note also Table 2. Credibility with regard to the idealised loci given in Figs 4, 5, 7, 9 and 10 is also to be restricted given the possible snap phenomenon associated with such studies. Formal definition of this snap requires identification of the idealised critical state, as yet only quantitatively evaluated in lateral pipeline buckling studies. 7 Noting this, it is d e e m e d appropriate to employ "permissible' rather than 'safe" with respect to the design interpretation of limiting temperature rise. Considering the whole set of imperfection ratios studied, then Tables 3 and 4 present data relating to the establishment of permissible temperature rises Tp. The cases given in Table 3 are those involving a maximum temperature rise state (Tin)--recall the 0.003 imperfection ratio loci given in Figs 4, 7 and 9---whilst those cases in which the first yield state is incurred statically, at temperature rise Ty with or without a prior snap, are listed in Table 4. There is some overlap between the tables due to the fact that there are essentially three possible configurations d e p e n d e n t upon the magnitude of the delineated imperfection ratio. First, there are those cases involving a Tm state in which the ensuing snap buckling results in achievement of a stable state which involves stresses in excess of first yield. These cases are associated with relatively low imperfection ratios and are given in Table 3 with T r = Tin. Second, there are those cases involving a Tm state in which snap buckling occurs but the post-snap stable state is sub-yield; these cases, involving middle-order imperfection ratios, are denoted in Tables 3 and 4 by superscript a, with Tp = Ty > Tm. Third, there are the cases relating to higher imperfection ratios which involve a fully stable path such that Tp = Ty; these cases are
Submarine pipeline buckling---imperfection studies
d
Key :
1\ \
(D
\
,-~ 75-
319
V - Vertical Mode
~ 70-
~ 6sE a_ 60-
55
_
i
0-002
_
_
,__
xx
_
i
_
_\1,~" !
i
i
i
i
0.004 0006 0.008 Imperfection RQtio Vom/Lo or Wom/L o
!
~
0.010
Fig. 11. Permissible temperature rise/imperfection ratio graph.
given in Table 4. The pairs of deformation values (Vm or Wm) given in conjunction with each Tm value in Table 3 correspond to the amplitudes associated with the temperature rise Tm pre- and post-snap. The singlevalued amplitudes given in Table 4 simply denote the first yield state. It can therefore be suggested that there are three imperfection range classifications: low, medium and high. Low imperfection ratios are associated with permissible temperature rises restricted, subject to safety factors, by maximum temperature rises associated with snap buckling. Medium ratios will involve careful consideration of sub-yield snap buckling whilst high ratios will afford the most stable and predictable basis for yield stress based permissible temperature rises. The imperfection ratio ranges associated with these classifications vary between the different modes and will be subject to individual pipeline parameters; note Table 1. The above considerations are illustrated in Fig. 11 and shown qualitatively in Fig. 12. A possible fourth classification, wherein snap and yield occur simultaneously, is also denoted in Fig. 12. Figure 11 is, subject to the imposition of safety factors, effectively a design chart for thermal submarine pipeline buckling with respect to the particular pipeline parameters employed. The general trends are that for
320
Neil Taylor, Aik Ben Gan
1
,lP y l --Ty
---
~j
~p
vm or w m
vm OF Wm
fm>!y
T m < ly
predominant
Low rQho E
I S
Ty only Gtobte p o s t buckling path
$nO,p involved ~-
r~
Vm Or Wm
Medium ratio
High rotio
L
I N
\ \ \
Vnq or wn/ *F', z [
t
"1
Imperfection Ratio Vorr,,'L o or Wom/L o
Fig. 12. Qualitative imperfection ratio relationships.
low imperfection ratios, the permissible temperature rise decreases with increasing imperfection ratio, whilst for high ratios, the permissible temperature rise increases with increasing imperfection ratio. For medium range imperfection ratios, the closely related vertical mode and lateral mode 1 loci possess a minimum turning point whilst lateral mode 2 generates a locus wherein the permissible temperature rise steadily increases with increasing imperfection ratio. Finally, noting the previous argument relating to the role of the respective idealised modal maximum compressive stress levels in determining the relative importance of the various lateral modes, it is suggested that for high imperfection ratios, the (stable) curves appertaining to lateral modes 3 and 4 appropriate to Fig. 11 will lie between those given for lateral modes 1 and 2 and above that for lateral
Submarine pipeline buckling--imperfection studies
321
mode 2 respectively. The corresponding infinity mode locus will then lie above that corresponding to lateral mode 4.
7 CONCLUSIONS An archetypal design chart for use in thermal submarine pipeline buckling has been produced. Clearly, such a chart could be prepared for alternative pipeline parameter and imperfection ratio data. The study has illustrated the effects of basic physical imperfections appropriate to the problem involved. The modes investigated have been considered upon the basis of their being the most sympathetic and susceptible to such localising physical 'triggers'. A yield stress based permissible temperature rise criterion, suited to both the design of new pipelines and the assessment of existing pipelines, has been suggested. This is considered to be superior to the somewhat vague safe temperature rise concept previously delineated despite the general absence of quantitative knowledge of the probable idealised critical state snap. That is, the idealised criterion offered little help in determining the appropriate post-buckling characteristics, making identification of the location of the pipeline, should buckling occur, impossible. The proposed approach affords formal post-buckling displacement characteristics, including information on the onset, or otherwise, of plasticity during buckling. Whilst it is not suggested that the present study totally overcomes such problems, difficulty lying with identifying any given physical imperfection, data trends can now be determined. Of additional interest is the possibility that recovery upon subsequent cooling would not be total, due to the non-conservative nature of the frictional forces involved. This leads to the realisation that, upon recovery, the imperfection ratio would have increased with a consequent change in post-buckling characteristics should thermal buckling recur. Such post-buckling considerations are perhaps particularly important with regard to trenched pipelines undergoing vertical buckling--'popups'--in view of the possibility of raised pipelines fouling, for example, anchor cables. Finally, it must be recalled that the effects of residual stresses remain to be considered, although their effects could perhaps be interpreted in terms of equivalent initial lack-of-straightness. Further, the eccentricity
322
Neil Taylor, Aik Ben Gan
of the frictional forces with respect to the pipe centreline has been neglected. Both features require further study.
ACKNOWLEDGEMENT The authors wish to thank Professor Alastair Walker and Dr Charles Ellinas of J. P. Kenny and Partners, London, for discussions held with regard to the contents of this article. However, the opinions expressed, and any errors incurred, are the authors' alone. REFERENCES 1. Hobbs, R. E., Pipeline buckling caused by axial loads, Journal of Constructional Steel Research, 1(2) (January 198 l) 2-10. 2. Hobbs, R. E., In-service buckling of heated pipelines, Journal of the Transportation Engineering Division, ASCE, 110(2) (March 1984) 175-89. 3. Taylor, N. and Gan, A. B., Regarding the buckling of pipelines subject to axial loading, Journal of Constructional Steel Research, 4(l) (1984) 45-50. 4. Taylor, N., Richardson, D. and Gan, A. B., On submarine pipeline frictional characteristics in the presence of buckling, Proceedings of the 4th International Symposium on Offshore Mechanics and Arctic Engineering, ASME, Dallas, Texas, 17-21 February 1985, 5(18-15. 5. Palmer, A. C. and Martin, J. H., Buckle propagation in submarine pipelines, Nature, 254 (March 1975) 46-8. 6. Bjerrum, L., Geotechnical problems involved in foundations of structures in the North Sea, Geotechnique, 23(1) (1973) 319-58. 7. Taylor, N. and Gan, A. B., Refined modelling for the lateral buckling of submarine pipelines, Journal of Constructional Steel Research, 6(2) 143--62. 8. Lyons, C. G., Soil resistance to lateral sliding of marine pipelines, 5th Offshore Technology Conference, OTC 1876, 2 (1973) 479-84. 9. Gulhati, S. K., Venkatapparao, G. and Varadarajan, A., Positional stability of submarine pipelines, IGS Conference on Geotechnical Engineering, 1 (1978) 430-4. 10. Anand, S. and Agarwal, S. L., Field and laboratory studies for evaluating submarine pipeline frictional resistance, Transactions of the ASME, Journal of Energy Resources Technology, 103 (September 198l) 250-4. l l. Marek, P. J. and Daniels, J. H., Behaviour of continuous crane rails, Journal of the Structural Division, ASCE, ST4 (April 1971) 1081-95. 12. Granstrom, A., Behaviour of continuous crane rails, Journal of the Structural Division, ASCE, ST1 (January 1972) 360-1. 13. Kerr, A, D., Analysis of thermal track buckling in the lateral plane, Acta Mechanica, 30 (1978) 17-50.
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14. Kerr, A. D., A model study for vertical track buckling, High Speed Ground Transportation Journal, 7 (1973) 351-68. 15. Kerr, A. D., The effect of lateral resistance on track buckling analyses, Rail International, 1 (1976) 30-8. 16. Tvergaard, V. and Needleman, A., On localized thermal track buckling, International Journal of Mechanical Sciences, 23(10) (1981) 577-87. 17. Timoshenko, S. P. and Gere, J. M., Theory of Elastic Stability, 2nd edition, McGraw-Hill--Kogakusha Ltd, New York, 1961, 76-81.