refined modelling for the lateral buckling of submarine pipelines

J. Construct. Steel Research 6 (1986) 14.3-162

Refined Modelling for the Lateral Buckling of Submarine Pipelines

Neil Taylor and Aik Ben Gan

Department of Civil Engineering, Sheffield City Polytechnic, Pond Street, Sheffield SI IWB, UK

SYNOPSIS

In-service buckling Of submarine pipelines can occur due to the institution of axial compressive forces caused bv the constrained expansions set up by thermal and internal pressure actions. An integral part of this area of study involves the resistance to movement provided by the supporting medium. Previous attempts at modelling the appropriate behaviour have employed fully mobilised friction forces. Herein presented is a set of analvses which incorporate consistent deformation-dependent resistance forces. This feature enables a more rational interpretation of submarine pipeline buckling behaviour to be established.

NOTATION

A a,

C, E F

I, I k k w-k~ L. LI

Cross-sectional area. Unprescribed coefficient. Axial compression of pipe element. Young's modulus. Shear force. Axial friction parameter. Lateral friction parameter. Second moment of area of cross-section. Constant of integration. Constants in Table 1. Buckle lengths.

143 J. Construct. Steel Research 0143-974X/86/$03.50 © Elsevier Applied Science Publishers Ltd. England. 1986. Printed in Great Britain

144 Veil Taylor, ,4zk Ben (ian

L~ M P P,~ P, q T T, Trnm

/2

l t ,

W

W m

142rn m

I,V m~

X

Slip length. Bending moment. Buckling force. Prebuckling axial force. Post-buckling axial force at state S. Submerged weight of pipeline per unit length. Temperature rise. Critical temperature rise. Minimum safe temperature rise. Axial deformation of the pipe. Resultant longitudinal movement at buckle/slip length interface. Fully mobilised axial displacement. Lateral deformation of the pipe. Maximum amplitude of the buckled pipe. Buckle amplitude at T ..... Post-buckling amplitude at state S. Fully mobilised lateral displacement. Spatial co-ordinate. Coefficient of linear thermal expansion. 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. ~ Analyses have been primarily oriented about thermal action with oil and gas temperatures potentially ranging up to I(R)°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

P,~ = A E o ~ T t l~

where A E is the axial rigidity of the pipe and ~ 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

Lateral buckling of submarine pipelines 145

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 employed the simplifying non-conservative assumption that th.e friction forces are fully mobilised throughout. Further. this precludes the ability to define a critical temperature rise at which idealised axial-flexural bifurcation occurs. In the following analvses a rational modelling of the deformation-dependent friction

x,7..! r , i C ] e 1

L s L L - s

- K,D~ c -

L~ L LI L L S

,d M c J e ~,

Fig. 1. Lateral buckling mode topologies.

146 Vezl Tavhm Azk Ben Gan

forces is unde r t aken employing recently established axial and lateral friction response l o c i . 4 As a result, a more logical interpretat ion of the submar ine pipeline buckling problem is made available. In addiuon, thi', will assist in the deve lopment of non-linear analvses involx ing physical imperfect ions .

The five established lateral buckling modes which relate to ~nakine m o v e m e n t s across the sea bed are illustrated in Fig. 1. ~: - \ t tention i, restr icted to these lateral modes given the less critical nature ,~f ~ert~cal buckl ing except in cases where trenching is i nvo lved . " Axial and lateral de format ions are deno ted bv u and w respectively. Fhe appropriate buckl ing lengths are deno ted by L and L, whilst L, represents the slip length. Axial friction resistance is generated through the slip length ~ hilst lateral friction resistance occurs through the buckling lengths. To datc. these friction forces have been assumed to be fully mobilised throughout their respective regions. ~-3 Material behaviour is taken to be elastic: gros~ sectional dis tor t ion as associated with laying operat ions ' is not a factor here.

2 G E O T E C H N I C A L F A C T O R S

Recen t geotechnical exper imentat ion ~ with respect to North Sea condi t ions ~ has provided information relating to the nature of the d e f o r m a t i o n - d e p e n d e n t friction force behaviour with regard to typical submar ine pipeline parameters . Based upon these findings, refined friction force characteristics are depicted in Figs 2 and 3 for axial (ut and lateral (w) pipeline movements respectively. The appropriate fully mobi l ised axial and lateral friction coefficients are given bv~b x and ,b~. The m o v e m e n t s corresponding to the a t ta inment of full mobilisation are d e n o t e d by u , and w,. The friction force parametersfA andf~, discussed in detail in the ensuing analyses, serve to complete the requisite loci definit ions. Suggested values for the parameters <b~. u~. aSt and w... for submar ine pipelines take the form ~ "

aS.~ = 0"7

u,b = 5mm ~ ~2t

rbl = I ' ( ) [

) w+ = 30 mm


. ~ ~, / 'Fu l ly Moblhsed LOCUS

f~- Smtall Score [×pertmental Locus 4

- ~uh ~coLe Des,gn Curve

~/%, : 1 - e-2%/"~

{ 2

t • 1 i i i

fAq , - ~ 0 ~ u

q : Submerged self-weight per un~! length ~ ppe

~onvemtlor

05 OE, 37 0 0 I0 u/u¢

Fig. 2. Generalised axial friction characteristics.

9%

'3 S

- 2

~ , / ~ J l } <'oh, seS Locus

/ / ~ : S ~ e r g e a se;f-~e,ght per und le-gth o '~ piDe

Cornver"i i o"

' ' ' ' ' ' ' O ' - • ' ' ' '~'- - ' ~ - S-3 3'-. } 5 E' 3 " O-S 3 9 1 .9

w/vv~)

Fig. 3. Oeneralised lateral friction characteristics.

148 .Vetl Taylor..4tk Ben Gan

and the design curves given in Figs 2 and 3 thereby empire

and

FL,"&I = l - e ...... ~ ,4.

noting the appropriate conventions given in the figures, fhesc expres- sions give good agreement with the designated non-dimenslonalised exper imental curves and. in addition, accord with the appropriate fullx mobilised asymptotes with f~,/cb~ ~fL/&t ~- 1 at u/u,,o = 1 and w;w~ = 1 respectively. It is to be noted that eqns (3) and (4) offer ~uperior modell ing characteristics to their precursors given e l s ewhere .

3 ESSENTIAL S T R U C T U R A L RELATIONSHIPS

Prior to the incorporation of eqns (3) and (4) within the proposed analyses, it is useful to list all the essential structural relationships, based on fullv mobilised friction resistance assumptions, obtained from previous submarine pipeline ~ ~ and related rail track ~'' studies. Noting the constants given in Table 1. the buckling force set up in the L. L, regions i~ given bv

E1 P = kl L?

T A B L E 1 C o n s t a n t s f o r L a t e r a l B u c k l i n g M o d e s

Mode kl k, k~ k4 k~ k.. (eqn (5)) (eqn (6)) (eqn (7)) teqn (7)~ (eqn fS)~ ceqn h~))

I 2[)-19 3 .851 2 × 10 ; 141 1-412-2 5 × ll) ¢ l-l} _-4B.~ "~ ~ 14} "

2 4rr-" 5 . 5 3 2 × 10 ~ 14) 8 .71~ × 14) 1 0 1 . ,4 . , t41

3 8 -515 0 -165 12 2 .588 1-787 8 x 10 ' 2 . 5 9 8 5-339 7 ~ 10

4 2 8 . 2 0 I . I I48 × 10 -" 1-61)8 2.771 6 × 141 s 1.6ll8 2 -143 - l(! ~'

-" N A N A ,~ce e q n tt~) :c n- 7 - 1 1 9 2 x 10 :

N A = n o t a p p l i c a b l e

Lateral buckling of subrnarine pipelines 149

where E 1 is the flexural rigidity of the pipe. The maximum amplitude of the buckle takes the form

chl qL 4 Wm = k , - - (6)

E l

where q is the submerged self-weight of the pipe per unit length. The resultant longitudinal movement at each buckle/slip length interface is given by

u, = k ~ ( P " - P ) L - k 4 ( - ~ I (7)

whilst the relationship between the pre-buckling force P,,. given by eqn ( 1 ). and the buckling force P for modes 1 to 4 is derived to be

P,, = + k , + , q z . 1 + 1 j (8)

For mode z, eqn (8) is replaced bv

(9)

As a corollary, it is instructive to assess the degree of full mobilisation actually involved in the above submarine pipeline studies. ~-' Noting the values for (b ~. 0St. u~ and w~ given in eqn (2). then Table 2 affords the

TABLE 2 Percentage of Region Subjected to Fully Mobilised Resistance Employing the Fully

Mobilised Friction Force Criterion

Mode % of buckle length L wherein w >_ w+ (w, = 30mm)

% of slip length L~ wherein u > - u , ( u , = 5ram)

1 85.5 76"6 2 88.0 75.7 3 98.5 74-1 4 98.0 73.7 x 05.8 NA

NA = not apphcable

150 Nell Taylor, Ark Ben Gan

appropriate information with regard to the state corresponding to the respective minimum safe temperature rise, ~ ' The most obvious deduction is that. employing the fully mobilised criterion, the axial friction modelling is relatively cruder than the lateral friction modelling throughout. Initially. therefore, it is proposed to re-define the axial friction force resistance employing the locus given in Fig. 2 a~ designated by eqn (3).

4 MODE 1 ANALYSIS: NON-LINEAR AXIAL RESISTANCE IN CONJUNCTION WITH FULLY MOBILISED LATERAL

RESISTANCE

The essential features of mode 1, with regard to established studies. ~ ~ are shown in Figs 4(a) and (b). The former figure details the topology whilst the latter depicts the axial force distribution within the pipe. Figures 4(c t and (d) depict the corresponding revised topology and axial force distribution.

Noting the non-linear axial force distributions in the slip length reglon~, the nature offx is detailed in Fig. 5(a) which illustrates a typical element of pipe within the slip length region (x> L). Axial compression is denoted by C,. Considering the longitudinal equilibrium of the elemental pipe 8x, then

8 C , = f A q a x ~ !~tl

so that

dC, - f , x q ~ilt

dx

Incorporating the appropriate thermal effect detailed in Fi~2. 5(b). compatibility gives

- u + d x = o~T(L, + L - r)

Differentiating with respect to x gives

( C, = AE oct dx ]


eL q

I ¢ ¢ ¢ '~ ½ ¢ ¢ ¢ ¢ I

L< L L L~

(a) Fudy Mob, hsed Topology

/ b ) Axtol Force Distr lbut lor"

eL q

2% J~ ~ e2'~J,'u¢) ~, z - , , CAq( 1 -

- , - # -~. - - W

: / Nor , -b~ecr ,t,~c, Res,stonce Topotogy

, J ~,~,3 =orce ~ istr D'dtlCn

Fig. 4. Fully mobilised and revised buckling topologies.

Substituting eqn (13) into eqn ( l 1 ) yields the field equation within the slip length region

d:u AE--~v,_ = - f ~ q L < - x < - L + L , (14)

The associated slip length boundarv conditions take the form

lim [u.du/d.r] = 0 ( 1 5 )

152 "Vetl T a y l o r . . 4 t k Ben Oan

i ~ - ~ , I

i

~ ~ c . ~

i ~ •

~ - - __ f - -

k

" t

Z

Z

Fig. 5. Slip length detail.

a n d

du ] _ ( P , , - P ) cLr ] L AE

(10)

Equat ion (15) can be considered to be a regularity condition which assists in precluding the prejudicial requirement of assuming some function u = f ( x ) .

Combining eqns (3) and (14) affords the non-linear slip length field equation

d2l t AEcLv2 = _d~.xq( I _ e 2 ...... b) ( I ' )

noting that the change of sign of the exponent is due to u and t , bem~4 co-oriented within the slip length region--note that Fig. 5 employs a different convention from Fig. 2. Employing the identity

d x : du L cLr j


then eqn (17) becomes

[ d u ) : 2~Aq {e-'~ "~ ) - - - u +k ~dx A E \ 5 (19)

where k is a constant of integration. Substituting eqn (15) into eqn (19) for the evaluation of k gives

u)] 1'2

(2O)

Equating this expression with the boundary condition given in eqn (16). and noting u]l. = u, yields

(P, P) = [26~qAE(e:%/5 u,)]~/2

Accepting that eqn (7) remains valid in view of the fact that the lateral friction force is fully mobilised, then. noting the values given in Table 1.

(P, -P)L 1.023× I0-~ ( ¢~Lq / 2L 7 (~'~) u , - , 4E \ t:! ] --

Solutions for ( P , - P) and u, are thereby obtained in terms of discrete values of L employing a computerised non-linear iterative algorithm involving eqns (21) and (22). Again noting that eqns (5) and (6) together with the respective values given in Table 1 remain valid, then locus 1. depicted in Fig. 6. is'produced by substituting the results for (P, ,- P) into eqns (I). (5) and (6). For this action-response locus, the external pipe diameter was taken to be 650 mm. the wall thickness 15 mm. the submerged self-weight 3-8 kN m -E. the Young's modulus 206 kN mm-: and the coefficient of linear thermal expansion 11 × 10-"°C-~; 1 the data given in eqn (2) complete the requisite numerical definition. The dashed portions of the locus denote computations involving buckling slopes in excess of 0.1 radians. Prior to discussing the implications of this analysis. it is proposed to briefly consider the equivalent analyses of the remaining lateral modes.

1 5 4 Veil Taylor, A rk Ben G a n

,~ 9 0

Q.

E

1

BO

i

I 'I//\ ! \ /

J I I I I I

~'~ ~ 15 .:' "~

Fig. 6. Comparison of lateral buckhng modes.

5 MODES 2, 3 AND 4: NON-LINEAR AXIAL RESISTANCE IN CONJUNCTION WITH FULLY MOBILISED LATERAL

RESISTANCE

Equations (10) through (21) remain valid for these three modes. Equation (7) is again employed in conjunction with Table I to generate an equivalent expression to that of eqn (22). The analvsis of mode ~:, in which there is no generation of axial friction force, remains as previously. '~ The appropriate behavioural loci are depicted in Fig. 6.

6 FURTHER CONSIDERATIONS

The action-response loci for modes 1. 2, 3 and 4 in Fig. ~ differ only slightly from their fully mobilised equivalents---mode ~, remains


unchanged. For example, with regard to mode 1, the minimum safe temperature rise (T,,.°) is reduced by only 0.03°C employing variable axial resistance. This should not detract from the study, however, in as much as the effect of employing the simpli~ing fully mobilised assumption has been shown to be reasonable from the findings of a rational model. Further, the ordering of the modes in terms of their respective minimum safe temperature rises is also shown to remain unchanged.

It should be noted that the solutions for the respective w = f ( x ) remain as previously. ~ As illustrated in Fig. 6, the behavioural loci approach the thermal axis asymptotically: this is a consequence of assuming the lateral friction resistance to be fullv mobilised even for vanishingly small displacements." In order to define a critical temperature rise, therefore, it is necessary to model the lateral friction force in a rational manner. Further, this will serve to provide a fully rational modelling of the buckling problem.

7 MODE 1: NON-LINEAR AXIAL AND LATERAL FRICTION RESISTANCES

In order to establish a critical temperature rise, To, it is necessary to incorporate the lateral friction-response locus given in Fig. 3 and designated by eqn (4). The appropriate topology and axial force distributions are depicted in Figs 4(c) and (d) except that the lateral resistance, d~Lq. is now replaced by d~Lq (1 - e-~W'w*).

The flexural parameters are in accordance with the detail illustrated in Fig. 7. Bending moment and shear force are denoted by M and F respectively. Applying transverse and rotational equilibrium, noting the self-compensating flexurai end-shortening effect, affords

dF - f l q (23) dx

and

dM dw \ - F = --~-r + P ~ ) (24)

Further. incorporating the appropriate moment-curvature expression, ~2

156 ,Ved Taylor..4tk Ben (;an

\ ,

1

~.1 .,~ U - ~-

7 "--~ ~ J, 4,

{ _ /

Fig. 7. Flexural topology

d i f f e r en t i a t i on of eqn (24) l inear i sed field equa t i on

d 4 w d e w E I - - ~ 4 + P d.r~ - t~ q

and combin ing with eqn {_.~) x ields the

!25',

No t ing eqn (4), eqn (25) can be rewri t ten

d a w d -~ w E l - - r - z - ~- P - ~& q(l - c ..... ')

d_v ~ C l _ l c

T h e c omplex i t y of this field equa t ion prevents a closed to rm solut ion be ing ob t a ined . A b o u n d a r y col locat ion p ro ced u re is t he re fo re ins t i tu ted e m p l o y i n g the assumed series

4

I*~2 m i ~ l l t )~ . t -~L , ~ _


where a, denotes a typical unprescribed coefficient. The symmetry of the buckling mode is incorporated directly in eqn (27). Accepting that relationships are sought between P, L and w,, as indicated by eqns (5) and (6) with regard to the previous study, then a seven point collocation procedure is required. Employing the following five conditions and identities

w I = 0 L

dw] = 0 dx L

dZw]' = 0 (transversalitv m) 7 1 t

I)

( El--~r4 +P--'-~ L = --~bLq(1--e°) = 0

t (28)

the respective coefficients of a, can be defined, such that

24 { x [ x ) li" = 14'm [ 1 -- T ~ Z , 7 ~L - ~ ( L ) 6 ''~- 3 ( L ~ 8 7 ~ L } ] (29)

Further. equilibrium demands that the shear force at the ends of the buckle length takes the form

_ Eld"wl f t -6""~) dx ckr 3 L = ~Lq (1--e (30) I

which gives

7cbLqLS ~, L vt'nl- 192E1 (l-e-6"'"6)dx (31)

158 ,Veil Taylor, Ark Ben Gan

and that the central bending moment affords

EI---~-5- ,) = d~LqL ,, ( I - e ...... ~)dx-(&q ,)

resulting in. noting eqn (31 I.

192EI P -

l

4 L f , ' ( 1 - e . . . . ~)cLr

(1 - e

~32

Slip length considerations are incorporated as previously employing cqn (2l). The matching compatibility condition at the ends of the buckle length takes the form "'

( P , ) - P ) L l f L ( d w ) - U{L=U'-- AE .~ ,) ~ ckv (34)

which yields, noting eqns (29) and (3l),

u, - A E 9,102× I()-4L~ [ E1 ] ( 1 - e ....... ~)d_v (35) a

( P , , - P) L

This represents the rational equivalent of the fully mobilised expression given in eqn (22). The solution is as previously; eqns (21) and (35) are solved iteratively, the integral steps being carried out numerically for any specified value for w~ (mm).

Figure 8 depicts the resulting action-response locus together with the formerly defined locus given in Fig. 6. Scaling factors preclude inclusion of those parts of the locus in the vicinity of the minimum safe temperature rise (w~ ~ 2.4 m as per Fig. 6). The revised curve becomes effectively, or graphically, indistinguishable from the former locus well before the minimum safe temperature rise state (Tram). At this state, the rational value is l% lower than that for the previous analysis (63.6°C). However. whilst the fully mobilised assumptions again appear to be validated, the" critical temperature rise has been established: this enables, in formal terms, superior definition of the post-buckling state.


7- %J0

~. ~_._ ~ ~ - Non-Ldneor A,IOI clr'd Full) ~ '~-- ~ Meb,l.sed Loterol Res,stances

2 ~×st on~- Loterol Res~stonce Curves

,% I 2 '~4' 3'C ~C 5~

Buckle Ampl,tude w m [mm)

Fig. 8. Rational modelling for mode 1.

i

6C "~C

8 COMMENTS AND CONCLUSIONS

The rational modelling has achieved two objectives. First. it supports the validity of the simplified fully mobilised studies. Second, a critical temperature rise has been established which is important, in formal or idealised terms, in that post-buckling characteristics can be assessed.

Figure 9 depicts the post-buckling mechanism. That is, an idealised pipeline will not buckle until the critical temperature rise, Tc as denoted in Fig. 9(a). is reached. The pipe then snap buckles to point S. The static topologies defined in Figs 1 and 4 only relate to this and ensuing states. Figure 9(b) depicts the corresponding buckling force characteristics. At the critical state denoted by Pc, the force in the pipeline is given by Pc = P,,. After snap. the force in the buckle reduces to the unique static value P -- P~. P, and location S being prescribed by the statically unique buckling amplitude Wm= W~. The chain-dotted locus defines a path which the idealised system cannot, in principle, follow due to snap at temperature rise To. This locus is. however, important in that it forms an upper bound to corresponding imperfection studies, particularly the theoretically stable portion between T~,n and state S. States S and P~ have

160 Ned Taylor, Ark Ben (Tan

A

i,

! <:m~- ~ L r ",

J¢

Cf \

Fig. 9. Post-buckling mechamsm

not previously been reported. The values for P,. P, and w,~, for mode I arc 52-48 MN, 386 kN and 124-5 m respectively. (T ...... = 63,1~C, wm~ - 2-4 m,)

In view of the importance of imperfections in realistic pipeline buckling performance, note below, and the general agreement of the rational loci with their fully mobilised equivalents, fully rational modelling of the remaining lateral modes has been left to further study. Further, such


study would benefit from the incorporation of geometric non-linearity as the point prescribed by w~, involves relatively large slopes. That is, the general post critical state agreement between the two mode 1 loci depicted in Fig. 8 shows that the fully rational modelling will also incur the onset of slopes in excess of 0.1 radians shortly after theoretical attainment of the minimum safe temperature rise state Tin,° (W~m -- 2"4 m)- -no te Figs 6 and 9. The locus is therefore conservative in this larger deformation range. ~ It should be noted that the value of Tc for any given analysis is sensitive to the lateral frictional curve employed: note the role of w~ in eqn (4). Such curves can be classified in terms of their respective fully mobilised displacements, w+. As w~ increases, the critical temperature rise. TL. will fall. Indeed, w~ can be such that T~.< T .... affording a completely stable post-buckling path with no T,,,° turning point.

A number of further observations can be briefly noted. First, given the validity of the fully mobilised analyses in predicting generalised behaviour, the finite length of slip, L,, can be determined, if required, from L, = ( P , - P,)/daAq for the post-buckling static state denoted by S. Associated studies ~ suggest that fully mobilised analyses underestimate the slip length at lower values of Wm (i.e. w,, < Wmm). Should the total length of pipe be less than the overall affected length, that is 2 (L + L,) with regard to the rational mode 1 modelling, the required temperature rise for state S will increase. ~ Further, post-buckling characteristics can be influenced by modal interaction. This is perhaps most important in cases ~.here trenching is involved in which vertical buckling ~ ' might occur initially. Fully rational vertical buckling studies are in preparation.

Finally. the influence of imperfections must be noted. Diagrammatic representations given elsewhere ~ indicate the importance of the minimum safe temperature rise state. However, thev also substantiate the importance of the w°,, state, to which all such imperfection loci should converge. Typical qualitative imperfection loci are illustrated in Fig. 9(a), one locus displaying snap buckling (type I), the other affording a stable post- buckling path (type 2). Of particular interest with respect to such loci is the fact 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, anv initial imperfection would be increased with a consequent change in post-buckling characteristics ~hould thermal buckling recur. For example, an imperfection locus type 1 could degenerate into a locus type 2. Imperfection studies are proceeding.

162 Ved Taylor, Ark Ben Gun

A C K N O W L E D G E M E N T

The authors wish to thank Professor Alastmr Walker and Dr Charlc~ Ellinas of J. P. Kennv and Partners, London, UK, for discussions held with regard to the contents of this article. However. the opinion~ expressed, and any errors incurred, are the authors" alone.

R E F E R E N C E S

1. Hobbs, R. E., Pipeline buckling caused bx axial Ioad~, ,I. (',,t~truct. Steel Res., 1 (1981) 2-10.

2. Hobbs, R. E., In-service buckling of heated pipelines. J Trat,sporr Et~e Div., ASCE, 110 ('1984) 175-89.

3. Taylor, N. and Gan, A. B., Regarding the buckling of plpehncs ,ublcct t,~ axial loading, J. Construct. Steel Res., 4 (1984} 45-50.

4. Taylor, N., Richardson, D. and Gan, A. B., On submarmc pipeline frictional characteristics in the presence of buckling. Proceedin¢s of the 4th International Symposium on Offshore Mechanics and Arctic En~,meerme. ASME, Dallas, Texas. 17-2l February 1985, 508--15

5, Palmer, A. C. and Martin, J. H., Buckle propagatum m submarine pipelines, Nature, 7,,$4 (1975) 46-8.

6. Bjerrum, k., Geotechnical problems involved m toundanon~ ,~f structure, in the North Sea, Geotechnique, 231 (1973) 319-58.

7 Lyons, C. G., Soil resistance to lateral sliding of marine pipelines. 5*/: Offshore Technology Conference. OTC 1876, Vol. 2, I~73. 470-ga

8. Gulhati, S. K., Venkatapparao, G and Varadarajan. A.. Posmona[ stability of submarine pipelines, IGS Conference ,m Geotechntcai Engineering, Vol. 1, 1078, 430--4.

9. Anand, S. and Agarwal, S. g., Field and laboratory ~tudles for evaluating submarine pipeline frictional resistance, Trans. AsME, J Enerev Resources Technol., 11}3 (19gl) 250-4.

10. Kerr, A. D., Analysis of thermal track buckling m the lateral plane, Acta Mech., 30 (1978) 17-5().

11. Kerr, A. D., On thermal buckling of strmght railroad tracks and the effect o~ track length on the track response, Rail International. 9 (1970) 750-6£.

12. Taylor, N. and Hirst. P. B., Regarding flexural curvature Pro< lnsm ('it' Engrs, 77 (1984) 399-400.

refined modelling for the lateral buckling of submarine pipelines

Documents