In-plane nonlinear localised lateral buckling of straight pipelines

Engineering Structures 103 (2015) 37–52

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In-plane nonlinear localised lateral buckling of straight pipelines Jianbei Zhu, Mario M. Attard ⇑, David C. Kellermann School of Civil and Environmental Engineering, The University of New South Wales, Sydney 2052, Australia

a r t i c l e

i n f o

Article history: Received 1 March 2014 Revised 26 August 2015 Accepted 28 August 2015

Keywords: Hyperelastic Straight pipelines Localised lateral buckling Critical length Safe temperature increment

a b s t r a c t The lateral buckling problem for thermally loaded pipelines is known to involve a localisation phenomenon within a limited region of the pipeline rather than an extensive global mode shape. In this paper, a strategy is presented to investigate the localised lateral buckling of pipelines under thermal loading and friction whereupon the constitutive relations are derived for thermal stress and finite strain based on a hyperelastic constitutive model. Using this hyperelastic formulation, we investigate the critical overall pipeline length above which localised buckling remains unchanged. The results show that increasing the length of the pipeline or changing the end boundary conditions when the pipeline length is greater than or equal to the critical overall length does not influence the localised buckling behaviour. The solutions to several examples are compared with the results in the literature and validated by use of the finite element package ANSYS. Parametric studies on diameter, imperfection, friction and shear deformation effects are subsequently performed on examples that identify which factors influence the localised buckling of thermal pipelines. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction If pipelines under thermal loading are situated on top of level land or submerged on a flat seabed, lateral buckling movements in the horizontal plane may occur if the axial friction restrains large enough axial compression under thermal loading in the pipeline. Many studies have been undertaken for the lateral buckling of pipelines. The most widely accepted formulations were proposed by Hobbs [1] whose solutions have been widely used by many researchers and engineers. In his analysis, Hobbs used the buckled and adjoining region assumptions that were made by Kerr [2] in his investigation of the buckling of railway tracks. In Hobbs’ analysis, it was assumed that the pipeline buckles in the lateral direction in a buckled region in which the axial compressive load was taken as constant, whereas the pipeline in the adjoining region only deformed axially. The axial resistance in the buckled region was neglected. Hobbs studied both the vertical and lateral buckling responses of perfectly straight pipelines and theoretical solutions to find the safe temperature increment were presented. Hobbs also further developed the formulations for thermal buckling of pipelines to cover a number of practical cases, in Hobbs [3], Hobbs and Liang [4], Ballet and Hobbs [5]. Hobbs’ method and solutions have been adopted by several of the studies on the buckling of pipelines Yun and Kyriakides [6], Orynyak and Bogdan [7], Orynyak and ⇑ Corresponding author. Tel.: +61 2 93855075. E-mail address: [email protected] (M.M. Attard). http://dx.doi.org/10.1016/j.engstruct.2015.08.036 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.

Bogdan [8], Taylor and Gan [9–12], Taylor and Tran [13,14], Kershenbaum and Harrison [15], Wang et al. [16], Shi et al. [17]. These researchers extended Hobbs’ method and made some improvements but the analysis was still based on the buckled and adjoining region assumptions or equivalent boundary condition assumptions. Yuan et al. who developed a model for landslide or debris flow impact on pipelines Yuan et al. [18,19] proposed that there is lateral displacement in the adjoining region. However, the lateral displacement was assumed to be very small and the lateral displacement in the adjoining region was just used to control the precision of the results. Taylor and Tran [13,14] carried out a series of experiments to evaluate the theoretical correlation for the upheaval buckling of single pipes with initial imperfections. The initial imperfections influenced the localised buckling significantly and lowered the safe temperature increment Croll [20], Tvergaard and Needleman [21], Nielsen et al. [22], Schaminee et al. [23]. Moreover, the localised buckling usually initiates at the region with imperfections as observed by Villarraga et al. [24] Zeng and Duan [25] have also investigated the lateral buckling of gas and oil submarine pipelines modelled by an axial compressive beam supported by lateral distributing nonlinear springs taking the soil berm effects in the horizontal plane. It was found that their model was governed by a time-independent Swift–Hohenberg equation. Karampour et al. [26] also investigated the lateral buckling of long subsea pipelines with nonlinear pipe-soil interaction and provided a new interpretation of localisation based on an isolated halfwavelength model. Karampour and Albermani [27] provided

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J. Zhu et al. / Engineering Structures 103 (2015) 37–52

Nomenclature

List of symbols A geometric area of cross-section Ce right Cauchy–Green deformation tensor D outside diameter of the pipeline E elastic modulus E 2G þ K f u; f v axial and lateral frictions, respectively f x; f y distributed loads in x and y directions, respectively unit base vectors e1 ; e2 F deformation gradient tensor FT thermal part of the deformation tensor mechanical part of the deformation tensor Fe Fe rotated mechanical part of the deformation tensor G shear modulus g1 ; g2 covariant initial base vectors in the undeformed state contravariant initial base vectors in the undeformed g1 ; g2 state ^1 ; g ^2 g covariant initial base vectors in the deformed state ^1 ; g ^2 g contravariant initial base vectors in the deformed state I identity tensor I second moment of cross-sectional area Je volume invariant imperfection parameter ki kou ; kov initial stiffness for axial and lateral frictions, respectively ku ; kv limiting value for axial and lateral frictions, respectively L overall length of pipelines Lcr critical length M cross-sectional bending moment N axial force defined perpendicular to the cross-sectional plane Nc central axial force ph hydrostatic stress components of load intensity in the directions of N and pn ; pt Q, respectively Q shear force along the cross-sectional plane R rotation tensor

experimental and finite element results on buckle interaction in subsea pipelines used to develop buckle interaction envelopes. Albermani et al. [28] conducted experiments used to investigate propagation buckling in deep subsea pipelines. The phenomenon of lateral buckling of pipelines under thermal loading is closely related to that of the lateral buckling of welded railway tracks on a hot day. The research on the buckling of railway tracks offers feasible methods that can be applied to the lateral buckling analysis of pipelines under thermal loading. Most of the early pipeline buckling formulations originated from the work related to railroad tracks, as in Kerr [2,29,30], Kerr and Accorsi [31], Donley and Kerr [32], Grissom and Kerr [33]. Kish et al. [34,35], Samavedam et al. [36], Kish and Samavedam [37] did a number of experiments to investigate buckling of railway tracks. It was found that the actual buckled mode shape was largely influenced by the imperfections and both straight and curved tracks typically buckle out in a symmetric mode. Similarly, we could postulate that pipelines would tend to laterally buckle in a symmetric mode. Nearly all the theoretical studies on the lateral buckling of pipelines or railway tracks have employed Hobbs’ and Kerr’s buckled and adjoining region assumptions. However, these assumptions introduce many boundary conditions that constrain

S S11 ; S12 ds d^s ds

DT DT l DT s U u u1 ; u2 uo

v vc v cs v co

x; y

ao b dij h

j K k1 ; k2 k1T ; k2T k10

t P

u /f

Lagrangian stress tensor physical Lagrangian stresses, normal and tangential to the cross-section, respectively differential length of the undeformed reference axis differential length of the deformed reference axis thermal differential length of the deformed reference axis temperature change limit temperature increment safe temperature increment strain energy displacement vector displacement components in the x and ydirections, respectively tangential displacement of the reference axis displacement of the reference axis in the y direction central lateral displacement central lateral displacement corresponding to the safe temperature increment initial central lateral displacement coordinates of the reference axis with respect to a rectangular coordinate system coefficient of linear thermal expansion uþh Kronecker delta bending rotation curvature Lamé constant stretches in the x and y directions, respectively nominated stretches in the x and y directions, respectively longitudinal stretch measured at the centroid Poisson’s ratio second Piola–Kirchhoff stress tensor shear rotation coefficient of friction

the deformation of the pipeline and over-simplify the buckling problem. The formulations based on these assumptions therefore overestimate the axial compression in the pipeline. In this paper, we propose a new approach for determining the nonlinear behaviour of pipelines or railway tracks under thermal loading that does not rely on the buckled and adjoining region assumptions used by Kerr and no assumption is made for the axial compressive force distribution or displacements. The critical overall length of a pipeline under thermal loading is first investigated for the localised lateral buckling problem. It is shown that localised buckling happens on long pipelines subjected to thermal loading and increasing the length or changing the boundary conditions, does not influence the localised buckling behaviour if the length of the overall pipeline is greater than or equal to a critical length. The results obtained in this paper compare very well with those of Kerr and Hobbs and are further validated by comparison to the results produced by the commercial finite element package ANSYS [38]. The results for the axial compressive force distribution during the formation of a localised lateral deformation along the pipeline is also examined and discussed. In the parametric studies, the outside diameter, imperfection effects, the influence of the frictional resistance and shear deformations within the pipeline are also investigated. It should be noted that the method proposed in this

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paper is also capable of being used to investigate the lateral buckling of railway tracks under temperature change if the effects of the ties and fasteners between the tracks are ignored and the rail-tie structure is considered as a beam. 2. Displacement model Here we consider the straight pipeline as a straight prismatic beam. The longitudinal axis of centroids of the undeformed beam is taken as the x or 1 axis and the principal axis in the plane of the cross-section is taken as the y or 2 axis. The deflected shape of the beam is characterised by the deflection of the centroidal axis and the rotation of the cross-sectional plane as in Attard and Hunt [39]. The initial axis system is a Cartesian rectangular system. The initial material lines are assumed to be parallel to the Cartesian coordinate system. The initial tangent base vectors in the undeformed state are aligned with the centroidal axis of the beam and the cross-section principal axes. It is assumed that the plane of the cross-section remains plane but not necessarily perpendicular to the centroidal axis during deformation– the Timoshenko beam approximation. In the deformed state, the angle between the material longitudinal axis and the undeformed longitudinal axis consists of a bending component hðxÞ (anti-clockwise) taken to be a function of the longitudinal centroidal coordinate x only, and a shear component defined by a shear rotation angle uðxÞ (see Attard and Hunt [39]). A rectangular element of a twodimensional plane continuum is shown in Fig. 1. Points O and C within this continuum define an initial line differential ds in the b and C b define the deformed state d^s. The undeformed state and O variables e1 and e2 are unit vectors in directions 1 and 2, respectively; gi and gi are the covariant and contravariant initial base vec^ i and g ^ i are the tors in the undeformed state, respectively; g covariant and contravariant initial base vectors in the deformed state, respectively; and angle b is defined in Fig. 1. Thus, the covariant tangent base vectors in the deformed state are given by:

ðcos h e1  sinh e2 Þ k1 cos u ð ð Þ sin u þ h e 1 þ cos ðu þ hÞ e2 Þ ^2 ¼ g k2 cos u ð1Þ

^1 ¼ k1 ðcos ðu þ hÞe1 þ sin ðu þ hÞe2 Þ g ^2 ¼ k2 ðsinh e1 þ cosh e2 Þ g

^1 ¼ g

in which k1 and k2 are the stretches in directions 1 and 2, respectively. The relationships of the stretches (k1 and k2 Þ and the angles (b and u) with the displacement gradients can be expressed as

k1 sin ðu þ hÞ ¼ u2;1  k2 sin h ¼ u1;2 where

k1 cos ðu þ hÞ ¼ 1 þ u1;1 k2 cos h ¼ 1 þ u2;2

ð2Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 ¼ ð1 þ u1;1 Þ2 þ u22;1 u2;1 tan ðu þ hÞ ¼ 1 þ u1;1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 ¼ ð1 þ u2;2 Þ2 þ u21;2 u1;2 tan h ¼  1 þ u2;2

ð3Þ

in which u1 and u2 are the displacement components in the x and y directions, respectively, and u1;1 symbolises differentiation with respect the 1 (xÞ axis. Manipulating Eq. (2) we can write for the normal and shear components of stretch:

k1 cos u ¼ ð1 þ u1;1 Þ cos h þ u2;1 sin h k1 sin u ¼ ð1 þ u1;1 Þ sin h þ u2;1 cos h k2 cos u ¼ ð1 þ u2;2 Þ cos ðu þ hÞ  u1;2 sin ðu þ hÞ

ð4Þ

k2 sin u ¼ ð1 þ u2;2 Þ sin ðu þ hÞ þ u1;2 cos ðu þ hÞ At the centroidal axis (y = 0) we define the displacements:

u1 ðx; 0Þ ¼ uo

u2 ðx; 0Þ ¼ v

ð5Þ

where uo is the longitudinal displacement (in direction 1 or xÞ and v the transverse (in direction 2 or yÞ displacement of the centroidal axis, respectively. Thus, the displacements are written as:

u1 ¼ uo  y sin h

u2 ¼ v þ yðcos h  1Þ

ð6Þ

Substituting Eqs. (6) into (4), we get:

k1 cos u ¼ k10 cos u  yh;x k1 sin u ¼ k10 sin u

ð7Þ

with:

k10 cos u0 ¼ ð1 þ uo;x Þ cos h þ v ;x sin h

k10 sin u0 ¼ ð1 þ uo;x Þ sin h þ v ;x cos h

ð8Þ

k10 defines the longitudinal stretch measured at the centroid given by:

k10 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ uo;x Þ2 þ v 2;x

ð9Þ

The deformation of the material can be characterised by the deformation gradient tensor F, which defines a linear mapping of the initial line differential ds in the undeformed state to that in the deformed state d^s associated with a displacement vector u (assumed to be smooth and differentiable), such that:

d^s ¼ ds þ du ¼ F  ds

ð10Þ

  ^ i  gi ¼ dij þ u j  gj  gi ¼ I þ r  u F¼g i

ð11Þ

 ^ i ¼ ðd:ij þ u j  Þ gj are the covariant tangent base vectors in where g i the deformed state, dij is the Kronecker delta, I ¼ gi  gi is the iden tity tensor, r  u ¼ u j i gj  gi is the grad of the displacement vec tor, and u j  represents the covariant derivatives of the u j vector i

Fig. 1. Two-dimensional deformed parallelogram with initial Cartesian coordinate.

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component with respect to the coordinate corresponding to the index i. Using Eqs. (1) and (11), the components of the associated deformation gradient tensor are therefore:

 F¼

k1 cos ðu þ hÞ k2 sin h k1 sin ðu þ hÞ

 ð12Þ

k2 cos h

For the thermal pipelines in which the deformation due to thermal loading are taken into account, the deformation tensor consists of two parts – the thermal and the mechanical parts. Under thermal loading, there is an expansion due to the temperature increase and the thermal differential ds in the deformed state can be expressed as:

ds ¼ FT  ds

ð13Þ

where FT is the thermal part of the deformation tensor. Under the mechanical load, the pipeline deforms from the state ds to the final deformed state d^s. Hence:

d^s ¼ Fe  ds

ð14Þ

where Fe is the mechanical part of the deformation tensor. Substituting Eqs. (10) and (13) in (14) gives the multiplicative decomposition of the deformation tensor:

F ¼ Fe  FT





1 þ ao DT

0

0

1 þ ao DT

ð16Þ

where ao is the coefficient of linear thermal expansion and DT is the temperature change. Based on Eq. (15), the mechanical part of the deformation gradient tensor Fe is derived as follows:

" k1 cosðuþhÞ

Fe ¼ F 

F1 T

¼

1þao DT

k2 sin h 1þao DT

k1 sinðuþhÞ 1þao DT

k2 cos h 1þao DT

#

ð17Þ

with volume invariant:

J e ¼ det Fe ¼

k1 k2 cos u

ð18Þ

ð1 þ ao DTÞ2

The deformation gradient tensor Fe can be simplified by decomposing it into the rotated deformation gradient tensor Fe and the rotation tensor R. Here, the mechanical deformation tensor Fe is rotated by -h and the rotated deformation tensor is therefore expressed as:

F ¼ e

R1 h

  Fe ¼

k1T cos u

0



k1T cos u k2T

ð19Þ

where

k1T ¼

U¼

1 1 2 Gðtr ðCe  IÞ  2 ln J e Þ þ Kðln J e Þ 2 2

k1 1 þ ao DT

k2T ¼

k2 1 þ ao DT

ð20Þ

The mechanical part of the deformation gradient tensor results in the stresses in the pipeline. The constitutive equations can be derived by substituting the mechanical part into the strain energy function, which is detailed in the following section. 3. Hyperelastic constitutive relations for stress and internal actions The strain energy density function U for a compressible isotropic neo-Hookean material Simo and Pister [40], Attard [41], Attard and Hunt [42] is employed here. Under thermal loading, the strain energy density function only contains the mechanical part of the

ð21Þ

where G is the shear modulus, K is the Lamé constant, tr symbolises the trace of a tensor and Ce is right Cauchy–Green deformation tensor. Using Eq. (21), the second Piola–Kirchhoff stress tensor P ¼ Pij gi  gj is given by (see Attard and Hunt [42]):

P¼2

@U ¼ GI  ph ðCe Þ1 @Ce

ph ¼ G  K ln J e ;

ð22Þ

where ph represents a hydrostatic stress. The physical Lagrangian stress system is used here, defined with respect to the crosssectional plane. Using Eq. (22), the Lagrangian stress tensor is therefore:

S ¼ P  FTe

ð23Þ

According to Eqs. (19), (21), (22) and (23), we get the constitutive law for the physical Lagrangian stresses as:

" S¼P

FTe

¼

h Gk1T cos u  k1T pcos u Gk1T sin u

ph sin u k2T cos u

ð15Þ

For isotropic materials, substances expand at the same rate in every direction. Hence, the thermal part of the deformation gradient tensor is:

FT ¼

deformation. Hence, the strain energy function is written in the following form:

#

Gk2T  kp2Th

ð24Þ

Therefore, the normal and tangential stresses on the cross-section plane are taken from the first row of Eq. (24). For long straight transmission pipelines, the strain in direction 2 is negligibly small compared with the strain in the longitudinal direction. Thus, the stretch k2 is considered as unity. Expanding the stresses to second order in terms of the deformations, we can write the normal and tangential stresses as:

S11 ¼ Eðk1T cos u  1Þ

S12 ¼ Gk1T sin u

ð25Þ

2Gt where E ¼ 2G þ K; K ¼ ð12 tÞ is the Lamé constant, G is the shear

modulus and t is the Poisson’s ratio. These stresses are consistent with Attard’s proposal for beam actions Attard [41], Attard and Hunt [43], Attard and Kim [44] when the temperature change is zero. The material parameter governing the normal stress is not the elastic modulus E because the assumed two dimensional displacements restrain the dilation of the cross-section which is associated with lateral stresses not present under a uniaxial stress state Attard [41], Attard and Hunt [43]. A further approximation in beam theory is to replace E by E. The constitutive relationships for the internal actions can then be determined by defining the internal actions as the stress resultants over the cross-section:

ZZ

ZZ

ZZ

S11 dA Q ¼

N¼ A

S12 dA M ¼ A

 yS12 dA

ð26Þ

A

Here, N is the axial force defined perpendicular to the crosssectional plane, Q is the shear force along the cross-sectional plane and M is the bending moment defined by the stresses normal to the cross-sectional plane. Substituting the constitutive relationships Eqs. (25) into (26), incorporating Eqs. (7) and (20) gives for the internal actions:

EA GA ðk10 cos u  1  ao DTÞ Q ¼ k10 sin u 1 þ ao DT 1 þ ao DT EI M¼ ð27Þ h;x 1 þ ao DT N¼

in which A and I are the geometric properties of the cross sectional area and second moment of area, respectively. If the temperature does not change (DT is zero), the above Eq. (27) are consistent with Attard’s formulations Attard and Hunt [43], Attard and Kim [44], in which the thermal loading was not considered.

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J. Zhu et al. / Engineering Structures 103 (2015) 37–52

4. Equilibrium The beam equilibrium equations derived by Reissner [45] based on a one-dimensional theory for the plane problem for originally straight beams are used for the bucking analysis of thermal pipelines here. The equilibrium equations are as follows:

dN dQ  Q j þ pn ¼ 0 þ Nj þ pt ¼ 0 dx dx dM ¼ Nk10 sin u  Qk10 cos u dx

ð28Þ

where j is the curvature of the beam and pn and pt are components of load intensity in the directions of N and Q respectively, which are presented using the distributed loads f x and f y in x and y directions as:

pn ¼ f x cos h þ f y sin h pt ¼ f x sin h þ f y cos h

ð29Þ

In this paper, the in-plane nonlinear buckling of pipelines is investigated in which the effects of shear deformations as well as geometric nonlinearity due to large deformations are taken into account. To solve the nonlinear problem, the equilibrium equations are transformed into three ordinary differential equations that can be solved using numerical methods. Substituting Eqs. (8) into the constitutive relationship Eq. (27) gives the transformed constitutive laws:

EA ðð1 þ uo;x Þ cos h þ v ;x sin h  1  ao DT Þ 1 þ ao DT GA ðð1 þ uo;x Þ sin h þ v ;x cos hÞ Q¼ 1 þ ao DT EI h;x M¼ 1 þ ao DT

N¼

ð30Þ

Substituting the above transformed constitutive equations into the equilibrium equations Eq. (28), gives the three transformed equilibrium equations:

EA ðv ;x h;x cos h  uo;x h;x sin h þ v ;xx sin h þ u;xx cos h  h;x sin hÞ 1 þ ao DT GA ðð1 þ uo;x Þ sin h þ v ;x cos hÞh;x þ f x cos h  1 þ a o DT þ f y sin h ¼ 0 ð31Þ EA ðð1 þ uo;x Þ cosh þ v ;x sinh  1  ao DT Þh;x  f x sinh þ f y cos h 1 þ ao DT GA ðuo;x h;x cosh  v ;x h;x sinh þ v ;xx cos h  uo;xx sinh þ 1 þ ao DT h;x cos hÞ ¼ 0 ð32Þ EIh;xx ¼

EA ðð1 þ uo;x Þ cos h þ v ;x sin h 1 þ ao DT 1  ao DTÞðð1 þ uo;x Þ sin h þ v ;x cos hÞ

this paper, the trapezoid method that uses Richardson extrapolation enhancement is adopted. In numerical analysis, the trapezoid method is a numerical method to solve ordinary differential equations derived from the trapezoidal rule for computing integrals. Richardson extrapolation is a sequence acceleration method, used to improve the rate of convergence of a sequence. Newton’s method is also involved to get a good estimate for the solution. Maple is employed here to perform the numerical strategy to solve the transformed equations. The detailed steps and the methodology for the nonlinear localised buckling analysis of pipelines are demonstrated in the following section of this paper. 5. Buckling of straight pipelines 5.1. Friction In this paper, it is assumed that a pipeline is subjected to a uniform temperature increase. Only in-plane buckling is analysed and the vertical deflections of the pipeline during buckling are assumed negligible. The friction resistance plays a signification role in the buckling of pipelines. It constrains the pipeline expansion in the axial direction which then induces axial compression. Also, it prevents the lateral deflection of the pipeline after it buckles. The friction resistance which is related to the self-weight of the pipeline and the displacements is usually assumed to be either constant Hobbs [1], Kerr [2], linear Chen and Baker [46] or bilinear Yuan et al. [18] to simplify the analysis. However, the practical frictional response is not strictly constant, linear or bilinear according to the experimental results Bruton et al. [47] White and Cheuk [48] Wagner et al. [49], Brennodden et al. [50]. In the experimental analysis, when displacements are small, the resistance usually rises with an increase of the displacement, but the rate usually becomes smaller and smaller. When the resistance rises to a certain value, the friction remains approximately constant. In reality, the friction is very complex and affected by a multitude of factors. Karampour et al. [26], Maltby and Calladine [51] observed that localised buckling in the presence of imperfections depended on the limiting friction drag force and was not influenced by the equation which describes the friction resistance. In this paper, a hyperbolic function as shown in Fig. 2, is used to model the axial and lateral frictional resistance between the soil and pipeline, which was also used in Donley and Kerr [32], Grissom and Kerr [33] to model the axial friction. This is a much more realistic model for the friction and provides a better fit to the experimental results. The friction is also divided into the axial direction f u and the lateral direction f v , which are defined by:

f u ¼ ku tanhðkou uo ðxÞÞ f v ¼ kv tanhðkov v ðxÞÞ

where ku and kv are the limiting value for the axial and lateral resistance and kou and kov determines the initial stiffness. A negative sign

GA ðð1 þ uo;x Þ sin h þ v ;x cos hÞðð1 þ uo;x Þ cos h 1 þ ao DT þv ;x sin hÞ ð33Þ 

In the above transformed equilibrium equations, the distributed forces f x and f y represent the friction response between a pipeline and the soil surface, which are the functions of the axial and lateral displacement uo ðxÞ and v ðxÞ, respectively. Therefore, the transformed equations are three ordinary differential equations with three variables consisting of uo ðxÞ; v ðxÞ and hðxÞ in terms of x. For specified loading and boundary conditions, these three ordinary differential equations can be solved using numerical strategy. In

ð34Þ

Fig. 2. Frictional resistance.

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J. Zhu et al. / Engineering Structures 103 (2015) 37–52

is used, as the direction of the resistance is always opposite to the movement. In the above Eq. (34) f u and f v are in the x and y directions, respectively. In the nonlinear buckling of pipelines, the friction is the only distributed load that the pipeline is subjected to. Therefore, f x and f y in Eq. (29) are replaced by f u and f v , respectively. 5.2. Length effects The lateral buckling problem for pipelines is known as a localisation phenomenon because a limited region of the pipeline buckles. In this section, localised buckling is investigated and a critical pipeline length is proposed. In order to solve the differential equations using numerical methods, the overall pipeline is first divided into two equal parts. A displacement is chosen which is used to control the solution of the differential equations to study nonlinear buckling. In these calculations, the selected prescribed displacement is the lateral displacement at the centre of the pipeline length denoted by v c . Once v c is prescribed, the corresponding temperature increase is numerically calculated and the other unknowns can be obtained. With different values of the central lateral displacement prescribed, the associated temperature increases and the nonlinear buckling behaviour is then determined. If, instead, the temperature increment is set first, there could be multiple corresponding displacements and the numerical analysis would have difficulty determining all possible solutions. The continuity conditions at the mid-point of the pipeline are as follows:

at x1 ¼

L & x2 ¼ 0 uo1 ¼ uo2 h1 ¼ h2 v 1 ¼ v c v 2 ¼ v c 2 duo1 duo2 dv 1 dv 2 dh1 dh2 ¼ ¼ ¼ dx1 dx2 dx1 dx2 dx1 dx2

where L is the length of the pipeline. The variables with the 1 subscript are for the first half of the pipeline and those with a 2 as subscript are for the second half. In the first example here, the pipeline is taken as clamped at both ends. The displacements and bending angle are all zero at both clamped ends. Therefore, the boundary conditions are defined by:

uo1 ¼ 0

v1 ¼ 0

h1 ¼ 0

uo2 ¼ 0

v2 ¼ 0

h2 ¼ 0

A numerical study is presented here to demonstrate the above procedure, to make comparisons with other published work and to carry out a parametric investigation. The pipeline example used here is based on the material properties used by Hobbs [1]. The pipeline has an outside diameter of D = 0.65 m and a wall thickness of 0.015 m. The Young’s Modulus is E ¼ 206 GPa and the Poisson’s ratio is 0.3. The coefficient of linear thermal expansion is

ao ¼ 11  106 =o C. The equivalent weight (including coating) is ð35Þ

at x1 ¼ 0 L at x2 ¼ 2

Fig. 3. Flow chart to get the critical length.

ð36Þ

With the continuity and boundary conditions, the transformed equilibrium equations can be solved numerically. Fig. 3 illustrates the main steps used to determine the critical length of the pipeline. Firstly, two different lengths for the pipelines are chosen. The material properties and friction response are specified. Secondly, the chosen initial lengths are substituted into the transformed equilibrium Eqs. (31)–(33) and two different temperature rise curves with respect to the lateral displacement at the centre are determined. The two temperature increment versus displacement curves are then compared. If the absolute difference of the safe temperature increment DT s (the lowest point) is bigger than the parameter g that controls the accuracy of the results, the smaller length is increased and the procedure repeated. If the absolute difference is not greater than the parameter g, the pipeline length is taken as the critical length. Beyond the critical length, increasing the pipeline length has no effect on the temperature displacement response. This critical pipeline length can be used to analyse the localised buckling for the pipelines with even larger length.

taken as 3800 N/m. The practical range of the coefficient of friction is 0:3 6 /f 6 0:7 Hobbs [1] and the lateral frictional coefficient is usually larger than the axial frictional coefficient. Thus, the friction coefficients /f are taken as 0.5 and 0.7 for the axial and lateral directions, respectively. The ku and kv parameters are 3800  0.5 N/m and 3800  0.7 N/m, respectively. A relatively large value of 50 N/m is taken for the kou and kov parameters. The results can be compared with Hobbs’ solutions in which a constant axial and lateral friction law was used. The results of the temperature increases versus the normalised lateral displacement at the centre of the pipeline are plotted in Fig. 4 for pipelines with different lengths. In Fig. 4, it is shown that changing the pipeline length influences the results significantly, when the length is small (50–1000 m). When the length of the pipeline is increased beyond 2000 m (for this example), any further increases in length do not make much difference on the temperature increment versus normalised displacement curves. In addition, if the length of the pipeline continues to increase beyond 2000 m, the safe temperature increment (the lowest point) DT s does not change according to Fig. 4. The safe temperature increment is of practical significance for the working design of pipelines and railway tracks. The safe temperature increment is the lower limit of the post-buckling path for an idealised straight pipeline or track. If the pipeline or track is idealised as straight, localised buckling will not occur when the temperature increment is lower than the safe temperature increment. In reality, imperfections exist and in pipelines this may be due to the pipelaying vessel sway motion during the installation process or foundation irregularities. Imperfections may be also induced because of some external factors likes the landside, debris flow Yuan et al. [18], Yuan et al. [19] or trawl gears in the fishing activity Herlianto et al. [52]. The imperfections could induce lateral movement so that the equilibrium path asymptotes to the post-buckling

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Fig. 4. Length effects.

equilibrium path as plotted in Fig. 4. Therefore, the safe temperature increment being the lowest point DT s on the idealised straight pipeline or track post-buckling path provides a conservative working temperature change limit if the imperfection is not known in advance. Fig. 5 shows the typical equilibrium paths for perfectly straight pipelines and straight pipelines with different imperfections. For small imperfections, the temperature increment rises dramatically under the pre-buckling curve but the lateral displacement does not increase much before the pipeline reaches a limit point. After the limit temperature increase, point DT l , there is a decrease with increasing lateral displacement with the response asymptoting with the solution for the perfectly straight pipeline. Snap-through buckling is possible with further increase of the temperature increment above the limit point DT l . The limit temperature increment DT l for small imperfections is much higher than the safe temperature increment DT s as shown in Fig. 5. The limit temperature increment provides an upper working temperature change if the imperfection is predicable, which is also very significant for practical pipelines. If the imperfection is considerably large, the response shows nonlinear behaviour and the limit temperature increment eventually disappears. The effects of the imperfection magnitude will be discussed in a later section.

Fig. 5. Typical equilibrium paths for straight pipelines.

The practical range of outside diameter for transmission pipelines is approximately from 0.1 m to 1.212 m. Four outside diameters 1.212 m, 0.65 m, 0.4572 m and 0.254 m are looked at in this section of the paper. The wall thickness mainly depends on the internal pressure and the corrosion of the working environment. A wall thickness of 0.015 m is used for all the pipe diameters investigated. Fig. 6 displays the safe temperature increment versus the length of pipelines with different diameters and the corresponding normalised lateral displacement v cs =D versus the length of pipelines plotted in Fig. 7. When the outside diameter is 0.65 m, it is obvious that the safe temperature increment and the corresponding central lateral displacement vary significantly when the pipeline lengths are small (50–1000 m). However, the safe temperature increment reaches a limit at a length of 1000 m. The corresponding normalised central lateral displacement reaches a limit at about 1500 m. After 2000 m, the corresponding central lateral displacement stays constant at 1.7 m (v cs =D= 2.6). In addition, the results for the temperature increment versus normalised lateral displacement curves do not vary much for lengths beyond 2000 m. Therefore, 2000 m is taken as the critical length Lcr for the pipeline with a diameter of 0.65 m. The critical length versus the outside diameter is plotted in Fig. 8. The critical length increases with increasing pipeline diameter. The critical lengths for the four diameters are 3000 m, 2000 m, 1500 m and 1000 m, respectively. The critical length shown in Fig. 8, varies almost linearly with pipeline diameter. A line of best fit is plotted in Fig. 8. For pipelines with a wall thickness of 0.015 m, the critical length can be estimated using the empirical formula given in Fig. 8 for outside diameters between 0.254 m and 1.212 m. Any pipeline with a length greater than the critical length has almost the same safe temperature increment. For pipelines with lengths smaller than the critical length, localised buckling can still occur but the safe temperature increment is influenced by the pipeline length and end boundary conditions. The safe temperature increment increases with increase of pipeline diameter, as show in Fig. 6. However, the safe temperature increment does not vary significantly. For the 0.254 m pipeline diameter, the safe temperature increment is 46.2 °C, while when the outside diameter is 1.212 m, the safe temperature increment increases to 50.7 °C. Therefore, for pipelines with lengths above the critical length and diameters between 0.254 m and 1.212 m, the safe temperature increment is between 46.2 °C and 50.7 °C.

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Fig. 6. Safe temperature increment versus the length of pipelines with different diameters.

Fig. 7. Corresponding normalised lateral displacement versus the length of pipelines with different diameters.

Fig. 8. Critical length versus the outside diameter of pipelines.

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The deformed shapes for the buckled pipeline with a length of 2000 m and an outside diameter of 0.65 m at different loading stages are plotted in Fig. 9. To highlight the deformed shapes, only a section of the pipeline near the centre is shown. In the figure, it is obvious that there is a localised symmetric buckled shape. Only a limited region at the middle of the pipeline buckles out due to the thermal loading, while the regions of the pipeline outside the central localised zone do not significantly deform laterally. With increasing load, the region where the localised buckling happens enlarges. It should be mentioned that when the length of the pipeline is small (50 m and 100 m), localised buckling does not occur and the overall pipeline deforms laterally. The localised deformed shape is similar to a cosine wave, but with decaying amplitude from the centre to both sides, which agrees with the experimental results for the railway tracks by Kish et al. [34], Kish and Samavedam [37]. Kish et al. found that straight and curved tracks typically buckle in a symmetric mode rather than in an antisymmetric mode. Therefore, as is shown in this example, the localised buckling occurs for relatively long pipelines subjected to thermal loading and that increasing the length of the pipeline beyond a critical length does not influence the localised buckling behaviour.

5.3. Boundary condition effects Fig. 10 shows the temperature increment versus normalised lateral displacement for the example pipeline with an overall length of 2000 m and an outside diameter of 0.65 m under three different boundary conditions. In the first case, the pipeline is clamped at both ends. The second case considered assumes that both supports are hinged. In the last case, the pipeline is clamped at one end and hinged at the other end. For the second and third cases, the continuity conditions are the same as in Eq. (35). However, the boundary conditions at the hinged support are as follows:

uo ¼ 0

v ¼0

dh ¼0 dx

ð37Þ

According to the results illustrated in Fig. 10, it can be seen that the responses are almost the same for the pipelines under different boundary conditions. This makes sense when one considers that the buckling mode is localised over a short region at the centre of the pipeline and is unaffected by the end boundary conditions.

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Therefore, we conclude that changes of the boundary conditions do not influence the localised buckling response as long as the length of the pipeline is larger than or equal to the critical length. 5.4. Comparison with results in the literature The nonlinear buckling results for pipelines with an overall length of 2000 m obtained using the methods described in this paper are compared with the results of Kerr [2] and Hobbs [1] and are shown in Fig. 11. In Kerr’s and Hobbs’ analysis, mode 1 and mode 3 are symmetric modes. The deformed shape of mode 3 is similar to the shape obtained in this paper and the experimental investigation by Kish et al. [34], Kish and Samavedam [37]. Because Hobbs also used the buckled and adjoining region assumptions made by Kerr, his results are almost the same as those of Kerr’s, as shown in Fig. 11. The numerical results presented in this paper agree with the initial declining part of the temperature increment versus normalised displacement curve presented by both Kerr and Hobbs. The safe temperature increment or lower limit for the temperature increment is almost the same. Beyond the lower limit, in the ascending part of the postbuckling curve, the present numerical results show a stiffer behaviour than those of Kerr or Hobbs. The reason for the difference is that Kerr and Hobbs assumed that there was no lateral deflection in the adjoining regions outside the localised buckled zone, which means that the pipeline can only deform axially and the axial compressive force in adjoining regions is overestimated. The overestimated axial compressive load results in overestimated displacements in the buckled region. Kerr and Hobbs also assumed that the axial resistance was negligibly small in the localised buckled region, which means that the axial compressive load is constant in this region. The pipeline has been shown in this paper, to deform significantly in the localised zone not only in the lateral direction but also in the axial direction. The axial friction resistance reduces the axial compressive load in the localised zone and the axial compressive load is not constant (this is discussed in the next section). Kerr and Hobbs’ overestimate of the axial compressive load in the buckled region leads to larger values for the lateral displacement. In the investigation by Zeng and Duan [25], the results also display a stiffer behaviour than those of Hobbs. In the analysis presented in this paper, no prior assumptions are made about the displacement or friction in the localised buckled region and/or the adjoining regions.

Fig. 9. Deformed shape of localised buckling.

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Fig. 10. Boundary condition effects.

Fig. 11. Comparison with published solutions.

5.5. Axial compressive force The axial compressive force distribution within the pipeline is investigated here. Fig. 12 shows the axial compressive force distribution near the centre of the pipeline for different values of the central lateral displacement and with an overall length of 2000 m. In the figure, it is found that the axial compressive force decreases from the both sides to the centre of the pipeline. The axial compressive force is almost constant in a very small limited region within the localised buckling zone. The region where the axial compressive force is nearly constant is much smaller than the length of the localised buckling zone as shown in Fig. 9. The axial compressive force at the centre of the pipeline versus the normalised central lateral displacement is plotted in Fig. 13 and compared with the results of Kerr and Hobbs who assumed that the axial compressive force is constant in the buckled region. In Fig. 13, the axial compressive force reduces with increase of the normalised central lateral displacement and the rate of change of the axial force also decreases. Comparing all the results, it is seen that the results of Kerr and Hobbs overestimated the axial

compressive force as discussed in Section 5.4, which results in higher lateral displacement in the buckled region. 5.6. Imperfection effects and validation using ANSYS The analysis so far has assumed that the profile of the pipeline is perfectly straight. Deviations from a straight profile for pipelines laid on the seabed are introduced by the pipe-laying vessel’s sway motion during the installation process and foundation irregularities. Imperfections may be also induced because of some external factors like landslides, debris flow Yuan et al. [18,19] or trawling gear due to fishing activity Herlianto et al. [52]. The imperfections can have a range of magnitudes considered to be anywhere between negligibly small or even considerably large. The outof-straightness or initial lateral imperfections means that any localised buckling initiates at the part of the pipeline with the critical imperfections. The imperfections can lower the safe temperature increment and affect significantly the nonlinear buckling behaviour. Therefore, it is important to investigate the imperfection effects on the localised buckling behaviour of pipelines. In this

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Fig. 12. Axial compressive force distribution near the centre of the pipeline.

Fig. 13. Axial compressive force at the centre of the pipeline versus normalised lateral displacement.

analysis, a lateral load imperfection is applied at the centre of the pipeline. The discontinuity shear forces at the middle of the pipeline are used to model the lateral force. The shear force is determined by the slope of the deformed pipeline. Therefore, a parameter ki is added in the slope continuity conditions as follows:

dv 1c dv 2c ¼ þ ki dx1 dx2

ð38Þ

The discontinuity of slopes results in a discontinuity of the shear forces at the middle of the pipeline. Fig. 14 presents the nonlinear buckling of pipelines with different values for imperfection parameter ki ranging from 0 to 1E4 and with an overall length of 2000 m. In Fig. 14, for small imperfections, the temperature increase of the pipelines rises dramatically under the prebuckling curve but the lateral displacement does not increase much before the pipeline buckles. After the temperature increase peaks DT l there is a decrease with the increasing lateral displacement with the response asymptoting with the solution for the perfectly straight pipeline. For small imperfections like ki = 1E6,

the safe temperature increase point is almost the same as for a perfectly straight pipeline. Between the peak point and the safe temperature increment point, snap-through type buckling is possible. Under the temperature control, if the temperature continues increasing after the limit temperature increment DT l , snapthrough buckling will occur. The limit temperature increment DT l for small imperfections is much higher than the safe temperature increment DT s . If the imperfections are predictable, the limit temperature increment provides an upper working-temperature change limit, which could be significant for practical pipeline design. With significant increase in the imperfection parameter, the temperature versus normalised displacement curve moves downward and the safe temperature increment also decreases. When the imperfection parameter increases to 3.14E5, the upper limit temperature increment goes downward to the same level (DT s ¼ 49:6  C) as the safe temperature increment for a perfectly straight pipeline. If imperfections are larger than 3.14E5, the safe temperature increment obtained from the idealised straight pipeline cannot be used as a conservative estimate of the limiting

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Fig. 14. Imperfection effects and validation with ANSYS.

temperature increment. When the imperfection parameter is set to a value at or above 5E5, there is no longer an upper limit point or a lower limit for the temperature increment and the normalised lateral displacement increases with increase of the temperature change. Fig. 15 presents the relationships between the initial central lateral displacement and the imperfection parameter when the temperature change is set to zero. The initial central lateral displacement is a relative measure of the initial imperfection. This chart can be used to relate the imperfection parameter being used in this study with a corresponding initial lateral out-ofstraightness. In order to validate the results, an elastic geometric nonlinear finite element model for the thermal buckling of pipelines was developed using ANSYS [38]. A 3D 2-node PIPE288 element was adopted to model the pipeline. CONTA175 element and TARGE170 element were employed to model the interaction between the pipelines and the foundation. In ANSYS, the imperfections were applied at the centre of the pipeline to induce localised buckling. Since only temperature control is possible in ANSYS for this

complex problem, the results for an imperfection parameter of 5E5 and 1E4 could only be obtained with the results shown in Fig. 14. The present results appear to agree very well with the results from ANSYS. Fig. 16 displays the axial compressive force distribution near the centre of the pipeline for different imperfections under the same temperature increment of DT s ¼ 49:6 °C (the safe temperature increment for an idealised straight pipeline). Imperfections values of 2E5 and 3.14E5, have been chosen. For the imperfection value of 2E5, there are two points on the temperature versus lateral displacement load path that correspond to a temperature increment of DT s ¼ 49:6 °C (see Fig. 14). For an imperfection of 3.14E5, a temperature increment of DT s ¼ 49:6 °C corresponds to the single limit point on the nonlinear path. Since there are two points on the equilibrium path for the same temperature increment when the imperfection is 2E5, these are labelled as 2E5 (1) and 2E5 (2) in Fig. 16. For the imperfections of 2E5 (1) and 3.14E5, the axial compressive force is almost constant as the lateral displacement is not significant. For the other two situations, the pipeline buckles

Fig. 15. Initial central lateral displacement versus the imperfection parameter.

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Fig. 16. Axial compressive force distribution for different imperfections under the same temperature increment.

out significantly and the axial compressive force at the centre of the pipeline is reduced significantly. Fig. 17 shows the relationships between the axial compressive force at the centre of the pipeline versus the temperature increment, for both the idealised straight pipeline and those with imperfections. It is found that the central axial compressive force in the pipeline with imperfections increases almost linearly with the temperature increment during the pre-buckling stage. This is because the strain is very small. Both the lateral and axial displacements are negligibly small and the pipeline constrained by the axial friction is similar to a beam with fixed end supports. Thus the axial compressive force grows with the temperature increment c linearly following the beam theory equation DT ¼ aN . There is an o EA upper limit for the axial compressive force for pipelines with imperfections. As discussed earlier, if a pipeline is subjected to increasing temperature, after the temperature increment limit point is reached, further temperature increase can only be achieved by snap-through buckling to a point on the equilibrium path where the response is stiffening. At this stage, the axial compressive force will decrease dramatically as shown in Fig. 17.

5.7. Friction effects The modelling of friction also plays an important role in the nonlinear buckling of pipelines. The axial friction prevents the pipeline expansion in the axial direction and induces the buckling problem. The lateral friction resists the lateral movement of the pipeline after buckling. The influence of the friction coefficient on the localised buckling of pipelines is investigated here. As mentioned by Hobbs [1], the practical range of the coefficient of friction is 0:3 6 /f 6 0:7. However, friction in reality is much more complex and affected by many factors. In this paper, the coefficient from 0.2 to 1 is investigated. Fig. 18 presents the buckling of straight pipelines with different coefficients of axial friction and with an overall length of 2000 m. In Fig. 18, it is found that the safe temperature increment drops with a decrease of the axial friction coefficient. The ascending part of the curve beyond the safe temperature increment also moves downward. The initial descending part is not greatly influenced by the change of the axial friction coefficient. The effects of changes in the lateral friction coefficient are illustrated in Fig. 19. Unlike the scenario under different axial

Fig. 17. Temperature increment versus central axial compressive force under different imperfections.

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Fig. 18. Localised buckling of pipelines with different axial friction coefficients.

Fig. 19. Localised buckling of pipelines with different lateral friction coefficients.

Fig. 20. Shear effects.

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friction coefficient, the whole curve drops with decreasing lateral friction coefficient. 5.8. Shear deformation effects Shear deformations can be important in buckling of columns Attard and Hunt [39], Goto et al. [53], Bazant and Cedolin [54], Bazˇant [55] Reissner [56], Kardomateas and Simitses [57], Attard et al. [58]. Usually, the shear effects become increasingly severe at very small slenderness. To demonstrate the effects of shear deformation on the localised buckling of pipelines, results are obtained for three axial to shear rigidity ratios of EA EA EA ¼ 2:6; GA ¼ 100 & GA ¼ 200, and are shown in Fig. 20; The overall GA length of the pipeline was 2000 m. An axial to shear rigidity ratios EA ¼ 2:6 represents an isotropic homogeneous solid. In Fig. 20, of GA the results for the different axial to shear rigidity ratios are almost identical. Shear deformation effects are not significant and can even be neglected for the localised buckling of the pipeline with the critical length. 6. Summary Constitutive relations that include the thermal effects for stresses and internal actions are derived using hyperelastic constitutive modelling for the in-plane lateral buckling of thermally loaded pipelines. The transformed equilibrium equations for straight pipelines under thermal loading and axial and lateral friction were obtained using the developed constitutive equations. A new nonlinear solution strategy has been presented to investigate the inplane nonlinear localised buckling of thermal pipelines that does not make any of the usual assumptions about the buckled and adjoining region adopted by Hobbs [1], Kerr [2]. The method proposed in this paper can also be employed to investigate the localised buckling of straight railway tracks. In the analysis, it has been shown that localised lateral buckling within the central region of the pipeline is evident for long pipelines subjected to thermal loading and friction. If the length of the pipeline is greater than or equal to a critical length, any changes to the length or changes of the end support boundary conditions do not influence the localised nonlinear buckling behaviour. The critical length increases with increasing pipeline outside diameter. The safe temperature increment also rises with an increase of the pipeline diameter, but the difference is not significant. For pipelines with diameters in the range of 0.254 m to 1.212 m, the critical length can be estimated by an empirical formula given in this paper. For pipelines with lengths greater or equal to the critical length and diameters in the range of 0.254 m to 1.212 m, the safe temperature increment is in the range of 46.2 °C and 50.7 °C. Comparisons were made with the results of Hobbs [1], Kerr [2]. The numerical results presented in this paper agreed with both Kerr and Hobbs for the initial declining part of the temperature increment versus normalised displacement curve. The present work predicted almost the same safe temperature increment or lower limit for the temperature increment. Beyond the lower limit, in the ascending part of the curve, the present numerical results showed a stiffer behaviour. The reason for the difference is the assumed deformed shapes and introduced boundary conditions by Kerr and Hobbs, which results in the overestimated lateral displacement in the buckled region. The axial compressive force decreases from both sides to the centre and stays nearly constant only in a very small region at the centre of the pipeline. In the parametric study, it was found that the imperfections influence the localised buckling significantly. The imperfections can lower the safe temperature increment and decrease the upper limit for the axial compressive force in the pipeline. The limit temperature increment DT l and

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the safe temperature increment DT s provide the upper and conservative working temperature change limits, respectively. The present localised buckling results have also been validated by the commercial finite element package ANSYS. The safe temperature increment decreases with the axial and lateral friction coefficient. Shear deformations have been shown not to influence the localised buckling of practical pipelines studied in this paper.

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