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12 vues14 pagesAnalysis of Buried Pipelines Subjected to Ground Surface Settlement and Heave

Analysis of Buried Pipelines Subjected to Ground Surface Settlement and Heave

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Analysis of Buried Pipelines Subjected to Ground Surface Settlement and Heave

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12 vues14 pagesAnalysis of Buried Pipelines Subjected to Ground Surface Settlement and Heave

Analysis of Buried Pipelines Subjected to Ground Surface Settlement and Heave

© All Rights Reserved

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ARTICLE

Analysis of buried pipelines subjected to ground surface

Can. Geotech. J. Editors' Choice Downloaded from www.nrcresearchpress.com by UNIVERSITY OF NEW SOUTH WALES on 04/02/19

George P. Kouretzis, Dimitrios K. Karamitros, and Scott W. Sloan

Abstract: This paper presents an analytical methodology for the calculation of internal forces and strains developing in

continuous buried pipelines that cross geotechnically problematic areas and are susceptible to permanent ground surface

settlement or heave. Material nonlinearity effects are introduced in the solution via an iterative procedure, while taking into

account the effect of pipeline elongation on its response. The use of a versatile bilinear expression to describe the stress–strain

response of the pipeline material renders the method appropriate for steel, high-density polyethylene (HDPE), concrete, and cast

iron pipelines alike. Comparison of the analytical results against those from benchmark ﬁnite element analyses highlights the

effectiveness of the simpliﬁed analysis. The method is a potential alternative to elaborate three-dimensional nonlinear numer-

ical analyses that are often used in pipeline design practice, and offers ease-of-use with no expense in accuracy, at least for

problems involving simple pipeline geometries.

Key words: buried pipelines, settlement, heave, reactive soils, ground subsidence.

Résumé : Cet article présente une méthode analytique de calcul des forces et contraintes présentes à l’intérieur des pipelines

continus enfouis qui traversent des zones géologiques problématiques et peuvent subir les effets des tassements et soulèvements

permanents de la surface du sol. Les effets matériels de non-linéarité sont incorporés à la solution par l’intermédiaire d’une

procédure itérative et tiennent compte de l’effet de l’élongation du pipeline sur la réponse de ce dernier. L’utilisation d’une

For personal use only.

expression bilinéaire polyvalente pour décrire la réponse contrainte–déformation du matériau constituant le pipeline rend la

méthode appropriée dans le cas des pipelines en acier, en polyéthylène haute densité (PEHD), en béton en fonte. La comparaison

des résultats analytiques avec ceux obtenus par la méthode de référence des éléments ﬁnis montre l’efﬁcacité de l’analyse

simpliﬁée. La méthode est une alternative possible pour élaborer des analyses numériques non linéaires tridimensionnelles, qui

sont souvent mises en pratique lors de la conception des pipelines, et offre une simplicité d’utilisation et une grande précision,

du moins dans le cas des problématiques liées aux pipelines à géométrie simple. [Traduit par la Rédaction]

Introduction A link between pipeline failures and the geotechnical risks re-

Rapid growth of buried pipeline networks inevitably leads to lated to permanent ground surface deformation has been identi-

routes that cross geotechnically problematic areas. In these cases, ﬁed in the recent literature. Gould et al. (2009) analysed data from

39 687 failures documented by Australian water authorities over a

permanent ground settlement or heave, which is not related to

10 year period. Field crews responsible for pipeline repairs veri-

pipeline construction activities, may result in the development of

ﬁed that a signiﬁcant number of these incidents were a result of

excessive pipeline strains and failure in the forms of cracking or

ground movement. Chan et al. (2007) identiﬁed a “clear relation-

buckling. Such problematic areas include, but are not limited to

ship” between City West Water Ltd. (Melbourne, Australia) pipe-

• Reactive soil deposits that undergo signiﬁcant volume changes line network failure rates and seasonal variation of soil moisture

due to environmental effects such as wetting–drying and freezing– content. Pratt et al. (2011) analysed a database of 8100 failures in

thawing cycles (Chan et al. 2007; Gould et al. 2009; Rajeev and cast iron and reinforced concrete water pipelines that occurred

Kodikara 2011). over a 10 year period in Western Australia, and reached similar

• Irrigated lands, where longer drought periods due to climate conclusions. In the USA, 10 143 serious incidents affecting on-

change may lead to aquifer overpumping; which in turn may shore pipeline systems transporting hazardous liquid and gas

result in nonuniform consolidation settlements (Budhu and were reported between 1993 and 2004, and were included in the

Adiyaman 2013; Wols and van Thienen 2014). US Pipeline and Hazardous Materials Safety Administration data-

• Areas susceptible to mine subsidence (Ho et al. 2007; ASCE-ALA base. Seven hundred ten (7%) of these cases are explicitly attrib-

2005; AS 2885.1 2012 (Standards Australia 2012)). uted to pipeline exposure to natural hazards, and resulted in

• Areas where ground surface subsidence is induced by near- 14 fatalities, 78 injuries, and property damage of 1.8 billion USD

surface tunnelling works (e.g., Wang et al. 2011). (source: PHMSA 2015 database). It must be stressed that, apart from

• Loose sand deposits that are susceptible to dynamic densiﬁca- their economic and social impact, pipeline failures may pose a

tion (Tokimatsu and Seed 1987). signiﬁcant environmental threat, as they may result in leakage of

G.P. Kouretzis and S.W. Sloan. ARC Centre of Excellence for Geotechnical Science and Engineering, Faculty of Engineering and Built Environment,

The University of Newcastle, Callaghan, NSW 2308, Australia.

D.K. Karamitros. Department of Civil Engineering, University of Bristol, Bristol, UK.

Corresponding author: George P. Kouretzis (e-mail: Georgios.Kouretzis@newcastle.edu.au).

Can. Geotech. J. 52: 1058–1071 (2015) dx.doi.org/10.1139/cgj-2014-0332 Published at www.nrcresearchpress.com/cgj on 5 December 2014.

Kouretzis et al. 1059

hazardous materials or chemicals and consequent contamination ground displacements of limited width, which will be addressed

of the soil or the aquifer. later in this paper. A step-like pattern for the differential ground

A number of studies in the recent literature deal with the stress surface offset (Fig. 1) is also considered. In reality, the shape of the

analysis of buried pipelines subjected to permanent ground set- deformed ground surface will be more complex, and vertical

tlement. The bulk of these studies focus on ground subsidence movements will develop progressively along a zone of ﬁnite

due to tunnelling where the axis of the tunnel crosses the pipeline width. Nevertheless, this “guillotine” type vertical offset is cer-

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route transversely. Analytical approaches (e.g., Klar et al. 2005, tainly conservative with respect to the calculated pipeline inter-

2007; Vorster et al. 2005; Wang et al. 2011) consider the pipeline as nal forces and strains. As per common practice in the design of

a beam on an elastic foundation or are based on a more rigorous buried pipelines (e.g., ASCE–ALA 2005), stress analysis of the pipe-

elastic continuum formulation. The effect of soil nonlinearity is line is performed via a beam–spring model and the soil–pipeline

introduced either by limiting the axial soil–pipeline friction in the interaction is quantiﬁed by replacing the soil surrounding the

elastic continuum solution (Klar et al. 2007) or by an equivalent pipeline with elastoplastic springs (Fig. 1). Three sets of springs are

linear approach, where the stiffness of the soil springs is a func- used to account for the different mechanisms governing the axial,

tion of the average deviatoric strain developing in the pipeline upwards, and downwards relative movement of the soil–pipeline.

(Vorster et al. 2005). Numerical parametric analyses have also The yield displacement and ultimate force of each set of springs

been employed for the statistical derivation of expressions for both depend on the geometry of the pipeline and properties of the

estimating the maximum bending strain developing in the pipe- soil. The out-of-plane horizontal relative displacement is taken as

line, based on the interpretation of ﬁnite element analyses results zero, so that transverse springs can be omitted from the analysis.

(Wang et al. 2011). These studies have been benchmarked against The in-plane vertical displacement is applied at the ﬁxed end of

relevant centrifuge tests presented by Vorster et al. (2005), the springs, resulting in the deformed pipeline axis shown in

Marshall et al. (2010), and others. Recently, Wols and van Thienen Fig. 1a for a pipeline subjected to differential ground surface set-

(2014) extended this concept to compute the pipeline stresses due tlement, and in Fig. 1b for a pipeline subjected to differential

to differential ground settlement along the pipeline route, using ground heave. The soil springs depicted in red (grey in the print

a Winkler-type model to simulate soil–pipeline interaction and version of this paper) in Figs. 1a and 1b are under compression (and

assuming elastic behaviour of the pipeline material. thus apply a distributed pressure on the pipeline), while the

Two simplifying assumptions are adopted in the abovemen- springs depicted in black are inactive, and correspond to sections

tioned studies, although their effect on the computed pipeline where no reaction load is applied on the pipeline.

stresses can be prominent in certain cases of practical interest. Figure 1 illustrates that when a simpliﬁed step-like pattern is

considered, the heave is actually the “mirror” problem of settle-

The ﬁrst is that the nonlinear pipeline material response is not

For personal use only.

ment, and vice versa. In both cases the pipeline reacts to the

introduced in the analytical solutions, even though it could be

imposed displacements predominantly via bending, and both

important for the rational design of steel and high-density poly-

deformation patterns intuitively should result in essentially the

ethylene (HDPE) pipelines in areas of considerable ground surface

same maximum internal forces and strains on the pipeline (for

deformations, where yield of the pipeline material could be ac-

the same relative offset magnitude). The difference is the location

ceptable as long as it does not lead to rupture or local buckling.

of the maximum strains along the pipeline length, which should

The second is that the pipeline elongation due to the vertical

be upstream of the point of application of the step-like displace-

ground surface offset is assumed to be negligible, in spite of the

ment for the settlement case, and downstream for the heave case.

fact that it can lead to the development of considerable axial

This is due to the fact that the downwards springs sustain a higher

strains, especially when the pipeline material reaches yield due to

yield force compared to the upwards springs, as given by the soil

bending.

failure surface that each deformation mode mobilizes (ASCE-ALA

To tackle these shortcomings, a new analytical method is pre-

2005, Kouretzis et al. 2014). Nevertheless, the exact location of the

sented for the estimation of internal forces and strains that develop

maximum strains is irrelevant to pipeline design.

in buried pipelines subjected to ground surface settlement along

Keeping the above in mind, an analytical solution will be for-

their route. This analysis is extended to also cover the scenario of

mulated to estimate the internal forces and strains in the pipe-

ground surface heave due to the pipeline route crossing reactive soil line, based on the same general concept of the ﬁnite element

deposits. The core of the method is based on the work of Karamitros nonlinear beam–spring model that is used in everyday design

et al. (2007) and Karamitros et al. (2011) for the stress analysis of practice. The solution could be considered as a variation of the

buried steel pipelines that cross strike–slip as well as normal and method of Karamitros et al. (2011) for normal faults, assuming

oblique faults, respectively. Nonlinearity of the soil and pipeline ma- that the deformation pattern imposed by the ground settlement is

terial is accounted for via a simple iterative procedure, which can be equivalent to that imposed by the rupture of a normal fault with

programmed easily in a spreadsheet. The results are compared against a dip angle 90° whose trace is perpendicular to the pipeline route.

those from benchmark nonlinear ﬁnite element analyses with a However, elongation of the pipeline due to the upwards or down-

beam–spring model, similar to the one described in the American wards displacement is taken into account here, and thus the cor-

Lifelines Alliance guidelines (ASCE-ALA 2005) for the design of buried responding axial pipeline force as a function of the width of the

steel pipes. To conclude, some practical implications and limitations settlement or heave zone is estimated. Albeit this elongation may

of the method are discussed, in view of its potential use in the prac- not be sufﬁcient to mobilize the full friction resistance at the

tical design of pipelines. soil–pipeline interface, as assumed in all published methodolo-

gies for the stress analysis of pipelines crossing active faults

Problem description (Newmark and Hall 1975; Kennedy et al. 1977; Wang and Yeh 1985;

Consider a straight continuous pipeline whose route crosses an Karamitros et al. 2007, 2011; Trifonov and Cherniy 2010), it will be

area of ﬁnite width where differential settlement or heave of the proven here that the axial tensile strains that are developed

ground surface may develop. It is assumed that no pipe bends or should not be neglected — at least for cases where the ground

anchoring points exist within the unanchored length of the pipe- surface offset is considerable compared to the diameter of the

line, which is deﬁned as the length where the relative axial soil– pipeline. In addition, the formulation for the heave problem will

pipeline displacement will take place due to the vertical offset of be provided, to prove the postulation that step-like settlement

the ground surface. This “free-to-slip” assumption does not apply and heave of equal magnitude will result in the same maximum

to the restraining effect on the relative axial movement between internal forces and strains on the pipeline. Certain parts of the

the soil and pipeline that is caused by a zone of differential analysis have already been presented in Karamitros et al. (2011),

1060 Can. Geotech. J. Vol. 52, 2015

Fig. 1. Pipeline subjected to step-like differential (a) settlement and (b) heave along its route. The corresponding idealized beam–spring

models are also shown, where the springs loaded by upwards and downwards movements are under compression and are depicted in red

(grey in the print version of this article) (the axial springs have been omitted for clarity).

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Fig. 2. Segmentation of the pipeline into three beams A=A, ABC, and CC=: (a) settlement scenario: (b) heave scenario.

For personal use only.

but are repeated here using the same nomenclature, where pos- trated in Fig. 2. Any soil movement in the horizontal plane is

sible, as they are essential to describe the analytical method disregarded in the following analysis, which suggests that it is not

completely. valid for cases of vertical offset developing as a result of certain types

of deformation — e.g., a rotational slope failure. Following the work

Methodology of Wang and Yeh (1985) and Karamitros et al. (2007, 2011), the analysis

A step-like vertical offset ⌬z of the natural ground surface will of the pipeline will be based on beam theory, while uncoupling the

result in the deﬂection of a straight continuous buried pipeline bending action from the axial elongation. To facilitate the analysis,

crossing the settling or heaving zone, following the pattern illus- the pipeline is segmented into three characteristic parts: A=A, ABC,

Kouretzis et al. 1061

and CC= (Fig. 2). Points A and C along the pipeline route deﬁne the Fig. 3. Positive sign convention for displacement, rotation, and

part ABC where it is presupposed that the maximum pipeline strains internal forces.

will develop. Its ends are the nearest points to the edge of the settle-

ment or heave zone where the relative vertical displacement be-

tween the soil and pipeline becomes zero. Point B is accordingly

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on the pipeline axis. Computation of the maximum values of the

pipeline internal forces and strains is performed in a ﬁve-step itera-

tive algorithm, which is described in the following.

Step 1

The case of the ground settlement proﬁle illustrated in Fig. 2a is

examined ﬁrst. Beams A=A (from –∞ to A=) and CC= (from C= to +∞)

are analysed by employing beam-on-elastic-foundation theory to

obtain the relation between the reaction force, V; bending mo-

ment, M; and rotation of the elastic line, , at points A and C. This

implies that parts A=A and CC= will behave elastically for the range

of the vertical ground surface displacements considered, and the

vertical beam deﬂection, w, will be limited so that geometric non-

The sign convention for the internal forces, displacements, and

linearity effects can be assumed to be negligible. In this case, the rotation follows Fig. 3, whereas in Fig. 2 the actual directions of

differential equation providing the elastic line of the beams is the internal forces and beam deformation are plotted.

given by One can easily visualize that the solution would be exactly the

same in the “mirror” problem of step-like ground surface heave

(1) E1Iw ⫹ Kw ⫽ 0 (Fig. 2b), if the order of parts A=A and CC= were reversed. Yet, to

maintain consistency throughout the presentation, the beam lo-

cated upstream of the edge of the heave zone will still be identi-

where E1 is the initial (elastic) Young’s modulus of the pipeline ﬁed as part A=A, and the beam located downstream of the edge of

material, I is the area moment of inertia of the pipeline cross- the heave zone as part CC=. In the latter case, the signs of eqs. (4)

For personal use only.

section, and K is the vertical spring constant. The latter is differ- and (5) are reversed.

ent for upwards and downwards movement (see, for example

Trautmann et al. 1985 and Kouretzis et al. 2014), but a mean value Step 2

will be assumed for these particular segments, for the sake of Segment ABC will be analysed as an elastic beam supported by

simplicity. The boundary conditions for eq. (1) are w = 0 for x = 0, as two rotational springs at its ends A and B. The constant of both

well as w = 0 for x ¡ –∞ or x ¡ +∞ for the beams A=A and CC=, these springs results from eqs. (4a) and (4b) as

respectively (Fig. 2). Under the above conditions eq. (1) yields

(6) Cr ⫽ M/ ⫽ 2E1 I

x

(2a) w ⫽ Ce sinx for beam A A

A prescribed displacement of magnitude ±⌬z is applied at the

(2b) w ⫽ Ce⫺x sinx for beam CC beam support C (Fig. 2), equal to the vertical soil settlement or

heave. A positive sign for ⌬z corresponds to settlement while a

negative sign designates heave, according to the convention

where shown in Fig. 3. Beam ABC of length L is accordingly partitioned

into two subsegments with lengths LAB and LBC, upstream and

(3) ⫽ 兹

4 K/(4E I)

1 downstream of the projection of the edge of the settlement or

heave zone on the pipeline axis, B. Lengths LAB and LBC are un-

known, and will result from the following analysis. Downwards

Differentiation of eqs. (2a) and (2b) results in the bending mo- vertical movement of support C will result in a negative ground

ment and reaction force at points A and C as reaction load, qAB, distributed along subsegment AB, and a posi-

tive ground reaction load, qBC, along subsegment BC. The direc-

(4a) MA ⫽ ⫺(2E1 I)〈 tion of the ground reaction distributed load, which follows the

convention of Fig. 3, is reversed in the case of upwards-vertical

(4b) MC ⫽ (2E1 I)C movement of support C (Fig. 2b).

For the analysis of beam ABC it is assumed that the positive

(5a) VA ⫽ MA reaction loads are uniform, and of a magnitude equal to the ulti-

mate value of the resistance force developed during pipeline up-

(5b) VC ⫽ ⫺ MC lift (ASCE-ALA 2005; Trautmann et al. 1985). This is due to the fact

that this ultimate force develops for relatively low relative move-

ments between the soil and pipeline, of the order of 0.01H to 0.02H

where A ⫽ wA and C ⫽ wC is the rotation of the elastic line of the

for pipelines buried in sand-backﬁlled trenches, with H being the

beam. These constitute the boundary conditions for the analysis

embedment depth of the pipeline measured from the its center-

of beam ABC during the next step of the algorithm. In order for line (ASCE-ALA 2005). Negative reaction loads, however, reach

these boundary conditions to apply, one should ensure the exis- their ultimate value, equal to the maximum resistance force

tence of an adequate attenuation length x = La on each side of the developed during relative movement of the pipeline vertically

beam ABC. The quantiﬁcation of this length is of special interest downwards, for much higher relative displacement values of the

in cases where the settling or heaving zone has a limited width, order of 0.1D, with D being the pipeline diameter (ASCE-ALA 2005).

and will be discussed in the following paragraphs. Therefore it is expected that the negative ground reaction will

1062 Can. Geotech. J. Vol. 52, 2015

Fig. 4. Estimation of ⌬zB in the case of differential ground (a) settlement and (b) heave.

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follow a triangular load distribution, reaching a peak value at Equations (9a) and (9b) allow ⌬z〉 to be estimated directly from

point B of the beam equal to Kdown⌬z〉 in the case of ground the input parameters of the problem. Thus the analysis can pro-

settlement, and to Kdown(⌬z – ⌬z〉) in the case of ground heave. ceed for elastic beam ABC, whose rotation is constrained by rota-

Here Kdown is the stiffness of the springs in the downward direc- tional springs at A and C, which is subjected to uniform loads qAB

tion (Fig. 1) and ⌬z〉 (in the case of settlement) or ⌬z – ⌬z〉 (in the and qBC and to a prescribed displacement of ⌬z at its support C.

case of heave) is the relative vertical displacement of the soil– Starting with the settlement cases, bending moments MA and MC

pipeline at point B. To facilitate the analysis, instead of using a

at points A and C are estimated from eqs. (10a) and (10b)

triangular load distribution an equivalent uniform distribution

(which has the same resultant) is considered for the negative

EI EI EI

ground reaction. This has the magnitude (10a) MA ⫽ 4 A ⫹ 2 C ⫺ 6 2 ⌬z

L L L

冉 冊

For personal use only.

冉 冊

(7a) 2 2 3

qABLAB LAB LAB qBCLBC LBC

along subsegment AB ⫹ 6⫺8 ⫹3 2 ⫺ 4⫺3

12 L L 12L L

(7b) qBC ⫽ [Kdown(⌬z ⫺ ⌬zB)]/2 for differential heave,

EI EI EI

along subsegment BC (10b) MC ⫽ ⫺2 A ⫺ 4 C ⫹ 6 2 ⌬z

L L L

line of subsegments AB and BC deforms as a circular arc (Fig. 4).

⫺

12

冉

2

qBCLBC

6⫺8

LBC

L

2

LBC

⫹3 2 ⫹

L 12L

3

冊

qABLAB

4⫺3

LAB

L

冉 冊

This implies that segment ABC behaves as a cable of zero bending

stiffness, as in the work of Kennedy et al. (1977). Nonetheless, this Combining eqs. (10a) and (10b) with eqs. (4a) and (4b) for bending

zero bending stiffness assumption is employed only for the esti- moments at A and C, resulting from the analysis of the beams A=A

mation of displacement ⌬z〉, and not for the subsequent struc-

and CC= in step 1, yields

tural analysis of the beam, as this would overestimate the pipeline

strains for low relative offset values. Given the above, the radii of

curvature of subsegments AB and BC can be calculated while con- [2 ⫹ (CrL/2EI)]MA⫽0 ⫹ MC⫽0

(11a) MA ⫽

sidering each circular arc to be part of the cross section of a 4 ⫹ (6EI/Cr L) ⫹ (Cr L/2EI)

“hollow cylinder”, with a radius equal to the radius of curvature of

the corresponding subsegment, subjected to uniform internal MA⫽0 ⫹ [2 ⫹ (Cr L/2EI)]MC⫽0

pressure qAB or qBC. Application of internal pressure to a hollow (11b) MC ⫽

4 ⫹ (6EI/Cr L) ⫹ (CrL/2EI)

cylinder will result in the development of a tensile hoop force, Fa,

translated as axial force for beam ABC, and

where

(8a) RAB ⫽ Fa /qAB

EI

MA⫽0 ⫽ ⫺6 2 ⌬z ⫹

L 12

2

qABLAB

6⫺8 冉

LAB

L

2

LAB

⫹3 2

L

冊

冉 冊

3

Accordingly, from the deformed shape illustrated in Figs. 4a qBCLBC LBC

⫺ 4⫺3

and 4b ⌬z〉 can be calculated from similar triangles as 12L L

(9a) ⌬zB ⫽

⫺qBC ⫹ 兹qBC(qBC ⫹ 2Kdown⌬z)

Kdown

(12b)

EI

MC⫽0 ⫽ 6 2 ⌬z ⫺

L 12

2

qBCLBC

冉

6⫺8

LBC

L

2

LBC

⫹3 2

L

冊

冉 冊

3

for differential settlement, Fig. 4a qABLAB LAB

⫹ 4⫺3

12L L

(9b) ⌬zB ⫽

⫺qAB ⫹ Kdown⌬z ⫺ 兹(qAB ⫹ Kdown⌬z) 2

⫺ (Kdown⌬z) 2

for differential heave, Fig. 4b considerations as

Kouretzis et al. 1063

(13a) VA ⫽

1

L

冋 冉

⫺MA ⫹ MC ⫺ qABLAB L ⫺

LAB

2

⫹ 冊 2

2

qBCLBC

册 about 60°–70°). This assumption, however, is not valid for the

settlement or heave problem, and might provide an explanation

for the discrepancies noted by Karamitros et al. (2011) between the

(13b) VC ⫽

1

L

冋 冉

⫺MA ⫹ MC ⫺ qBCLBC L ⫺

LBC

2

⫹ 冊 2

2

qABLAB

册 numerical and analytical results for locations near vertical fault

planes and large fault offsets. Here elongation of the pipeline due

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The locations of points A and C are not known a priori. Still, the L

(18) ⌬x ⫽ ⫺L

关 共 ⌬zL兲兴

length of each subsegment LAB and LBC and the total length of

beam ABC, L, can be computed iteratively, from the boundary cos arctan

conditions described in eqs. (5a) and (5b). For that an initial value

is assigned to lengths LAB = (5 to 10)D and LBC = (10 to 20)D, estimat-

Equation (18) is valid for both the settlement and heave scenar-

ing the internal forces at A and C from eqs. (11), (12), and (13), and

updating LAB and LBC during each iteration as ios. The maximum axial force that will develop on the pipeline

can be found based on the demand for compatibility between the

冉 冊

qBCLBC ⫹ VC ⫺ MA geometrically required elongation ⌬Lreq = ⌬x (eq. (18)) and the

(14a) LAB ⫽␣ ⫹ (1 ⫺ ␣)LAB available pipeline elongation ⌬Lavail, a concept originally intro-

qAB

duced by Newmark and Hall (1975). The latter is deﬁned as the

冉 冊

integral of the axial strains along the pipeline’s unanchored

qABLAB ⫹ VA ⫹ MC

(14b) LBC ⫽␣ ⫹ (1 ⫺ ␣)LBC length Lanch, i.e., the length along both sides of the projection

qBC of the edge of the settlement or heave zone on the pipeline axis

(point B) where some relative horizontal displacement occurs be-

where ␣ is an iteration coefﬁcient. tween the pipeline and its surrounding soil

The above procedure converges if a value of ␣ between 0.2 and

冕

Lanch

0.5 is selected. The maximum bending moment, which in the

differential settlement scenario will develop upstream of the edge (19) ⌬Lavail ⫽ 2 (L) dL

of the settling zone, is then calculated as

0

2

xmax VA

For personal use only.

(15) Mmax ⫽ MA ⫹ VAxmax ⫹ qAB where xmax ⫽ ⫺ where (L) is axial strain distribution along the pipeline.

2 qAB

All analytical methodologies for similar imposed-displacement

problems (Newmark and Hall 1975; Kennedy et al. 1977; Wang and

The procedure is essentially the same for the heave case, but the Yeh 1985; Karamitros et al. 2007, 2011; Trifonov and Cherniy 2010)

signs of the reaction moments and forces in eqs. (10) to (13) are assume, for simplicity, that the ultimate friction force between

now reversed. Lengths LAB and LBC are also reversed, so their initial the pipeline and surrounding soil is fully mobilized when calcu-

values to be entered into the iterative procedure should now be lating (L). This assumption, however, is not accurate for the

LAB = (10 to 20)D and LBC = (5 to 10)D. Moreover, the maximum problem examined herein. As discussed above, the geometrically

bending moment will now develop downstream of the edge of the required elongation due to settlement or heave, ⌬x, is consider-

heaving zone, and is calculated as ably smaller than the elongation induced by the rupture of a

major seismic fault, for which the abovementioned methodolo-

2

xmax VC gies were developed. Overlooking the elastic component of the

(16) Mmax ⫽ MC ⫹ VCxmax ⫹ qBC where xmax ⫽ ⫺ axial springs may result in a signiﬁcant overestimation of the

2 qBC

axial pipeline strain and, therefore, a reﬁnement in the calcula-

tion of the axial pipeline force is introduced in this study.

To this end, the maximum bending strain using elastic beam Starting with the case where the relative soil–pipeline displace-

theory can be computed as ment is not sufﬁcient to mobilize the ultimate axial friction force,

tu, the axial response of the pipeline is described by the following

Mmax D differential equation

(17) b ⫽

2E1I

dF(x) tu

(20) ⫽ t(x) ⫽ u(x)

Equation (17) implies that the pipeline response is linear elastic, dx xu

and as the pipeline material undergoes yield it will ovepredict the

bending strains. However, the required magnitude of settlement where axis x coincides with the pipeline axis (Fig. 5a), F(x) is the

or heave to result in yielding of a signiﬁcant portion of the pipe- axial force along the pipeline, u(x) is the axial displacement of the

line section (and consequently geometric nonlinearity effects be- pipeline relative to the surrounding soil, t(x) is the mobilized axial

coming prominent) is unrealistic. As discussed in the following friction force, and xu is the relative displacement required for the

step, the predominant mode of failure will be in bending, coupled development of the ultimate friction force tu. Given the small

with a low tensile force. Thus calculating the bending strains via magnitude of the required elongation, ⌬Lreq, it can be assumed

eq. (17) is a realistic approximation. that the relationship between axial stresses and strains along the

Step 3 pipeline remains elastic, thus

Vertical offset of the ground surface will result in some elonga-

du(x)

tion of the pipeline, ⌬x, and in the development of axial strains. (21) F(x) ⫽ E1 As(x) ⫽ E1As

Karamitros et al. (2011) ignored the elongation due to vertical fault dx

displacement, as it can be considered to be insigniﬁcant com-

pared to the elongation imposed by the horizontal displacement where As is the area of the pipeline cross section. Combining

component of a normal fault with a realistic dip angle (typically eqs. (20) and (21) yields

1064 Can. Geotech. J. Vol. 52, 2015

Fig. 5. Distribution of axial and friction forces along the pipeline for (a) ⌬Lreq < 2xu and (b) ⌬Lreq > 2xu.

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For personal use only.

(22a) ⫽ u(x) ⇒ u(x) ⫽ C1ex ⫹ C2e⫺x given as

dx2 E1 Asxu

where

冑

and the total elongation of the pipeline along length Lu will be

tu

(22b) ⫽

冕

E1 Asxu Lu

FoLu tuLu2

F(x)

(26) ⌬Lu ⫽ dx ⫽ ⫺

x⫽0 E1 As E1 As 2E1 As

Measuring x from the edge of the settlement or heave zone

(Fig. 5a), as x ¡ +∞ the displacement u(x) ¡ 0, and thus C1 = 0

and C2 = uo with uo being the relative displacement at the edge Further away from the edge, the pipeline behaviour may still be

described by eq. (22), after substituting x with x − Lu. To ensure

(point B of the pipeline). Furthermore, uo = ⌬Lreq/2 satisﬁes

compatibility between the two pipeline segments, where the ulti-

displacement compatibility, taking into account that the re-

mate friction force is or is not mobilized, the following force

quired elongation is distributed symmetrically along both sides

equilibrium equation needs to be considered:

of the settlement or heave zone edge. The corresponding axial

冑

force Fo at point B may be then calculated from force equilib-

rium considerations, as the integral of the friction forces along E1 Astu

(27) F(Lu) ⫽ Fo ⫺ tuLu ⫽ u(Lu)

the pipeline xu

(23) Fo ⫽ 冕

x⫽0

∞

t(x) dx ⫽ 冕 ∞

tuuo ⫺x

x⫽0 xu

e dx ⫽ 冑 E1Astu

xu o

u

Finally, to satisfy displacement compatibility

冑

(28) ⌬Lreq ⫽ 2(⌬Lu ⫹ xu)

E1Astu ⌬Lreq

⫽

xu 2

Combining eqs. (24) to (28) yields the maximum axial force

developed in the pipeline, at the edge of the settlement or heave

Figure 5a presents the resulting distribution of the axial pipe- zone

line force and soil–pipeline friction force, as calculated from

eqs. (20) to (22). Clearly eq. (23) is valid only when ⌬Lreq < 2xu. As (29) Fo ⫽ 兹E1 Astu(⌬Lreq ⫺ xu)

the required elongation ⌬Lreq = ⌬x increases, the ultimate soil

friction force will be eventually mobilized along a length Lu at

both sides of the edge, as shown in Fig. 5b. Length Lu can be As expected, when ⌬Lreq = 2xu, eqs. (23) and (29) yield the same

deﬁned via the yield criterion for the axial soil springs force at B, Fo. Furthermore, if xu = 0, eq. (29) reduces to the corre-

sponding formula proposed by Karamitros et al. (2007), who did

not take into account the elastic component of the axial soil

(24) u(Lu) ⫽ xu springs.

Kouretzis et al. 1065

Fig. 6. Distribution of axial and friction forces along the pipeline, in the case of settlement or heave zone of limited width, Ls.

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The above analysis essentially implies that the required elonga- Considering the friction force distribution shown in Fig. 6, the

tion is accommodated along an inﬁnite length of the pipeline. If ultimate friction force will be mobilized along a length Lu outside

the pipeline’s unanchored length where the relative axial dis- the settlement or heave zone edges, where the pipeline behaviour

placement between the soil and the pipeline should not be re- is described by eqs. (25) and (26). Further away from this length,

strained to avoid an increase in axial strain needs to estimated, the axial soil springs will behave elastically and the pipeline be-

the following expression can be used: haviour is described by eq. (22). Finally, within the settlement or

heave zone, where the ultimate friction is also mobilized, the

(30) Lanch ⫽ Fo /tu

axial force distribution will be as follows:

One common case where this unanchored length might not be (31) F(x ) ⫽ Fo ⫺ tux

available is when the anticipated settlement or heave occurs

within a limited zone of width Ls < 2Lanch, as shown in Fig. 6. A

rigorous analysis of the effect of this restraint on the pipeline where the axis x= has its origin at the edge of the zone, and its

strains would require distinguishing the following three cases: direction is opposite to that of the axis x (Fig. 6). The elongation of

the pipeline along each half of Ls (Fig. 6) is calculated as

1. The ultimate friction force is not mobilized along the pipeline

冕

length.

⌬Ls F(x )

Ls/2

FoLs tuLs2

2. The ultimate friction force is mobilized only outside the (32) ⫽ dx ⫽ ⫺

settlement or heave zone. 2 x ⫽0 E1 As 2E1 As 8E1 As

3. The ultimate friction force is mobilized both inside and out-

side the settlement or heave zone.

Finally, to satisfy displacement compatibility

Among these cases, only the third one is examined herein, as

the other two will have a negligible effect on axial pipeline (33) 2⌬Lreq ⫽ 2(⌬Lu ⫹ xu) ⫹ ⌬Ls

strains. Of course, due to symmetry, relative soil–pipeline dis-

placement becomes zero in the middle point of the settlement or

Combining the above equations yields

heave zone. This means that there must exist a region around this

冑

point where the relative soil–pipeline displacement is very small

and remains below xu, so that the ultimate friction force is not tu2Ls2 tuLs

mobilized. Nevertheless, in cases where the restriction of axial (34) Fo ⫽ E1 Astu(2⌬Lreq ⫺ xu) ⫹ ⫺

2 2

displacement is expected to become critical for pipeline design,

i.e., in cases of increased elongation and limited settlement or

heave zone width, the extent of this region is expected to have a For small values of Ls, eq. (34) predicts that the force Fo increases

trivial effect on the maximum pipeline strains that will develop for decreasing Ls values, as expected. Nevertheless, there is a crit-

near the edges of the permanent ground deformation zone. It ical Ls,cr value, above which this trend is reversed. Ls,cr is actually

therefore may be ignored, and the simpliﬁed distribution of the the maximum width of the settlement or heave zone that will

axial and friction forces along the pipeline is assumed as de- induce restraint on the axial force developed in the pipeline.

picted in Fig. 6. Therefore, the critical width Ls,cr can be determined as

1066 Can. Geotech. J. Vol. 52, 2015

(35)

dFo(Ls)

dLs

⫽ 0 ⇒ Ls,cr ⫽ 冑 2E1As(2⌬Lreq ⫺ xu)

tu

Fig. 7. Pipeline material bilinear stress–strain response.

value Ls > Ls,cr, then the axial force should be calculated using

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elongation ⌬Lreq = ⌬x compared to xu (Fig. 5). For narrower zones

where Ls < Ls,cr, eq. (34) should be employed, provided that ⌬Lreq =

⌬x > xu/2. When ⌬Lreq = ⌬x < xu/2 eq. (35) is not valid, as the

required elongation would be too small to mobilize the ultimate

friction force. In this case, the restraint effect of the settlement or

heave zone width on the axial forces is minimal, and need not be

taken into account during pipeline design. The axial force devel-

oped can be estimated directly from eq. (23).

The above formulation for considering the width of the settlement

or heave zone in the analysis is valid only when the half-width of

this zone is long enough to accommodate not only part BC of the

pipeline, but also an additional attenuation length La, thus ensur-

ing that the boundary conditions used in the analysis of part CC=

(eq. (1)) are satisﬁed. In other words, the half-width of the soil

deformation zone Ls/2 must be larger than Ls/2 > LBC + La. The

再

length LBC is calculated in step 2, while the attenuation length La

1 ⫹ E2( ⫺ 1) 0 ≤ ≤ 1

deﬁned in step 1 can be estimated by considering the equivalent

problem of a laterally loaded single pile in elastic homogenous (39) ⫽ E1 1 ≤ ≤ ⫺ 2

soil. The active length La, beyond which the behaviour of a later- ⫺ 1 ⫹ E2( ⫹ 1) ⫺ 2 ≤ ≤

ally loaded pile becomes independent of its length, was estimated

by Karatzia and Mylonakis (2012) as where ␣ is the axial strain; 1 and 1 are the yield stress and strain

(Fig. 7), respectively; and 1 deﬁnes the portion of the cross section

冉冊

For personal use only.

再

0.25

(36) ≈ (0.1 ⫺ 0.7 logn) the 1 line and below the –1 line)

Dp ES

冉1 ⫿ ␣

b

⬍ ⫺1冊

and Es are the stiffnesses of the pile and soil, respectively. Modi-

fying the above relationship to account for the hollow cylindrical (40) 1,2 ⫽ arccos 冉 1 ⫿ ␣

b 冊 ⫺1 ≤ 冉

1 ⫿ ␣

b

≤1 冊

section of the pipeline yields

0 1⬍ 冉

1 ⫿ ␣

b 冊

(37)

La

D

≈ (0.17 ⫺ 1.18 logn) 冉 冊

E1 t

KD

0.25

tudinal stresses over the cross section, as

where t is the thickness of the pipeline cross section.

冕

Considering a reasonable tolerance of 10% (n = 0.1), most buried

pipelines would have an attenuation length of the order La = (41) F⫽2 Rmt d ⫽ 2Rmt关E1␣ ⫺ (E1 ⫺ E2)(1 ⫹ 2)␣

(5 to 15)D. Nevertheless, even beyond the strict limits imposed by 0

the above derivation, the proposed methodology also provides ⫹ (E1 ⫺ E2)(1 ⫺ 2)1 ⫺ (E1 ⫺ E2)(sin1 ⫺ sin2)b兴

sufﬁciently accurate results for the intermediate case where LBC +

La > Ls/2 > LBC. The validity of this hypothesis is investigated via

where Rm = (D – t)/2 is the pipeline’s average radius. The axial

comparison with benchmark numerical analyses, and is discussed

strain, ␣, results from the requirement for compatibility of axial

in detail in the section titled “Comparison of the results against

forces calculated from eq. (41) and step 3. The solution is obtained

benchmark numerical analyses”.

iteratively, by employing a Newton–Raphson procedure. Starting

Step 4 with an initial value ␣ = 0, the axial strain is calculated in each

The maximum pipeline strain is now calculated from the re- successive iteration, k, as

quirement for stress equilibrium over the pipeline cross section, F共␣k 兲 ⫺ Fo

considering a bilinear stress–strain relationship for the pipeline (42) ␣k⫹1 ⫽ ␣k ⫺

material (Fig. 7). Section deformation effects are ignored, as they 具dF/d␣典␣⫽␣k

are not expected to be signiﬁcant for the range of ground move-

ments considered. Referring to the point where the maximum where

冋

bending moment develops, the longitudinal stress distribution

over the cross section can be drawn as shown in Fig. 8, assuming dF

(43) ⫽ 2Rmt E1 ⫺ (E1 ⫺ E2)(1 ⫹ 2)

an internal force combination Fo, Mmax that results in a certain d␣k

portion of the pipeline cross section undergoing yield.

The longitudinal strain, , and stress, , distribution can be

described via the polar angle of the cross section, , as

⫺ (E1 ⫺ E2)

d1

d␣冉⫹

d2

d␣ ␣ 冊

⫹ (E1 ⫺ E2)

d1

d␣

⫺

d2

d␣ 1 冉 冊

(38) ⫽ ␣ ⫹ b cos

⫺ (E1 ⫺ E2)

d1

d␣

cos1 ⫺ 冉

d2

d␣

cos2 b 冊册

Published by NRC Research Press

Kouretzis et al. 1067

Fig. 8. Strain and stress distribution over the pipeline cross section. The part of the cross section undergoing yield is depicted in red (i.e., the

portions above the 1 line and below the –1 line) and the part of the cross section remaining elastic is depicted in blue (i.e., the portions

between the 1 and –1 lines).

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再

and Fig. 9. Ramberg–Osgood stress–strain relation for API.5L-X65 steel

considered in the numerical analyses, plotted against the bilinear ﬁt

1 employed in the analytical solution.

± b sin1,2 ≤ ⫺0.01

b sin1,2

d1,2 ⫺100 ⫺0.01 ⬍ b sin1,2 ≤ 0

(44) ⫽

d␣ ±100 0 ⬍ b sin1,2 ≤ 0.01

1

± 0.01 ⬍ b sin1,2

b sin1,2

For personal use only.

Having calcutated both the axial strains from eqs. (42)–(44) and

bending strains from eq. (17), the maximum, max, and mini-

mum, min, longitudinal strains are derived from eq. (38), for

= 0 and = , respectively.

Step 5

The nonlinear behaviour of the pipeline material can now be

introduced into the solution by computing a secant Young’s mod-

ulus, Esec , compatible with the stress and strain distribution over

the pipeline cross section. This is calculated as

Mmax D

(45) Esec ⫽

2Ib

where the bending moment Mmax results from the stress distribu- Table 1. Summary of elastoplastic soil spring properties.

tion over the cross section as

Yield Yield

force displacement

冕

(46)

Mmax ⫽2 RmtRm cos d ⫽ 2Rm

2

t 冋E 2

1 b Spring

Axial (friction)

(kN/m)

18.60

(mm)

3.0

0 Vertical (upwards displacement) 31.91 2.3

⫺ (E1 ⫺ E2)(sin1 ⫺ sin2)␣ ⫹ (E1 ⫺ E2)(sin1 ⫹ sin2)1 Vertical (downwards displacement) 576.18 63.0

b

⫺ (E1 ⫺ E2)(1 ⫹ 2) ⫺ (E1 ⫺ E2)(sin21 ⫺ sin22)

2

b

4

册

linear large-deformation beam–spring numerical analyses, which

is the current state of practice for the design of buried pipelines

Steps 2–5 of the algorithm are repeated by updating the value of subjected to permanent ground deformations (ASCE-ALA 2005).

the Young’s modulus, until convergence is achieved. Via this pro- To investigate to what extent this objective has been met, the

cedure the nonlinear response of the pipeline material is taken maximum, minimum and axial pipeline strains calculated analyt-

into account (albeit approximately), while still using the mathe- ically are comapared against those predicted from the results of

matically convenient elastic theory for the analysis of beam ABC. three-dimensional ﬁnite element analyses with the commercial

code ANSYS.

Comparison of results against benchmark

To this end a typical straight, continuous 20 in. (1 in. = 25.4 mm)

numerical analyses natural gas pipeline, made of high-strength API.5L-X65 steel is

As stated earlier, the proposed analysis aims to provide results considered. The external diameter of the pipeline is D = 0.508 m

that match as closely as possible the results obtained from non- and its thickness t = 0.0079 m. The total length of the simulated

1068 Can. Geotech. J. Vol. 52, 2015

Fig. 10. Detailed results from the numerical analysis for differential heave offset ⌬z = 1.2D, focused on the area of interest where the

maximum strains develop. The sign of the vertical reaction forces follows the convention of Fig. 3.

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For personal use only.

Fig. 11. Comparison of axial, maximum, and minimum pipeline strains calculated with the proposed method against numerically computed

strains for relative settlement and heave. This set of results corresponds to the analysis for a settling or heaving zone of width Ls = 40 m.

pipeline is 2000 m, evenly divided along both sides of the settling racy of its results can be assessed. In doing this, the bilinear

or heaving zone. The total length of the model was proven ade- relationship of Fig. 7 is ﬁtted to the Ramberg–Osgood curve, as

quate to avoid any end effects on the results in the area of interest. depicted in Fig. 9.

Straight plastic beam elements (PIPE20) are used to discretize the The soil–pipeline interaction in the vertical and axial direc-

pipeline, each of which is 0.5 m long. A Ramberg–Osgood curve is tions is quantiﬁed via elastic–perfectly plastic spring elements

used to simulate the stress–strain response of the steel, as per (COMBIN39), connected to each node of the pipeline elements

common practice (two sets of spring elements per running metre). The properties of

the springs were derived according to the ASCE-ALA (2005) guide-

(47) ⫽

Ei冋 共

1⫹

b

r ⫹ 1 e 兲冉 冊 册

r

lines, assuming the pipeline is installed inside a trench of sufﬁ-

cient dimensions and backﬁlled with medium-dense sand up to a

level of 1.20 m, measured from the pipeline crown (Table 1).

where for X65 steel Ei = 210 GPa, e = 490 MPa, b = 38.32, and r = The sand unit weight is taken as ␥ = 18 kN/m3, its friction angle

31.5 (Fig. 8). In this way, the effect of considering a simple bilinear as = 36°, and the soil–pipeline interface friction angle as ␦ = 24°.

stress–stress relationship in the analytical solution on the accu- It is further assumed that the earth pressure coefﬁcient at-rest of

Kouretzis et al. 1069

Fig. 12. Comparison of axial, maximum, and minimum pipeline strains calculated with the proposed method against numerically computed

strains, for varying widths of the soil heave zone.

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For personal use only.

the backﬁll material is Ko = 1.0, to account for any accidental sufﬁciently wide to avert any interaction between the curved

compaction of the sand — although compaction of the backﬁll in parts of the pipeline located at the edges of the zone (Ls/2 > LBC + La

areas where relative ground surface deformations are expected from step 3). Detailed results of the analysis for ground heave with

should be avoided, as a denser backﬁll will introduce higher reac- an offset ⌬z/D = 1.2 are presented in Fig. 10, to assess pipeline

tion forces on the pipeline. Note that the uplift yield displacement response in light of the simpliﬁcations introduced in the analyti-

in Table 1 is derived by ﬁtting a bilinear force–displacement curve cal solution. The ﬁrst focus is on the pipeline deformed shape and

to the hyperbolic equation proposed by Trautmann et al. (1985). the reactions applied to the pipeline from the backﬁll in the axial

The free end of the spring elements is ﬁxed, and a step-like and vertical directions in the vicinity of the edge of the heave

prescribed displacement (Fig. 10) is applied to the ﬁxed nodes zone. It can be argued that the deformed shape of the pipeline is

located along the width of the settling or heaving zone. Analyses compatible with the deformation mode of the pipeline segments

for both ground settlement and heave are performed, with the depicted in Fig. 2b, while the vertical reaction forces along parts

absolute offset value ranging from 0.1016 to 1.016 m, which corre- AB and BC of the pipeline are in agreement with the shape and

sponds to normalized offset values ⌬z/D = 0.2 to 2.0. This range is magnitude assumed for loads qAB and qBC. It is also clear that the

expected to cover realistic ground surface offsets due to settlement or assumption of a uniform friction resistance developing along the

heave. full unanchored length of the pipeline is, as discussed earlier, not

First a pipeline crossing a settling or heaving zone of width Ls = totally accurate, as the friction forces diminish to zero at the

40 m (approximately 80D) is simulated, to check whether it is middle of the heave zone.

1070 Can. Geotech. J. Vol. 52, 2015

Fig. 13. Variation of axial, maximum, and minimum pipeline strains developing for vertical ground heave ⌬z/D = 1.0, with the half-width of

the heave zone Ls/2.

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Notice also that the location of maximum axial strains and design purposes, a bilinear ﬁt that adequately reproduces the

bending strains coincides, as implied in step 4 of the methodol- material’s behaviour within the region of the acceptable design

ogy. At this point the axial strains need to increase locally, so that limits is deemed sufﬁcient.

the integral of the longitudinal strains remains equal to the axial It must be stated, however, that the approach does not account

force. A more detailed discussion of this effect is provided by for the possibility of local buckling, due to excessive compressive

Karamitros et al. (2011), as well as by O’Rourke and Liu (2012). strains, which may cause early failure of the pipeline. For that

Given this agreement in capturing the elements of the pipeline aspect, the pipeline designer must compute a limiting compres-

response, it is no surprise that the proposed analytical method is sive strain that the pipeline section can sustain, which is a func-

able to provide accurate estimates of the pipeline strains, com- tion of the pipeline geometry (see, e.g., EN 1998-4 (CEN 2006)).

pared to those from the benchmark nonlinear ﬁnite element Furthermore, longitudinal strains due to internal pressure and

model. This comparison is illustrated in Fig. 11. Further to that, thermal effects were omitted from the analysis, as customarily

Fig. 11 provides additional evidence that both the settlement and they are expected to be an order of magnitude lower than the

heave cases essentially result in the same maximum pipeline strains induced by permanent ground deformation. The meth-

strains, though in different locations along the pipeline route. odology is applicable to straight, continuous pipelines, and to

To support the hypothesis that the proposed methodology settlement or heave zones that are wider than the curved length

For personal use only.

will provide accurate results when the half-width of the settling or of the pipeline, where bending strains develop. Unanchored

heaving zone Ls/2 is larger than the length of the beam BC plus the length estimates may be used to decide whether a more detailed

attenuation length La, Ls/2 > LBC + La, parametric analysis was analysis is required to take into account the restraint on the pipe-

performed where the half-width of the heave zone was decreased line axial deformation that is imposed by the existence of pipe

gradually from Ls/2 = 20 m (≈40D) to Ls/2 = 1 m (≈2D). The results of bends in the vicinity of the settlement or heave zone (these will

this parametric analysis are presented in Fig. 12, using the same lead to development of additional strains near the bends).

format as in Fig. 11. The analytically calculated strains match the

results of the ﬁnite element analyses, except for the case of the References

narrowest heave zone width Ls/2≈2D. This can be explained by ALA. 2005. Guidelines for the design of buried steel pipes. American Lifelines

means of Fig. 13. When the half-width of the ground offset zone Alliance, ASCE, New York.

becomes less than the length of the beam BC LBC, the high- Budhu, M., and Adiyaman, I. 2013. The inﬂuence of clay zones on land subsi-

dence from groundwater pumping. Ground Water, 51(1): 51–57. doi:10.1111/j.

curvature areas formed at the two edges of the heave zone (Fig. 10) 1745-6584.2012.00943.x.

overlap. This effect, which results in a substantial increase in CEN. 2006. Eurocode 8: Design of structures for earthquake resistance. Part 4:

bending strain, is not taken into consideration by the proposed Silos, tanks and pipelines. Standard EN 1998-4. European Committee for

methodology and the corresponding analytical predictions be- Standardization (CEN), Brussels, Belgium.

Chan, D., Kodikara, J.K., Gould, S., Ranjith, P.G., Choi, X.S.K., and Davis, P. 2007.

come nonconservative. On the other hand, it is worth noting that

Data analysis and laboratory investigation of the behaviour of pipes buried in

the analytical results match the numerical values fairly well in the reactive clay. In Proceedings of the 10th Australia New Zealand Conference on

intermediate cases where LBC + La > Ls/2 > LBC, even though the Geomechanics – Common Ground, Queensland, Australia.

theoretical boundary conditions considered for the analysis of Gould, S., Boulaire, F., Marlow, D., and Kodikara, J. 2009. Understanding how the

segment CC= in step 1 are violated. Australian climate can affect pipe failure. In Proceedings of OzWater 09,

Melbourne, March.

Ho, D.K.H., Dominish, P.G., and Tan, D.W. 2007. Rapid stress assessment of

Summary and practical considerations buried pipelines in mine subsidence region. In Proceedings of the Seventh

The method presented provides an attractive alternative to Triennial Conference on Mine Subsidence, Mine Subsidence 2007, Sydney,

New South Wales.

more rigorous (and complex) numerical analyses of pipelines that Karamitros, D.K., Bouckovalas, G.D., and Kouretzis, G.P. 2007. Stress analysis of

are subjected to permanent ground settlement and heave, offer- buried steel pipelines at strike-slip fault crossings. Soil Dynamics and Earth-

ing ease-of-use and a quick estimation of pipeline internal forces quake Engineering, 27(3): 200–211. doi:10.1016/j.soildyn.2006.08.001.

and strains. The simpliﬁcations introduced have a negligible Karamitros, D.K., Bouckovalas, G.D., Kouretzis, G.P., and Gkesouli, V. 2011. An

analytical method for strength veriﬁcation of buried steel pipelines at nor-

effect on the accuracy of the solution, at least for the realistic

mal fault crossings. Soil Dynamics and Earthquake Engineering, 31(11): 1452–

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