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AbstractIn off-shore plataform, petroleum from the oil

well may have to be heated up so that its density decrease,

making easier the pumping of petroleum along pipelines. Due

to temperature increase, such pipelines may be under thermal

dilatation and, consequently, under high compressive thermal

loading. There is a great difficult in finding the collapse load of

such submarine pipeline. An analytical method is presented in

this paper for the determination of the collapse load of

pressurized pipelines extended over large free spans. The

collapse load is determined from a closed solution equation.

Results of the presented formulation are compared withsophisticated finite element analyses carried out in ABAQUS

program. For the determination of the collapse load of

pressurized freespan pipelines under compression, non-linear

finite element analysis requires a lot of computer processing

while the present formulation takes practically no time to

assess a good approximation for the collapse load.

Key wordscollapse load and pipeline

NOTATION

Ao = Area of pipe cross-section

Ac = Section area under compressive stressAt = Section area under tensile stress

BP = Buried pipeline

E = Modulus of Elasticity

FSP = Freespan pipeline

Fc = Compressive cross-sectional force

I = Tensi le cross-sectional force

L = Moment of inertia of cross-section

M(x) = Length of freespan

My = Fully plastic bending moment

= Fully plastic moment capacity in the

present of P and .

P = Axial load at freespan ends

p = Internal/external pressure in pipe

PE = Euler critical load

Pca = Critical axial load for collapse

Pr = Force in centreline spring support

= FSP col lapse load

R or r = External or internal radius of pipe

= Centroid of Ac

Manuscript received March 09, 2011; revised April 18, 2011. This work

was supported by the Brazilian National Research Council, CNPq.

L. M. B. is with the University of Brasilia. Brasilia, DF 70910-900

Brazil (phone: 55-61-99819302, e-mail:[email protected]).

R. S. Y. C. S is with the University of Brasilia. Brasilia, DF 70910-900Brazil (phone: 55-61-81500286, e-mail: [email protected]).

(x,y,z) = 3D right-handed coordinate system

(u,v,w) = Displacements associated w/(x,y,z)

Z = Plastic section modulus

w = Beam's transverse displacement

= Arm inclination/for model without initial

imperfection (fig. 2)

= Arm inclination/for model with initial

imperfection (fig. 3)

= Arm lateral deflection at centerline

= Compressive longitudinal stress = P/A

E = Euler critical stress = P/A

c = Compressive longitudinal stress com As

t = Tensile longitudinal stress com As

y = Yie lding stress

u = Ultimate stress

= FSP collapse stress =P/A

1 = Von-Mises longitudinal stress

= Von-Mises circunferential stress

p = Radius of curvature of pipe bending

= Angle enveloping the area At (fig. 5)

= Angular cross-sectional coordinate

() = Non-dimensional hoop (longitudinal)stress

I. INTRODUCTION

He objective of this paper is to predict the behavior offreespan pipelines under compressive loads originatingfrom effects such as temperature differentials. The

paper deals exclusively with free span pipelines undercompressive load combined with internal pressure. Thecompressive load, P, is assumed to be applied at the ends ofthe pipe segment and to be collinear with a line through theend supports of the pipe segment. Consequently, the load is

considered to act along the chord connecting the two endsof the freespan segment, without change in the loaddirection. The collapse mechanics of a segment of a freespan pipeline (FSP) under compressive load is notnecessarily the same as for a buried pipeline (BP).Assuming a FSP under compression deforms as shown inFig.-1, the collapse mode of a FSP under compression,depends upon the length of the free span, and will bedifferent than for local wrinkle formation typicallyobserved in short segments of BPs. For short free spanlengths, the collapse mode of the FSP might be similar tothe local wrinkle formation mode observed in BPs. Forlong free spans, the collapse mode might be comparable tothe global buckling collapse mode observed in a structural

column.

Analytical and Numerical Solution for Collapse

Load in Oil Pipelines under Compressive Load

Luciano M. Bezerra, Ramon S. Y. C. Silva

T

Proceedings of the World Congress on Engineering 2012 Vol I

WCE 2012, July 4 - 6, 2012, London, U.K.

ISBN: 978-988-19251-3-8

ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

WCE 2012
mailto:[email protected]:[email protected]:[email protected]:[email protected]

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II. THEPIPE

Assuming small deformation theory, a long FSP, ifideally straight, elastic, and isotropic, loaded along thecentral axis, should behave like any long structural memberunder compression. The first model that comes to our mindis the buckling of the Eulers column. For all practical

purposes, the prescribed Eulers collapse load in Eq.(1) for apinned-end column is an upper limit of compressive loading

for an ideal FSP:

Figure 1: Freespan pipeline under compressive load and initial

imperfection

= 22 (1)To determine a more realistic behavior of FSPs under

compressive load it is necessary to admit the existence ofinitial imperfections, the possibility of inelastic behavior,

and the mobilization of fully plastic moment capacity of thepipe section. Let us consider the effect of initialimperfection and plastic deformations, using themechanical model shown in Fig.-2 [1]. Subsequently, it is

possible to examine for the effects of initial imperfectionand inelastic material behavior on the buckling behavior.The mathematical model consists of two rigid arms pinnedtogether at the span center-line at C. On the ends (A and B)they are pinned too, as in Fig.-3. A vertical spring, withstiffness K, is attached at C. Applying an increasinghorizontal axial force P at point A through the centroidalcross-sectional axis, with = 0, will make P reaches itscritical load, Pcr.

At this critical load, when buckling takes place, the

model forms a mechanism in which point C displaceslaterally through a distance , and the arm rotates - seeFig.-2b. Prior to instability, the force in the spring, Ps= 0.As soon as the instability takes place, P s=K - with K beingthe spring constant. From moment equilibrium of the armfrom A to C, about C, for Fig.-2b, we can write for smallangles

= 2 2 (2)In Fig.-2a, to simulate the Eulers buckling load, the

lateral defection will take place abruptly when the criticalload reaches the Eulers load. Substituting K for Psin Eq.

(2) and equating Pcr= PEyelds

= = 4 = 4 (3)

Figure-2: (a) An elementary buckling model and (b) free-body diagram.

Figure-3: The mechanical model with initial imperfection.

Up to now, the pre-buckling shape is a straight line,however, by now lets consider the existence of an initialimperfection o 0 in Fig.-3. Note that small initialimperfections will be amplified by the axial force. Themodel of the FSP with an initial imperfection o is in Fig.-3a. o exists initially, for P = 0 and Ps = 0. For P 0, theincremental displacement at the centerline increases by theamount due to the rotation of the arms see Fig.-3a. Thetotal displacement, due to the arm rotation, , becomes tot= 0 + - Fig.-3b. The moment equilibrium of arm A-C,about C, for Fig.-3b may be expressed by Eq.(4).

=

2 2

=

4 ;

= + (4)As PS = K and = tot 0, and using Eq.(3) for K;

Eq.(4) can be transformed into the following equation.

= 4 = 4 = 44 = ( )

(5)

= 1 The model for the FSP with an initial o will result in an

increase in bending moment at the center of the span as Pand tot increase. However, P in Eq. (5) indicates that thecompressive load for the model of the imperfect column (orFSP) will never reach the Eulers load, PE, but approachesPE asymptotically. In addition, the maximum bendingmoment that can arise at the central section cannot exceedthat associated with the fully plastic condition for the pipe.At the central section M = Ptot and the collapse load forcan be determined by the load P that produces the momentwhich, when combined with the axial effects, mobilizes thefully plastic capacity of the pipe section ( Mpc ). Tocompute the full plastic capacity of the pipe it is necessary ayield criterion.

Pressurized pipes are subjected to hoop and longitudinalstresses due to axial forces and transverse bending momentsacting on the pipe cross section. For a thin-walled pipe, thehoop stress is considered constant and stresses other thanhoop and longitudinal may be neglected. The longitudinalstress l and the hoop stress are identified as the

principal stresses 1 and 2, respectively. Using the Von-Mises-Hencky yield criterion (with y as the uniaxial yieldstrength) the maximum (and minimum) longitudinal stressesthat the fully-plastic pipe cross section can sustain on thecross section may be calculated as:

= 2

1 3

4 2

(6)

Eq.(6) is also valid for the ultimate stress (u) in theplace of the yielding stress (y). Eq.(6) also identifies the



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two values of l that produce yielding for a specified .One corresponds to a compressive stress (l= c), and theother, to a tensile stress (l= t) - Fig 4. These valuesrepresent the maximum longitudinal compressive andmaximum longitudinal tensile stresses that can be developedon the extreme fibers of the pipe cross-section for the given. If = 0, then, for yielding, l=c=t=y. If 0, then,the longitudinal stress l required to origin yield in tensionis l= twhich is different than that required in compression

(l = c) (See Fig.4). Naming = /l and = l /y, theVon-Mises-Hencky yield criterion is shown in Fig.-4 [4, 5].From Fig.-4 and Eq.(6), the extreme values for l and .For the determination of the fully-plastic capacity of the

pipe section, we will assume that the stress-strain curveshows a well defined yield-stress plateau. The yield stress isan important engineering property in order to establishlimits on the longitudinal and hoop stresses. The hoop stress is given by:

= () (7)The longitudinal stress acting on the pipe cross-section

will depend on the axial force P and the bending moment.The limiting combinations of axial force and bendingmoment that develop the fully plastic capacity of the pipesection can be presented on an interaction diagram due to [2,3]. In the following Section, the equations for the fully

plastic moment capacity of the FSP pipe section will bederived.

III. DEVELOPINGTHEFULLYPLASTICMOMENT

For a pipe, Fig.-5 shows the fully plastic stressdistribution, accounting for the effects of stresses t and c[2]. As the pipe is under compressive load P applied at the

pipe ends, the applied force is concentrically distributed onthe pipe end sections with area Ao giving rise to anequivalent longitudinal uniform stress = P/A0at points Aand B of Fig.-3, therefore

= = (22) (8)The stresses on the pipe section at the point of maximum

moment are in Fig.-5 which is a fully plastic condition. Atsuch a point, at the center of the span, we have acombination of stress from bending moment plus stress fromaxial loading. However, at the ends of the FSP (see Fig.-1and 3); the force P acts in concert with the transverse forceof P

s/2, and the combination of these loads must be

equilibrated by the stress distribution of Fig.-5 at thecenterline of the span. Therefore, at the point of maximummoment, the resultant longitudinal force given by thedifference between the tensile force Ft = tAt andcompressive force Fc = cAc, in Fig.-5, must be inequilibrium with the external applied force P at the ends ofthe FSP. The areas Ao, At, and Ac in Fig.-5 can beexpressed as

= 2 2 , = 2 (2 2)(9)

= (2

2

) 2

2 From Eqs. (8) and (9) we can write the following

longitudinal equilibrium equation

= 2 2 = (10)

= 2 2 2 2 (2 2) 2The angle can be calculated as a function of the

stresses , t, and cof Eq.(10).

= 2 + (11)A search for P (that causes the stress distribution

depicted in Fig.-5) is the same as a search for the equivalentstress = P/Ao at the end of the FSP. The arms ty & cy ofthe respective forces Ft and Fc, such forces are at thecentroids of the areas Atand Acin Fig.-5 - can be calculatedas:

=

422 sin

2

32(2

2

) (12)

Knowing At, Ac in Eq.(9); ty , cy in Eq.(12); and candt in Eq.(6); the maximum plastic resisting moment M

pc

can be determined due to the load P (or stress ) at the endsof the FSP. M

pc is in equilibrium with the moment caused

by the external force P and the eccentricity totof Fig.-3 andEq.(4), therefore

= + = + = (13) =

Using Eqs.(9), and (12) in Eq.(13), the expression for themaximum plastic bending moment is:

= 22 2 2 4333222 sin 2+ 2 (2 2) 4(33)3(22) sin 2 (14)Substituting into Eq.(14) the expression for the angle

from Eq.(11), we arrive at the following simplified versionof Eq.(14) which is an expression for the maximum momentcapacity for the FSP

= 2

3 ( )(3

3

) sin (

)

(+) (15)IV. THEPLASTICCOLLAPSE

The limiting fully plastic moment for the FSP asexpressed in Eq. (15) is an upper bound on the moment thatcan be developed before a plastic collapse bucklingmechanism occurs. For this mechanism to occur we notethat M

pc is a function of: (a) the maximum allowable

longitudinal stresses, t and c; and (b) the equivalentapplied stress (or load P, since = P/A0) applied at theends of the FSP. It is assumed that a structure with an initialimperfection and under increasing applied compressive loadwill deform until its fully plastic moment capacity is

developed. The expression for maximum moment capacityin Eq.(15) shows that for an increase in , there will be adecrease in M

pcat center span. The formulation contained,

herein, is based upon the argument that, to find thecompressive collapse stress of a FSP, the effect of out-of-



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straightness must be taken into account. In reality, everystructure has imperfections in geometry; but long structureslike FSP laid on rough terrains, are more susceptible. Theinitial imperfection 0 is taken into account in Eq.(5). Suchequation represents the behavior of the FSP in the elasticrange until the fully plastic stress distribution of Fig.-5 isdeveloped giving rise to Eq.(13). Note that Eq.(5), expressedin terms of Eulers critical stress E = PE/A0, can give anexpression for tot as

= = 1 = 1 (16)

= 1 Once yielding has fully developed, put M

pc from Eq.

(15) into Eq.(13) to get an expression for totas:

=

(17)

= 2( )(3 3)3(3 3) sin ( )( + )

Finally, by equating the right hand sides of Eq.(16) andEq.(17), we arrive at the following transcendental equationfor the determination of the collapse stress , which will bedesignated as

2(+)(33)3(22) 1 sin ()(+ ) = 0 (18)

Developing Eq.(18) into a Taylor series and keepingonly two terms of this series, one obtains:

2 + + = 0Where (19)

= 1,5 2 23 3 =

Figure 4: Von-Mises-HenckyYelding Criterion

Figure 5: Idealized fully plastic stress distribution

The solution for the collapse stress in Eq.(19) takesinto consideration: (a) the geometric properties of the pipesection; (b) the initial imperfection for the particular FSP;(c) an upper bound limit represented by the Eulers bucklingload; (d) the fully-plastic stress distribution, (Fig.-5); (e) thefully-plastic capacity depends on both the plasticity criterionand the hoop stress, which is a function of the appliedinternal pressure; (f) long structures, with initialimperfection, never reach the Eulers Load (which is an

upper bound limit); (g) the Eulers load (or stress) which is afunction of the modulus of elasticity, pipe cross-section

properties, and pipe length; and (h) the consideration ofinitial imperfection that is essential as it triggers the limitingfully-plastic moment capacity mechanism.

V. COMPARISONTOFINITEELEMENTANALYSES

For obtaining collapse load for FSPs as a function ofLi/D ratios; consider a typical pipeline for petroleumtransportation with a range of free spans Li. The materialand cross section properties of the pipeline are: (a) E=200000MPa, (b) y =448MPa, (c) u =531MPa, (d) D =323.85 mm, (e) t = 19.05 mm, (f) d = 285.75 mm, (g) R =D/2 (161.925 mm), and r =d/2 (142.875 mm), and nainternal pressure p = 10.2Mpa. The range of free span ratios(or Li/D) are shown in Table 1. For each FSP length Li, aninitial imperfection oiis assumed. In this paper, oiis takenas the transversal deformation of the FSP such that theextreme fibers of the pipe cross section are just reaching theonset of yielding. Any other value of oicould be arbitrarilyused. For simplification, and just to calculate an initialimperfection, it was assumed that on the onset of yieldingt=c=y. Each Lidetermines different Eulers load PE andstress E. The collapse loads of such FSPs without internal

pressure are readily obtained and reported in Table-1. Theanalytical solutions are compared to Finite ElementAnalyses using ABAQUS [6]. It is also noticed that the

analytical results reported in Table-1 consider the ultimatestress u in the place of yielding stress y into Eq.(6) - inthe ABAQUS runs and results the ultimate stress is reached.

Table-1: Comparison with FEM results



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VI. CONCLUSION

This paper presented a mathematical formulationregarding the investigation of compressive collapse loads of

pressurized FSPs. A strategy for obtaining collapse loads asa function of the span length, initial imperfection, and fully

plastic stress capacity has been presented and discussed.

Examples of collapse loads, for pressurized FSPs with avariety of lengths and initial imperfections, were comparedto the sophisticated FE results from the ABAQUS program.The numerical tests show that the proposed analyticalformulation represents a good approximation to freespansolutions. Instead of yield stress, the analytical solutionswere almost coincident with the collapse results generated

by ABAQUS FE analyses. Each complex nonlinear FE runin ABAQUS took approximately 5 hours of CPU on a SUNworkstation. Finally, it is noted that the scope of the presentformulation is not to propose a method to substitute preciseFEM modeling and analyses, but to provide an easy, fasterand practical way for a first assessment of compressivecollapse loads of pressurized FSPs for the petroleum

industry.

Figure 9: Comparison of analytical and FEM results for FSP

REFERENCES

[1] Shanley, F. R, Strength of Materials. McGraw-Hill BookCompany Inc. New York, 1957

[2] Popov, E., Mechanics of Solids. Prentice Hall. Englewood Cliffs,New Jersey, USA, 1998.

[3] Hoffman O., Sachs G., Theory of Plasticity, McGraw-Hill BookInc. New York, USA, 1953.

[4] Dorey, A.B., Critical Buckling Strains for Energy Pipelines. PhDThesis, University ofAlberta, Edmonton, AB, Canada, 2001.

[5] Mohareb, M., D. W. Murray, Mobilization of Fully PlasticMoment Capacity for Pressurized Pipes. Journal of Offshore

Mechanics and Arctic Eng., ASME. vol. 121. p. 237-241, 1999.[6] ABAQUS.,Standard Users Manual. Version 6.3. Hibbitt, Karlssonand Sorense, USA, 2000

.



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