Thin Walls to Ext Press

8/9/2019 Thin Walls to Ext Press

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Daniel Vasilikise-mail: [email protected]

Spyros A. Karamanos1

e-mail: [email protected]

Department of Mechanical Engineering,

University of Thessaly,

Volos 38334, Greece

Mechanics of ConfinedThin-Walled CylindersSubjected to External Pressure Motivated by practical engineering applications, the present paper examines the mechanicalresponse of thin-walled cylinders surrounded by a rigid or deformable medium, subjected to uniform external pressure. Emphasis is given to structural stability in terms of buck-ling, postbuckling, and imperfection sensitivity. The present investigation is computa-tional and employs a two-dimensional model, where the cylinder and the surroundingmedium are simulated with nonlinear finite elements. The behavior of cylinders made of elastic material is examined first, and a successful comparison of the numerical results isconducted with available closed-form analytical solutions for rigidly confined cylinders.Subsequently, the response of confined thin-walled steel cylinders is examined. The nu-merical results show an unstable postbuckling response beyond the point of maximum pressure and indicate severe imperfection sensitivity on the value of the maximum pres-sure. A good comparison with limited available test data is also shown. Furthermore, theeffects of the deformability of the surrounding medium are examined. In particular, soilembedment conditions are examined, with direct reference to the case of buried thin-walled steel pipelines. Finally, based on the numerical results, a comparison is attempted between the present buckling problem and the problem of “shrink buckling.” The differ-

ences between those two problems of confined cylinder buckling are pinpointed, empha-sizing the issue of imperfection sensitivity. [DOI: 10.1115/1.4024165]

1 Introduction

In several engineering applications, steel cylinders subjected toexternal pressure are surrounded by a confining medium. Under those conditions, the cylinders may buckle because of excessivehoop compression. Buried steel pipelines [1] under such loadingconditions can often fail in the form of structural instability; whenthe groundwater table is above the pipeline level, the water reaches the pipe through the permeable surrounding soil or con-

crete encasement, and hydrostatic pressure conditions developaround the pipeline, which may cause buckling of the steel pipe-line wall. In addition, thin-walled liners, made of steel or plasticmaterial, used to rehabilitate damaged pipelines [2], may also failunder similar loading conditions. Furthermore, tunnels and ductsthat transport gases or liquids in power plants are often lined withcylindrical steel shells [3], which may buckle because of externalpressure under lateral confinement. Finally, steel tubes employedas casing in oil and gas production wells [4] are also typical exam-ples of externally pressurized cylinders, which may fail under confined conditions.

In all the above engineering applications, significant hoopstresses develop in the cylinder wall due to hydrostatic pressureconditions because of the permeability of the surrounding me-dium. When these hoop stresses exceed a critical level, the cylin-der loses its structural stability and buckles. In such a case, due tothe surrounding medium, the cylinder wall is not free to deform inthe outward direction, and buckling occurs in the form of an“inward lobe,” as shown in Fig. 1 at a pressure level significantlyhigher than the one under unconfined conditions.

The present paper focuses on buckling and postbuckling of cylin-ders subjected to uniform external pressure, surrounded by a deform-able confining medium. It should be noted that buckling of confinedcylinders under hydrostatic pressure is different than buckling under thermal effects, sometimes referred to as “shrink buckling” [5 – 7].

For an extensive literature review on the shrink buckling problem,the reader is referred to the paper by Omara et al. [2]. In the lastpart of the present paper, a direct comparison between the presentproblem and the problem of shrink buckling is offered, based onnumerical simulation results.

The single-lobe mode of Fig. 1 has been observed in experi-ments and in real applications of externally pressurized cylindersunder confined conditions. The first analytical attempts to predictthis buckling behavior have been reported in the book of Feodo-

syev [8] and the paper of Glock [9]. Glock presented an energyformulation and solution of the hydrostatic buckling problem of rigidly confined cylinders made of elastic material, assuming nofriction between the ring and the nondeformable medium, as wellas no variation of stress and deformation in the axial direction of the cylinder, and resulted in the following closed-form expressionfor the buckling pressure:

pGL ¼ E1 2

t

D

2:2(1)

where E is Young’s modulus, is Poisson’s ratio, D is the cylin-der diameter, and t is the wall thickness. For a concise presenta-tion of the Glock’s solution, the reader is referred to the paper of Omara et al. [2]. It is interesting to note that for diameter-to-thick-

ness ( D=t ) values between 100 and 300, which are typical valuesfor buried pipelines and rehabilitation liners, the value of pGL issignificantly higher than the buckling (bifurcation) pressure pe of a long (free of boundary conditions) externally pressurized per-fectly round elastic cylinder under unconfined conditions, givenby the following formula [10]:

pe ¼ 2 E1 2

t

D

3(2)

The ratio of Eqs. (1) and (2), results in

pGL

pe¼ 0:5 D

t

0:8(3)

1Corresponding author.Manuscript received August 21, 2012; final manuscript received March 29, 2013;

published online November 26, 2013. Editor: Harry Dankowicz.

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It can readily verified that the ultimate value of pressure pGLcalculated from Eq. (1) is 20–48 times higher than the bucklingpressure pe under unconfined conditions. The validity of Glock’sEq. (1) in predicting the buckling pressure of rigidly confinedelastic cylinders has been verified by the finite element resultsreported by El-Sawy and Moore [11]. In that publication, El-Sawyand Moore also accounted for the presence of initial gap gbetween the cylinder and the rigid surrounding medium (“looselyfitted cylinders”) and resulted in the following empirical analyti-cal expression for the buckling pressure:

p EM ¼ 2 E1 2

t

D

3 25 þ 700 t = Dð Þ þ 315 g= Rð Þ0:15 þ 130 t = Dð Þ þ 1400 t = Dð Þ2þ145 g= Rð Þ

!

(4)

The last term in the right-hand side of Eq. (4) within the parenthe-sis expresses the increase of the classical elastic buckling pressure pe for unconfined conditions (see Eq. (2)) when rigid confiningconditions are imposed on the externally pressurized cylinder.Assuming a zero value of initial gap (g ¼ 0), the comparisonbetween Eq. (4) and Glock’s formula [1] shows that the former em-pirical formula can predict quite accurately the buckling pressure of tightly fitted elastic cylinders in a rigid cavity. The validity of Glock’s formula has also been tested against experimental data [12].

An enhancement of Glock’s solution [9] has been developed byBoot [13] to account for the presence of initial gap between thecylinder and the rigid medium, whereas notable contributions onthis subject have also been reported by Bottega [14] and Li andKyriakides [15], who examined analytically the behavior of twoconcentric, contacting elastic rings, subjected to external compres-

sive loading. The work in Ref. [15] has been extended to investi-gate buckling propagation in a long elastic cylinder in contactwith an outer elastic cylindrical shell [16].

The above works on confined cylinder buckling refer to cylin-ders made of elastic material. It is interesting to note that despitethe numerous publications on the mechanical behavior and buck-ling of elastic cylinders, significantly fewer investigations exist onthe corresponding buckling problem of steel cylinders, which isassociated with elastic-plastic material behavior. As a first approx-imation, the ultimate external pressure capacity can be estimatedas the pressure that causes first yielding at the outer fiber of thecylinder wall. Based on this assumption, Montel [17] employedTimoshenko’s solution for thin ring deflection [18] and experi-mental results [19] to develop a semi-empirical formula for the

buckling pressure of cylinders embedded in a stiff (nondeform-able) cavity, in terms of the material yield stress r y, the cylinder geometry D=t , the initial out-of-roundness with amplitude d0, andthe initial gap with maximum value g between the cylinder andthe rigid cavity:

p M ¼ 14:1r y

D=t ð Þ1:5 1 þ 1:2 d0 þ 2gð Þ=t ½ (5)

The above Eq. (5) has been proposed for a range of parameters,

namely 60 D=t 340, 250 MPa r y 500 MPa, 0:1 d0=t 0:5, and g=t 0:25. Clearly, the ultimate pressure p M of Eq. (5) is a decreasing function of both imperfection types d0 andg, as well as a decreasing function of the diameter-to-thickness ra-tio D=t . Furthermore, it can be readily shown that for D=t between100 and 300, the value of p M is well below the yield or plasticpressure p y of the cylinder, i.e., the nominal pressure that causesfull plastification of the cylinder wall. From thin-walled vesseltheory, and considering a von Mises yield criterion under planestrain conditions for the deforming cylinder cross section, theplastic pressure p y is readily calculated equal to

p y ¼ 2 r y ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ 2p

t

D

(6)

where r y is the yield stress of the material under uniaxial stressconditions, and factor 1=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ 2p accounts for increase of

yield stress in the hoop direction due to plane strain conditions,equal to 1.13 for ¼ 0:30. Note that the plastic pressure p y is adecreasing function of the D=t ratio. Dividing Eqs. (5) and (6),and assuming ¼ 0:30, one obtains for perfect cylinders(d0 ¼ g ¼ 0)

p M

p y¼ 6:24 t

D

0:5(7)

It is interesting to note that, for D=t values ranging between 100and 300, Montel’s equation predicts a pressure capacity p M rang-ing from 36% to 62.4% of the plastic pressure p y.

Failure at first yielding was also assumed by Amstutz [20] and

Jacobsen [21], who developed analytical expressions for the exter-nal pressure collapse of embedded rings. A comparison betweenMontel’s Eq. (5) and the methodologies proposed by Amstutz[20] and Jacobsen [21] is offered by Taras and Greiner [22], for the design of tunnel steel linings under external pressure. Yama-moto and Matsubara [23] reported a numerical solution for theultimate pressure sustained by a steel cylinder embedded in a rigidcavity. Kyriakides and Youn [24] and Kyriakides [25] conducteda more rigorous investigation of buckling and postbuckling behav-ior of confined cylinders under external pressure, using a semi-analytical formulation, which was based on nonlinear ring theory.El-Sawy, extending the work in Ref. [11], examined numericallythe buckling response of tightly fitted [26] and loosely fitted [27]steel cylinders, surrounded by a rigid boundary and subjected toexternal pressure. In a recent continuation of his work, El-Sawy

[28] examined the behavior of confined cylinders under externalpressure with particular emphasis on the stability of steel linerswith localized wavy imperfections, as well as the effects of liner’smaterial properties and geometrical parameters. Furthermore,buckling and postbuckling of vertical sandwich shells under lat-eral pressure has been investigated by Estrada et al. [29] through aparametric study that accounted for the influence of shell slender-ness, imperfection amplitude, and soil elasticity.

Vasilikis and Karamanos [30] investigated the structural stabil-ity of thin-walled elastic and steel cylinders, with diameter-to-thickness D=t ratio that ranges between 100 and 300, surroundedby an elastic medium, in terms of their structural stability under uniform external pressure using a nonlinear two-dimensional fi-nite element model. The numerical results have demonstrated that

Fig. 1 Schematic representation of the buckling problem of anexternally pressurized cylinder confined by the surroundingmedium

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the ultimate (buckling) pressure is substantially lower than theplastic pressure of the cylinder and is affected by the presence of initial imperfections in terms of cylinder out-of-roundness andgap between the cylinder and the medium. In a subsequent publi-cation, Vasilikis and Karamanos [31] proposed a methodology for the buckling design of steel cylinders confined by a rigid or deform-able medium, within the framework of the new European shell sta-bility design rules [32] and recommendations [33]. In both Refs.[30] and [31], special emphasis is given to the sensitivity of the ulti-mate pressure on the presence of initial imperfections, assumed inthe form of both out-of-roundness of the ring geometry and a smallgap between the ring and the elastic medium.

The present paper has a dual purpose. First, it offers an over-view of the hydrostatic buckling problem of confined cylinders. Inaddition, it provides a more-in depth investigation on several issuesof buckling behavior of confined cylinders, with emphasis on imper-fection sensitivity and the effects of deformable medium, extendingthe work presented in Refs. [30] and [31]. Towards the above pur-

pose, the interacting system of the cylinder and the surrounding-medium encasement is simulated through nonlinear finite elements.In addition, the problem of shrink buckling is modeled and the resultsare compared with available experimental data, as well as resultsfrom the corresponding hydrostatic buckling problem.

2 Finite Element Simulation

The structural behavior of confined cylinders under uniformexternal pressure is examined with nonlinear finite elements,

using the general-purpose finite element program ABAQUS [34].The analysis considers nonlinear geometry, through a large-straindescription of the deformable medium, as well as inelastic mate-rial behavior and it is similar to the one adopted in Refs. [30] and[31].

No variation of loading and deformation is assumed along thecylinder so that a two-dimensional finite element model of the cyl-inder is considered, with one element in the longitudinal directionof the cylinder, under plane-strain conditions. From the symmetryof the single-lobe postbuckling shape of the cylinder, half of the

cylinder cross section is analyzed, applying appropriate symmetryconditions at the h ¼ 0 plane. The thin-walled steel cylinder ismodeled with four-node reduced-integration shell elements (typeS4R), whereas eight-node reduced-integration solid elements(C3D8R) are used to simulate the surrounding medium, as shownin Fig. 2. The mesh dimensions L and H are chosen equal to 1.5and 3 cylinder diameters, respectively, following a short paramet-ric study, and a total of 150 shell elements around the cylinder half circumference have been found to be adequate to achieveconvergence of solution and accuracy of the numerical results.

A J 2 flow (von Mises) plasticity model with isotropic hardeningis employed in the analysis, to simulate inelastic behavior of thematerial of steel. The soil material is considered elastic in the ma- jority of cases examined in this paper. In a few cases, soil materialis also described through an elastic-perfectly plastic Mohr– Coulomb constitutive model, characterized by cohesion c, friction

angle /, elastic (Young’s) modulus E0, and Poisson’s ratio . Africtionless contact algorithm is employed for the interfacebetween the cylinder and the medium. Uniform external pressureis applied around the cylinder, and the nonlinear pressure-deflection ( p d) equilibrium path is traced using a Riks continu-ation algorithm.

The sensitivity of cylindrical response and strength on the pres-ence of initial imperfections are of particular importance in our work. Two types of initial imperfections are considered in thepresent study. The first type of imperfection is an initial gapbetween the confining medium and the cylinder. The gap is intro-duced in the model, assuming that the circular cavity of the me-dium has a radius slightly larger than the circular cylinder radius,and that the cylinder and the cavity are initially in contact ath

¼p (Fig. 3(a)) so that the maximum gap between the cylinder

and the medium occurs at h ¼ 0, and it is denoted as g . The sec-ond type of imperfection is a small initial “out-of-roundness”imperfection on the steel cylinder in the form of a small localizeddisplacement pattern at the vicinity of the h ¼ 0 location. It is animperfection of the shape of the buckling mode of the confinedcylinder (“single-lobe” mode). One way to impose this initial out-of-roundness in steel cylinders is through the consideration of asmall downward vertical load applied at the h ¼ 0 location. After the load is removed, and despite the elastic rebound of the steelcylinder wall, the cylinder at this location contains a small

Fig. 2 Finite element model of cylinder-medium system

Fig. 3 Schematic representation of a confined ring with (a ) gap-type initial imper-fection and (b ) out-of-roundness initial imperfection

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residual displacement d0, which is considered as the initial out-of-roundness amplitude, as shown in Fig. 3(b). Clearly, this type of method may not be suitable for elastic cylinders, due to the com-plete recovery of shape during the unloading step. Alternatively,this out-of-roundness imperfection can be imposed considering aninitial stress-free displacement pattern, in the form of a “single-lobe at the vicinity of the h ¼ 0 location, chosen in the form of the consecutive shapes of the perfect confined elastic cylinder under external pressure. This method of imposing initial out-of-roundness is suitable for both elastic and elastic-plastic (steel)

cylinders.

3 Mechanical Behavior of Elastic Cylinders

Thin-walled rigidly confined elastic cylinders with D=t valuesbetween 100 and 300 are analyzed in this section using the finiteelement simulation presented in Sec. 2. The cylinder material iselastic with modulus E and Poisson’s ratio equal to 210,000MPa and 0.3, respectively, whereas a frictionless interface is con-sidered between the elastic cylinder and the nondeformable con-finement medium. The cylinder capacity is compared withanalytical expressions and numerical results in Refs. [9] and [11].The modulus of the confinement medium E 0 has a value equal to21,000 MPa (one-tenth of the modulus of the steel material) sothat the confinement medium is practically nondeformable andmay be considered as rigid.

3.1 Perfect Elastic Cylinders. Figure 4 shows the variationof maximum pressure pmax in terms of the diameter-to-thicknessratio for rigidly confined cylinders. In this figure, the numericalresults (depicted with symbol~) are compared with the analyti-cal predictions pGL of Eq. (1). The comparison between numericalresults and predictions from Glock’s analytical solution is remark-able, and this offers a very good verification of the validity of Glock’s formula for the buckling pressure of imperfection-freeelastic confined cylinders in a stiff (nondeformable) medium. It isnoted that the buckling pressure p EM predicted by the closed-formexpression (4) for imperfection-free cylinders [11] offers verygood predictions as well.

The value of maximum pressure pmax is 20 to 48 times larger

than the value of the corresponding elastic buckling pressure peunder unconfined conditions, expressed by Eq. (2). The large val-ues of the pmax= pe ratio express quantitatively the very significanteffect of confinement on the buckling resistance. The structuralbehavior of the externally pressurized cylinder and the effects of confinement can be better understood if one follows the pressure-displacement curve of the deforming cylinder, assuming a smallgap between the cylinder and the rigid confining medium, in the

form depicted in Fig. 3(a). The corresponding results are shown inFig. 5; the cylinder is elastic with D=t ratio equal to 200 and thegap size g is less or equal to 2.7 10-3 times the cylinder radius R, i.e., less than 27% of the cylinder thickness. In particular, thebehavior under very low levels of external pressure is considered,as shown in Fig. 5(b); initially, the cylinder exhibits uniform con-traction, until the pressure corresponding to pe is reached, i.e., theelastic buckling pressure under unconfined conditions, calculatedfrom Eq. (2). At that pressure level, the cylinder buckles in anoval shape [10] but, very quickly, it accommodates itself within

the confinement boundary, and this is represented by the changeof slope in the pressure-displacement diagram, also shown in Fig.5(b). Therefore, the cylinder is able to sustain significant further increase of external pressure, as represented by the increase of pressure beyond the critical pressure level for unconfined condi-tions ( pe). Under those confined conditions, the top part of the cyl-inder, i.e., the part corresponding to the maximum gap location,behaves similar to an arch subjected to uniform external pressureand supported at the two “touchdown” points. This leads to buck-ling in the form of an inward single-lobe buckling mode, some-times referred to as “inversion buckling,” characterized by a limitpoint on the pressure-deformation equilibrium path and unstableresponse beyond the limit point represented by a rapid drop of pressure, as shown in Fig. 5(a). This “snap-through” behavior means that the secondary (postbuckling) equilibrium path may notbe reachable in a real pipe since the load is controlled by pressure.

The value of pressure at the limit point is referred to as maximumpressure pmax. For the particular case of imperfection-free cylin-ders it is also referred to as critical pressure, denoted as pcr .

Fig. 4 Comparison between numerical results and analyticalpredictions from Glock’s Eq. (1) and El-Sawy and Moore Eq. (4)for the buckling pressure of rigidly confined cylinders

Fig. 5 Structural response of rigidly confined elastic cylindersin the presence of small gaps; (a ) general response and (b ) ini-tial response for very low-pressure values




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Consecutive deformation configurations of an imperfection-freeelastic cylinder with D=t equal to 200 under external pressure areshown in Fig. 6(a), and the corresponding points on the pressure– deflection path are depicted in Fig. 6(b). The numerical resultsindicate that the maximum pressure pmax is equal to 1.95 MPa andoccurs at the stage where the local curvature at h ¼ 0 becomeszero (i.e., when “inversion” of the cylinder wall occurs). From Eq.(1), Glock’s prediction is equal to 1.99 MPa, very close to the nu-merical value. Using Eqs. (A3) and (A4) of the Appendix fromthe analytical solution of Glock [9] and conducting the appropri-ate differentiation of the radial displacement function wðhÞ, thelocal change of hoop curvature k 0 at the buckle location h ¼ 0 iscalculated as follows:

k 0 ¼ w00ð0Þ R2

¼ d2 R2

p

/ 2

(8)

Therefore, at the stage of maximum (critical) pressure, the changeof hoop curvature k 0;cr can be calculated analytically substitutingEqs. (A9) and (A10) into Eq. (8) to obtain:

k 0;cr ¼ 1:033 1 R

’ 1 R

(9)

Adding the value of Eq. (9) to the initial hoop curvature (equal to1= R), the curvature of the deformed configuration at the stage of buckling is readily computed equal to zero. This result fromGlock’s solution implies that the maximum pressure occurs at thestage where the local curvature at the h ¼ 0 location becomes

zero, i.e., the cylinder becomes locally flat and shows a very goodcorrelation with the numerical results.

Table 1 depicts the values of maximum pressure, as well ashoop stress, inward radial displacement at the critical location,and the size of the detachment zone at buckling stage, obtained byboth numerical analysis and Glock’s analytical solution. Thevalue of the nominal hoop stress rnom, corresponding to the fol-lowing formula:

rnom ¼ pD2t

(10)

from elementary mechanics of materials, is of particular interest.Furthermore, stresses, ra and rb refer to the membrane and bend-ing stress, respectively, and in Table 1 they are both normalizedby the value of rnom at buckling, i.e., considering p

¼ pmax in Eq.

(10). In general, a fairly good comparison has been obtainedbetween the numerical results and the analytical solution for elas-tic cylinders provided in Ref. [9].

An important observation refers to the magnitude of the hoopstresses. The above results demonstrate that Eq. (10) may not be areliable formula for computing the hoop stress on the pressurizedcylinder at the prebuckling stage. In addition, although the valueof membrane stress ra is comparable with the value of rnom, thevalue of rb is significantly higher, implying a substantial bendingdeformation before buckling. It is also noted that the above resultsassume elastic behavior of the cylinder material. Therefore, it isexpected that steel cylinders will have a reduced strength due toearly yielding caused by the development of significant bendingstress rb.

Fig. 6 Consecutive deformation shapes of a tightly fitted elastic cylinder; configuration (2) corresponds to the ultimate pres-sure stage

Table 1 Comparison between analytical results [9] and numerical results for rigidly confined elastic cylinders

D/t ¼ 100 D/t ¼ 200 D/t ¼ 300

Cylinder Analytical [9] FEM (present) Analytical [9] FEM (present) Analytical [9] FEM (present)

pmax (MPa) 9.187 9.066 1.999 1.948 0.819 0.788dcr /R 0.04563 0.04598 0.0262 0.0255 0.0189 0.0211/cr (rad) 0.4669 0.4674 0.3539 0.3591 0.3009 0.2952kcr 1.033 1.127 1.033 1.087 1.033 1.144rnom (MPa) 453.3 — 194.8 — 118.2 — ra / rnom 1.17 1.20 1.18 1.22 1.20 1.28

rb / rnom 5.09 5.78 5.93 6.47 6.51 8.44




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3.2 Imperfection Sensitivity in Elastic Cylinders. The nu-merical results obtained for perfect elastic cylinders indicate asubstantial drop of pressure in the equilibrium path beyond thecritical pressure point, which is a severe indication of imperfec-tion sensitivity. In the present study, two types of initial imperfec-tions are considered, namely initial gap between the cavity andthe outer surface of the cylinder and initial out-of-roundness in theform of a localized inward deformation at h ¼ 0.

The numerical results of Fig. 5 may offer a first indication of imperfection sensitivity in the presence of an initial gap between

the pressurized cylinder and the confining medium; a situationreferred to as loosely fitted cylinder. The results refer to elasticcylinders and indicate a significant effect of the gap on the maxi-mum pressure carried out by the elastic cylinder. In particular, agap size equal to only 6.7% of the cylinder thickness is responsi-ble for a 10.9% reduction of maximum pressure with respect tothe maximum pressure that an imperfection-free cylinder cansustain.

In the case of elastic cylinders, the initial out-of-roundnessimperfection is imposed assuming a stress-free initial configura-tion that follows the buckled shapes, as depicted in Fig. 6(a). Notethat the numerical results for g= R ¼ 5:4 103 (symbol n) are invery good agreement with the empirical Eq. (4) proposed byEl-Sawy and Moore [11]. Furthermore, Fig. 7 depicts thepressure-displacement curves of an elastic cylinder with D=t ratioequal to 200 and demonstrates the imperfection sensitivity of the

cylinder response. The values of initial out-of-roundness ampli-tude (d0= R) corresponds to the initial values of the pressure-displacement curves on the horizontal axis of the graph. Figure8(a) and Fig. 8(b) show the sensitivity of maximum pressure valueon the amplitude of both initial imperfections.

The above sensitivity on initial imperfections can be expressedin terms of the so-called “imperfection reduction factor” a, some-times referred to as “knock-down” factor, also adopted in Refs.[32,33] for the buckling load of imperfect elastic shells so that

pmax ¼ a pcr (11)

For the purposes of the present study, the reduction factor a isassumed in the following form:

a ¼ CD

m (12)

where D is an imperfection parameter that represents the size of the initial imperfection, considering both out-of-roundness andgap, and C; m; K are constant coefficients to be determined fromthe numerical results. The results in Fig. 8 indicate a dependency

of the imperfection sensitivity on the D=t value. Based on those

results, this imperfection parameter is considered in the followingform:

D ¼ d0 þ Kg R

ffiffiffiffiffiffiffiffiffiffiffi D

t

s (13)

In Eq. (13), coefficient K expresses the relative influence of thetwo forms of imperfections (gap and out-of-roundness) on the ulti-mate pressure p max. From the numerical results of Fig. 8, a valueequal to 3 is obtained for this coefficient ( K ¼ 3). Upon determin-ing the value of K , a standard curve fitting technique is employed;the values of C and m are calculated equal to 0.15 and 0.7, respec-tively, so that the elastic reduction factor becomes

a ¼0:15

D0:7 ¼

0:15

d0 þ 3g R

ffiffiffiffiffiffiffiffiffiffiffi D

t

s " #0:7 (14)

The results in Ref. [31] show that imperfection reduction factor predicted through Eq. (14) can provide lower-bound predictionsfor the ultimate pressure of externally pressurized elastic cylindersin the presence of initial imperfections.

Alternatively, the reduction of maximum pressure due to thepresence of initial out-of-roundness imperfections in externallypressurized elastic cylinders confined within a rigid cavity can bealso expressed with respect to the imperfection amplitude by thefollowing expression:

Fig. 7 Structural response of rigidly confined elastic cylindersin the presence of small initial out-of-roundness

Fig. 8 Buckling pressure of imperfect elastic cylinders overthe buckling pressure of the corresponding perfect elasticcylinders, confined within a rigid medium; (a ) effects of initialout-of-roundness and (b ) effects of initial gap




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pmax

pcr ¼ 1 C d0

t

n(15)

where pcr is the buckling pressure of the corresponding ‘‘perfect”cylinder (referred to as “critical pressure”), C is a positive con-stant that depends on the D=t ratio, and exponent n expresses therate of decay. This equation is considered in the same form as thesensitivity imperfection formula from general postbuckling theoryof elastic systems [35,36]. Figure 9 presents an attempt to fit theabove formula (15) with the finite element results. It is interesting

to note that for the range of small values of imperfection ampli-tude, the value of 2/3 on the exponent n results in good predictionsof the maximum pressure of imperfect cylinders.

4 Mechanical Behavior of Steel Cylinders

Using the above numerical models, the structural stability of externally pressurized confined steel cylinders is examined. Thevalues of pressure p are normalized by the yield pressure p y ¼ 2 1:13ð Þr yt = D, computed by Eq. (6), whereas the displace-ment d of point A at h ¼ 0 is normalized by the cylinder radius R.

4.1 Buckling of Steel Cylinders Surrounded by a RigidMedium. The response of a thin-walled metal cylinder with D=t

¼200 is shown in Fig. 10 for initial out-of-roundness imper-

fection and assuming a frictionless interface between the cylinder and the confinement medium. The values of initial out-of-round-ness amplitude (d0= R) correspond to the starting values of thepressure-displacement curves on the horizontal axis of the graph.The material of the cylinder is steel, with yield stress r y and ulti-mate stress ru equal to 313 MPa and 492 MPa, respectively,whereas postyield hardening is zero up to nominal strain equal to1.5%. A zero gap between the cylinder and the medium and a con-finement medium modulus E0 equal to 10% of E are assumed( E0 ¼ 21,000MPa). The value of E0 corresponds to practicallyrigid confinement (e.g., concrete encasement). Numerical resultswith higher values of E0 indicated no further influence on theresponse. The equilibrium curves in Fig. 10 represent the nonlin-ear relationship between the applied pressure and the downwarddisplacement of the cylinder point at h

¼0. The results demon-

strate that the value of the ultimate pressure pmax is substantiallysmaller than the yield pressure p y, even for negligible initialimperfection.

Comparison between the numerical results from the elastic case(Fig. 6) and those shown in Fig. 11 for steel cylinders shows that,

Fig. 9 Effects of small initial out-of-roundness imperfectionamplitudes on the buckling pressure of imperfect elastic cylin-ders; FEM results and predictions from the imperfection sensi-tivity formula of Eq. (15)

Fig. 10 Response of tightly fitted steel cylinders (g /R50),embedded in a rigid confinement medium for different values ofinitial out-of-roundness

Fig. 11 Postbuckling shapes of initially “perfect” steel cylinders with elastic-plastic material; configuration (2) corresponds tothe ultimate pressure stage




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in steel cylinders, the ultimate pressure capacity pmax of steel cyl-inders occurs at a lower level of pressure because of the signifi-cant effect of inelastic material behavior on the pressure capacityof the cylinder. Furthermore, the finite element results indicatethat the maximum pressure of elastic-plastic (steel) cylindersoccurs very soon after first yielding, and corresponds to a defor-mation stage before “flattening” of the cylinder wall occurs at theh ¼ 0 location. Figure 11(a) depicts the successive deformed con-figurations of the steel cylinder. A comparison between deformedshapes from elastic and inelastic cylinder behavior in Fig. 6(a)

and Fig. 11(a) shows that the postbuckling shape of inelastic cyl-inders is characterized by more abrupt changes of local curvatureat the symmetry point A and the touchdown points B and B0(Fig. 1), and this is attributed to the concentration of plastic defor-mation at those points and the formation of plastic hinges. In addi-tion, the buckled portion of the cylinder extends to a shorter circumferential area than in the case of elastic cylinders.

Upon reaching the maximum pressure pmax, the cylinder behav-ior becomes unstable, with significant drop of pressure capacity,and this is responsible for a severe sensitivity of the pmax value onthe presence of initial imperfections. The numerical results inFig. 10 for the cylinder under consideration show that initial out-of-roundness of amplitude less than 1% of the cylinder radiusresult in a 60% reduction of the ultimate pressure pmaxwith respectto the maximum pressure of the imperfection-free cylinder. Theunstable postbuckling behavior is due to the development of a

plastic collapse mechanism with one stationary plastic hinge atsymmetry point A and two moving hinges at the two touchdownpoints B and B’. Using a simple kinematic model [30], the follow-ing closed-form expression describing this mechanism has beenobtained:

p

p y¼ t

D

46 d

R

d

R

2 (16)

which verifies the unstable behavior beyond the maximum pres-sure, leading to plastic collapse of the pressurized cylinder. Amore detailed discussion on the plastic collapse mechanism isoffered in Ref. [28].

The presence of a small gap between the cylinder and the con-finement medium may also have significant effect on the maxi-mum pressure, as shown in Fig. 12. The cylinder has a D=t ratioequal to 200 and a steel material with yield stress r y equal to 313MPa. The gap size, denoted as g, is the maximum distancebetween the cylinder and the cavity inner surface at h ¼ 0, and itis normalized by the cylinder radius R. The numerical results inFig. 12, compared with the corresponding results of Fig. 10, indi-cate that in the presence of a rather small gap size equal to 0.27%of the cylinder radius R (or equivalently equal to 27% of the cylin-der thickness) and for zero initial out-of-roundness (d0= R ¼ 0),the ultimate pressure capacity is reduced by 40%. The maximumpressure is further decreased in the presence of initial out-of-roundness imperfections (d0= R). The effects of initial gap (g= R)and out-of-roundness (d0= R) imperfections on the maximum pres-

sure ( pmax= p y) are summarized in Fig. 13 for a cylinder with D=t ¼200 and r y ¼313 MPa. The finite element results indicatethat for values of initial out-of-roundness d0 greater than 3.5% of the cylinder radius R, the value of maximum pressure pmax is inde-pendent of the value of the initial gap size g.

One should note that the overall behavior of the pressurizedsteel cylinder is similar to the one described in the Sec. 3 for elas-tic cylinders. The cylinder is initially contracted due to externalpressure and “buckles” in an oval form at p= p y ¼0.016, whichcorresponds to the critical buckling pressure pe for elastic uncon-fined cylinders, calculated from Eq. (2). Nevertheless, the cylinder accommodates itself within the rigid cavity, allowing for signifi-cant further increase of external pressure, and behaving similar toan arch under external pressure that leads to “inversion buckling”

at a pressure equal to pmax and unstable response beyond the limitpoint. In the case of steel cylinders, the maximum pressure occursat a stage somewhat prior to the flattened configuration because of the presence of plastic deformation, as shown in Fig. 11(a).

The value of maximum external pressure that the cylinder cansustain depends on the value of the D=t ratio. Figure 14 shows thevariation of ultimate pressure pmax with respect to initial out-of-roundness for three values of D=t ratio ( D=t ¼ 100, 150, 200) andfor zero gap between the cylinder and the medium (g= R ¼ 0). Thenumerical results of Fig. 14 indicate similar imperfection sensitiv-ity for all three cases. It is also noted that the ultimate pressure pmax for the thicker cylinder ( D=t

¼100) is higher than the ulti-

mate pressure of the thin-walled cylinder ( D=t ¼ 200). On theother hand, the ultimate pressure pmax for the thin-walled cylinder is very low, significantly lower than the plastic pressure of the cyl-inder p y, even in the absence of initial imperfections.

For the purposes of describing buckling of confined cylinders inthe inelastic range in a simple and efficient manner, the so-called“shell slenderness” parameter is adopted, defined as follows:

k ¼ ffiffiffiffiffiffiffi

R pl

Rcr

r (17)

where R pl represents the load that causes full-plastic failure and Rcr the load corresponding to the elastic buckling condition of the

Fig. 12 Effects of initial out-of-roundness and initial gap onthe external pressure response of a confined steel cylinder em-bedded in a rigid confinement medium (E 0 / E 51021, D / t 5200)

Fig. 13 Effects of initial out-of-roundness and initial gap (g / R )on the maximum pressure sustained by a confined steelcylinder embedded in a rigid confinement medium (E 0 / E 51021,D / t 5200)




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perfect structure. The above definition is general and refers to anytype of loading. In the present case, the fully plastic pressure p y of Eq. (6) can be used for R pl, whereas Glock’s critical pressure pGLin Eq. (1) offers a very good analytical expression for the critical

pressure Rcr so that the slenderness parameter in Eq. (17) can bewritten as follows:

k ¼ ffiffiffiffiffiffiffi

p y

pGL

r ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2:26r yð1 2Þ

E

D

t

1:2s (18)

In the case of perfect elastic cylinders (g ¼ d0 ¼ 0), the numer-ical results in Fig. 4 show that the ultimate pressure pmax is equalto the one predicted by Eq. (1), i.e., equal to pGL. Therefore, com-bining Eqs. (1) and (18), one can write

pmax

p y¼ 1

k2 (19)

Neglecting strain hardening effects, the value of p y cannot beexceeded, i.e., pmax p y. Therefore, it would be tempting to arguethat for steel cylinders with slenderness values k less than unity,the maximum pressure pmax would be equal to p y, whereas Eq.(19) would express the maximum pressure of steel cylinders withslenderness values k greater than unity.

The validity of the above argument is examined using the pres-ent numerical tools, and the corresponding results are depicted inFig. 15, showing the variation of ultimate pressure p max in termsof k. The numerical results in Fig. 15 do not support the aboveargument; imperfection-free cylinders (g ¼ d0 ¼ 0) within a stiff confinement medium for three values of yield stress r y (235 MPa,313 MPa and 566 MPa, respectively) show that for k values lessthan 2.2, the buckling pressure pmax deviate significantly from Eq.(19). On the other hand, for k values greater than 2.2, the buckling

pressure pmaxcan be expressed quite accurately by Eq. (19), whichimplies that buckling occurs in the elastic range. In other words,the value of 2.2, denoted by k p, defines the transition betweenelastic and inelastic buckling regime.

For slenderness values k less than k p, i.e., for buckling in theinelastic range, the numerical results fall well below Eq. (19). Inparticular, the value of pmax approaches the plastic pressure p y for rather small values of k, equal to about 0.25. This is a characteris-tic slenderness value denoted as k0, and referred to as “squashslenderness.”

Based on the above results, it is possible to develop a bucklingcurve that expresses the maximum pressure pmax in terms of theslenderness value k of the cylinder. This curve should consider the general case of imperfect cylinders so that it is used for design

purposes. More specifically, taking into account the definition of the imperfection reduction factor a in Eq. (11), one can write thefollowing equation:

pmax

p y¼ a

k2 (20)

which is valid for the elastic buckling range, i.e., for k k p ¼ 2:2. The value of the imperfection reduction factor acan be computed from Eq. (14). For values of k less than theplastic limit slenderness (k p ¼ 2.2), buckling is associated withmaterial behavior in the inelastic range. In the absence of aclosed-form analytical expression for the buckling pressure in theinelastic regime, similar to the elastic buckling Eq. (1), the follow-ing expression, introduced in Refs. [32,33], is adopted:

pmax

p y¼ 1 b k k0

k pk0

g

(21)

where b is constant, g depends on imperfection parameter D, andslenderness k0 is the squash slenderness. Equation (21) is valid for intermediate values of cylinder slenderness k0 k k p. Finally,for slenderness values less than k0, the cylinder collapses due tothe development of excessive plastic deformation so that, neglect-ing strain hardening, one can write

pmax

p y¼ 1 (22)

It is important to note that the value of b in Eq. (21) is determinedequating expressions (20) and (21) for k ¼ k p, and one readilyobtains

b ¼ 1 ak2 p

(23)

Furthermore, the numerical results indicate a dependence of g onthe initial imperfection, which can be expressed as follows:

g ¼ 0:6 3D; g 0:3 (24)

Figure 16 shows the predictions of the above design methodologyfor imperfect steel cylinders against the numerical finite elementresults. The comparisons indicate that the proposed methodologyoffers an efficient approach for predicting the ultimate pressure of confined cylinders in both the elastic and the inelastic range. The

Fig. 15 Variation of maximum pressure p max steel cylinderswith no imperfections (g / R 50,d0 / R 50), embedded in a rigidconfinement medium with respect to the slenderness parameterk defined in Eq. (18)

Fig. 14 Effects of initial out-of-roundness and D / t ratio on themaximum pressure sustained by a confined steel cylinder em-bedded in a rigid confinement medium (E 0 / E 51021, g / R 50)




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methodology is fully compatible with the general methodologyfor shell buckling design [32,33] and could be used for designpurposes.

The predictions of Eq. (5), proposed by Montel [17], are com-pared with the present finite element results in Fig. 17. The com-parison shows that, despite its simplicity, the empirical formula

(5) can provide reliable, yet somewhat conservative, estimates of the maximum pressure sustained by a cylinder encased in a stiff

boundary, within a good level of accuracy, even beyond theapplicability ranges specified in the publication of Montel [17].Therefore, the formula can be used for the design of buried pipe-lines encased in concrete or other cylinders confined within a stiff medium.

A series of tests on small- and large-scale specimens of steelpipes confined in concrete have been conducted by Borot [19],also reported by Montel [17] and Taras and Greiner [22]. Themain geometric and material properties for each test specimen arereported in Table 2. For Test No. 4 with steel yield strength of 746 MPa, the value of r y in Eq. (5) was assumed equal to500 MPa, as proposed by Montel, because the real value of yieldstress exceeds the applicability range of the equation. The valuesof critical pressure are summarized and compared in Table 3. Ingeneral, there exists a good comparison between experimental

data, numerical results, and analytical predictions.

4.2 Steel Cylinders Surrounded by DeformableMedium. The results presented in Sec. 4.1 refer exclusively tothe case of steel cylinders enclosed within a nondeformable(rigid) cavity, considering an elastic confinement medium withhigh values of modulus E 0. However, quite often in buried pipe-line applications, the steel cylinder is embedded in a soft me-dium, which should be modeled as a deformable cavity. Previousnumerical results in Refs. [30] and [31] have indicated that themodulus E0 of the surrounding elastic medium has a significanteffect on the value of pressure capacity. In the present section,using the finite element tools, the influence of embedment flexi-bility on the mechanical response of externally pressurizedcylinders is examined. Motivated by the buckling problem of

externally pressurized buried pipelines, the top boundary of thefinite element model is free, whereas the nodes on the three other boundaries are fixed.

Figure 18 shows the response of a steel cylinder (r y ¼313 MPa, D=t ¼200) in terms of the pressure-deformation curves for differ-ent values of the confining medium modulus E0, with no imperfec-tions (d0 ¼ g ¼ 0). In this analysis, the gravity load of thesurrounding medium is not considered. The main observationfrom those results is the significant reduction of the pmax valuewith decreasing values of E0. Furthermore, with decreasing valuesof E0, the response becomes smoother, and it characterized by a“plateau” on the equilibrium path about the maximum pressure.

The effects of the E 0 value are shown in Figs. 19(a) and 19(b),which depict the variation of the ultimate pressure capacity with

Fig. 16 Variation of maximum pressure p max steel cylindersembedded in a rigid confinement medium with respect to theslenderness parameter k defined in Eq. (18); finite elementresults and predictions of Eqs. (20), (21), and (22)

Fig. 17 Comparison between numerical results and analyticalpredictions from Montel’s simplified Eq. (5) [17]

Table 2 Geometric and material characteristics of testspecimens

Test #1 Test #2 Test #3 Test #4

D / t ratio 66.7 66.7 133.3 142.8Out-of-roundness (d0= R) 0.009 0.009 0.005 0.007Initial gap (g= R) 0.0002 0.0002 0.00067 0.00067Yield strength (r y) (MPa) 304 304 451 746

Table 3 Comparison of critical pressure between experimen-tal, numerical, and analytical results

Maximum pressure

Test #1 Test #2 Test #3 Test #4

Experimental 6.1 MPa 6.7 MPa 2.9 MPa 2.6 MPaNumerical (FEM) 6.14MPa 6.14MPa 2.84MPa 2.88MPaDesign methodology 5.47MPa 5.47MPa 2.70MPa 2.57MPaMontel Eq. (5) 5.74 M Pa 5.74 M Pa 2.75 M Pa 2.42 M Pa




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respect to the relative stiffness of the medium E 0= E for imperfectsteel cylinders (r y ¼313MPa, D=t ¼200). The results indicatethat the response is imperfection sensitive, which becomes lesspronounced with increased flexibility of the confining medium.Comparison between the results from Fig. 19(a) and Fig. 19(b)indicate that the presence of gap affects the value of maximumpressure but this effect becomes less important increasing the flex-ibility of the surrounding medium.

Figure 20 shows the drop of the pmax value with decreasing val-ues of E0 for an elastic cylinder and a steel cylinder with D=t ¼200. It is interesting to note that for E0= E 3 104, thetwo curves coincide, indicating that buckling of steel cylinderswith D=t ¼200 in a highly deformable medium occurs in the elas-tic range. In other words, the effects of confinement are signifi-cantly reduced. The effect of E0= E in the present designmethodology can be taken into account introducing an appropriatereduction factor f [31] expressing the ratio of the maximum pres-sure pmax in a deformable medium over pmax;1, which is the maxi-mum pressure of the cylinder in a rigid confinement ( E0 ! 1):

f ¼ pmax pmax;1

(25)

Numerical results in Ref. [31] have shown that this factor is inde-pendent on the value of the D=t ratio. Based on the numericalresults, the following function is defined:

f ð xÞ ¼ 0:05 x2 þ 0:1 x þ 0:95 if 1 x 51 if x 1

(26)

where x is the modulus ratio parameter,

x ¼ log E0

E

(27)

which can be used for an efficient description of the effects of me-dium deformability on the ultimate pressure.

The previous results have been obtained without consideringgravity of the surrounding medium. In the following, the effectsof gravity on the structural response of the cylinder are examined.Motivated by the case of buried pipelines, soil conditions are con-sidered for the surrounding medium. Figures 21(a) and 21(b)show the pressure-displacements response of a steel cylinder with D=t ¼ 200, embedded in an elastic medium of density equal to 20kN/m3 and Young’s modulus ranging from 25 to 100 MPa, whichis typical for clays [37]. In this analysis, gravity of the cylinder/

soil system is considered as a first step, and subsequently, externalpressure has been applied on the outer surface of the cylinder. Theresults in Fig. 21(a) refer to ¼ 0:3 and indicate a high value of pmax at small values of E

0. This observation is more pronounced inFig. 21(b), which refers to ¼ 0:49, a nearly incompressible me-dium. This can be explained if the detachment w of the cylinder from the surrounding medium replaces the displacement d of thecylinder in the horizontal axis. The corresponding graphs areshown in Figs. 22(a) and 22(b) for low values of E 0. Due to grav-ity loading, the cylinder remains in contact with the medium up toa significant level of pressure and buckling is prevented. Immedi-ately after detachment, buckling occurs quite abruptly, associatedwith a rapid drop of pressure. This sudden collapse is more pro-nounced in the case of a nearly incompressible medium

Fig. 18 Structural response of perfect (g / R 50, d0 / R 50) steelcylinders (D / t 5200) for different values of confinement me-dium modulus (E 0 / E ); pressure versus deformation equilibriumpaths

Fig. 19 Effects of initial out-of-roundness and stiffness of con-finement medium (E 0 / E ) on the maximum pressure sustainedby a confined steel cylinder (g / R 50, D / t 5200)

Fig. 20 Comparison between elastic and steel cylinders(D / t 5200) with respect to the E 0 / E value for perfect cylinders(d0 / R 5g / R 50)




12/15

¼ 0:49ð Þ. In the case of a high value of E0, detachment occurs atrelatively low pressure levels and the behavior has similaritieswith the one described in Sec. 3 for rigid boundary.

Finally, the behavior of a steel cylinder with D=t ¼ 200, em-bedded in a deformable elastic-plastic medium is examined. Themedium is considered a soft-to-firm clay, which is modeledthrough a Mohr–Coulomb inelastic material model, with cohesionc ¼ 50kPa, friction angle u ¼ 0 deg, Young’s modulus E0 ¼ 25 MPa, and Poisson’s ratio ¼ 0.49. The effects of gravityhave been taken into account, as an initial loading step. Figure23(a) shows the structural response of the steel cylinder in termsof the pressure-displacement curve for the elastic-plastic medium,compared with the corresponding curve assuming elastic medium.The two curves practically coincide. They slightly deviate only af-ter significant deformation, well into the postbuckling range. Thecoincidence of the two curves is more pronounced when stiffer soil properties are employed. Figure 23(b) shows the distributionof plastic deformation in the deformable medium at a displace-ment value of 30 mm d= R ¼ 0:04ð Þ, well beyond the bucklingstage. Plastic deformation occurs at the touchdown points B andB0 (see Fig. 1), where the pressurized cylindrical arch is sup-ported. It is interesting to note that at the stage where bucklingoccurs, no plastic deformation is detected within the medium.

5 A Note on Shrink Buckling of Cylinders

The single-lobe buckling mode of a thin-walled cylinder or ringin a cavity can be obtained with two types of loading. The first

type of loading is external pressure on the cylinder outer surface,sometimes referred to as “hydrostatic pressure” problem. Alterna-tively, this buckling mode may be obtained under thermal loading,where the cavity prevents the extension of the encased cylinder.Similar to thermal loading, shrink buckling may also occur whenthe outer cavity contracts (shrinks), moving inwards and applyingexternal pressure to the encased cylinder, forcing it to buckle. Inseveral practical applications, sleeving the inside of cylinders canlead to shrink buckling as well. In all those cases, the resultantbuckling is often referred to as shrink buckling, as opposed to“hydrostatic buckling” presented in Secs. 3 and 4.

Early works on this subject have been conducted by Refs. [5],[6], and [7], pinpointing the importance of initial imperfections onthe maximum compression. Notable analytical contributions on

the problem of shrink buckling have been reported in Refs.[38 – 41], whereas Sun et al. [42] presented a thorough experimen-tal investigation of the problem, using a simple setup of com-pressed hemicircular very thin-walled rings ( D=t 400) within arigid cavity, focusing on the effects of initial imperfections.

The shrink buckling problem has several similarities but it isnot the same as the hydrostatic buckling problem. The main dif-ference is that in shrink buckling under thermal loading or sleev-ing, the buckled part of the cylinder is laterally free, whereas inhydrostatic buckling, pressure load is always present, applied onthe buckled portion of the cylinder in the postbuckling stage. Tounderstand the consequences of this difference on the mechanicalbehavior and strength of the loaded cylinder, a series of numericalsimulations are conducted in the course of the present study,

Fig. 21 Structural response of perfect steel cylinders for differ-ent values of confinement medium modulus; pressure versusdisplacement (d) equilibrium paths for (a ) m 50.3 and (b ) m 50.49

Fig. 22 Structural response of perfect steel cylinders for differ-ent values of confinement medium modulus; pressure versusdetachment (w ) equilibrium paths for (a ) m 50.3 and (b ) m 50.49




13/15

simulating the experimental setup of Sun et al. [42], shown in Fig.24(a). It consists of a thin-walled cylindrical specimen encasedwithin a rigid hemicircular cavity, compressed symmetrically atpoints B and B0. A small imperfection is assumed in the centralpoint A in the form of an inward localized displacement.

The numerical model employed for simulating this experimentis shown in Fig. 24(b) and it is very similar to the one described inSec. 2; shell elements and solid elements are used to model thesteel cylinder and the rigid cavity, respectively. The material of the cavity is considered isotropic elastic with Young’s modulus E0 ¼ 21,000MPa and Poisson’s ratio ¼ 0.30, corresponding torigid confinement conditions. Numerical results are obtained for the case of an elastic and a steel cylinder. Both cylinders have a

D=t ratio equal to 200, and the steel material of the second cylin-der has a yield stress equal to 313 MPa, similar to the materialused extensively in Sec. 4. The numerical results are presented inFig. 25, showing the maximum (buckling) acting stress at pointsB and B’ in terms of the size of initial imperfection. Both curvesare decreasing functions of the imperfection amplitude. The actingstress r is equal to the force F applied at points B and B0 dividedby the cross-sectional area of the ring model. For value of initialimperfection approaching zero, the curve for the elastic cylinder goes asymptotically to infinity. This implies the absence of buck-ling for a geometrically perfect system, a conclusion also reportedin early analytical works [40], as well as in the tests of Sun et al.[42]. The steel cylinder has a similar behavior, but the bucklingstress is significantly reduced with respect to the corresponding

stress of the elastic cylinder. In addition, for small values of imperfection amplitude (d0=t 0:4), the steel ring fails due toyielding at a stress level equal to the yield stress of the material r yunder plane strain conditions.

The absence of buckling in an imperfection-free cylinder con-stitutes the main difference between the shrink buckling and thehydrostatic buckling problem. As noted, this is attributed to thefact that hydrostatic pressure is always applied on the postbuckledportion of the cylinder, whereas external loading is released fromthe buckled portion in the case of shrink buckling.

6 Conclusions

The behavior of thin-walled cylinders, surrounded by an elasticmedium, is examined in terms of their structural stability under uniform external pressure using a nonlinear two-dimensional fi-nite element model. Numerical results for the ultimate pressure of cylinders of elastic material are found to be in very close

Fig. 23 (a ) Effect of elastic-plastic medium on the structuralresponse of confined steel cylinder, (b ) distribution of equiva-

lent plastic strain on the medium

Fig. 24 (a ) Schematic representation of test setup [42]; (b )finite element model of cylinder-medium system

Fig. 25 Maximum acting stress for rigidly confined elastic andsteel cylinders




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agreement with available closed-form analytical predictions.Confined thin-walled steel cylinders (100 D=t 300) are alsoanalyzed under external pressure.

The numerical results show a significant sensitivity of the ulti-mate pressure in terms of initial imperfections, in the form of bothout-of-roundness of the cylinder cross section and the presence of initial gap between the cylinder and the surrounding medium. It isalso demonstrated that reduction of the medium modulus resultsin a substantial reduction of the pressure capacity of the cylinder.The pressure-deflection equilibrium paths indicate a rapid drop of

pressure capacity after reaching the maximum pressure level, andthe postbuckling configuration is characterized by a three-hingeplastic collapse mechanism. A slenderness shell parameter is alsointroduced, which enables the presentation of maximum pressurein the inelastic range in a simple and efficient manner. The simpli-fied formula proposed in Ref. [17] is found to be quite close to thepresent numerical results and could be used for the prediction of buckling pressure of buried pipelines and other rigidly encasedsteel cylinders. Furthermore, a good comparison with limitedavailable test data is also shown.

The corresponding problem of shrink buckling has also beenexamined, through a rigorous simulation of experiments reportedelsewhere, and the main differences with the hydrostatic bucklingproblem have been identified.

Finally, it should be noticed that according to several designrecommendations, the external pressure capacity of steel pipelines

encased in a rigid cavity (e.g., concrete encasement) was takenequal to the nominal pressure that causes yielding of the cylinder,assuming that the rigid confinement prevents instability in theelastic range [1]. The results of the present study demonstratedthat this argument may be valid only for the case of shrink buck-ling. On the other hand, consideration of an ultimate pressureequal to p y, calculated by Eq. (6), leads to unsafe design of hydro-statically loaded cylinders. For the range of D=t ratio of interest,hydrostatically loaded steel cylinders encased in a rigid cavity areable to sustain only a portion of the yield pressure p y, even in theabsence of initial imperfections.

Appendix Brief Presentation of Glock’s

Analytical Solution [9]

Glock [9] developed a solution for external pressure buckling of elastic rings confined within a rigid cavity in the absence of initialimperfections. Kinematics was based on Donnell approximations of thin-ring equations [10], where the total hoop axial strain is given bythe following equation, as a sum of membrane and bending strain:

eh ¼ em þ kz (A1)The membrane and bending strain are given in terms of the radialand tangential displacements v and w of the ring reference line atmidthickness as follows:

em ¼ 1 R

v0 wð Þ þ 1 R2

w02 (A2)

k ¼ w00

R2 (A3)

Ring deformation consists of two parts, the “buckled” and the“unbuckled” portion (Fig. 1). An assumed shape function wðhÞ for the buckled region is considered in the following form:

wðhÞ ¼ d cos2 ph2/

(A4)

where / is the angle that defines the border between the buckledand the unbuckled ring portions so that / h /. Minimiza-tion of the total potential energy P with respect to both d and/ results in closed-form expressions for the pressure p, the ampli-tude of buckled shape d, and the axial force N in terms of angle /:

d

R¼ 6 pR

3

EI

/

p

4þ10 /

p

2(A5)

pR3

EI ¼ p

/

216

1

6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi16 80

3

EI

EAR2

p

/

5s 2435 (A6)

N ¼ 53

EI

R2p

/

2(A7)

Minimization of pressure in terms of angle /, results in finalclosed-form expressions for the critical pressure pGL, the corre-sponding angle /cr , and the corresponding amplitude of the buck-ling shape:

pGL R3

EI ¼ 0:969 EAR

2

EI

2=5(A8)

p

/

cr

¼ 0:856 EAR2

EI

1=5(A9)

d

R

cr

¼ 2:819 EI EAR2

2=5(A10)

For plane-strain conditions, Eq. (A8) can be written in the form of

Eq. (1).

References[1] Watkins, R. K., 2004, “Buried Pipe Encased in Concrete,” International Confer-

ence on Pipeline Engineering and Construction, San Diego, CA, ASCE.

[2] Omara, A. M., Guice, L. K., Straughan, W. T., and Akl, F. A., 1997, “BucklingModels of Thin Circular Pipes Encased in Rigid Cavity,” J. Eng. Mech.,123(12), pp. 1294–1301.

[3] Ullman, F., 1964, “External Water Pressure Designs for Steel-Lined PressureShafts,” Water Pow., 16, pp. 298–305.

[4] Ahrens, T., 1970, “An In-Depth Analysis of Well Casings and Grouting: BasicConsiderations of Well Design—Part II,” Water Well J., pp. 49–51.

[5] Chicurel, R., 1968, “Shrink Buckling of Thin Circular Rings,” ASME J. Appl.Mech., 35(3), pp. 608–610.

[6] Burgess, I., 1971, “The Buckling of a Radially Constrained Imperfect Circular Ring,” Int. J. Mech. Sci., 13(9), pp. 741–753.

[7] Bottega, W. J., 1989, “On the Behavior of an Elastic Ring Within a Contracting

Cavity,” Int. J. Mech. Sci., 31(5), pp. 349–357.[8] Feodosyev, V. I., 1967, Selected Problems and Questions in Strength of Materi-

als, Nauka Publishing House, Moscow.[9] Glock, D., 1977, “Überkritisches Verhalten eines Starr Ummantelten Kreis-

rohres bei Wasserdrunck von Aussen und Temperaturdehnung” (“Post-CriticalBehavior of a Rigidly Encased Circular Pipe Subject to External Water Pressure

and Thermal Extension”), Der Stahlbau, 7, pp. 212–217.[10] Brush, D. O., and Almroth, B. O., 1975, Buckling of Bars, Plates, and Shells,

McGraw-Hill, New York.[11] El-Sawy, K., and Moore, I. D., 1998, “Stability of Loosely Fitted Liners Used

to Rehabilitate Rigid Pipes,” J. Struct. Eng., 124(11), pp. 1350–1357.[12] Aggarwal, S. C., and Cooper, M. J., 1984, “External Pressure Testing of Insitu-

form Lining,” Coventry (Lanchester) Polytechnic, Coventry, UK, InternalReport.

[13] Boot, J. C., 1997, “Elastic Buckling of Cylindrical Pipe Linings With SmallImperfections Subject to External Pressure,” Tunnel. Undergr. Sp. Tech.,12(Suppl 1), pp. 3–15.

[14] Bottega, W. J., 1993, “On the Separation of Concentric Elastic Rings,” Int. J.Mech. Sci., 35(10), pp. 851–866.

[15] Li, F. S., and Kyriakides, S., 1991, “On the Response and Stability of Two Con-centric, Contracting Rings Under External Pressure,” Int. J. Solid. Struct.,27(1), pp. 1–14.

[16] Li, F. S., and Kyriakides, S., 1990, “On the Propagation Pressure of Buckles in Cylindrical Confined Shells,” ASME J. Appl. Mech., 57(4), pp.1091–1094.

[17] Montel, R., 1960, “Formule Semi-Empirique pour la Détermination de la Press-ion Extérieure Limite d’Instabilité des Conduits Métalliques Lisses Noyéesdans du B éton ,” La Houille Blanche, 15(5), pp. 560–568.

[18] Timoshenko, S., and Gere, J. M., 1961, Theory of Elastic Stability, 2nd ed.,McGraw-Hill, New York.

[19] Borot, H., 1957, “Flambage d’un Cylindre à Paroi Mince, Placé dans uneEnveloppe Rigide et Soumis à une Pression Extérieure,” La Houille Blanche,12(6), pp. 881–887.

[20] Amstutz, E., 1969, “Das Einbeulen von Schacht – und Stollenpanzerungen,”Schweizerische Bauzeitung, 87, pp. 541–549 (U.S. Dept. of the Interior, Trans-lation No. 825).



http://dx.doi.org/10.1061/40745(146)87http://dx.doi.org/10.1061/(ASCE)0733-9399(1997)123:12(1294)http://dx.doi.org/10.1115/1.3601259http://dx.doi.org/10.1115/1.3601259http://dx.doi.org/10.1016/0020-7403(71)90043-9http://dx.doi.org/10.1016/0020-7403(89)90059-3http://dx.doi.org/10.1061/(ASCE)0733-9445(1998)124:11(1350)http://dx.doi.org/10.1016/S0886-7798(98)00018-2http://dx.doi.org/10.1016/0020-7403(93)90044-Uhttp://dx.doi.org/10.1016/0020-7403(93)90044-Uhttp://dx.doi.org/10.1016/0020-7683(91)90141-2http://dx.doi.org/10.1115/1.2897636http://dx.doi.org/10.1051/lhb/1960048http://dx.doi.org/10.1051/lhb/1957061http://dx.doi.org/10.1051/lhb/1957061http://dx.doi.org/10.1051/lhb/1960048http://dx.doi.org/10.1115/1.2897636http://dx.doi.org/10.1016/0020-7683(91)90141-2http://dx.doi.org/10.1016/0020-7403(93)90044-Uhttp://dx.doi.org/10.1016/0020-7403(93)90044-Uhttp://dx.doi.org/10.1016/S0886-7798(98)00018-2http://dx.doi.org/10.1061/(ASCE)0733-9445(1998)124:11(1350)http://dx.doi.org/10.1016/0020-7403(89)90059-3http://dx.doi.org/10.1016/0020-7403(71)90043-9http://dx.doi.org/10.1115/1.3601259http://dx.doi.org/10.1115/1.3601259http://dx.doi.org/10.1061/(ASCE)0733-9399(1997)123:12(1294)http://dx.doi.org/10.1061/40745(146)87


15/15

[21] Jacobsen, S., 1974, “Buckling of Circular Rings and Cylindrical TubesRestrained Against Radial Displacement Under External Pressure,” Water Pow., 26, pp. 400–407.

[22] Taras, A., and Greiner, R., 2007, “Zum Gültigkeitsbereich der Bemessungsfor-meln für Druckschachtpanzerungen Unter Außendruck” (“Scope of the DesignAssumption for Pressure Tunnel Steel Linings Under External Pressure”),Stahlbau, 76(10), pp. 730–738.

[23] Yamamoto, Y., and Matsubara, N., 1982, “Buckling of a Cylindrical ShellUnder External Pressure Restrained by an Outer Rigid Wall,” Proceedings,Symposium on Collapse and Buckling, Structures; Theory and Practice, Cam-bridge University Press, London, pp. 493–504.

[24] Kyriakides, S., and Youn, S. K., 1984, “On the Collapse of Circular ConfinedRings Under External Pressure,” Int. J. Solid. Struct., 20(7), pp. 699–713.

[25] Kyriakides, S., 1986, “Propagating Buckles in Long Confined CylindricalShells,” Int. J. Solid. Struct., 22(12), pp. 1579–1597.

[26] El-Sawy, K., 2001, “Inelastic Stability of Tightly Fitted Cylindrical Liners Sub- jected to External Uniform Pressure,” Thin Wall. Struct., 39(9), pp. 731–744.

[27] El-Sawy, K., 2002, “Inelastic Stability of Loosely Fitted Cylindrical Liners,” J.Struct. Eng., 128(7), pp. 934–941.

[28] El-Sawy, K., 2013, “Inelastic Stability of Liners of Cylindrical Conduits With LocalImperfection Under External Pressure,” Tunnel. Undergr. Sp. Tech., 33, pp. 98–110.

[29] Estrada, C. F., Godoy, L. A., and Flores, F. G., 2012, “Buckling of VerticalSandwich Cylinders Embedded in Soil,” Thin Wall. Struct., 61, pp. 188–195.

[30] Vasilikis, D., and Karamanos, S. A., 2009, “Stability of Confined Thin-WalledSteel Cylinders Under External Pressure,” Int. J. Mech. Sci., 51(1), pp. 21–32.

[31] Vasilikis, D., and Karamanos, S. A., 2011, “Buckling Design of Confined Steel Cyl-inders Under External Pressure,” ASME J. Press. Vess. Tech., 133(1), p. 011205.

[32] Comité Europ éen de Nor malizati on, 2007, Strength and Stability of Shell Struc-tures, EN 1993-1-6 , Eurocode 3, Part 1–6, Brussels, Belgium.

[33] European Convention for Constructional Steelwork, 2008, Buckling of SteelShells, European Design Recommendations, 5th ed., J. M. Rotter, andH. Schmidt, eds., Brussels, Belgium, ECCS Publication No. 125.

[34] Hibbit, H. D., Karlsson, B. I., and Sorensen, P., 2007, Theory Manual, Version6.7, ABAQUS, Providence, RI.

[35] Koiter, W. T., 1963, Elastic Stability and Post-Buckling Behaviour , Proceedingsof the Symposium in Nonlinear Problems, University of Wisconsin Press, Madi-son, WI, pp. 257–275.

[36] Budiansky, B., 1965, “Dynamic Buckling of Elastic Structures: Criteria andEstimates,” Proceedings of the International Conference in Dynamic Stabilityof Structures, pp. 83–106.

[37] Jeyapalan, J. K., and Watkins, R. K., 2004, “Modulus of Soil Reaction ( E’) Val-ues for Pipeline Design,” J. Transp. Eng., 130(1), pp. 43–48.

[38] Hsu, P. T., Elkon, J., and Pian, T. H. H., 1964, “Note on the Instability of Circular Rings Confined to a Rigid Boundary,” ASME J. Appl. Mech., 31(3), pp. 559–562.

[39] Bucciarelli, L. L., Jr., and Pian, T. H. H., 1967, “Effect of Initial Imperfectionson the Instability of a Ring Confined in an Imperfect Rigid Boundary,” ASMEJ. Appl. Mech., 34(4), pp. 979–984.

[40] El-Bayoumy, L., 1972, “Buckling of a Circular Elastic Ring Confined to a Uni-formly Contracting Circular Boundary,” ASME J. Appl. Mech., 39(3), pp.758–766.

[41] Soong, T. C., and Choi, I., 1985, “Bucking of an Elastic Elliptical Ring Inside aRigid Boundary,” ASME J. Appl. Mech., 52(3), pp. 523–528.

[42] Sun, C., Shaw, W. J. D., and Vinogradov, A. M., 1995, “Instability of ConfinedRings: An Experimental Approach,” Exper. Mech., 35(2), pp. 97–103.

http://dx.doi.org/10.1002/stab.200710078http://dx.doi.org/10.1016/0020-7683(84)90025-8http://dx.doi.org/10.1016/0020-7683(86)90064-8http://dx.doi.org/10.1016/S0263-8231(01)00026-Xhttp://dx.doi.org/10.1061/(ASCE)0733-9445(2002)128:7(934)http://dx.doi.org/10.1061/(ASCE)0733-9445(2002)128:7(934)http://dx.doi.org/10.1061/(ASCE)0733-9445(2002)128:7(934)http://dx.doi.org/10.1016/j.tust.2012.09.004http://dx.doi.org/10.1016/j.tws.2012.05.010http://dx.doi.org/10.1016/j.tws.2012.05.010http://dx.doi.org/10.1016/j.ijmecsci.2008.11.006http://dx.doi.org/10.1115/1.4002540http://dx.doi.org/10.1061/(ASCE)0733-947X(2004)130:1(43)http://dx.doi.org/10.1115/1.3629687http://dx.doi.org/10.1115/1.3607866http://dx.doi.org/10.1115/1.3607866http://dx.doi.org/10.1115/1.3422785http://dx.doi.org/10.1115/1.3169094http://dx.doi.org/10.1115/1.3169094http://dx.doi.org/10.1007/BF02326466http://dx.doi.org/10.1007/BF02326466http://dx.doi.org/10.1115/1.3169094http://dx.doi.org/10.1115/1.3422785http://dx.doi.org/10.1115/1.3607866http://dx.doi.org/10.1115/1.3607866http://dx.doi.org/10.1115/1.3629687http://dx.doi.org/10.1061/(ASCE)0733-947X(2004)130:1(43)http://dx.doi.org/10.1115/1.4002540http://dx.doi.org/10.1016/j.ijmecsci.2008.11.006http://dx.doi.org/10.1016/j.tws.2012.05.010http://dx.doi.org/10.1016/j.tust.2012.09.004http://dx.doi.org/10.1061/(ASCE)0733-9445(2002)128:7(934)http://dx.doi.org/10.1061/(ASCE)0733-9445(2002)128:7(934)http://dx.doi.org/10.1016/S0263-8231(01)00026-Xhttp://dx.doi.org/10.1016/0020-7683(86)90064-8http://dx.doi.org/10.1016/0020-7683(84)90025-8http://dx.doi.org/10.1002/stab.200710078

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