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Available online at www.sciencedirect.com

Nuclear Engineering and Design 238 (2008) 1275–1285

Effect of local wall thinning on the collapse behavior of pipe elbowssubjected to a combined internal pressure and in-plane bending load

Jin-Weon Kim a,∗, Man-Gyun Na a, Chi-Yong Park b

a Department of Nuclear Engineering, Chosun University, 375 Seosuk-dong, Dong-gu, Gwangju 501-579, Republic of Koreab Nuclear Power Laboratory, KEPRI, 103-16 Munji-dong, Yusung-gu, Daejeon, 305-380, Republic of Korea

Received 29 January 2007; received in revised form 1 October 2007; accepted 21 October 2007

bstract

The objective of this study was to investigate the effect of local wall thinning on the collapse behavior of pipe elbows subjected to a combinednternal pressure and in-plane bending load. This study evaluated the global deformation behavior and collapse moment of the elbows, whichontained various types of local wall-thinning defects at their intrados or extrados, using three-dimensional elastic–plastic finite element analysis.he analysis results showed that the global deformation behavior of locally wall-thinned elbows was largely governed by the mode of the bendingnd the elbow geometry rather than the wall-thinning parameters, except for elbows with considerably large and deep wall thinning that showedlastic instabilities induced by local buckling and plastic collapsing in the thinned area. The reduction in the collapse moment with wall-thinning

epth was considerable when local buckling occurred in the thinned areas, whereas the effect of the thinning depth was small when ovalizationccurred. The effects of the circumferential thinning angle and thinning length on the collapse moment of elbows were not major for shallow wall-hinning cases. For deeper wall-thinning cases, however, their effects were significant and the dependence of collapse moment on the axial thinningength was governed by the stress type applied to the wall-thinned area. Typically, the reduction in the collapse moment due to local wall thinningas clearer when the thinning defect was located at the intrados rather than the extrados, and it was apparent for elbows with larger bend radius.

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2007 Elsevier B.V. All rights reserved.

. Introduction

Pipe bends and elbows are employed in nuclear power plantiping systems to allow modifications to isometric routings.hey also play an important role in maintaining the integrity ofiping systems under transient loading conditions by absorbingonsiderably large thermal expansions and seismic movements,nd by dissipating energy as a result of local plastic defor-ation (Martzen and Yu, 1998; Shalaby and Younan, 1999;hattopadhyay, 2002). Since the pipe bends and elbows must beesigned to avoid a collapse at any loading condition, their col-apse load must be accurately estimated to ensure the reliabilityf the piping systems during service.

Pipe bends and elbows in nuclear piping systems are sub-

ected to various degradation mechanisms. Carbon steel pipeends and elbows degrade greatly due to local wall thinningrom flow-accelerated corrosion (FAC) (Chexal et al., 1998;

∗ Corresponding author. Tel.: +82 62 230 7109; fax: +82 62 232 9218.E-mail address: jwkim@chosun.ac.kr (J.-W. Kim).

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029-5493/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.nucengdes.2007.10.017

uen and Yin, 1999). Local wall thinning in piping componentseduces the failure pressure, load-carrying capacity, deforma-ion ability, and fatigue resistance of the piping system (JAERI,993; Miyazaki et al., 1999; Ahn et al., 2002; Hasegawa et al.,002; Kim and Park, 2003). Thus, it is important to evaluatehe effect of local wall thinning on the structural integrity ofiping components and to develop an integrity evaluation pro-edure. Several experimental and analytical studies have beenarried out to investigate the effect of local wall thinning on thentegrity of nuclear piping components (JAERI, 1993; Miyazakit al., 1999; Wilkowski et al., 2000; Hasegawa et al., 2002; Kimnd Park, 2003, 2005; Shim et al., 2003), and integrity evalua-ion procedures for local wall-thinned piping components haveeen proposed based on these studies. However, most of therocedures focus on local wall thinning in straight piping com-onents. The effect of local wall thinning on the integrity of pipeends and elbows has not yet been systematically investigated,

ven though local wall thinning due to FAC occurs frequentlyChexal et al., 1998; Kuen and Yin, 1999).

The objective of this study was to systematically investigatehe effect of local wall thinning on the integrity of pipe elbows

1276 J.-W. Kim et al. / Nuclear Engineering and Design 238 (2008) 1275–1285

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Fig. 1. Dimensions of a wall-thinned pipe elbow.

nder a combined internal pressure and in-plane bending load.hus, nonlinear three-dimensional finite element analyses wereerformed on elbows containing various types of local wall thin-ing at their intrados or extrados. These elbows were subjectedo in-plane bending with a constant internal pressure. The effectsf local wall thinning on the global deformation behavior of thelbows were investigated by comparing the moment versus end-otation curves under various conditions. The collapse momentas also evaluated, and its dependence on the wall-thinningarameters, such as the thinning depth, length, circumferentialngle, and defect location, was investigated for different bendingodes and elbow geometries.

. Evaluation procedures

.1. Analysis conditions

Finite element analyses were performed on a 90◦ elbow withn outer diameter of 400 mm and a nominal thickness of 20 mm.wo bend radii, Rb/rm = 3 and 6, were considered to take intoccount the effect of the elbow geometry, where Rb is the bendadius and rm is the mean radius of the elbow, as depicted inig. 1. The wall-thinning location, which is governed by the flowatterns of the fluid in the pipe elbows, may influence the col-apse behavior of wall-thinned elbows. Therefore, we assumedhat local wall thinning was located at the extrados and intrados

enterlines of the elbow, and that the axial and circumferentialhapes of the defects were circular. The dimensions of the wall-hinning defects used in the analyses are listed in Table 1. Theseimensions indicate the areas over which the wall thickness was

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able 1atrix for the finite element analysis of a wall-thinned elbow

ocation Loading type Bend radius, Rb/rm Thinning length, L/D

xtrados Closing 3 0.25ntrados Opening 6 0.5

1.02.0

Fig. 2. Finite element model used in the analyses.

hinner than the minimum thickness (tmin) required by construc-ion codes (ASME, 1995a,c). In the table, L is the equivalentxial thinning length defined at the flank of the elbow, as shownn Fig. 1, and tnom and tp are the nominal thickness and minimumhickness of the wall-thinned area, respectively.

The combined internal pressure and bending load were con-idered in the analysis as applied loads. The magnitude of thenternal pressure was 10 MPa. Both closing- and opening-moden-plane bending were investigated.

.2. Finite element models

Fig. 2 illustrates the three-dimensional finite element modelsmployed in the analyses. The model consisted of 20 node-breaklements with a reduced integration order. Only one-fourth ofhe elbow was modeled by considering its geometrical sym-

etry. The elbow was connected to straight pipes with lengthsqual to 10 times the mean pipe radius (10 rm) to permit freevalization of the elbow end-section (Martzen and Yu, 1998;obertson et al., 2005). As shown in Fig. 2, the model used 20

lements along the circumference, 14 elements along the bend,elements along the straight pipe, and 3 elements across the

hickness. We assumed the end of the straight pipe was end-apped by a beam to apply a bending moment and an internal

o Maximum thinning depth, (tnom − tp)/tnom Thinning angle, θ/π

0.301 0.06250.417 0.1250.534 0.250.641 0.500.767

J.-W. Kim et al. / Nuclear Engineering a

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ig. 3. True stress vs. true strain curve used in the finite element analyses.

ressure, which was modeled using two layers of solid elements.ppropriate boundary conditions were used along the planes of

ymmetry at the edges of the one-quarter model. The bendingoment was applied as an in-plane rotational displacement to the

rtificial center node, which was assigned a multipoint constraintt the end-plane of the beam. Thus, the moment and end-rotationould be obtained from the reaction moment and rotational dis-lacement at this node. The internal pressure was applied to thenner surface of the elbow, attached pipe, and end-capped beams a distributed load.

The general-purpose ABAQUS finite element analysis pro-ram (Hibbitt et al., 2005) was used for this study. Botheometric and material nonlinearities were considered to modelhe large deformation due to local buckling and ovalizationt the bend region that contained the wall-thinning defectnd to model the plastic behavior of the material. A previ-us study demonstrated that geometrical nonlinearity in thenite element analysis are very important when attempting torecisely determine the pipe bend deflection for various com-

inations of closing- and opening-mode bending and internalressures (Chattopadhyay, 2002). The yield and ultimate tensiletresses of the selected elbow and attached pipe material were02 MPa and 452 MPa, respectively, while the elastic modu-

btdb

Fig. 4. Collapse moment of a pipe elbow: (a) “Twice the ela

nd Design 238 (2008) 1275–1285 1277

us and the Poisson ratio were 206 GPa and 0.3, respectively.ig. 3 shows the true stress versus true strain curve used in thenalyses.

.3. Definition of the elbow collapse moment

The collapse moment of elbows subjected to in-plane bend-ng can be defined by various methods (Shalaby and Younan,999; Yahiaoui et al., 2000; Robertson et al., 2005). We obtainedhe collapse moment using the “twice the elastic slope” (TES)

ethod from the moment (Mb) versus end-rotation (ρ) curves.hus, the collapse moment was determined from the intersec-

ion between the TES line and moment versus rotation curve.he TES line was defined as a straight line from the origin with

wice the slope of the initial elastic response of the moment ver-us rotation curve, as illustrated in Fig. 4(a). This is the easiestethod to use and its results are the most reproducible (Yahiaoui

t al., 2000). It is also recommended by the ASME B&PVode Sec. VIII Div. 2 (ASME, 1995b). When this intersec-ion was located beyond the maximum moment, the maximum

oment was taken as the collapse moment, as shown inig. 4(b).

. Effect of local wall thinning on the collapse behaviorf elbows

.1. Global deformation behavior of wall-thinned elbows

.1.1. Defect-free elbowsWe examined moment versus end-rotation curves of defect-

ree sound elbows subjected to in-plane bending with an internalressure to clarify the dependence of the deformation behaviorf pipe elbows on the elbow geometry and bending mode prior tonvestigating locally wall-thinned pipe elbows. Fig. 5 shows the

oment versus end-rotation curves for defect-free sound elbowsith Rb/rm = 3 and 6 during in-plane closing- and opening-mode

ending with a constant internal pressure of 10 MPa. The elas-ic response was almost the same for both bend radii, and theifference was mainly in the plastic response. For closing-modeending, the moment versus end-rotation curves became almost

stic slope” (TES) method and (b) maximum moment.

1278 J.-W. Kim et al. / Nuclear Engineering and Design 238 (2008) 1275–1285

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ig. 5. Dependence of the moment vs. end-rotation behavior of defect-free soundlbows on the bending mode and bend radius.

at after the applied bending rotation exceeded a certain limit.owever, when the same elbow was subjected to opening-modeending, the moment versus end-rotation curves had a positivelope that was greater for the elbow with a smaller bend radius.n the plastic region, the moment versus end-rotation curve waslways greater for opening-mode bending than for closing-modeending. In addition, the higher the bend radius was, the higherhe moments were for both modes of bending. The difference ofhe moment versus end-rotation curves between bending modesiminished as the bend radius increased. In all cases, no plasticnstabilities occurred over the range of bending deformationsonsidered.

According to prior studies (Martzen and Yu, 1998; Shalabynd Younan, 1998, 1999; Chattopadhyay, 2002), the saturationf the moment during closing-mode bending occurs becausehe cross-sectional area of the elbow is ovalized to flatten thehape such that its moment of inertia is reduced. The increasingoment during opening-mode bending occurs because the bend-

ng load opens the elbow so that its cross-sectional moment ofnertia becomes larger. Thus, the geometrical effect of the elbowauses it to weaken or stiffen when subjected to in-plane bend-ng. This effect decreases with increasing bend radius so thathe bending moment increases with the bend radius while theiscrepancy of the moment versus end-rotation curves betweenending modes is reduced.

.1.2. Locally wall-thinned elbowsFig. 6 shows moment versus end-rotation curves for

lbows with local wall thinning and dimensions of L/Do = 1.0,tnom − tp)/tnom = 0.534, and θ/π = 0.50. As shown in Fig. 6(a),xcept for the case with Rb/rm = 6 subjected to opening-modeending, the extrados wall-thinned elbow curves were similaro the defect-free sound elbow curves shown in Fig. 5, evenhough the moment was slightly lower in most cases. In the

ase of Rb/rm = 6 subjected to opening-mode bending, the plas-ic curve initially increased and then decreased above a criticalnd-rotation. Thus, instability occurred in the global deforma-ion behavior of the locally wall-thinned elbow. For intrados

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ig. 6. Moment vs. end-rotation curves for locally wall-thinned elbows: (a)xtrados and (b) intrados.

all-thinned elbows (see Fig. 6(b)), the overall patterns of theoment versus end-rotation curves were also similar to those

f the defect-free sound elbows. However, both elbows withb/rm = 3 during opening-mode bending and with Rb/rm = 6 dur-

ng closing-mode bending showed an instability in the momentersus end-rotation curve. This instability was also observedor other wall-thinning defect analysis cases with elbowsontaining longer, deeper, and larger circumferential angles sub-ected to a compressive stress and elbows containing shorter,eeper, and larger circumferential angles subjected to a tensiletress.

Usually, the plastic instability in elbows during in-planeending is a global structural instability associated with a majorhange in the shape of the cross-sectional area. It appears inigher deformed regions of defect-free sound elbows. How-ver, the instability appeared in much lower deformed regionsf elbows with local wall thinning. This plastic instability isssociated with local buckling at wall-thinned areas subjectedo compressive stresses and with local plastic collapsing at

all-thinned areas subjected to tensile stresses, as shown inigs. 7 and 8, which give the deformation shapes of a wall-

hinned elbow cross-sectional area at the extrados and intrados,

J.-W. Kim et al. / Nuclear Engineering and Design 238 (2008) 1275–1285 1279

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ig. 7. Deformation shapes of elbow cross sections containing local wall thinlosing-mode, Rb/rm=3.0; (b) closing-mode, Rb/rm=6.0; (c) opening-mode, Rb

espectively. In Fig. 7, the cross section buckled locally at thextrados wall-thinned area for Rb/rm = 6 during opening-modeending; the cross sections were ovalized for the other wall-hinned elbows. In Fig. 8, the cross section buckled locallyt the intrados wall-thinned area for Rb/rm = 6 during closing-ode bending, and locally collapsed (abnormal stretching of

lements) for Rb/rm = 3 during opening-mode bending. Thesendings are consistent with the instability observations for theoment versus end-rotation curves shown in Fig. 6.These results demonstrate that for in-plane bending with a

onstant internal pressure, the global deformation behavior ofocally wall-thinned elbows was mainly governed by the modef bending and the elbow geometry, as shown by the defect-freeound elbows. This was also true for most elbows with local wallhinning in the bend region. For elbows with considerably largernd deeper wall thinning, however, the plastic instabilities asso-iated with local buckling and plastic collapsing in the thinnedrea occurred in the early stages of the bending deformation andhus their global deformation behavior deviated from that of theefect-free sound elbows.

.2. Collapse moments of locally wall-thinned elbows

The collapse moments of the wall-thinned elbows were eval-ated for various thinning depths, lengths, and circumferential

fatt

t the extrados under closing- and opening-mode bending (ρ = 0.1 radian). (a).0 and (d) opening-mode, Rb/rm=6.0.

ngles to quantitatively investigate the effect of local wall thin-ing on the elbow integrity. The dependence of the collapseoments on the defect geometry was also investigated for dif-

erent thinning locations, bending modes, and bend radii. Theesults were quantified by the weakening factor (w), defined byhe collapse moment of a wall-thinned elbow (MC) normalizedy the collapse moment of a defect-free sound elbow (MC,NT)ith the same geometry

= MC

MC,NT(1)

his factor indicates the weakening effect of local wall thinningn the collapse of pipe elbows compared to defect-free soundlbows. The weakening effect of wall thinning is negligible as theeakening factor approaches one, and the effect is considerable

s the weakening factor approaches zero.

.2.1. Effect of the wall-thinning depthFig. 9 shows the weakening factor of elbows with local wall

hinning at their extrados and intrados as a function of thehinning depth. The weakening factor was evaluated for five dif-

erent thinning depths at a constant thinning angle (θ/π = 0.5)nd length (L/D0 = 1.0). Regardless of the thinning location,he weakening factor decreased parabolically with increasinghinning depths. A significant reduction occurred in the weak-

1280 J.-W. Kim et al. / Nuclear Engineering and Design 238 (2008) 1275–1285

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ig. 8. Deformation shapes of elbow cross sections containing local wall thinlosing-mode, Rb/rm=3.0, (b) closing-mode, Rb/rm=6.0, (c) opening-mode, Rb

ning factor with thinning depth for elbows with larger bendadii (Rb/rm = 6) for both bending modes. For the extrados wall-hinning cases, the effect of the thinning depth on the weakeningactor was clearer during opening-mode bending compared tolosing-mode bending, whereas this tendency was reversed forhe intrados wall-thinning cases. This indicates that the reductionn the weakening factor with thinning depth was more substan-ial when the wall-thinned area was subjected to a compressivetress rather than a tensile stress.

However, the influences of bending radius and bending modeere small, except for the wall-thinned elbow with Rb/rm = 6

hat was subjected to a compressive stress, in which the weak-ning factor was considerably reduced with increasing thinningepths. This reduction was associated with the occurrence ofocal buckling in the thinned area, as discussed in the previousection. According to prior studies on a wall-thinned straightipe (Kim and Park, 2003; Shim et al., 2003), the suscepti-ility to local buckling at a thinned area under compressivetress was appraisable for longer wall thinning (L/D0 ≥ 0.5),nd the reduction in the collapse moment induced by local buck-

ing was considerable compared to that induced by local plasticollapsing. Consequently, the collapse moment of wall-thinnedlbows decreased parabolically with increasing wall-thinningepth. The effect of the wall-thinning depth on the collapse

tndt

t the intrados under closing- and opening-mode bending (ρ = 0.1 radian). (a).0 and (d) opening-mode, Rb/rm=6.0.

oment was significant under conditions at which local buck-ing occurred in the thinned area. However, when ovalizationccurred in the thinned area, the effect was small and less sen-itive to the bending mode and bend radius.

.2.2. Effect of the circumferential wall-thinning angleFigs. 10 and 11 show the weakening factor for the L/D0 = 1.0

lbows with local wall thinning at their extrados and intrados,espectively, as functions of the circumferential thinning angleor three different thinning depths. For a small thinning angleθ/π = 0.0625), the weakening factors were similar for all thin-ing depths, regardless of the thinning location, bending mode,nd bend radius. This shows that the effect of local wall thin-ing on the collapse moment of an elbow is negligible for ann-plane bending load with a constant internal pressure unlesshe thinning defect has a sizable circumferential angle, evenhen the defect has considerable depth. For elbows with shal-

ow wall thinning at an extrados [(tnom − tp)/tp ≤ 0.534], shownn Fig. 10, the weakening factor decreased almost linearly ashe circumferential thinning angle increased from θ/π = 0.0625

o 0.5. The variation of the weakening factor with the thin-ing angle was small, less than 10%. However, for elbows witheeper wall thinning at an extrados [(tnom − tp)/tnom = 0.767],he weakening factor decreased parabolically with increasing

J.-W. Kim et al. / Nuclear Engineering and Design 238 (2008) 1275–1285 1281

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ig. 9. Variations in the weakening factor with the wall-thinning depth forifferent bending modes and bend radii: (a) extrados and (b) intrados.

ircumferential thinning angles. This reduction was comparableo that observed previously with the thinning depth. In particu-ar, the reduction with thinning angle was considerable duringpening-mode bending when a compressive stress was appliedo the thinned area. For the intrados wall-thinning cases, theverall dependence of the weakening factor on the circumfer-ntial thinning angle was also similar to that observed for thextrados wall-thinning cases (see Fig. 11). However, the effect ofhe thinning angle on the weakening factor was more consider-ble during closing-mode bending, in contrast with the extradosall-thinning cases.Previous studies showed that the effect of the circumfer-

ntial thinning angle on the failure of wall-thinned elbowsubjected to an internal pressure was negligible compared tohe effect of the thinning depth and length (Li et al., 2001; Kimt al., 2005). Thus, the circumferential thinning angle has beeneglected in failure pressure evaluation models (Li et al., 2001).

owever, the present results demonstrate that the effect of cir-

umferential thinning angle on the collapse moment of locallyall-thinned elbows subjected to in-plane bending was compa-

able to the effect of the thinning depth and length for deeper

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ig. 10. Variations in the weakening factor with the circumferential thinningngle for extrados wall-thinned elbows: (a) Rb/rm=3.0 and (b) Rb/rm=6.0.

all-thinning cases. The failure of wall-thinned piping com-onents subjected to internal pressure is dominated by hooptresses in the thinned area (Fu and Kirkwood, 1995; Li et al.,001), and thus the failure pressure is largely dependent onhe thinning length and depth rather than the circumferentialhinning angle. However, the collapse of an elbow subjectedo in-plane bending is dominated by deformations in the bendegion, which are governed by the stiffness of the elbow crossection. The stiffness of the cross section is influenced by theircumferential thinning angle as well as the thinning depthnd length. It changes dramatically when local buckling andocal plastic collapsing occur in the thinned area. Thus, theignificant reduction in the weakening factor with increasingircumferential thinning angle for the deeper wall-thinningases was related to the higher susceptibility to local buck-ing and plastic collapsing in the thinned area during in-planeending.

These results demonstrate that the effect of the circumfer-ntial thinning angle on the collapse moment of wall-thinnedlbows is considerable for deeper wall-thinning cases that are

1282 J.-W. Kim et al. / Nuclear Engineering and Design 238 (2008) 1275–1285

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ig. 11. Variations in the weakening factor with the circumferential thinningngle for intrados wall-thinned elbows: (a) Rb/rm=3.0 and (b) Rb/rm=6.0.

usceptible to local buckling and local plastic collapsing. Thisffect is comparable to that of the thinning depth and length.

hen considering in-plane bending as an applied load, there-ore, the circumferential thinning angle must be regarded as aey parameter, along with the wall-thinning depth and length,n integrity evaluations of local wall-thinned elbows.

.2.3. Effect of the wall-thinning lengthFigs. 12 and 13 show the variation of the weakening fac-

or with the axial thinning length at the extrados and intrados,espectively, of elbows with local wall thinning of θ/π = 0.5.s shown in Fig. 12, for elbows with shallow wall thinning

t their extrados [(tnom − tp)/tnom ≤ 0.534], the weakening fac-or decreased almost linearly with increasing axial lengths ofhe wall thinning, regardless of the bend radius and bending

ode. This decrease was less than 10%, even though the thinningength increased from L/Do = 0.25 to 2.0. However, elbows with

eeper wall thinning [(tnom − tp)/tnom = 0.767] had a differentependence on the axial thinning length. During opening-modeending, the weakening factor decreased exponentially as thexial thinning length increased. For elbows with Rb/rm = 6 sub-

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ig. 12. Variations in the weakening factor with the axial thinning angle forxtrados wall-thinned elbows: (a) Rb/rm=3.0 and (b) Rb/rm=6.0.

ected to closing-mode bending, the weakening factor initiallyncreased with the axial thinning length and then decreasedith further increases in the axial thinning length. Therefore,elow a critical length, the weakening effect of the wall-thinningefect was enhanced by decreasing the axial thinning length,lthough it was typically enhanced by increasing the thinningength.

For intrados wall-thinned elbows, shown in Fig. 13, the over-ll dependence of the weakening factor on the axial thinningength was similar to that obtained for extrados wall-thinnedlbows. The weakening factor decreased linearly with increas-ng axial lengths of shallow wall thinning. For deeper wallhinning [(tnom − tp)/tnom = 0.767], however, the weakening fac-or decreased exponentially with increasing thinning lengthsuring closing-mode bending, and increased initially and thenecreased with increasing axial thinning lengths during opening-ode bending, as observed for the extrados wall-thinned elbows

ubjected to closing-mode bending. Therefore, for both bendadii, the weakening factor during opening-mode bending wasess than that obtained during closing-mode bending for shorter

J.-W. Kim et al. / Nuclear Engineering a

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aicwteaathe equivalent stress distributions along the circumference ofa defect-free sound elbow with Rb/rm = 3 subjected to in-planeclosing- and opening-mode bending (ρ = 0.1 radian) with a con-stant internal pressure of 10 MPa. The equivalent stress in the

ig. 13. Variations in the weakening factor with the axial thinning angle forntrados wall-thinned elbows: (a) Rb/rm=3.0 and (b) Rb/rm=6.0.

all-thinning defects (L/Do < 1.0). This trend was reversed foronger wall-thinning defects (L/Do ≥ 1.0).

In Figs. 12 and 13, the weakening factor exponentiallyecreased with increasing axial thinning lengths when deeperall thinning was subjected to a compressive stress. Thisas associated with the susceptibility to local buckling in the

hinning area, as discussed in the previous section. This sus-eptibility was enhanced by increasing the axial wall-thinningength (Kim and Park, 2003; Shim et al., 2003). Also, the col-apse moment decreased considerably and then saturated withncreasing axial lengths of the wall thinning under conditionshat caused local buckling. Therefore, the exponential decreasen the weakening factor with the axial thinning length duringompressive stresses was related to this higher susceptibility ofocal buckling for longer wall-thinning areas. However, wheneeper wall thinning was subjected to tensile stresses, the weak-ning factor increased with the axial thinning length and then

ecreased above a critical wall-thinning length. This behavioras also been observed in straight pipes with deeper wall thin-ing (Kim and Park, 2003; Shim et al., 2003). Usually, the stressoncentration is minor in the wall-thinned area, but for a deeper

Fd

nd Design 238 (2008) 1275–1285 1283

all thinning, it becomes considerable with decreasing axialhinning lengths subjected to tensile stresses (Kim and Son,004). Therefore, the enhanced weakening effect below a criticalall-thinning length was related to higher stress concentrations

n the thinned area because the higher tensile stresses promotedocal plastic collapsing that considerably reduced the collapse

oment of the elbow.These results demonstrate that the collapse moment of shal-

ow wall-thinned elbow decreased almost linearly with the axialhinning length and the decreasing with thinning length wasmall, regardless of bending mode and bend radius. However, foreep wall-thinning cases, the effect of the axial thinning lengthn the collapse moment was dependent on the bending modend defect location. The variation in the collapse moment withhe axial length of the wall thinning was considerable. Unfortu-ately, the existing evaluation procedure for locally wall-thinnedlbows only considers the circumferential thinning angle andend radius as influencing parameters when determining thecceptance criteria (ASME, 2003). Therefore, improvements ofhe evaluation procedure are required for locally wall-thinnedlbows; the effect of the axial wall-thinning length on the col-apse behavior of the elbows observed in this study should beppropriately incorporated into the procedure.

.2.4. Effect of the wall-thinning location and bend radiusThe effects of the wall-thinning depth, circumferential angle,

nd length on the collapse moment of wall-thinned elbows werenvestigated in the previous sections. We found that the typi-al reduction in the collapse moment due to local wall thinningas more sensitive to the thinning dimensions when the wall-

hinning defect was located at the intrados rather than extrados,ven though it was dependent on the bending mode. This isssociated with the different stress states between the intradosnd extrados of elbows during in-plane bending. Fig. 14 shows

ig. 14. Stress distribution along the circumference of defect-free sound elbowsuring in-plane bending.

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284 J.-W. Kim et al. / Nuclear Engineer

ntrados region was greater than that in the extrados region,egardless of the bending mode, indicating that the intradosegion was weaker and more sensitive to the existence of a defect.herefore, when the wall thinning is located at the intrados, theeakening effect is greater and more sensitive to changes in the

hinning dimensions.For elbows with larger bend radii, the weakening effect due

o local wall thinning was significant and sensitive to the defecteometries. These are related to the different elbow geometries,hich govern the collapse behavior during in-plane bending.ypically, the geometric effects diminishes with increasing bendadii; that is, the weakening or stiffening effects during in-planeending decreases with increasing bend radii. For elbows witharger bend radii, the influence of local wall thinning on theollapse behavior is relatively clear, and the collapse moments sensitive to the wall-thinning geometry. However, for elbowsith shorter bend radii, the geometrical effects dominate their

ollapse behavior so that the effect of local wall thinning iselatively minor. Thus, the reduction in the collapse momentue to local wall thinning is less sensitive to changes in theall-thinning dimensions.

. Conclusions

Three-dimensional elastic–plastic finite element analysesere performed on locally wall-thinned elbows subjected to in-lane bending with a constant internal pressure to investigatehe effect of the wall thinning on the integrity of the elbows.he effects of wall-thinning parameters, such as the thinningepth, length, circumferential angle, and location, and the bendadius, on the collapse behavior of wall-thinned elbows werenvestigated. The following conclusions were drawn.

1) The global deformation behavior of locally wall-thinnedelbows was largely governed by the mode of the bending andthe elbow geometry, even when local wall thinning existed inthe bend region of the elbow. For elbows with considerablylarge and deep wall thinning, however, plastic instabilitiescaused by the global bending deformation occurred in theearly stages of the bending deformation. These were inducedby local buckling and plastic collapsing in the thinnedarea.

2) The collapse moment of wall-thinned elbows parabolicallydecreased with increasing wall-thinning depths, regardlessof the thinning location and bend radius. This decrease wasconsiderable when local buckling occurred in the thinnedareas. However, when ovalization occurred, the effect of thethinning depth was small and less sensitive to the bendingmode and bend radius.

3) For shallow wall-thinning cases, the collapse moment ofthe wall-thinned elbow decreased almost linearly withincreasing circumferential thinning angles and axial thin-ning lengths. The effects of the circumferential thinning

angle and thinning length on the collapse moment were notmajor.

4) For deeper wall-thinning cases, the effects of the circum-ferential thinning angle and thinning length on the collapse

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moment were considerable. In particular, the variation withthe axial thinning length was dependent on the bend-ing mode and thinning location. As the axial thinninglength increased, the collapse moment decreased expo-nentially when the elbow was subjected to a compressivestress in the thinned area. However, the collapse momentinitially increased and then decreased above a critical thin-ning length when the elbow was subjected to a tensilestress.

5) Typically, the reduction in the collapse moment due to localwall thinning was clearer when the thinning was locatedat the intrados rather than the extrados, even though it wasdependent on the bending mode. The weakening effect dueto local wall thinning increased with the bend radius of thepipe elbow.

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