Marine Structures 21 (2008) 138159Stress concentration factors at welds in pipelines andtanks subjected to internal pressure and axial forceInge LotsbergDet Norske Veritas, Veritasveien 1, 1322 Hvik, NorwayReceived 16 March 2007; accepted 13 December 2007AbstractIn this paper, analytical expressions for stress concentration factors in pipes subjected to internalpressure and axial force are derived for a number of design cases based on classical shell theory. Theeffect of fabrication tolerances in simple butt welds is assessed. Analyses based on classical mechanicsare compared with results from axisymmetric nite element analyses for verication of the presentedmethodology. Stress concentration factors are presented for circumferential butt welds in pipeswelded together from pipes with different thicknesses, welds at buckling arrestors, welds at angedconnections in pipelines, and welds at ring stiffeners on the inside and the outside of the pipes. It alsoincludes stress concentration factors at end closures in pipes for gas storage. Larger pipes arefabricated from plates with a longitudinal weld. This fabrication process introduces out-of-roundnessin the pipes. The actual out-of-roundness is a function of internal pressure. An analytical expressionfor the bending stress in the pipe wall due to this out-of-roundness is presented. The derived stressconcentration factors can be used together with a hot spot stress SN curve for calculation of fatiguedamage.r 2008 Elsevier Ltd. All rights reserved.Keywords: Fatigue; Stress concentration factors; Circumferential welds; Ring stiffeners; Flanged connections;Longitudinal welds; Pipes; Fabrication tolerances; Pipelines; Cylinders for compressed gas1. IntroductionA stress concentration factor can be dened as a stress magnication at a detail due tothe detail itself or due to a fabrication tolerance with the nominal stress as a referenceARTICLE IN PRESSwww.elsevier.com/locate/marstruc0951-8339/$ - see front matter r 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.marstruc.2007.12.002E-mail address: inge.lotsberg@dnv.comvalue. The maximum stress is often referred to as the hot spot stress that is used in relationwith SN data for fatigue life calculation. This hot spot stress is thus derived as the stressconcentration factor times the nominal stress. Derivation of stress concentration factors atbutt welds has been presented in the literature by Maddox [1], Connely and Zettlemoyer [2]and Lotsberg and Rove [3].A rather detailed assessment of stress concentration factors in pipes and tubularstructural members subjected to axial force was presented by Lotsberg [4]. Effects offabrication tolerances, thickness transitions and ringstiffeners and bulkheads on hot spotstress were investigated.The welding of circumferential welds in structural members is often performed fromoutside only, which is the case for the last circumferential welds made in fabrication ofjacket structures when welding braces to the stub sections in tubular joints. It wasARTICLE IN PRESSNomeclatureAr area of ring stiffenerD exural rigidity or stiffness of the shellE Youngs modulusM0 bending moment in the shell per unit circumferential length at x 0Mx bending moment in the shell per unit circumferential length at xN axial force per unit circumferential lengthQ0 shear force per unit circumferential length at x 0 acting in the radialdirection of the shellQx shear force per unit circumferential length at x acting in the radial direction ofthe shellSCF stress concentration factorW section modulus of plate and shell per unit widthd outer diameter of pipeexexp(x)le elastic lengthp internal radial pressure loadingr radius of the shell measured from the axis of the cylinder to the shell mid-surfacet thickness of shell or tubular or pipew radial deection of the shellwh homogenous part of the radial deection of the shellwpart particular part of the radial deection of the shellb, g, j, c coefcients introduced to simplify equations for SCFsdm eccentricity at butt weldsn Poissons ratioy rotation of shellst total stresssb bending stresssnominal nominal stressx reduced co-ordinateI. Lotsberg / Marine Structures 21 (2008) 138159 139concluded by Lotsberg [4] that transition in thickness on the outside of the pipewas preferred at circumferential welds between tubular sections with differentthickness in order to achieve good fatigue capacity for these welds. It was alsoshown that it is preferred to put ringstiffeners on the inside of the tubularmembers when the pipes are subjected to a longitudinal force range to achievelong fatigue lives. It might be questioned what is most efcient considering effect ofinternal pressure in this respect. The background for this together with analyticalexpressions for stress concentration factors in welded pipes is presented more in detail inthe following sections.Pipes are being used in pipelines for transportation of oil and gas and as cylindersfor storage of compressed gas. The internal pressure in the pipes gives stresses in thecircumferential direction of the pipes. The internal pressure also gives longitudinalstress in the pipe wall due to the end cap effect. This nominal longitudinal stressdue to internal pressure is only half that of the circumferential stress. However,increased wall thickness or stiffeners may be required at supports and at crackarrestors in deep water pipelines (e.g. [5]). This may introduce stress concentrationsthat make the stress cycling in the axial direction of the pipes important. It should also benoted that the longitudinal welds are normally made from both sides allowing for arather good SN curve while circumferential welds are normally made from the outsideonly and a less good SN curve apply to the root of the weld as compared withthe outer weld toe (refer Fig. 1). Reference is made to fatigue design standards such as IIW[6] and to DNV-RP-C203 [7] for description of applicable SN curves. Axial stresses in thepipes may also occur due to external forces such as wave loading and due to change intemperature.Pipes made of high strength steel can be subjected to large stress ranges and thewelds can only sustain a limited number of full stress cycles from lling and emptying ofthe pipes until failure occurs. In fabricated pipes there will always be some imperfectionsleft from fabrication. Residual out-of-roundness in the pipes after fabrication isan important parameter in this respect as it together with the membrane stress inthe circumferential direction in the pipe wall introduces local bending stresses. Themembrane stress and the local bending stress, which are acting normal to the longitudinalweld may be governing the fatigue lives of these elements due to stress cycles fromlling and emptying of the pipes. This problem is further considered in the last sectionof this paper.ARTICLE IN PRESSOutside of pipet

mWeld rootWeld toeLtdFig. 1. Section through a single side weld.I. Lotsberg / Marine Structures 21 (2008) 138159 1402. Classical shell theory2.1. Cylindrical shellThe differential equation for the cylindrical shell shown in Fig. 2 can be found in theliterature (e.g. [8])Dq4wqx4 Etr2 w px nrN (1)where D is the exural rigidity of the shell dened asD Et3121 n2 (2)where E is the Youngs modulus, n is the Possons ratio, t is the shell thickness, r is theradius of the cylindrical shell measured from the axis of the cylinder to the middle of theshell surface, p(x) is the internal radial loading and N is the axial force per unitcircumferential length.The solution of the homogenous part of the differential equation (1) can be expressed aswh l2e2DM0g4x Q0leg1x (3)where an elastic length is dened asle rtp31 n24_ (4)For steel with n 0.3 the elastic length becomesle 0:78rtp (5)and with reference to Fig. 2 the following denitions are made:x xleg1x excos xg2x exsin xg3x g1x g2xg4x g1x g2x 6ARTICLE IN PRESSNM0wQ0X, 2rdp (x)tFig. 2. Circular cylindrical shell loaded symmetrically with respect to its axis.I. Lotsberg / Marine Structures 21 (2008) 138159 141The particular part can be expressed for p(x) as a polynomial in less or equal thirddegree aswpart r2Etpx nrEtN (7)The total displacement, w, is the sum of the homogenous and the particular part. Thenfor constant pqwqx le2D2M0g1x Q0leg3x (8)The moment (per unit circumferential length) at a section x (Fig. 2) is calculated asMx Dq2wqx2 M0g3x Q0leg2x (9)and the shear force (per unit circumferential length) is derived asQx qMxqx 2M0g2xle Q0g4x (10)2.2. Spherical shellsReference is made to Flugge [9] and Spence and Tooth [10] for general theory onspherical shells. In the following the relations between the forces and rotations at the edgeof the sphere shown in Fig. 3 are presented. From these books the following equation isARTICLE IN PRESSM0sQ0spatsN = pa/2wN = pa/2Q0sM0sFig. 3. End sphere for gas storage pipe.I. Lotsberg / Marine Structures 21 (2008) 138159 142derived for radial displacement:ws 2Q0salEts 2M0sl2Ets(11)and for rotationqwsqx 2Q0sl2Ets 4M0sl3Eats(12)wherel 31 n24_ ats_ (13)The radial displacement due to pressure p iswp pa22Ets1 n (14)There is no rotation at the edge for a constant pressure p.3. Fabrication tolerances at a butt weld in a plateThe effect of fabrication tolerance at a simple butt weld in a plate with respect toadditional stress is illustrated in Fig. 4. The plate is subjected to an axial force per unitwidth of N sa*t. The fabrication tolerance results in a shift in neutral axis equal to dm atthe butt weld. This shift in neutral axis implies a bending moment in the plate at both sidesof the weld of M N*dm/2, as there will be an inection point in the beam model in thecentre of the weld. From Fig. 4 it is observed that the moment changes sign over a shortlength. The validity of classical beam theory might be questioned in this respect. Therefore,this has been investigated more in detail for plate structures by Lotsberg and Rove [3]. It isalso assessed further by a ne nite element mesh analysis reported in Section 4. Thesection modulus for a plate of unit width is W t2/6. Thus the bending stress in the plateat the weld becomessb MW Ndm=2t2=6 3Ndmt2 3sadmt (15)ARTICLE IN PRESSN

mM = N*m / 2Fig. 4. Moment distribution at a plate with an eccentric butt weld.I. Lotsberg / Marine Structures 21 (2008) 138159 143By denition the stress concentration factor at the weld toe in Fig. 4 becomesSCF 1 3dmt (16)where dm is the fabrication tolerance as shown in Fig. 4 and t is the pipe wall thickness.4. Circumferential butt welds in pipes at thickness transitions and with fabrication tolerancesAdditional stresses at butt welds in pipes resulting from thickness transitions andfabrication tolerances can be derived by use of shell theory as shown by Lotsberg [4]. Forthe outer side of a tubular subjected to axial stress, as shown in Fig. 5, a stressconcentration factor was derived asSCF 1 6dt111 t2=t12:5 ea(17)where d 12(t2t1) is the shift in neutral axis as shown in Fig. 5 anda 1:82Ld t1t1_ 11 t2=t12:5 (18)where L is the length for transition in thickness as shown in Fig. 5.For the inner sideSCF 1 6dt111 t2=t12:5 ea(19)Considering the axial stress in the pipe resulting from end cap effect from internalpressure (or from other axial force in the pipes) the following stress concentration factorcan be used for the weld toe side in Fig. 1SCF 1 3dmt et=dp(20)This equation is derived from Eq. (17) by putting L 0.91 times the width of the weld.This stress concentration factor should be used if the pipe is welded from both sides. It isconservative to use the same stress concentration factor for the root side in symmetricwelds (V-shaped weld groove) as explained in Lotsberg [4] and DNV-RP-C203 [7]. Thebending stress at the root of the weld in Fig. 1 is close to zero following the momentdistribution shown in Fig. 4. An axisymmetric model is made of a pipeline section withdiameter 4200 and wall thickness 31.0 mm in order to assess this further, refer to LotsbergARTICLE IN PRESSt2Outside 41Inside

LNeutralaxis

nominalt1Fig. 5. Preferred transition in thickness at tubular butt weld when subjected to external axial force.I. Lotsberg / Marine Structures 21 (2008) 138159 144and Holth [11]. Two different geometries were analysed as shown in Fig. 6: One single side welded connection with a fabrication tolerance equal to 2 mm as shownin Fig. 6a, which also shows a shift in neutral axis equal to 2 mm. One analysis with the same geometry at the notch region, but with added material onthe right pipe section such that the connection is without shift in neutral axis (Fig. 6b).ARTICLE IN PRESSB Ar = 1mm h = 0.05*tr = 1mm2 mmh = 0.05*t15t302 mmtB Ar = 1mm2 mm1530Fig. 6. Single side welded connections: (a) eccentric connection; and (b) symmetric connection (aligned neutralaxes).0.60.81.01.21.41.61.82.02.20 5 10 15 20 25 30 35Stress (MPa)Eccentric 2 mmSymmetricSCF eq. (17)Distance from inner side (mm)Fig. 7. Stress in axial direction in section AA.I. Lotsberg / Marine Structures 21 (2008) 138159 145The notch at the weld root and at the two weld toes in Fig. 1 is modelled by a radius1.0 mm. This requires modelling by a ne mesh typically that recommended for the notchstress methodology (ref. Appendix D of DNV-RP-C203 [7]). The connections weresubjected to a unit axial load over the 31 mm thick section. Stresses at different sectionsAA and BB are presented in Figs. 7 and 8, respectively.First the stress in the direction normal to line AA in Fig. 7 is considered. The left partof Fig. 7 shows stress at the inner side of the pipe and the right part shows stress at theouter side. At the outer side there is a peak stress resulting from the notch at the weld toe.More interesting in this respect is the stress at the inner side which is less affected bynotches (see Fig. 6). Here the stress is compared with that resulting from Eq. (17), where Lis put equal to the width of the weld. It is seen that the calculated stress for eccentricity2 mm is slightly larger than that from the linear stress distribution from Eq. (17). Thecalculated stress distribution at the inner side seems to be slightly disturbed by the presenceof the weld notch. The same disturbance is also present for the situation without shift inneutral axis, but with presence of the root notch (symmetric geometry). Based on this theglobal stress behaviour represented by Eq. (17) is considered to be in agreement with thenumerical results from the nite element analyses. The results using different equations forcalculation of stress concentration factors are shown in Table 1.Next the stress distribution through section BB is considered (refer Fig. 8). It isobserved that the gradients of both the stress distributions for the two geometries are smallaway from the notches indicating that a shift in neutral axis does not give signicant beamARTICLE IN PRESS0.60.81.01.21.41.61.82.02.22.42.60 5 10 15 20 25 30 35Distance from inner side (mm)Stress (MPa)Eccentric 2 mmSymmetricFig. 8. Stress in axial direction in section BB.Table 1Calculated stress concentration factorsMethodology used for calculation Stress concentration factorAxisymmetric nite element model 1.188SCF Eq. (16) 1.193SCF Eq. (17) 1.178SCF Eq. (20) 1.163I. Lotsberg / Marine Structures 21 (2008) 138159 146bending stress at the root that needs to be included in a fatigue analysis. However, thenotch stress due to the local root geometry needs to be accounted for. The SN curves inDNV-RP-C203 [7] for the root side that accounts for this effect can be used.Direct read out of the maximum stress in the root in the nite element modelrepresenting Fig. 6a gives snotch 2.62 MPa for unit axial load and the maximum stress inthe root in Fig. 6b gives snotch 2.45 MPa. (The maximum stress is a surface stress whichis slightly different from that of the axial stress presented in Figs. 7 and 8). This results in astress concentration factor SCF 2.62/2.45 1.069. As the geometry is the same exceptfor different shift in neutral axis one may use this result to assess an alternative equationfor stress concentration factor for the root asSCF 1 kdmt et=dp(21)Here only k is unknown and solving this equation gives k 1.2.From this it is seen that stress concentration factors due to eccentricity after fabricationat weld roots of single sided symmetric welds are signicantly less than for the weld toe(refer Eq. (20)).It is shown here that additional stress at the hot spot resulting from shift in neutral maybe accounted for by analytical expressions. Local stress increase due to local notchesshould be accounted for in the SN curve if not a notch stress approach is used such aspresented as alternatives in IIW [6] or DNV-RP-C203 [7]. Reference may also be made tolocal approaches presented by Radaj et al. [12].In general the stress concentration factor to be used is also linked to the SN curve thatis recommended for the fatigue assessment, refer also Lotsberg and Sigurdsson [13].5. Circumferential butt welds in pipes with different thicknessesLocal bending stresses in pipes will normally be present at connections welded togetherfrom pipes with different thickness such as at pipe supports and buckling arrestors. Awelded connection between pipes with different thickness is considered in the following(see Fig. 9). The following derivation is based on an assumption of a centric neutral axisgoing from one thickness to the next as shown in Fig. 9a. Then a shift in neutral axis isconsidered at the end of this section with thickness transitions on the outer side (Fig. 9b) orthe inner side of the pipe. For derivation of an equation for stress concentration factor atthe circumferential weld the following requirements are dened at the welded connection: The radial displacement of pipe no. 1 is equal to that of pipe no. 2. The rotation of pipe no. 1 is the same as for no. 2. The moments around the circumference in the two pipes are the same. There is continuity in shear force.Mathematically these requirements can be expressed asw1 w2 (22)qwqx_ _1 qwqx_ _2(23)ARTICLE IN PRESSI. Lotsberg / Marine Structures 21 (2008) 138159 147M01 M02 (24)Q01 Q02 (25)The axial force in the pipes due to internal pressure can be calculated as (end cap forcedivided by pipe area)N pr2 (26)Then the particular part from Eq. (7) can be expressed aswpart pr2Et 1 n2_ _ (27)From Eqs. (3), (22), (24), (25) and (27)w l2e12D1M0 Q0le1 pr2Et11 n2_ _ l2e22D2M0 Q0le2 pr2Et21 n2_ _ 28From Eqs. (8), (23)(25)qwqx le12D12M0 Q0le1 le22D22M0 Q0le2 (29)ARTICLE IN PRESSa) Centric neutral axispb) Shift in neutral axisat thickness transition pc) Sloped thickness transition14pt2 t1t1t2t2t1SupportFig. 9. Thickness transition in pipes at support and buckling arrestors: (a) centric neutral axis; (b) shift in neutralaxis at thickness transition; and (c) sloped thickness transition.I. Lotsberg / Marine Structures 21 (2008) 138159 148Then from Eqs. (28) and (29)M0 prt212 n431 n2_ g1t1 1t2_ _ (30)andQ0 2t2:51 t2:52 t2:52 t0:52 t21M0le1(31)whereg 2t2:51 t2:52 t2:52 t0:52 t211 t1t2_ _1:5_ _ t1t2_ _2 1 (32)The bending stress at the connection in the thinnest pipe is obtained assb M0W 6M0t21 prg2 n231 n2 1t1 1t2_ _ (33)It is assumed that the radius rbt1. The stress due to the end cap pressure is calculated assa pr2t1(34)The total stress at the inner side and the outer side is calculated asst sa sb (35)This equation can also be written asst sa 1 sbsa_ _ saSCF (36)where the stress concentration factor isSCF 1 sbsa(37)For the inner side of the pipeSCF 1 2 ng31 n2_ 1 t1t2_ _ (38)For the outer side of the pipeSCF 1 2 ng31 n2_ 1 t1t2_ _ (39)The equation for moment in Eq. (30) is valid for all thickness relations. The derivationof equations for stress concentration factors (Eqs. (38) and (39)) is based on calculation ofnominal stress in the pipe section with thickness t1. It should be noted that Eqs. (38) and(39) are derived based on a centric neutral axis as shown in Fig. 9a. Calculated stressconcentration factors using these equations are presented in Fig. 10. The results from oneARTICLE IN PRESSI. Lotsberg / Marine Structures 21 (2008) 138159 149geometry using nite element analysis using PREFEM [14] are included as check.It is observed that there is some difference in the calculated stress concentration factors.This may be explained by some difference in the stiffness properties of the nite elementmodel as compared with the classical analytical shell model. One 20-node isoparametricelement, representing a linear stress distribution, is used to represent one thickness inpipe 1. However, 2 elements are used to model pipe 2 in order to connect the two pipes.This gives a slightly more exible pipe 2 in the nite element model than in the analyticalapproach.It is observed from Fig. 10 that the largest stress is found at the inner side of the pipe.An eccentric thickness transition as shown in Fig. 9b also introduces local bending overthe pipe wall when subjected to axial forces as explained in Section 4. The informationARTICLE IN PRESS0.00.20.40.60.81.01.21.41.61.3 1.5 2.0 2.3 2.5Stress concentration factorInner sideOuter sideFE inner sideFE outer sideThickness ratio t2/t11.0 1.8Fig. 11. Stress concentration factor with step in thickness on outside.0.00.20.40.60.81.01.21.41.61.82.01.3 1.5 1.8 2.0 2.3 2.5SCF inner sideSCF outer sideFE inner sideFE outer sideStress concentration factorThickness ratio t2/t11.0Fig. 10. Stress concentration factor for longitudinal stress from internal pressure neglecting thickness transition(or shift in neutral axis).I. Lotsberg / Marine Structures 21 (2008) 138159 150from Section 4 can be used to derive stress concentration factors for eccentric thicknesstransitions by using the superposition principle.For the inner side of the pipe with thickness transition on the outsideSCF 1 2 ng31 n2_ 1 t1t2_ _ 3t2 t1t111 t2=t12:5 ea(40)The parameter a is dened by Eq. (18).For the outer side of the pipe with thickness transition on the outsideSCF 1 2 ng31 n2_ 1 t1t2_ _ 3t2 t1t111 t2=t12:5 ea(41)The calculated stress concentration factors using these equations are presented in Fig. 11together with results from a nite element analysis. It is seen from Fig. 11 that a transitionin thickness on the outside of the pipe reduces the stress concentration factor at the innerside as compared with that without shift in neutral axis shown in Fig. 10.For the inner side of the pipe with thickness transition on the inner sideSCF 1 2 ng31 n2_ 1 t1t2_ _ 3t2 t1t111 t2=t12:5 ea(42)The parameter a is dened by Eq. (18).For the outer side of the pipe with thickness transition on the inner sideSCF 1 2 ng31 n2_ 1 t1t2_ _ 3t2 t1t111 t2=t12:5 ea(43)It should be noted that these stress concentration factors applies only to the axial stressthat results from internal pressure. And the stress concentration factors are to be usedtogether with the nominal axial stress resulting from end cap force or internal pressureonly. For axial stress in the pipe wall resulting from external forces the stress concentrationfactors from Eqs. (17)(19) should be used. Thus for design it is necessary to separate thestress into these two stress components before the resulting stress range is calculated.Thus the following superposition of stress conditions should be used for derivation ofhot spot stress at the circumferential weld:Dshot spot DsInternal axial pressureSCFInternal pressure DsExternal axialSCFExternal axial 44where DsInternal axial pressure is the stress in the axial direction of the pipe due to internalpressure, DsExternal axial is the stress in the axial direction of the pipe due to external axialforce, SCFInternal pressure is the stress concentration factors from Eqs. (40) to (43) dependingon position of thickness transition and considered hot spot, SCFExternal axial is the stressconcentration factors from Eqs. (17)(19) depending on position of thickness transitionand considered hot spot.In many practical cases a slope transition equal to 1:4 is made for transition from onethickness to another as indicated in Fig. 9c. Finite element analyses of such geometriesshows that the effect of internal pressure on bending stress at the weld region reduces (referto Eqs. (38) and (39) representing effects from different pipe stiffness). Then the mainARTICLE IN PRESSI. Lotsberg / Marine Structures 21 (2008) 138159 151contribution to local bending stress is from axial force at thickness transitions andfabrication tolerances represented by Eqs. (17)(19).6. Ring stiffeners and buckling arrestors in pipelinesA ringstiffener in a cylindrical shell as shown in Fig. 12 is considered. This applies also toa short buckling arrestor in a pipeline and a bolted ange connection that is frequentlyused in risers for oil and gas production (refer Fig. 13, where the most critical hot spot isfound at point B). The inward displacement of a ring subjected to a radial force (per unitlength) 2Q0 on the outside can be calculated aswring 2Q0r2EAr(45)where Ar is the area of ringstiffener. (For a anged connection Ar is the area of the twoanges.)At a ring stiffener the rotation at the shell is zero due to symmetry (see Fig. 12), andfrom Eq. (8)qwqx 0 ) M0 Q0le2 (46)The displacement of the ring is equal that of the shell at the connection which gives fromEqs. (3), (27), and (45), for x 0.2Q0r2EAr l2e2DM0 Q0le pr2Et 1 n2_ _ (47)ARTICLE IN PRESS2rtArDeformed shapepFig. 12. Ring stiffener on pipe with internal pressure.BpFig. 13. Section through bolted ange connection.I. Lotsberg / Marine Structures 21 (2008) 138159 152Combining Eqs. (46) and (47) and inserting for D from Eq. (2) the moment in thecylindrical shell at a ring stiffener is obtained asM0 pr2t2Arle2 n8r2t3 12Arl3e1 n2(48)By dividing by the section modulus the bending stress in the shell is obtained assb 3p 2 n1 n2p 1bpr2t (49)where b is dened asb 1 2trtpAr31 n24_ (50)And the stress concentration at a ring stiffener is obtained asSCF 1 3p 2 n1 n2p 1b (51)where the plus sign applies to the inner side and minus to the outer side. This stress concentrationincludes effect from internal pressure and end cap pressure. It is to be used together with thenominal stress acting in the axial direction of the pipe wall due to end cap pressure.With n 0.3 for steel the expression for b becomesb 1 1:56trtpAr(52)and the stress concentration factor for the inner side becomesSCF 1 3:087b (53)and the stress concentration factor for the outer side becomesSCF 1 3:087b (54)Due to the notch of the weld itself the fatigue strength of the weld at the ring stiffeneritself becomes less than for the other shell side. As the stress is lesser on the outside than atthe inside it is thus recommended to place ring stiffeners on the outside of a shell structuresubjected to internal pressure. This is a different conclusion from that of ringstiffeners intubular members subjected to pure external axial force (see Ref. [4]).From Eq. (51) it is seen that the stress concentration factor at the inside of the pipe at aringstiffener becomes signicant. Example pipe for CNG with outer diameter 4200 andthickness 31 mm. With a atbar ringstiffener 30 200 the SCF on inside becomes 2.53,while for the outside it becomes 0.53.The same equations as used for a ring stiffener can also be used for assessment of stressconcentration factors at an open bulkhead by calculating an effective area asAr rtb1 n (55)where tb is thickness of the bulkhead.ARTICLE IN PRESSI. Lotsberg / Marine Structures 21 (2008) 138159 1537. Stress concentrations at transitions from pipe to end spheresThe following derivation is based on an assumption of centric neutral axes going fromthe pipe to a sphere. An eccentric transition from a pipe to a sphere can be included similarto that derived for thickness transitions in pipes (see Section 5). The methodology to derivestresses at the transition between the cylindrical pipe and the end spheres is similar to thatused to derive stress distributions at thickness transitions. This means compatibility indisplacements, rotations, bending moment and shear force (refer to Eqs. (22)(25)). Byputting a r and from Eqs. (3), (11), (14), (22), (24), and (25) the following equation isderived:w l2e2DM0 Q0le pr2Etc1 n2_ _ 2l2M0Ets 2lrQ0Ets pr22Ets1 n 56From Eqs. (8), (12), (23), (24) and (25) the following equation is derived:qwqx le12D2M0 Q0le 4l3M0Etsr 2l2Q0Ets(57)From Eqs. (56) and (57) the bending moment at the junction is derivedM0 pr331 n24_ cj1 nts 2 ntc_ _ (58)and the stress concentration factor for the inner side is obtained asSCF 1 31 n2_ cjr2tc1 nts 2 ntc_ _ (59)and for the outer sideSCF 1 31 n2_ cjr2tc1 nts 2 ntc_ _ (60)wherej r2c 1t2c 1t2s_ _ 2r1:5 1t1:5c 1t1:5s_ _ (61)andc t2c t2st2:5c t2:5s tstcr_ (62)where tc is the thickness of cylinder and ts is the thickness of end sphere.The equation for moment in Eq. (58) is valid for all thickness relations. The derivationof equations for stress concentration factors (Eqs. (59) and (60)) is based on calculation ofnominal stress in the pipe section without shift in neutral axis going from pipe to sphere.The resulting stress concentration factors are presented graphically in Fig. 14.ARTICLE IN PRESSI. Lotsberg / Marine Structures 21 (2008) 138159 154The shear force is obtained from Eqs. (56) and (57) asQ0 231 n24_ jr M0 (63)Reference is made to Section 5 for considerations of eccentric neutral axes. A similarterm as the last terms in Eqs. (40) and (41) may be added to Eqs. (59) and (60) as anapproximation to include the effect of an eccentric thickness transition.8. Longitudinal welds with bending stress over the pipe wall resulting from out-of-roundnessof fabricated pipesThe out-of-roundness of fabricated pipe elements results in increased stress due to abending moment over the wall thickness (see Fig. 15). The eccentricity due to out-of-roundness is a function of tension in the hoop direction of the pipe, which is reduced as theinternal pressure is increased and the hoop tension is increased. Thus the bending stressover the wall thickness is a nonlinear function of the internal pressure. Assuming that themoment M results from an eccentricity d, where hoop tension is accounted for in theanalysis, the following derivation of a stress concentration factor is performed. It isassumed that the out-of-roundness results in an eccentricity d0 without any hoop tensionforce from internal pressure. In order to simplify the calculation, the ring is transformed toan equivalent beam model as shown in Fig. 15. The circumference of a pipe between aninection point, where the bending moments are zero, and a symmetry point withmaximum moment is considered. Thus, the length l in the beam model becomes l pd/8.The relation between bending curvature and bending moment on a beam can be expressedaccording to classical beam theory asq2yqx2 MxEI (64)where E is the Youngs modulus and I is the moment of inertia of pipe wall sectionARTICLE IN PRESS0.00.51.01.52.02.51.00 1.25 1.50 1.75 2.00 2.25 2.50Stress concentration factorSCF Inner sideSCF Outer sideThickness ratio ts/tcFig. 14. Stress concentration factor at weld to sphere as function of thickness ratio.I. Lotsberg / Marine Structures 21 (2008) 138159 155The eccentricity along the beam can be expressed asdx d0l x (65)and the moment along the pipe wall beam isMx dx yN (66)By combining Eqs. (64)(66) this results in the following differential equation:q2yqx2 NEI y Nd0xEIl 0 (67)The general solution of Eq. (67) isy C1 elx C2 elx d0xl (68)wherel NEI_ (69)where N is the membrane stress in the circumferential direction times wall thickness forunit length of the pipe.The boundary conditions in the model shown in Fig. 15 are:y 0 for x 0qyqx 0 for x l 70These boundary conditions are used to determine the integration constants and theresults areC1 d0l1lell ell and C2 C1 (71)Maximum eccentricity at x l is then obtained asdmax d0 ymax d0ll tanhll (72)ARTICLE IN PRESSdInflection point

0Maximum bendingmomentl = d/8

0

maxyNXlFig. 15. Model for analysis of out-of-roundness.I. Lotsberg / Marine Structures 21 (2008) 138159 156The maximum stress at x l is obtained froms NA MW Nt 6Ndmaxt2 (73)This stress can also be presented ass Nt SCF (74)where the stress concentration factor isSCF 1 6dmaxt (75)where dmax is eccentricity as function of the axial force N as obtained from Eq. (72), and t isthickness of the pipe. In terms of out of roundness the equation for stress concentrationfactor can be derived asSCF 1 1:5dOORtll tanh ll (76)where the out of roundness is dened as dOOR dmaxdmin, l pd/8 and with l from Eq.(69), which is a function of the membrane hoop stress sm as follows:l 12smEt2_ (77)Results by use of this equation are derived for a pipe with 4200 diameter and wallthickness 31.8 mm using tolerance requirements from DNV-OS-F101 [15], which isreferred to in the DNV rules for compressed gas, DNV [16]. For standard dimensionalrequirements for line pipes the maximum allowable dOOR dmaxdmin 15 mm. Forenhanced dimensional requirements for line pipes the maximum allowable dOOR 10 mm.The results for this tolerance are shown in Fig. 16. The results have been compared withnite element analysis using the Abaqus program [17] where nonlinear geometry isaccounted for. It is observed that the analytical approach for derivation of bending stressin the pipe wall provides approximately the same results as derived from the Abaqusanalysis.The results are also compared with the following equation from [18]:sb sm1:5dmax dmintf1 0:5pm1 n2=Ed t=t3g(78)where pm is the maximum pressure at the operating condition being assessed. In section forremarks, it is stated that If under fatigue loading pm varies, use the mean value during thetime interval considered. sm is the membrane stress in the circumferential direction due tointernal pressure.Assuming that the results from Abaqus analysis are the most accurate, it is observed thatthe bending stress from BS 7910 is non-conservative using maximum pressure. However,using mean pressure, as recommended in the BS 7910 document, Eq. (78) provides saferesults close to that from the Abaqus analysis.ARTICLE IN PRESSI. Lotsberg / Marine Structures 21 (2008) 138159 1579. ClosureIn this paper, analytical expressions for stress concentration factors in pipes subjected tointernal pressure and axial force are derived for a number of design cases based on classicalshell theory. The effect of fabrication tolerances in simple butt welds is assessed.Analyses based on classical mechanics at circumferential welds are compared withresults from axisymmetric nite element analyses for verication of the presentedmethodology. The results from these analyses are found to be in good agreement with theproposed analytical equations for stress concentration factors that are derived based onclassical theory of elasticity.It is shown that stress concentration factors due to eccentricity after fabrication at weldroots of single sided symmetric welds are signicantly less than that for the weld toe.Stress concentration factors are presented for circumferential butt welds in pipeswelded together from pipes with different thicknesses, welds at buckling arrestors, welds atanged connections in pipelines, and welds at ring stiffeners on the inside and the outsideof the pipes.These stress concentration factors can be used for fatigue assessment of pipelines whenthe pipelines are subjected to variation of internal pressure such as at start and stops ofoperation. The same stress concentration factors can also be used for fatigue assessment ofpipes for gas transportation, where each lling and emptying of the pipes gives a stresscycle that contributes to fatigue damage accumulation. Also a stress concentration factorfor end closures in pipes for gas storage is presented.Larger pipes are fabricated from plates with a longitudinal weld. This fabricationprocess introduces out-of-roundness in the pipes. A pressure variation from internalpressure in the pipe leads to variation of local bending in the pipe wall resulting from thesefabrication tolerances. A design methodology to account for the resulting stresses has beenpresented. The analytical expressions for stress concentration factors can be used in fatiguedesign of pipelines and cylinders used for transportation of gas.ARTICLE IN PRESS0204060801001201401601802000 50 100 150 200 250 300Pressure (bars)Stress (MPa)Membran stressBending stress analyticalBending stress ABAQUSBS7910 mean pBS7910 max pFig. 16. Example of membrane and bending stress in hoop direction of 4200 diameter pipe.I. Lotsberg / Marine Structures 21 (2008) 138159 158The derived stress concentration factors can be used together with a hot spot stress SNcurve for calculation of fatigue damage.AcknowledgementsThe axisymmetric nite element analyses performed by Per Arvid Holth and AndrzejKludka are acknowledged. Also the nonlinear analysis of bending stress in the pipe due toout-of-roundness performed by Rikard Trnkvist is appreciated.References[1] Maddox SJ. Fitness for purpose assessment of misalignment in transverse butt welds subjected to fatigueloading. London: International Institute of Welding, IIW Document XIII-1180, 1985.[2] Connely LM, Zettlemoyer N. Stress concentration at girth welds of tubulars with axial wall misalignment. In:Proceedings of the international conference on tubular structures. London: E & F N Spon; 1993.[3] Lotsberg I, Rove H. Stress concentration factors for butt welds in stiffened plates. OMAE New Orleans,ASME, 2000.[4] Lotsberg I. Stress concentration factors at circumferential welds in tubulars. J Mar Struct 1998;11:20330.[5] Torseletti E, Marchesani F, Bruschi R, Vitali L. Buckle propagation and its arrest: buckle arrestor designversus numerical analyses and experiments. OMAE2003-37220. In: Proceedings of the 22nd internationalconference on offshore mechanics and arctic engineering, Mexico, 2003.[6] IIW. Fatigue design of welded joints and components. In: Hobbacher A, editor. Recommendations of IIWjoint working group, XIII-1539-96/XV-845-96. Abington Publishing, The International Institute of Welding,1996.[7] DNV-RP-C203 Fatigue strength analysis of offshore steel structures, August 2005.[8] Timoshenko SP, Woinowsky-Krieger S. Theory of plates and shells. 2nd ed. McGraw-Hill Book Company,Inc.; 1959.[9] Flugge W. Stresses in shells. 2nd ed. Berlin, Germany: Springer; 1973.[10] Spence J, Tooth AS, editors. Pressure vessel design concepts and principles. 1st ed. London: E & FN Spon,an inprint of Chapman & Hall 2-6 Boundary Row; 1994.[11] Lotsberg I, Holth PA. Stress concentration factors at welds in tubular sections and pipelines. In: OMAE,2007, OMAE paper no. 2007-29571.[12] Radaj D, Sonsino CM, Fricke W. Fatigue assessment of welded joints by local approaches. 2nd ed.Cambridge, England: Woodhead Publishing Limited; 2006.[13] Lotsberg I, Sigurdsson G. Hot spot SN curve for fatigue analysis of plated structures. In: OMAE-FPSO04-0014, international conference, Houston, 2004. Also in J Offshore Arctic Eng 2006;128:3306.[14] PREFEM, Preprocessor for generation of nite element models, users manual. DNV Sesam Report No. 95-7014/Rev.0, August 1, 1995.[15] DNV-OS-F101 Submarine pipeline systems, January 2000.[16] DNV Rules for Classication of Ships. Compressed natural gas carriers, Part 5, January 2005 [Chapter 15].[17] Hibbit, Karlson & Sorensen Inc. ABAQUS/standard users manual, vol. 1, version 6.4, 2003.[18] BS 7910:1999. Guidance on methods for assessing the acceptability of aws in metallic structures. BSI, 1999.ISBN:0580330818.ARTICLE IN PRESSI. Lotsberg / Marine Structures 21 (2008) 138159 159