a structural stress-based critical plane method for multiaxial...

A structural stress-based critical plane method for multiaxial fatiguelife estimation in welded joints

C. JIANG, Z. C. LIU, X. G. WANG, Z. ZHANG and X. Y. LONGState Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University,Changsha, China

Received Date: 27 July 2015; Accepted Date: 20 October 2015; Published Online: 2015

ABSTRACT This paper aims at proposing a new fatigue life estimation model that is preferablyadapted to welded joints subjected to multiaxial loading. First, a mesh-size insensitivestructural stress is defined that enables to characterize the stress concentration effect ap-propriately. Second, the multiaxial stress state and loading path influence are taken intoaccount in the lifetime prediction model by adopting a suitable critical plane method,originally proposed by Carpinteri and co-authors. Experimental verification is conductedfor a given welded joint geometry under different loading conditions, including uniaxial,torsional and multiaxial loads. The reliability and effectiveness of the new method arevalidated through substantive fatigue testing data.

Keywords critical plane method; fatigue; multiaxial load; structural stress; welded joint.

NOMENCLATURE fx′, fy′ = nodal forcesh = thickness of plate or tubek = negative inverse slope of the S–N curve

kσ, kτ = stress concentration factorsk0 = negative inverse slope of the standard torsional fatigue curvek1 = negative inverse slope of the standard uniaxial fatigue curveKτ = negative inverse slope of the modified Wöhler curve

lk, mk, nk = principal stress direction cosinesmx′, my′ = nodal moments

Mb = bending momentMt = torsion moment

NRef = reference number of cycles to failureNf,es = estimated number of cycles to failureNf,ex = experimental number of cycles to failureW(ti) = weight function at time instant tiWS = summation of the weights W(ti)

ϕ, θ, ψ = principal Euler anglesϕ , θ , ψ = weighted mean principal Euler angles

λ = distance from the reference plane to the weld toeρw = multiaxiality factorσhs = normal stress at the hot spotσm = membrane stressσb = bending stressσa = normal stress amplitude

σmean = mean normal stressσnpl = nonlinear stress peakσx = normal stress along the x directionσn = nominal stressσs = structural stress

Correspondence: X. G. Wang. E-mail: [email protected]

© 2015 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct 00, 1–12

ORIGINAL CONTRIBUTION doi: 10.1111/ffe.12369

σaf = fatigue limit for fully reversed normal stressσeq,ϕ = equivalent stress on the critical planeσn,max = maximum normal stress on the critical plane

τhs = shear stress at the hot spotτxy = shear stress on the x-y plane

τns,ϕ = shear stress on the critical planeτaf = fatigue limit for fully reversed shear stressτa = shear stress amplitudeτs = structural shear stressτm = constant component of shear stressτb = linear component of shear stress

τmean = mean shear stressΔσÂ = stress range perpendicular to the critical plane

ΔσIIW = fatigue class (FAT) value under the uniaxial loading modeΔτIIW = fatigue class (FAT) value under the torsional loading modeΔτRef = reference shear stress range

ΔτÂ,max = maximum shear stress range on the critical planeΔτÂ,eq = equivalent shear stress range on the critical plane

I NTRODUCT ION

Welded joints widely exist in engineering structures andplay important roles in many fields, such as shipbuilding,car manufacturing and nuclear industry. The weldedjoints in an in-service structure are often subjected tocomplex alternating loads and act in many cases as theweakest links for fatigue failure.1–3 Hence, it is of greatimportance to reliably evaluate the fatigue properties ofwelded joints under multiaxial loading.4–9 To this end,different fatigue assessment methods have been proposedin the literatures.10–14

According to the fatigue design recommendations15

established by the International Institute of Welding(IIW), the nominal stress (NS) method, hot spot stress(HSS) method and local stress (LS) method are consid-ered as the three routine methods for welded structurefatigue analysis. In the fatigue damage parameter formu-lation, the NS method considers only the NS range at theboundary of the welded joint, which shows, however, in-competent to reflect the LS state at the weld toe. TheHSS method allows taking into account of the stress con-centration effect at the weld toe, as the ‘hot spot’ stress.For the LS method, the weld toe is viewed as a notchby introducing the virtual radius concept, and the com-puted stress state at the notch is used for the damage pa-rameter formulation. In the References [15,16], detailedfatigue design procedures for different types of weldedjoints and components are provided with respect to eachof the three routine methods.

In general, the HSS method outperforms the NSmethod in terms of lifetime evaluation precision, as hav-ing been evidenced by extensive studies.15–20 Neverthe-less, the HSS method also has its shortcoming: the HSS

computation is sensitive to the mesh size in the finiteelement (FE) analysis. Thus, a mesh-size insensitivestructural stress definition was proposed by Dong21 totackle this problem. The new structural stress definitionenabled to provide an effective measure of a stress statethat pertains to fatigue behaviour of welded joints in theform of both membrane and bending components, andit is insensitive to the mesh size. The effectiveness of thismethod has been demonstrated in the works of Dong andhis collaborators22–24 and of other researchers25–27 withsubstantive experimental verifications. This method is in-teresting, because on the one hand, it enables to solve themesh-size sensitivity problem, and on the other hand, itallows the consideration of the stress concentration effectin welded joints. Alternative structural stress-based methods28,29

and other approaches such as the critical distance methods,30–32

fracture mechanics-based methods33–36 and thermo-graphic methods37–39 have been also developed for the fa-tigue analysis of welded structures. In the present study,the structural stress definition is referred to that in Dongmethod.21

In the most general case, the welded structures are sub-jected to multiaxial stress state in the real loading condi-tions. However, the routine fatigue assessment methodsand other improved structural stress methods normallyshow less satisfactory estimation ability for such applica-tion occasions. This point was particularly remarked in acomprehensive experimental investigation carried out byBäckström et al.40 It showed that the estimation resultsby applying the NS and HSS methods under the multiax-ial loading mode were always less reliable than that underthe uniaxial loading mode. This phenomenon can be at-tributed to the fact that the employed methods were inad-equate to reflect the multiaxial stress state of welded

2 C. JIANG et al.


joints. It is common for the current existing fatiguecriteria for welded structures. In a general sense, they lackthe consideration of the ‘critical planes’ that maximize thefatigue damage under the complex multiaxial loads. It isremarked by the authors that the well-developed criticalplane methods41–43 in multiaxial fatigue analysis has notyet been widely applied to the welded structures as a rou-tine approach. Some recent works show that the re-searchers began to draw attention to this researchdirection and have demonstrated some promising results.In particular, the multiaxial fatigue criteria proposed bySusmel,44 Carpinteri and co-authors45,46 were provedwell-adapted to the fatigue evaluation of welded struc-tures under complex loading conditions.

In this paper, a novel fatigue life estimation model isestablished, which is dedicated to the multiaxial fatigueproblem of welded joints. A critical plane method, incombination with a proper structural stress definition, isintegrated into the new model. The objective is to adaptthe fatigue analysis model to being more comprehensive,allowing accessing the true stress state and fatigue damageof welded joints under complex loading conditions. It isknown that the critical plane method is principally justi-fied for ductile and semi-ductile materials, in whichshorter fatigue lives are generally observed in case ofout-of-phase stress components. So the application ofthe developed model is mainly limited to these materials.

This paper is split into four parts. First, a pertinentstructural stress definition is introduced. Second, theadopted critical plane method is presented, and a new fa-tigue life estimation model dedicated to multiaxial fatigueis established. Then the new model is verified in the life-time estimation for a specific type of steel welded joint, incomparison with the experimental results. Finally, con-cluding remarks are drawn.

STRUCTURAL STRESS DEF IN I T ION

Welding is a very complex process involved with multiplephysical phenomena. It includes heat production, plasticdeformation and phase transformation that occur mani-festly in the vicinity of the heat affect zone. As a conse-quence, high-stress concentration takes place at theweld toe, which frequently is the preferential site for fa-tigue crack initiation. Hence, it is important to evaluateproperly the stress state at the weld toe, which is sensitiveto the geometric configuration of the welded joint. Thus,it necessitates taking into account the specific geometryof the welded joint in the stress analysis. In other words,a pertinent structural stress needs to be defined.

Before introducing the structural stress definitionadopted in the present work, it is necessary to first brieflypresent the HSS method, as the mostly used structuralstress method for fatigue analysis of welded structures.

In the HSS method, the HSS is defined through a lin-ear combination of two reference stresses, which are ob-tained through linear extrapolation to the weld toe ofstresses obtained at certain distances from the weld toe.This approach can be schematically illustrated in Fig. 1,concerning a plate-to-tube welded joint. The normalstress σhs and shear stress τhs at the hot spot are given by

σhs ¼ 1:67�σx 0:4hð Þ � 0:67�σx 1:0hð Þ (1)

τhs ¼ 1:67�τxy 0:4hð Þ � 0:67�τxy 1:0hð Þ; (2)

where h represents the thickness of the tube, σx(0.4h)stands for the normal stress at the 0.4h distance fromthe hot spot and σx(1.0h) the normal stress at the 1.0h dis-tance, τxy(0.4h) stands for the shear stress at the 0.4h dis-tance from the hot spot and τxy(1.0h) the shear stress atthe 1.0h distance. Here, it is worthy to note that the dis-tances 0.4h and 1.0h shown in Fig. 1 do not representtheir true values but are exaggerated ones, which areexpressed schematically for the purpose of a better pre-sentation on the HSS method.

As mentioned formerly, a major drawback of the HSSmethod lies in the fact that the defined stresses (σhs andτhs) are mesh-size sensitive. Thus, the reliability of the es-timation results could be debatable. Unlike the HSS def-inition, in Dong method21 the structural stress isevaluated using the integrated FE stress along the thick-ness direction. It avoids, therefore, the mesh-size sensi-tivity problem.

Taking the butt-welded joint of plates as an exampleas shown in Fig. 2, the LS at the welded joint can be sep-arated into three parts: the membrane stress, σm, thebending stress, σb, and the nonlinear stress, σnpl. The

Fig. 1 Definition of the hot spot stress by the HSS method in aplate-to-tube welded joint.

STRUCTURAL STRESS -BASED CR I T I CAL P LANE METHOD 3


three stress components are all along the thickness direc-tion, as schematically shown in Fig. 2.15 According toDong method,21 the membrane stress σm and bendingstress σb can be solved through the numerical integrationin the thickness direction, and more precisely

σm ¼ 1h∫h

0σx yð Þ�dy

σm�h2

2þ σb�h

2

6¼ ∫

h

0σx yð Þ�y�dyþ λ�∫h

0τxy yð Þ�dy

8>><>>: ; (3)

where σx is the applied normal stress along the x direc-tion, τxy is the applied shear stress on the x–y plane andλ represents the distance between Sections A-A and B-B.

The structural stress method may be based on anequilibrium effect of the stress and bending moment.Thus, the structural normal stress σs is defined as thesum of the membrane stress σm and bending stress σb.This definition is generally applicable to different typesof welded structures that may be modelled using plate el-ements, shell elements or others in FE simulation.21–24

Detailed calculation procedures can be found in Refer-ences [21,23], Here, we shall only present the final de-rived expression of the structural normal stress, whichtakes the form

σs ¼ σm þ σb ¼f y′hþ 6�mx′

h2; (4)

where fy represents the nodal force perpendicular to theweld line and mx′ the bending moment on the x′ axis, asillustrated in Fig. 3.

A stress concentration factor, kσ, is given with respectto the structural normal stress. It is defined as the ratiobetween the structural normal stress σs and the NS σnsolved at weld toe section, that is,

kσ ¼ σsσn

: (5)

Concerning the three-dimensional welded joint type(e.g. flange to tube), the shear stress is also necessary tobe taken into account in the structural stress evaluation.According to Dong,21 the structural shear stress can beobtained through

τs ¼ τm þ τb ¼ f x′h

þ 6�my′

h2; (6)

where τm and τb represent constant and linear distributionof shear stress through plate thickness, respectively, fx′represents the nodal force parallel to the weld line andmy′ the bending moment on the y′ axis, as shown in Fig. 3.

The stress concentration factor, kτ, with respect to thestructural shear stress is defined as the ratio between thestructural shear stress τs and the nominal shear stress τn,that is,

kτ ¼ τsτn: (7)

By employing the structural stress definition in Dongmethod,21 the problem of the mesh-size sensitivity of theHSS method can be solved, and more pertinent parame-ters for welded joint fatigue evaluation can be obtained. Itincludes the structural normal stress, structural shearstress and their corresponding stress concentrationfactors.

Under the uniaxial or torsional loading mode, themagnitude and direction of the normal stress (or shearstress) may alternate with the time, but the axis of theprincipal stress remains fixed. Thus, the stress concentra-tion effect can be invariably well reflected in spite of thestress variations. In contrast, under the multiaxial loadingmode, the normal stress and shear stress coexist, and theaxis of the principal stress may rotate with the time, sothe true stress state at the weld toe could not be wellreflected by the traditional definition of the structuralstress parameters without considering such an effect. Anappropriate adaptation of the structural stress methodfor its application to multiaxial fatigue problem is there-fore expected. The establishment of the new damage

Fig. 2 Nonlinear stress distribution separated to stress componentsin a butt-welded joint of plates, reproduced from Reference. 15

Fig. 3 Structural stress procedures for a shell/plate element adjacentto a weld, reproduced from Reference. 21

4 C. JIANG et al.


parameters and the refined fatigue life estimation modelis detailed in the following section.

MULT IAX IAL FAT IGUE L I FE EST IMAT IONMODEL

Critical plane definition

Multiaxial stress states are common to the in-servicewelded structures applied to engineering, which are fre-quently subjected to alternative loads. The multiaxialstress can be distinguished by a phase angle δ betweenthe normal stress and shear stress (the consideration refersto plate surfaces where the stress state is plane).41–43,47,48

In the multiaxial fatigue analysis, it is commonly acceptedthat the fatigue damage can reach its maximum when thephase angle δ equals to 90°. In other words, the fatiguedamage level increases with the increment of the angleδ, that is, the out-of-phase degree. It is the general philos-ophy of the critical plane method. Concerning the weldedjoints, here it is necessary to define some pertinent dam-age parameters that enable to characterize the multiaxialstress state appropriately. A straightforward way is toadapt a well-developed critical plane method for the ap-plication of the welded joint fatigue analysis.

Nowadays, there exists a considerable variety of criti-cal plane methods, for instance, the Findley critical,McDiarmid critical, Matake critical, Papadopoulos criti-cal, Dang Van critical and Carpinteri–Spagnoli (C-S)critical.43 In the present study, the C-S critical planemethod is adopted. The C-S method is originally pro-posed by Carpinteri and Spagnoli.42 It has been appliedsuccessfully in the fatigue analysis for the multiaxial load-ing mode42,45–48 and random loading mode. Moreover,References [45,46] show that the C-S method isfavourable for the fatigue life estimation of the weldedjoints subjected to multiaxial loading.

The general principle of the critical plane method canbe described as the following steps. First, a generic point Pof the analysis object is determined through the staticsanalysis by FEM. Second, a maximum damage planethrough the point P in a cycle is obtained according to agiven criterion. Finally, the combination of normal stressand shear stress on the maximum damage plane is formu-lated in a special form, serving as the fatigue damage indi-cator. Then, it can be connected with the fatigue life ofthe material or structure to establish a multiaxial lifetimeestimation model. Concerning the C-S method, it adoptsa series of rotation coordinate systems that allow describ-ing the multiaxial stress state in a precise manner. The ex-ecution steps of the C-S method can be summarized later.

First, a cycle period T of stress variation is divided inton times, and the matrixM of the principal stress directioncosines is calculated at each time ti(i = 1, 2, 3, …, n). The

relevant coordinate systems are schematically illustratedin Fig. 4. The 1-2-3 coordinate system of the principalstress at point P can be obtained by rotating the x-y-z co-ordinate system, and the Euler angles ϕ, θ and ψ repre-sent three counterclockwise sequential rotations aroundthe z-axis, y′-axis and 3-axis, respectively, which trans-form the x-y-z coordinate system into the 1-2-3 coordi-nate system. The matrix M can be expressed using theEuler angles ϕ, θ and ψ, as shown in the following:

M ¼l1 l2 l3m1 m2 m3

n1 n2 n3

264375

¼cφcθcψ � sφsψ �cφcθsψ � sφcψ cφsθsφcθcψ þ cφsψ �sφcθsψ þ cφsψ sφsθ

�sθcψ sθsψ cθ

264375;

(8)

where lk, mk and nk (k = 1, 2, 3) are the direction cosinevalues of the principal stress in the x-y-z coordinate sys-tem and cϕ = cosϕ, sϕ = sinϕ.

Furthermore, the Euler angles ϕ, θ and ψ are aver-aged through a weighting function, that is,

ϕ ¼ 1WS

∫T

0 φ tð Þ�W tð Þdt; (9)

θ ¼ 1WS

∫T

0 θ tð Þ�W tð Þdt; (10)

ψ ¼ 1WS

∫T

0 ψ tð Þ�W tð Þdt; (11)

where ϕ, θ and ψ are the weighted averages correspond-ing to ϕ, θ and ψ, respectively, W(t) is the weightingfunction and WS the summation of W(t).

Fig. 4 Principal stress directions 1, 2 and 3 described through theEuler angles ϕ, θ and ψ, reproduced from Reference [45].



The weighting function W(t) at the time ti takes theform

W tið Þ ¼ H σ1 tið Þ � σ1;max� �

; (12)

where σ1(ti) represents the maximum principal stress thatis time dependent and σ1,max stands for the maximumvalue of σ1(ti) during T. H(x) is the Heaviside functionthat can be expressed by

H xð Þ ¼ 1 x ≥ 00 x < 0

�: (13)

The equivalent stress σeq,ϕ, defined in the C-S criticalplane method,42 is given by

σeq;ϕ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ2n;max þ

σafτaf

� �2

τ2ns;ϕ

s; (14)

where σn,max represents the maximum normal stress onthe critical plane and τns,ϕ the shear stress acting on thecritical plane. σaf stands for the fatigue limit of the mate-rial determined under the uniaxial loading mode and τafthe one under the torsional loading mode, and ϕ is theangle between the axisb1 of the weighted average principalstress and the vector perpendicular to the C-S criticalplane.

From the earlier equation, it can be noted that the C-Smethod is an approach that adopts a nonlinear combina-tion of the normal stress and shear stress on the criticalplane. The formulated equivalent stress enables to pro-vide an adequate measurement of the fatigue damage un-der the multiaxial loading conditions, as evidenced byextensive studies, for instance, in References [42,45–48],It is expected to be integrated into a fatigue life estima-tion model, in conjunction with the formerly discussedstructural stress definition. This issue is addressed in thefollowing section.

Lifetime estimation model

A refined lifetime estimation model adapted for weldedstructures is built with explicit considerations of thestructural stress effect and multiaxial stress state. Here,we present our strategy to formulate such a new modelin the following.

According to Reference [15], the stress range is themost influential contributing factor for the fatigue dam-age of welded joints, which is considered more impor-tant than other parameters like the maximum stress.Thus, the original C-S method is suggested to be mod-ified as follows. The range of normal stress ΔσÂ over

the critical plane Â, as defined by the C-S method canbe obtained through

ΔσA ¼ σA tið Þ � σA tkð Þ�� ; (15)

where ti and tk are the corresponding times when therange of shear stress achieves its maximum value ΔτÂ,max during a period T.

The maximum range of shear stress ΔτÂ,max is calcu-lated by

ΔτA;max ¼ max0<ti<tk<T

τA tið Þ � τA tkð Þ�� : (16)

The stress parameters ΔσÂ and ΔτÂ,max constitute asthe basic elements for building an adequate equivalentstress that pertains to fatigue damage. These parametersare dependent on the loading history of the dangerouspoint at the weld toe. Hence, the real stress state of thewelded joint can be more correctly reflected.

A general relationship between a predefined equiva-lent stress range Δσeq and fatigue life N in the high cyclefatigue domain could be described through

N ¼ CΔσkeq

; (17)

where the parameter C is material constant and k thenegative inverse slope of the S–N curve.

In this work, the conventional Wöhler curve (Δσ-N orΔτ-N curve) for a given structure is first estimated in arelatively simplified way by the HSS method followingthe IIW recommendations.15 It serves as a reference.Then the modified Wöhler curve method44,49 isemployed in order to provide an improved Wöhler curveof the structure based on the estimated reference values,which should be able to characterize the internal stressstate and fatigue damage more approaching to the realones.First, a multiaxiality factor ρw is necessary to be cal-culated, as defined in the following49:

ρw ¼ ΔσAΔτA ;max

: (18)

Then, the reference shear stress range ΔτRef, underthe stress ratio ρw, can be evaluated through

ΔτRef ρwð Þ ¼ ΔσIIW2

� ΔτIIW

� �ρw þ ΔτIIW ; (19)

where ΔσIIW represents the reference fatigue limit, thatis, the fatigue class (FAT) value, under the uniaxial load-ing mode and ΔτIIW that is under the torsional loadingmode. The values of ΔσIIW and ΔτIIW adopted in thepresent study correspond to the survival probability of

6 C. JIANG et al.


50%. They were obtained based on the standard FATvalues with survival probability 97.7% given in Reference[15] via the transformation using the Gaussian log-normal distribution. Their corresponding fatigue lifeequals to 2 × 106 cycles,15 that is, NRef = 2 × 106.

The negative inverse slope of the modified Wöhlercurve, Kτ, under the multiaxiality factor, ρw, can be ob-tained as49

K τ ρwð Þ ¼ k1 ρw ¼ 1ð Þ � k0 ρw ¼ 0ð Þ½ �ρwþk0 ρw ¼ 0ð Þ;

(20)

where k0 represents the negative inverse slope of the Δσ-Ncurve under the torsional loading mode and k1 that ofthe Δτ-N curve under the uniaxial loading mode.

As elucidated in the aforementioned content, thestress range parameters ΔτÂ,max and ΔσÂ can be obtainedthrough the adapted C-S critical plane method, and thestress concentration factors kσ and kτ can be estimatedusing the structural stress method. Then a new equiva-lent stress ΔτÂ,eq can be defined by

ΔτA;eq ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikσ �ΔσA� �2 þ ΔσIIW

ΔτIIW

� �2

� kτ�ΔτA ;max

2s

: (21)

The proposed equivalent stress ΔτÂ,eq considers boththe stress concentration effect and time-dependent mul-tiaxial stress states. It is, therefore, applicable for the mul-tiaxial fatigue analysis of welded structures.

Finally, a new multiaxial fatigue life estimation modelcan be derived:

Nf ¼ ΔτRefΔτA ; eq

" #K τ ρwð Þ�NRef : (22)

It can be noted that in the new fatigue criterion, theexponent component is the parameter Kτ(ρw) that ismultiaxiality factor (ΔσÂ to ΔτÂ,max) dependent, and thefatigue life evaluation is based on both the parameterKτ(ρw) and the equivalent stress ΔτÂ,eq. The performanceof the proposed lifetime estimation model is verifiedthrough the available data of a large number of fatiguetests and is compared with the traditional HSS method.For the sake of convenience, the developed method in thiswork is named as SSCP (structural stress-based criticalplane) method for short.

EXPER IMENTAL VER I F ICAT ION ANDDISCUSS ION

The experimental data of substantive fatigue tests ap-plied to welded structures are available in References[49,50] The fatigue testing data are utilized in this studyfor the verification of the proposed fatigue life estima-tion model.

The testing object of the selected database in49,50 wasa flange-to-tube welded structure made of StE460 steelalloy. The geometrical dimensions of the welded struc-ture are illustrated in Fig. 5. The length–thickness ratioof the welded joint equals to 24 (= 240mm / 10mm); thus,the shell element was employed in the FE analysis. Thestatics analysis of the welded structure was implementedby Patran/Nastran commercial software. The FE modelis illustrated in Fig. 6 with 3129 four-node shell elementsand 3351 nodes. The coarse meshes were adopted be-cause the defined structural stress was mesh-size insensi-tive.21 The constraints were applied on the whole flange

Fig. 5 Geometrical dimensions of the investigated flange-to-tubewelded structure, reproduced from Reference [50] (unit: mm).

Fig. 6 Finite element model of the investigated welded structure.



surface, and the loads (bending and/or torsion) were in-troduced on the free end of the tube, as shown in Fig. 6.

First, the experimental data of eight representative fa-tigue tests with different parameters under the multiaxialloading mode are analysed as a preliminary step. Theyare primarily used for the determination of stress concen-tration factors. The loading conditions of the eightselected fatigue tests include two in-phase loading withδ = 0° and σmean = 0, two in-phase loading with δ = 0°and σmean≠0, two out-of-phase loading with δ =90° andσmean = 0 and two out-of-phase loading with δ =90° andσmean≠0.

For each fatigue test, it involves 13 times with avail-able stress state records in a cycle period T. This infor-mation can be used for the calculation of the nodalforces and bending moments. And then, according toEqs 3–7, the stress concentration factors kσ and kτ canbe obtained for each fatigue test, in which the NS wascalculated referring to the tube thickness of 10mm.The obtained parameters kσ and kτ, together with othernecessary loading information, for the eight concernedfatigue tests are reported in Table 1.

In principle, once the geometrical dimensions of thewelded joint are determined, the stress concentration fac-tors kσ and kτ are fixed accordingly. In order to verify thestability of the estimated parameters kσ and kτ under differ-ent loading conditions, their mean values for the eight se-lected fatigue tests in Table 1 were calculated. Theresults show that the upper bound of the deviation canreach 7.9% for kσ and 3.15% for kτ. This divergence levelis considered acceptable for the fatigue analysis of thewelded structures. In this case, the averaged kσ and kτ areadopted for the subsequent lifetime estimation analysis.

There are 62 fatigue tests in total in the adopted data-base, which are used for the fatigue life estimation in thisanalysis. The loading conditions of the 62 fatigue tests in-clude 25 uniaxial bending loading, 7 torsional loading, 14in-phase multiaxial loading and 16 out-of-phase multiax-ial loading. The experimental data of all the involved fa-tigue tests, including the loading conditions and fatiguelife results, are listed in Table 2 (the reference numbersof the eight selected fatigue tests in Table 1 are

underlined). The failure criterion with respect to the life-time determination was the crack growth through thewall of the tube in the tested specimens.50 In Table 2,σa stands for the normal stress amplitude and τa the shearstress amplitude, σmean represents the mean normal stressand τmean the mean shear stress and Nf,ex represents theexperimental number of cycles to failure.

First, the lifetime estimation was carried out by usingthe HSS method. The adopted HSS method in the pres-ent work was applicable to the multiaxial fatigue analysis,in which the critical plane was defined as the planeexperiencing the maximum shear stress amplitude. Andthe lifetime estimation was conducted using the modifiedWöhler curve method in terms of the HSS according toSusmel49 and Susmel and Tovo.51 The estimated resultsNf,es by the HSS method, in comparison with the experi-mental results Nf,ex, are illustrated in Fig. 7. Second, thedeveloped SSCP method in this work was utilized forthe lifetime estimation, and the obtained results werecompared as well with the experimental data, as shownin Fig. 8. In Figs 7 and 8, the scatter bands provided bythe uniaxial and torsional fatigue calibration tests are alsoplotted. The scatter band encircled by the continuouslines corresponds to the uniaxial loading condition andthat by the dashed lines corresponds to the torsional load-ing condition. These scatter bands refer to the survivalprobability ranging from 2.3 to 97.7%. The plotted lineswith survival probability 97.7% were obtained by calibrat-ing the criterion through standard fatigue curves with sur-vival probability 97.7% (given in IIW15) recalculated for asurvival probability equal to 50%.49 The adopted standardfatigue curves were the FAT 90 for the uniaxial loadingmode and FAT 100 for the torsional loading mode, whichwere considered appropriate for the present welded struc-ture according to Hobbacher.15

The effectiveness of the SSCP method is anticipatedto be validated by referring to the fatigue testing dataand in comparison with the estimated results by theHSS method. The comparisons are discussed as followswith respect to the loading mode; in each case, the tor-sional scatter band is taken as the reference of the lifetimeestimation measure, more precisely:

Table 1 Loading parameters of the eight selected specimens for the determination of stress concentration factors

No. Loading parameters σn (MPa) σs (MPa) kσ τn (MPa) τs (MPa) kτ1 δ = 0°, σmean = 0 156.1 199.2 1.276 156.1 211.9 1.6582 δ = 0°, σmean = 0 113.7 151.8 1.335 113.7 138.6 1.7193 δ = 0°, σmean≠ 0 177.8 251.9 1.417 177.8 234.8 1.7214 δ = 0°, σmean≠ 0 152.5 202.1 1.325 152.5 215.3 1.7125 δ = 90°, σmean = 0 109.2 164.1 1.403 109.2 139.6 1.6786 δ = 90°, σmean = 0 99.5 146.5 1.472 99.5 137.5 1.6827 δ = 90°, σmean≠ 0 155.3 251.2 1.417 155.3 269.6 1.7368 δ = 90°, σmean≠ 0 155.1 222.7 1.436 155.1 212.2 1.768

8 C. JIANG et al.


Table 2 Experimental parameters of the total 62 fatigue tests [49]

No. Loading type σa (MPa) σmean (MPa) τa (MPa) τmean (MPa) δ (Degree) Nf,ex (Cycles)

1 Uniaxial 408.5 0.0 0.0 0 0 52292 Uniaxial 356.1 0.0 0.0 0 0 73903 Uniaxial 278.6 0.0 0.0 0 0 22 6774 Uniaxial 281.1 0.0 0.0 0 0 32 2215 Uniaxial 204.9 0.0 0.0 0 0 86 6316 Uniaxial 199.8 0.0 0.0 0 0 108 6007 Uniaxial 200.2 0.0 0.0 0 0 208 0338 Uniaxial 155.9 0.0 0.0 0 0 261 8339 Uniaxial 150.9 0.0 0.0 0 0 448 75510 Uniaxial 151.5 0.0 0.0 0 0 598 98611 Uniaxial 123.8 0.0 0.0 0 0 390 61412 Uniaxial 120.1 0.0 0.0 0 0 520 73013 Uniaxial 120.5 0.0 0.0 0 0 582 00914 Uniaxial 120.3 0.0 0.0 0 0 1 012 57915 Uniaxial 120.6 0.0 0.0 0 0 1 053 90516 Uniaxial 119.6 0.0 0.0 0 0 1 901 27517 Uniaxial 103.5 0.0 0.0 0 0 1 856 64918 Uniaxial 309.5 309.5 0.0 0 0 745919 Uniaxial 253.9 253.9 0.0 0 0 15 80620 Uniaxial 206.4 206.4 0.0 0 0 26 95621 Uniaxial 127.8 127.8 0.0 0 0 364 09522 Uniaxial 82.3 82.3 0.0 0 0 1 147 06723 Uniaxial 104.2 104.2 0.0 0 0 1 612 41224 Uniaxial 93.8 93.8 0.0 0 0 2 077 03325 Uniaxial 72.4 72.4 0.0 0 0 1 381 35126 Torsional 0 0 198.5 0.0 0 15 03127 Torsional 0 0 199.2 0.0 0 21 19728 Torsional 0 0 160.4 0.0 0 196 40829 Torsional 0 0 151.4 0.0 0 202 64330 Torsional 0 0 151.4 0.0 0 225 92531 Torsional 0 0 120.4 0.0 0 730 69432 Torsional 0 0 120.2 0.0 0 816 68133 In-phase 156.1 0.0 156.1 0.0 0 23 33934 In-phase 113.7 0.0 113.7 0.0 0 23 42635 In-phase 135.1 0.0 135.1 0.0 0 48 19036 In-phase 103.5 0.0 103.5 0.0 0 125 54737 In-phase 93.9 0.0 93.9 0.0 0 332 44238 In-phase 83.6 0.0 83.6 0.0 0 479 49339 In-phase 72.6 0.0 72.6 0.0 0 1 385 77940 In-phase 177.8 177.8 177.8 177.8 0 15 37841 In-phase 152.5 152.5 152.5 152.5 0 40 28042 In-phase 126.9 126.9 126.9 126.9 0 79 89043 In-phase 103.2 103.2 103.2 103.2 0 108 58144 In-phase 80.9 80.9 80.9 80.9 0 389 46345 In-phase 91.9 91.9 91.9 91.9 0 572 60546 In-phase 71.2 71.2 71.2 71.2 0 641 43247 Out-of-phase 109.2 0.0 109.2 0.0 90 43 38648 Out-of-phase 99.5 0.0 99.5 0.0 90 123 36949 Out-of-phase 109.2 0.0 109.2 0.0 90 178 16250 Out-of-phase 99.8 0.0 99.8 0.0 90 238 99751 Out-of-phase 69.2 0.0 69.2 0.0 90 143 31952 Out-of-phase 69.5 0.0 69.5 0.0 90 393 06453 Out-of-phase 69.4 0.0 69.4 0.0 90 921 96154 Out-of-phase 80.0 0.0 80.0 0.0 90 961 99155 Out-of-phase 69.2 0.0 69.2 0.0 90 2 149 05556 Out-of-phase 155.3 155.3 155.3 155.3 90 23 23857 Out-of-phase 155.1 155.1 155.1 155.1 90 41 14458 Out-of-phase 133.9 133.9 133.9 133.9 90 47 78759 Out-of-phase 114.5 114.5 114.5 114.5 90 23 180

(Continues)



(1) Under the uniaxial loading mode, the estimated re-sults by both HSS and SSCP methods satisfy the97.7% confidence interval, but the scattering levelof the prediction results provided by the SSCPmethod is lower than that given by the HSS method.

(2) Under the torsional loading mode, the estimated re-sults of both HSS and SSCP methods satisfy the97.7% confidence interval as under the uniaxial

loading mode. Nevertheless, the scattering level ofthe results predicted by the SSCP method is visiblylower than that given by the HSS method.

(3) Under the in-phase multiaxial loading mode, the es-timated results of five fatigue tests in a total numberof 14 by the HSS method are beyond the 97.7% con-fidence interval, but for the SSCP method, the esti-mated result in only one fatigue test is beyond thesame confidence interval. It demonstrates that theSSCP method allows proving a more reliable estima-tion under the in-phase multiaxial load.

(4) Under the out-of-phase multiaxial loading mode, theestimated results of 14 fatigue tests in a total numberof 16 by the HSS method are beyond the 97.7% con-fidence interval. Under identical conditions, the esti-mated results in only eight fatigue tests by the SSCPmethod are beyond the same confidence interval. It iseven more remarkable that the scattering level of theresults, estimated by the SSCP method, shows dis-tinctly lower than that obtained by the HSS method.In addition, it can also be noted that the estimated re-sults by the SSCP method fall mostly on the conser-vative side, exhibiting the same favourable featurewith the HSS method.

In conclusion, the SSCP method outperforms theHSS method in terms of lifetime estimation precision,which is rather distinct for the multiaxial loading mode.Moreover, the reliability of the SSCP method is well val-idated through a large number of fatigue tests under dif-ferent loading conditions, including uniaxial, torsionaland in-phase and out-of-phase multiaxial loading modes.It is also remarked that the estimated results by the SSCPmethod exhibit conservative features, which fall mostly inthe survival probability range between 97.7 and 50%.This effect could be attributed to the fact that the newdamage parameter proposed in the present paper repre-sents a more comprehensive one that allows the consider-ation of both the geometry effect and multiaxial stressstate in welded joints. Consequently, it may lead to moreconservative estimations with shorter evaluated fatiguelives than those given by one approach with less consider-ation, such as the NS method. From a practical point ofview, the conservative feature of the SSCP method ispreferable for the fatigue design of welded structures inengineering applications.

Table 2. (Continued)

No. Loading type σa (MPa) σmean (MPa) τa (MPa) τmean (MPa) δ (Degree) Nf,ex (Cycles)

60 Out-of-phase 103.6 103.6 103.6 103.6 90 123 31361 Out-of-phase 93.5 93.5 93.5 93.5 90 330 33962 Out-of-phase 82.0 82.0 82.0 82.0 90 473 713

The underlined reference numbers correspond to the eight selected fatigue tests in Table 1.

Fig. 7 Comparison between the estimated results by employing thehot spot stress method and experimental data.

Fig. 8 Comparison between the estimated results by employing thestructural stress-based critical plane method and experimental data.

10 C. JIANG et al.


CONCLUS IONS

An original fatigue life estimation method, so-calledSSCP method, is developed in this paper. The SSCPmethod is well adapted for the fatigue evaluation ofwelded joints. Its unique advantages are mainly embodiedin the following aspects.

First, the stress concentration effect at the weld toe canbe evaluated in a proper manner by introducing the struc-tural stress definition in Dong method. It is improvedcomparing with the HSS method in the sense that the de-fined structural stress is mesh-size insensitive. Second, byadopting the C-S critical plane method, the influence ofthe loading path on fatigue damage accumulation is alsotaken into account in the new lifetime estimation model.Thus, the refined fatigue model allows treating the multi-axial loading problems.

The performance of the SSCP method is verified inthe fatigue life estimation of a flange-to-tube weldedstructure made of StE460 steel alloy. The estimated re-sults are compared with both the experimental fatiguetesting data and estimations given by the HSS method.The analysis shows that the SSCPmethod enables to pro-vide reliable and better estimations than the HSS methodunder different loading conditions. In particular, the esti-mated results are satisfactory in the multiaxial loadingmode, including both the in-phase and out-of-phasecases. In a perspective study, the SSCP method is antici-pated to be further developed for the application of fa-tigue estimation under more complicated and realisticloading conditions, for instance, the random loads.

Acknowledgements

This work is supported by the National Science Foundationfor Excellent Young Scholars (51222502), the NationalScience Foundation of China (11402297), the Fund for Dis-tinguished Young Scientists of Hunan Province (14JJ1016)and the National Excellent Doctoral Dissertation SpecialFund (201235).

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