Fatigue reliability analysis of kiln welded shell

Fatigue reliability analysis of kiln welded shellLi Xuejun' , Shen Yiping'.2 , Dhillon B. S. 2( 1. Hunan Provincial Key Laboratory of Health Maintenance for Mechanical Equipment,Hunan University of Science and Technology，Xiangtan, Hunan 411201，China; .2. Department of Mechanical Engineering, University of Ottawa, Ontario, Ottawa K1N6N5, Canada )Abstract: This paper presents a probabilistic reliability method for the welded shell during crack growth. The crackgrowth model incorporated with a failure assessment diagran( FAD )，which can provides a better estimation of the criti-cal crack length, is developed to describe fatigue failure. All variables for particular welded joints of the shell are stud-ied. Among them, the stress variables are based on the calculated stress by using the finite element ( FE) code AN-SYS. Fatigue reliability analysis of the welded shell is performed by using the Monte Carlo simulation method. The fail-ure probability curve of the example kiln is significantly useful to determine the repair schedule of shell cracks.Key words: fatigue reliability ; welded shell ;crack growth; FAD; Monte Carlo; ANSYSbecause of various stiffness ，different thicknesses along1 Introductionaxis direction, and complex loading condition. Hence,Rotary kiln shell is a large-scale welded structurethe FE code ANSYS is applied to perform stress analy-with length over 100 m and radius around 2 m, andsis. The analytical stress results together with otherproduced by welding thin cylindrical steel plate one byvariables are taken into consideration for fatigue relia-one. Padded plates are directly soldered to the shell inbility analysis. Consequently, the Monte Carlo simula-the supporting rollers places to reduce their concentrat-tion method is used to calculate shell fatigue reliabilityed stress. Cracks are often initiated at these weldedwith respect to service time. Taking a rotary kiln as anjoints，and the overlong circumferential cracks are pre-example，the fatigue failure probability curve of thevailing at weld joints near the supporting rollerswelded shell is obtained. The results are significantlyCracks can grow with complex overloading conditionsuseful to determine when and which cracks need to re-for over thousands of tons，and then result in prematurepair.shell failure. The effecting conditions for crack growth2 Fatigue cracks growth reliability analysisinclude material characteristics, initial crack size, andservice stresses. In general, all these conditions are2.1 Crack growth modelrandom. Therefore， failure probability of the weldedBased on the Paris linear crack growth relation-hell during crack growth needs to be quantified byship, the fatigue crack growth rate can be expressedusing probabilistic fracture mechanics.e dNAlthough fatigue analysis for the welded shell haveby!11]been performed 14，the use of probabilistic methodda= C( AK)"( 1)dNhas not been considered for the crack growth analysis ofthe welded shell. More recently， the probabilisticwhere, a is the crack length; N is the applied numbermethod of fatigue analysis based on fracture mechanicsof cycles; C and n are material constants and definedcrack growth model associated with failure criteria isexperimentally for a particular material, and AK is theprevailing for the welded structure' 5-10. Similarly , thisstress intensity factor range， which varies from thepaper uses Paris Law to describe the crack growth in-threshold stress intensity range ▲Kh to the fracturecorporated with a failure assessment diagram( FAD ) tostress intensity range △K。. Recent studies have reprovide a better estimation of the critical crack length.vealed that Eq.( 1 ) overestimates the crack growth inThis study also attempts to provide a better estimationthe vicinity of△K and underestimates it in the vicinityof all variables in the probabilistic model. However,of AK。. The simplification Eq.( 1 ) is tolerable forthe circumferential stress calculation is very diffcultpractical中国煤化工tire ranges.fYHCNMHGReceived 16 October 200952 Engineering SciencesThe stress intensity factor range AK can be ex-through Eq.(4 ) and Eq. (5 ), but it is not concernedpressed byabout crack growth 14. The method developed in thisOK = M,σ√πa(2)paper is based on the Paris Law crack growth modelwhere, Oσ is the applied stress range; M: is theand incorporated with the fracture event through anFAD.stress magnification factor. M is given as 1212.3Fatigue reliability analysisM。= ka-;(k。-1) .(3)Using Eq.(1 )to Eq.(3)，we get= C( YM,Aσ√πa)°(9)where, the relative crack length is"(≤0.2),k。isWith the applied stress, Eq. ( 9 ) solution yieldsthe concentrated stress coefficient in weld toes.In some references, fatigue reliability has beenthe average number of cycles N required to propagate adefined according to whether the crack has reached acrack from its initial value A, to failure value a; .critical lengthB. In fact, the Paris model is based onN,=1°_dx.(10)elasticity theory and omits plasticity inside the crack.CJonσ°( YM% Vπx)°Such assessment can provide optimistic analysis com-Where, ap can be calculated according to the fracturepared with the critical length determined by using frac-mechanics critical function Eq.( 8 ). The average num-ture mechanics criteria.ber of cycles N; can be changed into time( years ) by2.2 Critical crack length determinationdefining a variabletp =一Nyer is the annual num-Cracks in practical structures， given suffcientgrowth, will reach a certain limited length and thenber of operation cycles and is a deterministic variable.will cease to grow. Taking the crack termination intoConsequently, the failure probability at time t can beaccount the fracture possibility，numbers of researchersexpressed ashave incorporated fracture criteria with the_ crackP; = Pt≤t)=| f,(t)dt ( 11)growth model, such as those specified by R6' andBS 79106-10. In this paper， the BS 7910 Level 2AWhere,f( t) is the probability density function oftp.FAD is used to determine failure through fracture andThe integral for Eq.( 11 ) can be evaluated by usingplastic collapse, and expressed by 10.Monte Carlo simulations within Matlab.K. =(1 -0.14L2 )X0.3 +0.7e-0.6512 )(4)3 Study of variablesL,≤L,This section presents the variables used in fatigueK,=0 L, > Lx( 5)reliability analysis.where K, is the fracture ratio, a measure of the proximi-3.1 Stress variablesty to elastic fracture; L, is the load ratio, a measure ofIn industrial applications ，large-scale circumfer-the proximity to plastic collapse. The quantities K, andential cracks usually present in the welded joints nearL, are defined as followssupporting rollers due to the overlarge load distributedK,=A≤1(6)on each supporting structure. The load distribution be-tween supporting shelves is linearly related with axisL,=σ≤Lpx(7)line deflection and has been demonstrated inRef.[ 15 ]. However， the required circumferentialwhere K is the stress intensity factor; K。is the fracturestresses of shell， primary factor for crack growth, aretoughness; σ is the applied stress( function of the cracksill unavailable. In this study, the FE code ANSYS islength ), and σ is plastic yield stress.used to computer simulation stress.The FAD determines failure in terms of both brit-3.1.1 ANSYS modeling .tle fracture as well as ductile collapse. Thus, the stressANSYS is a finite element system developed byintensity factor should not exceed the fracture tough-ANSYS，Inc. ，Canonsburgh, PA，USA. The weldness, and the applied load should not exceed the plas-shell of rotary kiln under consideration is 4 m xtic yield stress. Furthermore, the failure crack length100 m, i.e. , the length is 100 m and the shell diame-a, corresponds to a point that relies on this criticalter is 4 m. The total weight of rotary kiln is aboutfunction.950x10* N and there are. 5 shelves located at dis-g(L,,K,) =(1 -0.14L2 )tance of中国煤化工;and 89. 1 m from(0.3 +0. 7e-0.654?)-K, =0 (8)kiln head.:fYHC N M H Gs with diferent lo-The failure probability can be directly exploredcations. Firstly, eight-node shell elements， shell 93 ,Vol. 8 No.4,Dee. 2010 53is used for modeling within ANSYS pre-processor.Symmetry condition is exploited so that only one quar-ter of weld shell is meshed.The second stage is load application including dis-placements and forces application. The shell sits loose-ly in 5 rollers. Rollers are supported by two wheels lo-1219 0.803E+07 0.161E:+08 0.241E:+08 0321E+08cated at a 30。angle from vertical line on each side.0.402E+07 0.121E+08 0.201E+08 0.281E+08Therefore, the radial displacement of these nodes inFig. 2 Shell resulting stress including deformationcontact with rollers is constrained. The axial displace-ment of these nodes at the second shelf is also con-strained because downward axial movement is restrictedthe closer the shell to the supporting rollers is, theby the catch on wheel. In regard to the field complexgreater the circumferential stress is， and the middleforces, except the concentrated load introduced by ap-section stress between two shelves becomes zero. Theparatus like driving gear and uniform load by liner,analytical results verify that overlong circumferential .and material， inconvenient concentration moment andcracks always present in the vicinity of supporting roll-non-uniformed loads are applied to the shell. Impera-ers. The fatigue reliability of the integral welded shellive transformation is conducted, and the transformedlies on cracks growth in the vicinity of supporting roll-ers, and the obtained stresses are shown in Fig.3.concentration and uniform loads are shown in Fig. 1.The uniform loads comply with cosine-shaped distribu-(x10*) .(x10*)tion across the shell 116- can be obtained by Eq.( 12 ).17480.H-cosa)6190π R。m4( 12)|H=-(_B+cos)2 sinβWhere, Q is vertical load per length along axialdirection; a is initial contact angle; R。is the mean ofx10*(a) The first shelf(x10(b) The second shelfinner and outer roller radius; H is a constant relatedwith contact angle β( π - a). For the loads applica-如啡tion, the ANSYS command language( ANSYS Paramet-ric Design Language APDL) is used to translate loads4020044nto formats that can be recognized by external pro-grams in pre-defined interfaces.IsT'"" 四”(0) The third shelf(d) The fourth shelfx101)山0 102030405060708090100(a) Transformned concentrated load distribution! 400包205515525335555DIsT。600(<) The fifth shelf目400200Fig. 3 Circumferential stresses across shell located950556065707580859) 95 100on supporting shelves(b) Transformned uniform load distributionFig.1 Transformed concentratedThe equivalent stress of circumferential stressand uniform load distributionσui applied on shell can be obtained from Fig. 3 byusing the following equation3.1.2 Stress result.(13)After programming, the von Mises stress contours中国煤化工Servilequivalent stressof the whole shell elements associated with the de-formed shape is shown in Fig. 2. It can be seen that:MYHC N M H Gicinity of 5 shelvesare calculated, as shown in Table 1. Since supporting54 Engineering Sciencesloads dynamically vary along the axis line deflection4.1 Monte Carlo simulation methodcaused by asymmetry abrasion and expansion, all fiveMonte Carlo simulation is based on statistical sam-stresses are assumed to be normally distributed with apling and used to assess reliability and availability. Athigh COV ( coffcient of variation ) of 0.3.any timet，the reliability of the weld shell can be cal-3.2 Other variables discussionculated by performing sufficient simulation cycles N .Table 1 presents the characteristics of all usedFor each simulating cycle, we perform following steps.variables. All variables except the applied stress for1 )Generate random numbers for given distribu-shell cracks in the vicinity of supporting rollers, are as-tions of all probabilistic variables.sumed to be the same.2 ) Determine the performance function of failureTable 1 Variables used from cracks nearcrack length a; according to critical failure functionsupporting rollersbased on FAD.VariableDiaributionMeanCOV3 ) Evaluate the probability performance function.CLognormal9.59x 10-120.64 )Count the failure numbers M to calculate theprobability of failure by usingPr = M/N.Determinate2.75In order to study failure probability during a longNormal1.1continuous service time, the corresponding program isKr /MPa96. 8performed within Matlab， and the flow diagram isσ/MPa230.1shown in Fig. 4.ao /mm0.4Constant: n N..NyearN._=..+Nwσ /MPa16.925.3Monte Carlo simulationσ2 /MPa23. 561Generate random mumbers ofC.k。σ3 /MPa23.597).3K。a,风and q, repetivelyσ4 /MPa15.573Compute a and then N13. 4900.N