Mean stress effects in fatigue of welded steel joints

Mean stress effects in fatigue of welded steel jointsDavid P.Kihl a ,Shahram Sarkani b,*aCarderock Division,Naval Surface Warfare Center,9500MacArthur Boulevard,West Bethesda,MD 20817-5700,USA bDepartment of Civil,Mechanical and Environmental Engineering,George Washington University,Washington,DC 20052,USAAbstractMean stress effects in steel weldments were examined under both constant and random narrowband amplitude fatigue loadings.The purpose of these tests was to provide experimental data with which to substantiate the use of analytical expressions to account for mean stress effects.Fatigue tests were performed under both tensile and compressive mean stress levels.Test results indicate agreement with the modi®ed Goodman equation to be favorable in accounting for the effect of tensile mean stresses on fatigue life.However,test results from high fatigue loadings (maximum stresses nominally above half ultimate)were found to possess better agreement with the Gerber formulation than with the modi®ed Goodman one.Behavior under compressive mean stresses indicated a linear correction relationship was required,which was less conservative than any of the relationships considered.Test results obtained under random amplitude fatigue loadings exhibited trends similar to those observed under constant amplitude loadings.This ®nding,along with supporting analysis,indicates that the same correction relationship can be used in the same manner for both constant amplitude and random (narrowband)amplitude loadings.q 1998Elsevier Science Ltd.All rights reserved.Keywords:Mean stress effects on fatigue;Fatigue loadings of welded joints;Constant and random amplitude fatigue;Modi®ed Goodman correction;Gerber correction;Soderberg correction1.IntroductionMany factors in¯uence the fatigue behavior of welded steel structures.While some are associated with the physical characteristics of the test specimen itself (geometry,surface condition,imperfections,residual stress),others are asso-ciated with the loading.Service loadings not only vary in type (axial,bending,etc.),but also in content (magnitude and frequency),whether deterministic or stochastic.The loading magnitude,however,may not always be referenced to a zero state of stress.This mean stress can manifest itself directly through the loading,i.e.applied mean stress,or indirectly through the physical characteristics of the test specimen or structure,i.e.residual stress.A widely-accepted observation is that fatigue strength decreases as tensile mean stress increases.In other words,to attain the same endurance under cyclic loads,the applied stress range must decrease as the applied tensile mean stress increases.Although the existence of mean stress effects is acknowledged,these effects are often ignored.Although stress range is typically considered the dominant stress contribution to fatigue damage and mean stress is often considered a secondary contributor,proper quanti®cationof these mean stress effects seems to remain a de®ciency in the area of structural fatigue assessment of welded steel structures.Admittedly,mean level effects can be dif®cult to quan-tify.Mean stresses may interact with stresses at or near the endurance limit or near the yield strength of the material in a way that is different from the way they interact with stresses in the linear (on a log±log plot)portion of the S /N curve.The way mean stresses manifest themselves can also be dif®cult to quantify.Mean stress can be applied,as when a stress offset is superimposed on an external loading.Exam-ples of this condition include stresses resulting from initial static loadings,thermal changes,or loading changes during operation.However,once quanti®ed,mean stresses asso-ciated with external loadings can often be assumed to remain constant during the period of evaluation.Mean stres-ses can also be residual,as when a stress arises from uneven cooling after welding or local yielding.Residual stresses may actually relieve themselves by gradually dissipating under sustained cyclic loads.The objective of this paper is to quantify,experimentally,the effects of mean level loadings which are superimposed on externally applied constant and random amplitude load-ings.Test specimens used are welded steel cruciforms.Results indicate that fatigue lives are reduced under tensile mean stress for both constant and random amplitudeProbabilistic Engineering Mechanics 14(1999)97±1040266-8920/99/$-see front matter q 1998Elsevier Science Ltd.All rights reserved.PII:S0266-8920(98)00019-8*Corresponding author.Tel.:11-202-994-6749;e-mail:sarkani@.loadings.Under compressive mean stress,fatigue lives are increased.2.BackgroundExperimental fatigue investigations date back nearly a hundred years.Early on,mean stress effects were recog-nised to have a potentially signi®cant effect on fatigue strength of structural members.These early efforts[1]led to analytical expressions for relating results of fatigue tests of plain(unwelded)specimens performed with a mean stress level to those performed without mean stress.These methods are referred to as mean stress correction equations. Such equations attempt to establish an`equivalent'zero mean stress cycle,S equ R 21,which would have caused the same amount of fatigue damage(cycles to failure)as the actual applied stress cycle,S ampl,superimposed on the nonzero mean stress.The R term in S equ R 21is the ratio of minimum applied stress to maximum applied stress,e.g.R 21indicates a fully reversed stress cycle with a zero mean stress.Typically,the tensile yield strength or tensile ulti-mate strength is used to normalize the mean stress which, together with the ratio of applied stress amplitude to equiva-lent R 21stress amplitude,can be expressed in a nondi-mensional form.The form of the equation is typically that of an interaction curve as shown below[2]:s ampl s equ R 211s means ult;yld23n1(1)When the exponent n equals one and the mean stress is normalized by the ultimate tensile strength,the expression is known as the modi®ed Goodman correction.When the exponent is two,the expression becomes the Gerber correc-tion.Similarly,when the mean stress is normalised by the yield strength and the exponent is one,the expression is known as the Soderberg correction.It should be noted that other mean stress correction models[3,4]exist but they rely heavily on parameters associated with localized strain beha-vior.Such models do not generally apply well to weldments, especially specimens having complicated geometry or uncertain material properties,i.e.at the heat-affected zone. Regardless of which correction equation is used,a zero-mean stress amplitude can be calculated to be equivalent to any combination of applied stress amplitude and mean stress (i.e.to result in the same number of cycles to failure).Mean stress corrections,if employed at all,are typically made for mean stress levels that are tensile.Increased fatigue strength associated with compressive mean stresses could be accounted for by using the modi®ed Goodman or Soderberg corrections,but little data are available with which to substantiate their use.The Gerber correction,because of the squared term containing the mean stress,would correct in the same manner for both tensile and compressive mean stress effects.There seems to be a lack of consistency as to which mean stress correction formulation,if any,would provide an accu-rate account of mean stress effects.Other uncertainties are associated with how to account for compressive mean stress effects,and whether mean stress effects quanti®ed under constant amplitude loadings apply to stochastic loadings. This experimental investigation was undertaken in an attempt to improve the general lack of experimental data (especially weldments)with which to support the use of a given mean stress correction relationship.3.Experimental investigationIn order to establish a data base from which mean stress effects could be quanti®ed,an experimental investigation was undertaken.The fatigue tests were conducted on welded steel cruciform-shaped specimens made of high strength low alloy(HSLA-80)steel.The specimens were 14in long,3.75in wide,and7/16in thick.Side attachments welded to both sides of the axially loaded member extended out2in.Although these side attachments were welded with full penetration®llet welds,they were not loaded.The®llet welds were deposited using a twin arc gas metal arc weld (GMAW)®xture with100s wire.A representative cruci-form specimen geometry is shown in Fig.1.The long continuous member of the test specimens was axially loaded by means of hydraulic grips.The fatigue tests were run in servo-hydraulic load machines in load control mode until either the specimen failed or the test was suspended.Under the concentrated stress,fatigue cracks generally started at the center of the weld toe adjacent to the loaded member and grew out toward the edges and through the thickness.Failure was de®ned when the compliance of the specimen at least doubled from its initial value.Failure sometimes resulted in the specimen completely separating into two pieces.Test specimens were examined to deter-mine the magnitude and distribution of residual stresses present.After a somewhat extensive investigation,it was concluded that magnitudes of residual stresses were not large enough to in¯uence the outcome of this study.This was not surprising,because high levels of residual stresses would not be expected in small specimens composed of thin plates.Both nonzero mean constant amplitude and random amplitude axial loads were used in this investigation.The mean stress levels were simply superimposed onto zero-mean cyclic waveforms.The random loadings contained peaks and troughs distributed according to a Rayleigh prob-ability distribution,and therefore could be associated with a narrowband response to a Gaussian process.To generate the random amplitude loadings,a single sequence of Rayleigh-distributed values was simulated using a®rst-order autoregressive technique[5].The sequence of10000values had a correlation coef®cient between consecutive values of0.95.A sequence of peaksD.P.Kihl,S.Sarkani/Probabilistic Engineering Mechanics14(1999)97±104 98and troughs was created by multiplying every other value by 21.A continuous waveform was then made by connecting the peaks and troughs by haversine curves.The continuous waveform had a unit root mean square (RMS)value and a zero mean value.For testing purposes,any simulated extrema that were greater than four times the RMS of the process were truncated,or clipped,to a chosen level of four times the RMS.Such clippings occurred no more frequently than once every 3000extreme.The ability to simulate a time history having the desired distribution of peaks,i.e.Rayleigh distribution,was assessed by comparing the ®rst 10moments of the simulated extrema with the theoretical values.The comparison showed the ®rst 10moments to be within 0.3%of the theoretical values before clipping and within 6%after clipping.Loadings associated with a given RMS value were obtained by multiplying the unit RMS loading by the desired load level.Load levels were determined by multi-plying the desired axial stress level by the average cross-sectional area of the continuous member of the specimen.The RMS levels described in this report correspond to a zero-mean process.Any mean level stress considered was superimposed onto the original random loading.4.Data analysisZero-mean and nonzero-mean constant amplitude test results are given in Tables 1and 2.Similarly,zero-meanD.P.Kihl,S.Sarkani /Probabilistic Engineering Mechanics 14(1999)97±10499Table 2Constant amplitude nonzero-mean fatigue test results Stress amplitude (ksi)Mean stress (ksi)Cycles to failure Exp #1Exp #2Exp #3Exp #41010958300132320011182800a 2451800102065070091140020425008864001050425800102680029078001535200151528150025520023660031810015302545001976002574002612003021015880018030017830033690030154240059500387003810030302550021300608002280030602210021300213001580040220135200110300128200113000aIndicates fatigue test was suspended withoutfailure.Fig.1.Typical cruciform-shaped test specimens.Table 1Constant amplitude zero-mean fatigue test results Stress amplitude (ksi)Cycles to failure Exp #1Exp #2Exp #3Exp #4Exp #51012634300290370026195300a 1680560025000000a 12775600373290011186008108001392000155720007795005150002292001071600306650061900706008280079100451450014800163002320018000aIndicates fatigue test was suspended without failure.and nonzero-mean random amplitude test results are provided in Tables 3and 4.Note that most of the zero-mean data were previously presented by the authors [7];they are included here for completeness.Data analysis [6]began with the construction of a constant amplitude S /N curve from the zero-mean tests.This was done by standard linear regression analyses.However,only the data at stress levels of 45,30,15and 12ksi were used in this analysis.Data at the 10ksi stress level,which contained runouts,were omitted from the analysis.The constant amplitude S /N curve and the resulting equation are shown in Fig.2.Constant amplitude data from tests run at different mean stress levels were then compared to data represented by theD.P.Kihl,S.Sarkani /Probabilistic Engineering Mechanics 14(1999)97±104100Table 3Random amplitude zero-mean fatigue test results Stress amplitude (ksi)Cycles to failure Exp #1Exp #2Exp #3Exp #44161131002415430050539400a 4407800526850005496200786320042400007.51504200111130011781001216300104880006867009017004630001593600112600128000141200aIndicates fatigue test was suspended without failure.Table 4Random amplitude nonzero-mean fatigue test results Stress amplitude (ksi)Mean stress (ksi)Cycles to failure Exp #1Exp #2Exp #3Exp #4525313140035742100a 1676340050016900a 1021567125004191400232680016798001020263100183100245200217500104072500161700123100301700aIndicates fatigue test was suspended withoutfailure.Fig.2.Constant amplitude S -N curve (fully reversed,R 21).zero-mean S/N curve.At each test condition,a total of four experiments were performed,from which a geometric mean life was calculated.An equivalent zero-mean applied stress amplitude was then determined from the S/N curve using the geometric mean life.Both the equivalent zero-mean stress amplitude and the loading with a mean stress would have the same average cycles to failure and therefore contribute the same fatigue damage.Similar calculations were performed for each data set and plotted on a nondimensional graph.Dividing the actual applied stress amplitude by the equivalent zero-mean stress amplitude made the ordinate axis nondimensional.Dividing the applied mean stress by the ultimate tensile or yield strength,as appropriate,made the abscissa nondimensional. For the random amplitude test results,an equivalent zero-mean RMS stress was determined using the Rayleigh approximation[8]equation given below:N pred2B=2s B x AG 12B=2 (2)This equation estimates the expected cycles to failure under a loading which has Rayleigh-distributed extrema. It assumes that linear cumulative damage theory[9,10] applies and that the constant amplitude S/N curve can be represented by a straight line on a log±log plot.Here,A and B are the life axis and slope of the constant amplitude S/iN curve,s x is the RMS stress of the narrowband random load-ing,and G(´)is the gamma function.Before using this equation to predict the failure life of the cruciform specimens,it must be modi®ed slightly to account conservatively for the truncation of the few extrema to four times the RMS:N predA22B=2s2B x G12B22G12B2;S2max2s2x23451S2B max exp2S2max2s2x23(3) where G(a,b)is the complementary incomplete gamma function and S max is the maximum(clipped)stress level:G a;b1bt a21e21d t(4)To determine the equivalent zero-mean RMS stress,the modi®ed Rayleigh approximation equation was solved for s x using the S/N curve parameters and the geometric mean fatigue life from the nonzero-mean random loading.Values of the yield and ultimate strengths used were taken from the material certi®cation sheets supplied with the original base plate,i.e.s yld 90.5ksi and s ult 98.5ksi.The resulting graph is shown in Fig.3.Also plotted on this graph are the two mean stress correction relationships discussed earlier, the modi®ed Goodman and the Gerber,along with stress range if no correction is made.A similar graph,shown in Fig.4,is made using the tensile yield stress to normalize the mean stress instead of the ultimate tensile strength.Also shown on this®gure is the Soderberg correction,and,as before,the horizontal line indicating no mean stress correc-tion(stress range).Data used to generate these graphs can be found in Table5.Results show that for tensile mean stresses,the constant amplitude data generally fall between the modi®edD.P.Kihl,S.Sarkani/Probabilistic Engineering Mechanics14(1999)97±104101Fig.3.Mean stress effects,test results normalized by ultimate tensile strength.Goodman line and the Gerber parabola.The data that corre-spond to lower stress levels tend to lie closer to the modi®ed Goodman line.The data corresponding to higher stress levels tend to lie closer to the Gerber parabola.Sarkani and Lutes [11]observed good agreement between the Gerber correction and test data at stress levels correspond-ing to the yield strength.Their investigation did not,however,include the lower stress levels considered here.Further,their specimens contained very high levels of resi-dual stress and were much thicker than the test specimens used here.It is interesting to note that the magnitude of mean stress,rather than the maximum stress level,seems to be the domi-nant variable,at least under tensile mean stresses.Speci-mens tested at the same mean stress but at different stress amplitude illustrate this point:essentially the same ratio of applied stress amplitude to equivalent zero-mean stress amplitude is obtained for a given mean stress level at differ-ent stress amplitude.Similar conclusions can be drawn based on data for the random amplitude loadings.Since the data tend to fall between the Goodman and Gerber curves,an appropriate choice for a mean stress correction relationship would be the Goodman relationship,the more conservative of the two.The Soderberg correction curve,although very close to the Goodman curve,is more conser-vative.Therefore nothing further is to be gained by its use.D.P.Kihl,S.Sarkani /Probabilistic Engineering Mechanics 14(1999)97±104102Fig.4.Mean stress effects,test results normalized by tensile yield strength.Table 5Constant amplitude equivalent zero-mean stress calculations Stress amplitude (ksi)Mean stress (ksi)Geometric mean life (cycles)Equivalent R 21stress (ksi)S ampl /S equ S mean /S yld S mean /S ult R 120129590011.848 1.0130.0000.0002115056270015.3640.9760.0000.000213007180029.179 1.0280.0000.000214501710045.6240.9860.0000.0002110101459500a 11.4170.8760.1100.1020151527120019.2870.7780.1660.152030302950038.4960.7790.3310.30501020101790012.7740.7830.2210.2031/3153024110020.0070.7500.3310.3051/330602000043.4510.6900.6630.6091/330154390034.0120.8820.1660.15221/31050118200012.1930.8200.5520.5082/34022012120024.788 1.61420.22120.203233021020360021.0891.42320.11020.10222S /N curve:log (N ) 9.5623.21£log (S ampl )S ult 98.5ksi S yld 90.5ksiaDoes not include suspended test results.For compressive mean stresses,the test results were found to lie above both the modi®ed Goodman and the Gerber curves.Data lying above the curves would indicate that values obtained from either correction curve would be conservative.In reality,a lower equivalent zero-mean stress would be indicated,and hence a longer fatigue life would be expected.In fact,regression analysis performed on the data indicate that the following straight line would provide an acceptable correction for compressive mean stresses:s amp s equ R 2113s means ult1(5)Similar conclusions can be drawn based on data from the random amplitude tests.Results of equivalent zero-mean RMS calculations for the random amplitude loadings can be found in paring the results of mean stress effects under constant amplitude with those under random amplitude load indicates that the same relationship could be used for both types of loading.This result is not entirely unexpected for the random loadings considered in this investigation,i.e.narrowband Gaussian(Rayleigh-distribu-ted extrema).It can be shown that for a process that has Rayleigh-distributed extrema,the constant amplitude S/N curve coef®cients can be related to the coef®cients of a random amplitude S/N curve using RMS stress instead of stress amplitude.Assuming that linear cumulative damage theory applies and the S/N curves can be represented by straight lines on a log±log plot,the following relationships can be established:A RMS A2B=2G 12B=2(6a)B RMS B(6b) Here,A RMS and B RMS are the life axis intercept and slope of the random amplitude S/N curve,shown below,which is analogous to the constant amplitude S/N curve de®ned by parameters A and B,de®ned previously:N A RMS s x B RMS(7) Mean stress corrections can be applied to the random amplitude loadings in a manner similar to the way they are analytically applied to constant amplitude loadings. The mean stress correction term,e.g.(12s mean/s ult)for a modi®ed Goodman correction,does not contain variables that are considered random.Furthermore,the equivalent stress amplitude is determined as the product of the correc-tion factor and the random amplitude stress.Note that the correction factor is a deterministic quantity.When dealing with narrowband Gaussian processes having a zero mean, the expected cycles to failure can be determined from Eq.(2).If,however,the same narrowband Gaussian process is offset by a tensile mean stress,it can be shown that the expected cycles to failure are determined in a manner analo-gous to that used under constant amplitude loads:N pred 12s means ult2B£2B=2s B x AG 12B=223(8)Noting the similarity between Eqs.(8)and(2),it can be concluded that the mean stress correction relationship expressed for constant amplitude loads,Eq.(1),can there-fore be applied in a similar manner to narrowband Gaussian random amplitude loads:s RMS nonzero means RMS equ zero mean1s means ult;yld23n1(9)Whether or not this relationship can be extended to nonnar-rowband loadings is deferred to a future investigation. Although nonnarrowband(Gaussian)loadings have an easily-de®nable probability density function(PDF)of stress peaks,the exact analytical relationship of that PDF to the PDF of stress cycles is not currently known.D.P.Kihl,S.Sarkani/Probabilistic Engineering Mechanics14(1999)97±104103 Table6Random amplitude equivalent zero-mean RMS stress calculationsRMS stress(ksi)Mean stress(ksi)Geometric mean life(cycles)Equivalentzero-meanRMSstress(ksi)S ampl/S equ S mean/S yld S mean/S ult4011971000a 3.75 1.0670.0000.000 504709700 5.020.9960.0000.000 7.5012441007.590.9880.0000.000 1006116009.47 1.0560.0000.000 150********.830.9480.0000.000 5257245200a 4.38 1.14220.05520.051 102153238300 5.63 1.77620.16620.152 102022510012.930.7730.2210.203 104014450014.840.6740.4420.406 S/N curve:log(N) 9.5623.21£log(S ampl)S ult 98.5ksiS yld 90.5ksia Does not include suspended test results.5.ConclusionsNonzero-mean fatigue test data associated with both constant amplitude and random amplitude loadings super-imposed upon a mean stress were generated to assess applied mean stress effects.Corresponding tests with zero-mean loadings had already been performed and were there-fore available for comparison.Both tensile and compressive mean loadings were considered.Equivalent zero-mean stress loadings were determined which produced the same number of cycles to failure as did the nonzero-mean loadings.Mean stress correction equations relating the equivalent zero-mean stress loadings to the nonzero-mean stress loadings in terms of the mean stress and ultimate or yield strengths were evaluated.Of the three mean stress correction equations evaluated, the test data indicated that the modi®ed Goodman equation produces generally conservative,if not accurate,agreement, especially at low tensile stress levels.At high levels of tensile stress,the Gerber correction proved to be a better mean stress correction relationship.Under compressive levels of mean stress,none of the relationships proved accu-rate.However,since the data under compressive mean stress appeared linear and since the modi®ed Goodman relation-ship worked well under tensile mean stress,a variation of the modi®ed Goodman correction was considered.The variation consisted of adjusting the slope parameter of the equation to®t the test data.This modi®cation was found also to work well for the narrowband random loading cases, although the correction would err slightly on the conserva-tive side.It should be noted that none of the test data supported the use of the stress range approach,which essentially ignores the presence of any mean stresses.Whether this conclusion applies only to weldments having little or no residual stres-ses,or can be extended to larger,more complicated weld-ments having high residual stress,is not known.In any event,the stress range approach(i.e.no correction)was unconservative under tensile mean stress conditions and was conservative under compressive loadings.References[1]Frost NE,Marsh KJ,Pook LP.Metal fatigue.New York:OxfordUniversity Press,1974.[2]Boresi AP,Sidebottom OM,Seely FB,Smith JO.Advancedmechanics of materials,3rd ed.New York:Wiley,1978.[3]Wehner T,Fatemi A.Effects of mean stress on fatigue behavior of ahardened carbon steel.International Journal of Fatigue 1991;13(3):241±248.[4]Lambert RG.Fatigue damage prediction for combined random andstatic mean stresses.Journal of the Institute of Environmental Sciences1993;May/June.[5]Sarkani S.Feasibility of auto-regressive simulation model for fatiguestudies.ASCE Journal of Structural Engineering Division 1990;116(9):2481±2495.[6]Standard practice for statistical analysis of linear or linearized stress-life(S±N)and strain-life(e±N)fatigue data.ASTM procedure E-739.Metals test methods and analytical procedures,vol.03.01,1988. 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