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    Co s t r u z io n e d i M a cch i n e Co s t r u z io n e d i M a c ch i n e

    Prof. Nicola BonoraProf. Nicola Bonora

    AAAA 2008/20092008/2009

    Lezione 4Lezione 4 Modi di rottura, fatica HCFModi di rottura, fatica HCF

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    Fatica: il conteggio dei cicli

    Nella realtNella realt operativa la forma dello spettro di sollecitazioneoperativa la forma dello spettro di sollecitazionenonnon quasi mai periodico nquasi mai periodico n assimilabile ad una sinusoideassimilabile ad una sinusoide

    Questo pone la questione di come si debbano conteggiare iQuesto pone la questione di come si debbano conteggiare icicli che effettivamente affaticano il materialecicli che effettivamente affaticano il materiale

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    Fatica: il conteggio dei cicli

    Diverse possibilitDiverse possibilit::

    PeaksPeaks countcount

    PeakPeak--betweenbetween--meanmean--crossingcrossing countcount

    LevelLevel crossingcrossing countcount

    FatigueFatigue metermeter countcount

    RangeRange countcount

    RangeRange--meanmean countcount

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    Fatica: il conteggio dei cicli

    1)1) CharacterizeCharacterize loadload curve incurve in termsterms ofof maximamaxima and minimaand minima onlyonly

    2) Introduce2) Introduceclassesclasses(stress(stress intervalsintervals)) suchsuch thatthat maximamaxima/minima/minima belongingbelonging totothethe samesame classclass areare countedcounted similarlysimilarly

    Signal may now be characterized in terms ofSignal may now be characterized in terms of peak counting: Histogram of maxima/minimapeak counting: Histogram of maxima/minima

    range counting: Histogram of ranges (differencesrange counting: Histogram of ranges (differences

    between adjacent maxima/minima)between adjacent maxima/minima)

    level crossing counting: Histogram of level crossingslevel crossing counting: Histogram of level crossings

    rainflowrainflow counting: Conversion of irregular time seriescounting: Conversion of irregular time series

    into a sequence ofinto a sequence of cyclescycles

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    Fatica: il conteggio dei cicli

    AllAll maximumsmaximums aboveabove thethe meanmean andand allall the minimumthe minimum belowbelow thethe meanmean areare countedcounted

    PeakPeak--countcount

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    Fatica: il conteggio dei cicliFatica: il conteggio dei cicli

    OnlyOnly thethe largestlargest peakpeak betweenbetween successivesuccessive crossingscrossings of theof the meanmean isis countedcounted

    PeakPeak--betweenbetween--meanmean countcount

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    Fatica: il conteggio dei cicli

    Al positiveAl positive slopeslope levellevel crossingscrossings aboveabove thethe meanmean and negativeand negative slopeslope levellevel

    crossingcrossing belowbelow thethe meanmean areare countedcounted

    LevelLevel crossingcrossing countingcounting

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    Fatica: il conteggio dei cicli

    RainRain flowflow methodmethod

    TheThe rainflowrainflow--counting algorithm (also known as the "raincounting algorithm (also known as the "rain--flow countingflow countingmethod") is used in the analysis of fatigue data in order to redmethod") is used in the analysis of fatigue data in order to reduce auce aspectrum of varying stress into a set of simple stress reversalsspectrum of varying stress into a set of simple stress reversals..

    Its importance is that it allows the application of Miner's ruleIts importance is that it allows the application of Miner's rule in order toin order toassess the fatigue life of a structure subject to complex loadinassess the fatigue life of a structure subject to complex loading.g.

    The algorithm was developed by Tatsuo Endo and M.The algorithm was developed by Tatsuo Endo and M. MatsuiskiMatsuiski in 1968.in 1968.

    Though there are a number of cycleThough there are a number of cycle--counting algorithms for suchcounting algorithms for such

    applications, theapplications, the rainflowrainflow method is the most popular as of 2008.method is the most popular as of 2008.

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    Fatica: il conteggio dei cicli

    RainRain flowflow methodmethod

    The algori thm

    1.1. Reduce the time history to a sequence of (tensile)Reduce the time history to a sequence of (tensile)peaks and (compressive) troughs.peaks and (compressive) troughs.

    2.2. Imagine that the time history is a template for a rigidImagine that the time history is a template for a rigidsheet (sheet (pagodapagoda roof).roof).

    3.3. Turn the sheet clockwise 90Turn the sheet clockwise 90 (earliest time to the(earliest time to thetop).top).

    4.4. EachEach tensile peaktensile peakis imagined as a source of wateris imagined as a source of waterthat "drips" down the pagoda.that "drips" down the pagoda.

    5.5. Count the number of halfCount the number of half--cycles by looking forcycles by looking for

    terminations in the flow occurring when either:terminations in the flow occurring when either:

    1.1. It reaches the end of the time history;It reaches the end of the time history;

    2.2. It merges with a flow that started at an earlierIt merges with a flow that started at an earliertensile peaktensile peak; or; or

    3.3. It flows opposite aIt flows opposite a tensile peaktensile peakof greaterof greater

    magnitude.magnitude.

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    Fatica: il conteggio dei cicli

    RainRain flowflow methodmethod

    The algori thm6.6. Repeat step 5 forRepeat step 5 for compressive troughscompressive troughs..

    7.7. Assign a magnitude to each halfAssign a magnitude to each half--cycle equal to thecycle equal to thestress difference between its start and termination.stress difference between its start and termination.

    8.8. Pair up halfPair up half--cycles of identical magnitude (butcycles of identical magnitude (butopposite sense) to count the number of completeopposite sense) to count the number of completecycles. Typically, there are some residual halfcycles. Typically, there are some residual half--cycles.cycles.

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    Fatica: il conteggio dei cicli

    RainRain flowflow methodmethod

    EXAMPLEThe stressThe stress historyhistory in Figure Ain Figure A isis reducedreduced toto peakspeaksandand troughstroughs in Figure B.in Figure B.

    HalfHalf--cyclecycle (A)(A) startsstarts atat tensiletensile peakpeak (1) and(1) andterminatesterminates oppositeopposite aa greatergreater tensiletensile stress,stress, peakpeak

    (2).(2). ItsIts amplitudeamplitude isis 1616 MPaMPa..

    HalfHalf--cyclecycle (B)(B) startsstarts atat tensiletensile peakpeak (4) and(4) andterminatesterminates wherewhere itit isis interruptedinterrupted byby a flowa flow fromfromanan earlierearlier peakpeak, (3),, (3), ItsIts amplitudeamplitude isis 1717 MPaMPa..

    HalfHalf--cyclecycle (C)(C) startsstarts atat tensiletensile peakpeak (5) and(5) andterminatesterminates at the end of theat the end of the timetime historyhistory..

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    Fatica: il conteggio dei cicli

    RainRain flowflow methodmethod

    EXAMPLE

    SimilarSimilar halfhalf--cyclescycles areare calculatedcalculated forforcompressivecompressive stressesstresses (Figure 4) and the(Figure 4) and the halfhalf--cyclescycles areare thenthen matchedmatched..

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    FaticaFatica multiassialemultiassiale

    Ci sono molte applicazioni pratiche inCi sono molte applicazioni pratiche incui lo stato di sforzo variabile nelcui lo stato di sforzo variabile neltempotempo multiassialemultiassiale

    FusolieraFusoliera

    Serbatoi e tubazioni in pressioneSerbatoi e tubazioni in pressioneAlberi di trasmissioneAlberi di trasmissioneEtc.Etc.

    A causa della natura dei carichiA causa della natura dei carichiesterni applicati e della loroesterni applicati e della lorofluttuazioni, gli stati di sforzo che sifluttuazioni, gli stati di sforzo che sigenerano possono fluttuare in fase ogenerano possono fluttuare in fase ofuori fasefuori fase

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    FaticaFatica multiassialemultiassiale

    PROPORTIONAL LOADINGPROPORTIONAL LOADING

    Per un campione sottoposto a faticaPer un campione sottoposto a faticamultiassialemultiassiale si dice chesi dice che inincondizioni di carico proporzionalecondizioni di carico proporzionale

    quando durante le variazioni deiquando durante le variazioni deicarichi esternamente applicati, lecarichi esternamente applicati, lecorrispondenti differenti componenticorrispondenti differenti componentidi sforzo variano in proporzionedi sforzo variano in proporzionecostante tra lorocostante tra loro

    Le costanti l possono variare daLe costanti l possono variare dapunto a punto ma rimangonopunto a punto ma rimangonocostanti nel punto durante la storiacostanti nel punto durante la storiadei carichidei carichi

    2 31 1 2

    1 1 = =

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    FaticaFatica multiassialemultiassiale

    Sforzo equivalenteSforzo equivalente

    Ipotesi generale:Ipotesi generale: possibilepossibilesupporre che lsupporre che leffetto di uno stato dieffetto di uno stato disforzo complesso in cui lesforzo complesso in cui le

    componenti sono perfettamente incomponenti sono perfettamente infase o fuori fase di 180fase o fuori fase di 180 siasiadescrivibile attraverso un criterio dides