MODELIZATION OF LOW CYCLE FATIGUE DAMAGE IN FRAMES

Transcription

1 MODELIZATION OF LOW CYCLE FATIGUE DAMAGE IN FRAMES Rcardo PERERA 1, Enrque ALARCON 2 And Albero CARNICERO 3 SUMMARY Damage models based on he Connuum Damage Mechancs (CDM) nclude explcly he couplng beween damage and mechancal behavor and, herefore, are conssen wh he defnon of damage as a phenomenon wh mechancal consequences. However, hs knd of models s characerzed by her complexy. Usng he concep of lumped models, possble smplfcaons of he coupled models have been proposed n he leraure o adap hem o he sudy of beams and frames. On he oher hand, n mos of hese coupled models damage s assocaed only wh he damage energy release rae whch s shown o be he elasc sran energy. Accordng o hs, damage s a funcon of he maxmum amplude of cyclc deformaon bu does no depend on he number of cycles. Therefore, low cycle effecs are no akng no accoun. From he smplfed model proposed by Flórez-López, s he purpose of hs paper o presen a formulaon ha allows o ake no accoun he degradaon produced no only by he peak values bu also by he cumulave effecs such as he low cycle fague. For, he classcal damage dsspave poenal based on he concep of damage energy release rae s modfed usng a fague funcon n order o nclude cumulave effecs. The fague funcon s deermned hrough parameers such as he cumulave roaon and he oal roaon and he number of cycles o falure. Those parameers can be measured or denfed physcally hrough he characerscs of he RC. So he man advanage of he proposed model s he possbly of smulang he low cycle fague behavor whou nroducng parameers wh no suable physcal meanng. The good performance of he proposed model s shown hrough a comparson beween numercal and es resuls under cyclng loadng INTRODUCTION Durng srong earhquakes, srucures are expeced o be subjeced o large laeral load reversals. Consequenly, relavely large nelasc cyclc roaons can be expeced. These roaons have an mporan nfluence on he overall behavor of he frames and on her dynamc response snce hey nvolve energy dsspaon. Fague damage ncreases wh appled cycles n a cumulave manner whch may lead o fracure. Palmgren [1924] suggesed he concep of lnear damage accumulaon rule whch was frs expressed n a mahemacal form by Mner n Snce hen, he reamen of cumulave fague damage has receved ncreasngly more aenon. As a consequence of, numerous papers have been publshed wh dfferen fague damage models [Faem and Yang, 1998; Soce and Morrow, 1976 ]. The problem of he esmaon of cumulave damage of a componen could be relavely easly solved under harmonc or block loadng usng he hypohess of fague cumulave damage proposed by Palmgren and Mner. Under random loadngs such as sesmc evens or wnd, cycles are no well-defned and hen developed cumulave hypohess canno provde sasfacory resuls due o a consderable varance of he esmaon range. Deparmen of Srucural Mechancs, Techncal Unversy (U.P.M.), Madrd, Span Emal: perera@esru.upm.es Deparmen of Srucural Mechancs, Techncal Unversy (U.P.M.), Madrd, Span Emal: perera@esru.upm.es Deparmen of Mechancal Engneerng, Ponfca Comllas Unversy, Madrd, Span Emal: carncero@dm.ca.upco.es

2 In hese cases, he mos mporan aspec s o coun closed hyseress cycles n he load hsory whch nvolves converng a random loadng hsory no an equvalen sum of cycles by a cycle counng mehod. Several mehods have been developed for cycle counng. The ranflow echnque developed by Masush and Endo [1968]s one of he mos commonly used cycle counng mehods. Ths echnque allows he converson of an rregular loadng hsory of random naure no a se of blocks of equvalen harmonc ampludes. In he same way, some varaons of he classcal ranflow mehod have been developed [Cacko,1992]. However, n he las years he fague sudy has been reorenaed hrough s ncorporaon n he Connuum Damage Mechancs (CDM) [Lemare and Chaboche, 1985; Lemare, 1993]. The same conceps used n CDM o model ducle falure can be exended o he low cycle fague damage processes, where plascy s he key mechansm for crack naon. Damage Mechancs deals wh damage as a connuum varable and, because of, CDM models ncludng plascy and damage can predc ducle crack naon. An exenson of hemselves ncludng he number of cycles could be suable o smulae he low cycle fague damage. Accordng o, Chaboche [1985] developed a formulaon for damaged maerals where he fague phenomenon was ncorporaed n he CDM. However, only harmonc loads were consdered beng he hypoheses of fague cumulave damage suable. In he presen work, a smplfed model for evaluaon of low cycle fague damage n frames s proposed. The proposed formulaon s based n a generalzaon of he lumped plascy models ncludng damage effecs accordng o he lnes of he CDM such as was developed n [Florez-Lopez, 1995; Cpollna e al, 1995]. I can be consdered as a smplfed damage mechancs ncorporang conceps of he CDM. A reformulaon of he model s developed n order o nclude he cumulave effecs produced n a low cycle fague process. The man advanage of hs model s he ably of represenng he cumulave fague damage n he classcal way used n he CDM whou necessy of ncorporang new rules. On he oher hand, he nonlnear cumulave damage s obaned drecly n a smple way avodng cycle counng echnques. Consuve Equaons ELASTOPLASTIC DAMAGE MODEL Damage n Connuum Damage Mechancs akes no accoun he degradaon of maerals resulng n a sffness reducon. Accordng o he Sran Equvalence Prncple proposed by Lemare (1971) and usng he Kachanov s defnon of effecve sress, he sffness of a damaged maeral can be obaned as E(1-d) beng E he nal Young s modulus and d a scalar represenng he soropc damage. Assumng a damaged elasc maeral, he sran due o damage can be obaned as [Orz, 1985; Ju, 1989]: d σd ε = E(1 d) whch s conssen wh he response of renforced concree under unaxal monoonc loadng. (1) Equaon (1) can be appled o a member of consan area A sujeced o an axal load: d NL da δ = EA 1 da where N s he axal force and δ d he elongaon due o he axal damage d a. (2) 2

3 M Elasc beam j M j N θ j θ ϕ L + δ Fgure 1: Generalzed sresses and srans for he model Equaon (2) can be generalzed n order o ake no accoun he flexural damage effecs n a frame member. For, we consder an elemen where he sress dsrbuon s descrbed by a hree componen vecor, q=[m,m j,n] T, collecng he bendng momens a he wo ends and he axal force (Fgure 1), whch s assocaed o he correspondng knemac varables u=[θ,θ j,δ] T. The consuve equaons expressng he relaons d d d d beween he flexural momens and he correspondng roaons due o damage, u [ θ, θ, δ ] T as: θ d d = 1 d L 4EI M =, are obaned d d j L θ j = M j (3) 1 d j 4EI beng d and dj he damage varables due o flexural effecs a boh ends of he member. Therefore, he damage vecor for each member wll be defned as D = (d d d ). j More deals abou he formulaon of he consuve equaons for hs model can be found n [Florez-Lopez, 1995; Perera e al, 1998]. Dsspave Poenals In order o specfy he complee se of equaons for a damaged maeral accounng he CDM, s necessary o defne wo dsspaon poenals, one for plascy and he oher for damage; no couplng beween boh poenals s assumed. Then he oal dsspaon poenal s gven as: F = f (q,r,x;d) + g(y; D) (4) where f s he dsspave poenal assocaed o plascy funcon of he acual sress ensor, q, and R and X are he soropc and knemac hardenng varables, respecvely.; g s he dsspave poenal assocaed o he damage process beng Y he nernal varable assocaed o damage (damage energy release rae) [Lemare and Chaboche, 1995]. Evoluon Laws The Prncple of Maxmum Plasc Dsspaon mples he normaly of he flow rules n generalzed sress space for plasc deformaons and n Y-space for damage varables: p p f d g du = dλ dd = dλ q Y where dλ p and dλ d are plasc and damage conssency parameers, respecvely. The expresson for hese wo funcons, f and g, s obaned from expermenal resuls and her consderaon wll be reaed n he nex secon. a j (5) 3

4 LOW CYCLE FATIGUE MODELLING Very usually, he Grffh creron s used as a damage dsspaon poenal. I s well known, neverheless, ha he Grffh creron canno descrbe crack propagaon under repeaed loads snce he maxmum energy released load remans consan n ha case. However, he applcaon of cyclc loadng acvaes he dslocaon moon ha wll lead o he formaon of a fague crack. Therefore, models based on he smple Grffh creron are no able o smulae he srengh degradaon due o fague effecs. Dfferen alernaves have been proposed n he leraure o perform he fague modellng [Chaboche, 1995; Margo, 1985; Suars e al, 1990]. In hs paper, s proposed a generalzaon of he Grffh creron n order o nclude he low cycle fague effecs. For, a new funcon affecng he crack ressance s nroduced. Ths funcon depends on he number of cycles and, so, an mplc evoluon of he nernal varables of he plasc and damage processes s obaned. Mner s Rule Under he condons of cyclc loadng, he analyss can be generally based on he prncple of lnear summaon of damage. Ths prncple, as appled o fague fracure was formulaed by Palmgren [1924] and Mner [1945]. Damage fracons due o each ndvdual cycle are summed unl fracure occurs. Falure s assumed o occur when hese damage funcons sum up o or exceed uny: D n n = N f 1 where n s he number of cycles for he curren amplude and N f s he number of cycles o falure for hs amplude The applcaon of he lnear cumulave damage model consss of converng random cycles no an equvalen number of consan amplude cycles. Technques lke ranflow [Masush and Endo, 1967] or range par [Dowlng, 1972] allow o perform hs converson. (6) σ ε εp ε Fgure 2: Toal and plasc sran amplude The quanfcaon of he number of cycles o falure N f s performed usually hrough he Manson-Coffn relaonshp [1953]: N p K f = C( ε ) where ε p s he plasc sran amplude of he hyserec cycles (Fgure 2) and C and K are parameers dependng on he maerals. Some auhors [Kunnah e al, 1997; Koh and sephen, 1991] suggesed he oal sran amplude could be used nsead of plasc sran. (7) 4

5 Formulaon From he Grffh creron, such as was defned n Florez-Lopez [1995], s proposed here he followng dsspaon poenal for low cycle fague damage: [ Y + Z( ) ξ( ω) ] g = Y cr D (8) where ω s a cumulave parameer and ξ(ω) s a funcon whch allows o nclude he fague effecs n he damage evoluon and whch has o sasfy he followng condons: ( ω) = 1 ω ωmn ( ω) = 0 ω = ωmax ξ ξ In he same way, s defned a plasc dsspaon poenal ncludng he new funcon as follows: ( M + ξ( ω) ) f = M X y R (10) The keypon n he smulaon of he fague phenomena s he choce of he fague funcon ξ(ω). From he Mner s rule and hrough dfferen numercal evaluaons a good correlaon has been obaned wh a funcon such as: ξ( ~ θ, θ ) = 1 N where ~ θ and f ~ θ θ ( θ ) 1 µ θ are he oal cumulave roaon and he oal roaon, respecvely, and µ s he ducly. Ths funcon s represened n he Fgure 3. In hs expresson, can be observed ha he cocen beween brackes can be consdered as a Palmgrem-Mner lke relaonshp. (9) (11) Fgure 3: Fague funcon The evaluaon of he number of cycles o falure N f n equaon (11) has been performed akng he expresson performed by Koh and Sephen [1991] whch s based on he oal sran: ε ε = 2 = ( 2N ) 3 f (12) Acualzaon of he number of cycles When we work wh cycles of non consan amplude, some nconssences due o a quck loss of srengh may appear n he model. The man reason of hs s he srong decrease of he value of he fague funcon when he cycle amplude s ncreased. Ths phenomenon s no conssen wh he expermenal ess. 5

6 To overcome hs problem, he oal cumulave roaon mus be recalculaed when he maxmum response s ncreased or decreased. For, he followng connuy condon has o be sasfed: ~ old old ~ new new ξ( θ, θ ) = ξ( θ, θ ) (13) from whch he new value s obaned new µ ~ old old ~ µ new new new N θ θ = f θ old old N f θ (14) RESULTS The model presened above s checked hrough some examples. Fgures 4a and b represen expermenal and numercal resuls usng he proposed dsspave funcons. Resuls from Fg. 10 are referred o a crcular cross secon renforced concree column [Kunnah e al, 1997] whch s subjeced o a consan axal load of 806 kn and laeral dsplacemen s conrolled. The numercal smulaon has be done wh he followng parameers: EI/L = 2.51E+7 Nm, M + - cr = M = + - cr knm, M p = M p = knm, M + u = M - u = knm, θ + pu = θ - pu = 0.029, α + = α - = Shear Force (N) ,08-0,04 0 0,04 0, Dsplacemen (m) Fgure 4: Expermenal es (lef) by Kunnah e al (1997) and numercal smulaon (rgh) Fgures 5a and b show expermenal and numercal resuls of a recangular cross secon renforced concree column wh moderae confnemen esed by Wehbe e al. [1996] Force (N) ,00E-01-1,50E-01-1,00E-01-5,00E-02 0,00E+00 5,00E-02 1,00E-01 1,50E-01 2,00E Dsplacemen (m) Fgure 5. Expermenal es (lef) by Wehbe e al. (1996) and numercal smulaon (rgh) 6

7 As n he prevous cases he column s subjeced o a consan axal load of 641 kn and laeral dsplacemen s conrolled. The numercal smulaon has be done usng he parameers: EI/L = 2.21E+7 Nm, M cr + = M cr - = 210 knm, M p + = M p - = 643 knm, M u + = M u - = 850 knm, θ pu + = θ pu - = 0.05, α + = α - = 1. The damage ndex evoluon n he numercal smulaon s represened n Fgure 6. 0,7 0,6 0,5 Damage 0,4 0,3 0,2 0,1 0, Pseudome Fg.6. Damage evoluon for example 3 CONCLUSIONS 1. The srengh degradaon due o low cycle fague has been formulaed hrough a suable choce of he dsspave poenals 2. Good correlaon beween expermenal and numercal resuls under cyclc loadng has been obaned 3. Damage ndex s assocaed wh he crackng level of he concree and plasc roaons are relaed o plasc deformaons n he renforcemen 4. Srengh degradaon due o fague effecs s assocaed wh he fague n he longudnal renforcemen 5. Ths model could be aken no accoun as a framework for sesmc rerofng decson makng of srucures. 6. The approach presened s amenable of furher generalzaons REFERENCES 1. Cacko, J. (1992), Smulaneous compuer smulaon of operaonal random processes and connual ranflow counng, In.J. Fague, 14, No.3, pp Chaboche, J. (1987), Connuum damage mechancs and s applcaon o srucural lfeme predcon, Rech. Aerosp., 4, pp Cpollna, A. López-Inojosa, A. and Flórez-López, J. (1995),. A smplfed damage mechancs approach o nonlnear analyss of frames. Compuers & Srucures Vol. 54, No.6, pp Dowlng, N.E. (1972), Fague falure predcons for complcaed sress-sran hsores, J. Maer., Vol.7, pp Faem, A. and Yang, L. (1998), Cumulave fague damage and lfe predcon heores: a survey of he sae of he ar for homogeneus maerals, In. J. Fague, Vol.20, No.1, pp Florez-López, J. (1995),. Smplfed model of unlaeral damage for RC frames. J. Sruc. Eng. Vol. 121, No.12,pp Ju, J. W. (1989), On energy-based coupled elasoplasc damage heores: consuve modelng and compuaonal aspecs. In. J. Solds Srucures. Vol. 25, No.7, pp Koh, S.K. and Sephen, R.I. (1991), Mean sress effecs on low cycle fague for hgh srengh seel, Fague Frac. Engng. Maer., Vol.14, pp

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