Fracture via a sequence of events: a saw-tooth softening model

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1 raure va a sequene o evens: a saw-ooh soenng model J.G. Ros Del Unversy o Tehnology, Del, The Neherlands S. Invernzz Poleno d Torno, Torno, Ialy. Also researh ellow a TUDel. B. Belle Unversy o Parma, Parma, Ialy. Also researh ellow a TUDel. ABSTRACT: The sequenally lnear analyss s a robus alernave o non-lnear ne elemen analyss o sruures when buraon, snap-bak or dvergene problems arse. The load-dsplaemen response s apured by a seres o lnear analyses as a sequene o evens. very even s a saled ral saes orrespondng o he reahng o some peak o some saw-ooh or some soenng elemen. In he presen paper, he approah s exended wh a rppled saw-ooh urve whh apples o any sress-sran dagram, nluded ompresson nonlneary and yeldng o renoremen. Several RC sruural examples demonsrae ha boh sharp snap-baks as well as dule alures an be handled orrely. INTRODUCTION Non-lnear ne elemen analyss s beomng a ommon ool or sudyng he behavor o renored onree sruures. Over he pas years, ehnques or non-lnear analyss have been enhaned sgnanly va mproved soluon proedures, exended ne elemen ehnques and nreased robusness o onsuve models. Neverheless, problems reman, espeally when rakng and rushng n realworld sruures s analyzed. The load-dsplaemen response o RC beams, plaes, shells and spaal sruures oen shows a number o loal peaks and snap-baks or valleys assoaed wh brle rakng [] and subsequen sress redsrbuon. In smulang hs behavor, one has o use soenng models. Unorunaely, hs nvolves negave angen sness whh may lead o numeral nsably and dvergene o he nremenal-erave proedure. To ry and solve suh problems, users have o resor o ar-lengh or ndre onrol shemes []. or prang engneers hs s umbersome and oen nadequae when he buraons are mulple, he peaks rregular or he snap-baks sharp [3]. These problems are ndependen o he ype o smeared rak ormulaon adoped, eher deomposed-sran, oal-sran, damage or plasy based rak models. In hs onrbuon, an alernave mehod s adoped [4]. The soenng dagram o negave slope s replaed by a saw-ooh dagram o posve slopes. The nremenal-erave Newon mehod s replaed by a seres o lnear analyses usng a speal salng ehnque wh subsequen sness/srengh reduon per ral elemen. I wll be shown ha hs even-by-even sraegy s robus and relable. The advanage s ha here s no suh hng as negave nremenal sness, as he sean lnear (sawooh) sness s always posve. The analyss always onverges. Mesh-sze obevy s aheved by keepng he raure energy nvaran. In he paper, deals are provded onernng he saw-ooh mplemenaon o he bas maerals n RC sruures, namely onree and seel boh n enson and ompresson. Subsequenly, varous renored sruures are onsdered: he renored enson-pull spemen, wo smply suppored deep beams, and one deep beam on hree suppors. In all ases, he response shows loal peaks and snapbaks assoaed wh he subsequen developmen o prmary raks sarng rom he rebar. Comparsons beween nremenal-erave soluons and sequenally lnear soluons are gven and he behavour s nerpreed n erms o rak spang and rak wdh. The model s demonsraed o be sable and robus and hereore appealng o prasng RC engneers. OVRALL VNT-BY-VNT PROCDUR The loally brle snap-ype response o many RC sruures nspred he dea o apure hese brle evens drely raher han ryng o erae around hem n a Newon-Raphson sheme. A ral even s raed and subsequenly a sean resar s made rom he orgn or rang he nex ral even. Hene, he proedure s sequenal raher han n-

2 remenal. The sequene o ral evens governs he load-dsplaemen response. To hs am, he soenng dagram s replaed by a saw-ooh urve and lnear analyses are arred ou sequenally [4]. The global proedure s as ollows. The sruure s dsrezed usng sandard elas onnuum elemens. Young s modulus, Posson s rao and nal srengh are assgned o he elemens. Subsequenly, he ollowng seps are sequenally arred ou: Add he exernal load as a un load. Perorm a lnear elas analyss. xra he ral elemen rom he resuls. The ral elemen s he elemen or whh he sress level dvded by s urren srengh s he hghes n he whole sruure. Calulae he rao beween he srengh and he sress level n he ral elemen: hs rao provdes he global load aor. The presen soluon sep s obaned resalng he un load elas soluon mes he global load aor. Inrease he damage n he ral elemen by redung s sness and srengh,.e. Young s modulus and ensle srengh, aordng o a saw-ooh onsuve law as desrbed n he nex seon. Repea he prevous seps or he new onguraon,.e. re-run a lnear analyss or he sruure n whh and o he prevous ral elemen have been redued. Trae he nex ral sawooh n some elemen, repea hs proess ll he damage has spread no he sruure o he desred level. The way n whh he sness and srengh o he ral elemens are progressvely redued onsues he essene o he model. In oher words, s neessary o provde a saw-ooh approxmaon o he onsuve sress-sran relaon [4-6]. In he presen paper a new generalzed ooh sze approah s presened, whh allows or a sraghorward unaon o saw ooh onsuve laws or onree n enson, onree n ompresson and seel n enson and ompresson..5 Lnear soenng Blnear Model Code 9 Proposed nonlnear soenng w / h w / h gure. MC9 ohesve law (blnear); lnear soenng and nonlnear soenng wh equal srengh and raure energy. 3 SAW-TOOTH CONSTITUTIV LAWS OR RC 3. Saw-Tooh Laws or Conree n Tenson The behavor o onree n enson s orrely desrbed by he Model Code 9 (MC9) blnear relaon [6]. A lnear relaon an be also adoped (g. ), whh preserves he ensle srengh and he raure energy, hough hs hoe urns ou o overesmae he mmedae pos peak behavor, and o underesmae he ulmae sran. 3. Saw-Tooh Nonlnear Tenson Soenng The ohesve relaon o he MC9 provdes he ensle sress σ ransmed by he rak as a unon o he rak openng w n he ollowng way: σ.85.5 σ w w w w ( w w).5 σ σ <.5 () The rak openng w and he ulmae rak openng w depend o he ensle srengh and raure energy: w w α G G.5 w () In absene o expermenal daa, he onree raure energy an be esmaed as a unon o he haraers ompressve srengh k :.7 m G G (3) Boh α and he bas raure energy o onree G are unons o he maxmum aggregae sze. The MC9 blnear expresson has been reenly moded by Belle, Ceron and Ior [7] n order o have a onnuous unon, whh s beer or our purpose. Ths expresson reads as ollows: w σ (4) w w w δ w δ where δ.75 s a parameer whh guaranees ha he area underneah he urve (.e. he raure energy) remans unhanged. The ollowng sep s o mplemen he above ohesve urve no a smeared-rak oal sran ormulaon. Thereore, s neessary o smear he rak

3 openng w over he rak band wdh or elemen sze h, and o express he rak sran as he derene beween he oal sran and he elas par as ollows: w εr h σ ε r ε (5) q. (5) an be subsued n eq. (4), provdng he ollowng quadra expresson: σ ε σ. (6) w σ w ε δ w δ h Aer some algebra manpulaon, and sne only he lower roo o he above equaon s physally meanngul, he sress sran relaon n erms o oal sran s he ollowng: σ ε B σ wh: A B C B 4 A C A ε < ε w h (7) ( δ w h w h) [ ( δ w h w h) ε w w w h] ( w w w h ε ) (8) q. (7) an now be adoped as a moher urve or he onsruon o he saw ooh approxmaon. Sne he soenng al s nonlnear, mplemenaon o prevous saw-ooh approahes [5-6] s no sraghorward. Thereore, a more general approah s proposed. The man dea s o dene a narrow band aross he moher urve, obaned by uplng and lowerng he soenng urve wh some quany proporonal o he ensle srengh (g. ). The upled soenng unon wll be he ollowng: B B 4 A C σ p (9) A where p s a perenage o he srengh. The nerseon beween he gener sean elas branh and he soenng al,.e. he arbrary ooh peak, s provded by he ollowng equaon: B B 4 A C ε p () A Aer some algebra manpulaon, q. () an be solved wh respe o he sran, gvng agan a quadra expresson, whh provdes only one physally meanngul soluon: b b 4 a ε () a The orrespondng srengh an be obaned easly by: ε () very me ha he elemen s ral, aordng o he overall even-by-even proedure, he sness and he srengh o he elemen mus be redued. The rule o apply s: p ε ε ; N (3) where - s he nerepon o he sean sness wh he lowered soenng urve. Ths rule an be appled sequenally, replang he nal soenng moher urve by he saw-ooh approxmaon (g. ). gure. Gener nerseon wh he upled soenng urve and saw-ooh onsuve law. Conrarly o prevous proedures [5-6], he hegh o eah rpple s onsan here and equal o we he upl amoun. The number o eeh o he saw-ooh approxmaon s equal o he number o repeons whh an be perormed unl - beomes negave,.e.: ( : ) N max > (4) Noe ha N orresponds o omplee damage. Thanks o he a ha every rpple has he same hegh, he blak rangles n g. are wo-by-wo equal o eah oher. Thereore, he area under he

4 saw-ooh urve s always equal o he raure energy dvded by he rak band wdh, regardless he elemen sze and/or he number o eeh n he dsrezaon. Ths provdes he saw-ooh approxmaon o be mesh-sze obeve. The above proedure s general and apples o any arbrary oal sran ormulaon. 3.3 Saw-Tooh Laws or Conree n Compresson The ompressve behavor o onree an be modeled by he smpled C b-lnear sress-sran relaon [9], where ε / and ε u3-3.5 / respevely, or haraers ylndral ompressve srenghs up o 5 MPa. ( p) ε (6) Thereore, analogously o q. (), he sran beomes: ( p) ε (7) Noe ha we use o quany he level o damage n ompresson, sne was used or damage n enson. The updaed (.e. degradaed) Young s modulus beomes: N ( p) p p ; ε p ε (8) A slghly deren reron s adoped o deermne he number o eeh; n a urns ou o be neessary o lm he ulmae sran o he sawooh dagram aordng o he moher urve: ( p) ε N ε u3 N p. (9) p nally, he number o eeh beomes: N INTlog ε ε 3 p u3 p ( p). () gure 3. Moher urve and saw-ooh approxmaons: onree n ompresson ; seel n ompresson and enson. Ths ase exhbs a plas behavor o onsan sress level nsead o a soenng degradaon (g.3a). Moreover, he C relaon s already expressed n erms o oal sran. The upled pos peak urve s obaned as ollows: ( p) σ (5) Where, s he ompressve srengh and p he perenage o srengh uplng. The nerseon beween he gener sean elas branh and he plas plaeau, provdes he ollowng equaon: 3.4 Saw-Tooh Laws or Seel n Tenson and Compresson An elas perely plas sress-sran dagram has been adoped or renorng seel (or enson and ompresson), aordng o C presrpons [9], see g. 3b. The proedure adoped s denal o he one used or onree n ompresson, wh he only derene ha n he ase o seel he same onsuve law wll hold or enson and ompresson: ( p) y σ () Where, y s he yeld srengh and p he perenage o srengh uplng. The nerseon beween he gener sean elas branh and he pos peak plas plaeau s gven by he ollowng equaon: ( p) y ε () Thereore, analogously o q. ():

5 ( p) y ε (3) nally, he updaed (.e. degraded) Young s modulus beomes: N ( p) y p y y p ; ε p ε (4) Also n hs ase s neessary o lm he ulmae sran o he saw-ooh dagram aordng o he moher urve. Consequenly, he number o eeh beomes: N INTlog εu ε y ( p) p p (5) Noe ha sne he renoremen s modeled wh one-dmensonal russ elemens, he ndex alone s suen o quany he damage level n boh enson and ompresson. 3.5 Orhorop xed Crakng The sepwse reduon o Young s modulus, as desrbed n he prevous seons, n a mples ha he sness s redued n all dreons,.e. sepwse sorop degradaon ours. Alhough hs soropy assumpon may work or ases o loalzed raure n unrenored ondons, a subsanal mprovemen s neessary when dealng wh renored onree []. Then, ompressve srus develop parallel o he raks, and he assumpon o soropy does no hold. Thereore, n analogy o he poneerng approah o Rashd [], he nal sorop sress-sran law an be replaed by an orhorop law upon rak ormaon. The axes o orhoropy are deermned aordng o a ondon o rak naon, beng n he dreon normal o he rak plane, and he dreon o he ompressve srus (.e. angenal o he rak plane). As ar as he presen work onerns, he rak plane s kep xed aer he rak s nuleaed. Reerrng o he plane sress suaon, and o a loal n, oordnae sysem orened along he rak plane, he ollowng onsuve relaon s assumed e.g. []: ν σ nn ν σ ν σ n ν ν ν ε nn ε ε n β G (6) where he redued Young s modulus n enson along he n-axs and he redued Young s modulus n ompresson along he -axs aordng o he above sequenally lnear sheme. Moreover, β s he so-alled shear reenon aor and G s he nal shear modulus. The equaon an be rewren n ompa orm as ollows: σ D ε (7) n n n Gven he ollowng ransormaons or he sran and sress veors: εn Tε ( φ ) ε xy (8) σ n Tσ ( φ ) σ xy q. 7 an be easly ransposed n erms o sress and sran global omponens by pre- and posmulplaon wh he ransormaon mares: σ xy Tσ φ ) DnsTε ( φ ) ( ε (9) xy The above orhorop sheme ombnes he deren saw-ooh laws or onree n enson and ompresson, and was mplemened n he overall even-by-even proedure. 4 SOM APPLICATIONS TO RINORCD CONCRT STRUCTURS Varous renored sruures are onsdered n hs Seon. very sruure has been modeled by ournoded plane sress elemens or he onree and wo-noded russ elemens or he renoremen. Pere bond was assumed beween he onree and renoremen. All he sequenally lnear analyses have been perormed by hoosng a srengh range perenage p%. Comparsons are made boh n erms o load-dsplaemen urve and rak paerns. gure 4. Load-elongaon response or enson-pull spemen ; rak paern.

6 4. Renored enson-pull spemen A long-embedmen enson-pull spemen s onsdered [3]. The spemen s 6 mm long and he square ransversal seon s 68x68mm, renored wh a Φ8mm rebar. The onree parameers were: Young s modulus 8 MPa, Posson s rao ν., ensle srengh.5 MPa, raure energy G 6N/m, shear reenon aor β.. The renorng bar was gven a Young s modulus s 93 MPa, and a yeld sress sy 4 MPa. g. 4a shows he numeral resuls obaned usng he saw-ooh enson soenng urve o g.. The sequenally lnear analyss shows abou ve loal peaks assoaed wh he subsequen developmen o ve prmary raks. Beyond hese peaks snap-baks appear auomaally (g. 4). The behavor s remarkably smlar o he expermen where veral umps our due o he use o dsplaemenonrol. Prese quanave omparsons have no been made, as hs would requre bond-slp o be nluded. The analyss also demonsraes ha renoremen plasy s apured orrely, due o he use o he saw-ooh urve o g. 3b or he seel. 4. Renored onree deep beam The deep RC beams, expermenally esed by Braam [4], have been analyzed. Beam #3 was loaded n our-pon bendng wh a span o 5m. The beam was 5.5m long wh reangular ransverse ross seon (3x8mm). ore [kn] xpermenal Sequenally lnear analyss 5 5 Dsplaemen [mm] gure 5. Comparson beween expermenal and numeral resuls or beam#3. The saw-ooh non-lnear enson soenng urve has been adoped or onree n enson and a sawooh elas-plas dagram or seel. Mehanal properes adoped or onree are he ollowng: Young s modulus 3 MPa, Posson s rao ν., ensle srengh 3 MPa, energy raure G 6 N/m. The longudnal renoremen o beam#3 s onsued o 4Φmm, he adoped Young s modulus s MPa and he seel yeldng s equal o sy 566 MPa. gure 6. Comparson beween expermenal rak paern and he onour o onree elemens whh reah omplee damage on her nal saw-ooh or beam#3. g. 5 shows he omparson beween expermenal and numeral resuls n erms o appled load versus mdspan deleon. The am o expermenal ess was o nvesgae he behavor a he serveably lm sae, so measuremens have been sopped beore he ulmae load was reahed, whle numeral resuls onnue unl he ollapse mehansm ours due o yeldng o rebars. g. 6 shows, or he zone beween he pon o applaon o he load and he mdspan o beam#3, he omparson beween expermenal rak paern and he onour o onree elemens whh reah he omplee damage, (.e. he maxmum number o eeh N). The loalzed rakng paern aompaned by loal peaks and snap-baks n he loaddsplaemen response s reprodued orrely. Please noe ha hs behavor s obaned ully auomaally, as a sequene o lnear analyses. Ths approah always onverges as he sean sysem marx s always posve dene. Inremenal-erave nonlnear proedures would enouner dules or even dvergene, espeally n he early sage o brle snap-bak rakng. 4.3 Renored onree deep beam on hree suppors DWT beam, esed by Leonhard and Walher [5], has been hosen or he numeral smulaon wh DIANA nonlnear models and wh sequenally lnear analyss. Ths beam s a wo span deep beam, o lengh 34 mm, deph 6 mm, and onsan hkness equal o mm, wh a 36 mm hk supporng member n he mddle (g. 7a). In he expermenal es, he alure o DWT deep beam ourred a he ulmae load equal o 46 kn, bu he measuremens were perormed up o a load level o kn NL analyss xplong symmery, only one hal o he beam has been analyzed. g. 7b shows ha also suppor plaens have been modeled by seel membrane elemens, rgdly xed o he onree elemens. NL analyses have been arred ou wh DIANA. A xed smeared rak model, based on he onep o oal-sran, was employed. The ompres-

7 son non-lneary o he onree has been gnored. Only ensle rakng has been nluded and elasplas behavor o he renoremen. gure 9. NLA: prnpal sran a sep 9 and 3. gure 7. Geomeral eaures o DWT deep beam and renoremen arrangemen [5] (dmenson n m.). lemen mesh wh deals o he suppor plaens. The mehansms ha ransm ores aross raks n RC have been modeled by an average enson-senng sress-sran relaonshp or onree n enson. The usual assumpon s ha he sress arryng apay o he renored onree gradually dereases and s exhaused one he renoremen sars yeldng. Ths mples ha he ulmae sran w /h o he proposed enson-senng urve equals he yeld sran y o he seel rebars. The nonlnear urve o g. has been approxmaed as lose as possble by adopng DIANA s mul-lnear opon. A onsan shear reenon aor equal o β. desrbes he shear behavor o xed raks. The load-deleon urve obaned wh he NL analyss (g. 8) exhbs a very sudden drop n sep 3. Here, he NL analyss dverged when a ull angen sness sheme based on he loal negave soenng slopes was employed. gure. NLA: seel sress a sep 9 and 3. Only by usng a non-onssen angen sness, a sruural level, based upon he posve sean sness o he sress-sran urves a loal level, he analyss ould be onnued, hough sll nsuen onvergene ourred. The onvergene has no been reahed aer eraons n nremen sep 3. In hs nremen sep, a rak besdes he supporng member suddenly appears, as shown n g. 9, where he prnpal srans and rak paerns are onoured, respevely. A he same me, yeldng o he renoremen Φ5 Type I ours over he mddle suppor, g.. 5 Toal load [kn 5 xpermenal 5 Sequenally lnear analyss NL analyss: xed rak model.5.5 Dsplaemen [mm] gure 8., Load-deleon dagram. NLA: sep 9 NLA: sep 3 NLA: sep 6 gure. Sequenally lnear analyss: onree damaged elemen a oal load 6 kn us beore snap-bak, a 75 kn n he valley o he snap-bak. Beyond hs ral pon, he analyss ould be parally onnued and he raks beome wder. The onluson s ha he sandard nremenalerave Newon-Raphson proedure s no apable o adequaely ahng he sudden, explosve rakng ha ourred n he expermen.

8 4.3. Sequenally lnear analyss As an alernave, he same beam wh he same parameers was analyzed n he sequenally lnear ashon. Saw-ooh approxmaons have been adoped or he non-lnear enson soenng urve or onree n enson and or he elas-plas dagram or seel. I s mporan o noe ha he expermenal es has been arred ou n load onrol whle NL and sequenally lnear analyses have been arred ou n dsplaemen onrol. or hs reason he expermenal urve shows a la plaeau n he zone where he NL analyss dverges. The sequenally lnear analyss learly reveals wha happens): shows a pronouned quas-sa snap-bak behavor (g. 8) revealng he very sudden and brle developmen o he maor veral rak(s) near he md-suppor. Ths snap-bak and also oher rpples appear auomaally due o he salng proedure. () gure. xpermenal rak paern, sequenally lnear analyss: onree damaged elemens and seel yelded elemens () a nal load 93 kn. g. a shows he expermenal rak paern a alure. gs. b show, n blak, onree elemens whh reah he omplee damage (.e. he maxmum number o eeh N), respevely or deleon values equvalen o nremen seps 9, 3 onsdered or NL analyss, and lose o alure. In g. yelded seel elemen are ndaed n red. sress-sran dagram, boh or onree n enson, onree n ompresson and seel n enson and ompresson. Resuls prove ha he model s apable o smulang brle snap-bak ype o rakng (ypal or RC) as well as dule plas response. The approah always onverges as he sean sawooh sness s always posve dene. The approah s sable and robus and hereore appealng o prang engneers. uure developmens are requred, e.g. owards non-proporonal loadng. RRNCS Carpner A. Inerpreaon o he Grh nsably as a buraon o he global equlbrum, n NATO Advaned Workshop on Applaon o raure Mehans o Cemenous Composes, d. P. Shah, Marnus Nho, 984: De Bors R. Compuaon o pos-buraon and pos-alure behavour o sran-soenng solds. Compuers and Sruures, 987; 5(): -4. Crseld MA. Aeleraed soluon ehnques and onree rakng. Compuer Mehods n Appled Mehans and ngneerng, 98, 33: Ros JG. Sequenally lnear onnuum model or onree raure. In raure Mehans o Conree Sruures, de Bors R, Mazars J, Pauder-Cabo G, van Mer JGM, Balkema AA (eds). Lsse: The Neherlands,, Ros JG, Invernzz S. Regularzed sequenally lnear sawooh soenng model. In. Journal or Numeral and Analyal Mehods n Geomehans, 4, 8: Ros JG, Invernzz S, Belle B, Saw-ooh soenng/senng a sable ompuaonal proedure or RC sruures. Compuers & Conree, 6, 3(4):3-33. CB-IB Model Code 99. Bullen No. 3/4 (993), Come Inernaonal du Beon. Belle B, Ceron R, Ior I. Physal approah or renored onree (PARC) membrane elemens. ASC Journal o Sruural ngneerng,, 7 (): 4-6. uroode : Desgn o onree sruuree Par : General rules and rules or buldngs. CN,. Ros JG, Invernzz S. Saw-ooh soenng/senng model, n raure Mehans o Conree Sruures, (eds. V.C. L, C.K.Y. Leung, K.J. Wllam, S.L. Bllngon), USA, 4, : Rashd YR. Analyss o presressed onree pressure vessels. Nulear ngng. And Desgn, 968, 7(4): Ros JG, Naua P, Kusers GMA, Blaauwendraad L. Smeared rak approah and raure loalzaon n onree. H- RON, 985, 3():-48. Gsbers BJ, Hehemann AA. Some ensle ess on renored onree. TNO-IBBC Repor BI-77-6, Rswk, 977. Braam CR. The behavor o deep renored onree beams xpermenal resuls. Repor /VA-, Sevn Laboraory, Del Unversy o Tehnology, Del, 99. Leonhard., and Waler R. Wandarge Trager. Deusher Ausshuss ür Sahlbeon, D.A.. S., He 78, rns & Sohn, Berln, Germay, CONCLUSIONS A sequenally lnear mehod or he analyss o RC sruures has been presened. The mehod replaes soenng urves o negave downward slope by posve sean slopes usng a saw-ooh rppled