New Approach for the Construction of Inelastic Stiffness Matrix for Dynamic Analysis. A. Shooshtari 1 and R. Khajavi 2

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1 Th 4 th World Conrnc on Earthua Enginring Octobr -7, 8, Bijing, China Nw Approach or th Construction o Inlastic Stinss Matrix or Dynamic Analysis A. Shooshtari and R. Khajavi Assistant Prossor, Dpt. o Civil Enginring, Frdowsi Univrsity o Mashhad, Iran Ph.D. Studnt, Dpt. o Civil Enginring, Frdowsi Univrsity o Mashhad, Iran ABSTRACT: Thr sms to b many situations whr an inlastic analysis nds to b prormd on a ram structur. In th last dcads, th subjct has rcivd much intrst rom many rsarchrs, spcially or sismic applications. Dirnt modls hav bn proposd or th ram lmnt to rprsnt its inlastic bhavior, among which, th on componnt modls with inlastic springs ar widly usd. In such modls, ach ram mmbr is rprsntd by an lastic lmnt, with svral nonlinar springs, mostly locatd at th nds o th lmnt. Ths modls sur svral dicincis, mntiond in th contxt. To ovrcom such dicincis, a nw approach is suggstd or th construction o inlastic stinss matrix o th ram lmnt. In th proposd mthod, som virtual inlastic springs ar attachd to th lastic lmnt. Such springs modl plasticity in th strain stats byond th rigid body mods o th lmnt. Th stinss matrix o th ram lmnt, obtaind through this procdur, is compard with th convntional ons or its priority. KEYWORDS: Fram lmnt, Inlasticity, Nonlinar analysis, Plastic hing, Strain stat, Virtual spring. INTRODUCTION Nw advancs in computing tchnology hav prsuadd structural nginrs to adopt mor icint mthodologis or th dsign o building rams, spcially against th hug arthua loadings. Such sophisticatd dsign approachs ar basd upon complicatd analyss which account or th inlastic bhavior o matrials as wll. Instancs to nam ar th two high-lvl mthods o Nonlinar static (Pushovr) and dynamic (Tim history) analyss, that hav put asid simpliying assumptions ndd in convntional analysis and dsign procdurs. Svral dirnt approachs hav bn dvlopd to modl th lastic-plastic bhavior o ram lmnts, among which, th lumpd plastic hing mthod is rgardd as bing o most us and popularity, crtainly du to its simplicity. To implmnt th mthod, zro-lngth plastic hings ar intgratd in sris to th rotational dos o bam nods, mainly at th two nds o th lmnt []. In this rsarch, a nw mthod is proposd to crat th lastic-plastic stinss matrix o a bam lmnt, bniting th concpt o lumpd plastic hing approach. It is basd on th nowldg o dirnt strain stats, th lmnt is capabl o modling. At th irst pac to commnt ormulation, w shall loo through th concpt o strain mods in glancs. Nxt sction will prsnt an acuaintanc with th currnt lastic-plastic stinss matrics, along with a rviw o thir dicincis and imprctions. Thn, th proposd mthod is introducd and applid to construct th lastic-plastic stinss matrix o th Eulr-Brnoulli bam lmnt, whil assssd and validatd in comparison with its countrparts.

2 Th 4 th World Conrnc on Earthua Enginring Octobr -7, 8, Bijing, China. FINITE EEMENT STRAIN STATES In init lmnt mthod, analysts ar sing or an unnown ild o displacmnt, strss, or strain ovr th structur domain. Th ild is assumd to b a linar combination o som basic sub-ilds. In displacmnt ormulation o FEM, Elmnt dormation is considrd as a suprposition o a st o basic dormations, nown as standard basis. From a mathmatical point o viw, ths basic dormations ar actually thos namd as shap unctions. Such unctions display how th lmnt dorms, whn subjctd to a unit displacmnt o a do, whilst all othrs ar constraind against movmnt. Th displacmnt unction is prsumd to b a linar combination o shap unctions, with th coicints turn out to b th nodal displacmnts: u ND (.) Hr, N is th matrix o shap unctions and D is th vctor o nodal displacmnts. This intrpolation rsults in a stinss rlation as: KD P (.) Whr K and P ar stinss matrix and nodal orc vctor in lmnt lvl, rspctivly. Th standard basis is just on among an ininit numbr o bass that can b considrd or th lmnt dormation spac. This basis is o practical us and intrst du to its coicints bing intrprtd as nodal displacmnts. Th lmnt dormation can b rprsntd by anothr usul basic unctions, honord with th titl o strain-stat basis: N u N [ N N N ] n (.) (.4) Th coicints in this linar combination, arrangd in vctor, ar not nodal displacmnts any longr, but rprsnt th portion o corrsponding basic unctions in th total dormation o th lmnt. In th ramwor o this nw basis, th stinss rlation rshaps into th orm o: K P (.5) This is a gnral transormation prototyp that corrlats th displacmnt and orc spacs at th lmnt lvl, according to th basis o N. En... is a spcial orm o this gnral ormulation, rproducd in th basis o N. An ininit numbr o rlations, as En..5., could b imagind, all o which rprsnt a uniu transormation, though in dirnt bass. All ths matrics o K ar mathmatically said to b similar transormations, which ar intr-rlatd as: K G T KG HD P G T P (.6)

3 Th 4 th World Conrnc on Earthua Enginring Octobr -7, 8, Bijing, China K H T K H D G P H T P (.7) Basic unctions o N can b slctd in such a way that no nrgy intraction occurs btwn dirnt strain stats o th lmnt. This status o nrgy orthogonality rstors a diagonal lmnt stinss matrix o K, bstowd th nam o principal stinss matrix, with diagonal ntris labld as ignstinsss: K λ λ O λn (.8) Each diagonal ntry, or ignstinss, is a rprsntativ o its corrsponding strain stat capacity to stor nrgy[].. UMPED PASTIC HINGE MODNG As mntiond bor, amongst dirnt mthods or modling th inlastic bhavior o ram lmnts, th lumpd plastic hing approach has arnd much intrst and applicability or its simplicity and low xpns. In this mthod, th inlastic bhavior o th bam is rprsntd by plastic hings attachd to spcial nods along th lmnt []. Th hings gt activatd as soon as th lmnt ntrs th plastic rang. Plastic hings, to rmar it clar, ar non-activatd ininit-stinss springs, awand whn plasticity is triggrd. Thy ar typically dind as lxurally opratd, though thos o shar and slip prormanc ar also applicabl, particularly in concrt rams. In this papr, lxural springs ar only rrrd. To implmnt th mthod, plastic springs ar constraind in sris to th rotational dos o th two nd nods o th lmnt. Analytically, this mans to add spring lxibilitis to th lastic lmnt lxibility matrix to gt that o th whol systm: F p pi pj (.) F s (.)

4 Th 4 th World Conrnc on Earthua Enginring Octobr -7, 8, Bijing, China F p pi 6 6 pj (.) Th invrs o th last matrix dnots th stinss matrix o th lastic lmnt-plastic spring systm as: 4 i ( j ) 4( j ) i (4 j ) Kp i j 4( j ) i (4 j ) A C C B sym. 4 j ( i ) 4( j ) i (4 j ) (.4) Th ntris o this matrix, as is only dind or th rotational dos, should b positiond in th complt stinss matrix o th lmnt: 6 K p - 6 A - 6 C - 6 sym. B (.5) Th abov stinss matrix surs dicincis, listd blow:. Th ormulation, though capabl o rprsnting lxural hings, is rustratd whn shar hings ar th cas.. It ails to rndr a corrct orm o stinss matrix whn on nd is hingd.. Th matrix ought to b abl to charactriz two rigid-body motions o translation and rotation, whil it lts to only on zro ignvalu, associatd to translation. 4. Th ormulation can only modl plasticity at th nd nods, and whn a sction in btwn gts hingd, it just izzls out. 5. It is undoubtdly not an asy practic to dtrmin th plastic spring coicint, so that assumptions should b mad to simpliy th procdur. Using th slop-dlction ormulation, Chn modiid th procdur to xhibit a sophisticatd dition o bam lmnt stinss matrics, thortically o grat avor, yt, ondd by cass, 4 and 5 o th abov mntiond [4]. 4. STRAIN STATE VIRTUA HINGE MODNG For th lumpd plastic hing approach, plastic springs ar st at th two rotational dos o th bam nds. Howvr, in th proposd mthod, som virtual springs ar appndd in sris to th strain mods o th lmnt. Th procdur is now implmntd or th wll-nown Eulr-Brnoulli bam lmnt, with an lastic stinss as blow [5]:

5 Th 4 th World Conrnc on Earthua Enginring Octobr -7, 8, Bijing, China (4.) K Th lmnt aords a stinss ual to th magnitud o an ignvalu, whn is xposd to a dormation o associatd ignvctor: (4.) [ ] [ ] [ ] Hnc, th principal lastic stinss matrix o th Eulr-Brnoulli bam lmnt is portrayd as: (4.) 6 4 K Two zro ignvalus stand or rigid-body mods, whil two othrs rprsnt constant and linar curvatur stats, as dpictd in Figur. A matrix layout o ignvctors is a transormation tool to mov btwn th two spacs o standard and strain mods: (4.4) G Now, virtual plastic springs ar attachd to th strain stats o th lmnt. As no nrgy is stord in th rigid body stats, thr is no nd to din plastic springs or thm. Howvr, or th two non-zro stinss stats, virtual plastic springs ar st, by adding ignlxibilitis o th lastic and plastic componnts:

6 Th 4 th World Conrnc on Earthua Enginring Octobr -7, 8, Bijing, China Figur. Strain stats o Eulr-Brnoulli bam lmnt p p p EA p (4.5) p4 4 p4 4 6 p4 4 6 p4 (4.6) I ths valus ar invrsd and organizd in an appropriat ram, th principal lastic-plastic stinss matrix o th Eulr-Brnoulli bam lmnt is obtaind: K p (4.7) Th non-diagonal ntry, 4, is also includd to rprsnt th intraction ct o constant and linar curvatur stats, as nrgy orthogonality may not happn in th plastic rang, whil is invitabl ovr th lastic domain. Th lastic-plastic stinss matrix o th lmnt is now rndrd by th us o th normalizd invrs o G as a transormation:

7 Th 4 th World Conrnc on Earthua Enginring Octobr -7, 8, Bijing, China K G - T K G p (4.8) This nw proposd stinss matrix is o grat rigor and robustnss that ovrcoms what has datd its ancstors:. Th stinss matrix can modl shar hing along th lmnt by a slction o virtual spring coicints as:, 4, 4 (4.9). It can rprsnt a corrct orm o stinss matrix whn idally lxural hings appar at ach nd o th lmnt, assuming: 5 6, 4, 4 (4.) or a hing at th irst nd, and 5 6, 4, 4 (4.) i a hing dvlops at th othr nd.. It has a corrct numbr o two zro ignvalus, to rprsnt rigid body motions. 4. It is capabl o modling plasticity and hing mrging procdur, occurring at any arbitrary sction along th lmnt. A cas in point is a hing apparing at th midspan o th girdr, with an lastic-plastic stinss matrix rndrd by spring coicints as:, 4, 4 (4.) 5. Th mthod suggsts a systmatic procdur to dtrmin th plastic spring coicints, through th application o strain mods to th lmnt, masuring th plastic dormations, and mploymnt o appropriat transormations. 5. CONCUSION This rsarch suggsts a nw mthod or th construction o lastic-plastic stinss matrix o ram lmnt, borrowing th convntional concpt o lumpd plastic hings. Th mthod has triumphd ovr th dcts xprincd by prvious approachs, by placing plastic springs, virtually, in strain stats, rathr than th rotational dos o th lmnt. Th mthod will ma it possibl to modl dirnt hings o lxur, shar, and vn slip prormanc, not only at th nd nods, but also at any arbitrary sction along th lmnt. Anothr appaling atur th mthod ualiis is its xtnsibility to othr varity o init lmnts, including plats and shlls. REFERENCES [] Elsiwat, J. (99). Ect o anchorag slip and inlastic shar on sismic rspons o rinorcd concrt rams, PhD Assrtation, Univrsity o Ottawa, Canada.

8 Th 4 th World Conrnc on Earthua Enginring Octobr -7, 8, Bijing, China [] Khajavi, R. (4). Partitioning th stinss matrix and th principal stinss matrix o init lmnt, MSc Assrtation, Frdowsi Univrsity o Mashhad, Iran. [] Nanaorn, P. (4). A Two-dimnsional Bam-column Finit Elmnt with mbddd rotational discontinuitis. Computrs and Structurs 8, [4] Arai Shahidi zadh, M. (). An invstigation o ram stability including th nonlinar bhavior o matrials, MSc Assrtation, Frdowsi Univrsity o Mashhad, Iran. [5] Rzai Pajand, M. (99). Thory o matrix analysis or structurs, Frdowsi Univrsity o Mashhad Prss, Iran.