STRUCTURAL DYNAMIC RESPONSE MITIGATION BY A CONTROL LAW DESIGNED IN THE PLASTIC DOMAIN

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1 STRUCTURAL DYNAMIC RESPONSE MITIGATION BY A CONTROL LAW DESIGNED IN THE PLASTIC DOMAIN 49 Alessandr BARATTA 1 And Oavia CORBI SUMMARY In revius aers [Baraa and Crbi, 1999] a cnrl algrihm has been rsed, referred an elas-lasic srucural mdel and based n he minimisain f a funcinal f he elas-lasic resnse, cnsrained by an uer bund n he cnrl energy; he funcinal has rved be a reliable index f he cnrl effeciveness in he lasic dmain, while he assumed algrihm, esed fr variusly shaed dynamic lads, has shwn achieve an exremely reduced srucural resnse when cmared he uncnrlled ne. In he aer ne cnsiders he ssibiliy relae he resuls deriving by he adin f such a rcedure hse ne can ge by fllwing an exac cnrl sraegy which minimises, fr he same assumed frcing funcin, he maximum value f he lasic dislacemen while cnaining he maximum cnrl frce under he highes value i reaches in he lasic-cnrlled srucural sysem numerical simulain. Furhermre, a cmarisn is rsed wih he resnse f he elas-lasic mdel cnrlled by means f a classical elasic nrm sraegy. The cnclusin is ha, whereas he lasic cmnen is cnsiderable, he deees miigain f he srucural resnse is bes accmlished by means f he lasic- cnrl rcedure. INTRODUCTION The rblem miigae he resnse f a srucure subjec a seismic acin can be arached by he imlemenain f acive cnrl echniques. In seismic engineering, due he ssibiliy ha he quake inensiy exceeds he design eak accelerain, even if an effecive cnrl acin is alied, an high rbabiliy exiss ha he srucural resnse exceeds he elasic hreshld. Fr he elasic-lasic mdel assumed fr he cnsidered s.d..f. srucure, a cnrl sraegy has been develed [Baraa and Crbi, 1999b] in rder aenuae he lasic resnse f he srucure, by fllwing he usual echnique fr bunding dislacemens in elasic lasic dynamics [see i.e. Maier, 1973 and Caurs, 1975]. The arximae echnique is rey reliable and aricularly desirable whereas large lasic resnse cmnens are sused ccur. THE STRUCTURAL MODEL AND THE EQUATIONS OF THE MOTION Le cnsider he equains f he dynamic equilibrium gverning he min f a s.d..f. srucural sysem [Figure 1] wih daming and frequency resecively ζ ed ω, subjec a frcing funcin f and cnrlled by a frce c, linearly deenden n he dislacemen variable u, sliable in an elasic ue and a urely lasic u cmnen in such a manner ha 1 Dearmen f Scienza delle Csruzini Universiy f Nales Federic II, Nales, Ialy alebara@unina.i Dearmen f Scienza delle Csruzini Universiy f Nales Federic II, Nales, Ialy crbi@unina.i

2 = + u u u e (1) And n is derivaive u by means f he cnrl arameers µ e ϖ. In he elasic field ne can wrie u E + ζωu E + ω ue + ce µ, ϖ = f ce µ, ϖ = µϖu E + ϖ ue u u ; u E = E = u ( ) ( ) where ue is he elasic resnse f, while, wheher he exciain rduces any excursin in he lasic field, he sandard equain f he min is exressed by u u Uuuu,,, u c, f, u u u Hu u u u Hu u u u ( ) ( ) ( ) + ζω + ω + µ ϖ = c µ ϖ = µϖ + ϖ [ ] [ ] = + + (3) where he erms U u u u = u + u u = u + u,, are given in Table 1, H(x) is he se funcin and ; (4) u u + u u u = u u = u u = (5) ( ) u = s / ω ; u = s / ω ; s, u > ; s, u < (6) wih s, s he values a he yielding hreshld f he variable [ ] = ω = ω s u u u e (7) which reresens he sandard (ha s say wih he hysical dimensins f an accelerain) shear sress in he iles. U, u, u values in he elasic and lasic field. Table 1: The Elasic field: Plasic field: Plasic field: u u u = = U = u u u u u u ; u u U = u = u = u u u u u ; u u U = u = u = u u u u Mrever [ ] s u = ω u u u (8) ( ) In he elas-lasic field, marking by s an admissible law s s s fllwing relains are me as well [ s s ] u [ s s ] s s f he sandard shear sress, he (9) [ ] (1) 49

3 [ ] [ ] s s s s u = (11) ha s say, frm eqn (9) eqn (11), he Drucker s sulae, he yielding cndiin and he rule f lasic flw. s ' f c s' > L' θ'= L' e e " u" < θ u' > u" θ u u' u θ"= L" s" < L". u u e A B Figure1: The cnsidered elasic erfecly lasic mdel a) f he s.d..f. sysem b), wih θ=ω ο. EVALUATION OF THE PLASTIC DISPLACEMENT IN A CONTROLLED SYSTEM. In rder evaluae he dislacemen induced in he srucural sysem during he seismic min, by fllwing he resnse bunding arach by Maier [Maier, 1973], le cnsider: 1. The linear elasic resnse u s which saisfies eqn.. A ficiius frcing funcin f, wih u, s. N N Furhermre, if ne cnsiders he ficiius min E N E N E, he same seismic exciain wih he same iniial cndiins, E N wih arbirary iniial cndiins un, u N, which he srucure resnds u = u + u ; s = s + s ; (1) and a ime hisry f he sandard shear sress s such ha = + s s s desn vilae eqn (1), ha s say he yielding cndiin, n he durain f he min T, and ha culd be e.g. [ ] [ ] [ ] [ ] s = s s H s s + s s H s s (13) (14) ne ges [ ] [ e ] [ ][ ] [ ][ e ] s s u u + u u s u s s u u s s u u s u (15) and, by he virual wrk rincile, afer sme algebra, 3 49

4 du τ ϖ 1 f u s u ϖ d f u T L {[ s s ] u N τ τ τ τ ϖ N τ τ τ τ τ τ τ} (16) dτ ω ω dτ whence ne realises ha he lasic dislacemen can be bunded ignring he addiinal frce f N and using he cnrl cmnen ϖ u as he ficiius frce. Being T ( τ ), L ( τ ) resecively he (siively defined) kineic and elasic energy f u ( τ) = u( τ) u ( τ) and L ( τ ) he ime derivaive f he laer, fr fn = and he same iniial cndiins fr he w mins, knwing ha u ( ) = and st ( ) s ( T) = =, ne ges 1 ϖ u s τ u τ dτ s τ u τ dτ. (17) Furhermre, wih sme addiinal algebra, alying he Schwarz inequaliy and nrmalising wih resec he energy f he frcing funcin, ne can wrie u f u τ dτ ( ) s d f f τ τ µ ϖ τ dτ ϖ f = F l, f = (18) F l (µ,ϖ f) has been shwn [Baraa and Crbi, 1999a] be a significan funcinal f he lasic resnse: acually, sme numerical ess have been exerienced n a s.d..f. srucure wih naural frequency ω =3 rad/sec, daming cefficien ζ=3% and limi shear sress values s = 4 cm / sec, s =4 cm / sec, subjec a sine wave funcin lasing 5 secs wih ulsain ω f =15 rad/sec and amliude f,. Such exerimen yields: i) a urely elasic min fr f = cm/sec, ii) an iniial lasic excursin fllwed by a seady-sae elasic sabilizain fr f = 3 cm/sec, iii) a sequence f elasic erfecly lasic cycles fr f > 4 cm/sec ; he resuls rve ha F l (µ,ϖ f) is an increasing funcin f he maximum lasic dislacemen u max in he range f he daming cnrl cefficien <µ<1 fr differen values f ϖ, whereas lasic excursins ccur [Figure.a]; ne can als nice ha he sle f he maximum lasic dislacemen versus he funcinal is small when he lasic cmnen f he resnse is marginal (i.e. f =3 cm/sec ), while i increases, when he lasic henmenn is very significan (i.e. f = 4, 5, 6 cm/sec ), ha s say ha he funcinal is rey sensiive sme increase in he resnse lasic cmnen. Mrever, as he insananeus lasic rae is rduced by direcly absrbing energy frm he grund accelerain 1 u d f τ τ τdτ f = (19) ne can ge he final exressin u s( ) d f Nl ( f τ τ = µ, ϖ ) ϖ N, f while Hence, a ssible cnrl sraegy culd be search fr he minimum f he funcinal l ( µϖ ) keeing he energy C( µϖ, f) f he cnrl frce c, nrmalised wih resec he exciain energy, under a desired hreshld C and he imum rblem can, finally, be defined as fllws find Nl ( µϖ, f ) = min (1) sub C µϖ, f C ( ) 4 49

5 .3. u max f =5 f =4 = 6 f ϖ = 5 ϖ =1 ϖ = 15 ϖ = f =5 f =4.E- 1.5E- 1.E- N el F l f =6 f =5 ϖ = 5 ϖ=1 ϖ = 15 f =4.1 f =3 5.E-3 f =6 f =4 f f =3 =5 f =4 µ. F l A B Figure : a)the max lasic dislac. u max fr µ [,1] vs he lasic funcinal F l fr varius ϖ and f. b)the elasic N el and lasic F l funcinals versus µ fr varius ϖ and f. EVALUATION OF THE OBJECTIVE FUNCTIONAL If he ficiius frcing funcin is equal zer, i.e. f N =, s = s E, he funcinal can be exressed as + Nl f s d f s d s d f ϖ ϖ + ( µϖ, ) = τ τ = τ τ τ τ where + [ ] [ ] [ ] s d { s s }[ s s ] d P [ s s ] d P s τ τ E τ τ E τ τ τ E τ τ τ E τ s dτ = > = Prb ~, ~ ~ ~ max + [ ] s d { s s }[ s s ] d P [ s s ] d P τ τ = Prb ~ E τ <, τ ~ E τ τ= τ ~ E τ τ τ ~ s E max τ s dτ (3) wih + and - he insans a which he lasic hreshld is exceeded in he w direcins. As ~ s + s h f 1 + s s h f 1 (4) τ ω τ ( τ γ ); ~ τ ω τ ( τ γ ) Emax c Emax c wih γ +, γ w design cefficiens suiably defined, ne ges ω N ( f) h f ( ) P d ( ) P l µϖ, = + + c 1 γ τ τ+ γ τdτ ϖ 1 + (5) u E, he marginal densiy f he dislacemen schasic variable ~u E, and u, u he dislacemen values a he yielding hreshld, ne can exress he rbabiliies P + and P - qued in eqn (5) Marking by ( u ) 5 49

6 + Prb { ~,} (, ) Prb{ ~ + u u u du s s, } P E > = u E E = E > = ; Prb { ~, } (, ) Prb{ ~ u u u du s s, } P E < = u E E = E < = u wih u( ue, ) having a suiable exressin 1 u E ( u ) e 1 1 uh u u E, = ρ if u < u E h h if u > u E h NUMERICAL RESULTS The srucural s.d..f. elas-lasic mdel f Figure 1, wih frequency ω = 3 rad sec and daming cefficien ζ=5%, and u = cm and u = cm, subjec an harmnic frcing funcin resuling by he cmbinain f en sine waves lasing 1 secs and acing a ω f =15,18,1,4,7,3,33,36,39,4 rad/sec 1 is cnsidered. One refers fr he uncnrlled resnse under he given exciain, and is cmarisn wih he lasic cnrlled resuls Table, which summarises sme reviusly fulfilled numerical invesigains [Baraa and Crbi, 1999b]. In Figure 4 he cmarisn f he al and lasic dislacemens ne ges by fllwing he abve rered rcedure wih hse deriving frm he adin f a nrm sraegy [Baraa and Viell, 1985] and an exac lasic sraegy is rsed. In deails, in he laer case, ne has assumed he bjecive funcin f he imum numerical rcedure cinciding wih he maximum lasic dislacemen ha ne ges by eraing he ime simulain (in ha he rcedure is exac ) n he linearly cnrlled mdel; he cnsrain cndiin cnsiss, in his case, f bunding he maximum read value f he cnrl frce under he maximum i reaches by ading he rsed arximaed lasic cnrl mehd; he rblem, is hence u 1 (6) (7) find u max ( µϖf) ( µϖ, ) sub f c c max, = min max (8) where c max is given in Table. I is clear ha, whereas he frcing funcin induces in he srucural sysem a main lasic cmnen, he bes erfrmance in erms f resnse reducin is accmlished by he lasic sraegy. µ Nrm cnrl µ Plasic cnrl ϖ = µ = ω ϖ = , µ = ω Figure 3: Cnsrained imisain fr he cnsidered cnrl araches. 6 49

7 Table : Perfrmance indexes fr he cnsidered cnrl araches. uncnrlled lasic crl u max exac lasic nrm crl u max u max c max 366= c max u(t) u (T) c(t) u, u.e+3 s. nrm crl lasic crl exac lasic crl 1.E+3-4..E u -8. nrm crl lasic crl exac lasic crl A B -1.E+3 -.E+3 Figure 4: Cnrlled resnse: a) al and lasic dislacemen; b) hysereic cycles. Acually he nrm sraegy, (an ineresing cmarisn is rsed in Figure.b beween F l and he elasic nrm N el, assumed as he bjecive funcin in a susedly urely elasic behaviur mdel [Baraa and Viell, 1985]), by nearby using he same cnrl frce (Figure 5b), seems fail when deeening in he lasic dmain (he seady-sae rin). On he her hand, he exac rcedure achieves a lighly lwer maximum dislacemen, while shwing a wrse erfrmance, in he average, n he ime vecr lengh, when alying a cnrl frce rey clse he lasic cnrl ne (Figure 5a). Frm Table ne can sill emhasise ha, fr he exac cnrl rcedure a small reducin in erms f he maximum values f he resnse variables resuls in higher residual values a he end f he min. CONCLUSIONS An imaliy crierin fr a linear cnrl sraegy in he elasic-lasic range has been rsed fr an elasicerfecly lasic s.d..f. srucural mdel subjec a dynamic frcing funcin. The arach, finalised reduce and miigae he excursins in he lasic dmain, is based n minimising he resnse exceeding he lasic hreshld, while keeing bunded he energy f he cnrl frce. The adin f he algrihm resuls in a dee reducin f he srucure duciliy demand and, if cmared wih cnrl sraegies assumed fr he linear mdel and usually alied nn-linear behaviur mdels, in a much higher ime and residual resnse miigain. An exac cnrl sraegy has als been cnsidered, in rder evaluae a ssible imrvemen rduced by he exac sluin f he differenial equains sysem when minimising he real maximum lasic dislacemen: even in his case, due he simly lcal acin f his arach, he lasic algrihm has shwn a visibly beer erfrmance and, definiively, an higher reliabiliy wih reference srucural sysems wih a main lasic cmnen resnse. 7 49

8 .E+3 c.e+3 c.e+.e+ -.E+3 -.E+3 exac lasic crl lasic crl -4.E A nrm crl lasic crl -4.E B Figure 5: The cnrl frce invlved in he cnsidered cnrl araches. ACKNOWLEDGMENTS Paer sured by grans f he Ialian Nainal Cuncil f Research (C.N.R.) and cfinanced by he Ialian Minisry f he Universiy and Scienific and Technlgical research (M.U.R.S.T.). REFERENCES Baraa, A. (1983), Plasic Defrmains in Srucures Under Randm Lading. Prceedings f he Inernainal Cnference n he Alicains f Saisics and Prbabiliy in Sil and Srucural Engineering. Baraa, A., Crbi, O. (1999a), An Oimal Arach Linear Cnrl f Elas-Plasic Srucures. Prceedings f he Inernainal Engineering Mechanics Secialy Cnference. Baraa, A., Crbi, O. (1999b), Prgeazine di un Algrim di Cnrll Aiv er Sruure a Cmramen Elas-Plasic. Prceedings f he Nainal Cnference L Ingegneria Sismica in Ialia. Baraa, A., Viell, G. (1985), Prgeazine di Sisemi di Cnrll Sruurale Anisimic mediane Sluzini in Nrma. Prceedings f he Nainal Cnference L Ingegneria Sismica in Ialia. Caurs, M. (1975), Dislacemen Bunding Princiles in he Dynamics f Elaslasic Cninua. Jurnal f Srucural Mechanics. Vl. 3, g Maier, G. (1969a), Cmlemenary Plasic Wrk Therems in Piecewise-linear Elas-lasiciy. Inernainal Jurnal f Slids and Srucures. Vl. 5, n.. Maier, G. (1969b), Sme Therems fr Plasic Raed and Plasic Srains. Jurnal de Mécanique. Vl. 8, n.1. Maier, G. (1973), Uer Bunds n Defrmains f Elasic Wrk-hardening Srucures in he Presence f Dynamic Secnd Order Effecs. Jurnal f Srucural Mechanics. Vl., g. 65. Marin, J.B. (1965), Dislacemen Bund Princile fr Inelasic Cninua Subjeced cerain Classes f Dynamic Lading. Jurnal f Alied Mechanics. Vl. 3, g.1-6. Marin, J.B, Pner, A.R.S. (197), Bunds fr Imulsively Laded Plasic Srucures. Jurnal f Engineering Mechanical Divisin. Vl. 98, g