Novel Technique for Dynamic Analysis of Shear-Frames Based on Energy Balance Equations

Transcription

1 Novel Technque for Dynamc Analyss of Shear-Frames Based on Energy Balance Equaons Mohammad Jall Sadr Abad, Mussa Mahmoud,* and Earl Dowell 3. Ph.D. Suden, Deparmen of Cvl Engneerng, Shahd Rajaee Teacher Tranng Unversy, Tehran, Iran.. Assocae Professor, Deparmen of Cvl Engneerng, Shahd Rajaee Teacher Tranng Unversy, Tehran, Iran. 3.Wllam Holland Hall Professor, Deparmen of Mechancal Engneerng and Maerals Scence, Duke Unversy, Durham, U.S.A. 3 Absrac In hs paper, an effcen compuaonal soluon echnque based on he energy balance equaons s presened for he dynamc analyss of shear-frames, as an example of a mul-degree-of-freedom sysem. Afer dervng he dynamc energy balance equaons for hese sysems, a new mahemacal soluon echnque whch s called Elmnaon of Dsconnuous Veloces s proposed o solve a se of coupled quadrac algebrac equaons. The mehod wll be llusraed for he free vbraon of a wo-sory srucure. Subsequenly, he damped dynamc response of a hree-sory shear-frame whch s subjeced o harmonc loadng s consdered. Fnally he analyss of a hree-sory shear-buldng under horzonal earhquake load, as one of he mos common problems n Earhquake Engneerng, s suded. The resuls show ha hs mehod has accepable accuracy n comparson wh oher echnques, bu s no only faser compared wh modal analyss bu also does no requre adjusng and calbrang he sably parameer as compared wh a mehod of me negraon lke he Newmark mehod. KEYWORDS: Numercal Technque; Dynamc Analyss; Shear-Frames; Energy Balance Equaons; Coupled Equaons; Elmnaon of Dsconnuous Veloces E-mal: m.jall@sru.edu, Phone: , Moble: , Fax: , P.O. Box: * Correspondng auhor; E-mal: m.mahmoud@sru.edu, Phone: , Moble: , Fax: , P.O. Box: E-mal: earl.dowell@duke.edu, Phone: ( , P.O. Box:

2 . Inroducon Generally, n all engneerng felds ha deal wh srucural desgn, undersandng he dynamc behavor of srucures s very mporan []. In hs conex, alhough he applcaons of srucural dynamcs n aerospace engneerng, cvl engneerng, engneerng mechancs, and mechancal engneerng are dfferen, he prncples and soluon echnques are bascally he same []. Accordngly, dynamc analyss plays a val role n analyzng he dynamc response of buldngs [3], dams [4,5], and brdges [6] o earhquakes. Conrol of very all and slender buldngs are among he mos mporan ssues for cvl engneerng researchers and have frequenly been nvesgaed n recen years. Almos all praccal srucures are mulple-degree-of-freedom (MDF sysems, because of he dsrbuon of dynamc properes such as mass n real sysems, so many DOFs are requred o deermne he vbraonal moon [7]. In addon, as we know, a greaer number of DOFs wll ncrease he complexy of solvng a vbraon problem. Thus, n engneerng applcaons, we prefer o work wh fewer DOFs whou losng oo much accuracy. For example, n he modelng of dynamcal sysems, smple srucures (such as a waer ank when he majory of he mass of he sysem s concenraed n an area of he srucure can be dealzed as a sysem wh a lumped mass (SDF 4 sysems [8]. Also, under some condon such as when a mahemacal funcon can express he varaon of he mass and sffness of srucure, he real sysem s consdered as a generalzed sngle-degree-of-freedom sysem [3]. Furhermore, here are oher echnques o reduce he dynamcal DOFs of large order sysems under some condons (e.g., refer o [9- ]. However, n many cases n praccal engneerng works here s no he possbly of smplfyng a real sysem o an SDF sysem, and performng an MDF dynamc analyss s essenal. There are varous mehods n order o evaluae he dynamc response of MDF sysems. For example, n some specal cases, by usng he mahemacal ools such as Fourer and Laplace negral ransforms he exac soluon of hese problems can be obaned [,3]. Moreover, modal analyss s a convenonal approach o evaluae he response of MDF srucures whch are subjeced o dynamc loads. One of he dsadvanages of hs mehod s s lmaon for srucures wh non-lnear behavor []. However some researchers have red o modfy he modal analyss n order o use for nonlnear analyses; bu, here s no sll a comprehensve mehod for modal analyss of nonlnear srucures (see, for example, [4-]. Even hough here exs some echnques o deermne he egenvalues and egenvecors of large order sysems (e.g. refer o [,]; as DOFs ncrease he calculaon of egenvalues and egenvecors s parcularly dffcul, and ha s anoher dsadvanage of hs approach. In engneerng analyses, he mos general soluon mehod for dynamc analyss s an ncremenal mehod or sepby-sep drec me negraon echnque n whch he equlbrum equaons are solved a mes,,3, ec. []. In hs caegory, Newmark [3], Houbol [4] and Wlson [5] are some common mplc mehods, and Cenral Dfference mehod s one of he well-known explc mehods [6]. Sably and accuracy of hese mehods are essenal n he praccal analyss [7-3]; herefore, s very mporan o use accurae and numercally effcen echnques n compuer programs [3]. As a resul of he large compuaonal requremens, can ake a sgnfcan amoun of me o solve srucural sysems wh jus a few hundred DOFs [6]. In addon, arfcal or numercal dampng mus be added o mos ncremenal soluon mehods o oban sable soluons. For hs reason, engneers mus be very careful n he nerpreaon of he resuls []. Here, should be noed ha he arfcal dampng, whch s defned as he reducon of he dsplacemen amplude wh me for an undamped sysem [33], s dfferen from he dampng propery of he srucures. Usng energy balance equaons, whch s proposed n hs sudy, can be an alernave approach o evaluae he dynamc response of a mul-dmensonal sysem. In hs conex, as an nsance, he energy conservaon and dsspaon properes of me-negraon mehods nvesgaed by Acary [34] for he non-smooh elasodynamcs wh conac. Even hough, several researchers n varous felds such as hydrodynamc [35], aerospace [36,37] and CFD 5 [38,39] have suded he energy mehod n order o deermne he response of her dynamc sysems, ye less aenon has been pad o hs opc n srucural dynamcs, excep for a few sudes ha ofen have red o use Hamlon s Prncple n order o calculae he frequency of smple SDF srucures (see, for example, [4,4]. Accordngly, hs sudy ams o presen a new numercal sep-by-sep mehod based on he energy equaons for MDF shear-frames. The man dea of hs approach s nroduced frs n [4] for lnear and nonlnear SDF sysems, and n he presen paper, hs echnque s nended o be generalzed o lnear MDF srucures. Ths mehod n he absence of dampng leads o he de-coupled quadrac equaons; and, when dampng s consdered, leads o a se of coupled quadrac equaon. Accordng o he quadrac form of he algebrac equaons a each me sep, a novel mahemacal echnque called Elmnaon of Dsconnuous Veloces s presened o deec he real velocy n every nsance. 4 Sngle-degree-of-freedom 5 compuaonal flud dynamcs

3 In hs sudy, shear frames are seleced o llusrae he proposed mehod. The mehod largely elmnaes he dsadvanages of oher mehods, such as mahemacal complexy and me-consumng calculaon of a modal marx n large order srucures, as well as sably concerns and adjusmen of he analycal coeffcens n numercal negraon mehods. I should be noed ha s possble o exend hs approach o oher mul-dmensonal srucures as well. Furhermore, whle n hs nvesgaon we assume ha he srucure wll behave lnearly; s possble ha he proposed mehod wh some smple modfcaons can be used for nonlnear analyses n fuure sudes.. Force vs. energy equaons In hs secon, shear-frames are nroduced n bref. Subsequenly, he mechancal energy relaonshps of hese sysems are expressed and by usng he prncple of conservaon of energy, he equaons of moons are derved from energy balance relaonshps of he sysem. The prncpal objecve of hese relaons s o prove he equvalence of he force and energy approaches n srucural dynamcs. Fnally, a he end of hs secon, he advanages and dsadvanages of usng hese mehods are compared wh each oher. Fg. depcs an n-sory shear-frame (or shear-buldng as one of he smples mul-degree-of-freedom sysems ha are wdely used n cvl engneerng. In hs dealzaon, he beams and floor sysems are rgd n flexure, and several facors are negleced lke axal deformaon of he beams and columns, and he effec of axal force on he sffness of he columns [8]. In hs respec, he defleced buldng wll have many of feaures of a canlever beam ha s defleced by shear force only, hence he name shear-buldng [43]. where x denoes he dsplacemen of h sory. Moreover, k and m are respecvely he sffness and mass of h sory. For hese srucures, he poenal energy of he sysem (E P can be expressed as follows, by assumng a lnear relaonshp beween force and dsplacemen. E m v m v... m v... m v m v ( K n n n n Also, he knec energy of he srucure regardng v dx / d (he velocy of h mass s gven by n n E k ( x ( T k x x m v ( By neglecng he effecs of energy dsspaons and usng he summaon noaon, he oal energy of he sysem (E T (sum of he poenal and knec energes can be wren as n n E k ( x ( T k x x m v (3 From he physcal pon of vew, he law of conservaon of energy saes ha he oal energy of an solaed sysem remans consan, s sad o be conserved over me [44]. Hence, dfferenang Eq.(3 wh respec o me, we oban de T (4 d alernavely, where a s he acceleraon of h mass,.e. n k x v k ( x x ( v v m v a n (5 a dv (6 Expandng he seres n Eq.(5 leads o k x v k ( x x v k ( x x v k ( x x v k ( x x v By facorng v, v, v 3,..., one can wre d m v a m v a m v a v k x k ( x x m a ] v k ( x x m a ]... v k ( x x m a ] (8 n n n n n n (7 3

4 Whch corresponds o he followng marx form m x k k k x m x k k k k x 3 3 k k 3 n m x k k x n n n n n (9 As we know, Eq.(9 represens he dynamc force equlbrum equaons of an n-sory shear-frame, and as prevously menoned, he prmary objecve of hs par was he proof of he equaly of energy and force balance equaons. As a resul, we can say ha force equlbrum equaons can be obaned from dervave of energy equaons and muually energy equaons mgh be derved from he negraon of force balance equaons. I mus be saed ha alhough bascally, hese wo equaons are he same bu each of hem has s own advanages and dsadvanages n pracce. As an llusraon, see Table. Here, s mporan o noe ha he presened mehod n hs sudy ncludes he lnear behavor of shear frames (as a smple srucural sysem. In oher words, nonlnear response analyss of general srucure lke sysems wh hyseress does no fall whn he scope of hs work. However, he dea presened n hs research can be he bass for he ulmae goal of dynamc analyss of large-scale srucures wh dfferen ypes of nonlneares. 3. Mehodology 3.. Dervaon of dscrezed energy balance scheme Now, he energy balance approach s exended for general forced vbraon problems o nclude he effecs of dampng, n whch dampng s assumed o be lnear regardng of velocy (vscous dampng n hs sudy. For hs purpose, consder he equaons of moon of an n-dof shear-frame as follows m x ( c c x c x ( k k x k x p m x ( c c x c x c x ( k k x k x k x p m x ( c c x c x c x ( k k x k x k x p ( m x ( c c x c x c x ( k k x k x k x p n n n n n n n n n n n n n n n n n m x c x c x k x k x p n n n n n n n n n n n where c and p denoe he dampng coeffcen and exernal force of h mass. Inegrang he equaon of Eq.( wh respec o x, we ge h m x ( c c x c x c x ( k k x k x k x dx p dx ( Each par of Eq.( accordng o he defnon of varous energes,.e., he area under he curve of he loaddsplacemen, mples he changes of a specfc ype of energy. ( ( m x dx c c x c x c x dx k k x k x k x dx p dx E E E K D P The frs negral n LHS represens he changes of knec energy E k, and by defnon of velocy, akes he followng form xdx vdv E m x dx m v dv Inegraon from zero o arbrary me gves K 4 E F ( (3 E m v m v (4 K (

5 The second negral n lef-hand-sde expresses he changes n damped energy ( E D, whch s somemes also called he energy loss, ha s E ( D c c x c x c x dx (5 Usng he defnon of velocy, Eq.(5 can also be wren as E ( c c v c v v c v v d D (6 Also, he change n poenal energy ( E P s E ( P k k x k x k x dx (7 By operaons equvalen o Eq.(7, can be shown ha E ( P k k x k x k x v d (8 and evenually, he changes n energy of he exernal loads E F s gven by E p dx (9 F Smlarly, n order o have n erms of velocy, Eq.(9 becomes E p v d ( F Therefore, energy balance equaon for he h mass s as follows E E E E ( k D P F Here, energy balance equaons are wren for all of he masses mv mv ( ( c c v cvv d ( k k x kx vd pv d ( ( ( m v m v c c v c v v c v v d k k x k x k x v d p v d m v m v ( ( ( c c v c v v c v v d k k x k x k x v d p v d ( n n n n( ( n n n n n n n n n ( n n n n n n n n n n n n n n ( n n n n n n n n n n n n m v m v c c v c v v c v v d k k x k x k x v d p v d m v m v c v c v v d k x k x v d p v d Consderng he h mass m v m v ( ( ( c c v c v v c v v d k k x k x k x v d p v d (3 In prncple, afer dscrezng Eq.(3 by usng numercal negraon mehods, such as Trapezodal and Smpson echnques [], he corresponden energy equaon of h mass would be evaluaed from Eq.(4. (See Appendx.A, for deals Av B v v C v v D v E (4 ( j ( j ( j ( j ( j ( j where, A, B, C, D, E are consan coeffcens ha are deermned from dscrezng of negrals n energy balance relaons; e.g., n he frs me sep ha we have o use he Trapezodal mehod, hese coeffcens ake he followng forms 5

6 A.5m.5 ( c c, B.5. c, C.5. c D.5 ( k k x k. x k. x p E ( ( ( (.5m v.5. v [( v c. v c. v ( x kx ( ( ( ( ( ( ( c c k k k x p ] ( ( In he me seps afer he prmary me sep, o ncrease he accuracy of negraon by usng he Smpson mehod, one can wre A.5 m ( / 3( c c, B ( / 3. c, C ( / 3. c D ( / 3 (k k x k. x k. x p ( j ( j ( j ( j E m v v c c v c v c v k k x k x.5 ( / 3 [(.. ( ( ( ( ( ( ( 4 v [( c c v c. v c. v ( k k x k x k x p ] ( ( ( ( ( ( ( ( v [( c c v c. v c. v ( k k x k x ( ( ( ( ( ( ( ( k x p k x p ( ( ( v [( c c v c. v c. v ( k k x k x k x p ] ( j ( j ( j ( j ( j ( j ( j ( j where j denoes he number of seps. 3.. Soluon procedure of coupled quadrac energy equaons As we saw n he prevous secon, afer dscrezng he energy balance equaons, we are encounered wh a se of equaons n he followng quadrac form a v c v v d v e a v b v v c v v d v e 3 ] ] (5 (6 a v b v v c v v d v e (7 a v b v v c v v d v e n n n n n n n n n n n a v b v v d v e n n n n n n n n Mahemacally, n solvng he prevous equaons here are wo man problems: A hese equaons are coupled and ha means hey are dependen on each oher and mus be solved smulaneously; n oher words, we canno calculae v from h equaon drecly. In addon, should be noed ha n he absence of dampng he equaons would be decoupled, vz., dampng s he reason for couplng he equaons. B The quadrac form of equaons mples ha more han one velocy a each me sep;.e., from he physcal pon of vew a every me sep hese relaons provde some unreal veloces n addon o he acual velocy of he srucure. To beer undersand he above expressons, suppose ha n a sample -DOF srucure n a gven me sep we wan o solve a mahemacal equaon of he form v v v v (8 v v v v If he erms of v v do no exs, one can oban wo values of each of v and v from solvng he wo uncoupled quadrac equaon. However, wh consderng he v v, by combnng he equaons ogeher and wre hem only n erms of one varable, we have 4 3 v v 6v 3v (9 4 3 v v 8v v 4 From Eq.(9, by solvng he wo fourh-degree equaon, s apparen ha four values for each of v and v wll be obaned. Noe ha a any momen he velocy of each mass s unque and only here s one value for real velocy of he sysem whle n hs case, hree unrealsc veloces for each mass have appeared n he equaon. Apparenly, hs 6

7 mehod (drec mehod canno be used o calculae he veloces a any nsans, especally n large DOF sysems, and we have o use a numercal mehod o calculae he veloces n each me sep. In hs conex, as s demonsraed n Appendx.B, well-known soluon echnques lke Newon-Raphson mehod are no effcen for he sysem of equaons under consderaon n hs sudy. Two man reasons for he defcency of hese mehods when appled o he consdered equaons n each me-sep can be expressed as follows: a Need for dervave of he sysem of equaons b Complex and me-consumng process of nverng he Jacob marx (specfcally n large-scale srucures Elmnaon of dsconnuous veloces echnque The problem of coupled equaons exss n many engneerng felds, parcularly n mul-dmensonal sysems, hence many researchers have suded how o solve hese equaons (e.g., refer o [45-5]. For he curren sudy, a novel numercal mehod s presened n whch he real veloces of he sysem a any me sep can be easly calculaed by removng he unrealsc veloces from he coupled equaons. In he proposed echnque, frs, he problem of coupled equaons s resolved by neglecng of he couplng erms (erms ha are he produc of wo dfferen veloces. In hs case, assumng a srucure wh n-dof, we are faced wh an n-quadrac equaon n erms of velocy. To solve he problem of non-lnear equaons (quadrac n erms of velocy and deec he acual veloces of sysem a any me, s assumed ha he varaon of veloces wh respec o me s connuous. Therefore, among he wo veloces obaned a any me from he quadrac equaons, he velocy closer o he velocy of he prevous me sep s seleced as he real velocy of he srucure. Therefore he name of he mehod s chosen as Elmnaon of Dsconnuous Veloces Technque. A he begnnng of hs procedure, he couplng erms are gnored o oban he veloces; here, he values of connuous veloces are subsued no hem, and hs eraon wll be connued unl he veloces n wo subsequen eraon approach o each oher. Table. gves a summary of he mehod. 4. Numercal examples and resuls Varous examples of mul-sory shear-frame srucures are analyzed by usng he energy mehod n hs secon. In he frs example, he vbraon of a smple wo-sory shear-buldng has been nvesgaed o descrbe he procedure of presened mehod n deal. In he nex examples, some mul-sory shear-frame srucures subjeced o harmonc and earhquakes loadngs have been suded. Moreover, he resuls were compared wh he exac soluon and oher common mehods Example.4.. The free damped vbraon of a wo-sory shear buldng. Fg. shows a -DOF shear-frame n whch for convenence he dynamc properes of he srucure are chosen as: m =m =k =k =, and c =.6, c =.6 (all uns assumed o be compable. Also, he followng nal condons wll be consdered n hs example. 3 x, v (3 4 In free-vbraon cases, he equaon of moon of hese srucures can be expressed as m x c x k x (3 where he mass, dampng, and sffness marces are..6 m c k.6.6 (3 Thus, by mulplyng he marces and vecors, we have he followng governng equaons x.x.6x x x (33 x.6x.6x x x Here, he Laplace ransform mehod s used o deermne he exac soluon of hs problem (for deals, see Appendx.C x.336e cos( e cos( ( x.e cos( e cos(

8 Applyng he energy mehod By subsung he assumed parameers of hs example no he Eq.(3, he energy balance equaons of he sysem would be as follows.5v.5v (. v.6v v d x x v d (35.5v.5v.6v.6v v d ( x x v d Now, he negrals n above equaons should be dscrezed o oban he algebrac equaons. Snce Smpson rule needs a leas hree-pon for negraon, canno be used n he frs me sep. Hence rapezodal mehod mus be used n he frs me sep. For he problem a hand, he sze of me nervals are assumed o be =.s, and accordng o Table., x and x (dynamc responses of floors a he me of =.s would be approxmaed by he Euler formula as follows x x v.3 (. ( ( (36 x x v.4 (. ( ( Dscrezed form of Eq.(36 s.5v.8v v.v (37.58v.8v v.55v Neglecng he couplng erm,.e. (-.8v v, we have wo quadrac equaons as follows.5v.v (38.58v.55v Roos of hese quadrac equaons are.5v.v v.9568, v.9763 (39.58v.55v v , v By comparng he roos obaned from Eq.(39 wh he velocy n he prevous sep, v =3, v =4, he closes veloces o prevous sep are seleced and oher ones are omed..5v.v v.9568, v.9763 (4.58v.55v v , v Now, new values x and x a =.s can be approxmaed by usng he average of he velocy of hs sep and prevous sep, hence v v ( (. x x.978 ( (4 v v ( (. x x.398 ( Here, he couplng erm (-.8v v, whch was negleced prevously, mgh be gven by he subsuon of v =.9568 and v = v.9568, v v v.848 (4 And new values for veloces of he sysem can be deermned as follows.5v.8v v.v vv.848.5v.v v.8v v.55v v.55v (43

9 Smlarly, he real veloces are.5v.v v.9867, v v.55v v , v (44 Now, he updaed couplng erm becomes v.9867, v v v.97 (45 By nroducng a relave error, e j, for veloces, as an absolue value of ( j j v v / v j for a convergence creron, where s he number of sory and j denoes he number of eraon. As shown n Table.3, he procedure can beer be monored by hs defnon. Noe ha n hs case olerance s chosen as -. Also, Fg.3 llusraes he process of convergence whch s shown he error of analyss vs. number of eraon. If we use a compuer program o connue he process o =s, we can ge he dynamc response of he sysem, as dsplayed n Fg.4. Ths fgure compares he obaned resuls of he presened mehod agans he exac soluon of he problem. As shown n Fg.4, nowhsandng he sze of me nervals =. seleced s no very small n hs analyss, can be seen ha he proposed mehod has excellen accuracy compared wh he exac soluon; n oher words, he numercal soluon can properly approach o he exac soluon of he problem n hs case. Now, by choosng a fxed me nerval, whch s delberaely no chosen oo small, ypcally =. here, we are gong o compare he accuracy and speed of analyss of presened mehod wh oher convenonal mehods such as modal [5], Newmark [3], and combned echnques lke modal-duhamel [53,54] and modal-newmark (see Fg.5 and Fg.6. Adjusmen facors n Newmark mehod, whch are used o mprove he accuracy and sably of he mehod, respecvely are seleced as: β=.5 and ϒ=/6 (ypcally hese values whch yeld he lnear acceleraon mehod, are used n pracce. In addon, combned modal mehods also by converng an n-dof srucure o n-sdf sysems and by usng numercal echnques such as Duhamel and Newmark he srucural analyss wll be performed. The olerance of he proposed mehod s consdered as -. From Fg.5 and Fg.6, can be observed ha he presened mehod compared o oher convenonal mehods used n he dynamc analyss of MDF srucure, has accepable accuracy. In fac, (consderng a consan among all mehods, combned Modal-Duhamel and he proposed mehod have been closer o he exac response of problem. Furhermore, n engneerng analyss, he me requred o calculae he soluon, or he speed of numercal echnque, s one of he facors nfluencng he choce of mehod. So n hs secon, n accordance wh he Table.4, he requred mes for analyss by he varous numercal mehods are compared. The resuls show ha he mehods usng he modal echnques are very me-consumng compared o oher mehods; For example, he compuaonal me for he proposed approach s less han half of he oher modal mehods. I mus be here saed ha alhough he whole of dampng marces consdered n hs sudy are of a classcal/proporonal ype, bu n general, for non-classcal dampng, s no easy o apply a convenonal modal mehod. Because n hs case he frequences, he shape-modes, and dampng raos n addon o he mass and sffness marces, dependng on he dampng marx of he sysem, and he complex modal coordnae mus be used (for more deals, see [55-57]. On he oher hand, s noeworhy ha he energy-based mehod presened n hs research has no any lmaons n hs regard and he classcal or non-classcal dampng wll be analyzed whou a parcular modfcaon ( s anoher advanage of hs echnque. Example.4.. The damped harmonc vbraon of a hree-sory shear buldng. A wo-dof shear-frame s depced n Fg.7 n whch, lke he prevous example, for convenence, he dynamc properes of he srucure are seleced as: m =m =m 3=k =k =k 3=, c =c =c 3=., and he zero nal condons wll be assumed n hs example. The srucure s subjeced o harmonc loads as: p =cos, p =cos, p 3=cos3. (All uns are compable In hs case, he equaon of moon s m x c x k x p (46 where he mass, dampng, and sffness marces are.. cos m c... k p cos.. cos 3 9 (47

10 Hence, he governng equaons of hs problem would be gven by x.x.x x x cos x.x.x.x x x x cos 3 3 (48 x.x.x x x cos As for he prevous example, he Laplace ransform mehod s used o deermne he exac soluon of he problem (for deals, see Appendx.D. x.3exp(. cos( exp(.775 cos( exp(.65 cos( cos( cos(.983. cos( x.383exp(. cos( exp(.775 cos( ( exp(.65 cos( cos( cos( cos(3.454 x.477 exp(. cos( exp(.775 cos( exp(.65 cos( cos( cos(.847.7cos(3 3.9 Applyng he energy mehod Usng Eq.(3, he energy balance equaons of hs sysem would be as follows.5v.v.v v d x x v d v cos d 3 3.5v.v.v v.v v d x x x v d v cos d v.v.v v d x x v d v cos 3 d Afer dscrezng he Eq.(5 and usng Table., he same procedure as n he former example, mus be performed. In hs case, Fg.8 hrough Fg. gves he obaned resuls where he dynamc response of floors, assumng =.s and e=., s ploed by usng he varous numercal mehods vs. exac soluon of he problem. Fg.8-Fg. demonsrae ha wh a fxed sze for me nervals, he proposed mehod n hs sudy ogeher wh Modal-Duhamel echnque are very close o he exac soluon of he problem, and he mehods usng Newmark echnque (wh β=.5 and ϒ=/6 are no appropraely converged. Here, smlar o he prevous example, he requred me for analyss of hs example s ndcaed n Table.5 where can be observed ha, lke former analyss, he modal echnques are very me-consumng compared wh ohers. In addon, noe ha alhough s rue ha Newmark mehod has a good speed, n hs case, s accuracy s no good compared o oher mehods. Afer observng Fgs.8-, gven ha he modal-duhamel mehod has been shown o be more accurae han oher approaches. Here, he effec of he numercal echnque used n he approxmaon of he Duhamel negral s nvesgaed. In hs regard, n addon o he Smpson rule whch was used a frs, he Trapezodal rule for compung he Duhamel negral s also provded n Fgs.-3. Moreover, mus be menoned ha o preven he cluered graphs he resuls of he Newmark and Modal-Newmark mehods are no represened n hese fgures. Generally, Fgs.-3 show ha he accuracy of he Duhamel mehod s srongly dependen on he numercal mehod (Smpson wh rapezodal used n he approxmaon of hs negral. As compared o he proposed mehod, he use of he Smpson mehod leads o more accurae resuls, and conversely, he applcaon of he Trapezodal rule leads o a reducon n he accuracy compared wh he presened mehod. In he followng, o examne he effcency of he proposed mehod n he case of large-scale srucures. A hghrse -sory shear-frame (as a generalzed sysem of he srucure suded n Example 4.. s consdered wh dynamc properes as below m, c., k, p cos,,,..., (5 Now, by choosng he las node above he srucure (as a conrol pon and hen employng he proposed mehod, f we plo he roof s velocy n seconds versus wo converged veloces: v roof ( and v roof ( a he end of eraons n quadrac energy equaon, Eq.(4 as dsplayed n Fg.4. (5

11 From Fg.4 can be seen ha n hs large-scale sysem, he roof velocy s properly calculaed from he selecon of rgh velocy based on he assumpon of connues veloces n me. Though, seems ha fuure works especally by consderng he nonlnear behavor n oher hgh-rse buldng sysems are essenal o verfy he effcency of he gven mehod n general problems of srucural dynamcs. Example4.3. The forced damped vbraon of a hree-sory shear buldng subjeced o an earhquake. Consder a hree-sory shear-frame ha s descrbed n Fg.5, and subjeced o ground moon, EL-Cenro earhquake (PGA 6 =.3g as shown n Fg.5. Also, he dynamc characerscs of he sysem are: m =m =m 3=, c =c =c 3=.5, and k =k =k 3=. Moreover, he zero nal condons are assumed n hs case. (All uns are compable In hs case, due o earhquake loadng, by defnon of effecve force (p eff he equaon of moon s mx cx kx p eff (5 where, p eff defnes as he negave produc of he mass marx m, nfluence coeffcen vecor acceleraon vecor of ground moon x g,.e. p m eff lxg For he problem a hand, nfluence coeffcen vecor s l, and he (53 T,, l (54 where, mass, dampng, and sffness marces are..5 m c.5..5 k.5.5 Applyng he energy mehod Accordng o Table., and wo examples menoned before, va he Energy mehod, and assumng pror assumpons, excep he value of olerance ha s equal o e= -4, he dynamc response of he srucure can be ploed. As we know, n hs knd of problems here s no a closed-form analycal soluon whch can be used o compare he resuls. Thus, only he resuls of varous numercal mehods (n a fxed me nerval equal o. are ploed n Fgs. 6 o 8. I should be noed ha n Fg.6-Fg.8 marked pons on he fgures are no ndcave me seps and hey are seleced o dsngush he resuls beer. In addon, he obaned resuls of Newmark and Modal-Newmark mehods are overlapped, and canno properly be denfed. Accordng o he fgures, an accepable agreemen beween he presened mehod and oher mehods can be seen. Hence, hs mehod can be used for long-me dynamc analyss of shear-frames such as sesmc analyses. Once agan for comparng he speed of analyses, Table.6 gves he requred me o analyze he hrd example of hs nvesgaon. Smlar o prevous examples, can be observed ha, n a consan me nerval, he proposed mehod regardng compuaonal me s n second place afer Newmark mehod. A frs glance, however, he mes gven n Table.6 may look grea for a small 3-DOF srucure. However, should be here menoned ha he mes presened n hs able nclude execung all he commands wren whn he MATLAB program (e.g., me-consumng synaxes lke (xlsread. In oher words, hese values do no ndcae he real me of he mplemenaon of he negraon schemes and are used only for comparson beween dfferen ypes of mehods. (55 6 Peak Ground Acceleraon

12 4.4. Sably and accuracy analyss Here, he effecs of me sep sze on he accuracy and sably of he presened mehod are dscussed. In hs regard, Bahe [58] has been proposed a echnque based on he free response analyss of a smple SDF sysem, as shown n Fg.9. For smplcy, f we assume he followng parameers n a compable un sysem: m=; k=4π ; x =; and, v =. The free response of hs sysem (exac soluon of he problem can be wren as follows x( (sn / (56 Wh respec o hs exac response, he values for he perod and amplude of he vbraonal moon are equal o T Exac = and A Exac =/π, respecvely. Obvously, he numercal soluon obaned from he presened mehod wll dffer wh hese values. So, would be approprae o defne wo followng parameers. and, R ( T T / T (57 T Exac Num Exac R ( A A / A (58 A Exac Num Exac where R T and R A respecvely represen he numercal error n perodc and amplude of he vbraonal sysem. T Numl and A Num also are he perod and amplude obaned from he numercal mehod, whch are funcons of he sze of he me sep,, used o dscreze he me. Thus, akng no accoun dfferen values for he me sep, one can plo he parameers R T and R A as shown n Fg.. Accordng o Fg., as he me sep ncreases (wh he rse of he numercal error n he sysem response, he accuracy of he soluon s reduced, as expeced. For example, when he me sep sze s =., he relave errors n he perod and amplude of he sysem respecvely are equal o abou.5% and 3%. In general, he resuls of hs secon show a greaer sensvy o he amplude of moon han he perod (hs s n lne wh he resuls of he reference [7]. Moreover, a key observaon can be made from hs fgure; ha s, by ncreasng he value of me sep numercal errors ncrease sgnfcanly a a ceran value (abou. o.5, ndcang ha nsably wll occur n he numercal soluon. For example, n he case of =.5, he use of abou 6-7 pons for he approxmaon of a complee sne wave creaed a sgnfcan error. Therefore, he selecon of he approprae value for s essenal n pracce. Snce large ones, by elmnang he precson of soluon, can lead o nsably. On he oher hand, small also ncrease he compuaonal me. Consequenly, an opmum sze for me sep should be used n praccal dynamc analyses. 5. Conclusons In hs paper, a novel sep-by-sep soluon echnque based on energy mehod s presened for he dynamc analyss of shear-frames, as one of he applcable srucures n pracce. In hs mehod raher han workng wh he equaon of moons, we solve he energy balance relaonshps whch have some advanage. For example, leads o a reducon of unknowns. The proposed mehod s performed on varous examples ncludng harmonc and earhquake loadng. The man mplcaons of he sudy can be lsed as follows: The resuls show ha he mehod has good accuracy compared wh oher common mehods (e.g., s more accurae vs. Newmark mehod. Anoher advanage of hs mehod compared o oher me negraon mehods such as Newmark s avodng selecng and calbrang he velocy and acceleraon adjusmen parameers such as,. Modal mehods whch have shown good accuracy n combnaon wh Duhamel s Inegral, has complex mahemac relaonshps, parcularly wh ncreasng he degrees of freedom of he srucure, and as was observed n hs sudy, hey are more me-consumng han oher echnques. The presened mehod, wh a smple mahemacal algorhm, has good accuracy and speed of analyss, and by selecng an allowable olerance (usually n he range of.-., can be used n praccal dynamc analyses of shear-frames. Fnally, should be noed ha he deas expressed n hs research have he capably o be appled o oher engneerng srucures and also non-lnear sysems wh some modfcaons

13 Appendx A. Dscrezaon of negral energy equaons. A.. Trapezodal rule The value of Recallng Eq.(3 f ( d can be evaluaed by he Trapezodal rule f ( d [ f ( f ( ] (A. m v m v ( ( ( c c v c v v c v v d k k x k x k x v d p v d (3-rep. Now, consderng Eq.(3 and he use of he rapezodal rule, he negrals n hs expresson can be dscrezed as follows. ( c c v ( c v ( v ( c v ( v ( f ( d {[( c c v c v v c v v ] [( c c v c v v c v ( ( ( ( ( ( ( ( ( ( f ( ( k k x ( k x ( k x ( v ( d f ( f ( { ( k k x k x k x v ( k k x k x k x v ( ( ( ( ( ( ( ( f ( f ( p ( v ( d { p v p v } ( ( ( ( f ( f ( f ( Subsung Eqns.(A.-(A.4 no Eq.(3, and rearrangng wh regard o veloces, yelds ( ( ( ( ( ( v } ]} (A. (A.3 (A.4 Av B v v C v v D v E (4-rep. where, A.5m.5 ( c c, B.5. c, C.5. c D.5 ( k k x k. x k. x p E ( ( ( (.5m v.5. v [( v c. v c. v ( x kx ( ( ( ( ( ( ( c c k k k x p ] ( ( (5-rep. 3

14 A.. Smpson rule Consderng j h me sep,.e., =j (j=,3,. In hs case, f ( d can be approxmaed by he compose Smpson rule as follows j f (d [f 4f 4f... 4f f ] (A.5 ( ( ( (( j ( j 6 Smlar o before, he dscrezed form of Eq.(3 usng Smpson rule s gven by j ( c c v ( c v ( v ( c v ( v ( d f ( {[( c c v c v v c v v ] 4[( c c v c v v c v 6 ( ( ( ( ( ( ( ( ( ( f (... [( c c v c v v c v v ]} j ( j ( j ( j ( j ( j f ( j ( k k x ( k x ( k x ( v ( d f ( { ( k k x k x k x v 4 ( k k x k x k x v 6 f ( ( ( ( ( ( ( ( ( f ( f (... ( k k x k x k x v } ( j ( j ( j ( j f ( j p ( v ( d { p v 4 p v... p v } ( ( ( ( ( j ( j 6 f ( f ( f ( f ( j Hence, by nserng Eqns.(A.6-(A.8 n Eq.(3 and smplfyng, one can oban he dscrezed form of energy equaons. ( ( ( ( ( ( v ] (A.6 (A.7 (A.8 Av B v v C v v D v E (4-rep. where, A.5 m ( / 3( c c, B ( / 3. c, C ( / 3. c D ( / 3 (k k x k. x k. x p ( j ( j ( j ( j E m v v c c v c v c v k k x k x.5 ( / 3 [(.. ( ( ( ( ( ( ( 4 v [( c c v c. v c. v ( k k x k x k x p ] ( ( ( ( ( ( ( ( v [( c c v c. v c. v ( k k x k x ( ( ( ( ( ( ( ( k x p k x p ( ( ( v [( c c v c. v c. v ( k k x k x k x p ] ( j ( j ( j ( j ( j ( j ( j ( j ] ] (6-rep. 4

15 Appendx B. The Newon-Raphson mehod for solvng Eq.(37. If we defne wo funcons (f,f, for each equaon n Eq.(37 as follows f.5v.8v v.v (37-rep. f.58v.8v v.55v By exendng he Newon-Raphson mehod o he sysem of equaons, o fnd he roo of f and f, one may wre he followng sequence [59] v v J f,,,... (B. Here, he subscrp,, represen he eraon number o acheve he convergence creron;{v}s he vecor of unknowns (veloces;{f}s he vecor of funcons; and, [J] denoes he Jacoban marx. f f v f v v v f J (B. v f f f v v In hs case, he nverse form of he Jacoban marx can be expressed as Smplfyng yelds.v.8v..8v J.8v.6v.8v.55 (B.3 5(8v 6v 55 4v D 4v (5v 4v 5 J D 488v 464v 597v v 865v 486v 75 Assumng he veloces of he prevous sep as he nal approxmaon, we can esmae he roos of Eq.(37 as 3 v (B.5 4 hen, Eq.(B.5 may be expressed as (B.4 v v J f (B.6 resuls n v v (B.7 By nserng he obaned value n he sequence of Eq.(B. and connung calculaons unl convergence s acheved, he accuracy of he soluon can be ncreased. Noe ha hs quany, [J] -, whn he Modfed Newon Raphson s deermned only once n he eraon and s assumed o be consan durng he nex eraons [6]. 5

16 Appendx C. Dervaon of he exac soluon of Example.4. by Laplace ransform Consderng Eq.(33 x.x.6x x x (33-rep. x.6x.6x x x Takng Laplace ransform, and le F=L(x and G=L(x, hen L s F sx x sf x sg x F G ( ( ( ( (C. s G sx x.6sf.6x.6sg.6x F G ( ( ( ( where s s a ransformed varable and he zero subscrp denoes he nal value of a =. Imposng he nal condons and solvng hs sysem algebracally for F and G, we oban 3 s 3.38s 4.96s 7.6 F 4 3 s.38s 3.96s.s (C. 3 s 4.76s 6.379s. G s s 3.96 s. s Eq.(C. may be wren n erms of paral fracons as follows F s ( s ( s ( s ( (C G s ( s ( s ( s ( Consequenly, by akng he nverse ransform and smplfyng, dsplacemens of he sysem (x,x wll be gven by x.336e cos( e cos( L (C x.e cos( e cos( Appendx D. Dervaon of he exac soluon of Example.4. by Laplace ransform The followng governng equaons wll be solved by he Laplace ransform mehod. x.x.x x x cos x.x.x.x x x x cos 3 3 x.x.x x x cos Le F=L(x, G=L(x, and H=L(x 3, hen akng Laplace ransform, we have s s F sx x.sf.x.sg.x F G ( ( ( ( s s s G sx x.sf.x.sg.x.sh.x F G H ( ( ( ( 3( s 4 s s H sx x.sg.x.sh.x G H 3( 3( ( 3( s 9 (48-rep. (D. 6

17 By mposng he zero nal condon and solvng for F, G, and H, we have s ( s.s F (.s G ( H s s (.s F ( s.s G (.s H s 4 s ( F (.s G ( s.s H s 9 (D. Rearrangng yelds, s 4s 73s 55s 888s 73s 4549s 98s 49s F A s 5s 55s 58s 643s 53s 6s 4s 6s G A s 5s 6s 4s 448s 73s s 3s 66s H A where, A s s s s s s s s s s 668s 8s 36 (D.3 (D.4 Eq.(D.4 may be wren n erms of paral fracons as follows.3s..868s.43.45s s.8.83s.546.s.6 F s.s.98 s.55s.555 s.35s 3.47 s s 4 s 9.383s..386s s.483.8s.9.549s.76.7s s.35s 3.47 s s 4 s 9 G s s s s (D.5.477s..696s.33.5s.5.944s.84.48s s.9 H s. s.98 s.55 s.555 s.35 s 3.47 s s 4 s 9 Fnally, by akng he nverse ransform and smplfyng, he dsplacemens of he sysem (x,x,x 3 wll be gven by x.3exp(. cos( exp(.775 cos( exp(.65 cos( cos( cos(.983. cos( x.383exp(. cos( exp(.775 cos( (49-rep exp(.65 cos( cos( cos( cos(3.454 x.477 exp(. cos( exp(.775 cos( exp(.65 cos( cos( cos(.847.7cos(

18 References [] Wlson, E.L. Dynamc Analyss, In Three-Dmensonal Sac and Dynamc Analyss of Srucures, 3rd Edn., pp , Compuers and Srucures, Berkeley, Calforna, USA (. [] Crag, R. and Kurla, A. Preface o Srucural Dynamcs, In Fundamenals of Srucural Dynamcs, nd Edn., pp. 3-4, Wley, New Jersey, USA (6. [3] Clough, R. and Penzen, J. Earhquake Engneerng, In Dynamcs of Srucures, 3rd Edn., pp , Compuers and Srucures, Berkeley, Calforna, USA (3. [4] Chopra, A.K. Earhquake analyss of arch dams: facors o be consdered, ASCE, 38(, pp. 5 4 (. [5] Chopra A.K. Earhquake analyss of concree dams: facors o be consdered, h U.S. Naonal Conference on Earhquake Engneerng, Anchorage, Alaska, USA, (4. [6] Feng, M.Q., Fukuda, Y., Feng, D., and Mzua, M. Nonarge vson sensor for remoe measuremen of brdge dynamc response, J. Brdg. Eng., (5, pp. (5. [7] Krodkewsk, J.M. Mechancal vbraon, Melbourne Unversy, pp. -47, Melbourne, Ausrala (8. [8] Chopra, A.K. Sngle degree-of-freedom sysems, In Dynamcs of Srucures, 4h Edn., pp. -37, Prence Hall, Berkeley, Unversy of Calforna, USA (3. [9] Makem, J.E., Armsrong, C.G. and Robnson, T.T. Auomac decomposon and effcen sem-srucured meshng of complex solds, Engneerng wh Compuers, 3(3, pp (4. [] Kougoumzoglou, I.A. and Spanos P.D. Nonlnear MDOF sysem sochasc response deermnaon va a dmenson reducon approach, Compu. Sruc., 6(, pp (3. [] Brahm, K., Bouhadd, N. and Fllod, R. Reducon of juncon degree of freedom n ceran mehods of dynamc subsrucure synhess, he nernaonal socey for opcal engneerng, Spe nernaonal socey for opcal, Bellngham, USA, pp (995. [] Kreyszg, E. Numercal analyss, In Advanced Engneerng Mahemacs, h Edn., Wley, pp , New York, USA (5. [3] Blahu, E.R. Fas algorhms for he dscree Fourer ransform, In Fas Algorhms for Sgnal Processng, pp. 68-4, Cambrdge Unversy Press, New York, USA (. [4] Elzalde-Sller, H.R. Non-lnear modal analyss mehods for engneerng srucures, Docoral dsseraon, Imperal College London, UK (4. [5] Wong, K.K.F. Nonlnear dynamc analyss of srucures usng modal superposon, ASCE Srucures Congress, pp , Las Vegas, Nevada, USA (. [6] Dou, S. and Jakob, S.J. Opmzaon of hardenng / sofenng behavor of plane frame srucures usng nonlnear normal modes, Compu. Sruc., 64, pp (6. [7] Dou, S. Graden-based Opmzaon n Nonlnear Srucural Dynamcs, PhD hess, Mechancal Engneerng, DTU, Denmark (5. [8] Dou, S. and Jakob, S.J. Analycal sensvy analyss and opology opmzaon of nonlnear resonan srucures wh hardenng and sofenng behavor 7h U.S. Naonal Congress on Theorecal and Appled Mechancs, Eas Lansng, Mchgan, USA, pp. -3 (4. [9] Chopra A.K. and Goel R.K., A modal pushover analyss procedure o esmae sesmc demands for unsymmerc-plan buldngs, Earhq. Eng. Sruc. Dyn., 33(8, pp (4. [] Chopra, A.K. and Goel, R.K. A modal pushover analyss procedure for esmang sesmc demands for buldngs, Earhquake Engneerng and Srucural Dynamcs, 3(3, pp (. 8

19 [] Cheng, F.Y. Egensoluon echnques and undamped response analyss of mulple-degree-of-freedom fysems, In Marx Analyss of srucural dynamcs, pp , Marcel Dekker, Rolla, Mssour, USA (. [] Bahe, K.J. Egen Problems In Fne Elemen Procedures n Engneerng Analyss, pp , Prence Hall, New Jersey, USA (996. [3] Newmark, N.M. A mehod of compuaon for srucural dynamcs, Journal of he Engneerng Mechancs, 85(7, pp (959. [4] Houbol, J.C. A recurrence marx soluon for he dynamc response of elasc arcraf, J. Aeronau. Sc., 7 (9, pp (95. [5] Wlson, E.L., Farhoomand, I. and Bahe, K.J. Nonlnear dynamc analyss of complex srucures, Earhq. Eng. Sruc. Dyn., (March 97, pp. 4 5 (973. [6] Bahe, K.J. and Wlson, E.L. Numercal mehods n fne elemen analyss, p.p. -58, Prence Hall, New Jersey, USA (976. [7] Bahe, K.J. and Wlson, E.L. Sably and accuracy analyss of drec negraon mehods, Earhq. Eng. Sruc. Dyn., (3, pp (97. [8] Felppa, C.A. and Park, K.C. Drec me negraon mehod n nonlnear srucural dynamcs, Compu. Mehod Appl. Mech. Eng., 7(8, pp (979. [9] Johnson, D.E. A proof of he sably of he houbol mehod, AIAA, 4(8, pp (966. [3] Gladwell, I. and Thomas, R. Sably Properes of he Newmark, Houbol and Wlson Mehods, In. J. Numer. mehods Anal. Mehods Eng., 4(Augus 979, pp (98. [3] Park, K.C. An mproved sffly sable mehod for drec negraon of nonlnear srucural dynamc equaons, J. Appl. Mech., 4(, pp (975. [3] Bahe, K.J. and Wlson, E.L. NONSAP - A nonlnear srucural analyss program, Nucl. Eng. Des., 9(, pp (974. [33] Paulre, P. Drec Tme Inegraon of Lnear Sysems, In Dynamcs of srucures. pp. 3-48, John Wley & Sons, Surrey, GBR (3. [34] Acary, V. Energy conservaon and dsspaon properes of me-negraon mehods for nonsmooh elasodynamcs wh conac, Journal of Appled Mahemacs and Mechancs, 96(5, pp (6. [35] Flemng, A., Peness, I., Macfarlane, G., Bose, N. and Dennss, T. Energy balance analyss for an oscllang waer column wave energy converer, Ocean Engneerng, 54, pp (. [36] Lee, T., Leok, M. and McClamroch, N.H. Geomerc numercal negraon for complex dynamcs of ehered spacecraf, Amercan Conrol Conference (ACC, San Francsco, Calforna, USA, pp (. [37] Dowell, E.H. Nonlnear aeroelascy, In A Modern Course n Aeroelascy, pp , Sprnger, NC, USA (5. [38] Sanon, S.C., Erurk, A., Mann, B.P., Dowell, E.H. and Inman, D.J. Nonlnear nonconservave behavor and modelng of pezoelecrc energy harvesers ncludng proof mass effecs, J. Inell. Maer. Sys. Sruc., 3(, pp (. [39] Hrsch C. An Inal Gude o CFD, In Numercal compuaon of nernal and exernal flows: The fundamenals of compuaonal flud dynamcs, nd edon, pp. -, Elsever, Ameserdam, Neherlands (7. [4] Yazd M.K. and Tehran P.H. The energy balance o nonlnear oscllaons va Jacob collocaon mehod, Alexandra Eng. J., 54(, pp (5. 9

20 [4] Mehdpour, I., Ganj, D.D. and Mozaffar, M. Applcaon of he energy balance mehod o nonlnear vbrang equaons, Curren Appled Physcs, (, pp. 4 (. [4] Jall Sadr Abad, M., Mahmoud, M. and Dowell, E.H. Dynamc analyss of SDOF sysems usng modfed energy mehod, ASIAN J. Cv. Eng., 8(7, pp (7. [43] Paz, M. and Legh, W. Srucures modeled as shear buldngs, In Srucural Dynamcs, pp. 5-3, Sprnger, Boson, USA (4. [44] Feynman, R.P., Leghon, R.B. and Sands, M. The Feynman Lecures on Physcs, Deskop Edon, Volume I, Basc books, (3. [45] Hall, K.C., Ekc, K., Thomas, J.P. and Dowell, E.H. Harmonc balance mehods appled o compuaonal flud dynamcs problems, In. J. Comu. Flud Dyn., 7(, pp. 5 67, (3. [46] Lgh, J.C. and Walker, R.B. An R marx approach o he soluon of coupled equaons for aom-molecule reacve scaerng, J. Chem. Phys., 65(, p.p (976. [47] Wang, Y., Dng, H. and Chen, L.Q. Asympoc soluons of coupled equaons of supercrcally axally movng beam, Nonlnear Dyn., 87(, pp (6. [48] Armour, E.A.G. and Plummer, M. Calculaon of he resonan conrbuon o Zeff(k usng close-coupled equaons for posron molecule scaerng, J. Phys. B A. Mol. Op. Phys., 49(5, (6. [49] Haojang, D. General soluons for coupled equaons for pezoelecrc meda, In. J. Solds Sruc., 33(6, pp (996. [5] Polyann A.D. and Lychev, S.A. Decomposon mehods for coupled 3D equaons of appled mahemacs and connuum mechancs: Paral survey, classfcaon, new resuls, and generalzaons, Appl. Mah. Model., 4(4, pp (6. [5] Mourad A. and Kamel, Z. An effcen mehod for solvng he MAS sff sysem of nonlnearly coupled equaons: Applcaon o he pseudoelasc response of shape memory alloys (SMA, IOP Conf. Ser. Maer. Sc. Eng., 3(, Mskolc-Lllafüred, Hungary, pp. -5 (6. [5] Nkas, N., Macdonald, J.H.G. and Tsavdards, K.D. Modal Analyss, Encycl. Earhq. Eng., (Dec, pp. (4. [53] Van Puen, M.H. Inroducon o Mehods of Approxmaon n Physcs and Asronomy, Sprnger, Sngapore (7. [54] Duhamel, J.M.C. Elemens de calcul nfnesmal, Malle-Bacheler, France 86. (In French [55] Gavn, H.P. Classcal dampng, non-classcal dampng, and complex modes, Dep. Cv. Envron. Eng. Duke Unv. NC, USA (6. [56] Ghahar, S.F., Abazarsa, F., and Tacroglu, E. Blnd modal denfcaon of non-classcally damped srucures under non-saonary excaons, Sruc. Conrol Heal. Mon., 4(6, pp (7. [57] Cruz, C. and Mranda, E. Evaluaon of he Raylegh dampng model for buldngs, Eng. Sruc., 38, pp (7. [58] Bahe, K.J. Soluon of equlbrum equaons n dynamc analyss, In Fne elemen procedures, nd Edn., pp , Prence Hall, New Jersey, USA (4. [59] Ben-Israel, A. A Newon-Raphson Mehod for he Soluon of Sysems of Equaons, J. Mah. Anal. Is Appl., 5, pp (966. [6] Rhenbold, W.C. Mehods of Newon Type, In Mehods for Solvng Sysems of Nonlnear Equaons. nd Edn., pp , Socey for Indusral and Appled Mahemacs (SIAM, Psburg, Pennsylvana, USA, (998.

21 Ls of Capons Fg.. Shear-frame srucure Fg.. Free vbraon of a wo-sory shear-frame Fg.3. The process of convergence n Example 4.. Fg.4. Comparson of he presened mehod wh exac soluon n Example 4.. Fg.5. Comparson of he dynamc response of he frs floor (x wh varous mehods ( =.s Fg.6. Comparson of he dynamc response of he second floor (x wh varous mehods ( =.s Fg.7. Three-sory shear-frame under harmonc loads Fg.8. Comparson of he dynamc response of he frs floor (x wh varous mehods ( =.s Fg.9. comparson of dynamc response of second floor (x wh varous mehods ( =.s Fg.. Comparson of he dynamc response of he hrd floor (x 3 wh varous mehods ( =.s Fg.. The accuracy of he proposed mehod compared wh wo numercal echnques o compue he Duhamel s negral for he frs floor Fg.. The accuracy of he proposed mehod compared wh wo numercal echnques o compue he Duhamel s negral for he second floor Fg.3. The accuracy of he proposed mehod compared wh wo numercal echnques o compue he Duhamel s negral for he hrd floor Fg.4. Convergence of veloces n a -sory shear-buldng consderng he conrol pon a he roof Fg.5. Three-sory shear-buldng under earhquake loadng Fg.6. Comparson of he dynamc response of he frs floor (x wh varous mehods ( =.s Fg.7. Comparson of he dynamc response of he second floor (x wh varous mehods ( =.s Fg.8. Comparson of he dynamc response of he hrd floor (x 3 wh varous mehods ( =.s Fg.9. An SDF sysem for accuracy and sably analyss of numercal analyses Fg.. The effecs of he dfferen me sep sze on he accuracy and sably of he presened mehod Table.. Comparson of he force and energy equlbrum equaons mahemacally Table.. Summary of sep-by-sep soluon procedure of he presened mehod Table.3. Convergence of veloces n he frs sep Table.4. requred me of analyss n Example.4. Table.5. requred me of analyss n Example.4. Table.6. requred me of analyss n Example.4.3

22 Fgure

23 Fgure Fgure 3 Fgure 4 3

24 Fgure 5 Fgure 6 Fgure 7 4

25 Fgure 8 Fgure 9 Fgure Fgure 5

26 Fgure Fgure 3 Fgure 4 6

27 Fgure 5 Fgure 6 7

28 Fgure 7 Fgure 8 Fgure 9 Fgure 8

29 Table Type of equaons Advanages Dsadvanages Force equlbrum Energy equlbrum *Excep n nonlnear analyss Quadrac (non-lnear erms do no exs n hese equaons * Frs order of dervave n he equaons ha leads o reducng he number of unknowns, ncludng: dsplacemen, velocy Second order of dervave n he equaons ha leads o ncreasng he number of unknowns, ncludng: dsplacemen, velocy and acceleraon Exsence of quadrac (non-lnear erms A. nal calculaons:. Form dynamc marces: mass m, dampng c, and sffness k Table. Form he vecors of nal condons: nal dsplacemens x, and veloces v 3. Selec he me sep 4. Selec he olerance for each eraon e s ( s s a posve neger number B. for each me sep: 5. Calculae a sarng vecor for x, dsplacemen vecor a he me of ( ( ( x x dx, dx v (The superscrps and subscrps refer o he number of eraon and me sep, respecvely 6. Calculae he coeffcen of energy balance equaon,.e. A, B, C, D, for all masses. Noe ha, he Trapezodal rule for he frs me sep and subsequenly Smpson rule mus be used. 7. Neglec he couplng erms,.e. B C 8. Solve he quadrac equaon of energy balance for he velocy of h mass, usng he correspondng, and v ( D / A 9. Selec a velocy whch s closer o he prevous me sep (call v (Elmnaon of Dsconnuous Veloces. Calculae a new approxmaed vecor for x, by usng he average of new obaned veloces and nal veloces. Deermne he couplng erms, whch were negleced a frs. Ierae hrough sep 6 o, excep sep 7, o convergence 3. Connue he procedure for subsequen me seps ( j ( j ( j x x dx, dx.5( v v, j, 3, Table 3 Number of eraons v e v e e. max