Chapter 2 Linear dynamic analysis of a structural system

Transcription

1 Chaper Lnear dynamc analyss of a srucural sysem. Dynamc equlbrum he dynamc equlbrum analyss of a srucure s he mos general case ha can be suded as akes no accoun all he forces acng on. When he exernal loads and dsplacemens are appled very slowly, a smplfed approach of he problem s done, leadng o he sac problem n whch some of he forces are negleced. hs dvson exclusvely depends on he value of he nera forces, equal o mass mes he acceleraon accordng o he Newon s second law. In he general case four erms are consdered n he equlbrum equaon: f + f + f = r (.) () I () D () S () where, f ()I f ()D f ()S r () s he vecor of nera forces s he vecor of vscous dampng forces s he vecor of nernal forces s he vecor of exernal loads hs equlbrum s vald for boh lnear and nonlnear sysems, f equlbrum s formulaed wh respec o he deformed geomery of he srucure. I s also vald when here s maeral nonlneary, and, n all cases, mus be fulflled a any me. In order o oban he fne elemen formulaon of he equlbrum, he behavour of he srucure s assumed o be lnear: Infnesmally small dsplacemens he naure of he boundary condons never changes durng he analyss he maeral s lnearly elasc, wh a consuve law of he form: σ = Dε+ σ 0 8

2 Lnear dynamc analyss of a srucural sysem Accordng o Bahe (98), hese hree assumpons lead o he expresson of he vecor of nernal forces as Ku. he general fne elemen formulaon of he nera forces s Mu and of he dampng forces s Cu, eher for lnear or nonlnear analyss. () () () Now we are able o rewre he equlbrum expresson as he second-order dfferenal equaon (.). Mu + Cu + Ku = r (.) () a () a () a () n whch M s he mass marx, C s he dampng marx ha approxmaes he energy dsspaon n he srucure, and K s he sac sffness marx. he me-dependan vecors, and u are he absolue nodal dsplacemens, veloces and acceleraons. u () a u () a () a For he sac approach hese las wo erms, velocy and acceleraon, are consdered o be very small wh regard o he dsplacemen, herefore he nera and dampng forces are negleced and he equlbrum s reduced o he smples expresson Ku = r whch s no me-dependan. If he analyss s quassac, r depends on me bu loads are appled so slowly ha he same approxmaons can be done, leadng o he expresson Ku. he choce for a sac or dynamc analyss s usually decded by () = r() engneerng crera, wh he am of reducng he analyss effor requred. However he assumpons done mus be jusfed, oherwse he resuls would be meanngless. he soluon o he dynamc equlbrum can be obaned wh sandard procedures for he soluons of dfferenal equaons wh consan coeffcens, also called Runge-Kua mehods, bu hey may become very expensve f he order of he marces s large (unless advanage s aken from any specal characersc of M, C or K). In he followng secons we wll concenrae n he effecve mehods, whch can be dvded n wo mehods of soluon: drec negraon and mode superposon. he am of boh mehods s he resoluon of a second-order dfferenal equaon, whch means ha wo nal condons, a me 0, are requred. I s mporan o remark ha only for wo of he varables (u,u and u ) he nal value s () () () mposed, snce he hrd one can be calculaed from he oher wo. For example f he nal acceleraon of he srucure (u ) s unknown, can be calculaed usng he (0) nal dsplacemens, veloces and he requremen of equlbrum fulflmen a me 0, leadng o a lnear sysem ha has o be solved. In order o have a less complex noaon of he dfferen procedures, he absolue me-dependan dsplacemens wll be wren as u nsead of u ()a. he same smplfcaon s made on he velocy and acceleraon vecors. 9

3 Lnear dynamc analyss of a srucural sysem. Drec negraon In drec negraon he equaons n (.) are negraed usng a numercal sepby-sep procedure. he erm drec means ha no ransformaon of he equaons s performed before he me negraon s carred ou. he drec negraon s based on wo deas: Insead of sasfyng equaon (.) a any me, wll be sasfed only a dscree nervals. A varaon of dsplacemens, veloces and acceleraons whn each me nerval s assumed. I s he form of hs assumpon ha deermnes he accuracy, sably and cos of he procedure. Nex, some of he effecve mehods wll be presened. Snce all of hem calculae he soluon of he nex me sep from he soluons a he prevous mes consdered, algorhms wll be derved assumng ha he soluons are known a mes 0,,,.., and he soluon a + s requred. herefore, he frs sep of he procedure wll be calculaed from he mposed nal condons on he srucure. he mos mporan choce n any drec negraon mehod s he value of he me sep, due o he cos of he analyss manly depends on. On one hand, he me sep mus be small enough o oban accuracy n he soluon, bu, on he oher hand, he me sep mus no be smaller han necessary, because hs would mean ha he soluon s more cosly han requred. he wo fundamenal conceps of he negraon scheme o be consdered are sably and accuracy, for more nformaon on hem we refer o Bahe (98)... he cenral dfference mehod In hs mehod he followng expressons for acceleraon and velocy are assumed: + u = { u u+ u } (.3) + u = { u+ u } (.4) he error n boh expansons s of order ( ). Subsung he relaons no he equlbrum equaon evaluaed a me he lnear sysem o solve n each sep s obaned, from whch + u can be calculaed. + + = M C u r K M u M C u (.5) I has o be remarked ha he fac of evaluang he equlbrum equaon a leads o an explc negraon mehod, n whch no facorzaon of he sffness marx s requred. On he oher hand, he followng mehods ha wll be consdered are mplc ones, evaluang equlbrum a me +. 0

4 Lnear dynamc analyss of a srucural sysem he effecveness of he procedure depends on he use of a dagonal mass marx and he neglec of dampng forces. hs frs dsadvanage s no very serous, snce usng a fne-enough mesh, good accuracy of he soluon can be obaned. A more mporan consderaon s ha he me sep mus be smaller han a crcal value cr, whch depends on he mass and sffness properes of he srucure. herefore hs mehod s condonally sable. If he used me sep s larger han cr he negraon s unsable, meanng ha any error from round-off or numercal negraon grow and make he resul worhless. he followng dscussed procedures are uncondonally sable. hus, any me sep can be chosen for carryng ou he analyss, and, n many cases, can be orders of magnude larger han wha cr would allow. However, as menoned above, hese mehods are mplc, so a rangularzaon of he effecve sffness marx s requred... he Houbol mehod As n he cenral dfference mehod, he Houbol negraon scheme uses sandard fne dfference expressons for he acceleraon and velocy. hese approaches are backward-dfference formulas, wh errors of order ( ) : u} (.6) u = { u 8 u+ 9 u u } (.7) u= { u 5 u+ 4 u + + he dfference from he prevous mehod s ha, o solve he dsplacemen feld a +, he equlbrum equaon (.) s evaluaed a me +. When subsung he expressons for + u and + u he lnear sysem n equaon (.8) has o be solved = + + M 6 C K u r M C u M C u M 3 C u (.8) I s shown n equaon (.8) ha u needs o be known n order o calculae he value of he ndependen erm n he lnear sysem. Alhough he knowledge of 0 u, 0 u and 0 ü s useful o sar he Houbol mehod, usually specal sarng procedures are employed. An example s o calculae u and u by means of a condonally sable scheme, such as he cenral dfference mehod. I should be remnded ha he evaluaon of equaon (.) a + yelds o an mplc mehod, n whch no resrcon for exss bu he effecve sffness marx, defned as he marx facor of + u, M+ C+ K n hs case, has o be 6 facorzed.

5 Lnear dynamc analyss of a srucural sysem I can be seen n equaon (.8) ha when neglecng mass and dampng effecs (M=0 and C=0) he Houbol mehod reduces drecly o a sac analyss for medependan loads. In hs parcular case he cenral dfference mehod can no be used...3 he Wlson θ mehod In he Wlson θ mehod a lnear varaon of he acceleraon from me o me +θ s assumed, where θ >. If he uncondonal sably of he mehod s requred, θ mus be larger or equal o.37; hus, θ=.40 s usually employed. Denong τ as he me ncrease, he prevous menoned lnear varaon s expressed as: τ u = u + ( u u ) (.9) θ + τ + θ Inegrang hs equaon, formulas for velocy and dsplacemen varaon are derved: + τ τ u = u + u τ + ( u u ) + θ (.0) θ 3 + τ τ u= u+ u τ + u τ + ( + θ u u ) (.) 6θ Evaluang hese expressons a τ = θ s possble o solve + θ u and + θ u n erms of + θ u, wh he resul shown n equaons (.) and (.3). u 6 θ 6 = ( ) θ u u θ u u (.) 3 u = ( u u) u u (.3) θ + θ + + θ + θ θ o oban he soluon for he dsplacemens, veloces and acceleraons a me +, he equlbrum equaons (.) are consdered a +θ. Consequenly, he load vecor has o be lnearly nerpolaed. + θ + θ + θ + θ M u + C u + K u= r (.4) + θ + r = r+ θ r r) (.5) ( Subsung equaons (.) and (.3) no equaon (.4) a lnear sysem s obaned and + θ u can be solved. Nex, hs value s used n equaon (.) o oban + θ u whch s employed n equaons (.9), (.0) and (.) wh τ =. As menoned n secon.., he Wlson θ mehod s an mplc procedure, because he sffness marx s a coeffcen of he unknown dsplacemen vecor + θ u. I mgh also be noed ha no sarng procedures are needed, snce he varables a + are only expressed n erms of he same quanes a me.

6 Lnear dynamc analyss of a srucural sysem..4 he Newmark mehod In hs mehod he followng assumpons for he dsplacemen and velocy vecors are used: ( δ) + u = u + u + δ + u (.6) u= u+ u + α u + α u (.7) + + where α and δ are parameers ha can be deermned o oban negraon accuracy and sably. he scheme wll be uncondonally sable when δ > ½ and α > 0.5(δ + 0.5). Newmark (959) orgnally proposed as an uncondonally sable scheme he consanaverage-acceleraon mehod (also called rapezodal rule), n whch α = ¼ and δ = ½, ha has he mos desrable accuracy characerscs. o solve he dsplacemen, veloces and acceleraons a me +, he equlbrum equaons (.) are evaluaed a he same me M u + C u + K u= r (.8) Workng wh equaons (.6) and (.7) s possble o oban + u and + u n erms of + u only. hese relaons are subsued no equaon (.8) and + u s obaned solvng a lnear sysem. he Newmark mehod has smlar characerscs as he Wlson θ mehod, as s an mplc negraon scheme and no sarng procedures are needed..3 Mode superposon A very rough esmaon on he number of operaons needed o solve one me sep n a drec negraon mehod, usng a dagonal mass marx and neglecng dampng, s nm k, where n s he order of K and m k s s half-bandwdh. Moreover, n he mplc mehods an nal cos for he facorzaon of he effecve sffness marx s added. If a conssen mass marx s used and dampng s aken no accoun, he oal number of requred operaons s abou αnm k s, where α > and s s he number of me seps. Due o hs consderaon s expeced effecveness of a drec me negraon only for shor duraons (few me seps). If he negraon has o be done over a longer nerval may be more effecve o frs ransform he equlbrum equaons (.) no a form wh less cosly sep-by-sep soluon. In parcular, a reducon of he halfbandwdh m k would be successful. We may noe ha hs bandwdh depends on he nodal numberng, bu modfyng hs feaure here s a mnmum value for m k ha can no be reduced. 3

7 Lnear dynamc analyss of a srucural sysem.3. Change of bass o modal generalzed dsplacemens A more effecve form of he equlbrum equaons s obaned by usng he followng ransformaon on he nodal dsplacemens vecor: u () = Px() (.9) where P s a square marx, whch s sll unknown and wll be deermned, and x () s he me-dependan vecor conanng he generalzed dsplacemens. I s remarkable ha he marx P mus be non-sngular, n order o have a unque relaon beween u () and x (). Subsung equaon (.9) no equaon (.) and pre-mulplyng by P he new equlbrum equaon (.0) s derved. P MPx + P CPx + P KPx = P r () () () () Mx + Cx + Kx = r () () () () (.0) where he new sffness, mass and dampng marces have smaller bandwdh han he orgnal ones. Many dfferen marces P can be used, bu an effecve one s se by usng he dsplacemen soluons of he free vbraon equlbrum equaons wh dampng negleced. Mu + Ku = 0 (.) () () Consderng he soluon of he form u() = φsnω 0, where φ s a vecor of order n and ω represens he frequency of vbraon (rad/s) of he vecor φ, he generalzed egenproblem s derved: K φ= ω Mφ (.) he egenproblem leads o n egensoluons (ω,φ ), (ω,φ ),, (ω n,φ n ) where he egenvecors are M-orhonormalzed: ( ) and 0 < ω < ω < < ω n = ; = j φ Mφ j (.3) = 0; j he vecor φ s called he h-mode shape vecor and ω s he correspondng frequency of vbraon (rad/s). Wh hs group of egenvalues wo useful marces are defned. Φ s a marx whose columns are he egenvecors φ and Ω s a dagonal marx sorng he values ω. ω ω Φ = [ φ, φ,..., φ n ] Ω = (.4) ωn 4

8 Lnear dynamc analyss of a srucural sysem Snce he egenvecors are M-orhonormal, Φ MΦ = I, and wrng he n soluons as KΦ = MΦΩ, he followng expresson s obaned: Φ KΦ = Ω (.5) I s apparen ha he marx Φ s suable as he ransformaon marx P, u() = Φx(). hs yelds o he equlbrum equaon for modal generalzed dsplacemens: x C x x (.6) () + Φ Φ () + Ω () = Φ r() he nal condons on x () are obaned from u () usng he expressons: x = Φ Mu x = Φ Mu (.7) (0) (0) (0) (0).3. Analyss wh dampng negleced he eases problem ha can be solved s neglecng he dampng effecs on he srucure. hen, he analyss reduces o he soluon of he dfferenal equaon (.8). x (.8) () + Ω x() = Φ r() where mus be noed ha all equaons are decoupled, so hey can be solved ndvdually. Each h-equaon correspond o he equlbrum equaon of a sngle degree of freedom sysem wh un mass and sffness ω : x + ω x = r = φ r (.9) () () () () Equaon (.9) can be solved usng eher he prevous dscussed drec negraon procedures or he numercal evaluaon of he Duhamel negral: x = r ( τ)snω ( τ) dτ + α snω+ β cosω () ω 0 (.30) where α and β have o be deermned from he nal condons on dsplacemen and velocy. For he complee response of he srucure he dfferenal equaon (.9) has o be solved for =,,, n. hen, he me-hsory of he oal nodal dsplacemens s obaned by he superposon of each mode as shown n equaon (.3). n u = φ x (.3) () () = 5

9 Lnear dynamc analyss of a srucural sysem.3.3 Analyss ncludng dampng When dampng effecs on he srucures are consdered, s also desred o calculae havng decoupled equlbrum equaons. Bu, s only possble f he assumpon of proporonal dampng can be made, n whch case φ Cφj = ωξδ j (.3) where ξ s he modal dampng parameer and δ j s he Kronecker dela. herefore he egenvecors φ are also C-orhogonal and he equlbrum equaon reduces o n decoupled dfferenal equaons of he form: x + ωξ x + ω x = r (.33) () () () () he soluon of he dfferenal equaon can be calculaed usng he same procedures as n he negleced dampng case. hese are eher drec me negraon, mplc or explc, or evaluaon of he Duhamel negral: x r e d e + (.34) ( ) { ω } ξω ( τ) ξω () = ( τ) snω τ τ αsnω βcos ω + 0 where ω s defned as condons. ω = ω ξ and α and β are obaned from he nal he assumpon of proporonal dampng leads o wo remarkable consderaons: he oal dampng n he srucure s he sum of ndvdual dampng n each mode. here s no need o calculae he marx C for he complee srucure, snce only he modal dampng rao (ξ ) s used n he decoupled equaons. Les make he assumpon ha n some parcular case he drec sep-by-sep procedure s more effecve, and p modal dampng raos are known (ξ,, ξ p ). In consequence he marx C has o be explcly evaluaed o carry ou he analyss. If p = =, Raylegh dampng can be assumed, whch s of he form C= αm + βk (.35) where α and β are consans o be deermned from wo gven dampng raos and wo assocaed frequences of vbraon, workng wh expressons (.3) and (.35). ( ) φ φ φ φ (.36) C = αm+ βk = α + βω = ωξ 6

10 Lnear dynamc analyss of a srucural sysem whch gves wo equaons o solve he wo consans α and β. If p > a possble opon o deermne he C marx s o manan he assumpon of Raylegh dampng and use wo pars of average values (ξ, ω ) and (ξ, ω ). Anoher possbly s o use a more complcaed dampng marx, ha sasfes he relaon (.37). p k C= M a k M K (.37) k = 0 where he coeffcens a k, k =,, p, are calculaed from he p smulaneous equaons n (.38) a ξ = + aω + a ω a 0 3 p ω p 3 ω (.38) I has o be noed ha for p = reduces o he Raylegh dampng. An mporan consderaon s ha for p > 3 he marx C s, n general, a full marx, whch means ha he cos of he analyss s sgnfcanly ncreased. herefore, n order o work wh a banded C-marx, n mos analyses Raylegh dampng s assumed. A dsadvanage of he Raylegh dampng s ha he hgher modes are consderably more damped han he lower modes. he reason s ha, once he coeffcens have been deermned usng he lowes modes, any oher requred dampng rao s obaned from equaon (.39), whch provdes hgher values of he dampng rao for hgher egenfrequences. α + βω ξ = (.39) ω In pracce, approxmaely he same values for α and β are used n smlar srucures. However, he dampng characerscs of some srucures are no well approached by usng proporonal dampng. An example s he analyss of foundaonsrucure neracon problems, where more dampng may be observed n he foundaon han n he surface. For a case lke hs one, s reasonable o assgn dfferen Raylegh coeffcens (α and β) o dfferen pars of he srucure. Anoher example of nonproporonal dampng s he exsence of concenraed dampers, e.g. a he suppors of he srucure. he consderaon of nonproporonal dampng does no affec he soluon of he equlbrum equaons by a drec negraon scheme. No modfcaons are requred n he procedures prevously dscussed. On he oher hand, orgnaes a bg change n he mode superposon analyss, snce Φ CΦ s a full marx, equaon (.40 a). I means ha he equlbrum equaons n (.6) are no longer decoupled. Assumng ha he response of he srucure s manly conrolled by he frs p modes (φ,, φ p ), only he frs p equaons need o be consdered. herefore, he assumpon ha x, =,, p and x, =p+,, n are decoupled s done, equaon (.40 b), and he frs p equaons, shown n expresson (.40 c), can be solved by means of a drec negraon procedure. 7

11 Lnear dynamc analyss of a srucural sysem c, c, p c, p+ c, n c p, c p, p c p, p+ c, Φ CΦ = C p n = c p+, c p+, p c p+, p+ c p+, n cn, cn, p cn, p+ c n, n c, c, p 0 0 c p, c p, p 0 0 C = 0 0 c p+, p+ c p+, n 0 0 cnp, + c nn, (.40 a) (.40 b) 0 c r, c, p ω 0 x+ x + x= 0 c r p, c p, p 0 ω p p (.40 c).4 Comparson of soluon procedures he unque dfference beween a mode superposon and a drec negraon analyss s ha, before he me negraon s done, a change of bass s carred ou, from he fne elemen coordnae bass o he bass of egenvecors. Snce he same space s spanned by he n egenvecors as by he n nodal pon fne elemen dsplacemens, he same soluon mus be obaned n boh cases. herefore, he choce of mehod wll be only decded by effecveness crera. I s for hs reason ha anoher aspec of he mode superposon has o be dscussed, because can sgnfcanly decrease he cos of he analyss. Consder an appled load of he form r M, where f() s an arbrary = φ f () j () funcon of, and he nal condons u = 0, u = 0. In such a case, and workng (0) wh he decoupled equaons (.33), only he componen xj() s nonzero. Also he frequency of he loadng deermnes f he h equaon wll have an mporan conrbuon o he response. herefore, x () s relavely large f he excaon frequency of r, defned n (.9), les near ω. he concluson s ha frequenly only a small number of he decoupled equaons need o be consdered n order o oban a good approxmaon o he response of he sysem. he mos usual case s o use he equaon (.33) for =,, p, where p<<n, whch also means ha only he p lowes egenvalues and correspondng egenvecors have o be calculaed. hs reducon n he number of equaons can make he mode superposon much more effecve han drec negraon, bu he menoned (0) 8

12 Lnear dynamc analyss of a srucural sysem effecveness depends on he number of modes ncluded n he analyss, as well as her relaon wh he appled load (orhogonaly and frequency). When obanng he behavour of he srucure no only he dsplacemens are obaned, he elasc forces over he srucure members are also of neres. hese forces are calculaed accordng o expresson (.4). p () = () = ω () = S ω x() = f Ku Mu M φ (.4) Because each modal conrbuon s mulpled by he square of he modal frequency, s evden ha he hgher modes are of greaer sgnfcance n defnng he forces han n he dsplacemens. As a consequence, wll be necessary o nclude more modal componens o defne he forces o any desred degree of accuracy han o defne he dsplacemens. In any numercal analyss s useful o know he orgn of he errors, so he correc decsons can be aken o reduce hem. In a drec negraon hey appear because a oo large me sep s used, whereas n he mode superposon he reason s ha no enough modes have been used, assumng ha he p decoupled equaons have been accuraely solved. We may also remark ha, when dealng wh a nonlnear analyss, he only possbly s o perform a drec me negraon. Bu none of he here dscussed sepby-sep mehods are vald, snce hey were derved for a lnearly elasc maeral of whch he sffness properes,.e. K-marx, do no vary durng he whole analyss. Dealed nformaon n nonlnear dynamc analyss can be found n Bahe (98) and Clough (975). 9