DAMPING AND ENERGY DISSIPATION

Transcription

1 DAMPING AND ENERGY DISSIPATION Liear Viscous Dampig Is A Propery Of The Compuer Model Ad Is No A Propery Of A Real Srucure INTRODUCTION I srucural egieerig, viscous, velociy-depede dampig is very difficul o visualize for mos real srucural sysems. Oly a small umber of srucures have a fiie umber of dampig elemes where real viscous dyamic properies ca be measured. I mos cases modal dampig raios are used i he compuer model o approximae ukow oliear eergy dissipaio wihi he srucure. Aoher form of dampig, ha is ofe used i he mahemaical model for he simulaio of he dyamic respose of a srucure, is proporioal o he siffess ad mass of he srucure. This is referred o as Rayleigh dampig. Boh modal ad Rayleigh dampig are used i order o avoid he eed o form a dampig marix based o he physical properies of he real srucure. I rece years, he addiio of eergy dissipaio devices o he srucure has forced he srucural egieer o rea he eergy dissipaio i a more exac maer. I his chaper, he limiaios of modal ad Rayleigh dampig will be discussed. I addiio, deailed algorihms will be preseed for umerical soluio, by ieraio, for several differe ypes of oliear eergy dissipaio devices.

2 2 DYNAMIC ANALYSIS OF STRUCTURES DAMPING IN REAL STRUCTURES I is possible o esimae a effecive viscous dampig raio direcly from laboraory or field ess of srucures. Oe mehod is o apply a saic displaceme by aachig a cable o he srucure ad he suddely removig he load by cuig he cable. If he srucure ca be approximaed by a sigle degree of freedom, he displaceme respose will be of he form show i Figure For muli-degreeof-freedom srucural sysems, he respose will ivolve he respose of more modes ad he es ad he aalysis mehod required o predic he dampig raios will be more complex. I should be poied ou ha he decay of he ypical displaceme respose oly idicaes ha eergy dissipaio is akig place. The cause of he eergy dissipaio may be due o may differe effecs such as maerial dampig, joi fricio ad radiaio dampig a he suppors. However, if i is assumed ha all eergy dissipaio is due o liear viscous dampig, he free vibraio respose is give by he followig equaio: ξω u () = u() 0 e cos( ω ) D (19.1) 2 where ω D = ω 1 ξ Figure Free Vibraio Tes of Real Srucures, Respose vs. Time

3 DAMPING AND ENERGY DISSIPATION 3 Equaio (19.1) ca be evaluaed a ay wo maximum pois m cycles apar ad he followig wo equaios are produced: u( 2π) = u = u( 0) e ξω 2π/ ω D u( 2π( + m)) = u = u( 0) e + 1 ξω2π ( + m)/ ω D (19.2) (19.3) The raio of hese wo equaios is u u + 1 = e 2π mξ 2 1 ξ (19.4) Takig he aural logarihm of his raio ad rewriig produces he followig equaio: ξ l( u / u+ m ) ξ or, ξ = ξ 0 1 ξ ( 2π m = i 1) (19.5) This equaio ca easily be solved for he dampig raio ξ by ieraio. For example, if he decay raio u + m is equal o 1.25 bewee wo adjace maximums, hree ieraios yield ξ = Field esig of real srucures, subjeced o small displacemes, idicaes ypical dampig raios are less ha wo perce. Also, for mos srucures, he dampig is o liear ad is o proporioal o he velociy USE OF VISCOUS DAMPING IN ANALYSIS I he elasic dyamic aalysis of mos srucures subjeced o earhquake moios i is very commo o use five perce dampig for all modes. However, his value, i mos cases, has very lile experimeal or heoreical jusificaio. Also, for muli degree-of-freedom sysems, he use of modal dampig violaes dyamic equilibrium ad he fudameal laws of physics. For example, i is possible o calculae he

4 4 DYNAMIC ANALYSIS OF STRUCTURES reacios, as a fucio of ime, a he base of a srucure by he followig wo mehods: Firs, he ieria forces, a each mass poi, ca be calculaed i a specific direcio by he muliplicaio of he absolue acceleraio i ha direcio imes he mass a he poi. I he case of earhquake loadig, he sum of all hese forces mus be equal o he sum of he base reacio forces i ha direcio sice o oher forces ac o he srucure. Secod, he member forces, a he eds of all members aached o reacio pois, ca be calculaed as a fucio of ime. The sum of he compoes of he member forces i he direcio of he load is he base reacio force experieced by he srucure. I he case of zero modal dampig hese reacio forces, as a fucio of ime, are ideical. However, for ozero modal dampig, hese reacio forces are sigificaly differe. These differeces idicae ha liear modal dampig iroduces exeral loads, acig o he srucure above he base, which are physically impossible. This is clearly a area where he sadard sae of he ar assumpio of modal dampig eeds o be re-examied ad a aleraive approach mus be developed. Eergy dissipaio exiss i real srucures. However, i mus be i he form of equal ad opposie forces bewee pois wihi he srucure. Therefore, a viscous damper, or ay oher ype of eergy dissipaig device, coeced bewee wo pois wihi he srucure is physically possible ad will o cause a error i he reacio forces. There mus be zero base shear for all ieral eergy dissipaio forces. Aoher ype of eergy dissipaio ha exiss i real srucures is radiaio dampig a he suppors of he srucure. The vibraio of he srucure srais he foudaio maerial ear he suppors ad causes sress waves o radiae io he ifiie foudaio. This ca be sigifica if he foudaio maerial is sof relaive o he siffess of he srucure. A sprig, damper ad mass a each suppor ofe approximae his ype of dampig.

5 DAMPING AND ENERGY DISSIPATION NUMERICAL EXAMPLE I order o illusrae he errors ivolved i he use of modal dampig a simple sevesory buildig was subjeced o a ypical earhquake moio. Table 19.1 idicaes he values of base shear calculaed from he exeral ieria forces, which saisfy dyamic equilibrium, ad he base shear calculaed from he exac summaio of he shears a he base of he hree colums a each ime icreme. I is of ieres o oe ha he maximum values of base shear, calculaed from wo differe mehods, are sigificaly differe for he same compuer ru. The oly logical explaaio is he exisece of exeral dampig forces ha exis oly i he mahemaical model of he srucure. Sice his is physically impossible, he use of sadard modal dampig ca produce a small error i he aalysis. TABLE Compariso of Base Shear for Seve Sory Buildig Dampig Perce Dyamic Equilibrium BASE SHEAR (kips) Sum of Colum SHEARS (kips) ERROR Perce Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec I is of ieres o oe ha he use of oly five perce dampig reduces he base shear from 371 kips o 254 kips for his example. Sice he measureme of dampig i mos real srucures has bee foud o be less ha wo perce he selecio of five perce reduces he resuls sigificaly STRUCTURES WITH LINEAR VISCOUS DAMPERS I is possible o model srucural sysems wih liear viscous dampers a arbirary locaios wihi a srucural sysem. The exac soluio ivolves he calculaio of complex eigevalues ad eigevecors ad a large amou of compuaioal effor. Sice he basic aure of eergy dissipaio is o clearly defied i real srucures ad viscous dampig is ofe used o approximae oliear behavior, his icrease

6 6 DYNAMIC ANALYSIS OF STRUCTURES i compuaioal effor is o jusified sice we are o solvig he real problem. A more efficie mehod o solve his problem is o move he dampig force o he righ had side of he dyamic equilibrium equaio ad solve he problem as a oliear problem usig he FNA mehod STIFFNESS AND MASS PROPORTIONAL DAMPING A very commo ype of dampig used i he oliear icremeal aalysis of srucures is o assume ha he dampig marix is proporioal o he mass ad siffess marices. Or, C= η M+ δ K (19.6) This ype of dampig is ormally referred o as Rayleigh dampig. I mode superposiio aalysis he dampig marix mus have he followig properies i order for he modal equaios o be ucoupled: T 2ω ζ = φ C φ (19.7) Due o he orhogoaliy properies of he mass ad siffess marices, his equaio ca be rewrie as 2 2ωζ = η+ δω or, ζ 1 ω η ω = + δ 2 2 (19.8) I is appare ha modal dampig ca be specified exacly a oly wo frequecies i order o solve for η ad δ i he above equaio. I addiio, he assumpio of mass proporioal dampig implies he exisece of exeral suppored dampers ha are physically impossible for a base suppored srucure. The use of siffess proporioal dampig has he effec of icreasig he dampig i he higher modes of he srucure for which here is o physical jusificaio ad ca resul i sigifica errors for impac ype of problems. Therefore, he use of Rayleigh ype of dampig is difficul o jusify for mos srucures. However, i coiues o be used by may compuer programs i order o obai resuls for umerically sesiive srucural sysems.

7 DAMPING AND ENERGY DISSIPATION NONLINEAR ENERGY DISSIPATION Mos physical eergy dissipaio i real srucures is i phase wih he displacemes ad is a oliear fucio of he magiude of he displacemes. Neverheless, i is commo pracice o approximae he oliear behavior wih a equivale liear dampig ad o coduc a oliear aalysis. The major reaso for his approximaio is ha all liear programs for mode superposiio or respose specrum aalysis ca cosider liear viscous dampig i a exac maer. This approximaio is o loger ecessary if he srucural egieer ca ideify where ad how he eergy is dissipaed wihi he srucural sysem. The FNA mehod provides a aleraive o he use of equivale liear viscous dampig. I his secio various oliear devices will be discussed ad heir ieraive soluio algorihm will be summarized. Base isolaors are oe of he mos commo ypes of predefied oliear elemes used i earhquake resisa desigs. Mechaical dampers, fricio devices ad plasic higes are oher ype of commo oliear elemes. I addiio, gap elemes are required o model coac bewee srucural compoes ad uplifig of srucures. A special ype of gap eleme, wih he abiliy o crush ad dissipae eergy, is useful o model cocree ad soil ypes of maerials. Cables, ha ca ake esio oly ad dissipae eergy i yieldig, are ecessary o capure he behavior of may bridge ype srucures BILINEAR PLASTICITY ELEMENT The geeral plasiciy eleme ca be used o model may differe ypes of oliear maerial properies. The fudameal properies ad behavior of he eleme are illusraed i he figure show below:

8 8 DYNAMIC ANALYSIS OF STRUCTURES f d y k y k e k e Figure Fudameal Behavior of Biliear Plasiciy Eleme Where k e = iiial liear siffess k y = Yield siffess d y = Yield deformaio The force-deformaio relaioship is calculaed from f = k y d + ( k e - k y ) e (19.9) Where d is he oal deformaio ad e is a elasic deformaio erm ad has a rage ± d y. I is calculaed a each ime sep by he umerical iegraio of oe of he followig differeial equaios: d If If de & 0 e & = (1 - e & d )d y (19.10) & & & de < 0 e = d (19.11) The followig fiie differece approximaios, for each ime sep, ca be made: &d = d - d - Ad &e = e - e - The umerical soluio algorihm (six program saemes) ca be summarized a

9 DAMPING AND ENERGY DISSIPATION 9 he ed of each ime icreme, a ime " " for ieraio "i ", i Table Table Ieraive Algorihm For Biliear Plasiciy Eleme 1. Chage i deformaio for ime sep a ime for ieraio i v = d - d- 2. Calculae elasic deformaio for ieraio i (i-1) if v e 0 e = e + v (i-1) if v e > e = e + (1 - e d ) v if e > d y e = dy if e < -d y e = -dy 3. Calculae ieraive force: f = k d + (k - k )e y e y y Noe ha he approximae erm e d - y is used from he ed of he las ime icreme e raher ha he ieraive erm. This approximaio elimiaes all problems d y associaed wih covergece for large values of. However, he approximaio has isigifica effecs o he umerical resuls for all values of. For all pracical purposes, a value of equal o 20 produces rue biliear behavior DIFFERENT POSITIVE AND NEGATIVE PROPERTIES The previously preseed plasiciy eleme ca be geeralized o have differe posiive, d P, ad egaive, d, yield properies. This will allow he same eleme o model may differe ypes of eergy dissipaio devices such as he double diagoal Pall fricio eleme. For cosa fricio he double diagoal Pall eleme has k e = 0 ad 20. For small forces boh diagoals remai elasic, oe i esio ad oe i compressio. A some deformaio, d N, he compressive eleme may reach a maximum possible

10 10 DYNAMIC ANALYSIS OF STRUCTURES value. Fricio slippig will sar a he deformaio u P afer which boh he esio ad compressio forces will remai cosa uil he maximum displaceme for he load cycle is obaied. This eleme ca be used o model bedig higes i beams or colums wih osymmeric secios. The umerical soluio algorihm for he geeral biliear plasiciy eleme is give i Table Table Ieraive Algorihm For No-Symmeric Biliear Eleme 1. Chage i deformaio for ime sep a ime for ieraio i v = d - d- 2. Calculae elasic deformaio for ieraio i (i-1) if v e 0 e = e - + v (i-1) if ve > 0 ad e > 0 (i-1) if ve > 0 if e > d y e = dy if e < -d y e = -dy 3. Calculae ieraive force a ime : f = k d + (k - k )e e = e - + (1 - e d ) v y ad e < 0 e = e - + (1 - e d ) v y e y THE BILINEAR TENSION-GAP-YIELD ELEMENT The biliear esio oly eleme ca be used o model cables coeced o differe pars of he srucure. I he rerofi of bridges his ype of eleme is ofe used a expasio jois o limi he relaive moveme durig earhquake moios. The fudameal behavior of he eleme is summarized i Figure The posiive umber d 0 is he iiial pre-sress deformaio. A egaive umber specifies he axial deformaio associaed wih iiial cable sag. The permae eleme yield deformaio is u p.

11 DAMPING AND ENERGY DISSIPATION 11 f d 0 d y d p k y k ee k e d Figure Tesio-Gap-Yield Eleme The umerical soluio algorihm for his eleme is summarized i Table Noe ha he permae deformaio calculaio is based o he coverged deformaio a he ed of he las ime sep. This avoids umerical soluio problems. Table Ieraive Algorihm -Tesio-Gap-Yield Eleme 1. Updae Tesio Yield Deformaio From Previous Coverged Time Sep y = d + d d 0 y d p p = if y < he d y 2. Calculae Elasic Deformaio for Ieraio d = d + d0 e = d d p e if y > d he e = dy 3. Calculae Ieraive Force: f = k ( d d ) + (k - k )e if f i y e y 0 >0 he f i =0

12 12 DYNAMIC ANALYSIS OF STRUCTURES NONLINEAR GAP-CRUSH ELEMENT Perhaps he mos commo ype of oliear behavior ha occurs i real srucural sysems is he closig of a gap bewee differe pars of he srucure; or, he uplifig of he srucure a is foudaio. The eleme ca be used a abume-soil ierfaces ad for modelig soil-pile coac. The gap/crush eleme has he followig physical properies: 1. The eleme cao develop a force uil he opeig d 0 gap is closed 2. The eleme ca oly develop a compressio force 3. The crush deformaio d p is always a moaomic decreasig egaive umber The umerical algorihm for he gap-crush eleme is summarized i Table Table Ieraive Algorihm To Model Gap-Crush Eleme 1. Updae Crush Deformaio From Previous Coverged Time Sep y = d + d + d if y 0 y > d he d = y 2. Calculae Elasic Deformaio: e = d + d d if e < -d y he e = -dy 3. Calculae Ieraive Force: f = k ( d d ) + (k - k )e if f i y y e y 0 >0 he f i The umerical covergece of he gap eleme ca be very slow if a large elasic siffess erm k e is used. The user mus ake grea care i selecig a physically realisic umber. I order o miimize umerical problems, he siffess k e should o be over 100 imes he siffess of he elemes adjace o he gap. The dyamic coac problem bewee real srucural compoes ofe does o have a uique soluio. Therefore, i is he resposibiliy of he desig egieer o selec maerials =0

13 DAMPING AND ENERGY DISSIPATION 13 a coac pois ad surfaces o have realisic maerial properies ha ca be prediced accuraely VISCOUS DAMPING ELEMENTS Liear velociy-depede eergy-dissipaio forces exis i oly a few special maerials subjeced o small displacemes. I erms of equivale model dampig, experimes idicae ha hey are a small fracio of oe perce. Maufacured mechaical dampers cao be made wih liear viscous properies sice all fluids have fiie compressibiliy ad oliear behavior is prese i all mamade devices. I he pas i has bee commo pracice o approximae he behavior of hese viscous oliear elemes by a simple liear viscous force. More recely, vedors of hese devices have claimed ha he dampig forces are proporioal o a power of he velociy. Experimeal examiaio of a mechaical device idicaes a far more complex behavior ha cao be represeed by a simple oe-eleme model. The FNA mehod does o require ha hese dampig devices be liearized or simplified i order o obai a umerical soluio. If he physical behavior is udersood i is possible for a ieraive soluio algorihm o be developed which will accuraely simulae he behavior of almos ay ype of dampig device. I order o illusrae he procedure le us cosider he device show i Figure f = k d ( i ) p p f = f + p s f k p I k s c J f s ( i = k ( d e ) = sig( e& ) e& s ) N c Figure Geeral Dampig Eleme Coeced Bewee Pois I ad J I is appare ha he oal deformaio, e, across he damper mus be accuraely calculaed i order o evaluae he equilibrium wihi he eleme a each ime sep. The fiie differece equaio used o esimae he damper deformaio a ime is

14 14 DYNAMIC ANALYSIS OF STRUCTURES e = e + e& d = e + ( e& + e& τ τ ) 2 (19.12) A summary of he umerical algorihm is summarized i Table Table Ieraive Algorihm For Noliear Viscous Eleme 1. Esimae damper force from las ieraio: ( i ) f = k( d e 1 ) c 2. Esimae damper velociy: 1 fc N e& = ( ) sig( f c c 3. Esimae damper deformaio: 1 e = e + ( e& ) + e& ) 2 4. Calculae oal ieraive force: f = k d + k ( d e ) p s ) SUMMARY The use of liear modal dampig, as a perceage of criical dampig, has bee used o approximae he oliear behavior of srucures. The eergy dissipaio i real srucures is far more complicaed ad eds o be proporioal o displacemes raher ha proporioal o he velociy. The use of approximae equivale viscous dampig has lile heoreical or experimeal jusificaio. I is ow possible o accuraely simulae he behavior of srucures wih a fiie umber of discree eergy dissipaio devices isalled. The experimeal deermied properies of he devices ca be direcly icorporaed io he compuer model.