21. NONLINEAR ELEMENTS

Transcription

1 21. NONLINEAR ELEMENTS Earhquake Reia Srucure Should Have a Limied Number o Noliear Eleme ha ca be Eail Ipeced ad Replaced aer a Major Earhquake INTRODUCTION { XE "Eerg:Eerg Diipaio Eleme" }{ XE "Noliear Eleme" }Ma diere pe o pracical oliear eleme ca be ued i cojucio wih he applicaio o he Fa Noliear Aali mehod. The FNA mehod i ver eecive or he deig or reroi o rucure o rei earhquake moio becaue i i deiged o be compuaioall eicie or rucure wih a limied umber o predeied oliear or eerg diipaig eleme. Thi i coie wih he moder philooph o earhquake egieerig ha eerg diipaig eleme hould be able o be ipeced ad replaced aer a major earhquake. Bae iolaor are oe o he mo commo pe o predeied oliear eleme ued i earhquake reia deig. I addiio, iolaor, mechaical damper, ricio device ad plaic hige are oher pe o commo oliear eleme. Alo, gap eleme are required o model coac bewee rucural compoe ad upliig o rucure. A pecial pe o gap eleme wih he abili o cruh ad diipae eerg i ueul o model cocree ad oil pe o maerial. Cable ha ca ake eio ol ad diipae eerg i ieldig are ecear o capure he behavior o ma bridge pe rucure. I hi chaper he behavior o everal o hoe eleme will be preeed ad deailed oluio algorihm will be ummarized.

2 21-2 STATIC AND DYNAMIC ANALYSIS 21.2 GENERAL THREE-DIMENSIONAL TWO-NODE ELEMENT The pe o oliear eleme preeed i hi chaper i imilar o he hreedimeioal beam eleme. However, i ca degeerae io a eleme wih zero legh where boh ed are locaed a he ame poi i pace. Thereore, i i poible o model lidig ricio urace, coac problem ad coceraed plaic hige. Like he beam eleme, he uer mu deie a local reerece em o deie he local oliear eleme properie ad o ierpre he reul. A pical eleme, coeced bewee wo poi I ad J, i how i Figure z d 5, 5 d 4, 4 d 2, 2 d 1, 1 d 3, 3 J d 6, 6 L I x Figure 21.1 Relaive Diplaceme - Three-Dimeioal Noliear Eleme I i impora o oe ha hree diplaceme ad hree roaio are poible a boh poi I ad J ad ca be expreed i eiher he global X-Y-Z or local reerece em. The orce ad diplaceme raormaio marice or hi oliear eleme are he ame a or he beam eleme give i Chaper 4. For mo eleme pe, ome o hoe diplaceme do o exi or are equal a I ad J. Becaue each hree-dimeioal eleme ha ix rigid bod diplaceme, he equilibrium o he eleme ca be expreed i erm o he ix relaive diplaceme how i Figure Alo, L ca equal zero. For example, i a

3 NONLINEAR ELEMENTS 21-3 coceraed plaic hige wih a relaive roaio abou he local 2-axi i placed bewee poi I ad J, ol a relaive roaio d 5 exi. The oher ive relaive diplaceme mu be e o zero. Thi ca be accomplihed b eig he abolue diplaceme a joi I ad J equal GENERAL PLASTICITY ELEMENT { XE "Plaici Eleme" }The geeral plaici eleme ca be ued o model ma diere pe o oliear maerial properie. The udameal properie ad behavior o he eleme are illuraed i Figure d k k e k e Figure 21.2 Fudameal Behavior o Plaici Eleme d where k e k iiial liear ie Yield ie d Yield deormaio The orce-deormaio relaiohip i calculaed rom: k d +(ke - k) e (21.1) Where d i he oal deormaio ad e i a elaic deormaio erm ad ha a rage ± d. I i calculaed a each ime ep b he umerical iegraio o oe o he ollowig diereial equaio:

4 21-4 STATIC AND DYNAMIC ANALYSIS I I e de & 0 e(1- & ) d & (21.2) d d & e< 0 e & d& (21.3) The ollowig iie dierece approximaio or each ime ep ca be made: d - d d & - ad e - e- e & (21.4a ad 21.4b) The umerical oluio algorihm (ix compuer program aeme) ca be ummarized a he ed o each ime icreme, a ime or ieraio i, i Table Table 21.1 Ieraive Algorihm or Plaici Eleme 1. Chage i deormaio or ime ep a ime or ieraio i v d - d - 2. Calculae elaic deormaio or ieraio i (i-1) i v e 0 e e + v (i-1) i v e > 0-3. Calculae ieraive orce: k d + (k - k )e - e e + (1 - e d ) v i e > d e d i e < -d e -d e e - Noe ha he approximae erm d e icreme raher ha he ieraive erm d i ued rom he ed o he la ime. Thi approximaio elimiae all problem aociaed wih covergece or large value o. However, he approximaio ha iigiica eec o he umerical reul or all value o

5 NONLINEAR ELEMENTS For all pracical purpoe, a value o equal o 20 produce rue biliear behavior DIFFERENT POSITIVE AND NEGATIVE PROPERTIES { XE "Algorihm or:biliear Plaici Eleme" }The previoul preeed plaici eleme ca be geeralized o have diere poiive, d P, ad egaive, d, ield properie. Thi will allow he ame eleme o model ma diere pe o eerg diipaio device, uch a he double diagoal Pall ricio eleme. Table 21.2 Ieraive Algorihm or No-Smmeric Biliear Eleme 1. Chage i deormaio or ime ep a ime or ieraio i v d - d - 2. Calculae elaic deormaio or ieraio i (i-1) i v e 0 e e + v (i-1) i v e > 0 (i-1) i v e > 0 - ad e ad e e 0 e e- +(1- ) v d p > i e > d p i e < - d 3. Calculae ieraive orce a ime : k d + (k - k )e < 0 - e d p e d e e e + (1 - e d ) v - For coa ricio, he double diagoal Pall eleme ha k e 0 ad 20. For mall orce boh diagoal remai elaic, oe i eio ad oe i compreio. A ome deormaio, d, he compreive eleme ma reach a maximum poible value. Fricio lippig will ar a he deormaio d p aer which boh he eio ad compreio orce will remai coa uil he maximum diplaceme or he load ccle i obaied.

6 21-6 STATIC AND DYNAMIC ANALYSIS Thi eleme ca be ued o model bedig hige i beam or colum wih o-mmeric ecio. The umerical oluio algorihm or he geeral biliear plaici eleme i give i Table THE BILINEAR TENSION-GAP-YIELD ELEMENT { XE "Teio-Gap-Yield Eleme" }The biliear eio-ol eleme ca be ued o model cable coeced o diere par o he rucure. I he reroi o bridge, hi pe o eleme i oe ued a expaio joi o limi he relaive moveme durig earhquake moio. The udameal behavior o he eleme i ummarized i Figure The poiive umber d 0 i he axial deormaio aociaed wih iiial cable ag. A egaive umber idicae a iiial pre-re deormaio. The permae eleme ield deormaio i d p. d 0 d d p k k e k e d Figure 21.3 Teio-Gap-Yield Eleme The umerical oluio algorihm or hi eleme i ummarized i Table Noe ha he permae deormaio calculaio i baed o he coverged deormaio a he ed o he la ime ep. Thi avoid umerical oluio problem.

7 NONLINEAR ELEMENTS 21-7 { XE "Algorihm or:teio-gap-yield Eleme" }Table 21.3 Ieraive Algorihm or Teio-Gap-Yield Eleme 1. Updae Teio Yield Deormaio rom Previou Coverged Time Sep d d 0 d i < he d d p p 2. Calculae Elaic Deormaio or Ieraio d d d 0 e d d p e i > d he e d 3. Calculae Ieraive Force: k ( d d 0 ) + (k - k )e i 0 he 0 < e 21.6 NONLINEAR GAP-CRUSH ELEMENT Perhap he mo commo pe o oliear behavior ha occur i real rucural em i he cloig o a gap bewee diere par o he rucure; or, he upliig o he rucure a i oudaio. The eleme ca be ued a abume-oil ierace ad or modelig oil-pile coac. The gap/cruh eleme ha he ollowig phical properie: 1. The eleme cao develop a orce uil he opeig d 0 gap i cloed. A egaive value o d 0 idicae a iiial compreio orce. 2. The eleme ca ol develop a egaive compreio orce. The ir ield deormaio d i peciied b a poiive umber. 3. The cruh deormaio d c i alwa a moooicall decreaig egaive umber. The umerical algorihm or he gap-cruh eleme i ummarized i Table 21.4.

8 21-8 STATIC AND DYNAMIC ANALYSIS { XE "Algorihm or:gap-cruh Eleme" }Table 21.4 Ieraive Algorihm or Gap-Cruh Eleme 1. Updae Cruh Deormaio rom Previoul Coverged Time Sep: d + d0 + d i > d c he d c 2. Calculae Elaic Deormaio: e d + d o d c i e < -d he e -d 3. Calculae Ieraive Force: k ( d + d 0 ) +( k e - k i 0 he 0 > )e The umerical covergece o he gap eleme ca be ver low i a large elaic ie erm k e i ued. The uer mu ake grea care i elecig a phicall realiic umber. To miimize umerical problem, he ie k e hould o be over 100 ime he ie o he eleme adjace o he gap. The damic coac problem bewee real rucural compoe oe doe o have a uique oluio. Thereore, i i he repoibili o he deig egieer o elec maerial a coac poi ad urace ha have realiic maerial properie ha ca be prediced accurael VISCOUS DAMPING ELEMENTS { XE "Algorihm or:dampig Eleme" }{ XE "Algorihm or:vicou Eleme" }Liear veloci-depede eerg-diipaio orce exi i ol a ew pecial maerial ubjeced o mall diplaceme. I erm o equivale modal dampig, experime idicae ha he are a mall racio o oe perce. Mauacured mechaical damper cao be made wih liear vicou properie becaue all luid have iie compreibili ad oliear behavior i pree i all mamade device. I he pa i ha bee commo pracice o approximae he behavior o hoe vicou oliear eleme b a imple liear

9 NONLINEAR ELEMENTS 21-9 vicou orce. More recel, vedor o hoe device have claimed ha he dampig orce are proporioal o a power o he veloci. Experimeal examiaio o a mechaical device idicae a ar more complex behavior ha cao be repreeed b a imple oe-eleme model. The FNA mehod doe o require ha hoe dampig device be liearized or impliied o obai a umerical oluio. I he phical behavior i uderood, i i poible or a ieraive oluio algorihm o be developed ha will accurael imulae he behavior o almo a pe o dampig device. To illurae he procedure, le u coider he device how i Figure p I k d p p k p k c J ( k ( d e ) ig( e& ) e& i) N c Figure 21.4 Geeral Dampig Eleme Coeced Bewee Poi I ad J I i appare ha he oal deormaio, e ), acro he damper mu be accurael calculaed o evaluae he equilibrium wihi he eleme a each ime ep. The iie dierece equaio ued o eimae he damper deormaio a ime i: e i ( ) e e& d e ( e& e& + τ τ + + ) (21.5) 2 A ummar o he umerical algorihm i ummarized i Table (i

10 21-10 STATIC AND DYNAMIC ANALYSIS { XE "Algorihm or:noliear Dampig" }Table 21.5 Ieraive Algorihm or Noliear Vicou Eleme 1. Eimae damper orce rom la ieraio: ( i 1) k ( d e ) 2. Eimae damper veloci: e& ( i ( c ) ) 1 N ig( 3. Eimae damper deormaio: e ) ( + ( e& ) + e& 2 e i) 4. Calculae oal ieraive orce: () i k d + k ( d e ) p ) 21.8 THREE-DIMENSIONAL FRICTION-GAP ELEMENT { XE "Fricio-Gap Eleme" }Ma rucure have coac urace bewee compoe o he rucure or bewee rucure ad oudaio ha ca ol ake compreio. Durig he ime he urace are i coac, i i poible or ageial ricio orce o develop bewee he urace. The maximum ageial urace orce, which ca be developed a a paricular ime, are a ucio o he ormal compreive orce ha exi a ha ime. I he urace are o i coac, he ormal ad he urace ricio orce mu be zero. Thereore, urace lip diplaceme will ake place durig he period o ime whe he allowable ricio orce i exceeded or whe he urace are o i coac. To develop he umerical algorihm o predic he damic behavior bewee urace, coider he coac urace eleme how i Figure The wo urace ode are locaed a he ame poi i pace ad are coeced b he gap-ricio eleme ha ha coac ie k i all hree direcio. The hree direcio are deied b a local, ad +90 o reerece em. The eleme

11 NONLINEAR ELEMENTS deormaio d, d ad d +90 are relaive o he abolue diplaceme o he wo urace. d d Figure 21.5 Three-Dimeioal Noliear Fricio-Gap Eleme Durig he ime o coac, he orce-deormaio relaiohip or he riciogap eleme are: Normal Force: Maximum Allowable Slip Force: Tageial Surace Force: kd (21.6a) a µ (21.6b) k( d ) ig( ) or, (21.6c) a The coeicie o lidig ricio i deigaed b µ. The urace lip deormaio i he direcio i. The ieraive umerical algorihm or a pical ime ep i ummarized i Table To miimize umerical problem, he ie k hould o be over 100 ime he ie o he eleme adjace o he gap.

12 21-12 STATIC AND DYNAMIC ANALYSIS { XE "Algorihm or:fricio-gap Eleme" }Table 21.6 Ieraive Algorihm or Fricio-Gap Eleme 1. I i1, updae lip deormaio rom previoul coverged ime ep a ad ( ) ( ) 2. Evaluae ormal ad allowable lip orce i d > 0 0 i d 0 kd µ a 3. Calculae urace orce a ad i d > 0 0 i 0 d k( d ) i > a ig( 4. Calculae lip deormaio a ad i d > 0 d i d k a / ) a 21.9 SUMMARY The ue o approximae equivale liear vicou dampig ha lile heoreical or experimeal juiicaio ad produce a mahemaical model ha violae damic equilibrium. I i ow poible o accurael imulae he behavior o rucure wih a iie umber o dicree gap, eio ol, ad eerg diipaio device ialled. The experimeall deermied properie o he device ca be direcl icorporaed io he compuer model.