On displacement-based plastic design of parallel chord vierendeel girders

Transcription

1 t J Adv Struct Eg (0) 6:6 DO 0.007/ OGNAL ESEACH O dplacemet-baed platc deg of parallel chord veredeel grder ark Grgora eceved: 3 arch 0 / Accepted: 8 Jue 0 / ublhed ole: 9 July 0 The Author() 0. Th artcle publhed wth ope acce at Sprgerlk.com Abtract The paper troduce the prcple of dplacemet-baed platc deg (DBD) ad t applcato to the effcet deg of parallel chord teel veredeel grder uder ormal odal force. A mplfyg aumpto ha bee made that the mathematcal model compoed of magary, p coected module that ft wth the bay of the prototype. The ue of th modelg cocept cojucto wth the applcato of the uform tregth theory lead to the developmet of a algorthm that deally uted for maual, mmum weght deg of teel veredeel grder uder ay dtrbuto of vertcal odal force. The reultg oluto are exact ad uque ad led themelve well to DBD ad mmum weght treatmet. DBD whch ak to performace cotrol, member tregth ad tffee are aged rather tha teted. Several geerc example have bee provded to llutrate the applcato of the propoed deg procedure. The umercal reult of thee example have bee verfed through log had ad computer method of aaly. A exteve proof of the propoed method of approach ha bee provded the Appedx. Keyword Veredeel grder latc deg mum weght Dplacemet at cpet collape troducto latc deg the ba of a umber of recetly developed deg methodologe for momet frame uder. Grgora (&) GA Structural Egeerg Coultat c., Gledale, CA, USA e-mal: markarja@aol.com lateral loadg, (azzola ad luo 997; Goel et al. 00; Grgora ad Grgora 0). The techcal mert ad ecoomc beeft of platc deg have bee amply documeted the lterature a well a uch authortatve text uch a Baker et al. (956), Beedle (958), Neal (963), Nethercot (00). latc deg alo the corertoe of the preet cotrbuto. The ovelty of the preet cotrbuto that t corporate the magtude ad locato of maxmum platc deformato a part of the member zg crtera. A pror kowledge of the elatoplatc performace of ay tructure at dtct tage of loadg, uch a at frt yeld ad cpet collape, may help deg the ytem a a tructure of uform tregth (US). Structure of US led themelve well to DBD, C ad drect materal optmzato. A large umber of regular tructural framework uch a momet frame, (Wog 009; Grgora ad Grgora 0a), true, grllage, veredeel grder (VG) etc., ca be deged ad/or optmzed a tructure of US. Sce the demad capacty rato cotat for the cottuet member of tructure of US, they ted to fal a tate of over-complete collape where all platc hge form multaeouly. Whle the tate of over-complete collape aocated wth mmum weght deg, the correpodg dtrbuto of platc hge ca be looked upo a a admble template for mpog ay order of propagato of platcty wthout affectg the ecoomc ad/or the ultmate carryg capacty of the ytem. The curretly avalable deg ad optmzato techque are hghly teratve ature ad ca bet be utlzed f the correpodg oftware acceble. All uch procedure beg wth a tal deg followed by umerou terato, utl covergece take place. The problem cotrat cot uually of et of code ad purpoepecfc codto. Sce both the orgal ad the fal 3

2 6 age of t J Adv Struct Eg (0) 6:6 (optmzed) deg are addreed through teratve procee, the combed effort may be vewed a propectve ature ad vetgatve effort, ad a uch apply to almot all tradtoal deg for cotemporary tructure. cotrat to propectve procee that eek to atfy et of precrbed crtera, kowledge-baed methodologe are expected to meet certa repoe objectve. vetgatve procedure, theore of tructure ad materal cece are followed rather tha appled. t mght teret the reader to ote that the makg of atural tructure, uch a tree, kowledge duced rather tha teted. The purpoe of th paper to troduce DBD a a obervato-baed approach for the effcet deg of certa type of momet frame uch a parallel chord VG ad mlar tructure, (Grgora 0). erformace-oreted deg are kowledge-baed tratege that am at provdg ght to the repoe of certa type of tructure uder pecfc loadg codto. th cotext, obervato baed refer to relace o theoretcal ad/or expermetal data for realtc deg purpoe. DBD a recet developmet that allow for the mpoto of the requred a well a derable deg requremet, a oppoed to vetgatg the ame a reult of teratve computato. The derable requremet may clude the ablty to cotrol the equece of formato of the platc hge, lmt the maxmum dplacemet at cpet collape, optmze the elf weght of the tructure, prevet premature collape, etc. DBD heretly capable of reultg mmum weght elatoplatc oluto for ductle momet frame ad a uch provde a opportuty for the effcet deg of teel VG uder -plae loadg. The paper beg by revewg the fudametal cocept volved modular aaly of regular momet frame ad tructure-pecfc fdg that help formulate the propoed oluto. The lmt tate load carryg charactertc of parallel chord VG of US uder arbtrary dtrbuto of ormal odal force are the preeted a a prelude to the determato of the effcet deg of VG through everal geerc example. Thee example preet a umber of ueful formulae for the prelmary deg of VG. All ymbol are defed a they frt appear the text. The paper doe ot addre the vbratory behavor of VG (Grgora 970). The modelg cocept t ha bee demotrated by Grgora (993), Grgora ad Grgora (988), Taraath (998) ad Bozorga ad Bertero (00) that mathematcal model of uprght, regular rectagular momet frame ca be cotrued a beg compoed of hypothetcal, rgdly joted, rectagular module tacked o top of each other ad upported o bae level boudary module. A mlar approach ha bee adopted here to tudy the effcet deg of VG a horzotal momet frame, mply upported at both ed ad ubjected to a mootocally creag dtrbuto of ormal odal force appled alog the top ad/or bottom chord of the grder. The effect of hear, pael zoe, axal load ad other ecodary pheomea o flexural repoe of the ytem have bee dcarded for the ake of mplcty. Stra hardeg ad platc hge offet have alo bee gored favor of clarty a well a hgher load factor agat falure. The poeerg tude pertag to platc deg of VG are due to Hedry (955) ad orovch (958) who troduced the applcato of mple platc theory to the deg of mld teel parallel chord VG. The le dagram of a geeralzed parallel chord VG uder a arbtrary dtrbuto of ormal odal force preeted Fg. a. The correpodg mathematcal model compoed of magary rectagular module coected to each other ere by mea of hypothetcal rgd lk how Fg. b. Coordate = 0,, refer to the locato of the jot of the grder alog t horzotal ax. The etre tructure compoed of? vertcal, ad horzotal member. The locato of each module detfed by the coordate of t rght had jot. Here, J ad, ad N ad tad for ecto modul ad bedg momet, repectvely, of the beam ad pot of the orgal tructure. Smlarly, J ad ; ad N ad repreet the ecto modul ad bedg momet, repectvely, of the beam ad pot of magary module of Fg. b. Thee module are forced to deform compatbly ad to rema equlbrum throughout the loadg htory of the tructure. Obvouly, whe the magary module are merged to recotruct the prototype, the hypothetcal lk dappear ad the momet of erta ad bedg momet of the beam ad colum of the reaembled tructure become: ¼ þ þ ; N ¼ N ¼ ad ¼ þ þ ; J ¼ J ðþ By the ame toke, the platc momet of retace of the mergg module or the reaembled tructure may be expreed a: ¼ þ þ ad N ¼ N ¼ ðþ t follow therefore that 0 ¼ N ¼ ; ¼ N ¼ ; J ¼ J ; 0 ¼ ; J ¼ J ad ¼. The bac module The preme of the cocept that f the dvdual module ca be elected to atfy the precrbed performace crtera, the ther aemblage hould alo be coformace 3

3 t J Adv Struct Eg (0) 6:6 age 3 of 6 J 0 a 0 J N N J W J =V L 0 J b J N N + =V L V V c V d h 0 θ N Δ N + φ Δ φ e 0 0 L Fg. a arallel chord veredeel grder. b magary p coected model. c Shear force dagram. d ackg momet dagram. e Falure mecham; veredeel grder of uform tregth cludg actve ope crcle ad flled crcle actve hge wth the tpulated deg codto. other word, f each module deged to be of mmum weght wth repect to the expected mode of repoe, the ther aembly would alo be of mmum weght wth repect to the ame performace requremet. Some of the fudametal obervato, leadg to the developmet of the propoed algorthm have bee ummarzed a follow, that; cae where all deg varable ca be calculated from the codto of US, the US deg ad mmum weght/volume deg cocde (Bradt 98). odular modelg ca reduce the tak of otherwe complex tructural aaly to mple maual computato (Bozorga ad Bertero 00). Doubly ymmetrc rectagular frame module, ubjected to uform odal hear ca be deged a mmum weght, US ytem (Grgora 993). ectagular module of US may be combed horzotally ad/or vertcally to form ew load bearg aemble of US, (Grgora ad Grgora 0b, c). dvdual module are qua-determate ub-aemble ad provde great ght to the performace of the tructure durg the deg proce. The ed ecto of the member of the module ca be treated uch a way a to prevet or force the formato of platc hge at thoe locato, e.g., ug reduced beam ecto, added flage plate, hauche, etc. Th mple that the locato ad equece of formato of the platc hge ca be cotrolled a teded. The momet ad deformato of the elemet of the VG of US at ay ordate may be related to the local rackg momet ¼ V L, Fg. d, of the correpodg module, e.g., ¼ =; where V the total exteral hear actg o jot, Fg. c. The tffee of the dvdual module ca be elected uch a way a to cotrol the maxmum dplacemet of the frame throughout the elatoplatc loadg htory of the frame. Due to double ymmetry, the pot of flecto of the vertcal elemet of parallel chord grder occur at ther md heght. tructure of US, the redtrbuto of momet through formato of platc hge force the correpodg pot of flecto to move toward the md pa of the beam. VG of US may fal through a tate of over collape, where all cottuet module fal together. Whle teel momet frame uder lateral loadg are mot utable for platc deg, they are alo uceptble to ufavorable effect of drft hftg, epecally at cpet collape. Hamburger et al. (009) ad Grgora (03b) have reported that a reducto dfferetal drft ca help mprove rackg tablty all categore of momet frame. Hece, a oberved aerto wa made that pot of flecto of all member of VG of US occur at ther md pa durg the elatc a well a platc tage of loadg. 3

4 6 age of t J Adv Struct Eg (0) 6:6 Th a acceptable preumpto wth coequetal marg of error ce the oluto become more accurate ad evetually exact a platcty pread over all member of the tructure. Thee obervato form the eece of the optmzato phloophe that lead to the mmum weght deg of regular momet frame cludg parallel chord VG. Lkewe, the dpeble aocato of the md-pa pot of flecto wth the exact mmum weght deg codto mmeely facltate the compreheo ad mathematcal formulato of the problem. addto, a obervato ha bee made that almot all type of tradtoally deged framework, the drft agle due to rackg effect geerally a maxmum ear the upport ad a mmum ear the locato of maxmum dplacemet (Taraath 998). Th ugget that t would be prudet to frt allevate the poblty of premature formato of platc hge wth the frt ad lat bay of the frame, mmze large drft agle ear the upport ad the work deway to cotrol the dplacemet of the remag bay or module. The atcpated gle collape mecham of tructure of US have more tha oe degree of kematc freedom ad ca be cotrued a beg compoed of everal mergg falure patter. Structure of US almot varably fal through a tate of over-complete collape. The lkelhood of over-complete collape, ugget that the equece of formato of the platc hge may be cotrolled uch a way a to prevet the formato of platc hge the vertcal member utl the frt et of hge form the dered locato of the horzotal elemet. Smlarly t would be poble to delay the formato of platc hge at the crtcal ed of the horzotal member utl the complete falure of all vertcal member. omet cotrol techologe, uch a added flage plate ad/or reduced beam ecto ca be employed to duce ay mode of collape ad equece of formato of platc hge, wthout otceably affectg the ultmate load carryg ad/ or dplacemet developmet capacty of the tructure. Theoretcal developmet The purpoe of th ecto to utlze the formato preeted the precedg ecto to deg a geeralzed parallel chord VG of US uder a arbtrary dtrbuto of ormal odal force. Th acheved by agg a much tregth ad tffe a demad mpoed o each module of the tructure. The tregth N of a module defed a t maxmum carryg capacty at falure. The mmum tffe of the module drectly related to t target drft rato h at cpet collape. h, ad t compatble couterpart /, (Fg. e) are the two drft agle that uquely defe the locato of the maxmum odal dplacemet D at collape a well a the correct collape mode of the framework. Sce h = h for 0 B B ad h = / = / for B B the h ad / may be related to each other through D ¼ h ¼ L ¼ / ¼þ L : Capacty agmet, tregth deg erhap the mot teretg facet of modular tructure of US that the ultmate carryg capacte of ther member ca be etmated almot by pecto (Grgora ad Grgora 0a, d, e). Th lead to the tatemet that: the maxmum carryg capacty of ay module of a veredeel grder of US ca be related drectly to the ultmate rackg momet actg o that module. other word, oce the exteral hear ad rackg momet dagram of a geeralzed, parallel chord veredeel grder uch a thoe depcted Fg. c, d are kow, the correpodg exact capacte may be computed a: N ¼ ¼ V L : ð3þ The valdty of the dtrbuto of N may be verfed by the um ¼ V ¼ ðn =L Þ¼0. Whle there o eed to ubtatate the oude of Eq. (3), t tructve to acerta the plaublty of the potulated collape mecham. The falure mode of Fg. e dcate that due to the developmet of ( - ) actve platc hge alog the chord of the tructure, the grder may ted to fal through a tate of over-complete collape wth oly two et of actve platc hge alog the chord of the tructure. The actve hge are how a black old crcle. Th allow the vrtual work equato of the elected falure cearo to be expreed a: X ¼0 or, a: X ¼0 W D ¼ hn þ X /N þ þ h W D ¼ X ¼0 þ X ½ð d Þ ½ ¼þ ¼0 þ d N h þ / X ¼þ þ d þ N X / ¼ h ¼ ðþ ð5þ where the Kroeckar delta d ha bee troduced to overrde the mathematcal complexte arg from abrupt chage platc rotato h ad /. Here, d = for =, d = 0 for =, ( - d ) = 0 for = ad ( - d ) = for =. The vrtual work equato correpodg to the ame falure mecham but term of the gle varable of the collapg module may be expreed a: 3

5 t J Adv Struct Eg (0) 6:6 age 5 of 6 X ¼ W D ¼ X ¼ h ¼ X ¼0 ð þ þ Þh ¼ X ¼ h ð6þ Sce, Eq. (5) ad (6) are detcal the the modular cocept ca be expected to yeld the ame exact reult a the tradtoal method of approach. Demotratve example Deg a geeralzed veredeel grder of US wth detcal bay of heght h ad legth L uder a uform dtrbuto of ormal odal force W. Soluto whle there o eed to coduct a vrtual work aaly to determe the platc momet of retace of each module, a udertadg of the deformed hape of the tructure eetal to developg ght to the performace of the tructure ad etmatg the correpodg dplacemet at cpet collape. The admble collape mecham of the ubject VG for eve ad odd umber of bay are depcted Fg. a, b, repectvely. Becaue of double ymmetry oly the oluto of the left half of the grder addreed the preet ecto. The exact mmum weght oluto for both the eve ad odd umber of bay ca be wrtte dow, ether by pecto or followg the gudele of the precedg ecto. The vertcal hear V utaed by each module may be expreed a a cotuou fucto of the varable a: V ¼ð þ Þ W th gve: for ¼ ; ; 3...: ð7þ 0 ¼ 0 ¼ N ¼ ¼ ¼ N ¼ N ¼ ¼ V L for ¼ ; ; 3...: ¼ð Þ WL 8 ¼ð þ Þ WL 8 ð8þ ð9þ p ¼ þ þ WL ¼ðÞ for ¼ ; ; 3... ð0þ The valdty of oluto (8), (9) ad (0) may be verfed through the correpodg work equato; W ext. = W t.. Aumg D = 9 D ad, D = L 9 h, t gve: = W ext: ¼ X WD ¼ þðþ WLh 8 ; ¼ = W t: ¼ X h ¼ h þðþ WL 3 ; ðþ ¼ Demotratve example Deg a geeralzed VG of US wth detcal bay of heght h ad legth L uder a cocetrated ormal odal force at a arbtrary jot j. Soluto the potulated collape mode of the grder uder coderato preeted Fg. 3. The problem eetally that of olvg a tructurally determate ytem. However, aumg that the potulated falure mecham of Fg. 3a correct the the correpodg rackg momet dagram may be preeted a Fg. 3b, whece; ¼ N ¼ ¼ V L ð Þ ¼ h 0 L ðþ other word ¼ N ¼ = ¼ ð ÞL= for C C ad ¼ N ¼ = ¼ L= for (? ) C C. The complete oluto for the vertcal member may ow be ummarzed a: p ¼ þ þ ð Þ ¼ h 0 h þ 0 L for ¼ ; ; 3;... ð3þ Here, the tep fucto h 0 ¼ 0 for \ ad h 0 ¼ for [. For the partcular cae =, Eq. () reduce to ¼ N ¼ L=8: Fg. egular parallel chord veredeel grder uder uform loadg. a Admble falure mecham, eve. b Admble falure mecham, odd a 0 3 W W W W W W W W W W 0 = / = ( )/ b ( +) / W Fg. 3 egular parallel chord veredeel grder uder arbtrarly placed pot loadg. a Geometry ad falure mecham. b ackg momet dagram a L L 0 b 3

6 6 age 6 of t J Adv Struct Eg (0) 6:6 h λ > N N J J L V λ < J J V λ > λ < V V N J N J J λ = N / ρ = L / Jh a b c d Fg. a, b ectagular cloed loop module at collape, c, d ortal type module at collape Dplacemet at cpet collape The cluo of maxmum dplacemet at cpet collape help develop ght to the repoe of the tructure at cpet collape. The deformed hape of the collapg VG of Fg. e may be cotrued a beg compoed of a umber of compatble deformed module that ft to the bay of the tructure. Comparg the geometrc dpoto of the module of Fg. e wth the deformed hape of Fg., t may be ee that the two adjog module at jot ted to chage hape a ed retraed module of Fg. c, d, wherea all other module ted to deform a uretraed rectagular frame uch a thoe depcted Fg. a, b. t ha bee how, through the ue of Eq. ( 3), that the computatoal model of Fg. b ca be utlzed to accurately etmate the ultmate load carryg capacte of parallel chord, VG of US, wthout reortg to complcated tructural aaly. A attempt ha bee made the preet ecto to how that a mlar approach may be adopted to tudy the correpodg maxmum ormal odal dplacemet at cpet collape. Grder dplacemet at cpet collape The geeralzed force deformato relatohp of typcal VG, uch a that how Fg. a or e, term of module tffe rato q ¼ L =hj ; ad corer momet or may be expreed a: D ¼ X L þ þ L þ 6EJ ¼ q EJ q þ ð þ ÞhL ðþ Eð þ þ Þ or, D ¼ ð þ ÞhL þ þ þl þ þ Eð þ þ Þ EJ þ q þ þ X L þ ð5þ 6EJ q ¼þ other word, f t ca be how that each term of Eq. (, 5) defe the lateral dplacemet of a depedet module, the t may be uggeted that; the maxmum ormal odal dplacemet of VG of US at cpet collape may be computed a the um of the correpodg dplacemet of the dvdual module ether to the left or rght of the jot of maxmum platc rotato. Th mple that both the left ad rght had egmet of the collapg grder may be ued a utable prmary tructure for dplacemet aaly (Hayma 96, 97; Hore 979). A formal proof of Eq. (, 5), together wth a example utlzg both egmet of the grder for dplacemet aaly alo preeted the Appedx. Computato baed o Eq. (, 5) atfy the precrbed yeld crtera, tatc equlbrum, boudary upport codto a well a the kematc of the potulated falure mecham, ad a uch atfy the requremet of the uquee theorem ad mmum weght deg due to Foulke (953, 95). odule dplacemet at cpet collape The tudy of the maxmum dplacemet of bac module a cottuet elemet of VG at cpet collape may help apprecate the eece of the propoed procedure. Two local deformablty codto affect the global dplacemet of the veredeel frame. Depedg upo ther locato, eghborg module ca deform ad rotate ether the ame or oppote drecto. odule rotato are caued due to chage platc curvature or a dctated by the boudary upport codto. The umeral ad the purely way collape mecham of Fg. refer to the equece of formato of the platc hge. The dahed le of Fg. e, c, d repreet pecfc locato where the curvature ted to revere drecto. The dahed le elemet ted to rema elatc ad reder the module effectvely fx eded. For fxed ed module, the equece of formato of the hge deped o the over-tregth factor k a well a the relatve tffe q of the module. cae of module of US, 3

7 t J Adv Struct Eg (0) 6:6 age 7 of 6.e., k =, platc hge ca form at the four corer multaeouly. Uretraed rectagular module Uretraed module ca deform freely accordace wth falure mode (a) ad (b) of Fg.. For module rotatg the ame drecto, falure may occur through the multaeou formato of platc hge ether the vertcal or horzotal member of the module. A pecto of the frt ad lat term of Eq. (, 5) dcate that the dplacemet of doubly ymmetrc rotatoally uretraed module at cpet collape may be expreed a: d ¼ þ L q 6EJ ; ¼ ¼ VL for k ad ¼ N ¼ VL for k\ ð6þ Ed retraed rectagular module artally ed retraed ad fxed bae portal module may be ued advatageouly to addre the platc compatblty codto where curvature chage g. The way mecham of Fg. c, d depct two dtct poblte depedg upo the relatve value of q ad k. The dahed le Fg. e repreet the locato of two back to back, magary fxed bae portal frame o ther de. However, ce platc rotato chage g at the de of the jot that cota the maxmum platc dplacemet, the commo vertcal pot at rema elatc ad caue the adjog module to act a rotatoally retraed horzotal portal frame. However, for retraed ed module of US, wth k = the maxmum lateral dplacemet at collape may be computed ug the lat two term of Eq. (),.e., d ¼ L EJ þ q þ ð þ ÞhL Eð þ þ Þ ð7þ The ecod term of Eq. (7) cota the form ad magtude of the curvature of the commo vertcal member eeded to mata deformato compatblty betwee the two adjog module. Two partcular cae are frequetly ecoutered. Frt, ¼ þ ¼ ; or þ ¼;.e., cae that decrbe the dplacemet of fully fxed portal frame at cpet collape, whece: d ¼ þ L q EJ ; ¼ ¼ VL ð8þ ð þ kþ Secod, þ ¼ þ ¼ 0; reduce Eq. (7) to(6). A detaled accout of the platc dplacemet aaly of portal frame uder combed gravty ad lateral force, cludg the p delta effect, wth dfferet boudary upport codto may be foud Grgora ad Grgora (0b, e). Demotratve example 3 Etmate the maxmum ormal odal dplacemet of the veredeel grder of example at cpet collape. Aume the chord ad the vertcal of the grder are cotructed out of ecto of form momet of erta J ad = ad = for all other, repectvely, ad that US acheved by approprately detaled reduced beam ecto egmet. Soluto t coveet to olve for eve ad odd cae eparately. Sce for = eve, =? ad = (? - )WL/8, Eq. () gve: D ¼ X L þ þ =L þ 6EJ q EJ q ¼ ¼ WL3 þ ð9þ 96EJ q q By the ame toke, ce for = odd? =? = 0, Eq. () yeld X L D ¼ þ ¼ WL3 þ 6EJ q 9EJ q ¼ Demotratve example ð0þ Etmate the maxmum ormal odal dplacemet of the VG of example at cpet collape. ember properte are the ame a for example 3 above. Soluto ce = = L( - )/ ad, Eq. () gve, upo ubttuto: D ¼ L3 96EJ ð Þ þ q Dplacemet-baed platc deg q q ðþ A mportat apect of DBD t ablty to cotrol ad relate the performace of the tructure to t maxmum dplacemet at cpet collape. Whle t poble to cotrol the dplacemet of VG through the ue of Eq. () ad (5), practcal or other coderato may tpulate the eed for cotat ecto or uform ter-pael drft rato addto to other lmtato. A method of cotrollg the drft rato preeted a follow. Stffe agmet, dplacemet cotrol Aumg that the chord of the grder rotate through cotat drft agle h = h ad / = /, at collape, the the maxmum odal dplacemet at may be computed a: D ¼ h ¼ L ¼ / ¼þ L, or D ¼ ¼ d ¼ ¼þ d : Ad, f D aged a target value D = D, 3

8 6 age 8 of t J Adv Struct Eg (0) 6:6 the h ¼ h ¼ D= ¼ L ad / ¼ / ¼ D= ¼þ L : Coequetly, to mmze the ter-pael drft, the dvdual momet of erta of the dvdual module may be elected a; J ¼ þ L q 6Eh ad J ¼ L Eh J ¼ þ q þ L q þ q þ for þ þ ð þ Þ ð þ g Þq for 6E/ ad J þ ¼ þl þ E/ ð þ Þ ð þ g þ Þq þ where, g =? / ad g? = /?. Demotratve example 5 ðþ ð3þ Select the properte of a three bay regular VG of US ubjected to a gle cocetrated ormal odal force uch a way that maxmum drft doe ot exceed the precrbed value c. Soluto for = 3 ad =, Eq. () gve; = L/ ad = 3 = L/. A detaled aaly of the momet of the curret example provded a part of example 7 of the Appedx. For q =, Eq. (7)or() gve, d = D left = 3L 3 /88. Coequetly, = = 3L 3 /88cE. Smlarly, from Eq. (7), d = 5L 3 /88, therefore = 5L /88cE = 5/3. Next, Eq. (8) gve d 3 = 8L 3 /88 3, whch tur yeld 3 = 8L / 88cE = 8/3. The valdty of the propoed oluto may be verfed by obervg that D rght = d? d 3 = L / 88cE. The VG of example may be categorzed a a tructure of uform repoe where uform tregth alo aocated wth uform drft (Grgora 03a, b). Cotrol of equece of formato of platc hge A the tate of over-complete collape aocated wth the multaeou formato of all platc hge, t ca be ued a a roadmap to duce a cotrolled, equetal formato of platc hge wthout compromg the lmt carryg capacty or the mmum weght tatu of the framework. jot loaded VG, the drft agle due to rackg effect geerally a maxmum ear the upport ad a mmum wth the pa. Th mple that t would be prudet to prevet the poblty of premature formato of platc hge ear the upport. The preet ecto propoe a mple method for delayg the formato of platc hge ear the upport utl the ret of the tructure ted to become a mecham. Th acheved by acceleratg the equece of formato of platc hge by the mplemetato of reduced beam ecto a calculated maer ad the compeatg for the lo of the global carryg capacty of the tructure. For tace, f t dered to accelerate the formato of the platc hge of a rotatoally actve member, uch a the horzotal elemet to the left of jot or the vertcal elemet at jot of Fg. e, the t would be uffcet to reduce the platc momet of retace of thee member by mall percetage r ad r - repectvely,.e. from N to ð r ÞN ad from - to ( - r - ) -, repectvely. partcular, f t tpulated to eforce a tate of equetal falure, tartg from, ay, jot of Fg. a ad progreg gradually toward the upport, the t would be uffcet to ag a momet reducto quotet r to the actve ed of the horzotal member meetg at ad r to each vertcal member, uch that r \r ; ð r Þ \ð r þþþ ad that r =0 = 0. The effect of reducto of local momet of retace of the actve member o the global carryg capacty of the grder may be aeed by comparg the magtude of the teral work capacte of the grder before ad after BS treatmet. The compeato for uch a reducto may the be cluded a part of the tpulated load factor. The followg example ha bee provded to llutrate the applcato of the propoed method of cotrollg the equece of formato of the platc hge VG of US. Demotratve example 6 edeg the VG of example for = 6 uch a way that the equece of formato of the actve platc hge would follow the order how Fg. a. Soluto let r 0 = 0, r = 0.05, r = 0.0 ad r 3 ¼ 0:5: Th reduce to ( - r )(6 - )WL/. From Eq. (), W t. = 9WLh. The effect of r o the teral work may be computed a; " # W t:after ¼ ð r = ÞN= þ X ð r Þ h ¼ 9WLh 0:75WLh ¼ 8:55WLh 0 ðþ Th mple that the orgal load factor may be creaed by a compeato factor of 9/8.55. Cocluo Th artcle ha troduced a umber of mple dea that lead to the effcet deg of parallel chord teel VG, amely, the troducto of the magary Bac module, dplacemet-baed platc deg ad a method of 3

9 t J Adv Struct Eg (0) 6:6 age 9 of 6 computg platc dplacemet at cpet collape. A aalytc procedure ha bee provded to facltate ad revve teret platc deg of effcet VG. The phloophcal dfferece betwee platc deg aaly ad deg led platc aaly have bee dcued ad demotrated through geerc ad umercal example. deg led platc aaly, collape mode ad tablty codto are mpoed rather tha vetgated. The theory of tructure appled rather tha followed. Several umercal oluto have bee preeted to demotrate the accuracy ad applcato of the propoed cocept ad deg formulae. The propoed formulae are etrely utable for maual a well a preadheet ad electroc computato. The propoed deg atfy all codto of the uquee theorem, ad a uch, caot be far from mmum weght oluto. However, they are lmted to regular veredeel frame where all member ca develop full platc momet at collape. t ha bee demotrated, cotrary to the commo belef, that dplacemet aocated wth platc deg, ca be computed maually ad utlzed advatageouly wthout reortg to legthy aaly or hgh power computer. The propoed methodologe offer dplacemet cotrol opto at frt yeld ad cpet collape. The determato of maxmum platc dplacemet of the cla of grder dcued the paper ha bee reduced to the ummato of maxmum dplacemet of mple module. t ha bee how that predetermed equece of formato of platc hge could be arraged through ratoal electo of the tregth ad tffee of the cottuet module of the framework. The advatage of the propoed cocept over tradtoal method of deg t ablty to provde ecoomc/optmzed oluto whle matag dplacemet cotrol at ultmate loadg. The propoed procedure are partcularly ueful for the prelmary deg of regular teel VG. Author cotrbuto The author ha bee volved tructural reearch for well over 0 year. H latet cotrbuto clude the troducto of erformace Cotrol a a alteratve deg phloophy for ductle tructure. Coflct of teret teret. The author declare that he ha o competg Ope Acce Th artcle dtrbuted uder the term of the Creatve Commo Attrbuto Lcee whch permt ay ue, dtrbuto, ad reproducto ay medum, provded the orgal author() ad the ource are credted. Appedx A bref decrpto of the formulato of the maxmum ormal odal dplacemet of VG of US at cpet collape preeted th appedx. Aumg that the grder rema kematcally table utl the formato of the lat et of platc hge at each de of jot, the the geeralzed elatc platc ormal odal dplacemet of the ame jot may be computed by performg the vrtual work ummato over all member of the framework,.e., D ¼ X h þ X ð=þ m d ðaþ where: ¼ ¼ ¼ or ¼ ad m " ¼ ð L L Þ L # 0 L L L L ðaþ are the corer momet of the th module due to the exteral odal force ad the ut load actg o jot, repectvely. Here, L; L ad L are the total pa legth, ad dtace to jot ad from the left had upport, repectvely, ee Fg. 6b, c. t tructve to ote that m ca be expreed term of two cotat multpler ad the wdth of each module L, thu: m ¼ ð L L ÞL L ¼ k left L ; for ad m ¼ L L L ¼ k rght L for þ ða3þ Sce at the oet collape, the platc hge aocated wth rotato h are jut formg, they may be et to zero to compute the dered dplacemet (Hayma 96, 97; Hore 979). Coder the geeralzed curvature caued by the rackg momet of dvdual module of Fg. d ad the dtrbuto of the vrtual momet decrbed by Eq. (A), the the correpodg vrtual work equato, aumg jm j[ jm þ j, may be formulated a: D ¼ X m L þ þ m L þ 6EJ q 6EJ q ¼ þ h ð þ Þðm m þ Þ 6E ð þ þ Þ þ þm þ L þ þ þ X 6EJ þ q þ ¼þ m L 6EJ þ q ðaþ The complexte aocated wth Eq. (A) may be allevated by reducg t to the ymbolc form; D ¼ðA þ B Þk left þ C ðk left k rght ÞþðD þ þ E Þk rght ða5þ where, the frt ad lat ummato term, A ad E decrbe the accumulatve cotrbuto of the dvdual module to the left ad rght de, repectvely, of the module that hare the commo vertcal at jot. The 3

10 6 age 0 of t J Adv Struct Eg (0) 6: ( + + ) ( + ) E( + + ) + + ( + ) E( + + ) a b c + + ( ) Fg. 5 a, b Curvature dagram of module harg the jot of maxmum dplacemet before mergg, c curvature dagram of the ame module after mergg ( + ) E( + + ) m m m + (b) (c) m + (a) + + m + m m + m + + a b c Fg. 6 Curvature dagram for adjog module that hare the ame maxmum dplacemet. a Curvature dagram reflectg compatble deformato for the vertcal member at jot. b, c Ut load momet for depedet module ad?, repectvely C + (b) B D C + E (c) (d) (e) a b c Fg. 7 Example 7, three bay parallel chord veredeel grder uder a gle ormal odal force. a Collape mode ad properte. b latc bedg momet. c ctoral terpretato of Eq. (A8) ecod ad fourth term, B ad D?, decrbe the cotrbuto of the, th ad? th module of Fg. d, repectvely, le the cotrbuto of ther commo vertcal elemet at. The remag term C cota the cotrbuto of the vertcal member at, where the curvature chage g. The latter term alo decrbe the commo curvature hared by module ad?. A pctoral terpretato of the compatblty codto betwee the adjog module at preeted Fg. 5, 6. Equato (A5) may be further mplfed a: D ¼ðA þ B þ C Þk left þðc þ D þ þ E ¼ D left þ D rght Þk rght ða6þ Sce k left? k rght = / the D left = k left D ad D rght = k rght D. Therefore; D = (A? B? C )/ ad D ¼ ðc þ D þ þ E Þ=: Coequetly; D left ¼ D ¼ X ¼ L 6EJ þ ð þ ÞhL Eð þ þ Þ þ q D rght ¼ D ¼ ð þ ÞhL þ Eð þ þ Þ þ X L þ 6EJ q ¼þ þ L þ EJ q ða7þ þ þl þ þ EJ þ q þ ða8þ Sce each term of Eq. (A7) ad (A8) repreet ether the dplacemet of dvdual or ummato of dplacemet of dvdual module the t may be argued that; the maxmum ormal odal dplacemet of VG of US at cpet collape may be computed a the um of the 3

11 t J Adv Struct Eg (0) 6:6 age of 6 correpodg dplacemet of the dvdual module ether to the left or rght of the jot of maxmum platc rotato. Example 7: ue Eq. (A7) ad (A8) to etmate the maxmum ormal odal dplacemet of the veredeel Grder of US of Fg. 6a at cpet collape. L = L = L 3 = h = L. The momet of erta ad platc retace of the member are alo how Fg. 7a. Soluto puttg q = for =, ad 3 ad ubttutg for the pertet value of, h, L, ad J from Fg. 6 to Eq. (A7) ad (A8), t gve: D left ¼ 0 þ 3 ð ÞL ¼ 3 L ad D rght ¼ ð ÞL ¼ 3 L repectvely. eferece þ ð ÞL þ 6 L þ 8 L ða9þ ða0þ Baker ALL, Hore, Hayma J (956) The Steel Skeleto, vol ad. Cambrdge Uverty re, UK Beedle LS (958) latc deg of teel frame. Wley, USA Bozorga Y, Bertero VV (00) Earthquake egeerg. CC, Florda Bradt A (98) Crtera ad method of tructural optmzato. Cluver Academc, USA, p 9 Foulke J (953) mum weght deg ad the theory of platc collape. Q Appl ath 0: Foulke J (95) The mmum-weght deg of tructural frame. roc Soc Lod Ser A 3:8 9 Goel SC, Lao WC, Bayat, Chao SH (00) erformace-baed platc deg (BD) method for earthquake retat tructure: a overvew. Struct De Tall Spec Buld 9( ):5 37 Grgora (970) Vbrato of veredeel grder. J Soud Vb (): 7 Grgora (993) O the lateral repoe of regular hgh-re frame. Struct De Tall buld 3:33 5 Grgora (03a) A troducto to performace cotrol for momet frame of uform repoe uder lateral loadg. Aa J Cv Eg ():3 3 Grgora (03b) A ew approach to platc deg ad optmzato of parallel chord veredeel grder. t J Optm Cv Eg 3(3): Grgora (0) erformace cotrol for effcet deg of double layer grd uder uform loadg. t J Adv Struct Eg. 6:5 (uder pre) Grgora, Grgora C (988) relmary mmum weght deg of momet frame uder lateral loadg. ASC Eg J 5:9 36 Grgora, Grgora C (0) erformace cotrol for emc deg of momet frame. J Cotr Steel e 67:06 Grgora, Grgora C (0a) erformace cotrol: a ew elatc platc deg procedure for earthquake retg momet frame. J Struct Dv ASCE 38(6):8 8 Grgora, Grgora C (0b) Lateral dplacemet of momet frame at cpet collape. J Eg Struct :7 85 Grgora, Grgora C (0c) A ew performace baed deg method for earthquake retg momet frame. Ca J Cv Eg 39: Grgora, Grgora C (0d) A troducto to the methodology of earthquake retat tructure of uform repoe. Buldg :07 5 Grgora, Grgora C (0e) ecet developmet platc deg aaly of teel momet frame. J Cotr Steel e 76:83 9 Grgora, Grgora C (03) Drft cotrol for multtory momet frame uder lateral loadg. t J Hgh-e Buld (): Hamburger O, Krawkler H, alley JO, Ada S (009) Semc deg of teel pecal momet frame: NEH emc deg techcal bref o., Natoal ttute of Stadard ad Techology, Gatherburg, D, NST GC Hayma J (96) O the etmato of deflecto elatc platc frame. roc t Cv Eg 9:89 Hayma J (97) latc deg of frame. Cambrdge Uverty re, UK Hedry AW (955) latc aaly ad deg of mld teel veredeel grder. Struct Eg 33:7 Hore (979) latc theory of tructure. ergamo, Lodo azzola, luo V (997) latc deg of emc retat teel frame. Earthq Eg Struct Dy 6():67 9 orovch A (958) Applcablty of platc deg to the veredeel true. Tulae Uverty of Louaa, USA Neal BG (963) The platc method of tructural aaly. Chapma ad Hall, Lodo Nethercot DA (00) Lmt tate deg of tructural teelwork. Spo, UK Taraath BS (998) Steel, cocrete, ad compote deg of tall buldg, d ed. cgraw Hll, USA, pp 7 77 Wog B (009) latc aaly ad deg of teel tructure. Elever, UK ark Grgora The author a teratoal coultat, vetgator ad lecturer. He ha more tha 5 year of experece tructural deg, educato, code ad tadard developmet, forec tude, reearch ad project maagemet. He hold B.Sc. ad.sc. degree Structural Egeerg from the Uverty of acheter (UST) ad a D.hl. degree egeerg from the Uverty of Oxford. He a Calfora lceed Cvl ad Structural Egeer ad a actve member of the Structural Egeer aocato of Souther Calfora (SEAOC). He a foudg member ad the frt charma of the Faculty of Structural Egeerg (Sazeh) of Sharf Uverty ra. Curretly he the Chef Structural Egeer of GA Structural Egeerg Coultat c. baed Gledale Ca. addto to h techcal actvte, he mata a actve ocal role that focue o artal Art ad Huma Behavor. H latet book, Search of the d, ad Karate-Do a Way of Lfe, deal maly wth behavoral apect of huma lfe ad ocal codto. He ca be reached at (88)--675 ad (88)6-33, E-mal, markarja@aol.com. Correpodece addre 589 Treto Ave. Gledale, Ca. 906, US. 3