PROBABILITY-BASED OPTIMAL DESIGN OF MOMENT-RESISTING FRAMES USING KRIGING APPROXIMATION MODEL

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1 9 th Worl Cogre o Structural a Multcplary Optmzato Jue 13-17, 2011, Shzuoa, Japa PROBABILITY-BASED OPTIMAL DESIGN OF MOMENT-RESISTING FRAMES USING KRIGING APPROXIMATION MODEL Jgyao Zhag 1, Maoto Oha 2 1 Rtumea Uverty, Kuatu, Japa, zhag@fc.rtume.ac.jp 2 Hrohma Uverty, Hgah-Hrohma, Japa, oha@hrohma-u.ac.jp 1. Abtract I th tuy, we preet a optmzato approach for probablty-bae eg of momet-retg teel frame, where ucertate materal properte are tae to coerato. The tructural relablty efe by probablty of atfyg pecfc tructural performace evaluate by Mote Carlo mulato (MCS). Becaue MCS ee a large umber of tructural repoe correpog to poble materal properte, we aopt Krgg metho to prect thee repoe ug oly a lmte umber of tructural aalye. Smulate aealg, whch a heurtc metho for combatoral optmzato problem, aopte to earch for the optmal combato of ecto from a pecfe lt of avalable ecto, o a to have the mmum weght for the tructure uer cotrat o the tructural relablty. 2. Keywor: Relablty-bae eg; Steel frame; Ucertaty; Krgg metho; Smulate aealg. 3. Itroucto A tructure that cotructe the fel of bulg egeerg eetally fferet from that ege, becaue of the extece of ucertaty geometrcal properte a materal properte etc. To tae accout of thee ucertate, a emprcal coeffcet calle afety factor trouce covetoal (etermtc) eg proceure: omal value of the tructural parameter are reuce to epeable value by vg by afety factor. The bac ea beh th proce tee to eg a coervatve tructure through precto of the wort value of repoe, uch a maxmum placemet a tertory rft agle, by the combato of the wort cae of put the epeable value of the tructural tffe/tregth parameter le tha thoe of the real tructure, a the exteral loa greater tha that mght pobly occur. However, coervatve eg by ug afety factor ca oly be guaratee tatc cae, a may mlea eger yamc cae. For example, the emc repoe of a tructure gfcatly epe o charactertc of repoe pectra of the put moto rather tha tffe of the tructure; moreover, le tregth ome tructural member may lea to le repoe of the tructure owg to more platc eergy pato. Thu, covetoal eg proceure may ot lea to coervatve eg a expecte, a coul e up wth overetmatg capacty of the tructure, whch cate that more ophtcate approache to coerg ucertaty are eceary. To evaluate yamc charactertc of a bulg tructure ubjecte to evere earthquae, tme htory aaly (THA) regare a the mot relable approach, though at the ame tme, much hgher computato cot eceary comparo to evaluato ug tatc puhover aaly. Moreover, the framewor of performace-bae egeerg, yamc performace of a tructure ubjecte to poble ucertaty volve tructural aaly houl be evaluate the framewor of probablty theory. For uch purpoe, Mote Carlo mulato (MCS) the mot traghtforwar way; however, t ulely to be rectly applcable to complex ytem. Th becaue that expeve THA for a large umber of poble value of the tructural parameter ha to be coucte, whch reult uacceptably expeve computatoal cot. Itea of a large umber of THA by the rect approach, MCS ca alo be coucte by approxmato approache, whch are to etmate the tructural repoe by ug a lmte umber of tructural aalye. Th woul of coure lea to le accuracy; hece, o that a trae-off betwee accuracy a computatoal cot ha to be mae. Metamoel are uch of approxmate approache, terpolatg the reult (yamc repoe) obtae prelmary expermet (tructural aalye) wth mooth olear fucto. The repoe for the parameter value, for whch expermet have ot bee carre out, are precte from the olear fucto. There have bee a umber of metamoel evelope o far, uch a repoe urface approxmato, raal ba fucto, artfcal eural etwor, Krgg metho a multvarate aaptve regreo ple. Amog thee, Krgg metho ha gae much atteto egeerg lterature becaue of t hgh accuracy a low computatoal cot [1]. 1

2 A the tructural relablty avalable term of probablty, we coer the problem of fg the momet-retg teel frame wth mmum weght, whch aemble by the avalable ecto. For th typcal combatoral optmzato problem, the mulate aealg (SA) aopte th tuy. Followg th troucto, Secto 4 gve a bref troucto to Krgg moel for repoe precto, a a bref ummary of SA; Secto 5 coer the probablty-bae optmal eg of a two-meoal momet-retg frame to emotrate the avalablty of the propoe approach; a Secto 6 coclue the tuy. 4. Approxmate repoe a optmal tructure Th ecto gve a bref ecrpto of Krgg metho for prectg tructural repoe from a lmte umber of umercal expermet, a mulate aealg for fg the optmal combato of avalable ecto to have the leat-weght tructure whle atfyg cotrat o tructural relablty Krgg metho The olear urrogate fucto Krgg metho cotructe by mmzg the mea error of the weghte um of repoe at the amplg pot, at whch expermet are coucte. Krgg metho wa tally evelope for tattcal evaluato of mg ata, a gae further a much wer applcato other egeerg fel from the e of I th ubecto, we brefly ummarze the bac equato of Krgg metho a ecrbe [2] for the completee of the tuy. Suppoe that we coer the ucertaty tructural parameter, a carry out prelmary aalye at the amplg pot ( 1,..., ), at whch the true repoe are eote by y. The precto pot, at whch repoe are to be precte, are eote by x. Let R eote the correlato matrx, ecrbg correlato of the repoe at the amplg pot, a rx ( ) the correlato vector for amplg pot a precto pot: the th etry of rx ( ) the correlato betwee the precto pot x a the amplg pot, a the (, j)-etry of R the correlato betwee the two amplg pot a j. The ormalze value ŷ or of the precte (approxmate) repoe ŷ at the precto pot x eterme a follow by mmzg the mea quare error a ug the bet lear ubae prector: T 1 ˆ T 1 yˆ ˆ or ( x) r( x) R ( y ) wth ˆ R y T 1, (1) R where every etry equal to oe, a y the ormalze vero of y. Thu, the precte repoe ŷ compute by yˆ y yˆor y, (2) where y a y are the taar evato a mea of the repoe at the amplg pot, repectvely. The correlato uually efe a a fucto of correlato parameter a tace betwee the relevat pot: the (, j)-etry R( θ,, ) of R a the th etry r( θ, x, ) of r wrtte a where, j a amplg pot a tace j j j 1 R( θ,, ) R(, ), 1 r( θ, x, ) r(, ), (3) are repectvely the th etre of correlato parameter vector θ, tace j betwee betwee amplg pot a precto pot. The correlato parameter θ are uow a are eterme by mmzg precto error at pecfc (verfcato) pot th tuy. More amplg pot are eee, f the repoe caot be precte from the extg et of amplg pot wth hgh eough accuracy. There are everal approache for ag ew amplg pot a ummarze [2], a we aopt the approach that a the pot havg the maxmum mea quare error of precto to the et of amplg pot Optmzato metho Ug the approxmate repoe precte by Krgg metho, the tructural relablty term of probablty ca be ealy compute by carryg out MCS wth aumpto o probablty ete of ucerta parameter the tructural aaly. The tructural relablty term of yamc performace ca be efe varou maer. I a umercal example Secto 5, the performace of a momet-retg frame efe by the probablty of maxmum tertory rft agle to excee the pecfe value. 2

3 Satfyg certa yamc performace a cotrat, our ext tep to f the optmal tructure wth the mmum weght. To coer the eg problem practce, the member are electe from a gve lt of avalable ecto. The member are clafe to m group, where member each group have the ame ecto. The eg varable vector eote by J ( J1,, J m ), whch ha teger value. For example, f J, the ecto of the th group ha the th ecto of the lt. Let F( J ) eote the total volume of teel materal, equvaletly tructural weght, of the momet-retg frame. The maxmum value of the maxmum tertory rft agle amog all tore eote by G( J ) wth the pecfe upper bou G. The lower bou P gve for the probablty Pr[ G( J ) G] of G( J ) to excee G. The the optmzato problem formulate a Mmze F( J) ubject to Pr[ G( J ) G] P J {1,, }, ( 1,, m) (1) where the umber of amble value for J. Th a typcal combatoral optmzato problem, a we aopt mulate aealg a ummarze the ext ubecto Smulate aealg metho A t ame mple, mulate aealg (SA) explot a aalogy betwee the metal aealg proce a the proce of earchg for the bet oluto a optmzato problem [3]. Graet of the objectve or cotrat fucto are ot eceary, a the major avatage of SA over other heurtc approache the ablty to f the global optmum. There are total fve procee volve SA: (a) tal oluto, (b) local earch, (c) trato of oluto, () coolg cheule, a (e) termato coto. The typcal flowchart for thee procee how Fgure 1. Fg. 1: The flowchart of clacal mulate aealg metho. Amog thee procee, oluto trato the ey for jumpg out from a local optmum, ce t eure that acceptace of o-mprovg oluto alo poble. To be pecfc, oluto trato wll occur f a raomly geerate umber P (0,1) le tha the probablty P of trato calculate by / P m{1, e f t }, (4) where f the creae of the objectve value for a mmzato problem a t the temperature at the curret terato. It obvou from Eq. (4) that, trato to a mprovg oluto alway accepte, a trato to a o-mprovg oluto poble but become more a more ffcult alog wth the cotuouly ecreag temperature accorg to the coolg cheule. 5. Numercal example A three-pa e-tory plae teel frame a how Fgure 2 coere a the example tructure Moel ecrpto The heght a wth of the frame are 34.7 m a 12.8 m, repectvely. The frame ha momet-retg jot, a rgly upporte at the colum-bae. The weght of the roof a the other floor, clug tructural a otructural compoet, are 4.6 N/m 2 a 3.3 N/m 2, repectvely, where the epth of 6.4 m aume for th plae frame moel. The covetoal aumpto of rg floor ue. The mae are cocetrate at the beam-to-colum coecto (oe). 3

4 exteror colum teror colum Fg. 2: A three-pa e-tory plae teel frame. The member are eparate to twelve group;.e., m = 12: the exteror a teror colum (or beam) every three tore. The ecto of the colum are electe from 113 avalable ecto, a thoe of beam from 191 avalable ecto. The artfcal emc moto a how Fgure 3 geerate by the taar uperpoto metho of uoal wave, correpog to the lfe-afe performace level urg the very rare earthquae pecfe Notfcato 1461 a 1457 of the Mtry of La Ifratructure a Traport, Japa. The phae fferece pectrum of El-Cetro 1940(EW) ha bee ue. To coer extreme loa, the emc moto cale by three a apple at the bae horzotal recto. The eformato ue to elf weght ot coere for mplcty. The member are clafe to group to preerve the ymmetry of tructure. THA carre out ug the ope ource olver OpeSee [4]. The effect of geometrcal olearty tae to coerato. Raylegh ampg aopte for THA, wth the ame ampg rato h=0.02 for both of the 1t a 2 moe. The tme tep for tegrato by the Newmar- metho ( 0.25 ) 0.01 eco. 5.2 Ucertaty a optmal tructure The teel materal are ealze by a blear cottutve moel efe by Youg moulu, yel tre a hareg coeffcet, of whch ucertate are coere. Dampg rato of the tructure alo a mportat tructural parameter for yamc aaly, but wll ot be coere th example becaue t ha mootoc relato to the maxmum repoe of the tructure: the larger the ampg rato, the le the repoe Accelerato (m/ 2 ) tme () Fg. 3: The artfcal emc moto. 4

5 8 The omal value of the yel tre, Youg moulu a hareg coeffcet are N/m 2, N/m 2 a 1/100, repectvely. The upper a lower bou for ucertate are repectvely et a 1.2 a 0.8 tme of ther omal value. Furthermore, to carry out MCS, thee tructural parameter are uppoe to have uform probablty trbuto ete. To apply Krgg metho for repoe precto, we tart from e tal amplg pot, whch are the combato of the upper a lower bou of ay two of the three parameter, ato to the amplg pot wth omal value. New amplg pot, whch lea to the maxmum reucto of MSE are coecutvely ae a amplg pot orer to refe the urrogate fucto. Relablty of the tructure evaluate by the probablty of exceeace of a pecfc tertory rft agle, ay 5% th example. The probablty houl be le tha a pecfc target value to guaratee a afe tructure, whch et a 10% a a cotrat coto. Moreover, we ue the Gaua patal correlato fucto, whch preferable for a fferetable repoe fucto [5]; the lower a upper bou of the correlato parameter are age a 0.1 a 10.0, repectvely. The correlato parameter are fou by mmzg the precto error at pecfe verfcato pot, ug the optmzato tool fmco( ) prove MATLAB [6]. The tal temperature for SA age a 3.0, a coeffcet for the lear coolg proceure 0.9. The proce of fg ew oluto wll be termate whe the temperature le tha Covergece performace of earch proceure llutrate Fgure 4. It how that SA ha ablty of jumpg out from local optmum, a graually coverge at (early) global optmum. The trbuto of cro-ectoal area of the tal a optmal tructure are llutrate Fgure 5. To llutrate the trbuto clearer the fgure, wth of each member plotte proporto to t cro-ectoal area Volume Iterato Fg. 4: Covergece performace. (a) Ital tructure (Volume: 4.25m 3 ) (b) Optmal tructure (Volume: 2.98m 3 ) Fg. 5: The optmal cro-ectoal area. 5

6 6. Cocluo I th tuy, we have tue the relablty-bae eg methoology for momet-retg frame, ubjecte to poble ucertaty volve the parameter of tructural aaly. The approach ca be apple to other tructural ytem, for example two-meoal arch moel [7] a three-meoal gle-layer lattce hell [8] our prevou tue. The tuy ha how that ucertate materal properte ca be ealy coere by Krgg metho to evaluate relablty of a tructure agat pecfc exteral loa. However, exteral loa, epecally grou moto, are hghly ucerta, a hece, ther fluece o yamc repoe houl alo be carefully vetgate, whch the future topc of the tuy. Moreover, we have coere oly uform probablty trbuto for ucertate materal properte the umercal example; but ay other type of probablty trbuto ca be corporate, wth mor mofcato applyg MSC. Furthermore, other tha the gle-objectve optmzato the example, more tructural performace meaure a more objectve fucto coul be coere. 6. Referece [1] Saata S, Aha F a Zao M, Structural optmzato ug rgg approxmato, Comput. Metho Appl. Mech. Egrg., 192, pp , [2] Lee TH a Jug JJ, Krgg metamoel bae optmzato, Optmzato of Structural a Mechacal Sytem, ete by Arora JS, Worl Scetfc, [3] Krpatrc S, Gelatt CD Jr. a Vecch MP, Optmzato by mulate aealg, Scece, 220:4598, , [4] Neuehofer A a Flppou FC, Geometrcally olear flexblty-bae frame fte elemet, Joural of Structural Egeerg, 1998, 124:6, [5] Mtchell TJ a Morr MD, The patal correlato fucto approach to repoe urface etmato, Wter Smulato Coferece archve Proceeg of the 24th coferece o Wter mulato table of cotet, Arlgto, Vrga, Ute State. pp , [6] Bore GJ, Numercal Metho wth MATLAB. Iteratoal Thomo Publhg Ic [7] Zhag JY a Oha M, Relablty-bae optmzato of patal tructure ug approxmato moel. Proc. Iteratoal Aocato for Shell a Spatal Structure, Valeca, Spa, Sep [8] Zhag JY a Oha M, Relablty-bae eg of gle-layer lattce hell ug Krgg approxmato. Proc. Iteratoal Aocato for Shell a Spatal Structure, Shagha, P.R. Cha, Nov