EFFECT OF MODAL TRUNCATION IN MULTIPLE SUPPORT RESPONSE SPECTRUM ANALYSIS OF BRIDGES.

Transcription

1 he 4 th World Coferece o Earthquae Egeerg October -7, 008, Bejg, Cha EFFEC OF MODAL RUNCAION IN MULIPLE SUPPOR RESPONSE SPECRUM ANALYSIS OF BRIDGES K. Koal ad A. Der Kuregha Doctoral Studet, Dept. of Cvl Egeerg, Uverty of Calfora, Bereley Profeor, Dept. of Cvl Egeerg, Uverty of Calfora, Bereley Emal: atera_@bereley.edu, ad@ce.bereley.edu ABSRAC: he queto of how may mode to clude the dyamc aalye of multply-upported tructure ubjected to patally varyg groud moto codered th paper. he ue vetgated wth the framewor of the Multple Support Repoe Spectrum (MSRS) method developed by Der Kuregha ad Neuehofer (99). he orgal MSRS rule exteded to approxmately accout for the cotrbuto of the trucated hghfrequecy mode. A brdge tructure deged by the Calfora Departmet of raportato (Caltra) ued a a example applcato. he modal cotrbuto to varou repoe quatte are examed cojucto wth meaure of the partcpatg modal ma ad the mprovemet acheved wth the exteded MSRS rule. KEYWORDS: brdge; coherece; modal trucato; MSRS rule; patal varablty; wave paage.. INRODUCION Proper emc deg of exteded tructure, uch a brdge, requre accoutg for the patal varablty of emc groud moto due to the lo of coherecy of emc wave, the wave paage effect ad the dfferece the local ol codto. Dfferetal upport moto ca ehace or decreae the emc demad depedg o the charactertc of the tructure ad the groud moto feld. he MSRS rule, developed by Der Kuregha ad Neuehofer [], evaluate the tructural repoe term of the repoe pectra of the upport moto ad a coherecy fucto that characterze the patal varablty of the groud moto radom feld. h method baed o modal aaly ad the fudametal prcple of radom vbrato theory, ad accout for the cro-correlato betwee the mode of the tructure a well a betwee the upport moto. A mportat practcal problem that ha ot bee properly addreed the developmet of a relable gudele to defe the umber of mode that hould be cluded the MSRS aaly. he crtero ued covetoal emc aaly baed o the percetage of tructural ma repreeted by the mode. However, th meaure doe ot accout for dfferetal upport moto, or doe t accout for the effect of cloely paced mode. Kaha [4] propoed a meaure of the partcpatg modal ma for the lmtg cae of totally coheret groud moto. he aforemetoed meaure are examed th paper coderg the model of a extg brdge Calfora. Furthermore, a exteded vero of the MSRS rule developed to accout for the qua-tatc cotrbuto of the trucated mode. Numercal vetgato how that the exteded rule mprove the repoe etmate whe the cotrbuto of hgher mode gfcat.. SRUCURAL RESPONSE O DIFFERENIAL SUPPOR MOIONS Coder a N -degree-of-freedom lumped-ma lear tructural model ubjected to m upport moto. Let x be the N -vector of total dplacemet at the ucotraed degree of freedom ad u be the m -vector of precrbed upport dplacemet. he total dplacemet vector decompoed to peudo-tatc ad dyamc d compoet, x= x + x. he peudo-tatc compoet, x, the repoe of the ytem whe dyamc effect

2 he 4 th World Coferece o Earthquae Egeerg October -7, 008, Bejg, Cha are eglected ad related to the upport dplacemet through a fluece matrx, R,.e. x = Ru. he th colum of the fluece matrx, deoted r, repreet the dplacemet at the ucotraed degree of freedom whe the th upport degree of freedom tatcally dplaced by a ut amout whle other upport degree of freedom rema fxed. Let φ, ω ad ζ, =,..., N deote, repectvely the mode hape, atural frequece, ad modal dampg rato of the fxed-bae tructure. Aumg clacal dampg ad eglectg the dampg force aocated wth the cotraed degree of freedom, the total dplacemet vector ca be decompoed a m m N x () t = r u () t + φ β () t = = = (.) where β = φ Mr / φ Mφ the modal partcpato factor whch M the N N ma matrx aocated wth the ucotraed degree of freedom, ad () t the th ormalzed modal repoe to bae moto u () t obtaed a the oluto to u t + ζω + ω = ( ) (.) A geerc repoe quatty of teret, zt ( ), uch a a odal dplacemet or a teral force compoet, ca be wrtte a a lear combato of the dplacemet at the ucotraed degree of freedom ad the upport dplacemet,.e. zt () = pu() t + qx() t. Equvaletly, oe ca wrte m m N (.3) zt () = au + b () t = = = whch a = p +qr ad b = q φβ. he coeffcet a ad b deped oly o the properte of the tructure (ot the groud moto). 3. HE MSRS RULE Ug the decompoto of zt ( ) Eq. (.3) ad aumg jotly tatoary, zero-mea upport moto, Der Kuregha ad Neuehofer [] developed the MSRS (Multple Support Repoe Spectrum) combato rule: m m m m N E[max z(t) ] = aalρ u max max max ( ) uu l, ul, + abljρu u lj, D l ωj, ζ j = l= = l= j= / m m N N + bb ljρ ( ) ) D lj ω, ζ D( l ωj, ζ j = l= = j= (.4) Accordg to the above rule, the mea pea repoe gve term of the tructural properte a reflected the coeffcet a ad b, the mea pea groud dplacemet, u, max, ad the ordate of the mea dplacemet repoe pectrum, D ( ω, ζ ), for each upport moto, ad three type of cro-correlato coeffcet: he cro-correlato coeffcet ρ uu decrbe the correlato betwee the upport dplacemet; they oly l deped o the auto- ad cro-power pectral dete (PSD) of the upport moto. he cro-correlato coeffcet ρ decrbe correlato betwee the repoe of two mode ( ad j ) to two upport moto lj ( ad l ), whle the cro-correlato coeffcet ρ u decrbe correlato betwee the modal repoe ad lj the upport moto. he latter coeffcet are fucto of the auto- ad cro-psd of the upport moto a well a the modal frequece ad dampg rato. he cro-psd of the upport moto are gve term of the auto-psd ad a coherecy fucto that characterze the patal varablty of the groud moto feld.

3 he 4 th World Coferece o Earthquae Egeerg October -7, 008, Bejg, Cha Whe the upport moto are decrbed through repoe pectra, cotet auto-psd are determed from relato gve []. hu, the et of pea groud dplacemet ad repoe pectra for the upport degree of freedom together wth a coherecy fucto provde complete pecfcato of the groud moto radom feld for MSRS aaly. 4. EXENDED MSRS RULE ACCOUNING FOR RUNCAED HIGH FREQUENCY MODES For large ω, the lat term the left-had de of Eq.. domate. Hece, for uch cae, a approxmato of the th ormalzed modal repoe to bae moto u () t () t ω u. Ug th relato to approxmate the th modal repoe for < N, we ca wrte m m m (.5) zt () au + b () t du() t = = = = Oe ca how that the coeffcet d oly deped o the dyamc properte of the frt mode ad ca be obtaed a d b K Mr (.6) ω = q = where K the tffe matrx aocated wth the ucotraed degree of freedom. Employg the approxmato decrbed Eq. (.5), the exteded MSRS rule that approxmately accout for the cotrbuto of the trucated mode m m m m E[max zt ( ) ] = aa lρu,max,max,max (, ) uu l ul + ab ljρu u lj Dl ωj ζ j = l= = l= j= + m m = l= = j= bbρ D( ω, ζ ) D( ω, ζ ) lj lj l j j m m m m + ddρ u u adρ u,max u,max l uul,max l,max l uu l l = l= = l= m m bdρ D( ω, ζ ) u = = l= l u l l,max / (.7) I the above expreo, u,max the mea pea groud accelerato at the th upport degree of freedom ad ρ, uu ρ l uu ad ρ l u are three ew type of cro-correlato coeffcet reflectg, repectvely, the correlato betwee the groud accelerato at tato ad l, the correlato betwee the groud dplacemet at tato ad the groud accelerato at tato l, ad the correlato betwee the th modal repoe at tato ad the groud accelerato at tato l. Smlar to the orgal MSRS rule, the cro-correlato coeffcet the exteded rule are determed term of the modal properte of the tructure ad the groud exctato, the latter decrbed term of the mea repoe pectra, the mea pea dplacemet ad accelerato at each upport degree of freedom, ad the coherecy fucto characterzg the patal varablty. 5. MEASURES OF PARICIPAING MODAL MASS he meaure commoly ued egeerg practce to defe the umber of mode requred the dyamc aaly of ordary tructure

4 he 4 th World Coferece o Earthquae Egeerg October -7, 008, Bejg, Cha r U M γ = = (.8) M where M = φ Mφ, γ = ( φ M )/ M, ad M the total tructural ma aocated wth the ucotraed degree of freedom. h rato a meaure of the accuracy the etmate of bae hear force for the cae of uform upport moto wth oly mode cluded the aaly. Kaha [4] developed a meaure of the accuracy the bae hear force for the cae of totally coheret groud moto: m M I = = m rmr = r = β (.9) he dervato of th meaure baed o the orgal MSRS formulato, where the cro-modal correlato a well a the cro-correlato betwee the modal repoe ad upport dplacemet have bee eglected. 6. APPLICAION O EXAMPLE BRIDGE 6. Decrpto of the tructure A a example applcato, we coder the model of a extg brdge deged by Caltra (Calfora Departmet of raportato). he elevato, pla ad grder cro-ecto are how Fgure. Each bet cot of a gle colum wth crcular cro ecto of dameter D =.3 m. he tructure made of cocrete wth a omal compreve tregth of 5 MPa for the colum ad 8 MPa for the grder. Momet-curvature aaly dcated that the effectve flexural tffe of the colum 5% of the ucraced flexural tffe. he toroal momet of erta of the colum reduced to 0% of t ucraced value. No tffe reducto requred for the pretreed cocrete box grder. he colum are codered rgdly coected to the dec at the top ad fxed at the bottom. he repoe of the eat abutmet at the two ed of the brdge modeled through two tralatoal prg, oe logtudal ad oe travere. he tffe of the logtudal prg calculated by adjutg the tal embamet fll tffe proportoal to the bacwall heght. he tffe of the omal travere prg equal to 50% of the travere tffe of the adjacet bet. Vertcal tralato at the ed upport are fully cotraed, but free rotato are allowed all drecto. Elevato Pla Vew Cro Secto of the Grder Fgure Brdge model (Dmeo are gve meter) he fte elemet model of the brdge cot of 3 elemet per bet ad 6, 8, 8 ad 4 elemet the t, d, 3 rd ad 4 th pa from the left, repectvely. he logtudal ax of a grder elemet pae through the cetrod of the grder. hree vertcal rgd frame elemet are ued for the coecto of the upper colum elemet wth the grder elemet. Codeg out the rotatoal degree of freedom, the tructure modeled wth 03 tra-

5 he 4 th World Coferece o Earthquae Egeerg October -7, 008, Bejg, Cha latoal ucotraed degree of freedom ad 5 tralatoal upport degree of freedom. he fudametal perod of the tructure =.39 ec. All mode are aumed to have 5% modal dampg. 6. Repreetato of the groud moto he upport are ubjected to tralatoal groud moto the logtudal, X, travere, Y ad vertcal, Z, drecto (ee Fgure ). he groud moto thee drecto are codered to be tattcally depedet. he horzotal groud moto at each upport are decrbed by the Accelerato Repoe Spectrum (ARS) curve recommeded by Caltra for the correpodg ol type, aumg a pea roc accelerato of 0.3 g ad a momet magtude M w = 6.5 for the maxmum credble earthquae. he vertcal groud moto decrbed by the model propoed by Bozorga ad Campbell [], whch defe the vertcal pectral accelerato term of the horzotal pectral accelerato, the ource-to-te dtace ad the local te codto. I th example we coder the brdge to be located 0 m away from the ource. For the pectral value at low frequece, we aume a pectral hape varyg quadratcally wth frequecy. he PSD of groud accelerato that are cotet wth the pecfed ARS are obtaed accordg to the procedure decrbed [] for each ol type ad are how Fgure Horzotal Groud Moto 0.03 Vertcal Groud Moto ol type B ol type B 0.05 ol type C ol type D 0.05 ol type C ol type D Accelerato PSD, m /ec Accelerato PSD, m /ec ω/π, Hz ω/π, Hz Fgure PSD of Groud Accelerato for ol type B, C ad D he patal varablty of the groud moto decrbed by the coherecy model developed [3] wave paage te repoe γ ( ω ) = γ ( ω) exp{[ θ ( ω ) +θ ( ω ) ]} (.0) l l l l coherece wave paage whch γl ( ω ) = exp[ ( αd lω /v ) ] decrbe the coherece compoet, θ l (ω) = L te repoe ωd l / vapp the phae hft due to the wave-paage effect, ad θ l (ω) = ta {Im[ h (ω) hl ( ω)] / Re[ h (ω) hl ( ω)]} the phae ft due to the te-repoe effect. I thee expreo, α a coherece parameter, d l the dtace betwee upport ad l, v the hear wave velocty of the L groud medum, dl the projected horzotal dtace the logtudal drecto of wave propagato, v app the urface apparet wave velocty, ad h ( ω ), =, l, the frequecy repoe fucto for the abolute accelerato repoe of the te aocated wth the th upport degree of freedom. I the curret example we ue the frequecy repoe fucto h ( ω + ζ ωω ω) = (.) ω ω + ζ ω ω able Flter Parameter cotet wth Accelerato Repoe Spectra Repoe pectrum ω (rad/ec) ζ Horzotal, ol type B (roc) 6.5 π 0.8 Horzotal, ol type C (oft roc) 5.0 π 0.8 Horzotal, ol type D (frm ol) 4.5 π 0.8 Vertcal (ol type B, C, D) 3.0 π 0.8

6 he 4 th World Coferece o Earthquae Egeerg October -7, 008, Bejg, Cha whch aume that the ol colum behave a a gle-degree-of-freedom ocllator wth frequecy ω ad dampg rato ζ. I the ubequet aaly we ue α / v = /600 ad v app = 400 m/. he value of ω ad ζ cotet wth the repoe pectra Fgure 4 are how able. he wave are aumed to propagate the drecto of the X ax. 6.3 Reult Fgure 3 how the meaure of the partcpatg modal ma of the example brdge for the cae of uform ad U I totally coheret upport moto, r ad r, repectvely, a a fucto of the umber of mode codered. he fgure ugget that the cotrbuto of hgher mode more gfcat the cae of dfferetal upport moto. I the ubequet aaly, we aume that cludg the frt 35 mode yeld exact reult r uform upport moto totally coheret upport moto Fgure 3 Partcpatg modal ma for uform ad totally coheret upport moto Fgure 4 how the rato of the etmate of elected repoe whe the frt =,..., 0 mode are cluded over the correpodg exact etmate. he repoe quatte codered are the dplacemet, axal ad hear force, ad toroal ad bedg momet at the mddle of the t ad d pa. he dplacemet are gve wth repect to the global X, Y ad Z axe, wherea the elemet force ad momet are gve wth repect to the local axe x, y ad z of the grder cro ecto. Reult baed o the orgal MSRS rule a well a the exteded MSRS rule are preeted. he reult ugget that or 3 mode are uffcet to accurately evaluate the dplacemet repoe. Hgh accuracy for almot all repoe quatte acheved whe the frt 8 mode are codered. he correpodg U I meaure of the partcpatg modal ma are r 8 = 0.80 ad r 8 = Whe fewer mode are cluded, the exteded MSRS rule provde a mproved approxmato mot cae. However, ome cae, t gfcatly overetmate the repoe whe very few mode are cluded. he pea the curve for Fy at = 7 are due to the large value of the correpodg b coeffcet for the vertcal upport moto. We ote that, for each vertcal upport moto, the coeffcet b of the cloely paced 7 th ad 8 th mode have value of mlar magtude ad oppote g. hu, the cluo of the 8 th mode balace the overetmato by the 7 th mode. h obervato ugget that par of cloely paced mode hould ot be eparated the aaly, epecally whe ther correpodg coeffcet have oppote g. b he precedg aaly ugget that for tructure ubjected to o-uform upport moto atfactory accuracy acheved wth a percetage of partcpatg modal ma maller tha that requred the aaly of tructure ubjected to uform upport moto. h becaue the repoe to o-uform upport moto clude a gfcat peudo-tatc compoet, whch ot affected by the modal trucato. herefore, order to obta a better udertadg of the modal cotrbuto for o-uform upport moto, we eed to exame the dyamc compoet of the total repoe,.e. coder oly the trple ad fourfold um the MSRS formula, Eq. (.4). Fgure 5 compare the modal cotrbuto the total repoe ad the dyamc

7 he 4 th World Coferece o Earthquae Egeerg October -7, 008, Bejg, Cha compoet of the repoe to o-uform exctato wth the modal cotrbuto the repoe to uform exctato. We oberve that although dfferetal upport moto ca creae the cotrbuto of hgher mode, depedg o the repoe codered, better accuracy may be acheved the cae of o-uform exctato tha the cae of uform exctato, wth the ame umber of mode cluded. Mddle of pa orgal MSRS exteded MSRS dp X / dpx dp Y / dpy dp Z / dpz F x / Fx F y / Fy F z / Fz M x / Mx M y / My M z / Mz Mddle of pa orgal MSRS exteded MSRS dp X / dpx dp Y / dpy dp Z / dpz F x / Fx F y / Fy F z / Fz M x / Mx M y / My M z / Mz Fgure 4 Repoe etmate baed o the orgal ad exteded MSRS rule

8 he 4 th World Coferece o Earthquae Egeerg October -7, 008, Bejg, Cha Mddle of pa F z / Fz 0.4 M y / My Uform No-uform, total repoe No-uform, dyamc compoet Uform No-uform, total repoe No-uform, dyamc compoet Fgure 5 Comparo of modal cotrbuto for uform ad o-uform exctato 7 CONCLUSIONS A evaluato of the modal cotrbuto the repoe of tructure ubjected to dfferetal upport moto preeted. A example applcato demotrated that order to acheve a dered level of accuracy, fewer mode are uually requred the cae of o-uform exctato tha the cae of uform exctato. h becaue o-uform upport moto duce gfcat peudo-tatc cotrbuto the tructural repoe, whch ot affected by modal trucato. Improved reult ca be obtaed wth the exteded MSRS rule that approxmately accout for the cotrbuto of the trucated mode wth mall addtoal computatoal effort. 8 ACKNOWLEDGMEN he preet tudy upported by the Calfora Departmet of raportato uder Cotract No. 59A0580. h upport gratefully acowledged. he frt author whe to alo gratefully acowledge poorhp from Oa Foudato. he author tha Paul Chug for valuable atace developg the brdge model. 8 REFERENCES. Campbell K.W. ad Bozorga Y. (003). Updated Near-Source Groud-Moto Relato for the Horzotal ad Vertcal Compoet of Pea Groud Accelerato ad Accelerato Repoe Spectra. Bull. Sem. Soc. Am., 93, Der Kuregha A. ad Neuehofer A. (99). Repoe pectrum method for multple-upport emc exctato. Earthquae Eg. Struct. Dy.,, Der Kuregha A. (996). A coherecy model for patally varyg groud moto. Earthquae Eg. Struct. Dy., 5, Kaha M. (996). Ifluece of emc wave patal varablty o brdge: a etvty aaly. Earthquae Eg. Struct. Dy., 5,