Uniform Response Spectrum Method for Seismic Travelling Responses Analysis of Long-span Arch Bridges

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1 Uniform Rpon Spctrum Mthod for Simic Trvllin Rpon Anli of Lon-pn Arch Brid M. Lou & Y. Tn Stt K Lbortor of Ditr Rduction in Civil Eninrin, Tonji Univrit, Shnhi, Chin SUMMARY: A nw mmtricl nd nti-mmtricl uniform xcittion rpon pctrum mthod i propod for tudin th imic rpon of th lon-pn rch brid. Firt, th mmtricl nd nti-mmtricl cclrtion tim-hitor r obtind dpndin on th brid pn nd th imic pprnt vlocit. Thn bd on th four tp of ninrin it, both th mmtricl nd nti-mmtricl uniform xcittion rpon pctr conidrin th wv p ffct r tblihd rpctivl. At lt, th imic rpon of lon-pn rch brid undr th uniform xcittion nd th trvllin xcittion r nlzd. Th numricl rult how tht th mthod propod in thi ppr i impl nd ffctiv tchniqu to vlut th imic rpon of th rch brid uin convntionl rpon pctrum whn th wv p ffct mut b conidrd. Kword: Lon-pn brid; wv p ffct; rpon pctrum mthod; uniform xcittion; tim-hitor mthod 1. INSTRUCTIONS A on of th mot importnt iu of tructurl dnmic nli, th ffct of round motion vrition on th imic rpon of tructur h bn tudid b mn rrchr. Accordin to thir tudi, th vrition of round motion m includ th wv p ffct, th prtil cohrnc ffct, th ffct of wv ttnution nd vn th locl it ffct. Epcill th wv p ffct hould not b inord whn th pn of tructur i no l thn qurtr of th wvlnth. A th ffct of wv p h inificnt influnc on th imic rpon of lon-pn tructur, corrctl formultin th t of motion t upport point for th lon-pn tructur i importnt. Rpon pctrum mthod (RSM) i world-widl doptd b th imic din cod of diffrnt tructur, but th convntionl rpon pctrum mthod cnnot conidr th multi-upport xcittion. So tht dvlopin th nw rpon pctrum mthod conidrin multi-upport xcittion (MSRM) i th im of mn cholr. Th tpicl work i th rrch of Kiurhin nd Nunhofr. Thouh th utd MSRM ffctivl ccount for th contribution of th pudo-ttic nd dnmic componnt of rpon wll thir covrinc, it qution mk th clcultion too complictd tht it i not t ccptd b prcticl ninr. Bd on th convntionl rpon pctrum mthod (RSM) nd th tructurl mmtr, nw RSM i dvlopd for imic rpon nli of lon-pn rch brid undr th multi-upport xcittion in thi ppr.. S/A-RSM FOR MULTI-SUPPORT SEISMIC RESPONSE ANALYSIS OF ARCH BRIDGE.1. Simplifid mthod for imic rpon nli of rch brid W tk impl rch brid hown in Fi..1 which i xcitd b round motion 1 () t to formult mthodolo tht convrtin th imic rpon clcultion of th whol brid undr nd () t

2 multi-upport xcittion into tht of th mmtricl nd nti-mmtricl mi-rch brid undr th uniform xcittion () t nd () t rpctivl. Fiur.1. Dcompoition mthod drwin Bd on th mmtricl chrctritic of th rch brid, th cclrtion xcittion () t cn b obtind b () t nd 1 t t 1 t t 1 () [ ( ) ( t)] 1 t) () [ ( ( )] (for horizontl xcittion) (.1) or 1 t t 1 t t 1 () [ ( t) ( )] 1 ) () [ ( ( t)] (for vrticl xcittion) (.) whr ubcript nd dnot mmtricl nd nti-mmtricl componnt rpctivl, i () t i th input cclrtion t upport-i of th brid. Whn th uniform xcittion i tkn into ccount, 1 () t () t. If th wv p ffct i onl conidrd, ( t ) t (.3) ( ) vpp vpp whr dnot th horizontl ditnc btwn th upport 1 nd, v i th urfc pprnt 1 vlocit. () t i th input imic cclrtion nd 1 () t () t. Thn Eqn..1 nd Eqn.. cn b rwrittn follow pp () t (, t 1, vpp ) () t (, t 1, vpp ) (.4) whr (, t 1, vpp ) nd (, t 1, vpp ) dnot th mmtricl nd nti-mmtricl componnt of input motion. Th qution of motion for th mmtricl nd nti-mmtricl mi-rch brid ubjct to () t nd () t rpctivl cn b writtn in th mtrix form

3 [ M ]{ u( t)} [ C]{ u ( t)} [ K]{ u( t)} [ M ]{ } ( t) [ M ]{ u( t)} [ C]{ u ( t)} [ K]{ u( t)} [ M]{ } ( t) (.5) whr { u ( t )} nd{ u ( t)} r th diplcmnt of th mmtricl nd nti-mmtricl mi-rch brid, [M], [C] nd [K] r th m, dmpin nd tiffn mtric rpctivl. Th mthodolo mntiond bov i n ccurt clcultion mthod tht cn b pprovd b th mchnicl thor nd numricl rult. Tht i, whn mmtricl tructur i undr n rbitrl two-upport imic xcittion, it i fibl to divid th imic xcittion into two prt: th mmtricl xcittion (SE) nd th nti-mmtricl xcittion (AE). Thu, th imic rpon of th whol tructur cn b dcribd th um of th rpon of th mmtricl mi-tructur undr th uniform xcittion SE nd tht of th nti-mmtricl on undr th uniform xcittion AE... Smmtricl nd nti-mmtricl uniform xcittion rpon pctr Obvioul, two qution of motion in Eqn..5 cn b olvd b th convntionl rpon pctrum mthod bcu th qution dcrib th tructurl dnmic rpon undr th uniform imic xcittion. If th uniform xcittion rpon pctrum corrpondin to th mmtricl nd nti-mmtricl componnt () t nd () t i tblihd rpctivl, it will b ir to timt th imic rpon of th lon-pn rch brid thn th multi-upport xcittion rpon pctrum mthod. A prcticin ninr hv bn o fmilir with th ppliction of th uniform xcittion rpon pctrum mthod, th mthod utd in thi ppr will not hv difficult to ppl in ninrin. Accordin to th dfinition of rpon pctrum, th rpon pctr corrpondin to nd () () t t rpctivl r clld th mmtricl or nti-mmtricl uniform xcittion rpon pctrum. Obvioul, unlik th trditionl rpon pctrum, th pctr nmd S (,, t) nd S (,, t) rpctivl r th function of prmtr Δt=Δ 1 /v pp. How to tblih th cclrtion pctr S (,, t) nd S (,, t) will b tudid in trm of prcticl imic wv from diffrnt it in th nxt ction..3. Qulittiv nli of mmtricl nd nti-mmtricl rpon pctr Th qulittiv nli of rpon pctrum i ttiticl nli bd on th chrctritic of rthquk rthr thn tht of tructur. Hr, two principl r tkn into ccount: 1 th doptd prcticl imic wv i obtind ccordin to th Chin Cod for Simic Din of Buildin (GB ), in which four tp of oil it r clifid includin Sit I (V >760 m -1 ), Sit II (760 V >360 m -1 ), Sit III (360 V 180 m -1 ) nd Sit IV (180>V m -1 ); th numbr of prcticl imic wv from ch it i nrl th m. Thn, 16 domtic nd ovr tpicl imic rcord litd in Tb..1 r ud in thi tud. In ordr to mk th comprion mon tho pctr mor ffctivl, th mplitud of rcordd imic wv ud in thi tud r djutd to 1. Dfinin tn vlu of prmtr t =0 (qul to th uniform xcittion), 0.4, 0.6, 0.8, 1.0, 1.5,.0,.5, 3.0, 4.0 nd 5.0. Tbl.1. Simic rcord from four tp of oil it Sit tp I II III IV Simic rcord F1, F, N1 F3, F4, F5, N F6, F7, F1, N3 F8,F9,F10,F11, N Smmtricl nd nti-mmtricl dnmicl mnifiction fctor curv A dnmicl mnifiction fctor (DMF) curv μ(ω i ) cn b dfind th rtio of cclrtion mplitud ut () mx obtind from inl dr of frdom (SDOF) tm dividd b th input cclrtion mplitud i () t mx. If SDOF tm i linr, it cn b xprd b

4 u i () t mx S ( i, ) (.6) ( i ) () t () t mx mx whr, S( i, ) i th cclrtion rpon pctrum of SDOF tm; i of vibrtion nd ζ i th dmpin rtio. i th nturl frqunc Fi.. onl iv th mmtricl nd nti-mmtricl DMF curv of F1 from Sit I du to th ppr pc limittion. Hr, ζ=0.05. Th x-xi dnot nturl vibrtion priod T from 0.01 to 10; th -xi dnot th dnmicl mnifiction fctor curv μ(t). In Fi.. () nd Fi.. (b), th mmtricl nd nti-mmtricl DMF curv with diffrnt Δt includin Δt =0 which rprnt th uniform xcittion DMF curv r ivn rpctivl. () Anti-mmtricl DMF curv of F1with diffrnt Δt (b) Smmtricl DMF curv of F1with diffrnt Δt Fiur.. Th mmtricl nd nti-mmtricl DMF curv of F1 from Sit I In Fi.., it i clr to tht th mmtricl nd nti-mmtricl DMF curv ( ) t 0 T with diffrnt prmtr Δt r lmot of imilr hp th uniform xcittion DMF curv ( ) t 0 T nd th vlu of ( ) t 0 T fluctut round tht of ( ) t 0 T. Th mmtricl or nti-mmtricl DMF vr curv t 0, i( T ) nd t 0, i( T ) with diffrnt Δt r computd for four it rpctivl. A wll th uniform xcittion DMF vr curv t 0, i( T ) i clcultd for comprion. Whn Δt=1,, 3, th computd rult t 0, i( T ) nd t 0, i( T ) of Sit I r hown in Fi..3 rprntd b dhd lin nd dh-dot lin rpctivl. Mnwhil, blck olid lin rprnt th t 0, i( T ). It i obviou from Fi..3 tht th diffrnc btwn t 0, i( T ) nd t 0, i( T ) i mll nd cn b conidrd tht t 0, i( T ) nd t0, i( T ) r m for ninrin ppliction. Th concluion for othr thr it i m, o tht th rult r not litd in th ppr. () t 1 (b) t (c) t 3 Fiur.3. Avr dnmicl mnifiction fctor curv of Sit I.3.. Smmtricl nd nti-mmtricl mplitud ttnutin curv Accordin to Eqn..6, th input cclrtion mplitud of th mmtricl nd nti-mmtricl () t nd () t r th k to mk th currnt RSM fibl for th dnmic nli of th rch

5 brid undr two-upport xcittion. Th mplitud A, ( t) nd A, ( t) of mmtricl () t nd nti-mmtricl () t for four it cn b obtind. Whr, i dnot Sit i (I-IV); j dnot th nm of wv. Du to limitd ppr pc, onl on wv from four it i hown in Tb... Tbl.. Smmtricl nd nti-mmtricl mplitud (cm/ ) Δt SitⅠ SitⅡ Sit Ⅲ Sit Ⅳ ( ) AI,F1 t I,F1 A ( ) II,F3 t A AII,F3 t A ( ) III,F6 t AIII,F6 t AIV,F10( t) AIV,F10( t) i j i j In ordr to mk th dt from th tbl bov b comprd with ch othr, th hould b trnformd b Eqn..7 nd Eqn..8. Firt, dividin th mmtricl nd nti-mmtricl cclrtion mplitud ( t 0 ) b th uniform cclrtion mplitud ( t 0 ), nd thn clcultin / th vr of A i, j ( t) from ch it nd till tkin t vribl. A ( t) A (.7) / / i, j i, j( t) / Ai, j ( t 0) n / A i, j( t) / i 1 A i ( t) (.8) n whr, A i function of t / i, j ( t) nd bin with 1; n i th numbr of wv from th m it. Th followin fiur how th vr mplitud of mmtricl nd nti-mmtricl input componnt of four it. Dhd lin tnd for mmtricl componnt nd olid lin tnd for nti-mmtricl componnt. () Sit I (b) Sit II

6 (c) Sit II Fiur.4. Avr mplitud curv (d) Sit IV Fi..4 illutrt tht: 1 in nrl, th mmtricl nd nti-mmtricl vr mplitud of four it dcr whn Δt incr; whn t =0~1, th curv dcr hrpl whil th pitch bcom lowl whn t >1. In ordr to mk mplitud ttnution curv ir for ppliction, th fittin ttnutin qution i, ( t) nd i, ( t) of th mmtricl nd nti-mmtricl vr mplitud for four it r writtn follow. I, ( t) 0.5 t 0.8 t 0.009t 0.09t0.7 0 t 1 t 1 (.9) I, ( t) 0.6 t 0.9 t t 0.06 t t 1 t 1 (.9b) II, ( t) 0.6t 0.95t t t 0.04 t 0.33 t 1 (.10) II, ( t) 0.6t 0.95t t 0.005t 0.05t0.34 t 1 (.10b) III, ( t) 0.6t 1.1t t t 1 (.11) III, ( t) 0.6t 1.1t t t 0.05 t 0.45 t 1 (.11b) IV, ( t) 0.5t 0.9t t 0.01 t t 0.4 t 1 (.1) IV, ( t) 0.55t 0.95t t 0.01 t 0.09 t 0.33 t 1 (.1b).3.3. Smmtricl nd nti-mmtricl rpon pctrum Th im of introducin mplitud ttnutin curv i to corrct th currnt RSM which i onl fittd for th uniform xcittion rpon pctrum nli nd cnnot conidr th wv p ffct. Th mplitud ttnutin curv cn b ud kind of corrction fctor to provid implifid RSM conidrin th wv p ffct for trvlin imic rpon nli of th tructur with

7 onl two upport. Th rpon pctr of mmtricl nd nti-mmtricl input componnt of th trvlin imic input cn b obtind b multiplin th uniform rpon pctrum S (, ) b mplitud ttnutin fctor i, ( t) nd i, ( t) rpctivl. Tht i, S (,, t) ( t) S (, ) (.13) i, S (,, t) ( t) S (, ) (.13b) i,.3.4. S/A-RSM for trvlin imic rpon nli of rch brid Bd on Eqn..13, th imic rpon v mx nd vmx of th rch brid undr th mmtricl nd nti-mmtricl xcittion cn b obtind b RSM. Th totl imic rpon v mx of th rch brid undr th trvlin imic input cn b clcultd b th qur root of th um of th qur (SRSS): v ( v ) ( v ) mx mx mx (.14) Th implifid nli procdur utd bov i clld th mmtricl nd nti-mmtricl uniform xcittion rpon pctrum mthod (S/A-RSM). 3. A CASE STUDY FOR S/A-RSM 3.1. Projct profil nd finit lmnt modl Th brid hown in Fi. 3.1 i prtrd concrt (PC) T-irdr tructur. Th min pn of th rch with 14 brid opnin i m=49.4m lon. Th bm lmnt modl for th brid i hown in Fi. 3.. Fiur 3.1. Prtrd concrt brid Fiur 3.. Structurl finit lmnt modl 3.. Simic rpon of th brid Thr imic rcord of Sit IV includin Wtrn Whinton wv (th durtion tim i 89.16), Wtmorlnd wv (th durtion tim i 88.44) nd Colin wv (th durtion tim i 65.0) r doptd in th tud. In th clcultion, v pp = (uniform input), 000m/,1000m/ nd 500m/ (tht i Δt=0, 0.1, 0.4, 0.84) r conidrd. Th pk vlu of cclrtion ( m/ ) nd diplcmnt (d m) of th brid ubjct to th thr wv obtind b S/A-RSM r hown in Tb In th Tbl, HD nd VD dnot horizontl dirction nd vrticl dirction rpctivl. In th computtion, th rpon pctrum i clcultd dirctl from th thr imic wv rpctivl. Th rltiv rror of th pk vlu comprd with rult computd b tp-b-tp mthod (THM) r lo litd in th Tbl. In th followin work, th tndrd pctrum in Chin cod i ud intd of th pcil rpon pctrum of th thr imic wv. Th pk vlu of th imic rpon of th brid obtind b S/A-RSM r litd in Tb. 3.4 whn input imic cclrtion pk vlu i qul to 0.. Th pk vlu of th imic rpon computd b THM

8 r th vr vlu from th rult whn djutin mplitud of th thr imic wv to 0.. Tbl 3.1. Pk vlu of th rch obtind b S/A-RSM ubjct to Wtrn Whinton wv Pk vlu of imic rpon Rltiv rror comprd to THM (%) Rpon Loction Uniform 000m/ 1000m/ 500m/ Uniform /4-pn, HD (m/ ) 1/4-pn, VD /-pn, HD /-pn, VD /4-pn, HD d (m) 1/4-pn, VD /-pn, HD /-pn, VD Tbl 3.. Pk vlu of th rch obtind b S/A-RSM ubjct to Wtmorlnd wv Rpon Loction Pk vlu of imic rpon Rltiv rror comprd to THM (%) Uniform 000m/ 1000m/ 500m/ Uniform /4-pn, HD (m/ ) 1/4-pn, VD /-pn, HD /-pn, VD /4-pn, HD d (m) 1/4-pn, VD /-pn, HD /-pn, VD Tbl 3.3. Pk vlu of th rch obtind b S/A-RSM ubjct to Colin wv Rpon Loction Pk vlu of imic rpon Rltiv rror comprd to THM (%) Uniform 000m/ 1000m/ 500m/ Uniform /4-pn, HD (m/ ) 1/4-pn, VD /-pn, HD /-pn, VD /4-pn, HD d (m) 1/4-pn, VD /-pn, HD /-pn, VD Tbl 3.4. Avr rpon vlu of th brid obtind b S/A-RSM Rpon Loction Spctrum Pk vlu of imic rpon Error comprd to THM (%) Uniform 000m/ 1000m/ 500m/ Uniform /4-pn, AS HD SS /4-pn, AS (m/ ) VD SS /-pn, AS HD SS /-pn, AS VD SS /4-pn, AS HD SS /4-pn, AS d (m) VD SS /-pn, AS HD SS /-pn, AS VD SS

9 Th rult indict tht thr r om diffrnc btwn th clcultd rror obtind b S/A-RSM nd th currnt RSM, but th r in th m rror lvl, pcill mximum rror ppl in RSM. So th S/A-RSM utd in th ppr i till ccptbl in ninrin ppliction. 4. CONCLUSION Bd on th mmtricl chrctritic of th rch brid, th mmtricl nd nti-mmtricl uniform xcittion rpon pctrum mthod (S/A-RSM) i utd in th ppr for conidrin th wv p ffct on th imic rpon of lon-pn rch brid. Th rult how tht: (1)Th mmtricl nd nti-mmtricl rpon pctr r th function of th imic pprnt vlocit nd th brid pn. Th lonr th pn i, th ffct of trvllin wv on th lon-pn tructur bcom mor inificnt nd cnnot b inord. ()Th mmtricl nd nti-mmtricl dnmicl mnifiction fctor curv bd on th four tp of ninrin it hv nic rmnt with th uniform xcittion DMF curv, which mn tht th RSM introducd in th din cod cn till b doptd for dnmic rpon nli of tructur undr multi-upport xcittion. (3)Th numricl rult how tht th S/A-RSM i n ffctiv implifid RSM tht it cn b il ud in ninrin, nd it i lo uful for nlzin th trvlin imic rpon nli of othr lon-pn tructur with two upport. AKCNOWLEDGEMENT Thi rrch w ponord b Stt K Lbortor Bic Thor Foundtion of th Minitr of Scinc nd Tchnolo of Chin throuh rnt SLDRCE08-A-07 nd Ntionl Nturl Scinc Foundtion of Chin throuh rnt Th upport r rtfull cknowldd. REFERENCES H, Q. (009). Rviw of tructurl imic nli of trvllin wv ffct. Journl of Erthquk Eninrin nd Structurl Vibrtion. 9:1, Zhon, W. (000). Dvlopmnt nd tndnc of th ntiimic din of th lon-pn tructur. Scinc &Tchnolo Rviw. 1:3, Kiurhin, A.D. nd Nunhofr, A. (1996). A cohrnc mthod for ptill vrin round motion. Erthquk Eninrin nd Structurl Dnmic. 5:1, Kiurhin, A.D. nd Nunhofr, A. (199). Rpon pctrum mthod for multi-upport imic xcittion. Erthquk Eninrin nd Structurl Dnmic. 1:8, Hrdi-Zvoni, E. nd Vnmrck, E.H. (1994). Simic rndom-vibrtion nli of muti-upport tructurl tm. Journl of Eninrin Mchnic. ASCE. 10:5, Lou, M. nd Go, S. (009). Th implifid mthod for th vrticl imic rpon nli of th rch brid undr th multipl-upport xcittion. Journl of Ditr Prvntion nd Mitition Eninrin. 9:6, Biot, M.A. (1941). A mchnicl nli for prdition of rthquk tr. Bull. Sim. Soc. Am. 31:1, Chin Acdm of Buildin Rrch. (010). Cod for imic din of buildin GB Bijin: Chin Architctur & Buildin Pr. Xi, L. nd Zhi, C. (003). Stud on th mot unfvorbl round motion in imic din. ACTA Simoloic Sinic. 5:3, Clouh, R. W. nd Pnzin, J. (1993). Dnmic of Structur. Nw York: McGrw-Hill Inc.