SEISMIC RESPONSE ANALYSIS FOR TELECOMMUNICATION TOWERS BUILT ON THE BUILDING

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1 SEISMIC RESPONSE ANALYSIS FOR ELECOMMUNICAION OWERS BUIL ON HE BUILDING 534 K KANAZAWA And K HIRAA SUMMARY he em reone erum mehod for eondry yem develoed, o onder dynm neron wh he rmry yem. he rooed em reone evluon mehod dvded no wo e. he fr e o rnform he em nu uh floor reone er vlue he erod of he eondry yem from he efed degn er he ground. In h e, we reen he modl ynhe mehod n whh he modl roere of he ombned yem re deermned from he modl hrer of he rmry yem nd he equvlen ngle-degreeof freedom ollor of he eondry yem. Furhermore, he mehod onder rnen effe of he rmry yem on he reone of he eondry yem. On he oher hnd, he eond e he em reone lulon of he eondry yem ung he relve eleron reone er vlue. o onder rgd body mode nd loed ed mode, he reen modl ombnon rule nlude wo knd of orrelon oeffen beween he nu eleron nd relve eleron nd mong he relve eleron. o llure he reen mehod, em reone nly of ower-buldng model rred ou, nd he ury of he mehod dued. INRODUCION For he em degn of he eondry yem, uh eleommunon ower or equmen mouned on he buldng (he rmry yem), reone erum mehod ofen ued. In h mehod, he fr e, ommon o evlue he em floor reone er (FRS) from he efed degn er of he ground moon. he FRS re he mxmum reone ere of ngle-degree-of-freedom (S-DOF) ollor whh hve dfferen dmng ro nd nurl frequene nd whh umed o be mouned on he floor of he rmry yem. From he genered FRS, he eond e, he reone of eondry yem re evlued by men of ome modl ombnon rule. In order o obn em reone of he eondry yem urely, he FRS hould be evlued exly he fr e, nd he ol reone hould be ombned from eh modl reone reonbly he eond e. In hee on of vew, mny mehod were develoed follow. In he evluon of FRS, he omble ower erum deny funon (PSDF) luled from he degn er, hen he PSDF of he floor he eondry yem hed onvered from he ground PSDF nd he modl roere of he ombned yem, fnlly he FRS evlued from he floor PSDF. he omble mehod beween PSDF nd degn er (or beween PSDF nd FRS) w rooed by Kul (978) or Unruh nd Kn (98). her mehod re bed on he onry roe, nd lo onder nfne erhquke duron by dumen o dmng for of S-DOF ollor [Rouhenbluh, 969]. In urn, n he PSDF rnformon from ground o floor, he ombned modl roery hould be evlued urely. For exmle, when he wegh of eondry yem relvely lrge, he dynm neron beween rmry ndeondry yem gnfn. From h on of vew, mny modl ynhe mehod were rooed, he modl roere of ombned yem re luled from rmry nd eondry one ndvdully e.g. [Sngh nd Surez, 986,987]. A menoned bove, n he revou ude, n he evluon of FRS dynm neron beween rmry nd eondry yem ondered erfely, however, non-onry or rnen reone effe re negleed. Cenrl Reerh Inue of Eler Power Indury, Abko, Jn. Eml: knzw@re.denken.or. Geoehnl & Erhquke Engneerng Dermen, Cenrl Reerh Inue of Eler Power Indury, Abko, Jn.

2 (gven) (gven) Model Proere (gven) Degn Ser R g rnform Mehod Effeve Duron d Exeedne Prob. e Prmry Sy. Synhe Mehod Seondry Sy. Power Ser S g Predon Mehod Modl roere of Combned Sy. Mxmum Ael. Vlue Correlon Coeffen Abolue Ael. Relve Ael. Inu-Reone Reone of Eh Order δ Reone-Reone ε Rgd Reone R Relve Reone R Modl Combnon Rule ol Reone of Seondry Syem R Fg. Flow Chr of Sem Anly for Seondry Syem heorelly, he umon of onry reone gve overemed reul omred wh he rel reone, nd gnfn when he nurl erod of ombned yem long, or when he erhquke duron mll. On he oher hnd, n he modl ombnon rule of reone, he qure roo of he um of he qure (SRSS) rule [Goodmn e l., 953] well known nd ll ued que wdely, bu no deque n he e of he reone wh loely ed frequene. he double um (DSUM) rule [Roenblueh, 969] nd he omlee qudr ombnon (CQC) rule [Der Kureghn, 98] gve reonble emon even when eh mode loed, bu no deque when he nu moon nrrow-bnd roe or when he rgd body mode nfluenl. o olve hee roblem, Hdn (98) nd Sngh nd Meh (983) rooed lernve rule ung relve eleron reone er nd mxmum nu eleron (or ZPA; zero erod eleron). Alo, A.K.Gu e.l.(984) rooed rule n whh ol reone re ynhezed from he rgd body mode nd he dmed erod mode. Der Kureghn nd Nkmur (993) mroved he orgnl CQC rule, o vod he runon error. In he e of he eondry yem, he nu moon he reone he floor of he rmry yem, herefore, generlly beome nrrow-bnd roe. In h er, em reone erum mehod for eondry yem rooed, ondered dynm neron roblem wh rmry yem. Frly, we reen mehod o red he mxmum reone for he rmry-eondry yem (ombned yem), onderng nfne erhquke duron or rnen reone. In h mehod, he mxmum bolue or relve eleron reone of he ombned yem n be obned, under he umon of he onry ground moon. Seondly, ung hee vlue, he modl ombnon rule for he mxmum reone of he eondry yem reened. h mehod bed on he modl ueroon heory nd lo ued for mxmum vlue of he relve eleron nd nu eleron. he em nly for he eondry yem n h udy hown n Fg.. Egenvlue Anly MAXIMUM RESPONSE OF HE COMBINED SYSYEM Ex formulon of he egenvlue roblem re hown by Sngh nd Sure (987), for mul-degree of freedom (M-DOF) rmry yem wh S-DOF eondry yem. In h er, fr, h formulon exended o M-DOF rmry yem wh M-DOF eondry yem. When N SDOF ollor equvlen o eondry yem re hed o he M-DOF rmry yem (number of freedom: N ) nd he ombned yem ube o ground moon x g (), he equon of moon beome M M C x M x + x K = x M xg () 534

3 or, Dz + Ez = z G () M C K M = C [ vv] K [ vv] = N + = N + h m m mu v,, ω ω (3) = = n whh M, C nd K re he m, dmng nd ffne mrx of he ombned yem, he mre of ubr re hoe of he rmry yem, ω, h nd m re he nurl frequeny, dmng ro nd modl m of he -h mode of he eondry yem. denoe dgonl mrx, nd ueroed denoe rnoe. And, he veor v defned follow f he -h mode ollor umed o h o node k of he rmry yem: k k k N N N v = {,,,,,,,, } he mgnude of modl m m gven, m = N = M βφ n whh M m of node, β he -h ron for nd φ re he mode he vlue of he eondry yem, reevely. he egenvlue equon of equon () beome, ( D + E) φ = {}, = ` ( N + N ) (6) φ o obn he modl roere of he ombned yem, we need o olve equon (6). However, here re wo roblem, olvng h egenvlue equon. One he numerl nury whh ould our n he oluon of equon (6) due o ll-ondonng of he mre ued by he lghne of he eondry yem. And he oher he number of he order n equon (6) wo me lrge he number of degree of freedom of he rmry nd eondry yem, herefore, he omuonl o o olve equon (6) relvely lrge. In re, he lower mode of he ombned yem whh ffe o he reone n be ynhezed from he fr few mode of he rmry nd eondry yem ndvdully. hu, o vod hee dffule, we n nrodue he followng rnformon n equon (6): φ Φ φ = = φ φ U, U = U (7) u m v n whh Φ he mode mrx of he rmry yem n whh rrnged he lower mode (number of mode: m ), nd whh normlzed by Φ M Φ = I (I un mrx). Alo, he number of ollor he eondry yem ondered he fr lower mode (number of mode: m ). In re, m nd m n be eleed o nlude n he rnge of frequeny nere. Subung equon (7) n equon (6), remullyng by nd ulzng equon (4) nd he orhonorml roere of Φ, we obn ( ) D + E φ = {} (8) I D = D = u h m ω v I h ω + ww = (9) E = E = -I u ω v + m = m ω w w (4) (5) (9b) 3 534

4 w = v U = { φ,, φ,,,, - m,, } M M+ k km P M+ Solvng he omlex egen equon n equon (8), we n obn he egenvlue nd egen-veor Φ, Φ of he ombned yem. h, he modl roere of he ombned yem re ynhezed from gven rmry nd eondry one ndvdully. () he Momen Funon of Abolue nd Relve Aeleron Reone In eon 3, modl ombnon rule for he reone of he eondry yem derbed, n whh wo knd of mxmum reone vlue re ued: hoe re he mxmum bolue eleron of he floor he eondry yem hed, nd he mxmum relve eleron of eondry yem. o evlue hee mxmum vlue, we wll derve he momen funon of he bolue nd relve eleron n h ubeon. Ung hee momen funon, mxmum eleron reone re evlued by men of mehod hown n he nex ubeon. he equon of moon () n be deouled wh he hel of he ndrd rnformon, x = Φξ() = Φξ () () x n whh ξ he veor of he rnl oordne. Subung equon () no equon () nd remullyng by Φ, we obn ( m + m ) deouled equon ( ) d ξ () + e ξ () = Γ x (), g = ` m + m () Γ = φ M (3) n whh Γ he omonen veor of he ron for of he ombned yem. I n be hown h he bolue eleron reone ubeed o he ground moon node k of he ombned yem gven follow: k = Re φαξ k (4) = nd he fr dervve of he reone wh ree o me k gven follow: Re x k = φkα ξ + φkαβ g (5) = = α = e d, β = Γ d (6) = +. n whh m he ol number of he mode of he ombned yem ( m m m) he momen funon of he bolue eleron reone wll be reened below. Now, ung he rmeer n equon (6), he un mule reone funon of ξ n equon () n be obned follow: [ α ] h() = βex (7) By men of equon (4) or (5), (7), umng he ground moon x g onry roe. he fr hree momen funon re derved m m + σ () = [ k() ] = φkφkαα A() Sg ( ω) dω E Re (8) = = + σ () = [ k () k () ] = φφαα k k A() φφααβ k k Bg() Sg ( ω) dω E Re (9) - = = = = 4 534

5 + [ ] Re k k + k k ( g + g ) σ () = E k () = φ φ α α A () φ α β φ α B () B () = = = = A () = + m φαβ k Sg( ω) dω () = ( α ω)( α + ω) ββ ( Ex[ - ω] - Ex[ -α ] )( Ex[ ω] - Ex[ - α ] ) β ( Ex[ (-α ω) ] ) Bg = Ex (-α ω) () ( [ ] ) β Bg () = +, () - (b) α ω α + ω n whh S g ( ω ) he PSDF of x g, nd σ (), σ () nd σ () re he -h, - nd -nd momen funon of he bolue eleron reone node k of he ombned yem, reevely. he hree momen funon re deenden of me, nmely, he bolue reone he non-onry roe rnen reone. Prlly, negrl n equon (8) hrough () re luled numerlly n he rnge of frequeny of nere. he fr hree momen of he relve eleron reone lo n be derved n he me wy he bove formulon. When modl m of eondry yem node hed o he floor of rmry yem r node k n he ombned yem, he relve eleron reone k of node n erm of node k r k k ( = = Re φ φk ) αξ () = he reul of he fr hree momen funon re obned by relng φ k o φ φk n equon (8) hrough (), reevely. Predon of Mxmum Reone he fr hree momen funon of he rnen reone re formuled n he revou ubeon. In h ubeon, he redon mehod of he mxmum reone of he non-onry roe reened. In he rndom vbron heory, u nd down-rong of reone level ()=± b our n ordne wh Poon roe. For non-onry reone, f he reone ( ) ymmer nd zero-men rndom roe, he drbuon of mx n be wren follow [Amn nd Gungor,97] d Prob[ mx b, d] = e( d) = ex + vb ( ) d (3) n whh d he duron, e he exeedene robbly nd v + () b n u-rong re. he u-rong re v + () b n be luled from on deny funon of reone nd dervve me follow [Amn nd Gungor, 97]: + ν b () = f ( b, ; ) d (4) n whh f he on robbly deny funon, nd he robbly deny funon of ( ) lo defned f ( ; ). If he f nd f re Gun funon, equon (4) derved + b b vb = b ( ) + σ () σ () () ρ ()ex ρ () ex (5) π σ () ρ () σ () () () 8π σ σ σ () ρ () = (6) σ () σ () 5 534

6 n whh ρ he orrelon oeffen for ( ) nd dervve me. In he reedng ubeon, σ, σ nd σ re rereened n equon (8) hrough (), nd hee funon mu be evlued numerlly. hu, ubung equon (5) no equon (3), he u-rong lebel b he gven exeedne robbly e nno be obned loed-form. We hould, herefore, olve he nonlner equon b, follow: d gb ( ) = ex + b ( d ) ( e( d) ) ν (7) A redon of he mxmum reone b, when gb ( ) equl o he gven e. Menwhle, he uhor olved h nonlner equon by men of he Newon-Rhon heme n he ler exmle. MODAL COMBINAION RULE Bed on he modl ueroon heory, we rooe modl ombnon rule, follow: R = ε R R + δ R R + R (8) n whh R ol reone, R nd R re he -h nd he -h modl reone luled from relve eleron er, R rgd reone luled from he mxmum nu eleron. ε modl orrelon oeffen beween he -h nd he -h relve modl reone, nd δ modl orrelon oeffen beween he -h relve modl reone nd he nu eleron. hee modl orrelon oeffen re defned follow: d σ () ε = d (9) σ () σ () d σ () δ = d (3) σ() σ () n whh σ () nd σ ( ) re he momen funon of relve eleron reone, nd σ () he momen funon of nu eleron (or eleron reone he floor he eondry yem hed). hee funon re obned n eon. Alo, n he me wy of eon, σ nd σ n be obned. ILLUSRAIVE EXAPLES he numerl exmle re reened for n exng eleommunon ower eondry yem hed o he SDOF rmry yem hown n Fg.. he ower modeled hree-dmenonl ymmerl ru ruure wh lner member, nd he fr hree rnlonl undmed nurl erod re.6,.7 nd. eond. o onder vrou degn ondon, nurl erod of he rmry yem re e.5,.3,.6,. nd.5 eond rmeer orreondng o he nurl erod re of he rmry o eondry yem (/) o /4, /, /, / nd 4/. M re of he rmry o eondry yem fxed o, nd dmng ro of he rmry nd eondry yem wh ree o ll mode re e o.5 nd., reevely. o evlue lbly of he rooed roh, he reul re omred wh he me hory nlye of he ombned yem. he degn erum for he rooed mehod nd mle of rfl erhquke re hown Fg. 3 nd 4. he effeve duron of erhquke d e o 8. eond from he Hud lo [A.K.Gu, 99], n whh he re of rong moon onrbuon defned 9%. Alo, he Mone Crl mulon (MCS) of me hory nlye were rred ou ung rfl erhquke moon, o evlue he ury of he rooed mehod. In he fr e of he rooed mehod, he FRS for eondry yem re evlued from he efed degn er, he effeve duron of he ground moon nd he rmry nd eondry egen roere. In he below exmle, he exeedne robbly e e o.5 nd he effeve duron d were fxed hrough he nlye. Fg.5 how he -h momen funon of buldng reone normlzed by he ymoe, luled n equon (8). I ler h he ymo level rehed muh more lowly n he rmry yem wh lrger nurl erod, eelly n he e of b/=4/ he reone of buldng doe no rehed o onry n he duron. ble how he mxmum eleron of nu o eondry yem (h he mxmum bolue eleron reone of he rmry yem) nd he - nd -nd relve eleron reone of o 6 534

7 Aeleron [gl] Fg. A ower-buldng Model men-ƒð men-σ men men+ƒð Perod [e] Fg 3. he Degn Ser eondry yem, nd re he omron of he reul by he rooed mehod wh hoe by mehod umng onry reone nd MCS. he onry mehod ued he omble reone/ower erum rnformon rooed by Unruh nd Kn (98) nd he onry rnfer funon of he ombned yem. he reul of he rooed mehod how good greemen wh he verge of MCS exe he - nd -nd relve eleron n /=/ nd /4. Eelly, for he mxmum nu eleron o eondry yem, he ury of he rooed mehod hgher hn h of he onry mehod. Exeon our n he e n whh he modl frequeny beween he eondry nd rmry yem loe, nd n h e he rooed mehod gve onervve reul. In he eond e, he reone of he eondry yem re evlued from he nu eleron for he fr e. Fg.4 nd 5 how he mxmum eleron reone nd xl re of bre member of he ower luled n equon (8) hrough (3). o evlue he effeny of he modl orrelon oeffen, he reul of he modl ombnon rule whou ε nd whou δ re lo derbed. In /=4/, / nd /, he reone obned by he rooed mehod gree well wh he one of MCS. Bu, n he e of /=/ nd /4, he rooed mehod gve onervve reul of MCS. he mn reon of hee no greemen eem o be overemon of FRS fr e. On he oher hnd, onern he modl orrelon, hown n Fg.4 nd 5 (eelly n he e of b/=/ or /), evluon of he ol reone wh ll erm gree well wh hoe of MCS, however, he evluon whou δ gve erroneou reul. hu he oeffen δ effeve o mrove he ury of evlued ol reone. In urn, he oeffen ε no evlued n he exmle beue he loed mode of he ower do no ex. Bu, he oeffen ε beome muh effeve n he e of he yem wh loed e mode [Wlon, e l., 98]. CONCLUSIONS me [e] Sem reone erum nly for eondry yem reened, whh dvded no wo e. he fr e o be evlued em nu FRS for eondry yem. Ung he rooed mehod, one n obn Aeleon [gl] Fg 4. A Smle of Arfl Erhquke 8.8 b/ /4.6 /.4 /. / 4/ me [e] Fg 5. he -h Momen Funon of Buldng σ ()/σ ( ) Reone ( Normlzed by Aymoe) ABLE. CALCULAED FRS / *) Inu eleron[gl] Prooed Sonry me Anly**) / /79/88 / /46/6 / 47 86/4/3 / /5/ 4/ /53/6 / he relve eleron[gl] Prooed Sonry me Anly**) / /397/459 / /565/65 / /565/97 / /3/3 4/ 4 9/3/7 / he nd relve eleron[gl] Prooed Sonry me Anly**) / /74/45 / /45/67 / 8 9 7/3/35 / /7./8.3 4/...9/./.6 *) he erod re of rmry o eondry yem **) men-ƒð /men/men+ƒð 7 534

8 he mxmum eleron of he ombned yem em nu for eondry yem, onderng he nfne erhquke duron or he rnen reone under he onry ground moon. he eond e 5 Hegh [m] A [gl] A [gl] A [gl] A [gl] A [gl] () / =/4 (b) / =/ () / =/ (d) / =/ (e) / =4/ Fg 5. Mxmum Aeleron of ower 5 Hegh[m] Sre [kgf/m] Sre [kgf/m] Sre [kgf/m] Sre [kgf/m] Sre [kgf/m] () / =/4 (b) / =/ () / =/ (d) / =/ (e) / =4/ Fg 6. Mxmum Sre of Bre Member of ower Preen Mehod Preen Mehod wh wh ll ll ll erm erm e whou del δ del whou e ε Mone Clro Clro Sm. Sm. men-σ men men+ Preen Preen Mehod Mehod 9wee wh ll ll erm 9.wee whou del δ 9.wee whou e ε Mone men- Clro Sm. men-σ men men+ on of he modl ombnon rule o evlue mxmum reone of eondry yem from em nu gven by he fr mehod. he rooed rule ued for he evluon of he relve modl eleron nd he nu eleron, nd wo knd of orrelon re ondered: one mong he relve modl reone nd one beween he relve modl reone nd nu eleron. o demonre he rooed mehod, he uhor led he rooed mehod o eleommunon ower bul on he buldng. he reul how h he numerl oluon by he rooed mehod mlr o he reul of Mone Clro mulon (MCS) exe he e n whh he modl frequeny beween he eondry nd rmry yem loe. In he exeon e, he mehod gve onervve reul. A hown n reul, he orrelon oeffen beween he relve reone nd nu eleron effeve o evlue ol reone of eondry yem urely. REFERENCES Amn, M. nd I. Gungor (97): Rndom vbron n em nly-an evluon, Pro. ASCE Nl meeng ruurl Engneerng, MD, 9-3. Gu, A. K. (99): Reone Serum Mehod, Blkwell Senf ublon. Hdn, A. H. (98): Sem reone of ruure by he reone erum mehod, Nuler Engneerng nd Degn, 66,.79-. Kul, M. K. (978): Soh hrerzon of erhquke hrough her reone erum, Erhquke Engneerng nd Sruurl Dynm, Vol.6, Der Kureghn, A. (98): A reone erum mehod for rndom vbron nly of MDF yem, Erhquke Engneerng nd Sruurl Dynm, Vol.9, Der Kureghn, A. nd Y. Nkmur (993):CQC Modl Combnon Rule for Hgh-Frequeny Mode, Erhquke Engneerng nd Sruurl Dynm, Vol., Roenblueh, E. nd J. Elorduy (969): Reone of lner yem o ern rnen durbne, 4h World Conferene on Erhquke Engneerng, Vol., Sngh, M. P. nd K. B. Meh (983): Sem Degn Reone by n Alernve SRSS Rule, Erhquke Engneerng nd Sruurl Dynm, Vol., Sngh, M. P. nd L. E. Surez (986): A erurbon nly of he egenroere of equmen-ruure yem, Nuler Engneerng nd Degn, 97, Sngh, M. P. nd L. E. Surez (987): Sem reone nly of ruure-equmen yem wh non-ll dmng effe, Erhquke Engneerng nd Sruurl Dynm, Vol.5, Unruh, J. F. nd D. D. Kn (98): An erve roedure for he generon of onen ower/reone erum, Nuler Engneerng nd Degn, 66, Wlon E.L., A. Der Kureghn nd E.P. Byo (98): A relemen for he SRSS mehod n em nly, Erhquke Engneerng nd Sruurl Dynm, Vol.9,