An Analysis of Load Uncertainty on Axisymmetric Structures: A Case study on Conical Water Tanks.

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1 ISSN N: Ittil Jul f Advcd Egiig d Glbl Tchlgy Vl-2, Issu-6, Ju 24 A Alysis f Ld Uctity Axisytic Stuctus: A Cs study Cicl Wt Tks. Mh.G Pfss d Hd, Dptt f Civil Egiig, Mtusi Egiig Cllg, Hydbd, Idi. Eil: hgp@yh.c R Rddy.G Pfss d Hd, Dptt f Civil Egiig, CMR Istitut f Tchlgy, Hydbd, Idi. Eil: gtu_24@yh.c.i Abstct Cicl wt tk ws lysd usig fuzzy fiit lt thd i d t vlut th ffct f ld uctity th stuctul sps. Mthd f clcultis d sults discussd. A tigul bship fucti is chs t dfi ld uctity. Sttic sps f th tk t th fuzzy ldig ws btid i ts f fuzzy displcts, d th sults w futh pcssd t bti fuzzy itvl vlus f hp fcs, idil fcs d sh fcs. Ssitivity lysis ws pfd t vlut th ltiv uctitis f stuctul sps qutitis, viz. displcts, ttis d fcs t giv d s wll s v st f giv ds. Th pst wk dstts th ffctivss f fuzzy fiit lt dl t vlut th stuctul sps i th psc f ld uctity. Th usfulss f dtild ssitivity lysis i d t udstd th ltiv ssitivity f stuctul sps is ls highlightd. Kywds: Displcts, fcs, ssitivity, Stiffss tix, d uctity. Itducti Alysis d dsig f stuctus ccupis iptt plc i th fild f civil giig. Md dy stuctus usully cplx i gty d d f cbiti f svl tils, such s cct d stl. Dsig f stuctul syst quis tht th pfc f th syst b gutd v its lifti. Accdigly, th bjctiv f stuctul lysis is t pdict th bhvi f th stuctu duig its stitd liv pid ccutly. Thugh clssicl fiit lt lysis is pstly th st ppul tl f th lysis f stuctus, it is t suitd t hdl uctitis i th dsig vibls f th stuctul syst. Ay viti i th sps f th stuctu pdictd usig th thticl dl d th sps f th physicl stuctu culd b du t uctitis ivlvd i til d gtic pptis, svic lds d budy cditis. Th s du t bv uctitis c ith b hdld liitd by th us f clssicl fiit lt lysis. Thus, th is d t ipv th xistig thticl dl f th stuctu i d t big its bhvi s cls s pssibl t th ctul bhvi f th physicl stuctu t hc th fuctility d dpdbility f th thticl dl f th stuctu. Clssicl fiit lt dl f lysis ds t tuly pst th dg f ipcisi uctity ivlvd i th physicl dl f th pbl. Thf, it is cssy t iclud ths uctitis i lysis i d t vlv fid d libl dl. Thus, th is d f uctity-bsd dl f lysis d f giig stuctus. Littu Rviw Muh d Mull hdld uctity i chics pbls by usig itvl-bsd ppch. Elt-by-lt (EBE) tchiqu ws plyd t bti vy shp clsu f th fuzzy sluti by liitig th sucs f vstiti. A cbl-styd bidg ws dld d lysd usig fuzzy-fiit lt lysis by R R d Rsh Rddy 2. A ctiuu stuctu c b csidd f studyig th sps f uctity f ipsd lds. A cicl wt tk ws chs s xpl 766

2 ISSN N: Ittil Jul f Advcd Egiig d Glbl Tchlgy Vl-2, Issu-6, Ju 24 pbl. Uctity f ipsd ld (wt) ws itducd by tigul bship fucti. Sttic sps f th tk t th fuzzy ldig is btid i ts f fuzzy displcts. Th sults dstt th ffctivss f itducig fuzziss i th lysis f difft pfils f th cicl tks. I fiit lt lysis f pblic wt tk, Mh d Pksh R 4, discussd i dtil but th stuctul bhvi d its ssitivity. Th ccpt f hdlig ipsd ld uctity usig fuzzy-fiit lt thdlgy is pplid i th cs f th lysis f cicl wt tks. Th cut wk psts study f th ffct f fuzziss i ipsd ld f cicl tk i ts f displct, tti d fcs, by tkig up cs study. Th ssitivity f stuctul sps t th viti f ld uctity is xpld. Fulti f li itvl qutis Th stiffss qutis i fuzzy stuctul syst with cisp til ppty d is subjctd t fuzzy itvl ipsd lds t lvl β giv by [K] {U β } = {P β } () Wh [K] is cisp stiffss tix,{u β } is th fuzzy-itvl displct vct cspdig t liv ld uctity t lvl β d {P β } is th fuzzy-itvl ipsd ld vct t th lvl f uctity β. Th st f qutis pstd by Eq. () givs syst f li itvl qutis i th psc f liv ld (Muh d Mull, 999). Sluti f this syst f itvl qutis dictly by Gussis liiti lds t vstiti. Svl thds vilbl i littu t vid vstiti i th sluti f li itvl tix qutis (Hs, 965, Jss, 99, Muh d Mull, 999). I pticul, Muh d Mull, 999 hdld th pbl f dpdcy by ultiplyig ll th itvl vlus fist d pfig ptis fuzzy vlus s fil stp. Th itvl bdy-fc vct {P β } is csidd s th pduct f -fuzzy tix [M] d itvl lt liv ld vct {q β } s {P β } xl = [M] x {q β } xl (2) Wh is th kitic idticy f th stuctu d is th ub f lts i th stuctu. Th j th clu f [M] tix ctis th ctibuti t lt fcs du t uit liv ld th j th lt f th stuctu. Substitutig Eq.(2) i Eq.() [K] {U β } = [M] {q β } (3) Th fuzzy dl displct vct {U β } is btid s {U β } xl = [[K] - x [M] x ] {q β } xl (4) Thus ll -itvl vlus ultiplis fist, d th lst ultiplicti ivlvs th fuzzy-itvl qutitis. If this d is t itid, th sultig itvl sluti will t b shp (Muh d Mull 999). I cs th stuctu is ctd up by st f fuzzy pit lds, th pit ld vct {P } xl is giv by {U β } xl = [K] - x {P β } xl (5) Wh P i = [P il, P iu ] is th fuzzy pit ld pplid t th i th dg f fd. Pst Pcssig f Sluti F pst-pcssig f th sluti, th fllwig ppch is ppsd i th pst study: Aft slvig f th fuzzy displct vct {U β } xl (piy ukw), th vlus f fuzzy xil fc, fuzzy sh 767

3 ISSN N: Ittil Jul f Advcd Egiig d Glbl Tchlgy Vl-2, Issu-6, Ju 24 fc d fuzzy bdig t (scdy ukws) btid by pst-pcssig th sluti t th lt lvl. Csidig th quti f quilibiu t th lt lvl, [K] () d x d [T] () d x d {U β } () d x l = {P β } () d x l + {g} () d x l (6) Wh d is th ub f dgs f fd f th lt [T] () is th tsfti tix f th lt d {g} () is th vct f lt-d fcs. I cs f pl f lt, {g} () csists f {-A, -V, -M, A 2, V 2, M 2 } T wh A, A 2, V, V2, M, M 2 th xil fcs, sh fcs d bdig ts t th fist d d th scd d f th lt spctivly. Nw, th displcts pppit t giv lt xtctd s {U β } () d x l = [L] d x + {U β } xl (7) Wh [L] d x is Bl cctivity tix ctiig s d s. Substitutig Eq. (4) d Eq. (7) i Eq. (6) f pit lds, [K] () [T] () [L] [K] - [M] {q β } = {P β } () + {g} () (8) Substitutig Eq. (5) d Eq. (7) i Eq. (6) f pit lds, [K] () [T] () [L] [K] - {P } = {P β} () + {g} () (9) F Eq. (8) d Eq.(9) {g} () is cputd d th fuzzy vlus f A, A 2, V, V 2, M, M 2 btid f giv lt. Cs study Th cs study csidd h is vhd wt tk with vibl cicl ut shll. Th stuctul lts f th tk csist f cicl f, tp ig b, cicl shll, btt ig b, cicul d d cylidicl shll. Th gty d th pts f tk with 3 kl w shw i Fig.. Th dt ws pstd i Tbl. Tbl. Disis f th cicl tks f 3-killit cpcity (Rf Fig.) SN Shll () Dpth f wt H () Agls f icliti with hiztl i dgs θ θ 2 θ 3 θ 4 Disis f ig bs, Tp Btt Bx5 Bx

4 ISSN N: Ittil Jul f Advcd Egiig d Glbl Tchlgy Vl-2, Issu-6, Ju 24 Z t 4 3 B 2 t z III t 6 H t2 z II.75 t t 3 z I B Figu. Pts f cicl wt tks (Axisytic) (H t =5, t 2 = 2, t 3 =3, d t 4 =t 5 = t 6 =5 ) Z = 5 = 6 = 7 Figu 2. Pfils (Gty) f cicl wt tks Th ut cicl shll ws pstd by x =, wh csidd s shll cstt, is dil xis d z is xi-sytic xis. Difft pfils f th tks w lyzd by chgig th cicl shll cstt f 5 t 7 t itvl f, kpig th pfils f th cpts ultd. Th pfils s shw i Fig.2. Th dpths f wt (H) quid f difft pfils f tks w clcultd 5. Th dpth f wt chgs with th pfil f ut shll. A typicl dl f th tk is shw i Fig

5 ISSN N: Ittil Jul f Advcd Egiig d Glbl Tchlgy Vl-2, Issu-6, Ju 24 Z H Figu 3. Typ f Shll Stuctu usd i th wk f th lysis 36 Th tks w lysd 6,7,8 f slf-wight s wll s ipsd fcs. F xpl, th wt dpth f 5.7 (H) f tp ig b is csidd f th tk with x = 6 s its ut shll pfil f 3 killit cpcity. Mbship fucti f ipsd ld uctity β dptd f th tk is idictd i Fig Bt Nlisd Ld Bt Hp Fc (kn/) Fig.5 psts th bship fucti f hp fcs t d 34, t which xiu hp ccus. As 77

6 ISSN N: Ittil Jul f Advcd Egiig d Glbl Tchlgy Vl-2, Issu-6, Ju 24 th xiu hp idtifid t d 34, th th spss studid t th s d ly. Fig.6 psts th bship fucti f idil fcs i d 34. Th sh fcs t d 34 w btid with gligibly sll qutitis Bt Midil Fc (kn/) Figu 6. Mbship Fucti f Midil fc i 34 th lt.6e-3.75 Bt E-4.2E E-4.9E E-4.26E-3 8.3E-4.32E- 7.5E-4 9.5E-4.5E-3.35E-3 Di l t i Fig.7 psts th bship fucti f displct i hp dicti t d 34. It is bsvd i ll th css tht th bship fuctis tigul. Ssitivity Alysis Ssitivity Alysis is t lyzig th ltiv viti f stuctul sps t giv viti f stuctul chctistics. Fig.8 dpicts th ltiv ssitivity f fcs t 34 th lt ud th cti f ipsd ld with spct t pctg viti f wt ld but its vlu. 77

7 ISSN N: Ittil Jul f Advcd Egiig d Glbl Tchlgy Vl-2, Issu-6, Ju 24 v i t i b u t Mxiu Hp fc Pctg viti f ld but Figu 8. Ssitivity Alysis f fcs V i ti b u t Pctg viti f ld but Hiztl displct Figu 9. Ssitivity Alysis f Displcts Fig.9 shws th ltiv ssitivity f displcts t 34 th lt ud th cti f wt ld with spct t pctg viti f wt ld but its vlu. Gtst ssitivity t th viti f th ld (f th ) is xhibitd by sh fc d tti whil; fcs d displcts i hp dicti xhibit th lwst ssitivity

8 ISSN N: Ittil Jul f Advcd Egiig d Glbl Tchlgy Vl-2, Issu-6, Ju 24 v i ti b u t 3 2 Shll t = 5 Shll t = 6 shll t = Pctg viti f ld but Figu. Ssitivity Alysis f Hp Fc i 34 th Elt V i ti bu t Shll t = 5 Shll t = 6 Shll t = P ctg viti f ld but Figu b. Ssitivity f Displcts i Hp Dicti V i ti b u Shll t = 5 Shll t = 6 shll t = t Figu. Ssitivity Alysis f Hp fc, Displcts & Midil fc t 34 th lt

9 ISSN N: Ittil Jul f Advcd Egiig d Glbl Tchlgy Vl-2, Issu-6, Ju 24 Fig. psts th ssitivity f difft pfils i hp fc & its displcts d idil fc f 3 kl cpcity f th tk. It is ls bsvd tht th pfil f x = 6 is shwig lwst ssitivity, cpig with th th pfils. Cclusi I this pp, th itvl-bsd ppch ppsd by Muh d Mull hs b dptd t tk it ccut th uctity i wt ld. Th thd is illusttd by pplyig it t th pbl f cicl wt tk. A w thdlgy is ppsd f pst-pcssig th sluti. Ssitivity lysis is fud t b usful tl t vlut th ltiv ssitivity f stuctul sps i th psc f ld uctity. Rfcs. Muh R.L. d Mull R.L., 2, Uctity i Mchics Pbls-Itvl-Bsd- Appch, Jul f Egiig Mchics, Vl.27, N.6, pp R R M.V. d Rsh Rddy R., 24, Alysis f Cbl-Styd Bidg by Fuzzy Fiit Elt Mdlig, Ittil Cfc Advcs i Cct d Cstucti, Hydbd, Idi, Dc Jss. C., 99, Itvl Li Systs with sytic tics, skw-sytic tics, d dpdcis i th ight hd sid, Jul f Egiig Cputig, Vl 46, pp Mh G, d Pksh R D.S., Fuzzy Fiit Elt Alysis f Pblic Wt Tks, Ntil Cfc Egig Tchlgis d Dvlpts i Civil Egiig, Avti, Mhst, Idi, 2-23, Mch,27 pp v Gwl, G.S, High giig thtics, Jh Wily d Ss, Zikiwicz, O.C d Tyl, R.L, Fiit lt thd, McGw Hill Publictis, Nw Yk, 2, pp Kishthy, C.S., Fiit lt lysis (thy d pgig), Tt McGw Hill Publishig Cpy Ltd. Nw Dlhi, 2 pp Ck, R.D., Mlkus, D.S., d Plsh, M.E, Ccpts d pplictis f fiit lt lysis, Jh Wily d Ss, 997, pp