Southrn Cross Unvrsty Publcatons@SCU 23rd Australasan Confrnc on th Mchancs of Structurs and Matrals 24 Spctral stochastc fnt lmnt analyss of structurs wth random fld paramtrs undr boundd-but-uncrtan forcs D M. Do Unvrsty of Nw South Wals W Gao Unvrsty of Nw South Wals C M. Song Unvrsty of Nw South Wals Publcaton dtals Do, DM, Gao, W, Song, CM 24, 'Spctral stochastc fnt lmnt analyss of structurs wth random fld paramtrs undr boundd-but-uncrtan forcs', n ST Smth (d.), 23rd Australasan Confrnc on th Mchancs of Structurs and Matrals (ACMSM23), vol., Byron Bay, NSW, 9-2 Dcmbr, Southrn Cross Unvrsty, Lsmor, NSW, pp. 825-83. SBN: 978994528. Publcatons@SCU s an lctronc rpostory admnstrd by Southrn Cross Unvrsty Lbrary. ts goal s to captur and prsrv th ntllctual output of Southrn Cross Unvrsty authors and rsarchrs, and to ncras vsblty and mpact through opn accss to rsarchrs around th world. For furthr nformaton plas contact pubs@scu.du.au.
23rd Australasan Confrnc on th Mchancs of Structurs and Matrals (ACMSM23) Byron Bay, Australa, 9-2 Dcmbr 24, S.T. Smth (Ed.) SPECTRAL STOCHASTC FNTE ELEMENT ANALYSS OF STRUCTURES WTH RANDOM FELD PARAMETERS UNDER BOUNDED-BUT- UNCERTAN FORCES D.M. Do* School of Cvl and Envronmnt Engnrng, Th Unvrsty of Nw South Wals Sydny, NSW, 252, Australa. duy.m.do@gmal.com (Corrspondng Author) W. Gao School of Cvl and Envronmnt Engnrng, Th Unvrsty of Nw South Wals Sydny, NSW, 252, Australa. w.gao@unsw.du.au C.M. Song School of Cvl and Envronmnt Engnrng, Th Unvrsty of Nw South Wals Sydny, NSW, 252, Australa. c.song@unsw.du.au ABSTRACT Th nvstgaton of ntrval tchnqu arsng n th contxt of th spctral stochastc fnt lmnt mthod (SSFEM) s nsprd by th substantal ffct producd by unavodabl dffrnt typs of uncrtants on structural bhavour. Ths papr prsnts th study on non-dtrmnstc problms of structurs wth random fld and ntrval paramtrs undr uncrtan-but-boundd forcs. ntrval approachs ar ntgratd nto SSFEM to handl th mxd uncrtants from structural paramtrs and loads. Th probablstc framwork s xtndd to account for ntrval forcs. Man valu and standard dvaton of th random ntrval structural rsponss ar not dtrmnstc valus but ntrvals. Quas-Mont Carlo mthod s ncorporatd wth SSFEM to dfn th chang rangs of statstcal momnts of th structural rsponss. Th fasblty and ffctvnss of th proposd mthod ar llustratd by numrcal xampls. KEYWORDS SSFEM, random-fld paramtrs, mxd uncrtants, uncrtan-but-boundd forcs. NTRODUCTON n an ffort to stablsh th mathmatcal modl for structural analyss and dsgn n accordanc wth th undrlyng ral-world nformaton, uncrtanty consdraton for dsgn paramtrs must b takn nto account n practcal dsgn of ral ngnrng problms. Whl th probablstc mthod s th ratonal tratmnt of thos uncrtants, applcaton of a random fld smulaton allows th spatal fluctuaton of th structural proprts to b prformd n a mor ralstc approach. A powrful tool n computatonal stochastc mchancs adoptng th framwork of a random fld modl s th spctral stochastc fnt lmnt mthod, dnotd as SSFEM, ntroducd by Ghanm and Spanos (23). Th ffcncy of ths mthod s provd by many studs n ngnrng problms and vdncd n a gnral rvw wth rgards to ts dvlopmnt n th past, prsnt and futur (Stfanou 29). n many ral safty dsgns, ngnrng problms can lad to varatons for varous structural paramtrs. Rcntly, dffrnt proprts of structurs hav bn charactrzd as randomnss nvolvng random varatons of multpl matral proprts and/or gomtrc proprts (.g. Graham and Dodats 2; Stfanou and Papadrakaks 24). Ths studs hav nsprd th uncrtanty modllng for dffrnt Ths work s lcnsd undr th Cratv Commons Attrbuton 4. ntrnatonal Lcns. To vw a copy of ths lcns, vst http://cratvcommons.org/lcnss/by/4./ 825
paramtrs n SSFEM to construct approprat mathmatcal modl corrspondng to ral ngnrng problms. Morovr, th ssu of mployng sutably mathmatcal modl dpnds on assssmnt of th sourcs of th uncrtanty n ach partcular cas. Th nvstgaton of ntrval tchnqu arsng n th contxt of th spctral stochastc fnt lmnt mthod (SSFEM) s motvatd n th cas that som dsgn paramtrs can b modlld by stochastc fld wth suffcnt statstcal nformaton whl ntrval arthmtc s mployd to dscrb uncrtanty for th othrs du to th lack of nformaton n rgards to charactrzng randomnss. A nw hybrd modl, n whch th ntrval arthmtc s combnd wth stochastc functon n th contxt of SSFEM, s proposd n ths papr to handl th mxd uncrtants from structural paramtrs and loads whch has not bn studd bfor. Th probablstc framwork s xtndd to account for ntrval forcs. Thus, man valu and standard dvaton of th random ntrval structural rsponss ar not dtrmnstc valus but ntrvals. Quas- Mont Carlo mthod, dnotd as QMCM s ncorporatd wth SSFEM to dfn th chang rangs of statstcal momnts of th structural rsponss. Th nxt scton prsnts th ncorporaton of th gnral formulaton for th hybrd modl and th numrcal xampl thn s mplmntd to nvstgat th possbl rang for th output of ntrval-ssfem. Th prsntaton nds wth a concludng dscusson. NTERVAL SPECTRAL STOCHASTC FNTE ELEMENT ANALYSS Ths scton prsnts th gnral mathmatcal xprssons for th nw hybrd modl of uncrtanty n th contxt of spctral stochastc fnt lmnt mthod n whch th random fld smulaton s combnd wth ntrval arthmtc to accommodat th uncrtanty paramtrs. For matral proprty consdrd as uncrtanty, consttutv matrx s hrn xamnd nto th cas of combnaton of dffrnt uncrtanty modls. t not only taks nto consdraton th spatal varaton of th lastc modulus, but also th uncrtanty of th Posson s rato. Spcfcally, th uncrtanty for consttutv E x, and ntrval rang matrx s prformd by th random fld for lastc modulus, symbolzd as for Posson s rato, dnotd as. s th ntrval varabl of Posson s rato and ts chang rang s rstrctd by th lowr bound valu and uppr bound valu. Thus, th consttutv matrx bcoms a hybrd functon, D x. Th random fld for lastc modulus s assumd as a Gaussan fld prsntd by usng K-L xpanson: Ex, E W x E x () whr E s th man valu of lastc modulus;,2, ar uncorrlatd random varabls; and x ar th corrspondng gnvalus and gnfunctons of th autocovaranc functon C x, x 2 whch s dfnd on th doman. quaton and x can b obtand by mans of th ntgral C x, x2 x dx x 2 (2) Consdrng th combnaton btwn random fld and ntrval uncrtanty for th consttutv matrx, th stffnss matrx of lmnt s a hybrd matrx of uncrtanty xprssd as, k B D x Bd T T T B D B d W x B D B d k k whr D s th ntrval consttut matrx corrspondng to th man valu of lastc modulus; k s th ntrval man stffnss matrx wth rspct to D and dtrmnd by (3) k ar th ntrval stffnss matrcs ACMSM23 24 826
k T W x B D B d (4) By dnotng and assmblng lmnt stffnss matrx, th ntrval stochastc global stffnss matrx can b xprssd as E E K k k K (5) Th systm of quatons rsultng from a lnar fnt lmnt analyss can b wrttn n th followng form K U F (6) Th appld forcs consdrd hrn ar charactrzd as uncrtanty but ntrval. F s th ntrval vctor of appld forcs. Du to th hybrd modl of uncrtanty from matral proprty and loads, th dsplacmnts ar accordngly rprsntd by th hybrd modl combnd btwn th ntrval uncrtanty and randomnss. n th sprt of SSFEM, th ntrval random dsplacmnt fld s dscrbd by usng th Polynomal Chaos Expanson (PCE) whr j U U j j (7) dnots th srs of random Hrmt polynomals whl U j coffcnts for PCE. n ths work, U j Substtutng Eq. 5 and Eq. 7 nto Eq. 6 ylds j s th vctor of conssts of th boundd-but-uncrtan paramtrs. K U j j F (8) j Consdrng K trms for th KL xpanson and ordr p th for th PCE, and thn mnmzng th rsdual n th man squar sns (Ghanm and Spanos 23), Eq. 8 bcoms whr cjk and Fk ar spcfd as K P cjk K U j Fk, k,, P (9) j cjk E j k () F k () E F k whl c jk ar avalabl n tabulatd form (Ghanm and Spanos 23). Du to th mxd uncrtants comprsng stochastc functon and ntrval uncrtanty, th rsponss of th hybrd modl n ths work ar th ntrval man valu and ntrval covaranc matrx of dsplacmnts spcfd as E U U (2) P 2, j j Cov U U E U U (3) Th Quas-Mont Carlo mthod, dnotd as QMCM, n whch samplngs of smulatons gnratd by usng th low-dscrpancy squncs, s ncorporatd wth nw hybrd modl of SSFEM to prdct th chang rangs of rsponss n ths papr. NUMERCAL EXAMPLE n ths scton numrcal tst s prformd usng th algorthms prsntd n th prvous scton. Th covaranc functon for th random fld s xprssd as th followng C x x x x2 y y2 2 Lcx Lcy, 2 (4) T ACMSM23 24 827
.44.44.8.8 whr, L cx and L cy ar accordngly th standard dvaton, corrlaton lngth n x-drcton and corrlaton lngth n y-drcton, dfnd as L cx = L cy =.4 (m). Th coffcnt of varaton consdrd n ths tst cas s.5. A closd-form rprsntaton of th gnvalus and gnfunctons of abov covaranc functon s avalabl n th ltratur (Ghanm and Spanos 23). 4 trms n KL xpanson and ordr 3 th n PCE ar consdrd for ths numrcal tst. Th gomtry of th plat subjctd to th ntrval forc and ts mshng ar shown n Fgur. Th doman of plat s dscrtsd ovr a msh wth 9 nods and 72 four-nodd quadrlatral lmnts. A sampl ralzaton showng th spatal varatons of random fld s dscrbd n Fgur 2. Th Posson s rato of plat s assumd as ntrval varabls 6 pos pos 6 n Fgur (b). All of pos, dscrtsd nto mshng of th plat and numbrd from to hav th sam chang rang dfnd as [.27,.3]. Dtrmnstc valu of q s - 8 MPa and ts ntrval chang rato s.25. Th targt solutons nvstgatd hrn s th man valu and standard dvaton (STD) of dsplacmnts at pont A, B, and C n X and Y drctons, dnotd as AX, AY, BX, BY, CX, and CY. Th rsults drvd by mans of smulatons for QMCM ar shown n Tabl whl th dformd shap of plat corrspondng to th lowr and uppr man valus of som targt ponts ar prsntd n Fgurs 3 and 4. smulatons for Mont Carlo smulaton mthod, dnotd as MCSM ar also mplmntd to vrfy th rsults drvd by QMCM..48 A B C q 4 4 4 2 2 2 2 3 3 2 2 2 2 3 3 3 2 2 2 2 3 3 3 3 4 4 4 5 5 5 5 6 6 6 4 4 4 5 5 5 5 6 6 4 4 4 5 5 5 5 6 6 6 6 (a) Gomtry of th plat. Unt (m) (b) Dscrtzaton of th doman Fgur. Numrcal modl Fgur 2. A ralzaton of th random fld ACMSM23 24 828
Tabl. Prdcton of chang rangs for targt solutons. Unt (mm) Mthod QMCM MCSM QMCM MCSM Targt soluton Pont A Drcton X X Y Y Man Uppr -.77556 -.7755.2237.223796 Lowr -.29393 -.29462.73993.732376 STD Uppr.5298.5379.39577.39654 Lowr.94.993.83566.83587 Targt soluton Pont B Drcton X X Y Y Man Uppr -.5694 -.57.8275.835 Lowr -.86277 -.86337.7862.7866 STD Uppr.88.95.36237.36326 Lowr.65.653.8634.8587 Targt soluton Pont C Drcton X X Y Y Man Uppr -.2857 -.28539.5574.5676 Lowr -.47694 -.47729.69273.69267 STD Uppr.5753.57554.3393.3444 Lowr.34455.3449.8276.826 (a) Lowr man valu - AX (b) Uppr man valu - AX (a) Lowr man valu - AY (b) Uppr man valu - AY Fgur 3. Boundd man valu of dsplacmnts of pont A. Unt (mm) ACMSM23 24 829
(a) Lowr man valu - CX (b) Uppr man valu - CX CONCLUSONS (a) Lowr man valu - CY (b) Uppr man valu - CY Fgur 4. Boundd man valu of dsplacmnts of pont C. Unt (mm) Ths papr prsnts th novl mxd modl of uncrtanty n whch stochastc functon combnd wth ntrval arthmtc, contrbutng to th dvlopmnt of challngng n rgards to complx modl for stochastc mchancs and thus allowng th structurs to b smulatd n a mor ralstc approach. Th chang rangs of statstcal momnts of th structural rsponss, namly, man valu, and standard dvaton ar vard du to th mxd uncrtanty of paramtrs nput. Th Quas-Mont Carlo mthod s ffctvly ncorporatd wth proposd framwork of study to nvstgat thos statstcal rsponss whn th rsults drvd from QMCM ar n good agrmnt wth thos from MCSM for th sam smulatons. t s suggstd to mploy QMCM rathr than MCSM as QMCM ylds much bttr rror bounds compard to th probablstc Mont Carlo rror bounds and usually lads to a fastr rat of convrgnc than a corrspondng Mont Carlo mthod (Ndrrtr 992). Ths smulatons wr carrd out on a computr quppd wth a 3.3 GHz 5-25 CPU. Th tm consumng of QMCM and MCSM for targt rsponss ncludng th lowr and uppr bounds ar smultanously 7557.6 sconds and 776.66 sconds. Futur work can xplor th possblty of xtndng th analyss wth advancd mthods to mtgat th tm consumng compard to QMCM or MCSM. REFERENCES Ghanm, R. G. and Spanos, P. D. (23) Stochastc fnt lmnts: a spctral approach, Courr Dovr Publcatons. Graham, L. L. and Dodats, G. (2) "Rspons and gnvalu analyss of stochastc fnt lmnt systms wth multpl corrlatd matral and gomtrc proprts." Probablstc Engnrng Mchancs, Vol. 6, No., pp. -29. Ndrrtr, Harald. (992) Random numbr gnraton and Quas-Mont Carlo mthods, SAM. Stfanou, G. (29) "Th stochastc fnt lmnt mthod: past, prsnt and futur." Computr Mthods n Appld Mchancs and Engnrng, Vol. 98, No. 9, pp. 3-5. Stfanou, G. and Papadrakaks, M. (24) "Stochastc fnt lmnt analyss of shlls wth combnd random matral and gomtrc proprts." Computr Mthods n Appld Mchancs and Engnrng, Vol. 93, No. -2, pp. 39-6. ACMSM23 24 83