Reliability of a silo structure with initial geometric imperfections loaded with pressure below atmospheric and wind

Transcription

1 Rliability of a silo structur with initial gomtric imprfctions loadd with prssur blow atmosphric and wind Magdalna Gołota, Karol Winklmann, Jarosław Górski and Tomasz Mikulski Dpartmnt of Structural Mchanics and Bridgs, Gdańsk Univrsity of Tchnology ul Narutowicza 11/12, Gdańsk, Poland -mail: maggolot@pggdapl Abstract A mthod of rliability stimation of shll structurs with random initial gomtric imprfctions and random loading is considrd An xampl of calculations of a silo structur is prsntd As th silo cylindrical constructions can b loadd with prssur blow atmosphric any gomtric discrpancis ar vry dangrous and can significantly rduc thir capacitis Th initial discrpancis can b dscribd using dtrministic rlations and random non-homognous corrlation functions, th paramtrs of which ar chosn on th basis of th EU cods and according to th nginring knowldg Wind loading is also dscribd as random Its magnitud and othr charactristic paramtrs and dirction ar stimatd on th basis of th allowabl data Th ralizations of th random filds and paramtrs ar gnratd using a conditional rjction mthod To calculat th silo rliability a vrsion of a stratifid sampling mthod is applid Kywords: rliability, silo structurs, random initial gomtric imprfctions, wind loading 1 Introduction Silo structurs can b mad of vry thin mtal shting Bcaus of th sht slndrnss thy can asily los both gnral and local stability Among various rasons which can lad to th rduction of th silo load capacity, initial gomtric imprfctions ar on of th most important To dscrib th filds dtrministic and random initial gomtric imprfctions wr gnratd Th silo structurs wr loadd with prssur blow atmosphric Th ngativ prssur can b causd by a suddn drop of th arching matrial or a malfunction of roof filtrs whil mptying th silo [3] Wind loading was also dscribd using probabilistic functions Its magnitud and dirction ar considrd as random On th basis of th Mont Carlo rduction mthods th silo s rliability was stimatd 2 Silo structur 3 A cylindrical aluminium silo of V = 324m capacity was considrd Th hight of th silo shll is 25m Th shll compriss 10 rings, ach 2500mm high Th silo shts thicknss ar 105mm at th bottom and 4mm at th top (total of 10 diffrnt valus s Fig 1) Th intrnal diamtr of th shll is th sam vrywhr and quals Dw = 4000mm Th roof forms a dom with th maximum hight of 536mm and 55mm of thicknss Th bottom of th silo forms a hoppr that is 3340mm high and 105mm thick Th silo s shll and th roof ar mad of AlMg3 aluminium mtal shts with th plasticity limit R,min = 70MPa whras th silo s hoppr is mad of AlMgSi1 aluminium, with th plasticity limit R,min = 120MPa Th program MSC Visual Nastran [4] was usd to carry out th analysis of th silo Four-nod shll lmnts of siz mm wr applid whr th first valu corrsponds to th vrtical dirction Figur 1: Silo gomtry (a) and th sht thicknss (b) In th aras of th silo s supporting construction, th roof and th hoppr, th finit lmnt msh sizs wr appropriatly smallr In total, thr wr shll lmnts and nods Th aluminium lasticity modulus E = 69GPa and Poisson s ratio ν = 03 wr assumd A corrsponding modl was computd in a commrcial program SOFiSTiK [5] to assur that th numrical rsults ar trustworthy Th SOFiSTiK modl consists of shll lmnts and nods

2 3 Gomtric imprfctions 31 Dtrministic dscription of imprfctions in Polish Standards (PN) and in Eurocod 3 (EC) Th first stp in th gnration of any imprfctions in a structural dsign is to mak initial assumptions about thir siz Th adaptation of imprfctions on an idalizd latral surfac of th structur nds to rly on a st of normativs, which rgulat th probabl dimnsions of imprfctions that can naturally appar on a calculatd silo Initially, th first assumptions for numrical computations hav to b stimatd on th basis of formulas givn in th dsign guidlins Currntly, thr is a possibility to altrably conduct th dimnsioning basing on Polish Standards (PN) or th EC 3 To masur th indntations on can us simpl survys in crtain positions of both main dirctions of silo s latral surfac ithr mridional or circumfrntial dirction In calculations don on th basis of dsign cods it is initially assumd, that th limit siz of a dnt t v0 quals 2cm, whras th maximum lngth of covrag of this indntation l m quals 2m, as suggstd by th formula: tv0 = 001 lm In cas of aras, whr th mridian comprssiv strsss appar, indntation masurmnts along th cylindr s gnratrix (in mridian dirction) hav to b conductd in both of th main dirctions, using survys of a lngth dtrmind by th formula (1): lgx = 4 rt (1) Th dpth of th initial dnt is assssd as dpndant on th rlativ indntation paramtr U, dfind by th formula (2): U w = U 0x 0 x gx Thrfor: ω = 2cm (as assumd in th first stimation) So, th lngth of an indntation along th cylindr s gnratrix quals, for xampl: = 4 Rt = 4 2 0,004 = 0,357m dla t = 0,004m Aftr calculating th rvisd dpth of th initial indntation ω, on can obtain: w U, rsulting in th data for a givn sction of th silo, as prsntd in Tabl 1 Thus, according to Eurocod 3 basd calculations, on obtains a maximum lngth of a mridian indntation qual to: ω, C class = 0,57cm 0,93cm for a silo of a C class rction standard, which is a lssr valu, than th initially assssd on In cas of aras, whr th circumfrntial comprssiv strsss or shar strsss appar, indntation masurmnts along th cylindr s primtr (in circumfrntial dirction) hav to b carrid out using survys of a lngth dtrmind by th formula (3): 2 0,25 2,3 ( Rt), but with rspct to: g R (3) = whr: is th mridian lngth of shll s sgmnt, = 2πR = 2π 2m= 12,566m According to PN-B-03202:1996 th calculations mntiond abov ar don using a diffrnt formula (4): 2,3R lm = R (4) R R l t θ (2) Yt, this is just a rcast vrsion of th Eurocod formula, and it s giving idntical rsults Th dpth of th initial dnt is in this cas assssd as dpndant on th rlativ indntation paramtr U0 θ, dfind by th formula (5): w0 θ U0 θ = U (5) Aftr calculating th rvisd dpth of th initial indntation ω 0θ, on may obtain: w0 θ U, rsulting in th data for a givn sction of th silo, as prsntd in Tabl 2 for th first sction of construction in qustion Thus, according to Eurocod 3 basd calculations, on obtains a maximum lngth of a circumfrntial indntation qual to: ω0 θ, C class = 3,90cm 4,97cm for a silo of a C class rction standard, which is a gratr valu, than th initially assssd on Summarizing, th mthods of calculation of th indntations dimnsions ar idntical in Polish and Europan dsign cods and ar giving th sam rsults Howvr, in th Eurocod 3 additional conditions that may incras th dpth of th indntation ar statd (showing mor consistncy with th ralitis of nginring) Thrfor it is shown, that th first stimation th dpth of th dnt t v0 qual to 2cm and th indntation s scop lngth in both main cylindr dirctions rspctivly qual to 06m and 2m may b implmntd in th procss of computational modls gnration in numrical programs, as a fin starting point for furthr, mor accurat dlibrations 32 Random dscription of imprfctions Gomtric initial imprfctions of any shll or plat can b dscribd by mans of two-dimnsional random filds Hr, thr random filds of initial imprfctions wr analyzd: noncorrlatd fild (th whit nois fild), homognous corrlatd fild, as wll as non-homognous corrlatd fild Du to th lack of data dscribing filds of ral imprfctions, th paramtrs of th corrlat functions wr assumd a priori; howvr, ths rflct th standard valus In th first cas th initial imprfctions wr dscribd by mans of non-corrlatd random fild (i th whit nois) Gomtric dviations ar dfind in ach point irrspctivly of th rmaining ons by mans of uniform distribution For th purpos of th calculations, thr valus of th dviations t v0 in rlation to th cylindrical surfac of th shll wr assumd: 50, 25, and 20mm Th roof s and hoppr s gomtry, as wll as th lowr dg of th silo blow th lvl of th support ara wr assumd as idal Standard dviation of th fild dscribd by th uniform distribution can b calculatd according to th formula (6), (for t v0 = 25mm): 2t v σ = = = 00144m (6) Th modl including such initial imprfctions is purly thortical as, in rality, a modl containing dformations of such scal and intnsity would nvr b approvd for xploitation Howvr, an analysis of such a construction nabls us to dfin diffrncs in limit load, which appar with diffrnt modls of imprfctions

3 Tabl 1: Calculatd dpth of th initial dnt ω sct t [cm] R [m] [cm], prsntd for th first sction of th silo (from 0m to 25m blow th top dg) A class silo: B class silo: C class silo: U ω U ω U ω cm cm cm Tabl 2: Calculatd dpth of th initial dnt ω, prsntd for th first sction of th silo (from 0m to 25m blow th top dg) A class silo: B class silo: C class silo: sct t [mm] R [m] [cm] p [cm] U ω0θ U ω0θ U ω0θ cm cm cm Gnrating corrlatd random filds rquirs an advancd softwar A mthod proposd in [6] that nabls gnrating two-dimnsional and thr-dimnsional truncatd Gaussians random filds was usd Th filds can b dscribd by mans of homognous and non-homognous corrlatd functions In th algorithm th squnc mthod that uss an xpanding ara of gnratd points random variabls was applid In this way, discrt filds of larg ara (thousands of points) can b gnratd Conditional probability distributions as wll as th rjction mthod wr mployd Th input data of th algorithm ar: a covarianc matrix, xpctd valus of random variabls as wll as th nvlop of th random valus dtrmind in vry point of th fild Th nvlop of th random fild usd in this algorithm nabls gnrating sts of random variabls with boundary conditions corrsponding prcisly to th analyzd nginring objct also on th basis of xprimntal data or masurmnts of actual constructions Th algorithm gnrats circular data, which is of particular importanc whn analyzing a silo a) b) Figur 2: Ralizations of th silo initial of th initial gomtric imprfction: a) homognous fild, b) non-homognous fild In this papr two random filds wr gnratd For lack of th appropriat data th corrlation function is usually chosn arbitrarily Apparntly th function should confirm that th corrlation btwn random variabls vanishs whn th random point distanc incrass Th initial gomtric imprfctions wr dscribd by mans of th following homognous corrlatd function (7): 2 2 ( ) 2 (, ) σ xp ( β ) ( γ ) K x x = x x (7) whr: x1 and x2 ar distancs btwn th points of th fild along th horizontal and vrtical axis, σ is a standard dviation dscribing th variability of th fild, and β and γ ar th dcay cofficints In th xampl th following 1 paramtrs wr assumd: σ = m, β = γ = 2,2m For comparison th sam standard dviation as in th noncorrlatd fild (6) was takn Th following rang of truncation of th Gaussian distribution was applid: ± 3σ =± = 00433m (8) A st of ralizations initial gomtric imprfctions of th silo wr gnratd Th fild of imprfctions, for which th calculations wr mad, is illustratd by Figur 2a Th introducd dformations can dscrib th dntd mtal shting of th construction It should b pointd out, though, that th imprfctions as shown in th illustration ar of biggr scal, as, in rality, only in som placs, thir amplituds xcd th valu of 40mm (8) For such typ of imprfctions th maximum valu of ngativ prssur phom = 2282kPa Th scond analysd fild was non-homognous fild dscribd by th following function (9): K x, x = σ cos α x xp β x γ x (9) ( ) ( ) ( ) whr th following paramtrs wr assumd: σ = 00144m, 1 1 α = 02m, β = γ = 0005m In this cas th circular shap of th construction is takn into account by mans of th cosin function A modl of a fild of imprfctions is prsntd in Figur 2b On th basis of gnratd imprfctions on can conclud that this kind of a fild nabls modlling dformations rlatd to th joints mad, fusions in th mtal shting and damags xisting on whol part of th tank 2000 ralizations of initial imprfctions wr gnratd Thn, thy wr classifid according to th avrag amplitud of th displacmnts For th purposs of th calculations thr filds wr chosn: on with th lowst amplitud, on with an avrag amplitud and on with th highst avrag amplitud

4 4 Ngativ prssur loading acting on silos with gomtrical imprfctions On of th variants of th load which was adoptd for analysis is th prssur blow atmosphric, an important factor with a strong tndncy to wakn th silo s shll structur and is, in som xtrm cass, vn abl to lad to th loss of silo s stability Th ngativ prssur in th silo chambr may occur du to various diffrnt rasons: a suddn subsidnc of matrial rsiding at th top lvls of th chambr, roof filtrs malfunction during an mptying of a lak proof silo or an xplosion of dust from matrials stord insid th chambr Th valu of ngativ prssur cratd bcaus of th dust xplosion is a sum of influncs of many various factors, such as th matrial typ, its concntration and th ignition nrgy of its dust, also th plac whr th xplosion took plac, th typ and siz of th scurity flaps and th xplosion prssur growth factor Th first stag of th analysis involvd calculations mad for a silo of a prfct shll gomtry Th valu of th maximum prssur that was obtaind, i pidal = 2717kPa constitutd a rfrnc point for furthr calculations, in which diffrnt typs of gomtric imprfctions cratd additional factor rducing th load capacity of th idal shll Th nxt stag was to analyz th influnc of dviations in th form of local dnts of th shll on its load capacity It was assumd that th rmaining part of th shll was not subjctd to any dformations Th dnt was stimatd according to th standard dsign cod formula (1) (4) whr lm = 2000mm and th dpth of th dnt corrsponding to this dformation was tv0 = 20mm Calculations wr mad for tn diffrnt hight options, on which th dformation was locatd Th middl points of th dnts wr placd on th following hights: 25, 625, 875, 1125, 1375, 1625, 175, 1875, 200, 215, 225, and 235m Th lowst load valu p dnt1 = 2155 kpa was obtaind whn th dnt is placd on th hight of 215m It can b xplaind by th fact that this ara of th shll is built of th thinnst mtal shting Bsids that, th strngthning influnc of th silo s roof is not dominating For cass whr th dnt is of lliptic shap additional calculations wr mad Th sam lngth of th dnt along th primtr l m = 2000 mm as wll as a much shortr gnrating lin of th shll clos to th standard valu of 600 mm wr assumd Th calculations wr mad for thr localisations of th dnts: 200, 215, and 225 m Th maximum valus of ngativ prssur wr: p wg 2a = 2375, p wg 2b = 2355 and pwg 2c = 2388kPa, so thy wr highr than in th cas of a symmtric dformation but lowr than in th cas of th idal shll Th nxt option introducd an initial dformation in th form of two dnts situatd on th hight of h = 215 m, whr th placd dformation was last favourabl Th dnts wr locatd on two opposit points of th cross sction Th maximum valu of ngativ prssur rachd pwg3 = 2173 kpa, so it appard highr than in th cas of a singl dnt A dtaild analysis of imprfctions including a singl dnt or a group of a fw dformations rquir making an invntory of ral cass of such dformations Th maximum ngativ prssur that was obtaind for noncorrlatd fild of imprfctions, with th maximum valu of imprfctions ±50 mm, rachd p 1 = 7390 kpa, and for wn ±25 mm pwn2 = 4704 kpa Additionally, calculations for thr silo modls with maximum imprfctions qualling 20 mm, bing clos to th ons dscribd by norms wr mad: p wn3a = 41709, p wn3a = 42038, pwn3c = kpa In all th cass mntiond abov, th valu of th ngativ prssur was much highr than in th cas of th idal surfac of th shll This bing so, on can conclud that that chaotically distributd dnts strngthn th shll For th cas whn th imprfctions fild is dscribd by a homognous corrlation function th calculations ar prformd 3 tims Th ngativ prssur, which th silo is abl to withstand, with such indntations initially input, was qual to p = 2282kPa Slightly gratr valus ar obtaind for th imprfctions fild dscribd by a htrognous corrlation function 2000 gnrations of imprfctions valus ar gnratd Ths rsults ar thn classifid in trms of th avrag amplitud of displacmnts Thr filds of th lowst, avrag and maximum man amplitud ar chosn for th calculations Th limit ngativ prssur valus ar as follows: p1 = 2,55kPa, p2 = 2,544kPa, p3 = 2,4704kPa 5 Wind loading acting on silos with gomtrical imprfctions Scond typ of loads, analyzd in th prsnt work is th prssur/suction of th wind Wind load may b, according to EC3 normativs, adoptd for th calculations in thr forms as an accurat loading, xprssd with a trigonomtric functions sris formulas (a); as a sgmnt loading, whr th prvious variant is simplifid to thr zons of diffrnt valus of wind prssur p i (b); or xprssd as a loading of only on simplifid valu p 0, indpndnt from th angl of wind prssur dirction (c) Ths approachs ar illustratd in Fig 3 Figur 3: Typs of wind loading [1]: a) xact dscription trigonomtric functions, b) sgmnts loading, c) simplifid homognous loading Th first typ of loading which is bing considrd is th wind loading on th idal surfac of th silo, with no dnts In this xampl, on a modl with an idalizd gomtry, thr forms of wind loading wr imposd: an xact trigonomtric dscription, sgmnt and simplifid A st of quations ndd for th accurat modl of th wind, which was usd in th most xact numrical calculations ar as follows: Dfining th wind load charactristic valu as: p = q β C C (10) k k a) if z 0 m;10m, thn th quation stats as follows: CC( zα, ) = 1,0 ( 0, ,322cosα + 0,636cos 2α + + 0,501cos3α + 0,058cos 4α 0,128cos5α 0,034cos6 α) (11)

5 Tabl 3: Calculatd dpth of th initial dnt ω, prsntd for th first sction of th silo (from 0m to 25m blow th top dg) variant load form f crit [-] p crit [kpa] A xact loading (trigonomtric function) B sgmnt loading (3 valus of th load) C simplifid loading (1 uniform load valu) b) if z 10 m;20m, thn th quation stats as follows: CC( z, α) = (0,8 + 0,02 z) ( 0, ,322cosα + 0,636cos 2α + + 0,501cos3α + 0,058cos 4α 0,128cos5α 0,034cos6 α) (12) c) if z 20 m;30m, thn th quation stats as follows: CC( z, α) = (0,9 + 0,015 z) ( 0, ,322cosα + 0,636cos 2α + + 0,501cos3α + 0,058cos 4α 0,128cos5α 0,034cos6 α) (13) Th rsults of calculations of prcding formulas ar prsntd in Tabl 3 It is obsrvd, that th chang of th form of th load dos not significantly affct th rsults, which partly confirms th dsign standards procdurs, and also partly indicats th rationality of an assumption of th asist mathmatical form of wind prssur in th dsign procss Howvr, in computations of lss complx structurs or fragmnts of largr structurs on can us mor accurat, dtaild wind load forms, as it is shown that thy do not chang th duration of th modl calculations In furthr calculations, which ar taking into account th possibility of imprfction occurrnc, as suggstd by th mntiond normativs, a full dscription of th function of th wind was usd Anothr goal of th structural analysis, prsntd in this articl, is to analyz th critical prssur of a silo undr th wind load acting dirctly on th gomtric cntr of th st point imprfction An lliptical dnt of fixd dpth (2cm) and fixd siz (60cm high, 200cm wid) will b st in this sris of computations on th latral surfac of th silo, on som xplicit hights abov th ground lvl, similar to th xampls from th abov paragraphs, for a bttr corrlation of rsults Critical prssur as dpndnt on th location of th dsign cod suggstd dnt on slctd hights is shown in Fig 4 0 Whn obsrving th valus of critical prssur for this spcific numrical xampl, obtaind from th calculation dscribd abov, on can s that a part of th structur, locatd in th ara btwn 3 and 5 mtr blow th uppr dg of th jackt is th most snsitiv ara to th initial imprfction occurrnc This is surly bcaus of th smallst thicknss of mtal shting in ths sctions of th silo, as wll as th fact that th wind loading on ths hights has th gratst valus Dnts blow th 13 mtr (counting from th uppr dg of th jackt) dos not affct th carrying capacity of th silo significantly It bhavs undr wind loading practically in an idntical way, albit th critical prssur valu is significantly lowr than for th construction of an idal gomtry Th smallst valu of critical prssur is p = 2,02kPa in this cas Anothr important calculations sris, which ar prsntd in this articl is th analysis of th critical prssur chang in a silo undr th wind loading of a fixd valu, which is acting on th latral surfac of th silo with a singl imprfction on it Th imprfction is st on on lvl (at 350cm blow th top of th silo jackt), but it will chang its angular position in rlation to th wind prssur dirction Th indntation s location will b masurd as an angl btwn th dirction of th gomtric cntr of th lliptical indntation and th dirction of th wind Th dnt is charactrizd by th sam dimnsions as in th prvious xampls Th lvl of imprfction location has bn chosn dlibratly th impact of th wind at this godtical lvl givs th slightst critical prssur ratio Th rlationship btwn th location of a dsign cod corrsponding dnt and th silo s critical prssur is prsntd in Fig5 3 mtrs blow th uppr dg of th silo critical prssur valu critical prssur valu 1875 Figur 4: Critical prssur, as dpndnt on th location of th dsign cod suggstd indntation on slctd hights angl of dirction of imprfction's localization Figur 5: Th rlationship btwn th location of a dsign cod corrsponding dnt and th silo s critical prssur

6 Whn obsrving th valus of critical prssur for th xampl dscribd abov on can s that thr ar two zons on th silo s structur whr th wind intracts with th indntation placd on a diffrnt angl than th wind load dirction s angl clarly to th disadvantag of th construction Ths ar th rangs whr th wind blows dirctly in th cntr of th indntation, on its dg or narly byond it (angular o o diffrnc of th dirctions quals 0 40 ) and in th rang whr th wind suction is drastically rducd (angular o o diffrnc of th dirctions quals ) For othr rangs of angls btwn th dnt dirction and wind dirction no clar ffct of gomtric imprfctions on capacity of th silo is visibl th silo bhavs th sam way as a dsign with a prfct gomtry Th smallst valu of prssur which th construction is abl to withstand quals to p = 1, 21kPa, which is a valu significantly lssr than thos obtaind in th prvious sris of calculations Th final stag of th analysis of th silo will considr whthr thr is a discrnibl influnc th valu of wind loading (controlld with its rlvant componnts of charactristic load) and th rsults of th critical prssur Th following valus ar dtrmind to b variant: q k, β and C Th loadd silo has a prfct gomtry For a full mapping of th xtrm situations th load factor 10 is takn as an quity for th surfac wind loading producing a critical prssur valu qual to p = 2,59kPa Th critical prssur valus, as dpndnt on th valu of th wind load, normalizd on th basis of prvious assumptions is shown in Fig 6 normalizd critical prssur lvl Conclusions Th prsntd rsults of th numrical calculations rvald that th initial gomtric imprfctions significantly influnc th mchanical bhavior of silo structurs Th imprfctions can b modld using data dscribd by th dsign cods and probabilistic mthods Th calculations, aidd by th statistical analysis of masurd wind paramtrs hav provd th capability of th stochastic mthods Th formulation of a mthodology for th idntification and dscription of th initial gomtric imprfctions and random wind loading can nsur a bttr and much safr dsign Rfrncs [1] Dsign of stl structurs Strngth and Stability of Shll Structurs, PN-EN , 2007 [2] Gołota, M, Górski, J, Mikulski, T and Winklmann K, Influnc of gomtric imprfctions on capacitis of silo structurs loadd with prssur blow atmosphric, Shll Structurs: Thory and Applications Procdings of th 9th SSTA Confrnc, Gdańsk, Poland, CRC Prss/Balkma, Vol 2, pp , 2010 [3] Hotała, E, Load capacity of unribbd cylindrical stl silo sits, Wroclaw Univrsity of Tchnology, 2003 (in Polish) [4] MSC Nastran for Windows, Los Angls: MSC Softwar Corporation, 2001 [5] SOFiSTiK AG for Windows Structural Analysis Programs Vrsion 23 Obrschlisshim, 2007 [6] Walukiwicz, H, Bilwicz, E and Górski, J, Simulation of nonhomognous random filds for structural applications Computrs and Structurs, 64, 1-4, pp , wind load multiplir Figur 6: Critical prssur, as dpndnt on th valu of th wind load, normalizd on th basis of propr assumptions Whn obsrving th valus of critical prssur acquird in th calculations illustratd abov, on can s that th in th rang of th possibl wind loading valus, th silo s critical prssur is narly fully proportional to th lvl of th normalizd load multiplir Whn th load multiplir is rducd a hundrd tims in comparison with th normalizd factor 10 th failur stat causing prssur is so larg that any furthr rduction in th valu of th wind factor has no ffct on th stability of th structur Howvr, at ach stp of a succssiv wind valu incras th critical prssur, causing th failur of th structur can b xpctd to lssn