Analysis of Truss Structures with Uncertainties: From Experimental Data to Analytical Responses

Transcription

1 179 Analyss of ss Stctes wth Uncetantes: Fom Expemental Data to Analytcal Responses P. Longo 1), N. Mage 1), G. Mscolno ) and G. Rccad ) 1) Depatment of Engneeng, Unvesty of Messna, Vllaggo S. Agata, Messna, taly, {plongo, nmage}@nme.t ) Depatment of Engneeng and nte-unvesty Cente of heoetcal and Expemental Dynamcs, Unvesty of Messna, Vllaggo S. Agata, Messna, taly, {gmscolno, gccad}@nme.t Abstact: hs pape deals wth the analyss of tss stctes wth ncetan elastc modls nde detemnstc loads. he ncetan paametes, whch chaacteze the elastc modls, ae detemned by examnng the expemental data obtaned fom tensle tests on seveal steel bas pefomed n the Laboatoy of Stctes and Mateals of the Depatment of Engneeng (Unvesty of Messna). Analyzng the expemental data, both the pobablstc and non-pobablstc models ae examned. n the fst case the andom ncetantes ae completely chaactezed thogh the knowledge of the pobablty densty fncton, whch s detemned by applyng the maxmm entopy appoach and compaed wth Gassan model. n the second case the nteval model s adopted and the ncetantes ae chaactezed by the mdpont and devaton vales. Fnally, n ode to compae the popagaton of these two models of ncetantes, the esponse of a benchmak tss stcte s evalated and the eslts n tems of dsplacements ae compaed. Keywods: pobablstc ncetantes, maxmm entopy appoach, nteval analyss, atonal sees expanson 1. ntodcton n ecent yeas t has been ecognzed that the analyss of stctal systems shold take nto accont all the elevant ncetantes pesent n the analyzed poblem. Uncetantes assocated wth an engneeng poblem, de to the dffeent soces, can be dvded nto two man gops: andom ncetantes and epstemc ncetantes (Elshakoff and Ohsak, 1). he andom ncetantes ae completely chaactezed thogh the knowledge of the fll set of ts statstcs (whch cold be moments, cmlants o othe deved qanttes) o, whch s the same, thogh the knowledge of ts pobablty densty fncton (PDF). Despte the sccess, nfotnately the pobablstc appoaches gve elable eslts only when sffcent expemental data ae avalable to defne the PDF of the flctatng popetes. f avalable nfomaton ae fagmentay o ncomplete so that only bonds on the magntde of the ncetan stctal paametes ae known, non-pobablstc appoaches can be altenatvely appled. n the famewok of nonpobablstc appoaches the nteval model, whch stems fom nteval analyss (see e.g. Mooe, 1966; Mooe et al., 9), may be consdeed as the most wdely sed analytcal tool among non-pobablstc 16 by athos. Pnted n Gemany.

2 18 P. Longo, N. Mage, G. Mscolno and G. Rccad methods (Mhanna and Mllen, 1; Moens and Vandeptte, 5). Accodng to ths appoach, the flctatng stctal paametes ae teated as nteval nmbes nsde the lowe and ppe bonds. n the famewok of pobablstc appoaches sally the ncetantes ae assmed as stochastc vaables modelled by Gassan dstbtons. Howeve, often, ths dstbton does not eflect the actal one. As a conseqence the nmecal eslts obtaned by assmng the Gassan appoxmatons cold be vey fa fom the effectve ones. On the contay n ths pape, statng fom data obtaned fom expements on seveal steel bas pefomed n the Laboatoy of Stctes and Mateals of the Depatment of Engneeng (Unvesty of Messna), the PDF of elastc modls of the mateal s deved by applyng the maxmm entopy appoach poposed by Alband and Rccad (8). hen the pobablstc esponse of a benchmak tss stctes s detemned once the nvese of the global stochastc stffness matx s evalated n appoxmate explct closed fom by applyng the ecently poposed Ratonal Sees Expanson (RSE) (Mscolno and Sof, 13; Mscolno et al., 14). So opeatng a sbstantal comptatonal savngs ove classcal Monte Calo Smlaton (MCS) s obtaned. n the famewok of nteval analyss the mdpont and devaton vales of the ncetan elastc modls ae detemned by analyzng expemental data. hen, appoxmate explct expessons of the bonds of the nteval nodal dsplacements of the benchmak tss stctes ae deved by applyng the so-called nteval Ratonal Sees Expanson (RSE) (Mscolno and Sof, 13; Mscolno et al., 14) ecently poposed to evalate the explct nvese of the global stffness matx wth nteval modfcatons.. Pelmnay concepts and defntons.1. EQUAONS GOVERNNG HE PROBLEM OF RUSS SRUCURES t s well known that the eqlbm eqatons of a tss stcte wth n nconstaned nodal dsplacements and m elements, sbected to known statc loads, can be wtten as follows: C Q = f ; eqlbm eqaton; Q = R q ; constttve eqaton; C U = q. compatblty eqaton. (1a,b,c) whee U s the s the vecto, of ode n, of nodal dsplacements; f s the vecto, of ode n, collectng the extenal loads appled at the nodes; Q and q ae the vectos, of ode m, of ntenal foces and defomatons espectvely; C s the eqlbm matx and R s the m m dagonal ntenal stffness matx. Let s now ndcate wth E A L the axal stffness of the -th element, whee E, A and nm L ae the Yong elastc modls, the aea and the length of the -th element, espectvely. Let s assme now that m elements possess ncetan elastc modls. Denotng wth the dmensonless flctaton of the -th ncetan elastc modls aond the nomnal vale, that E E, 1, one gets: E,, of the -th element, sch

3 181 Analyss of ss Stctes wth Uncetantes: Fom Expemental Data to Analytcal Responses E 1 A L (1 ) (),, whee, E, A L s the nomnal vale of the axal stffness of the -thelement wth 1,,, m. hen, the ntenal stffness matx whee α R( α) can be wtten as: E, E, 1 R( α) = R + l l, (3) s the vecto collectng the ncetan dmensonless flctatons, ntenal stffness matx and l E, othe ones eqal to zeo. Notce that the dyadc podct s a vecto of ode n wth only the -th element eqal to l l E, E, R s the dagonal nomnal, and the gves a change of ank one to the nomnal ntenal stffness matx. Afte smple sbstttons nto Eqs. (1) the solvng eqlbm eqaton, n the famewok of the dsplacement method, can be wtten as: K( α) U( α) f (4) whee U( α ) s the vecto, of ode n, of the nknown nodal dsplacement dependng on ncetantes and K( α) s the ncetan stffness matx whch, by means of Eqs. (1), can be wtten as: hen, accodng to Eq. (3), the stffness matx ewtten as: K( α) C R( α) C (5) K( α) 1, whch possesses ncetan paametes. can be K( α) K K K K α (6) whee K s the nomnal stffness matx and K s a ank-one matx defned espectvely as: K C R C K v v (7a,b) ; wth the vecto v gven as: Fnally, the solton of Eq. (4) can be fomally wtten as: v C l E, (8) U( α) K( α) 1 f (9) Becase of the pesence n Eq. (9) of the vecto α, collectng the ncetan dmensonless flctatons, the solton of pevos eqaton can be obtaned effcently f explct expessons of the nvese of the andom stffness matx K( α ) ae known. o do ths, n the next sbsecton a new sees expanson s descbed.

4 18 P. Longo, N. Mage, G. Mscolno and G. Rccad.. EXPLC NVERSE OF HE SFFNESS MARX FOR SRUCURAL SYSEM WH RANK-ONE MODFCAONS n ode to deve the explct expesson of the nvese of the stffness matx, n ths secton a ecently poposed sees expanson, called Ratonal Sees Expanson (RSE), s descbed (Mscolno and Sof, 13; Mscolno et al., 14). he RSE has been obtaned by popely modfyng the Nemann sees expanson n the case of stctal systems wth moe ank-one modfcatons n the stffness matx. So opeatng an appoxmate explct expesson of the nvese of an nvetble matx wth modfcatons was deved. n patcla, fo tss stctes, the matx, whch collects the ank- change n the K α stffness matx, can be wtten as the speposton of ank-one matces as follows: whee the vecto ncetantes ae lesse than one, that s v 1 K α v v (1) has been defned n Eq. (8). Moeove, snce the flctatng dmensonless nvese of stffness matx by etanng only the fst ode tem as follows: whee the followng qanttes have been ntodced: s 1, t s possble to evalate n explct fom the appoxmate d K α K v v K D (11) d ; Notce that Eq. (11) cetanly holds f the followng condton s satsfed: v K v D K v v K. (1a,b) d <1. (13).3. EXPLC MEAN-VALUE VECOR AND COVARANCE MARX FOR SOCHASC UNCERANES hs secton addesses the poblem of statc analyss of stctes n whch the ncetantes ae modelled as zeo-mean stochastc ndependent vaables, wth assgned Pobablty Densty Fncton (PDF) p ( x), collected n the vecto α. n ths case the solton of eqlbm eqatons depend on stochastc vaables, that s: K α U α f (14) whee the tlde denotes a stochastc qantty. By applyng Eq. (11), the nvese of the stochastc matx K α can be evalated as: d K α K v v K D (15)

5 183 whee Analyss of ss Stctes wth Uncetantes: Fom Expemental Data to Analytcal Responses K s the stffness matx of the nomnal system whle d and D have been defned n Eq. (1). Accodngly, the solton of the set of lnea stochastc Eq. (14) can be wtten n the followng appoxmate explct fom: U α U K α f K f D f (16) 1 1+ d 1 1 Fnally, snce zeo-mean stochastc vaables ae ealstcally assmed ndependent ones, the meanvale vecto and the covaance matx of the stochastc esponse vecto U can be evalated, espectvely, as follows: U 1 E U K f E D f; 1+ d 1 E UU E E D f f D 1 1+ d 1+ d U U U (17a,b) whee E s the stochastc opeato defned as: x x E p ( x) d x; E p ( x)d x. 1+ d 1+ x d 1+ d 1+ x d (18a,b) Obvosly, f the stochastc vaable s defned n a fnte nteval ab, the pevos elatonshps can be ewtten as: b b x x E p ( x)d x; E p ( )d. x x 1+ d 1+ xd 1+ d 1+ x d a a (19a,b) he pevos eqatons povde sbstantal comptatonal savngs ove classcal MCS method snce they 1+ d wthot eqng the nveson of the st nvolve the statstcs of the andom vaables global stochastc stffness matx. Fthemoe, the closed-fom expesson of the andom esponse n Eq. (16) enables one to evalate hghe-ode statstcal moments sefl to detemne the PDF of the esponse..4. EXPLC BOUNDS OF HE RESPONSE FOR NERVAL UNCERANES Let s consde now the case n whch the ncetantes ae modelled wth ncetan-bt-bonded paametes modeled as nteval vaables. Accodng to nteval analyss, the vecto α, of ode, n ths α α, α. n the followng wth the apex s denoted an nteval qantty. he case has to be defned as: vecto α, collects the ncetan-bt-bonded symmetc flctatons of axal stffness aond the nomnal vale and defnes a -dmenson bonded convex set-nteval vecto of eal nmbes, sch that

6 184 ααα P. Longo, N. Mage, G. Mscolno and G. Rccad whose -th element s. Wthot loss of genealty t s assmed the mdpont vale vecto,, eqals to. hen the devaton ampltde vecto, α, whch collect the flctatons aond the mdpont s gven as: 1 α α α α α α (a,b) whee the symbols α and α denote the lowe and ppe bond vectos espectvely. As a conseqence of Eqs. (), the followng elatonshp holds fo the genec nteval vaable whee eˆ 1, 1 1,,, e (1) ˆ s the so-called Exta Untay nteval (EU) (Mscolno and Sof, 1; Mscolno and Sof, 13). Fo detemnstc statc loads and ncetan-bt-bonded paametes, the eqlbm Eq. (5) can be ewtten as: K ( ) ( K ) ( ) f () t follows that the stffness matx, depends only on devaton ampltde vale of the ncetanbt-bonded paametes and accodng to nteval fomalsm s wtten as: whee: ˆ he goal s now to fnd the naowest nteval ˆ e e 1 1 K K v v K K (3) K = v v (4) contanng all possble esponse vectos, satsfyng the eqlbm Eq. (), when the vecto α assmes all possble vales nsde the nteval vecto poblem s fomally solved as: 1 α α. he K α f (5) Snce n stctal engneeng the stffness matx s egla and t can be assmed that the ncetantes ae not lage, so that 1, the nvese nteval of the stffness matx, by applyng the mpoved nteval Analyss (Mscolno and Sof, 1), can be detemned by the applyng so called nteval Ratonal Sees Expanson (RSE) (Mscolno and Sof, 13; Mscolno and al., 14) as (see Eq. (11)): ˆ e ˆ e 1 11 ˆ ed K α K v v K D (6)

7 185 Analyss of ss Stctes wth Uncetantes: Fom Expemental Data to Analytcal Responses Obvosly, the accacy of Eq. (6), whch gves the explct nvese of a matx wth flctatng paametes, depends on the magntde of the flctatons. Altenatvely, Eq. (6) can be ewtten n the so-called affne fom (Mscolno and Sof, 13) as: K α K ˆ ˆ e v v K a, a e D (7) 1 1 whee a, and a ae gven by: hen the solton of Eq. () s gven espectvely as: a d a d d ;., a ˆ, a e 1 (8a,b) K α f K f D f (9) De to the monotoncty of the components of the vecto wth espect to the genec lowe and ppe bonds of dsplacements can be evalated espectvely as: whee the followng vectos ae ntodced: α ( ), the ; (3a,b) 1 a, ; a 1 1 K f D f D f (31a,b) wth the symbol whch denotes the component wse absolte vale. 3. Pobablty densty fncton deved by expemental data n the famewok of stctal engneeng sally the ncetantes ae assmed as stochastc vaables modelled by Gassan dstbtons. Howeve, often, ths dstbton does not eflect the actal one. As a conseqence the nmecal eslts obtaned by assmng the Gassan appoxmatons cold be vey fa fom the effectve ones. o ovecome ths dawback, n ths secton, a method to deve the dstbton coheent n some way wth the hstogam obtaned analysng the eslts of a set of expemental data s pesented. he method s based on the maxmm entopy pncple poposed by Alband and Rccad (8), whch deved the effectve PDF coheent wth expemental data n tems of moments. pˆ X () ab, whch can be wtten as speposton of bass PDF x; x, h Let denote wth nteval, x the appoxmatng PDF of the gven andom vaable X, defned n a fnte N () 1 : X pˆ x p x; x, h (3) X X

8 186 () whee the -th bass PDF x; x, h P. Longo, N. Mage, G. Mscolno and G. Rccad, havng ntay aea n the nteval doman [, ], depends on the X locaton x and bandwdth h. he locaton paametes ae N ponts belongng to the doman [, ], chosen fo sake of smplcty wth a constant step x x 1 x, (wth 1,,, N 1). n a smla way t has been assmed a constant bandwdth paamete h h q x, a good choce s q 3. he speposton of bass PDF (Eq. (3)) epesents a PDF f and only f the coeffcents followng condtons: p 1, 1 N N p 1 1 ab ab p satsfy the (33) Eqatons (3) and (33) show that a genec PDF can be expessed as a lnea convex combnaton of smple PDFs, whose coeffcents have the meanng of pobabltes. n ode to evalate the pobabltes, t s sefl to ewte Eq. (3) as follows: p pˆ x φ x p (34) X whee (1) () ( N ;,, ;,,, ) φ x x x h x x h x; x, h and,,, X X X 1 X X N p p1 p p N ae vectos of ode N. Mltplyng both sdes of Eq. (34) by (wth k,1,,, M ). and ntegatng ove the doman, takng nto accont Eqs. (33), the followng system of eqatons s obtaned: x k whee 1 s a nt vecto of ode N, M () of ode k of the -th bass PDF x; x, h : X 1p1 Mp = μ s a matx of ode M N, whose elements, m k (35a,b), ae the moments b k () k X a m x x; x, h dx (36) whle μ s a vecto of ode M collectng the k-th moment deved fom expemental data. n the system of eqatons (Eq. (35)) the nmbe of moments M gves the data nfomaton. Hee t s assmed that only the lowe sx moments ( M 6 ) can be deved wth good accacy fom expemental data. he nmbe N of kenel denstes gves the esolton fo the ecovey of the taget PDF x ; as mch as N nceases, comptatonal complexty gows; t s a good choce to select N n the ange -1, beng geneally N lowe than the N s sample data. o solve the system (Eq. (35)) the Maxmm Entopy method s adopted, that leads to fnd the nqe ME mnmm of the fee fnctonal H H1,,, M, whee λ s the -th Lagange mltple, defned as (Alband and Rccad, 8): px

9 187 Analyss of ss Stctes wth Uncetantes: Fom Expemental Data to Analytcal Responses whee ME H H,,, (37) 1 M k k k1 M 1,,, ln exp N M k M k x 1 k1 s the nomalzaton constant, that can be expessed as a fncton of λ 1,λ,., λ M, and (38),,, k k 1 M ME he fee fncton,,, H H 1 M N M k k x exp x 1 1 N exp M 1 1 x k (39) s convex wth espect to the Lagange mltples 1,,, M and, as a conseqence, t has an nqe mnmm, whch can be obtaned thogh a standad convex optmzaton algothm, wth a lmted nmbe of teatons. he coespondng coeffcents p can be compted as: p M k exp k x k1 (4) whee the Lagange mltples ae solton of the Maxmm Entopy optmzaton poblem (Alband and Rccad, 8). 4. Nmecal eslts vess expemental data Am of ths stdy s to pefom the analyss of tss stctes wth ncetan Yong elastc modls nde detemnstc loads by applyng both pobablstc and non-pobablstc appoaches. o do ths the Yong elastc modls s detemned by seveal expements on steel bas pefomed n the Laboatoy of Stctes and Mateals of the Depatment of Engneeng (Unvesty of Messna). ensle stength tests, accodng to UN EN SO , wee pefomed on 18 specmens, sng nvesal machne, Galdabn VB47, Qasa 1 and elastc modls was compted by electonc extensomete mcon moto (class.5 accodng to UN EN SO 9513). he man statstcs of expemental data: Coeffcent of Vaaton (CoV), 3 4 / ; skewness coeffcent,, and excess ktoss, ( / ) 3, ae epoted n able. 3, / n ths secton, the descbed pocede s appled to the benchmak tss stcte depcted n Fge 1. he coss-sectonal aeas and Yong s modl of fve bas ae A 1=A =A 3=A 4=A 5=.9 [m ] and E,1=E,=E,3=E,4=E,5= [N/mm ] espectvely. n patcla, fst, n the famewok of pobablstc appoaches, the statstcs of the esponse ae evalated by applyng the poposed fomlaton and compaed 4,

10 188 P. Longo, N. Mage, G. Mscolno and G. Rccad wth the same obtaned by Monte Calo Smlaton (MCS). hen, by applyng the mpoved nteval Analyss, the bonds of nodal esponse n tems of dsplacements ae evalated. able. Statstcal eslts fo Yong Elastc Modls fom expemental data. N. Samples 18 Mean: [N/mm ] Standad Devaton: [N/mm ] Mnmm [N/mm ] Medan [N/mm ] Maxmm [N/mm ] CoV.7 Skewness Coeffcent.4188 Excess Ktoss F=1kN y A x 4 B L= 3 m Fge 1. Sketch of benchmak tss system PROBABLSC APPROACH n the famewok of the pobablstc appoach and accodng to the Chavenet cteon fo the selecton of the effectve expemental eslts (Babato et al., 11), the Kenel PDF of the Yong elastc modls s evalated n the nteval doman [a,b] of exstence of PDF whch s chosen as 4., 4. ; then, to avod nmec nstablty, the nteval s nomalzed nto the doman [,1]. Fnally the Kenel PDF s detemned, accodng to Eq. (3), as a lnea combnaton of 3 nomal bass kenel denstes, whose

11 189 Analyss of ss Stctes wth Uncetantes: Fom Expemental Data to Analytcal Responses coeffcent p ae compted by Eq. (4) once the Maxmm Entopy optmzaton poblem s solved (Alband and Rccad, 8). Kenel PDF Gassan Fge. Compason between Kenel PDF, Gassan PDF and expemental data. he Kenel PDF s depcted n Fge togethe wth the hstogam of expemental data and the Gassan PDF havng same mean vale and standad devaton of expemental data. Clealy ths fge shows as the Kenel PDF bette fts expemental data. he nomnal dsplacement vecto N s evalated as follows : N 1 K F (41) obtanng: x N y mm ; (4) x y o evalate n explct fom the fst two statstcs of the esponse, the Kenel PDF, to satsfy the condton (13), has to be nomalzed nto a lesse than 1 doman. hs nomalzaton has been pefomed by means of the followng tansfomaton: e (43) whee s the mean vale of Kenel PDF. So opeatng the nomalzed Kenel PDF of ncetan elastc modls, p e, epesented n Fge 3, les nto the nteval doman.8,.8 E.

12 19 P. Longo, N. Mage, G. Mscolno and G. Rccad Fge 3. Nomalzed Kenel PDF. o test the accacy of poposed method, n able ae epoted the mean vales, devatons,, and the standad, of dsplacements of stded stcte, evalated by means of Eqs. (17) and compaed wth the ones comng fom MCS. n able the pecentage eos ae gven, compang the analytcal data wth the eslts obtaned fom MCS of 5, 5, 1 samples. Neglgble pecentage eos confm the accacy of poposed method, povded a consdeable edcton of the comptatonal effot. able. Compason between stochastc eslts and Monte Calo Smlaton (Kenel Densty). Paamete analytcal MCS 5 Eo [%] MCS Eo [%] MCS 1 6 Eo [%] [mm] 5 µ x µ y µ x µ y σ x σ y σ x σ y he same analytcal appoach has been appled assmng a Gassan PDF. n able the pecentage eos between analytcal eslts and MCS wth 1.. samples ae epoted, confmng the accacy of poposed method.

13 191 Analyss of ss Stctes wth Uncetantes: Fom Expemental Data to Analytcal Responses able. Compason between stochastc eslts and Monte Calo Smlaton (Gassan PDF). Paamete[mm] analytcal MCS1 6 Eo[%] µ x µ y µ x µ y σ x σ y σ x σ y able V smmazes the 3 d and 4 th ode cental moments soted by MCS sng 1 6 samples, fo the adopted denstes. able V. Compason between 3 d and 4 th ode cental momentsfo both PDF. Paamete Kenel PDF Gassan PDF 3,x1 [mm 3 ] ,y1 [mm 3 ] ,x [mm 3 ] ,y[mm 3 ] ,x1 [mm 4 ] ,y1 [mm 4 ] ,x [mm 4 ] ,y[mm 4 ] Sch cental moments combned wth the MCS mean vales and standad devatons ae sed to evalate 3 4 the CoVs, /, the skewness coeffcents, 3, /, and excess ktoss, ( 4, / ) 3, of both Kenel PDF and Gassan PDF n tems of dsplacements. De to nonlnea flteng of npt data, t s well known that the expected stctal esponse has popety of non Gassanty. Adoptng the kenel densty fncton, whch takes n accont of hghe ode

14 19 P. Longo, N. Mage, G. Mscolno and G. Rccad statstcs of the npt data, the stctal esponse statstcs of ode hghe than the second gves eslts whch ae dffeent fom the ones soted by smply assmng the Gassanty of npt data (.e. takng nto accont the npt data statstcs p to the second ode). Sch consdeaton stfes the need to take nto accont of npt data statstcs of hghe ode to the second, when t s nteestng to well catch the non Gassan chaacte of stctal esponse. On the othe hand, the non Gassanty of the esponse n both cases s hghlghted by slghtly ghttaled shape wth espect to the mean of esponses, as ndcated by skewness coeffcent, gven n able V whee t s clealy evdent that, adoptng the Kenel PDF, the sensble hghe vale of excess ktoss s povded. able V. C.O.V., skewness coeffcents and excess ktoss of the esponses. Paamete Kenel PDF Gassan PDF x1 x1 y1 1 x x y y 3 3, x1 x1 3 3, y1 y , x x , y y 4, x1 4 x1 4, y1 4 y1 4, x 4 x 4, y 4 y NERVAL ANALYSS t s well known that when the nfomaton on expemental data s ncomplete o fagmentay, the nteval analyss s a vey effcent method to evalate the popagaton of the ncetantes on stctal esponse. o defne the nteval of ncetanty, the knowledge of the dstbton fncton s not eqed bt ts bonds only. Fthemoe, accodng to the phlosophy of nteval analyss, the meased data defne an nteval wth fll confdence that the vale s wthn the nteval, and not otsde t. hat s, t s not a

15 193 Analyss of ss Stctes wth Uncetantes: Fom Expemental Data to Analytcal Responses confdence nteval o cedblty nteval. Rathe, the nteval epesents se bonds of the measement, wth fll degee of confdence on expemental data (Feson et al., 7). he statng pont n sng a bonded nteval to model the measement ncetanty s to acknowledge the ntnsc mpecson n measement. n the stded tss stcte, the poplaton of expemental data s nmeos, so that elable eslts can be obtaned by applyng the pobablstc appoach, descbed n the pevos secton. he am of ths secton, s to compae the eslts povded by the pobablstc model wth the ones evalated by applyng the mpoved nteval Analyss. Fo ths ppose, the fst step s to defne a elable nteval of the ncetan elastc modls. A easonable choce seems to be a nomalzed nteval contanng all expemental data,.e..8,.8, whch has been chosen fo Kenel PDF evalaton. n able V, the chosen vales of lowe bond (LB),, and ppe bond (UB),, as well as the mdpont,, devaton, epoted., and Coeffcent of nteval Uncetanty (C..U.), able V. nteval paametes fo Yong Elastc Modls fom expemental data (1% of expemental data). [N/mm ] [N/mm ] [N/mm ] [N/mm ] C.. U. he coespondng mdpont dsplacement vecto and the lowe and the ppe bonds of dsplacements vectos calclated by Eqs. (3) and (31) ae gven espectvely as: mm mm ; mm ; he dffeence between mdpont vales and explct mean vale vectos fo Kenel PDF and Gassan PDF, epoted n able and able, s epoted n able V. n Fges 4-7 the bonds of nteval esponses, n tems of dsplacement, soted by mpoved nteval Analyss, ae compaed wth the eslts obtaned n tems of confdence ange dsplacements, calclated as the mean vale ± thee tmes standad devaton of stochastc eslts, comng ot by assmng maxmm entopy appoach and adoptng the nomal dstbton. hese Fges show that fo the analysed tss stcte, the confdence ange le nto the bonds defned by nteval Analyss, f an nteval that ncldes all expemental data s chosen., ae

16 194 P. Longo, N. Mage, G. Mscolno and G. Rccad able V. Compason between mean vales and mdpont dsplacements. Paamete [mm] μ U μ U Dffeence [%] (Kenel PDF) Dffeence [%] (Gassan PDF) (Kenel PDF) (Gassan PDF) µ x µ y µ x µ y Fge 4. x1: nteval analyss bonds and Kenel and Gassan PDF 3. Fge 5.y1: nteval analyss bonds and Kenel and Gassan PDF 3.

17 195 Analyss of ss Stctes wth Uncetantes: Fom Expemental Data to Analytcal Responses Fge 6. x: nteval analyss bonds and Kenel and Gassan PDF 3. Fge 7. y: nteval analyss bonds and Kenel and Gassan PDF Conclsons Statng fom tensle tests pefomed on steel bas, whee elastc modl ae meased, the PDF of ncetan elastc modls was ecoveed by maxmm entopy appoach. hs fncton was adopted fo the statc analyss of a tss stcte to evalate stochastc stctal esponse by means the Ratonal Sees Expanson technqe. he compason wth Monte Calo Smlaton eslts confmed the accacy of method. n ode to compae the eslts povded by the pobablstc model wth the ones evalated by applyng the mpoved nteval Analyss, the confdence ange dsplacement, calclated as the mean vale ± thee

18 196 P. Longo, N. Mage, G. Mscolno and G. Rccad tmes standad devaton of stochastc eslts, s detemned. hen the bonds of the esponse nteval by applyng the mpoved nteval Analyss ae evalated. he compason of two esponse ntevals showed that the selected confdence ange les nto the bonds defned by nteval Analyss f, fo the ncetan elastc modls, the nteval that ncldes all expemental data s chosen. Refeences Alband U. and G. Rccad. Effcent evalaton of the pdf of a andom vaable thogh the kenel densty maxmm entopy appoach. ntenatonal Jonal fo Nmecal Methods n Engneeng, 75: , 8. Babato G., E. M. Ban, G. Genta and R. Lev. Feates and pefomance of some otle detecton methods. Jonal of Appled Statstcs, 38: , 11. Elshakoff. and M. Ohsak. Optmzaton and Ant-optmzaton of Stctes nde Uncetanty, mpeal College Pess, London, 1. Feson S., V. Kenovch, J. Haagos, W. Obekampf and L. Gnzbg. Expemental Uncetanty Estmaton and Statstcs fo Data Havng nteval Uncetanty. SANDA REPOR: SAND 7-939, 7. Moens, D. and D. Vandeptte. A Svey of Non-Pobablstc Uncetanty eatment n Fnte Element Analyss. Compte Methods n Appled Mechancs and Engneeng 194: , 5. Mooe, R. E. nteval Analyss, Pentce-Hall, Englewood Clffs, Mooe, R. E., R. B. Keafott and M. J. Clod. ntodcton to nteval Analyss, SAM, Phladelpha, USA, 9. Mhanna, R. L. and R. L. Mllen. Uncetanty n Mechancs: Poblems-nteval-Based Appoach. Jonal Engneeng Mechancs ASCE 17: , 1. Mscolno G. and A. Sof. Stochastc Analyss of Stctes wth Uncetan-bt-Bonded Paametes va mpoved nteval Analyss. Pobablstc Engneeng Mechancs 8:15-163, 1. Mscolno, G. and A. Sof. Bonds fo the Statonay Stochastc Response of ss Stctes wth Uncetan-bt-Bonded Paametes. Mechancal Systems and Sgnal Pocessng 37: , 13. Mscolno G., R. Santoo and A. Sof. Explct feqency esponse fnctons of dscetzed stctes wth ncetan paametes. Compte and Stctes, 133:64-78, 14.