3rd WSEAS International Conference on APPLIED and THEORETICAL MECHANICS, Spain, December 14-16,

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1 3d WSEAS Intnational Confnc on APPLIED HEOEICAL MECHANICS, Spain, Dcmb 4-6, Hybid Modl fo Stuctual liability With Impci Pobabilitic Chaactitic Lihong Ma, Zhiping Qiu, Xiaojun Wang Intitut of Solid Mchanic Bijing Univity of Aonautic Atonautic Bijing 00083, P.. China Abtact: -A hybid modl of a pobabilitic non-pobabilitic liability thoy i dvlopd to pdict th tuctual liability whn th pobabilitic chaactitic paamt of tuctual popti a impcily known. In thi tudy, th paamt a dcibd by appopiat llipoid. Intval of th tuctual liability will b ought, i.. timating th wot poibl liability th bt poibl liability by llipoid-bound convx modl mthod. A numical xampl of a 60-ba pow pagoda i ud to illutat th faibility validity of th popod thoy. Ky-Wod: - Stuctual liability; liability intval; Hybid modl; Pobabilitic; Non-pobabilitic liability; Ellipoid-bound convx modl Intoduction Stuctual liability analyi play an impotant ol in th analyi dign of tuctu. In th ngining application, pobabilitic liability thoy appa to b pntly th mot impotant mthod [-3]. h cnt ach how that th liability nitivity of tuctu ytm tongly dpnd on th paamt of th pobability modl [4-8]. Howv, vy oftn, ufficint infomation on th pobabilitic chaactitic i abnt, a a ult, on may po only impci o limitd data on th pobabilitic chaactitic. h natual qution ai a to how to dal with thi ituation which i almot invaiably ncountd in an ngining pactic. h a val altnativ appoach to dal with th poblm, including pobabilitic (om dciption, fuzzy t dciption convx t dciption. But almot total lack of communication. Stochatician almot xcluivly utiliz pobabilitic mthod thi modl a gnally infomativ-intniv [,3,9], analyt of fuzzy t mploy fuzzy logic [0], wha invtigato daling with anti-optimization (i.. convx modling, intval analyi utiliz modl bad on unknown-but-boundd quantiti [5,8]. hfo, th dvlopmnt of igouou mathmatical mthod of combining th xiting infomation fo obtaining gnal timat of th liability of th nti ytm pnt an actual poblm [5,6]. Such a combination of th pobabilitic non-pobabilitic analyi appoach wa pfomd in th huttl application by Elihakoff, Lin Zhu [] a wll a fo th analyi of unctainty in pang aicaft dign fo compoit matial by Elihakoff, Li Stan []. hi analyi i fd to alo a obut unctainty modling o info-gap unctainty [3,4]. In thi tudy, w will combin llipoid-bound convx modl analyi mthod with pobabilitic mthodology to valuat th low bound upp bound of th tuctual liability indx liability. h bound will b vy uful in pactic could pdict th imum imum liability whn th xpimntal data a vy limitd. Pobabilitic hoy in Stuctual liability liability analyi i to analytically fomulat th failu givn a failu citia o failu mod. A a implification, it i aumd that all tat of th tuctu a dividd into two tat: failu tat af tat. Dfin th limit tat function o failu function that pnt th woking tat of tuctu a M = g( X ( wh X = ( X, X,, X n i an n-dimnion om vaiabl vcto. If M > 0, th tuctu i in th af tat, if M < 0, th tuctu i in th failu tat. M = 0 dfin th limit tat ufac which paat th failu gion fom th af gion. Givn a limit tat function M = g( X a joint dnity function f ( x of th om vcto X

2 3d WSEAS Intnational Confnc on APPLIED HEOEICAL MECHANICS, Spain, Dcmb 4-6, X = ( X, X,, X n, th tuctual liability o pobability of uvival i computd by f ( x dx ( = g( X > 0 X h pobability of failu P F i th complmnt of i computd a (. In gnal, an analytical valuation of th intgal givn by Eq ( i not poibl du to th complxity of both f X ( x g( X. Alo, if th numb of om vaiabl i lag, a numical intgation of th poblm i not faibl. hfo, appoximat mthod, fit od liability mthod (FOM, i ud to obtain th failu pobability. h dciption of thi mthod can b dividd into th tp. In th fit tp, th vcto of baic vaiabl X = ( X, X,, X n i tanfomd into an indpndnt tad (zo man unit tad dviation nomal vcto = (,,, n uing a pobability pving tanfomation. In th cond tp, th failu ufac in th y-pac i linaly appoximatd. In th final tp, th pobability contnt of th y-pac can b xactly computd fo th lina domain. h tuctual liability i givn by = P{ g( X > 0} (3 wh X = ( X i,, X k i th baic vaiabl g( X i th nonlina failu function. h tuctual liability could b timatd by th following pntd mthod. Suppo that th xit a tanfomation = (X, which can tanfom th vaiabl X = ( X i,, X k to th indpndnt tad nomal vaiabl = ( i,, k. hn Eq.(3 can b tanfomd into = P( M > 0 = P( g( X > 0 (4 = Pg ( ( ( > 0 = Ph ( ( > 0 wh h ( = g ( (. If FOM i ud, by vitu of th linaization h ( + α th tanfomation Z = α, thn Z bcom a tad nomal vaiabl, can b appoximatd a = P( h( > 0 P( Z > =Φ ( (5 wh Φ ( i th tad nomal ditibution x function Φ ( x = xp( t dt π th o-calld liability indx i dfind a μ M = (6 M h on-on lation btwn i givn by th following quation = Φ ( o =Φ ( (7 Stuctual liability li on two kind of dign paamt: on i th t o ditotion of th tuctu o componnt pat caud by th vaiou xtnal load, which i calld t ultant, dnotd a ; th oth i th capacity of nduing th load fo tuctu, componnt pat o thi matial, which i calld itanc o intnity, dnotd a. In th contuction tim vic tim of tuctu, thy xit in th mann of ith Saf o Failu. In thi pap, th limit tat function M = g( X i takn a th lina function of. M = g( X = (8 In th analyi of pobabilitic liability, th t ultant itanc a uppod to b om vaiabl, hnc M i alo a om vaiabl. In th limit tat function (8, it i aumd that oby th am kind of pobabilitic ditibution, thi man valu tad vaianc a, pctivly, μ, μ,. Hnc, th ditibution of th limit tat function M = i th am with, it man valu tad vaianc a, pctivly, μm = μ μ, M = + (9 h tuctual liability indx th tuctual liability can b obtaind by u of Eq.(5, (6 (9 μm μ μ = = (0 M + μ μ =Φ ( =Φ ( + Eq.(0 ( a th two baic fomula fo calculating th tuctual liability indx liability. Fom th abov analyi, w can conclud that th valuation of th liability indx hing on th calculation of th man valu tad vaianc of th tuctual t intnity. 3 Limitation of Pu Pobability

3 3d WSEAS Intnational Confnc on APPLIED HEOEICAL MECHANICS, Spain, Dcmb 4-6, h mathmatical thoy of pobability ha povn uful in many tchnological application. Howv, it ha limitation which, whn claly idntifid, facilitat ou undting of th non-pobabilitic altnativ. On main citicim of pobabilitic concpt of unctainty ai in dicuion of pio pobability Bayian infnc dciion thoy. A claical objction to Bayian tatitic hit at th ouc of th pio ditibution utility function. Uniqun i fomulating pio ditibution i illuiv: a givn quantity of pio infomation i oftn not pntd by a uniqu pio pobability ditibution. h difficulty of quantifying pio knowldg i n quit claly in uch quai a th 3-box iddl [4], th pion dilmma imila poblm wh altnativ dciion ach m fully conitnt with th initial infomation. Pobabilitic modl hav bn ud in cnt dcad to pnt th unctainty in ngining. h concn about th modl ai fom th fact that a tochatic modl pnt typical vnt much mo liably than a vnt, pcially whn th modl i bad on limitd infomation. a vnt in pobabilitic modl a dcibd by th tail of th ditibution, whil pobability ditibution a uually pcifid in tm of man man-vaiation paamt. hi mak pobabilitic modl iky dign tool, inc it i a vnt, th catatophic on, which mut undli th liabl dign. Whn a pobabilitic dciption of th unknown lmnt i at h, on i natually ld to conid tochatic modl. Whn only patial infomation, o no infomation at all, i availabl, howv, th i undtably a luctanc to ly on uch modl. In pug that pobability ditibution xit thy m inhntly midictd. It can b n that on thinking about unctainty can, omtim hould b non-pobabilitic. Futhmo, accoding to th law of lag numb [5], only if th numb of obvation in th ampl bcom lag that i, n appoach infinity, th ampl man vaianc convg to th al man valu vaianc. Howv, in pactic, n may b impoibl to b lag nough to obtain th xact man vaianc valu o that th oftn xit unctainti in th paamt of pobabilitic chaactitic. 4 Hybid Modl fo Stuctual liability Indd, th indtacy about th unctain vaiabl involvd could b tatd in tm of th vaiabl blonging to ctain t, uch a h unctain paamt x i boundd, x a ( h unctain function ha nvlop bound, xlow ( t x( t xupp ( t (3 wh xlow ( t xupp ( t a dtitic function which dlimit th ang of vaiation of x( t. h unctain function ha an intgal qua bound, + x ( tdt a (4 h unctain paamt vcto ha an intantanou llipoid-bound, ( x x W( x x α (5 0 0 wh W i a poitiv dfinit matix, x 0 = ( x 0i i th noal valu of th vcto x = ( x i, α i th adiu of th convx t. Conid a alitic ituation whn on on h th man valu μ μ th tad dviation of th itanc th t ultant a unctain, but on th oth h, inufficint infomation i availabl on μ, μ, to jutify th abov pobabilitic famwok. It i aumd that w po only cac infomation on th pobabilitic chaact μ, μ,, namly, th unctainti in th man valu μ, μ th tad dviation, a boundd t a th fouth kind of convx modlling mntiond abov ( μ μ0 ( μ μ0 (6 ( 0 ( 0 g g wh μ 0, μ 0 0, 0 a th cnt point of th llipoid, a chon a th noal o typical input;, g, g a th adii of th llipoid; θ θ a poitiv contant dt th iz of th llipoid. h valu a all bad on availabl limitd infomation on th itanc th t ultant.

4 3d WSEAS Intnational Confnc on APPLIED HEOEICAL MECHANICS, Spain, Dcmb 4-6, Lt u conid th tuctual liability ( ubjct to th containt condition (6. Fo convninc, th containt condition (6 may b wittn a follow Z ( μ, μ,,, θ ( μ μ0 ( μ μ0 (7 = ( μ, μ: Z (,, g, g, θ ( 0 ( 0 (8 = (, : g g h a many application in Eq.( with th containt condition (6 o (7 (8, wh th man valu μ μ th tad dviation a not pcily known. Bcau th man valu μ μ th tad dviation a unctain but boundd, th aociatd pobabilitic liability of th tuctu imilaly contitut th boundd vaiabl. hat i to ay, th pobabilitic liability of th tuctu with boundd pobabilitic chaactitic will bcom a t a follow μ μ Γ= =Φ ( : =, + ( μ μ0 ( μ μ0, (9 ( ( 0 0 g g W hould t that Γ may b gnally of complicatd gomtic hap o that it may b uually impactical to ty to olv thm. Intad, in thi tudy, w a inttd in th t o th intval containing th tuctual pobability liability with unctain but boundd pobabilitic paamt. hfo, it i a common pactic to k th intval of th pobabilitic liability I = [, ] = [, ] (0 wh =, = ( which i th mallt width intval ncloing all poibl pobabilitic liability valu. i th wot poibl liability o imum valu i th bt poibl liablity o imum. Obviouly, th imum valu poblm th imum valu poblm in Eq.(0 a global optimization poblm. 5 Dtation fo Intval of Stuctual liability Indx liability In thi ction, th intval of th tuctual liability indx liability will b computd. Bad on Eq.(7 th monotonicity of function (, th xtm valu poblm of th tuctual liability (0 can b tanfomd into th following xtm valu poblm of th tuctual liability indx μ μ Η= : =, + ( μ μ0 ( μ μ0, ( ( ( 0 0 g g hu, th t o intval of th tuctual liability indx can b xpd a I = [, ] = [, ] (3 wh =, = (4 Und th condition that μ a tatitically indpndnt, lt u conid th xtm valu poblm of th tuctual liability indx. Claly, bad on th multi-objctiv optimization thoy, th imum valu th imum valu can b, pctivly, xpd a ( μ ( μ =, = (5 ( ( wh μ = μ μ = +. hu, th xtm valu poblm of th tuctual liability indx i tanfomd into th xtm valu poblm of μ. Fo th xtm valu poblm xtm μ = μ μ (6 ubjct to ( μ μ0 ( μ μ0 (7 Accoding to th optimization thoy, inc μ i a lina function of unctain paamt μ μ, Z ( μ, μ,,, θ i a convx t, th xtm valu of μ will occu on th bounday of th t Z μ, μ,,,. h bounday of th t ( θ

5 3d WSEAS Intnational Confnc on APPLIED HEOEICAL MECHANICS, Spain, Dcmb 4-6, Z ( μ, μ,,, θ pnt an llipoidal hll, i.., S ( μ, μ,,, θ ( μ μ0 ( μ μ0 (8 = ( μ, μ: + = θ By th mthod of Lagang multipli, th Lagangian function can b wittn a ( μ μ0 ( μ μ0 L = μ μ + λ ( + θ (9 wh λ i th Lagang multipli. Ncay condition fo taking th xtm valu a L ( μ μ0 = + λ 0 μ (30 L ( μ μ0 = + λ 0 μ (3 ulting in μ μ0 = λ (3 μ μ0 = λ (33 Subtitution of Eq.(3 (33 into th following containt condition ( μ μ0 ( μ μ0 + = θ (34 h Lagang multipli i obtaind a + λ = ± θ (35 Subtituting Eq.(35 into Eq.(3 (33 yild th xtm point of th man valu μ μ μ = μ0 θ + (36 μ = μ0 ± θ + (37 So th imum valu th imum valu of th man valu μ μ can b xpd a ( μ 0 ( μ = μ + = μ 0 ( μ 0 ( μ = μ + = μ 0 θ + θ + θ + θ +, (38, (39 hu, th imum valu th imum valu of μ can b dtd by ( μ = ( μ ( μ = μ0 μ0 + θ + (40 μ (4 Fo th xtm valu poblm xtm = + (4 ubjct to ( 0 ( 0 (43 g g In vitu of th following vaiabl tanfomation ( = ( μ ( μ = μ0 μ0 θ + u =, u = ( h optimization poblm (4 can b convtd into th following fom xtm = ( u ( u + 0 (45 = ϕ0 + g δ + δ δ ubjct to th containt condition u u (46 g g wh ϕ0 = ( 0, 0, g = ( 0, 0, δ = ( u, u. Fo convninc, th containt condition (46 can b wittn δ Ωδ θ (47 wh Ω= diag(, g g.

6 3d WSEAS Intnational Confnc on APPLIED HEOEICAL MECHANICS, Spain, Dcmb 4-6, Obviouly, th xtm valu poblm ( xtm i quivalnt to th xtm valu poblm ( ϕ + g δ + δ δ, i.., 0 xtm ( = ( ϕ + g δ + δ δ (48 xtm 0 xtm Namly, th following xtm valu poblm will b olvd w= ϕ0 + g δ + δ δ (49 ubjct to th containt condition (47. By vitu of th mthod of Lagang multipli, dfin th Lagangian function a L( δ, η = ϕ0 + g δ + δ δ + η( δ Ωδ θ (50 wh η i th Lagang multipli. Accoding to th xtm condition, w can obtain L = g + δ + η Ω δ = 0 (5 δ Moov, th containt condition mut b atifid δ Ωδ θ (5 Sinc Eq.(5 i an inquality, th Lagang multipli mut atify on of th following lation[5] η = 0 if δ Ωδ < θ (53 η 0 if δ Ωδ = θ (54 Fom Eq.(5 (53, w can obtain δ = g (55 Subtitution of Eq.(55 into Eq.(49 yild ( wxt = ϕ0 g g = 0 (56 By olving Eq.(5 Eq.(54, w can obtain two Lagang multipli η η. By ubtituting thm into Eq.(49, two xtmum valu can b obtaind a follow, pctivly ( wxt = ϕ0 g ( I + ηω g (57 + g ( I + ηω ( I + ηω g ( t wxt = ϕ0 g ( I + ηω g (58 + g ( I + ηω ( I + ηω g Fom th abov quation (56, (57 (58, th poibl xtmum valu a obtaind. By chooing th imum valu fom th th poibl xtmum valu a th upp bound of th objctiv function, th imum valu a th low bound, o w hav ( ( ( t w = ( wxt, wxt, wxt (59 ( ( ( t w = ( wxt, wxt, wxt (60 hn ( = w (6 ( = w (6 In vitu of Eq.(5, fom Eq.(40, (4 (6, (6, th imum valu o upp bound th imum valu o low bound of th tuctual liability indx can b obtaind a follow μ0 μ0 + θ + = = w (63 μ0 μ0 θ + = = w (64 Conquntly, accoding to th on-on lation (7, th bt poibl valu o th upp bound th wot poibl valu o th low bound of th tuctual liability can b calculatd, pctivly =Φ ( (65 = Φ ( (66 6 Numical Exampl Dicuion Conid a 60-ba pow pagoda a hown in Fig.. h lngth of od can b obtaind fom Fig.. wo hoizontal load P = 40000N a applid at nod pctivly. h man valu of oung modulu co-ctional aa of th ba a, pctivly, μ E =. 0 N / m 3 μ A =.0 0 m. h unctain paamt vcto = ( EA, i nomally ditibutd with a cofficint of vaiation h allowabl t i nomal ditibutd, th ditibution fom i 8 6 ~ N( μ, = N(.5 0, h 0 0 man valu tad vaianc of th applid 8 t a, pctivly, μ 0 =.35 0, 6 0 = h ult of Eq.(0 ( calculatd by dtitic pobabilitic chaactitic paamt a: tuctual liability indx, =.6794, liability, = 95.3%. ak into account th unctainty in th pobabilitic chaactitic μ, μ, of th applid t allowabl t. Suppo that thi man valu th tad vaianc chang in th following llipoid, pctivly,

7 3d WSEAS Intnational Confnc on APPLIED HEOEICAL MECHANICS, Spain, Dcmb 4-6, ( μ μ ( μ μ +, ( αμ ( αμ 0 ( ( + ( α ( α 4 0 wh α, α, α 3 α 4 a unctain cofficint of th llipoid mi-adiu. In thi numical xampl, thy a chon in uch a mann that th vaiation at mot will copond to ± 5% vaiation of ach paamt. In pactic, th bound of th unctainti will li on th maud xpimnt data o xpinc. ult of th calculation of th tuctual liability indx liability vu paamt α bad on Eq.(63, (64 (65, (66 a hown in abl, abl Fig.-Fig.5, wh a th upp bound low bound of th tuctual liability indx, a th upp bound low bound of th tuctual liability coponding to diffnt ca of cofficint α, i =,,3,4, pctivly. abl how th i upp bound low bound of th tuctual liability indx vu α α, fo α3 = α4 = 0.0, namly, th man valu a only conidd to b unctain th tad vaianc a dtitic. Wha in abl, th tad vaianc a unctain, th man valu a dtitic, namly α = α = 0.0 α3 = α In abl 3, not only th man valu but alo th tad vaiabl a unctain. Fig.-Fig.5 dpict th vaying cuv of with th vaiation of unctain facto α i ( i =,,3,4. h numical ult indicat that unctainti in pobabilitic chaactitic popti hav ignificant ffct on th tuctual liability. Fom th tabl figu, w can found that intval [, ] of th tuctual liability indx intval [, ] of th tuctual liability bcom wid with th incaing of unctainti of pobabilitic chaactitic paamt. 7 Concluion Impci o unctain pobabilitic popti which a ult fom mall ampl om tt o incomplt infomation on unctain vaiabl a conidd. A hybid modl of a pobabilitic non-pobabilitic tuctual liability thoy i pntd in thi tudy to pdict th vaiation of th tuctual liability with unctain pobabilitic chaactitic paamt. h pobabilitic chaactitic paamt a aumd to b dcibd by appopiat llipoid. Unctainti in pobabilitic chaactitic popti hav ignificant ffct on th tuctual liability. It i makabl that thi modl i abl to pdict th wot poibl valu th bt poibl valu of th tuctual liability indx liability du to unctainty. h liability intval will b vy uful in pactic could b dictly incopoatd into dign whn xpimntal data a vy limitd th convntional pobabilitic liability appoach cannot b utilizd. h hybid modl bidg th communication btwn tochatician analyt of t. Conquntly, th ituation that thy almot xcluivly utiliz pobabilitic mthod o anti-optimization mthod will b bokn. fnc: []obt, M. B., Bnja, S. D., homa, L. B. Statitical Mthod fo Engin Scintit. Nw ok: Macl Dkk, Inc., 975. [] hoft-chitnn, P. Bak, M. J. Stuctual liability thoy it application: Sping-Vlag, 98. [3] Elihakoff, I. Pobabilitic mthod in th thoy of tuctu. Nw ok: Wily, 983. [4] Bn-Haim,. A non-pobabilitic mau of liability of lina ytm bad on xpanion of convx modl. Stuctual Safty, 995, 7: [5] ao, S. S. Bk, L. Analyi of unctain tuctual ytm uing intval analyi. AIAA Jounal, 997, 35: [6] Utkin, L. V., Guov, S. V., Shubinhy, I. B. liability of Sytm by Mixtu Fom of Unctainty. Mico-lctonic liability, 997, 35(5: [7] Guov, S. V. Utkin, L. V. Sytm liability und Incomplt Infomation. St. Ptbug: Lubavitch Publihing, 999. [8] Wu, J. S., Apotolaki, G. E., Oknt, D. Unctainti in Sytm Analyi: Pobabilitic vu Non-pobabilitic hoi. liability Engining Sytm Safty, 990, 30: [9] hoft-chitnn, P. Muotu,. Application of Stuctual Sytm liability hoy. Blin: Sping Vlag, 996.

8 3d WSEAS Intnational Confnc on APPLIED HEOEICAL MECHANICS, Spain, Dcmb 4-6, [0] Pantlid, C. P. Ganzli, S. Compaion of Fuzzy St Convx Modl hoi in Stuctual Dign. Mchanical Sytm Pocing, 00, 5: [] Elihakoff, I, Lin,. K., Zhu, L. P. Amtdam: Elvi Scinc Publih, 994. [] Elihakoff, I., Li,. W., Stan, J. H. Non-Claical Poblm in th hoy of Elatic Stability: Cambidg Univity P, 00. [3] Bn-Haim,. Infomation-Gap Dciion hoy: Dciion und Sv Unctainty. San Dgo: Acadmic P, 00. [4] Bn-Haim,. obut liability in th mchanical cinc. Blin: Sping-Vlag, 996. [5] ic, J. A. Mathmatical tatitic data analyi. Bijing: China Machin P, 004. P P m m m m m m m Fig. A 60-ba Pow pagoda α abl h tuctual liability indx vu α α whn ( α3 = α4 = 0.0 α abl h tuctual liability indx vu α 3 α 4 whn ( α = α = 0.0 α 3 α

9 3d WSEAS Intnational Confnc on APPLIED HEOEICAL MECHANICS, Spain, Dcmb 4-6, abl 3 h tuctual liability indx vu α ( α = α = α = α3 = α4 α : α =α =0.0, α 4=0.0 : α =α =0.0, α 4=0.03 : 3 α =α =0.0, α 4= : α =0.00, α 3 =0.0, α 4= : α =0.0, α 3 =0.0, α 4=0.0 3 : α =0.03, α 3 =0.0, α 4= : α =0.04, α 3 =0.0, α 4= α α 3 Fig. h tuctual liability vu Fig.4 h tuctual liability vu α ( α3 = α4 = 0.0 α ( α = α = : α =0.0, α 3 =0.0, α 4= : α =0.0, α 3 =0.0, α 4=0.0 3 : α =0.03, α 3 =0.0, α 4=0.0 4 : α =0.05, α 3 =0.0, α 4= α : α =α =0.0, α 3 = : α =α =0.0, α 3 =0.03 : 3 α =α =0.0, α 3 = α 4 Fig.3 h tuctual liability vu Fig.5 h tuctual liability vu α ( α3 = α4 = 0.0 α ( α = α = 0.0 4