WEIBULL FUZZY PROBABILITY DISTRIBUTION FOR RELIABILITY OF CONCRETE STRUCTURES

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1 Enginring MECHANICS, Vol. 17, 2010, No. 5/6, p WEIBULL FUZZY PROBABILITY DISTRIBUTION FOR RELIABILITY OF CONCRETE STRUCTURES Zdněk Karpíšk*, Pr Šěpánk**, Pr Jurák* Basd on h fuzzy probabiliy disribuion and is propris, h papr dfins h fuzzy rliabiliy and is characrisics for h doubl-sa probabiliy modl of objc. Two fuzzy rliabiliy modls ar dscribd ha ar basd on h Wibull fuzzy disribuion. Th rsuls can b applid o drmining h rliabiliy of ral objcs in cass whr pr-failur ims ar of a vagu numrical yp. Kywords : fuzzy probabiliy disribuion, fuzzy rliabiliy, Wibull fuzzy disribuion, concr srucurs 1. Inroducion Th dsign of concr srucurs and hir mahmaical modling is rahr subjciv in is naur. Ordrd incrasingly wih rspc o h cos of inpu informaion acquisiion, a comparison shows ha drminisic analyss ar h chaps, howvr, hir rsuls ar of limid validiy. Probabilisic analyss provid h dsignr wih xnsiv informaion including h disribuion of h sough quaniis, howvr, h inpu daa acquisiion is considrably xpnsiv and, in som cass such as h dsign or calculaion of rsidual lifim of uniqu srucurs, is us is irrlvan du o h lack of knowldg abou h inpu paramrs. Whras rliabiliy hory only dfins h liklihood of oucoms (dos an vn occur?), fuzzy logic is an xclln mans o dscrib h xn or frquncy o which an vn occurs also if fw sampls ar availabl. Evaluaion of h safy lvl in concr srucurs should b carrid ou considring h sochasic bhavior of h main paramrs involvd, no only undr ulima condiions, bu also undr srvicabiliy and durabiliy condiions. Paricularly in concr srucurs, h larg variabiliy of mchanical and rhological paramrs may giv ris o significan dviaions from h xpcd bhavior if a drminisic approach is usd. On h ohr hand, i is wll known ha h probabiliy dnsiy funcion and is paramrs canno b univocally dfind. To ovrcom his problm, h fuzzy probabilisic hory may b usd in h procssing of sochasic paramrs, aking ino accoun hir fuzzy naur. In ordr o prsrv also h lss probabl daa ims in addiion o h complly ru ons, w can dscrib hm using ools of h hory of fuzzy ss. Thy paricularly includ h dscripion of ach quaniaiv pic of daa in h form of a fuzzy numbr whr h mmbrship funcion valu corrsponds o h probabiliy of h quaniy masurd. * doc. RNDr. Z. Karpíšk, CSc., Ing. P. Jurák, Ph.D., Insiu of Mahmaics, Faculy of Mchanical Enginring, Brno Univrsiy of Tchnology, Tchnická 2, Brno, Czch Rpublic ** prof. RNDr. Ing. P. Šěpánk, CSc., Insiu of Concr and Masonry Srucurs, Faculy of Civil Enginring, Brno Univrsiy of Tchnology, Vvří 95, Brno, Czch Rpublic

2 364 Karpíšk Z. al.: Wibull Fuzzy Probabiliy Disribuion for Rliabiliy of Concr Srucurs This approach hn maks i possibl o modl h rliabiliy of an objc as a rliabiliy sysm wih diffrn mmbrship dgrs, using fuzzy rliabiliy, basd on h noion of fuzzy probabiliy 1, 4]. Hnc, for xampl, w can obain h fuzzy hazard funcion and h fuzzy man valu of h im o failur. Evn hough a rvrs inrpraion dos no lad o a focusing of h rsuling informaion, i dos mak i possibl o assss h valus of h characrisics analyzd in a maximum rang including h dgrs of mmbrship of hir individual valus as masurs of hir rliabiliy. Our fuzzificaion of h sochasic modl of rliabiliy is basd on h following noions 1, 4]. L Ω b a univrsal s (basic spac). A fuzzy s à = (Ω,μ A )wihaborl masurabl mmbrship funcion μ A is calld fuzzy vn. Evry crisp random vn A Ω is vidnly a fuzzy vn. W dfin h rlaionships,, = and opraions,,,, + wih fuzzy vns also for fuzzy ss. A nonmpy s Σ of fuzzy vns à =(Ω,μ A is calld a fuzzy Borl fild of fuzzy vns ) ovr h univrsal s Ω, if Σ has h following propris : 1. A Σ, α 0, 1] αa Σ. 2. Ã Σ Σ. Ā 3. à 1,à 2,... Σ Ã i Σ. i=1 4. à 1,à 2 Σ Ã 1 à 2 Σ. L Ω = R m (R dnos h s of all ral numbrs) whr m N (N dnos h s of all naural numbrs) b a univrsal s, Σ a crisp Borl fild of random vns ovr Ω, Π a nonmpy s of probabiliy masurs P on (Ω, Σ ), Σ a fuzzy Borl fild of fuzzy vns on Ω, and à =(Ω,μ A ) Σ a fuzzy vn. L, furhr, =(Π,μ P ) b such a fuzzy bunch P on Π ha P Πandμ P (P ) = 1. Thn h fuzzy bunch is calld a fuzzy probabiliy on Ω and h fuzzy probabiliy of a fuzzy vn à is h fuzzy P s P(Ã) =(0, 1],μ P )) (A whr μ P )(p) = sup μ P ) (A (P R P μ A dp = p Ω for p 0, 1]. If no masur P Π xiss such ha Ω μ A dp = p, w pu μ P ) =0. Th (A ripl (R m, Σ,P) is calld a fuzzy probabiliy spac on R m. Th rsricion Ω = R m is givn by h applicaion framwork and by h possibiliy of spcifying probabiliy masurs by disribuion funcions of random variabls. Evry crisp probabiliy P is vidnly a paricular insanc of a fuzzy probabiliy P. Usually, w xprss h fuzzy probabiliy by mans of fuzzy ral numbrs 4]. Th propris of fuzzy ral numbrs and h xndd binary arihmic opraions,,, ar dscribd in 2, 3]. 2. Fuzzy probabiliy disribuions If probabiliy masurs P for a fuzzy probabiliy ar givn by disribuion funcions F (x) =P (X <x)whrx ar random variabls on P R m, h noion of a random variabl can b xndd. For simpliciy, w pu m = 1. Th horms in his par dscrib h fundamnal aribus of fuzzy probabiliy disribuion. Proofs of hs horms and addiional proposiions ar shown in 6].

3 Enginring MECHANICS 365 Dfiniion 1. L X b a nonmpy s of random variabls X on R, Φ a s of hir disribuion funcions and Σ a fuzzy Borl fild ovr Ω = R. L =(Φ,μ F b a fuzzy F ) bunch of disribuion funcions F on R such ha F Φandμ F )=1for x R. Th (F fuzzy s =(X, μ X )whrμ X (X) =μ F ), for X, is calld a fuzzy random variabl (F and is fuzzy disribuion funcion is h fuzzy bunch =(Φ,μ F Th pair,f) is calld F ). a fuzzy disribuion of probabiliy. ( Rmark 1. By h fuzzy valu of a fuzzy random variabl w undrsand an arbirary fuzzy vn from h fuzzy Borl fild Σ. Thn h fuzzy probabiliy of a fuzzy random variabl assuming h fuzzy valu à =(R,μ A ) is h fuzzy probabiliy P(Ã) =(0, 1],μ P )) (A whr μ P )(p) = sup μ F (F ). (A R R μa F df = p In paricular, for x R, hvalu of h fuzzy disribuion funcion F(x) =P( <x)is h fuzzy probabiliy Ỹ =(0, 1],μ Y )whr μ Y (y) = sup μ F (x)). F (F F (x)=y If all h disribuion funcions F ar absoluly coninuous, hn h fuzzy random variabl is coninuous and can b spcifid by h so-calld fuzzy dnsiy of probabiliy, ha is, by h fuzzy bunch f of dnsiis of probabiliy f. In his cas, h Lbsgu-Siljs ingral is qual o h Rimann ingral. Dfiniion 2. L P, Q b fuzzy probabiliis. W say ha is qual o or lss han Q P wriing Q if q Kr(Q) such ha, for p Kr(P), p q. By analogy, w dfin h P rlaions Q, Qand Q. For crisp probabiliis P, Q h rlaions and hy P P P ar idnical, c. Dfiniion 3. L =(R,μ x b a fuzzy numbr. W say ha h valu of a fuzzy x ) random variabl is lss han a fuzzy numbr wriing if is valu is a fuzzy vn =(R,μ x < x B )whr B { 1 μx (x), x (, inf Kr(x)], μ B (x) = 0 ohrwis. W say ha h valu of a fuzzy random variabl is grarhanorqualoa fuzzy numbr x wriing x if is valu is a fuzzy vn B =(R,μ B )whr μ B (x) = { μx (x), x (, sup Kr(x)], 1 ohrwis. By analogy, w dfin h rlaions and x. x > Thorm 1. If =(R,μ x ), x1 =(R,μ x 1 x )andx2 =(R,μ x ) ar fuzzy numbrs, hn, for 2 h fuzzy valu of h fuzzy disribuion funcion F(x) x), h following holds : a) 0 F(x) x) =1 F(x) for all fuzzy =P( numbrs < x, b) F(x1) F(x2) 1andP( for arbirary fuzzy numbrs x1, x2 such ha inf Kr(x1) < inf Kr(x2) and inf Supp(x1) < inf Supp(x2).

4 366 Karpíšk Z. al.: Wibull Fuzzy Probabiliy Disribuion for Rliabiliy of Concr Srucurs Corollary 1. I follows from Thorm 1 ha, for h fuzzy valu of a fuzzy disribuion funcion and ral numbrs x, x 1 and x 2, h following holds : a) 0 F F(x) x) x) =1 F(x) for all x, 1andP( =P( b) F(x 1 ) F(x 2 ) for all x 1 <x 2. Thorm 2. For any wo coninuous fuzzy numbrs x1, x2 such ha inf Kr(x1) < < inf Supp(x2), w hav P(x1 x2) 2 1), < =P(B B and μ B 2 B1(x) =μ B (x) μ 2 B (x) for x 1 R whr h fuzzy vn B 1 corrsponds o h rlaion x1 and h fuzzy vn 2 corrsponds o h rlaion x2. < B < Using h xnsion principl, w dfin h fuzzy numrical characrisics of h fuzzy random variabl. Dfiniion 4. L b h fuzzy disribuion funcion of a fuzzy random variabl, E(X) h mans of h F random variabls X from h fuzzy bunch and F hir disribuion funcions. Th fuzzy s )=(R,μ E )) whr Ẽ( (X μ E (X ) = sup F E(X)=x μ F (F ) is calld h fuzzy man of h fuzzy random variabl. If a random variabl X has no man or no random variabl X xiss such ha E(X) =x, w pu μ E ) =0. (X In a similar way, also ohr momn fuzzy characrisics ar dfind of h fuzzy random variabl X and also of a fuzzy random vcor whr m>1 using h fuzzy man of g ) ( whr g(x) is a Lbsgu masurabl funcion dfind on R m. Thorm 3. If a fuzzy random variabl has h fuzzy man ), hn Ẽ(a )= = a ) for any wo ral numbrs a, b. Paricularly, for Ẽ( h fuzzy numbr b ), Ẽ(a bẽ( ) is also a fuzzy numbr. Ẽ( b Dfiniion 1 admis h following fuzzificaion of a crisp disribuion funcion using a fuzzy paramr. Th firs fuzzy probabiliy disribuion modl is basd on a paramric sysm of disribuion funcions of h class of probabiliy disribuion F (x, s). Rplacing h paramr s R k whr k N by a fuzzy paramr =(R S k,μ S yilds a fuzzy disribuion funcion wih h fuzzy paramr F(x, S) puing μ F (F )=μ S for x R and )wihkr(s) (s) s R k. Thus, h fuzzy random variabl is dfind in h sns of Dfiniion 1. Thn h fuzzy probabiliy of h fuzzy random variabl assuming h fuzzy valu à =(R,μ A ) is h fuzzy probabiliy P(Ã) =(0, 1],μ P )) whr (A )(p) = sup μ S (s) μ P (A R R μa s df =p and h valu of h fuzzy disribuion funcion F(x, S) <x) is h fuzzy probabiliy Ỹ =(0, 1],μ =P( Y )whr μ Y (y) = sup μ S. s (s) F (x, s)=y Similarly, w obain a fuzzy man )=(R,μ E )) whr Ẽ( (X μ S. (s) μ E )(x) = (X sup s E(X)=x

5 Enginring MECHANICS 367 A coninuous fuzzy random variabl may also b dfind by fuzzifying h probabiliy dnsiy f(x, s) using h fuzzy probabiliy dnsiy wih h fuzzy paramr f(x, S). In his cas, w pu μ f (f) =μ S for x R and s R (s) k. Th scond fuzzy probabiliy disribuion modl is again basd on a paramric sysm of disribuion funcions, bu only on is subs rlad o h paramr. This is a fuzzy probabiliy disribuion of h yp F(x, S) =F η(x) η(x) indicas h fuzzinss of h main probabiliy disribuion (x) S valu F (x). whrhrms Thorm 4. L F(x, S) =F (x) η(x) b a fuzzy disribuion funcion whr F (x) is h crisp disribuion funcion of h random S variabl X, F (x) andη(x) 0 ar coninuous funcions for x R. L h fuzzy paramr b a coninuous fuzzy numbr wih h main valu s =0andμ S S =0fors/ 1, 1]. Thn (s) η(x) 1 R(x) =F (x) for x (,x M ], η(x) R(x) for x x M, + ), lim η(x) = lim η(x) =0 x x + whr x M signs h mdian of h random variabl X wih h rliabiliy funcion R(x) = =1 F (x). Rmark 2. If, for h funcion η(x) from Thorm 4, R(x 2 ) R(x 1 ) η(x 2 ) η(x 1 ) F (x 2 ) F (x 1 ) for x 1 <x 2, hn all h funcions F (x)+sη(x) ar vidnly non-dcrasing for s 1, 1]. Corollary 2. Th fuzzy valu of h fuzzy disribuion funcion F(x, S) =F (x) η(x) from Thorm 4 is a coninuous fuzzy numbr for x R, and hus, using α-cus, w obain S F(x, S) = αf (x)+s 1α η(x),f(x)+s 2α η(x)] α 0, 1] whr s 1α is h minimum and s 2α h maximum roo of h quaion μ S =α. Th (s) funcion F (x) is h main valu of F(x, S) and ( ) y F (x) μ S, y F () η(x),f()+η(x)], μ F = η(x) (x)(y) 0 ohrwis. for x R. For x,orx +, whavμ F (0) = 1, or μ F = 1 rspcivly and (1) μ F =0ohrwis. (y) Thorm 5. L a fuzzy disribuion funcion F(x, S) =F η(x) hav h propris from Thorm 4 and a crisp random variabl X wih h disribuion (x) S funcion F (x) hava man E(x). Thn h fuzzy man of h fuzzy random variabl is h coninuous fuzzy numbr )=E(X) ψ, Ẽ( S whr ψ = + η(x)dx. Paricularly, ( ) x E(X) μ E ) = (X μ S, y E(X) ψ, E(X)+ψ], ψ 0 ohrwis.

6 368 Karpíšk Z. al.: Wibull Fuzzy Probabiliy Disribuion for Rliabiliy of Concr Srucurs 3. Fuzzy rliabiliy modls wih Wibull fuzzy disribuion W assum ha h objc (lmn or sysm) undr invsigaion is ihr in a failurfr or in a failur sa. Th failur-fr sa im is a random variabl T,whichassums valus 0, + ). Nx w assum ha only h ransiion from a failur-fr sa o a failur on is possibl. Th crisp rliabiliy funcion of objc is R() =(P (T ) =1 F () whr F () is h disribuion funcion of T,andF() =0for (, 0]. Th fuzzy probabiliy modl of rliabiliy prsums ha h im of such a ransiion is a fuzzy random variabl, which dscribs h vagunss of h ransiion im and h uncrainy of h probabiliy disribuion. W dfin h fuzzy rliabiliy by mans of h fuzzy disribuion funcion of fuzzy random variabl (s Scion 2). Dfiniion 5. Th fuzzy rliabiliy (fuzzy lifim, fuzzy survival) funcion is h fuzzy probabiliy R() ) for 0, + ). =P( Rmark 3. I follows from Corollary 1 ha R() ) =1 F(), F() =1 R() for (, 0]. In addiion R(0) = 1, and R(+ ) =P( =0isvidn. Dfiniion 6. L F() b a fuzzy disribuion funcion, whr all h disribuion funcions F () ar absoluly coninuous ino 0, + ), and f() = df ()/d h dnsiis of probabiliy. Th fuzzy hazard funcion (fuzzy failur ra) is h fuzzy funcion λ() =(0, + ),μ λ 0, + ) )whrfor μ λ (λ) = sup F f 1 F =λ μ F (F ). Rmark 4. W canno wri = f or = f λ R λ (R 1 ) as h fuzzy funcions f and R ar dpndn. Dfiniion 7. Th fuzzy man im o failur is h fuzzy man Ẽ() of fuzzy random variabl. Rmark 5. If R() =1 F() is a fuzzy rliabiliy funcion, whr h all disribuion funcions F () ar absolu coninuous ino 0, + ), hn h fuzzy man im o failur whr Ẽ() = + 0 μ E )( (T )= R()d =(0, + ),μ E )) (T + R 0 sup F R()d= μ F (F ), sinc + df = + R d and μ 0 0 R =μ F (F )forr =1 F. (R) A crisp random variabl T wih h wo-paramr Wibull probabiliy disribuion W (b, ) whrb>0ishshap paramr, >0ishscal paramr, and 0, ) has hs funcional and numrical characrisics : ( ) ] b disribuion funcion F () = 1 xp, rliabiliy funcion R() = 1 F () = xp ( ) ] b,

7 Enginring MECHANICS 369 hazard funcion λ() = b ( ) b 1, ( ) 1 man (xpcd valu) E(T )=Γ b +1 whr Gamma funcion Γ(z) = 0 y z 1 xp( y)dy, and h P -prcnil P = ln(1 P )] 1 b for P 0, 1). Fuzzy rliabiliy modl A. This modl rsuls from h firs modl of fuzzy probabiliy disribuion (s Scion 2). W assum ha h valus of a fuzzy random variabl ar h fuzzy numbrs =(0, ),μ = whr is h obsrvd valu of a crisp random variabl T and is a so-calld )and vagunss κ cofficin. Th vagunss cofficin is a ral κ riangular fuzzy numbr =(0, ),μ κ κ )wihh main valu κ = 1 and mmbrship funcion Fig.1 κ κ min, κ κ min, 1], 1 κ min μ κ = κ κ max (κ), κ 1,κ max ], 1 κ max 0 ohrwis whr 0 <κ min 1 κ max, and boundary valus κ min, κ max hy ar givn by an xpr s sima. Fig. 1 shows graph of μ κ (κ). If a random variabl T has a crisp Wibull probabiliy disribuion W (b, ) hn h corrsponding fuzzy random variabl wih h Wibull fuzzy probabiliy disribuion (b, ) has h following fuzzy characrisics. For 0, ) h fuzzy disribuion funcion W F() =1 )] xp{ /(κ b },sohafor α 0, 1] h α-cus of fuzzy disribuion funcion F α () =F 1α (),F 2α ()] = ( ) ] b ( ) ]] b = 1 xp, 1 xp (1 κ max ) α + κ max ] (1 κ min ) α + κ min ]. For 0, ) h fuzzy rliabiliy funcion R() =xp{ /(κ )] b },sohafor α 0, 1] h α-cus of fuzzy rliabiliy funcion R α () =R 1α (),R 2α ()] = ( ) ] b ( ) ]] b = xp, xp (1 κ min ) α + κ min ] (1 κ max ) α + κ max ]. For 0, ) h fuzzy hazard funcion λ() =b b 1 /(κ ) b,sohafor α 0, 1] h α-cus of fuzzy hazard funcion λ α () =λ 1α (),λ 2α ()] = b b 1 ((1 κ max ) α + κ max ]) b, b b 1 ] ((1 κ min ) α + κ min ]) b.

8 370 Karpíšk Z. al.: Wibull Fuzzy Probabiliy Disribuion for Rliabiliy of Concr Srucurs Th fuzzy man of fuzzy random variabl is riangular fuzzy numbr Ẽ() Γ(1/b+1) whr =κ κ min Γ ( 1 b +1) (1 κ min ) Γ ( 1 b +1), κ min Γ ( 1 b +1),Γ ( 1 b +1)], μ E )() = κ max Γ ( 1 b (T +1) (1 κ max ) Γ ( 1 b +1), Γ ( 1 b +1),κ max Γ ( 1 b +1)], 0 ohrwis. Th fuzzy P -prcnil is P = κ ln(1 P )] 1 b,for P 0,1), whr κ min ln(1 P )] 1 b, (1 κ min ) ln(1 P )] 1 b μ () = κ max ln(1 P )] 1 b P, (1 κ max ) ln(1 P )] 1 b 0 ohrwis. ] κ min ln(1 P )] 1 b, ln(1 P )] 1 b ] ln(1 P )] 1 b,κ max ln(1 P )] 1 b Fuzzy rliabiliy modl B. This modl rsuls from h scond modl of fuzzy probabiliy disribuion (s Scion 2). L a random variabl T hav again h crisp Wibull probabiliy disribuion W (b, ), bu wih h corrsponding fuzzy random variabl having a fuzzy Wibull probabiliy disribuion (b, ) wih h fuzzy disribuion funcion W ( ) ] b 1 xp S η(), 0, ), F(, ) = b 0 ohrwis whr h funcion { xp( r b ) xp (r + q) b ], 0, ), η() = 0 ohrwis conains ral consans r, q. Icanbshownha,forr b, ) andq 0, b ), η() has h propris shown in h proposiion of Thorm 4 and Rmark 2 so ha F(, S) is a fuzzy disribuion funcion for h coninuous ral fuzzy numbr =( 1, 1],μ S ), μ S = 1. S (0) Fig. 2 shows h graph of η() whr h numbr M = ln 2/b is h mdian of random variabl T. Th fuzzy disribuion funcion F(, S) has h main valu 1 xp (/) b ]and,, Fig.2

9 Enginring MECHANICS 371 μ S y 1+xp ( ) ] b y 1 xp ( ) ] b η(),, μ F S) = η() (, 1 xp ( ) ] ] b + η(), 0 ohrwis for (0, ). For (, 0] w hav μ F S = 1, and μ F (, S)(y) =0holdsfor (, )(0) y 0. For α 0, 1] h α-cus of h fuzzy disribuion funcion F(, S) ( ) ] b ( ) ] ] b F α (, S) = 1 xp + s 1α η(), 1 xp + s 2α η() whr s 1α,s 2α ]arhα-cus of h fuzzy paramr S. By analogy, w acquir h mmbrship funcion and α-cus of h fuzzy rliabiliy funcion R(). Th α-cus of fuzzy hazard funcion λ() can b numrical calculad by mans of inrval arihmic 2, 3, 5]. Sinc E(T )= Γ(1/b +1)and ψ = + η()d = ( ) ( ) r 1 b (r + q) 1 1 b Γ b +1 w g h fuzzy man ( ) 1 ( ) ( ) Ẽ() = Γ b +1 r S 1 b (r + q) 1 1 b Γ b +1, whr μ E )() = (T ( ( Γ 1 b μ +1) ) S ψ 0 ohrwis., Γ ( 1 b +1) ψ, Γ ( 1 b +1) + ψ ], For P 0, 1) and 0, ) wghα-cus of fuzzy P -prcnil from h nonlinar quaions ( ) ] b xp s 1α η() =1 P, ( ) ] b xp s 2α η() =1 P. In h abov-mniond modls A and B, h slcion of h mmbrship funcion of h vagunss cofficin κ or of h fuzzy paramr S is of a subjciv naur. For applicaions, an xpr s sima of his funcion is usd as a bas, is form bing chosn o b as simpl as possibl and rspcing h xpr s valuaions of h accuracy dgrs of h valus obsrvd. In h modl B consans r, q can furhrmor b slcd for h norm of η() o b minimal or maximal 8]. 4. Conclusion Th fuzzy-s approach dscribd abov allows for a logical and sysmaic analysis of uncrainis. Th modls of fuzzy rliabiliy wih Wibull fuzzy disribuions could b,

10 372 Karpíšk Z. al.: Wibull Fuzzy Probabiliy Disribuion for Rliabiliy of Concr Srucurs usd for fuzzy rliabiliy drmining of a concr srucur cross scion or of a concr mmbr. Uncrain paramrs (lik rinforcmn ara wih influnc of corrosion, chang of concr srngh du o dgradaion/carbonaion/sulphaion in im) can b xprssd as fuzzy ss. I is possibl o procss h fuzzy uncrainis in rliabiliy analysis of a concr mmbr. Fuzzy uncrainy could b incorporad in h simad probabiliy of failur. Thn h opimizaion modls dsignd for rliabl dsign problms 8] can b modifid by includd fuzzy paramrs and h a posriori vrificaion of rsuls can also b gnralizd in h similar way 9]. Th approach allows an assssmn of h liklihood ha a paricular concr cross-scion (or concr mmbr sudid) will hav a highr failur probabiliy han h failur probabiliy of h drminisic dsignd cross scion or mmbr 7]. Acknowldgmn Th papr was suppord by grans of Gran Agncy of h Czch Rpublic (Czch Scinc Foundaion) Rg. No. 103/05/0292 Dsign opimizaion of progrssiv concr srucurs and Rg. No. 103/08/1658 Advancd opimum dsign of composd concr srucurs, by projc from MSMT of h Czch Rpublic No. 1M06047 Cnr for Qualiy and Rliabiliy of Producion, and rsarch projc No. 3 Managmn Suppor of Small and Middl-Sizd Firms Using Mahmaical Mhods of Acadmy Sing, Businss Collg in Brno. Rfrncs 1] Zadh L.A.: Probabiliy Masurs and Fuzzy Evns, J. of Mah. Analysis and Applicaions, 23(2), p ] Klir G.J., Yuan B.: Fuzzy Ss and Fuzzy Logic, 1s d., Prnic Hall, Nw Jrsy, 1995, ISBN ] Marš M.: Compuaion ovr Fuzzy Quaniis, CRC Prss, Boca Raon, Florida, ISBN ] Karpíšk Z.: Fuzzy Probabiliy and is Propris, In MENDEL 00, 6h Inrnaional Confrnc on Sof Compuing, Brno 2000, p , ISBN ] Karpíšk Z., Pospíšk M., Slavíčk K.: Propris of a Crain Class of Fuzzy Numbrs, In Procdings Eas Ws Fuzzy Colloquium 2000, 8h Ziau Fuzzy Colloquium, Ziau, 2000, p , ISBN X 6] Karpíšk Z.: Fuzzy Probabiliy Disribuion Characrisics and Modls, In Procdings Eas Ws Fuzzy Colloquium 2001, 9h Ziau Fuzzy Colloquium. Ziau, 2001, p , ISBN ] Šěpánk P.: Nw Mhods and Trnds for Srnghning of Concr and Masonry Srucurs, In WTA Almanach 2008 Rsauraion and Building-Physics, Munchn, 2008, p , ISBN ] Plšk J., Šěpánk P., Popla P.: Drminisic and Rliabiliy Basd Srucural Opimizaion of Concr Cross-scion, Journal of Advancd Concr Tchnology, Vol. 5(1), 63 74, ] Žampachová E., Popla P., Mrázk M.: Opimum Bam Dsign via Sochasic Programming, Kybrnika, Vol. 46(3), pp , 2010 Rcivd in dior s offic : April 1, 2010 Approvd for publishing : January 13, 2011 No : This papr is an xndd vrsion of h conribuion prsnd a h inrnaional confrnc STOPTIMA 2007 in Brno.