JOURNAL OF THEORETICAL AND APPLIED MECHANICS 42, 3, pp. 651-666, Warsaw 2004 STRUCTURAL RELIABILITY – FUZZY SETS THEORY APPROACH Ewa Szeliga Civil Engineering Faculty, Warsaw University of Technology e-mail: e.szeliga@il.pw.edu.pl In the paper two kinds of uncertainty: randomness and imprecision are proposed to be considered in a structure description. Imprecise experts’ opinions canbedescribedusing fuzzynumbers.Asa result, the reliability analysis of a structure canbebasedon the limit state functionwith fuzzy parameters. As a consequence, the structural failure or survival can be treated as fuzzy events. The probabilities of these fuzzy events can be the upper and the lower estimations of the structural reliability. They can be achieved using well-known reliabilitymethods (e.g. Hasofer-Lind index andMonteCarlo simulations). They can be used as a base for the calibration of partial safety factors in design codes. Key words: probability theory, fuzzy sets theory, reliability of structure 1. Introduction There are two kinds of uncertainty associated with civil engineering (Bloc- kley, 1980) – and all engineering activities – randomness and imprecision (Gasparski, 1988). The randomness is the unpredictability of events. The randomness is de- scribed by the probability distributions based on the observation of the event frequency. It is a task of the probability theory. The imprecision is a lack of certainty of experts’ assessments. An expert of a given domain formulates his opinion arbitrary based on his knowledge, experience and intuition, using words like ”big”, ”small”, ”medium” instead of precise numbers.Modelling andprocessing of imprecise data is a task of the fuzzy set theory. 652 E.Szeliga As it is known, according to the fuzzy set theory (Zadeh, 1965, 1978) the two-value logic is extended to themulti-value logic. As a result, the conventio- nal notion of a set A (a crisp set) is extended to a fuzzy set Aby the extension of the two-value characteristic function, Eq. (1.1), to themulti-valuemember- ship function, Eq. (1.2). The notion of a real number is extended to a fuzzy numberwhich is defined as the fuzzy set satisfying several conditions (Piegat, 1999). Fuzzy numbers can be processed using the rule of extension (Kacprzyk, 1986) • cA(x) : R→{0,1} ∀x∈R cA(x)= { 1 for x∈A 0 for x /∈A (1.1) • χA(x) : R→< 0,1> ∀x∈R χA(x)=    1 if x belongs to A α∈ (0,1) if x belongs to A to some degree 0 if x does not belong to A (1.2) Similarly, the conventional notion of an event A (a crisp event), which can be defined as the crisp subset of the sample space X and described by the characteristic function, Eq. (1.1), is extended to a fuzzy event A described by the membership function, Eq. (1.2). The fuzzy event can occur, or not occur and also can occur to some degree because any element x of the sample space X can match up to a given event, or not, and also can match up to it to some degree. In otherwords, the fuzzy event is a fuzzy subset of the sample space X. The boundary between that event and its complement is fuzzyfied, not crisp. Two kinds of probability of fuzzy events are defined (Kacprzyk, 1986): • the probability according to Zadeh – a real number from the interval < 0,1> – for a continuous random variable defined as follows P(A)= ∫ A f(x) dx= ∫ R χA(x)f(x) dx (1.3) where f(x) is the probability density function of the randomvariable X Structural reliability – fuzzy sets theory approach 653 • the probability according to Yager – a fuzzy subset of the interval < 0,1 > – for a continuous random variable defined as a set of pro- babilities of crisp events Aα P(Aα)= ∫ Aα f(x) dx (1.4) with the following membership function χP(A) =α (1.5) where Aα is α-level of a fuzzy set A defined as follows ∀α∈ (0,1> Aα = {x∈R : χA(x)­α} (1.6) The probabilities mentioned above make the measure of two kinds of un- certainties: the randomness of the variable X and the imprecision (fuzziness) of the event A definition, Fig.1. Fig. 1. Randomness and fuzziness As it is known, according to the reliability theory, only uncertainty due to randomness is expressed (PN-ISO 2394, 2000; Nowak andCollins, 2000). The 654 E.Szeliga structural failure or survival is treated as an event. The probability of failure is a measure of structural reliability and is calculated as follows Pf =P(g(X1, ...,Xn)< 0)= ∫ . . . ∫ g(x)<0 f1...n(x1, ...,xn) dx1...dxn (1.7) where X = {X1, ...,Xn} – n-dimensional randomvariable,which representsna- tural randomness of loads, environmental influences, material properties, geometry etc. f1...n(x1, ...,xn) – joint probability density function of the n- dimensional random variable X g= g(X1, ...,Xn) – limit state function of load capacity or serviceabili- ty, which divides the n-dimensional sample space X into the following subsets: • the area of structural failure – g(X1, ...,Xn)> 0 • the area of structural survival – g(X1, ...,Xn)< 0 • the limit state – g(X1, ...,Xn)= 0. That probability is used e.g. for the calibration of partial safety factors in design codes. In the paper, two kinds of uncertainty will be modelled and taken into consideration in the limit state function (Szeliga, 2000): • the randomnesswill be still represented by randomvariables and proba- bility density functions • the imprecision will be represented by fuzzy numbers and membership functions. As a result, the structural failure or survivalwill be treated as fuzzy events. Partial safety factors will include ”variability of fortune” and mistakes of experts’ opinions (Fig.2). The purpose of the paper is not to prove that the structural reliability problem in fuzzy sets approach is better – but to show that it is possible. 2. Reliability of structure as a fuzzy event Let us consider a linear limit state function. The resistance consists of two parts: the random part X1 of normal distribution N(µ1,σ1) and the imprecise part described by a fuzzy number a1 = {about a 0 1 in }, Structural reliability – fuzzy sets theory approach 655 Fig. 2. Probability of a structural failure or survival as fuzzy events a01 =0.The loadalso consists of the randompart X2 of thenormaldistribution N(µ2,σ2) and the imprecise part a2 = {about a 0 2 in }, a 0 2 =0 g(X1,X2)= (X1+a1)− (X2+a2) a1 ∈ a1 a2 ∈ a2 (2.1) where a1, a2 – fuzzy numbers of membership functions χa1, χa2. The limit state function can be expressed as follows: g(Z)=Z+ b b∈ b (2.2) where Z =X1−X2 is the random safety margin, b= a1 −a2 and b – fuzzy number of membership function χb. b= {about b0 in } b0 =0 (2.3) If the parameter b was not fuzzy number, the following three crisp sets could be found in the axis Z: the failure area F , the limit point L and the survival area S (Fig.3a). They could be described – by the characteristic functions – as follows cF(z)= { 1 for Z < 0 0 for Z ­ 0 cL(z)= { 1 for Z =0 0 for Z 6=0 (2.4) cS(z) = { 1 for Z > 0 0 for Z ¬ 0 656 E.Szeliga Because of fuzzy numbers in the limit state function, the axis Z is divided into 3 fuzzy areas: the failure area F , the limit state area L and the survival area S (Fig.3c) χF(z)=    1 for Z } (3.3) β0 = µZ + b 0 σZ =0 β− = µZ + b − σZ β+ = µZ + b + σZ Thus, one Cornell reliability index (Fig.3b) is replaced by a fuzzy set of such indexes (Fig.3d). Each of them represents the limit state to some degree χβ. Structural reliability – fuzzy sets theory approach 659 The probabilities according to Zadeh of the fuzzy events F and −S can be calculated as follows (see Fig.3f) PF =P(Z ′ ∈F)= +∞∫ −∞ χF(z ′)ϕ(z′) dz′ =Φ(−β+)+ −β0∫ −β+ χF(z ′)ϕ(z′) dz′ (3.4) P−S =P(Z ′ ∈−S)= +∞∫ −∞ χ−S(z ′)ϕ(z′) dz′ =Φ(−β0)+ −β−∫ −β0 χ−S(z ′)ϕ(z′) dz′ The fuzzy probabilities according to Yager can be calculated as follows PFα =P(Z ′ ∈Fα)= +∞∫ −∞ cFα(z ′)ϕ(z′) dz′ =Φ(−β+)+ −β(α)∫ −β+ ϕ(z′) dz′ (3.5) P−Sα =P(Z ′ ∈−Sα)= +∞∫ −∞ c−Sα(z ′)ϕ(z′) dz′ =Φ(−β0)+ −β(α)∫ −β0 ϕ(z′) dz′ and χPF =α χP−S =α (3.6) The following equivalent reliability indexes can be also defined βF =−Φ −1(PF) β−S =−Φ −1(P−S) (3.7) They determine two crisp areas in the axis Z ′: (−∞,−βF > and (−∞,−β−S > so that the random safetymargin Z hits themwith the proba- bilities PF and P−S, respectively. Because these areas satisfy (−∞,−β F >⊂F ⊂ (−∞,−β−S > so βF ­β ­β−S (3.8) The equivalent indexes βF and β−S are the upper and the lower esti- mations of the reliability index β. That ”estimation” not ”calculation” is a consequence of the fuzzy numbers introduced to the limit state function. Let us consider an n-dimensional limit state function with fuzzy parame- ters. Let us take the following assumptions: • the random variables Xi have normal distributions N(µi,σi) and are uncoreleted 660 E.Szeliga • the limit state function is linear and one of its parameter, not multi- plied by the random variable Xi, is a fuzzy number described by the membership function χan+1 g(X)= a1X1+ ...+anXn+an+1 an+1 ∈ an+1 (3.9) where an+1 is the fuzzy number of membership function χan+1. The limit state function determines the fuzzy sets of limit states surfaces parallel to each other (Fig.5a). Each of these surfaces describes the limit state more or less precisely. It represents the limit state to some degree χg(x)=0. According to the extension rule, each of these surfaces represents the limit state to the same degree as each number of fuzzy set an+1 represents the parameter an+1 χg(x)=0 =χan+1 (3.10) The X space is divided into 3 fuzzy areas: the failure F , limit states L and survival S area χF(x)=    1 for χg(x)=0 =0 g(X)< 0 1−χg(x)=0 for χg(x)=0 6=0 0 for χg(x)=0 =0 g(X)> 0 χL(x)=χg(x)=0 (3.11) χS(x)=    1 for χg(x)=0 =0 g(X)> 0 1−χg(x)=0 for χg(x)=0 6=0 0 for χg(x)=0 =0 g(X)< 0 Now, themeasure of reliability cosists of the probabilities of hitting by the random variable X the following areas: • the fuzzy area of failure F • the fuzzy area of no survival −S (Fig.5a) χ−S(x)= 1−χS(x) (3.12) The probabilities according to Zadeh are the following PF =P(X ∈F)= +∞∫ −∞ ... +∞∫ −∞ χF(x)f1...n(x) dx (3.13) P−S =P(X ∈−S)= +∞∫ −∞ ... +∞∫ −∞ χ−S(x)f1...n(x) dx Structural reliability – fuzzy sets theory approach 661 Fig. 5. Fuzzy areas of the failure F , limit state L and survival S in the variables X1,X2 and standard variables U1,U2 coordinate systems 662 E.Szeliga They can be calculated by transformation of the problem of n randomva- riables to theproblemof one variable.After standardisation of the variable X, the equation of limit state takes the following form G(u)=α1u1+ ...+αnun+β=0 β ∈β (3.14) where αi = aiσi√ n∑ i=1 (aiσi) 2 i=1, ...,n (3.15) β = n∑ i=1 aiµi+an+1 √ n∑ i=1 (aiσi)2 an+1 ∈ an+1 and β, an+1 – fuzzy numbers ofmembership functions χβ,χan+1, respectively. Equations (3.14) and (3.15) describe the fuzzy set of the limit state surfaces parallel to each other (Fig.5b) of a distance β ∈ β to the origin of coordian- tes U. Thus, one reliability Hasofer-Lind index is replaced by the fuzzy sets of indexes. Each of these indexes represents the limit state surface in coordi- nates U to some degree χβ. According to the extension rule, each of these indexes represents the limit state to the same degree as each number of fuzzy set an+1 represents the parameter an+1 χβ =χan+1 (3.16) After rotation of the U1, ...,Un axes around the point of origin into Z′,Z′′, ...,Z(n) axes so that Z′ axis is perpendicular to the limit state sur- faces (3.14) and others are parallel to them (Fig.5b), the probabilities of the fuzzy events F and −S according to Zadeh can be calculated according to (3.4), and the fuzzy probabilities according to Yager – according to (3.5). The linear limit state function with at least one variable of X1, ...,Xn multiplied by the fuzzy number g(X)= a1X1+ ...+anXn+an+1 aj ∈ aj (3.17) describes the fuzzy set of surfaces (line in Fig.5c) and aj – fuzzy numbers of membership functions χaj, j=1, ...,n+1. Each of them represents the limit state to some degree χg(x)=0 = max g(x)=0 ( min j=1,...,m χaj) (3.18) Structural reliability – fuzzy sets theory approach 663 The limit state surfaces (Fig.5d) are not parallel to each other, so the probabilities calculated using the Hasofer-Lind index are approximated. The nonlinear limit state function of n random variables X1, ...,Xn and m fuzzy parameters a1, ...,am g(X)= g(X1, ...,Xn,a1, ...,am) aj ∈ aj (3.19) can be approximated by fuzzy sets of linear functions, see Fig.5e,f (aj – fuzzy numbers of membership functions χaj, j =1, ...,n+1). Thus, the structural reliability analysis based on the limit state function with fuzzy parameters is similar to well known methods. In some cases, the Hasofer-Lind indexes can be used. 4. Fuzzy Monte-Carlo methods Monte-Carlo simulations can also be used in the case of fuzzy events. The probabilities according to Zadeh can be estimated as follows P̂F = NF N P̂−S = N−S N (4.1) where N is a number of experiments. NF and N−S are numbers of hitting the fuzzy areas F and −S in particular trials calculated as follows NF =NF +χF(u) ϕn(u,0, I) ϕn(u,U ∗, I) (4.2) N−S =N−S +χ−S(u) ϕn(u,0, I) ϕn(u,U ∗, I) Themembership functions χF(u) and χ−S(u) are equal to degrees of hitting the fuzzy areas F and −S by the n-dimensional standardised random va- riables u generated according to the importance sampling method (Melchers, 1987; see Fig.6). 664 E.Szeliga Fig. 6. Generation of random values according to the importance sampling method These functions can be defined as follows χF(u)=    0 for 0¬G0 1−χG(u)=0 for G 0 < 0¬G+ 1 for G+ < 0 (4.3) χ−S(u)=    0 for 0¬G− χG(u)=0 for G − < 0¬G0 1 for G0 < 0 where χG(u)=0 is the membership function of a fuzzy value of the limit state function G(u) with fuzzy parameters (Fig.7) G(u)= {about G0 in } (4.4) Fig. 7. A fuzzy number as a value of the limit state function with fuzzy parameters Structural reliability – fuzzy sets theory approach 665 The probabilities according to Yager can be estimated as follows P̂Fα = NFα N χPF =α P̂−Sα = N−Sα N χP −S =α (4.5) where NFα and N−Sα are the numbers of hitting the α-levels of the fuzzy areas F and −S calculated as follows NFα =NFα+1 · ϕn(u,0, I) ϕn(u,U ∗, I) (4.6) N−Sα =N−Sα+1 · ϕn(u,0, I) ϕn(u,U ∗, I) References 1. PN-ISO 2394, 2000,General Principles on Reliability for Structures 2. Blockley D., 1980, The Nature of Structural Design and Safety, Wiley, Chichester 3. Gasparski W., 1988, Expertness of Design-Elements of Design Knowledge, WNT, Warsaw 4. Kacprzyk J., 1986, Fuzzy Sets in a System Analysis, (in Polish), PWN, Warsaw 5. Melchers R.E., 1987, Structural Reliability Analysis and Prediction, Wiley, Chichester 6. Nowak A.S., Collins K.R., 2000, Reliability of Structures, Mc-Graw Hill, NewYork 7. Piegat A., 1999, Fuzzy Modelling and Control, (in Polish), Akademicka OficynaWydawnicza EXIT,Warsaw 8. Szeliga E., 2000, Structural reliability in uncertainty conditions according to the fuzzy sets theory, (in Polish), Ph.D. Thesis, Warsaw University of Tech- nology 9. Zadeh L.A., 1965, Fuzzy sets, Information and Control, 8, 338-353 10. Zadeh L.A., 1978, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1, 3-28 666 E.Szeliga Niezawodność konstrukcji w ujęciu teorii zbiorów rozmytych Streszczenie W niniejszej pracy proponuje się uwzględnić w opisie konstrukcji dwa rodzaje niepewności: losowość i nieprecyzyjność. Nieprecyzyjne oceny ekspertów dotyczące konstrukcji proponuje się opisywać za pomocą liczb rozmytych.W rezultacie, nieza- wodność konstrukcji określać się będzie w oparciu o funkcję stanu granicznego z roz- mytymi parametrami. W konsekwencji, awarię konstrukcji lub jej brak traktować się będzie jako rozmyte zdarzenia losowe. Prawdopodobieństwa tych zdarzeń stano- wić będą dolne i górne oszacowanie niezawodności konstrukcji. Można je wyznaczać za pomocą metod stosowanych już w niezawodności (np. wskaźnik Hasofera-Linda lub metodyMonte Carlo).Mogą one służyć jako podstawa kalibrowania częściowych współczynników bezpieczeństwa w normach projektowych. Manuscript received January 26, 2004; accepted for print March 24, 2004