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Hybrid probabilistic fuzzy and non-probabilistic model of structural reliability
Computers & Industrial Engineering 58 (2010) 463–467
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Hybrid probabilistic fuzzy and non-probabilistic model of structural reliability q Zao Ni *, Zhiping Qiu Institute of Solid Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100083, PR China
a r t i c l e
i n f o
Article history: Received 2 July 2009 Received in revised form 10 November 2009 Accepted 10 November 2009 Available online 15 November 2009 Keywords: Structural hybrid reliability Fuzzy Random Non-probabilistic Distribution function Membership function
a b s t r a c t This paper presents a new hybrid reliability model which contains randomness, fuzziness and non-probabilistic uncertainty based on the structural fuzzy random reliability and non-probabilistic set-based models. By solving the non-probabilistic set-based reliability problem and analyzing the reliability with fuzziness and randomness, the structural hybrid reliability can be obtained. The presented hybrid model has broad applicability which can handle either linear or non-linear state functions. A comparison among the presented hybrid model, probabilistic and non-probabilistic models, and the conventional probabilistic model is made through two typical numerical examples. The results show that the presented hybrid model, which may ensure structural security, is effective and practical. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction With the continuous development of technology, the complexity of engineering structural systems gradually increases so that the anticipated influence of the uncertainty on them becomes more and more profound. Over the past few decades, probability theory has made great progress in the reliability estimate of all kinds of industrial systems. The probabilistic reliability method has become the most universal method of dealing with uncertainties. The experimental data is usually scant; therefore, the applicability conditions of the probabilistic model tend to be insufficiently substantiated. Many scholars doubt the applicability of the probabilistic model; they consider that the non-probabilistic set-theoretic model is more appropriate for describing uncertainty when lacking statistical information. Some non-probabilistic methods for analyzing reliability have been brought forward by Ben-Haim and Elishakoff (1990), Ben-Haim (1994, 2006) and Elishakoff (1995a, 1995b). Elishakoff proposed the notion of the non-probabilistic safety factor, which is defined as the ratio of the yield stress – in case it is a deterministic quantity – by the upper bound of stress. In the opposite case in which the stress is deterministic but the strength is a non-probabilistic variable, this safety factor equals the lower bound of the strength divided by the stress. In the general case in which both the stress and strength are interval variables, the non-probabilistic safety factor is defined as the ratio of q
This manuscript was processed by Area Editor E.A. Elsayed. * Corresponding author. Tel.: +86 10 82328498. E-mail address: [email protected] (Z. Ni).
0360-8352/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2009.11.005
the lower bound of the strength by the upper bound of the stress. Ben-Haim proposed the concept of the non-probabilistic reliability of structures, which he later referred to as robust reliability. Robust reliability uses the maximum degree of the uncertainty that allowed by the structural system to measure the structural reliability. Furthermore, in 1996, he presented ‘‘info-gap decision model” for reliability analysis, in which the magnitude of info-gap is utilized to scale the reliability the structures. Based on interval arithmetic, Guo, Lu, and Feng (2001) and Guo and Lu (2001) quantified the uncertain structural parameters as interval variables and proposed another measure of ‘‘non-probabilistic reliability”, which is taken as the shortest distance g from the origin to the failure surface. This was done by performing a comparison between the intervals of the structural stress and strength. The reliability indexes discussed by Ben-Haim (1994, 2006), Guo et al. (2001) and Guo and Lu (2001) both use convex model to bound the uncertainty and provide the measurement of the safety degree of the non-probabilistic structures. Wang and Qiu (2003) and Qiu, Mueller, and Frommer (2004) extensively revisited the concept of ‘‘non-probabilistic reliability’’ in the critical light. They suggested a non-probabilistic convex reliability model by using the partial order relation of the superscripted hyper-rectangle or interval vectors. They put forward a non-probabilistic set-based model for structural reliability in which the ratio of the volume of the safe region to the total volume of the region constructed by the basic interval variables is suggested as the measure of structural nonprobabilistic set-theoretic safety (Wang, Qiu, & Elishakoff, 2008). The simulation of uncertainty can either implement the use of a probabilistic model or non-probabilistic model. When probabilistic
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Z. Ni, Z. Qiu / Computers & Industrial Engineering 58 (2010) 463–467
and non-probabilistic variables appear in the same problem, the failure probability can be provided by the hybrid reliability model. A combination of stochastic variables and uncertain-but-bounded variables has been suggested for applications in such circumstances (Elishakoff & Colombi, 1993; Oberkampf, Helton, Joslyn, & Wojtkiewicz, 2004). Some numerical methods, including the function approximation technique by Penmetsa and Grandhi (2002), the iterative rescaling method by Hall and Lawry (2004) and the probability bounds (p-box) approach by Karanki, Kushwaha, Verma, and Ajit (2009), have been proposed for the lower and upper bounds estimation of the structural reliability in the presence of both stochastic and interval variables. Similar problems have also been studied by Guo and Lu (2002), Berleant, Ferson, Kreinovich, and Lodwich (2005), Utkin and Kozine (2005), Kreinovich, Xiang, Starks, and Longpre (2006) and Luo, Kang, and Li (2009). As the literature survey reveals, a number of attempts have been made for mixed uncertainty quantification. However, most of the existing papers focus on the combination of stochastic randomness and interval set with ignoring the existence of fuzziness. Fuzziness is usually involved in the basic random variables of structures, and the fuzziness of parameters can make the safe state fuzzy. It is thusly irrational to consider the randomness while ignoring the existence of fuzziness (Reddy & Haldar, 1990). Moreover, most of them can only deal with structures whose state function is linear. In order to overcome the abovementioned insufficiencies and reflect the practical working state of structures more objectively, a new hybrid reliability model which contains randomness, fuzziness, and non-probabilistic uncertainty is proposed based on the conventional probabilistic model (Ditlevsen & Madsen, 1996) and non-probabilistic set-based model (Wang et al., 2008). Moreover, the presented hybrid model has broad applicability which can handle either linear or non-linear state functions. 2. Structural fuzzy random reliability model
ð1Þ
is the state function of structures, and Z = g(X) = 0 is the limit state function of random variable space, which is also called the failure surface. The basic variable space can be divided into two parts which are failure region and safe region by the failure surface. The reliability of the structure can be written as
Ps ¼
Z
fX ðxÞdx ¼
Z>0
ZZ
Z
ðx1 ; x2 ; . . . ; xm Þdx1 dx2 dxn
ð2Þ
z>0
where fX(x1, x2, . . . , xm) is the joint distributional density function of the basic random variables X1, X2, . . . , Xm. The reliability can be rewritten as
Ps ¼
ZZ
z>0
Z
f ðX 1 Þðx1 ÞfX2 ðx2 Þ fX n ðxn Þdx1 dx2 dxn
~ ¼ gðX ~ 1; X ~2; . . . ; X ~ mÞ ¼ 0 Z ¼ gðXÞ
ð3Þ
when X1, X2, . . . , Xm are mutually independent random variables. Fuzziness is usually involved in the basic random variables. For instance, structural stress is correlated with loads, geometry size, and supporting conditions; the stress is fuzzy if any of these variables is fuzzy.
ð4Þ
~ is the fuzzy random vector, X ~ 1; X ~2; . . . ; X ~ m are the basic fuzwhere X zy random variables, and the membership functions of them are lX~ ðXÞ and lX~ 1 ðx1 Þ; lX~ 2 ðx2 Þ; . . . ; lX~ n ðxn Þ, respectively. The formula of the probability for the fuzzy random event (Onisawa, 1989; Zadeh, 1968) is
~ ¼ PðXÞ
Z Z>0
lX~ ðxÞdP ¼ E½lX~ ðxÞ
ð5Þ
where E½lX~ ðxÞ is the mean value of lX~ ðxÞ. Based on Eq. (5), the fuzzy random reliability of the structure can be written as
Z ~ ¼ E½l ~ ðxÞ ¼ Ps ¼ PðXÞ lX~ ðxÞfX ðxÞdx X Z>0 Z Z Z lX~ ðx1 ; x2 ; . . . ; xm Þ ¼ z>0
fX ðx1 ; x2 ; . . . ; xm Þdx1 dx2 dxm
ð6Þ
The fuzzy random reliability can be rewritten as
Ps ¼
Z Z
z>0
Z
lX~ 1 ðx1 ÞfX 1 ðx1 ÞlX~ 2 ðx2 ÞfX2 ðx2 Þ . . . lX~ m
ðxm ÞfX m ðxm Þdx1 dx2 dxm
ð7Þ
where X1, X2, . . . , Xm are mutually independent random variables. 3. Structural non-probabilistic set-based reliability model Assuming that Y = (Y1, Y2, . . . , Yn)T is the basic interval variable vector, by use of the interval notations in interval mathematics (Alefeld & Herzberger, 1983; Moore, 1979), Yi can be expressed as
Y i 2 Y Ii ¼ ½Y i ; Y i i ¼ 1; 2; . . . ; n
Probabilistic reliability can typically be measured by the probability of structural functions which satisfy certain requirements. The structural function can be expressed by the state function, which is determined by the failure criteria. Assuming that X = (X1, X2, . . . , Xm)T is the m-dimensional random variable vector denoting the various factors that affect the structural functioning, such as the state of load, material properties, environmental factors, size, surface coarseness degree, and stress concentration and so on. Then,
Z ¼ gðXÞ ¼ gðX 1 ; X 2 ; . . . ; X m Þ
The fuzzy failure surface can be written as
ð8Þ
The state function of the structure is expressed as follows:
Z ¼ gðYÞ ¼ gðY 1 ; Y 2 ; . . . ; Y n Þ
ð9Þ
Given the specified values of Yi (i = 1, 2, . . . , n), it is possible to judge the structural state in terms of it being in the state of safety or the state of failure. The basic variable space will be divided into two parts, namely the safe region and the failure region, by the failure surface, i.e.,
Z ¼ gðYÞ ¼ gðY 1 ; Y 2 ; . . . ; Y n Þ ¼ 0
ð10Þ
The positive value of Z indicates the safe region of basic variables, while the negative value of Z represents the failure region. We shall first take the 2-dimensional state function Z = R S (R e RI, S e SI) as an example to define the non-probabilistic setbased reliability, and then we will give the general definition for multi-dimensional case. The possibility that the strength R is larger than the stress S can be expressed as
gðgðR; SÞ > 0Þ > 0
ð11Þ
where g(T) represents the possibility of the event T. It is instructive to represent the stress and strength in a plane as shown in Fig. 1. The solid rectangle shows the region of variation of both stress and strength. It is being crossed by the failure plane R = S. The safe region is hatched, whereas the failure region is unshaded. The possibility that Eq. (11) holds or the possibility that the strength is larger than the stress will be referred by us the non-probabilistic set-based safety measure (non-probabilistic set-based reliability), which can be defined as the ratio of the area of safe region to the total area of basic variables region, i.e.,
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Z. Ni, Z. Qiu / Computers & Industrial Engineering 58 (2010) 463–467
R
R R=S
R safe region
safe region
A( S , S )
R B( R , R )
S
failure region O
S
S
S
Fig. 1. Scheme for safe region and failure region.
gðgðR; SÞ > 0Þ ¼
Asafe Atotal
ð12Þ
Asafe ¼1 Atotal
ð13Þ
The state when the upper bound S of the stress is equal to the lower bound R of the strength could be called ‘‘critical state’’ as shown in Fig. 2. For the general non-linear failure surface as shown in Fig. 3, the above concept of the non-probabilistic safety measure can still be applied as a ratio of the appropriate areas. When the limit state function is the function of multi-dimension interval variables such as Eq. (10), the multidimensional region or hyper-rectangle enclosed by basic interval variables will be divided into safe region and failure region by a hyper-surface. So the non-probabilistic set-based safety measure (non-probabilistic set-based reliability) can be defined by the ratio of the hyper-volume of safe region to the hyper-volume of basic interval variables region, which is
gðgðY 1 ; Y 2 ; . . . ; Y n Þ > 0Þ ¼
S
S
Fig. 3. Nonlinear structural state function.
When the interference between the stress and the strength does not take place or when the maximum value or upper bound of the stress is equal to or smaller than the minimum value or lower bound of the strength, the event that the stress is bigger than the strength is impossible. In other words, the possibility that the strength is bigger than the stress is equal to 1, i.e.,
gðgðR; SÞ > 0Þ ¼
failure surface
R
failure region
O
R
V safe V total
ð14Þ
4. Hybrid probabilistic fuzzy and non-probabilistic reliability model If both fuzzy random variables and non-probabilistic interval variables are contained in the basic variables which relate to the state function, the failure surface can be expressed as
~ YÞ ¼ gðX ~ 1; . . . ; X ~ m; Y 1; . . . ; Y nÞ ¼ 0 Z ¼ gðX;
ð15Þ
~ ¼ fX ~1; . . . ; X ~ m g is the m-dimensional fuzzy random variable where X vector and Y = {Y1, . . . , Yn} is the n-dimensional, non-probabilistic interval variable vector. ~ is taken as a constant vector, the hyIf fuzzy random vector X brid model then transforms into the non-probabilistic one. Similarly, the hybrid model will transform into the fuzzy random model when the interval vector Y is considered as a constant one. Thus, we find that the hybrid probabilistic and non-probabilistic model is a special case of the hybrid probabilistic fuzzy and non-probabilistic model. Above all, assuming that one implementation ~ x is taken to re~ while ignoring the fuzziness place the fuzzy random vector X and randomness, according to the non-probabilistic set-based model, the non-probabilistic reliability g can be written as
g ¼ gð~xÞ
ð16Þ
where the non-probabilistic reliability g is the function of ~x. Then the structural hybrid reliability can be expressed as follows: utilizing the distributional density function and membership function of ~ the vector X.
~ Pr ¼ E½gðXÞ
ð17Þ
Then, the computational process will be described by the following linear state function:
~ bY Z ¼ aX
ð18Þ
~ is the fuzzy random variable, its membership function is where X lX~ ðxÞ and the distributional density function is f(x); Y 2 Y I ¼ Y; Y is the non-probabilistic interval variable; it is assumed the coefficients a, b > 0. From Eq. (14), the structural non-probabilistic reliability is
~ ¼ gðXÞ
~ Y r Y c þ ab X r 2Y
ð19Þ
Utilizing Eqs. (14) and (18), when the failure surface intersects the ~ is interval YI, the span of X
R
b c b ðY Y r Þ; ðY c þ Y r Þ a a
R=S
R safe region
ð20Þ
~ is involved in the interval expressed by Eq. (20), the hybrid When X reliability of this part is
R critical state
P0r ¼
O
S
S
Fig. 2. Scheme for critical state.
S
Z
bðY c þY r Þ a bðY c Y r Þ a
lX~ ðxÞgðxÞf ðxÞdx
ð21Þ
When x 2 1; ba ðY c Y r Þ , g(x) = 0, then the hybrid reliability of this part is
P00r ¼ 0
ð22Þ
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Z. Ni, Z. Qiu / Computers & Industrial Engineering 58 (2010) 463–467
b
Moreover, when x 2 a ðY c þ Y r Þ; þ1 , g(x) = 1, then the hybrid reliability of this part is
P000 r ¼
Z
þ1
bðY c þY r Þ a
lX~ ðxÞf ðxÞdx
ð23Þ
Thus, the hybrid probabilistic fuzzy and non-probabilistic reliability of the structure can be written as
Z
~ ¼ Pr ¼ E½gðXÞ ¼
Z
1 bðY c Y r Þ a
1
þ ¼
P0r
1
Z
lX~ ðxÞgðxÞf ðxÞdx
lX~ ðxÞgðxÞf ðxÞdx þ
Z
bðY c þY r Þ a
bðY c Y r Þ a
Fig. 5. Dovetail membership function
lX~ ðxÞgðxÞf ðxÞdx
þ1
lXðxÞ ~ gðxÞf ðxÞdx
bðY c þY r Þ a
þ P00r þ P 000 r
ð24Þ
which is the sum of the hybrid reliabilities in three parts.Substituting Eqs. (22) and (23) into Eq. (24), we can obtain
Pr ¼
Z
bðY c þY r Þ a
bðY a
c
r
Y Þ
lX~ ðxÞgðxÞf ðxÞdx þ
Z
þ1
bðY c þY r Þ a
lX~ ðxÞf ðxÞdx
ð25Þ
5. Numerical examples Example 1. A freely-supported beam subjected to uniform load is shown in Fig. 4. The length l, section width b, section height h, and the load q are all basic interval variables, that is l e lI = [3880, 4120] mm, b e bI = [110, 130] mm, h e hI = [225, 255] mm and q e qI = [201, 219] ~ is a N/mm2. The material of the beam is 45-steel whose strength R fuzzy random variable, its distribution form is normal, that is R N(550, 202) MPa, and the membership function lR~ ðrÞ graphed as Fig. 5 is
8 0 > > > > r480 > > < 40 lR~ ðrÞ ¼ 1 > > 660r > > > 80 > : 0
r < 480
0:75ql bh
480 6 r < 520 520 6 r < 580
ð26Þ
580 6 r < 660 r > 660
2
ð27Þ
2
Then, the failure criterion of the freely-supported beam can be defined as 2
~ Z¼R
0:75ql 2
bh
According to the presented hybrid model, the hybrid probabilistic fuzzy and non-probabilistic reliability of the freely-supported beam is 0.978011; if the fuzziness of the strength is not taken into account, that is lR~ ðrÞ ¼ 1 ð1 < r < 1Þ, the hybrid probabilistic and non-probabilistic reliability of the freely-supported beam is 0.999827. Furthermore, if the basic interval variables lI, bI, hI and qI are transformed into normal random variables according to ‘‘ 3r criterion” in probability theory (Rao, 1984), that is, l N(4000, 402) mm, b N(120, (10/3)2) mm, h N(240, 52) mm and q N(210, 32) N/mm2, the structural reliability can be obtained from the conventional probabilistic reliability model, which is Pprob 1. Example 2. Consider a case featuring a non-linear state function of ~ is a fuzzy two variables. We suppose that the material strength R random variable, its distribution form is normal, that is R N(550, 202) MPa, and the membership function lR~ ðrÞ graphed as Fig. 6 is ðr100Þ2 10
lR~ ðrÞ ¼ e
1
ð29Þ I
The working stress interval is S = [85 b, 85 + b] MPa where b is the control parameter which is introduced here to describe the possible scatter in the working stress. The state function of the structural member is taken as
Z ¼ R 2 S2
According to the fundamental theory of material mechanics, the maximum stress of the freely-supported beam is
S¼
lR~ ðrÞ.
¼0
ð28Þ
ð30Þ
The presented hybrid model and the conventional probabilistic model are used to compute the reliability of the structural member. When utilizing the conventional probabilistic model, the working stress interval variable should be transformed into normal random variables according to ‘‘3r criterion”, that is S N(85, (b/3)2) MPa. Fig. 7 shows the variation in the reliabilities obtained by the two models with the parameter b. We can see from Fig. 7 that in the case of the specified stress interval, if the strength interval is widening, the interference region between the stress and strength also becomes wider. As a result, the reliability of the structural member decreases. We also find that Pr1 (which represents the reliability obtained by conventional probabilistic model) is larger than Pr2 (which represents the reliability obtained by the presented hybrid model). These results
l
h b Fig. 4. Force diagram of a free beam.
Fig. 6. Normal membership function
lA~ ðzÞ.
Z. Ni, Z. Qiu / Computers & Industrial Engineering 58 (2010) 463–467
0.985
References Pr1
0.980
Pr2
0.975 0.970
Pr 0.965 0.960 0.955 0.950
467
10
11
12
β
13
14
15
Fig. 7. Structural member reliability versus b.
can be anticipated qualitatively. We provide the quantitative assessment of this fact. For example, for b = 5, Pr1 = 0.98276 and Pr2 = 0.97454, whereas for b = 15, Pr1 = 0.96682 and Pr2 = 0.95117. From the above numerical examples, it can be seen that the reliability obtained by the presented hybrid model is lesser than those obtained by both the hybrid probabilistic and non-probabilistic model and conventional probabilistic model. That is to say, the reliability obtained by the presented hybrid model is more conservative and can ensure structural security to a larger extent. 6. Conclusions In structural analysis and design, it is necessary to deal with the uncertainty that affects the performance of structures properly. In order to reflect the practical working state of structures more objectively, a new hybrid reliability model containing randomness, fuzziness and non-probabilistic uncertainty is proposed in this paper. The presented hybrid model has broad applicability which can handle both linear and non-linear state functions. Numerical examples show that the reliability obtained by the presented hybrid model is lesser than those obtained by both the hybrid probabilistic and non-probabilistic model and conventional probabilistic model. That is to say the reliability obtained by the presented hybrid model is more conservative. We can thusly draw the conclusion that the presented hybrid model can utilize available data and the disadvantages inherent in each individual model fully and effectively and may reflect, truly and objectively, the real state of structural safety. Acknowledgements The work is supported by the Aeronautical Science Foundation of China (Grant No. 2007ZA51003), the National Natural Science Foundation of China (Grant Nos. 90816024, 10872017, and 10876100), the Astronautical Technology Innovation Foundation of China and the 111 Project (Grant No. B07009).
Alefeld, G., & Herzberger, J. (1983). Introductions to interval computations. New York: Academic Press. Ben-Haim, Y., & Elishakoff, I. (1990). Convex models of uncertainty in applied mechanics. Amsterdam: Elsevier Science. Ben-Haim, Y. (1994). A non-probabilistic concept of reliability. Structural Safety, 14, 227–245. Ben-Haim, Y. (2006). Info-gap decision theory: decisions under serve uncertainty (2nd ed.). New York: Academic Press. Berleant, D. J., Ferson, S., Kreinovich, V., Lodwich, W. A. (2005). Combining interval and probabilistic uncertainty: Foundations, algorithms, challenges – An overview. In Fourth international symposium on imprecise probabilities and their applications. Pittsburgh, Pennsylvania. Ditlevsen, O. D., & Madsen, H. O. (1996). Structural reliability method. New York: John Wiley & Sons. Elishakoff, I., & Colombi, P. (1993). Combination of probabilistic and convex models of uncertainty when scarce knowledge is present on acoustic excitation parameters. Computer Methods in Applied Mechanics and Engineering, 104(2), 187–209. Elishakoff, I. (1995a). Essay on uncertainties in elastic and viscoelastic structures: From A.M. Freudenthal’s criticisms to modern convex modeling. Computers & Structures, 56(6), 871–895. Elishakoff, I. (1995b). Discussion on: A non-probabilistic concept of reliability. Structural Safety, 17(3), 195–199. Guo, S. X., Lu, Z. Z., & Feng, Y. Y. (2001). A non-probabilistic model of structural reliability based on interval analysis. Chinese Journal of Computational Mechanics, 18(1), 56–60 [in Chinese]. Guo, S. X., & Lu, Z. Z. (2001). Procedure for the robust reliability design of structures. Chinese Journal of Applied Mechanics, 22(5), 451–453 [in Chinese]. Guo, S. X., & Lu, Z. Z. (2002). Hybrid probabilistic and non-probabilistic model of structural reliability. Journal of Mechanical Strength, 24(4), 524–526 [in Chinese]. Hall, J. W., & Lawry, J. (2004). Generation combination and extension of random set approximations to coherent lower and upper probabilities. Reliability Engineering and System Safety, 85, 89–101. Karanki, D. R., Kushwaha, H. S., Verma, A. K., & Ajit, S. (2009). Uncertainty analysis based on probability bounds (p-box) approach in probabilistic safety assessment. Risk Analysis, 29(5), 662–675. Kreinovich, V., Xiang, G., Starks, S. A., & Longpre, L. (2006). Towards combining probabilistic and interval uncertainty in engineering calculations: Algorithms for computing statistics under interval uncertainty, and their computational complexity. Reliable Computing, 12, 1385–3139. Luo, Y. J., Kang, Z., & Li, A. (2009). Structural reliability assessment based on probability and convex set mixed model. Computers & Structures, 87(21–22), 1408–1415. Moore, R. E. (1979). Methods and applications of interval analysis. London: PrenticeHall, Inc. Oberkampf, W. L., Helton, J. C., Joslyn, C. A., & Wojtkiewicz, S. F. (2004). Challenge problems: Uncertainty in system response given uncertain parameters. Reliability Engineering and System Safety, 85, 11–19. Onisawa, T. (1989). Fuzzy theory in reliability analysis. Fuzzy Sets and Systems, 29, 250–251. Penmetsa, R. C., & Grandhi, R. V. (2002). Efficient estimation of structural reliability for problems with uncertain intervals. Computer & Structures, 80, 1103–1112. Qiu, Z. P., Mueller, P. C., & Frommer, A. (2004). The new non-probabilistic criterion of failure for dynamical systems based on convex models. Mathematical and Computer Modeling, 40(12), 201–215. Rao, M. M. (1984). Probability theory with applications. New York: Academic Press. Reddy, R. K., & Haldar, A. (1990). A random-fuzzy reliability analysis. In B. M. Ayyub (Ed.), Proceedings of the first international symposium on uncertainty modeling and analysis (pp. 161–166). Los Alamitos: IEEE Computer Society Press. Utkin, L. V., & Kozine, I. O. (2005). Computing system reliability given interval-value characteristics of the components. Reliable Computing, 11, 19–34. Wang, X. J., & Qiu, Z. P. (2003). Robust reliability of structural vibration. Journal of Beijing University of Aeronautics and Astronautics, 29(11), 1006–1010 [in Chinese]. Wang, X. J., Qiu, Z. P., & Elishakoff, I. (2008). Non-probabilistic set-theoretic model for structural safety measure. Acta Mechanica, 198, 51–64. Zadeh, L. A. (1968). Fuzzy algorithm. Information and Control, 12, 94–120.
Contents lists available at ScienceDirect
Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Hybrid probabilistic fuzzy and non-probabilistic model of structural reliability q Zao Ni *, Zhiping Qiu Institute of Solid Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100083, PR China
a r t i c l e
i n f o
Article history: Received 2 July 2009 Received in revised form 10 November 2009 Accepted 10 November 2009 Available online 15 November 2009 Keywords: Structural hybrid reliability Fuzzy Random Non-probabilistic Distribution function Membership function
a b s t r a c t This paper presents a new hybrid reliability model which contains randomness, fuzziness and non-probabilistic uncertainty based on the structural fuzzy random reliability and non-probabilistic set-based models. By solving the non-probabilistic set-based reliability problem and analyzing the reliability with fuzziness and randomness, the structural hybrid reliability can be obtained. The presented hybrid model has broad applicability which can handle either linear or non-linear state functions. A comparison among the presented hybrid model, probabilistic and non-probabilistic models, and the conventional probabilistic model is made through two typical numerical examples. The results show that the presented hybrid model, which may ensure structural security, is effective and practical. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction With the continuous development of technology, the complexity of engineering structural systems gradually increases so that the anticipated influence of the uncertainty on them becomes more and more profound. Over the past few decades, probability theory has made great progress in the reliability estimate of all kinds of industrial systems. The probabilistic reliability method has become the most universal method of dealing with uncertainties. The experimental data is usually scant; therefore, the applicability conditions of the probabilistic model tend to be insufficiently substantiated. Many scholars doubt the applicability of the probabilistic model; they consider that the non-probabilistic set-theoretic model is more appropriate for describing uncertainty when lacking statistical information. Some non-probabilistic methods for analyzing reliability have been brought forward by Ben-Haim and Elishakoff (1990), Ben-Haim (1994, 2006) and Elishakoff (1995a, 1995b). Elishakoff proposed the notion of the non-probabilistic safety factor, which is defined as the ratio of the yield stress – in case it is a deterministic quantity – by the upper bound of stress. In the opposite case in which the stress is deterministic but the strength is a non-probabilistic variable, this safety factor equals the lower bound of the strength divided by the stress. In the general case in which both the stress and strength are interval variables, the non-probabilistic safety factor is defined as the ratio of q
This manuscript was processed by Area Editor E.A. Elsayed. * Corresponding author. Tel.: +86 10 82328498. E-mail address: [email protected] (Z. Ni).
0360-8352/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2009.11.005
the lower bound of the strength by the upper bound of the stress. Ben-Haim proposed the concept of the non-probabilistic reliability of structures, which he later referred to as robust reliability. Robust reliability uses the maximum degree of the uncertainty that allowed by the structural system to measure the structural reliability. Furthermore, in 1996, he presented ‘‘info-gap decision model” for reliability analysis, in which the magnitude of info-gap is utilized to scale the reliability the structures. Based on interval arithmetic, Guo, Lu, and Feng (2001) and Guo and Lu (2001) quantified the uncertain structural parameters as interval variables and proposed another measure of ‘‘non-probabilistic reliability”, which is taken as the shortest distance g from the origin to the failure surface. This was done by performing a comparison between the intervals of the structural stress and strength. The reliability indexes discussed by Ben-Haim (1994, 2006), Guo et al. (2001) and Guo and Lu (2001) both use convex model to bound the uncertainty and provide the measurement of the safety degree of the non-probabilistic structures. Wang and Qiu (2003) and Qiu, Mueller, and Frommer (2004) extensively revisited the concept of ‘‘non-probabilistic reliability’’ in the critical light. They suggested a non-probabilistic convex reliability model by using the partial order relation of the superscripted hyper-rectangle or interval vectors. They put forward a non-probabilistic set-based model for structural reliability in which the ratio of the volume of the safe region to the total volume of the region constructed by the basic interval variables is suggested as the measure of structural nonprobabilistic set-theoretic safety (Wang, Qiu, & Elishakoff, 2008). The simulation of uncertainty can either implement the use of a probabilistic model or non-probabilistic model. When probabilistic
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and non-probabilistic variables appear in the same problem, the failure probability can be provided by the hybrid reliability model. A combination of stochastic variables and uncertain-but-bounded variables has been suggested for applications in such circumstances (Elishakoff & Colombi, 1993; Oberkampf, Helton, Joslyn, & Wojtkiewicz, 2004). Some numerical methods, including the function approximation technique by Penmetsa and Grandhi (2002), the iterative rescaling method by Hall and Lawry (2004) and the probability bounds (p-box) approach by Karanki, Kushwaha, Verma, and Ajit (2009), have been proposed for the lower and upper bounds estimation of the structural reliability in the presence of both stochastic and interval variables. Similar problems have also been studied by Guo and Lu (2002), Berleant, Ferson, Kreinovich, and Lodwich (2005), Utkin and Kozine (2005), Kreinovich, Xiang, Starks, and Longpre (2006) and Luo, Kang, and Li (2009). As the literature survey reveals, a number of attempts have been made for mixed uncertainty quantification. However, most of the existing papers focus on the combination of stochastic randomness and interval set with ignoring the existence of fuzziness. Fuzziness is usually involved in the basic random variables of structures, and the fuzziness of parameters can make the safe state fuzzy. It is thusly irrational to consider the randomness while ignoring the existence of fuzziness (Reddy & Haldar, 1990). Moreover, most of them can only deal with structures whose state function is linear. In order to overcome the abovementioned insufficiencies and reflect the practical working state of structures more objectively, a new hybrid reliability model which contains randomness, fuzziness, and non-probabilistic uncertainty is proposed based on the conventional probabilistic model (Ditlevsen & Madsen, 1996) and non-probabilistic set-based model (Wang et al., 2008). Moreover, the presented hybrid model has broad applicability which can handle either linear or non-linear state functions. 2. Structural fuzzy random reliability model
ð1Þ
is the state function of structures, and Z = g(X) = 0 is the limit state function of random variable space, which is also called the failure surface. The basic variable space can be divided into two parts which are failure region and safe region by the failure surface. The reliability of the structure can be written as
Ps ¼
Z
fX ðxÞdx ¼
Z>0
ZZ
Z
ðx1 ; x2 ; . . . ; xm Þdx1 dx2 dxn
ð2Þ
z>0
where fX(x1, x2, . . . , xm) is the joint distributional density function of the basic random variables X1, X2, . . . , Xm. The reliability can be rewritten as
Ps ¼
ZZ
z>0
Z
f ðX 1 Þðx1 ÞfX2 ðx2 Þ fX n ðxn Þdx1 dx2 dxn
~ ¼ gðX ~ 1; X ~2; . . . ; X ~ mÞ ¼ 0 Z ¼ gðXÞ
ð3Þ
when X1, X2, . . . , Xm are mutually independent random variables. Fuzziness is usually involved in the basic random variables. For instance, structural stress is correlated with loads, geometry size, and supporting conditions; the stress is fuzzy if any of these variables is fuzzy.
ð4Þ
~ is the fuzzy random vector, X ~ 1; X ~2; . . . ; X ~ m are the basic fuzwhere X zy random variables, and the membership functions of them are lX~ ðXÞ and lX~ 1 ðx1 Þ; lX~ 2 ðx2 Þ; . . . ; lX~ n ðxn Þ, respectively. The formula of the probability for the fuzzy random event (Onisawa, 1989; Zadeh, 1968) is
~ ¼ PðXÞ
Z Z>0
lX~ ðxÞdP ¼ E½lX~ ðxÞ
ð5Þ
where E½lX~ ðxÞ is the mean value of lX~ ðxÞ. Based on Eq. (5), the fuzzy random reliability of the structure can be written as
Z ~ ¼ E½l ~ ðxÞ ¼ Ps ¼ PðXÞ lX~ ðxÞfX ðxÞdx X Z>0 Z Z Z lX~ ðx1 ; x2 ; . . . ; xm Þ ¼ z>0
fX ðx1 ; x2 ; . . . ; xm Þdx1 dx2 dxm
ð6Þ
The fuzzy random reliability can be rewritten as
Ps ¼
Z Z
z>0
Z
lX~ 1 ðx1 ÞfX 1 ðx1 ÞlX~ 2 ðx2 ÞfX2 ðx2 Þ . . . lX~ m
ðxm ÞfX m ðxm Þdx1 dx2 dxm
ð7Þ
where X1, X2, . . . , Xm are mutually independent random variables. 3. Structural non-probabilistic set-based reliability model Assuming that Y = (Y1, Y2, . . . , Yn)T is the basic interval variable vector, by use of the interval notations in interval mathematics (Alefeld & Herzberger, 1983; Moore, 1979), Yi can be expressed as
Y i 2 Y Ii ¼ ½Y i ; Y i i ¼ 1; 2; . . . ; n
Probabilistic reliability can typically be measured by the probability of structural functions which satisfy certain requirements. The structural function can be expressed by the state function, which is determined by the failure criteria. Assuming that X = (X1, X2, . . . , Xm)T is the m-dimensional random variable vector denoting the various factors that affect the structural functioning, such as the state of load, material properties, environmental factors, size, surface coarseness degree, and stress concentration and so on. Then,
Z ¼ gðXÞ ¼ gðX 1 ; X 2 ; . . . ; X m Þ
The fuzzy failure surface can be written as
ð8Þ
The state function of the structure is expressed as follows:
Z ¼ gðYÞ ¼ gðY 1 ; Y 2 ; . . . ; Y n Þ
ð9Þ
Given the specified values of Yi (i = 1, 2, . . . , n), it is possible to judge the structural state in terms of it being in the state of safety or the state of failure. The basic variable space will be divided into two parts, namely the safe region and the failure region, by the failure surface, i.e.,
Z ¼ gðYÞ ¼ gðY 1 ; Y 2 ; . . . ; Y n Þ ¼ 0
ð10Þ
The positive value of Z indicates the safe region of basic variables, while the negative value of Z represents the failure region. We shall first take the 2-dimensional state function Z = R S (R e RI, S e SI) as an example to define the non-probabilistic setbased reliability, and then we will give the general definition for multi-dimensional case. The possibility that the strength R is larger than the stress S can be expressed as
gðgðR; SÞ > 0Þ > 0
ð11Þ
where g(T) represents the possibility of the event T. It is instructive to represent the stress and strength in a plane as shown in Fig. 1. The solid rectangle shows the region of variation of both stress and strength. It is being crossed by the failure plane R = S. The safe region is hatched, whereas the failure region is unshaded. The possibility that Eq. (11) holds or the possibility that the strength is larger than the stress will be referred by us the non-probabilistic set-based safety measure (non-probabilistic set-based reliability), which can be defined as the ratio of the area of safe region to the total area of basic variables region, i.e.,
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R
R R=S
R safe region
safe region
A( S , S )
R B( R , R )
S
failure region O
S
S
S
Fig. 1. Scheme for safe region and failure region.
gðgðR; SÞ > 0Þ ¼
Asafe Atotal
ð12Þ
Asafe ¼1 Atotal
ð13Þ
The state when the upper bound S of the stress is equal to the lower bound R of the strength could be called ‘‘critical state’’ as shown in Fig. 2. For the general non-linear failure surface as shown in Fig. 3, the above concept of the non-probabilistic safety measure can still be applied as a ratio of the appropriate areas. When the limit state function is the function of multi-dimension interval variables such as Eq. (10), the multidimensional region or hyper-rectangle enclosed by basic interval variables will be divided into safe region and failure region by a hyper-surface. So the non-probabilistic set-based safety measure (non-probabilistic set-based reliability) can be defined by the ratio of the hyper-volume of safe region to the hyper-volume of basic interval variables region, which is
gðgðY 1 ; Y 2 ; . . . ; Y n Þ > 0Þ ¼
S
S
Fig. 3. Nonlinear structural state function.
When the interference between the stress and the strength does not take place or when the maximum value or upper bound of the stress is equal to or smaller than the minimum value or lower bound of the strength, the event that the stress is bigger than the strength is impossible. In other words, the possibility that the strength is bigger than the stress is equal to 1, i.e.,
gðgðR; SÞ > 0Þ ¼
failure surface
R
failure region
O
R
V safe V total
ð14Þ
4. Hybrid probabilistic fuzzy and non-probabilistic reliability model If both fuzzy random variables and non-probabilistic interval variables are contained in the basic variables which relate to the state function, the failure surface can be expressed as
~ YÞ ¼ gðX ~ 1; . . . ; X ~ m; Y 1; . . . ; Y nÞ ¼ 0 Z ¼ gðX;
ð15Þ
~ ¼ fX ~1; . . . ; X ~ m g is the m-dimensional fuzzy random variable where X vector and Y = {Y1, . . . , Yn} is the n-dimensional, non-probabilistic interval variable vector. ~ is taken as a constant vector, the hyIf fuzzy random vector X brid model then transforms into the non-probabilistic one. Similarly, the hybrid model will transform into the fuzzy random model when the interval vector Y is considered as a constant one. Thus, we find that the hybrid probabilistic and non-probabilistic model is a special case of the hybrid probabilistic fuzzy and non-probabilistic model. Above all, assuming that one implementation ~ x is taken to re~ while ignoring the fuzziness place the fuzzy random vector X and randomness, according to the non-probabilistic set-based model, the non-probabilistic reliability g can be written as
g ¼ gð~xÞ
ð16Þ
where the non-probabilistic reliability g is the function of ~x. Then the structural hybrid reliability can be expressed as follows: utilizing the distributional density function and membership function of ~ the vector X.
~ Pr ¼ E½gðXÞ
ð17Þ
Then, the computational process will be described by the following linear state function:
~ bY Z ¼ aX
ð18Þ
~ is the fuzzy random variable, its membership function is where X lX~ ðxÞ and the distributional density function is f(x); Y 2 Y I ¼ Y; Y is the non-probabilistic interval variable; it is assumed the coefficients a, b > 0. From Eq. (14), the structural non-probabilistic reliability is
~ ¼ gðXÞ
~ Y r Y c þ ab X r 2Y
ð19Þ
Utilizing Eqs. (14) and (18), when the failure surface intersects the ~ is interval YI, the span of X
R
b c b ðY Y r Þ; ðY c þ Y r Þ a a
R=S
R safe region
ð20Þ
~ is involved in the interval expressed by Eq. (20), the hybrid When X reliability of this part is
R critical state
P0r ¼
O
S
S
Fig. 2. Scheme for critical state.
S
Z
bðY c þY r Þ a bðY c Y r Þ a
lX~ ðxÞgðxÞf ðxÞdx
ð21Þ
When x 2 1; ba ðY c Y r Þ , g(x) = 0, then the hybrid reliability of this part is
P00r ¼ 0
ð22Þ
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b
Moreover, when x 2 a ðY c þ Y r Þ; þ1 , g(x) = 1, then the hybrid reliability of this part is
P000 r ¼
Z
þ1
bðY c þY r Þ a
lX~ ðxÞf ðxÞdx
ð23Þ
Thus, the hybrid probabilistic fuzzy and non-probabilistic reliability of the structure can be written as
Z
~ ¼ Pr ¼ E½gðXÞ ¼
Z
1 bðY c Y r Þ a
1
þ ¼
P0r
1
Z
lX~ ðxÞgðxÞf ðxÞdx
lX~ ðxÞgðxÞf ðxÞdx þ
Z
bðY c þY r Þ a
bðY c Y r Þ a
Fig. 5. Dovetail membership function
lX~ ðxÞgðxÞf ðxÞdx
þ1
lXðxÞ ~ gðxÞf ðxÞdx
bðY c þY r Þ a
þ P00r þ P 000 r
ð24Þ
which is the sum of the hybrid reliabilities in three parts.Substituting Eqs. (22) and (23) into Eq. (24), we can obtain
Pr ¼
Z
bðY c þY r Þ a
bðY a
c
r
Y Þ
lX~ ðxÞgðxÞf ðxÞdx þ
Z
þ1
bðY c þY r Þ a
lX~ ðxÞf ðxÞdx
ð25Þ
5. Numerical examples Example 1. A freely-supported beam subjected to uniform load is shown in Fig. 4. The length l, section width b, section height h, and the load q are all basic interval variables, that is l e lI = [3880, 4120] mm, b e bI = [110, 130] mm, h e hI = [225, 255] mm and q e qI = [201, 219] ~ is a N/mm2. The material of the beam is 45-steel whose strength R fuzzy random variable, its distribution form is normal, that is R N(550, 202) MPa, and the membership function lR~ ðrÞ graphed as Fig. 5 is
8 0 > > > > r480 > > < 40 lR~ ðrÞ ¼ 1 > > 660r > > > 80 > : 0
r < 480
0:75ql bh
480 6 r < 520 520 6 r < 580
ð26Þ
580 6 r < 660 r > 660
2
ð27Þ
2
Then, the failure criterion of the freely-supported beam can be defined as 2
~ Z¼R
0:75ql 2
bh
According to the presented hybrid model, the hybrid probabilistic fuzzy and non-probabilistic reliability of the freely-supported beam is 0.978011; if the fuzziness of the strength is not taken into account, that is lR~ ðrÞ ¼ 1 ð1 < r < 1Þ, the hybrid probabilistic and non-probabilistic reliability of the freely-supported beam is 0.999827. Furthermore, if the basic interval variables lI, bI, hI and qI are transformed into normal random variables according to ‘‘ 3r criterion” in probability theory (Rao, 1984), that is, l N(4000, 402) mm, b N(120, (10/3)2) mm, h N(240, 52) mm and q N(210, 32) N/mm2, the structural reliability can be obtained from the conventional probabilistic reliability model, which is Pprob 1. Example 2. Consider a case featuring a non-linear state function of ~ is a fuzzy two variables. We suppose that the material strength R random variable, its distribution form is normal, that is R N(550, 202) MPa, and the membership function lR~ ðrÞ graphed as Fig. 6 is ðr100Þ2 10
lR~ ðrÞ ¼ e
1
ð29Þ I
The working stress interval is S = [85 b, 85 + b] MPa where b is the control parameter which is introduced here to describe the possible scatter in the working stress. The state function of the structural member is taken as
Z ¼ R 2 S2
According to the fundamental theory of material mechanics, the maximum stress of the freely-supported beam is
S¼
lR~ ðrÞ.
¼0
ð28Þ
ð30Þ
The presented hybrid model and the conventional probabilistic model are used to compute the reliability of the structural member. When utilizing the conventional probabilistic model, the working stress interval variable should be transformed into normal random variables according to ‘‘3r criterion”, that is S N(85, (b/3)2) MPa. Fig. 7 shows the variation in the reliabilities obtained by the two models with the parameter b. We can see from Fig. 7 that in the case of the specified stress interval, if the strength interval is widening, the interference region between the stress and strength also becomes wider. As a result, the reliability of the structural member decreases. We also find that Pr1 (which represents the reliability obtained by conventional probabilistic model) is larger than Pr2 (which represents the reliability obtained by the presented hybrid model). These results
l
h b Fig. 4. Force diagram of a free beam.
Fig. 6. Normal membership function
lA~ ðzÞ.
Z. Ni, Z. Qiu / Computers & Industrial Engineering 58 (2010) 463–467
0.985
References Pr1
0.980
Pr2
0.975 0.970
Pr 0.965 0.960 0.955 0.950
467
10
11
12
β
13
14
15
Fig. 7. Structural member reliability versus b.
can be anticipated qualitatively. We provide the quantitative assessment of this fact. For example, for b = 5, Pr1 = 0.98276 and Pr2 = 0.97454, whereas for b = 15, Pr1 = 0.96682 and Pr2 = 0.95117. From the above numerical examples, it can be seen that the reliability obtained by the presented hybrid model is lesser than those obtained by both the hybrid probabilistic and non-probabilistic model and conventional probabilistic model. That is to say, the reliability obtained by the presented hybrid model is more conservative and can ensure structural security to a larger extent. 6. Conclusions In structural analysis and design, it is necessary to deal with the uncertainty that affects the performance of structures properly. In order to reflect the practical working state of structures more objectively, a new hybrid reliability model containing randomness, fuzziness and non-probabilistic uncertainty is proposed in this paper. The presented hybrid model has broad applicability which can handle both linear and non-linear state functions. Numerical examples show that the reliability obtained by the presented hybrid model is lesser than those obtained by both the hybrid probabilistic and non-probabilistic model and conventional probabilistic model. That is to say the reliability obtained by the presented hybrid model is more conservative. We can thusly draw the conclusion that the presented hybrid model can utilize available data and the disadvantages inherent in each individual model fully and effectively and may reflect, truly and objectively, the real state of structural safety. Acknowledgements The work is supported by the Aeronautical Science Foundation of China (Grant No. 2007ZA51003), the National Natural Science Foundation of China (Grant Nos. 90816024, 10872017, and 10876100), the Astronautical Technology Innovation Foundation of China and the 111 Project (Grant No. B07009).
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