Discussion on: A non-probabilistic concept of reliability

Structural safety ELSEVIER

Structural Satety 17 (1995) lt)5 199

Discussion on:

A non-probabilistic concept of reliability Y. B e n - H a i m

Structural Safety, 14 (1994) 2 2 7 - 2 4 5 Isaac E l i s h a k o f f Department ~Mecham~ al En,~,im'ering, Florida .qthmtic Umrersity, Boca Raton, FL 33431-0991, USA

In the recent interesting article [1] Ben-Haim proposes a non-probabilistic concept of reliability. Let us recall the classical definition of reliability [2]: "Reliability: the probability that a

component part, equipment, or system will satisfactorily perform its intended function under given circumstances, such as em'ironmental conditions, limitations as to operating time, and frequency and thoroughness of maintenance, for a specified period of time". The article [1] does not give an alternative definition of the (non-probabilistic) reliability. However, a possible definition can be inferred from Eq. (60), which reads r = 1 - am~,×/tt(r~ )

(1)

where r stands for non-probabilistic reliability, "~ is the maximum radial deviation (imperfection) the ensemble of shells may experience, g(fi) is the minimum buckling load for shells whose allowable radial tolerance is ~. The quantity A..... is introduced through the following question [1]. " H o w large a radial tolerance is acceptable, when the shell will bear static axial loads up to the value A,,~,~?". Moreover, the radial tolerance is chosen in such a m a n n e r so that the axial load does not exceed the least buckling load 0 < Am~×//_t(r~)~ 1,

O_
(2)

It appears to this writer that the non-probabilistic concept of reliability is not necessary. Indeed, there exists a universally accepted probabilistic definition of reliability. The alternative, non-probabilistic definition of the reliability is not formulated in [1]. What one may need, however, is the non-probabilistic concept of sal'ety factor. Let us recall that Freudenthai's achievement [3,4] was an establishment of a possibility to directly connect the classical (deterministic) concept of safety factors with the probabilistic concept of reliability. Safety factor is one of the quantitative characteristics of the structural safety. Likewise, instead of suggesting a non-probabilistic concept of reliability, it appeared logical to this writer to suggest a non-probabilistic concept of safety-factor [5]. This safety-factor is introduced as follows: consider, for example, a system subjected to combination of loads Elsevier Science B.V.

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which belong to some convex set ~ (since often an energy bound [6,7] is utilized, the set is denoted by ~). Assume that the mathematical model is convex too. By utilizing convex analysis we determine the interval of variation of the stress E. In order to emphasize the d e p e n d e n c e of the results upon the convex set ~ , we use the following notation for stress ~ ( ~ ) : :~(~) = [,~(~'), ~ ( ~ ) ] (3) where _o-(~) is the minimum stress the system may experience, when loads vary in ~ , whereas (7(~) is the maximum stress the system may assume the loads vary in ~ . Let the yield Zy stress be also an interval uncertain variable

Y~y=[o-r,~y ]. Then the safety-factor

(4) St

can be defined as ratio of two interval variables

=

(s)

The safety-factor turns out to be an interval variable

Sr(~ ) = [_st(N), .£,(~)1.

(6)

Let both ~-(~) and Y.~ take on positive values. T h e n

_sr(~) = _o,/~(~ ~).

~ = ~r/_O-(U).

(7)

Whereas this definition strongly resembles the classical one, Eq. (7) does not coincide with the latter. Indeed, the values o-(N) and Y(~U) are d e t e r m i n e d through convex or general, non-convex, s e t - u a l u e d modeling; moreover, the uncertainty is fully accounted for through specification (via experimental data) of the convex set ~. The new definition of the (non-probabilistic) safety factor given in [5] is closely related to the theoretical estimate of knockdown factor K introduced in [6] for imperfection-sensitive structures:

(s) where Pd is the classical buckling load found through the linearized theory; /~(~) is the maximum load the system can carry when the norm of initial imperfections is b o u n d e d by 4, and is d e t e r m i n e d through use of the nonlinear theory a n d / o r numerical analysis. Engineers are accustomed to highly reliable structures, with reliabilities of order 1 - 10 -7, which has a frequency interpretation if the ensemble of produced structures is sufficiently large. A natural question arises: Will non-probabilistic reliability take on comparable values? Two alternative techniques, non-probabilistic, set-valued modelling of uncertainty and probabilistic reliability modelling must lead to the interpretation of the familiar concept of safety-factor. We should note that the two paradigms, set-valued and probabilistic ones should not replace each ,~ther. As it was shown recently, the same problem may contain both probabilistic and r m-probabilistic variables. In these circumstances one should combine (non-probabilistic) s t-valued (and in particular convex) and (probabilistic) reliability analyses to arrive at the hybrid probabilistic-convex treatment [8-10]. R e m a r k 1. Caq our non-probabilistic safety factor be associated with Ben-Haim's concept of "non-probabilistic reliability"? Moreover, can one offer a formal d e f i n i t i o n of non-probabilistic reliability. We propose the following definition: r = 1 - 1/sf (9)

l)l~cu,sion,~ /" Stru~ tura/ 3"al~'t) 17 ~1995) 195 199

197

for safety-factors exceeding unity. For the non-linear stability problems exhibiting limit loads one can define r=t(,

0
(10)

where K is the knockdown factor. With Eq. (8) taken into account we arrive at r = ~z(~)/~..

(11)

For symmetric [11] imperfection-sensitive structures. /.t(~) _< Pd; hence 0 < r < 1.

Remark 2. From Eq. (9) wc arrive at thc fl)llowing conclusion:

qsf= 1

(12)

where q = 1 - r is the non-probabilistic unreliability. Thus one could define a non-probabilistic

unreliability as the recq~roc'al to the non-prohahilistic sal'ety-lactor. The definition of the non-probabilistic reliability can be recast, if we first define the non-probabilistic safety margin M ( Z ) = \' - - E ( / ) . Then Eq. (9) can be rewritten as r( -U,") = :]4( f ) / ' <

(13)

i.e. the non-probabilistic reliability is the ratio of the safety margin (expressed in terms of the stress) to the yield stress. Since x£(~,) and xL, are interval variables, so are the safety margin M and the non-probabilistic reliability r. For design purposes it appears reasonable to utilize the lower b o u n d of the reliability _r(g')= m(~)/ff,.. This is not unlike the belief function in the D a m s t e r - S h a f f e r ' s theory of lower and upper probabilities [12]. Even though the present Note offers a definition of the non-probabilistic reliability (absent in [1]) it is still felt that reliability should be considered in the realm of probabilistic paradigm. Safety-factor. however, must be an attendant result of each uncertainty analysis, be it of probabilistic, set-valued or f u z z - s e t s - b a s e d nature.

Acknowledgement Author thanks Professor Toshiaki Hisada of the University of Tokyo. His questions during lectures in D e c e m b e r 1992, when author was a Fellow of the Japan Society of Promotion of Science, led to the introduction by this writer of safety factor in the context of convex modelling of uncertainty. Discussion with Professor Yakov Ben-Haim in D e c e m b e r 1993, when the author was serving as a Visiting Professor in the Technion - Israel Institute of Technology is acknowledged. Constructive comments of reviewers of this manuscript are sincerely appreciated. The work reported herein is supported by grant MSM-9215698 of the National Science Foundation (Dr. K.P. Chong - Program Director). Opinions, findings, conclusions and recommendations, expressed in this paper are those of the writer and do not necessarily reflect the views of the sponsoring organization.

Dts~'ussions / Structural Sat?ty l 7 (1905) 195-199

19S

References [1] [2] [3] [4] [5]

Y. Bcn-Haim, A non-probabilistic concept of reliability, Struct. Safety, 14 (1994) 227-245. McGraw-Hill Dictionao' of Scientific and Technical Terms, New York (1975). A.M. Frcudenthal, Safety of structures, Trans. ASCE, 112 (1947) 125-180. A.M. Freudenthal, Safety and probability of failure, Trans. ASCE, 121 (1956) 1337-1375.

I. Elishakoff, A new safety factor based on convex modelling, in: B.M. Ayyub and M.M. Gupta (Eds.), Uncertainty Modelling and Analysis: Theory and Applications, North-Holland, Amsterdam (1994) 145-171. [6] Y. Ben-Haim and I. Elishakoff, Com'ex Models of Uncertainty in Applied Mechanics, Elsevier, Amsterdam (1990). [7] 1. Elishakoff, Y.W. Li and J.H. Starnes, Jr., A deterministic method to predict the buckling load variability due to uncertain elastic moduli, Comput. Methods Appl. Mech. Engrg., 111 (1994) 155-167. [8] I. Elishakoff and P. Colombi, Combination of probabilistic and convex models of uncertainty when limited knowledge is present on acoustic excitation parameters, Comput. Methods" Appl. Mech. Engrg., 104 (1993) 187-209. [9] I. Elishakoff, On the uncertainty triangle, The Shock and Vibration Digest, 22 (10) (1990) 1 (editorial). [10] I. Elishakoff, Y.K. Lin and L.P. Zhu, Probabilistic and Com,ex Modeling of Acoustically Excited Structures, Elsevier, Amsterdam (1994). [11] I. Elishakoff, Probabilistic Methods in the Theory ~t" Structures, Wiley, New York (1983). [12] P. Walley, Statistical Reasoning with Imprecise Probabilities, Chapman and Hall, London (1991).

Authors' reply Prof Elishakoff writes that "It a p p e a r s . . , that the non-probabilistic concept of reliability is not necessary. Indeed, there exists a universally accepted probabilistic definition of reliability. The alternative, non-probabilistic definition of reliability is not formulated in the paper. W h a t one may need, however, is the non-probabilistic concept of safety factor." Later in his discussion Prof Elishakoff nonetheless extends an example of non-probabilistic reliability from my article and writes that the "non-probabilistic reliability is the ratio of the safety margin (expressed in terms of the stress) to the yield stress." Prof Elishakoff thus introduces a linguistic quibble, first insisting that reliability should only be probabilistic, since that is the classical definition, then recognizing that other quantifications can in fact be useful. A primary point of my article [1] is that "Reliability has a plain lexieal meaning" (p. 227) which is logically and etymologically prior to the probabilistic model referred to by Elishakoff. Once one recognizes this, many different quantifications of reliability b e c o m e possible. For instance, one can measure reliability in terms of the robustness of the system to uncertainty. A system is reliable if it is robust with respect to uncertainty; it is unreliable if it is fragile with respect to unvertainty. This idea is developed non-probabilistically in detail in a later paper [2], and is inherent in many quality-assurance theories such as Taguchi's [3, p. 3]. Robustness-based ideas can, of course, also be developed probabilistically, As mentioned above, Elishakoff in his discussion develops a non-probabilistic measure of reliability based on one of my examples, on the safety-factor concept and on interval arithmetic. Other quantifications can be imagined, and will be useful in various situations. While Elishakoff proposes that only probabilistic reliability is useful, I am not arguing for the intellectual validity of a particular model of reliability. I am adcovating intellectual pluralism: the co-existence of distinct ideas, based on distinct, even contradictory axioms, definitions and

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primitive intuitions. The natural/technological world is complex and uncertain, and our information is often fragmentary and peculiar. It behooves us to seek new and innovative ways to represent and exploit that information.

References [1] Y. Ben-Haim, A non-probabilistic concept of reliability, Struct. Safi'ty, 14 (1994) 227-245. [2] Y. Ben-Haim, A non-probabilistic measure of reliability of linear systems based o n e x p a n s i o n o f c o n v e x models, Struct. Safety, 17 (1995) 91-109. [3] G. Taguchi, E.A. Elsayed and T.C. Hsiang. Quality En~,,meering in Production Systems, McGraw-Hill, New York (1989).