Issue 66

R. B. P. Nonato, Frattura ed Integrità Strutturale, 65 (2023) 17-37; DOI: 10.3221/IGF-ESIS.66.02









  , , Z X X X ,

   , , , , , X Z X X X X 





 , ,

k

i

i

n

k

i

i

n

1

1

1

1



(16)

ik EE

i

i



in which  is a vector of values within the interval associated with the UIQ i X , and k Z is the SRQ. One of the variance-based methods, the first-order sensitivity index, can be defined by the variance of the conditional expectation of each UIQ related to the variance of the common SRQ. Mathematically, Eqn. 17 provides the ik -th first order index corresponding to the effect of the UIQ i X related to the SRQ k Z , where   . V and   . E mean, respectively, the variance and the expected value operators. This can be understood as the variance of the expectation of the k -th SRQ conditioned to the occurrence of the i -th UIQ related to the entire variance of the SRQ.          | k i ik k V E S V Z X Z (17) his section associates the maximum entropy principle (MEP) with UQ. The general framework of MEP applied to known support and moments of a distribution is presented in order to evidence the particular case where only interval and mean of data are concomitantly known, a situation applied in this paper to represent the UIQs. MEP in the context of uncertainty In the case where insufficient number of probabilistic distribution moments are the only information available, the MEP can shape the most unbiased distribution, meeting this given data [42]. The contributions strictly related to structural systems has an ever-growing potential, which has been highlighted by the greater attention deserved through last years. Based on the MEP, a belief reliability distribution was proposed to measure system performance from the influences of design margin, aleatory and epistemic uncertainties [43]. PDFs were estimated using more than four moments of distribution and a reliability-based design optimization was performed using the MEP and compared to that originated by the finite difference method [44]. The MEP was also applied to obtain the reliability bound with respect to the first moment truncated for the first time. The reliability of machine components via MEP was studied too [45], in which a PDF and a failure probability model were established. A structural reliability analysis was performed based on the MEP using polynomial chaos expansion, e.g. [46]. In addition, the MEP and the Dempster-Shafer theory were combined to perform a reliability analysis [47]. General framework of MEP applied to known support and moments of a distribution In the case where a continuous PDF of a variable is unknown and a set of its parameters is available (support and/or moments distributions), it is possible to apply the MEP in order to find the distribution using the restrictions provided by this information, according to the processes proposed in [16, 48]. Let i X be a random variable, which has   i X i f x as its PDF and   i X i F x as its CDF. The entropy H is the quantity of uncertainty inherent to a process result. Let i X be delimited by a support interval     , LB UB i i X X , which can be the only available information or one item of a set of information about the distribution. Therefore, the entropy is quantified in Eqn. 18: T M AXIMUM ENTROPY PRINCIPLE

    X i

    X i

UB

UB

X

X

 





i

i

 H f



x log f

x dx

log f

x dF .

(18)

X i

i

X

LB

LB

X

X

i

i

i

i

i

i

For example, H can be interpreted as the average information provided by the realization of an experiment or simulation. It is the average uncertainty removed or average amount of information aggregated by observing the outcomes of i X . In addition, if information about the moments of the PDF is also available,  im X , then it is given by Eqn. 19. The m -th moment

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