A new structural reliability index based on uncertainty theory

Wang, Pidong; Zhang, Jianguo; Zhai, Hao; Qiu, Jiwei

doi:10.1016/j.cja.2017.04.008

Cited by 34 publications

(26 citation statements)

References 25 publications

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“…In this case, the reliability index is actually the Cornell reliability index which just considers the effect of aleatory uncertainty. Besides, when the uncertain random variable degenerates to an uncertain variable τ = f ( τ 1 , …, τ n ), the reliability index can degenerate to the reliability index based on the uncertainty measure, and then according to the definition of γ Ch , the system reliability index γ U is defined as

\begin{matrix} lefttrue \\ γ_{U} = \frac{E (f (τ_{1} \dots τ_{n}))}{\sqrt{V ((), f ((),, τ_{1} \dots τ_{n}))}} \\ \begin{array}{c}  \end{array} = \frac{\int_{0}^{1} f ({normalϒ}_{1}^{- 1} (α) {normalϒ}_{2}^{- 1} (α) \dots {normalϒ}_{k}^{- 1} (α) {normalϒ}_{k + 1}^{- 1} (1 - α) {normalϒ}_{k + 2}^{- 1} (1 - α) \dots {normalϒ}_{n}^{- 1} (1 - α)) dα}{\sqrt{\int_{0}^{1} {(f ({normalϒ}_{1}^{- 1} (α) {normalϒ}_{2}^{- 1} (α) \dots {normalϒ}_{k}^{- 1} (α) {normalϒ}_{k + 1}^{- 1} (1 - α) {normalϒ}_{k + 2}^{- 1} (1 - α) \dots {normalϒ}_{n}^{- 1} (1 - α)) - e)}^{2} italicdα}} \end{matrix}

…”

Section: The New Reliability Index Based On Chance Measuresupporting

confidence: 89%

“…The uncertainty distribution of an uncertain variable τ is defined by Equation for any real number x . For example, the linear uncertain variable τ ~ L ( a , b ) has an uncertainty distribution

normalϒ ((), x) = ({, \begin{array}{l} 0, \begin{matrix} center \\ if & x \leq a \end{matrix} \\ \begin{matrix} center \\ (x - a) / (b - a), & \begin{array}{c} italicif & a \leq x \leq b \end{array} \end{matrix} \\ \begin{matrix} center \\ 1, & \begin{array}{c} italicif & x \geq b \end{array} \end{matrix} \end{array})

where a and b are real numbers with a < b , as shown in Figure . Besides, the zigzag uncertain variable τ ~ Z ( a , b , c ) has an uncertainty distribution

normalϒ ((), x) = ({, \begin{array}{l} \begin{matrix} center \\ 0, & if & x \leq a \end{matrix} \\ \begin{matrix} center \\ (x - a) / 2 (b - a), & if & a \leq x \leq b \end{matrix} \\ \begin{matrix} center \\ (x + c - 2 b) / 2 (c - b), & if & b \leq x \leq c \end{matrix} \\ \begin{matrix} center \\ 1, & if & x \geq c \end{matrix} \end{array})

…”

Section: Preliminarymentioning

confidence: 99%

“…If f ( η 1 , η 2 , …, η m , τ 1 , τ 2 , …, τ n ) is strictly increasing with respect to τ 1 , τ 2 , …, τ k and strictly decreasing with respect to τ k + 1 , τ k + 2 , …, τ n , then the uncertain random variable

ξ = f ((),,,,,,, η_{1} η_{2} \dots η_{m} τ_{1} τ_{2} \dots τ_{n})

has a chance distribution

normalΦ ((), x) = \int_{ℜ^{m}} F (();,,, x y_{1} y_{2} \dots y_{m}) d Ψ_{1} ((), y_{1}) d Ψ_{2} ((), y_{2}) \dots d Ψ_{m} ((), y_{m})

where F ( x ; y 1 , y 2 , …, y m ) is the uncertainty distribution of the uncertain variable f ( y 1 , y 2 , …, y m , τ 1 , τ 2 , …, τ n ) for any given real numbers y 1 , y 2 , …, y m . According to the definition of the uncertainty distribution in the article, when ϒ 1 , ϒ 2 , …, ϒ n are continuous uncertainty distributions, F ( x ; y 1 , y 2 , …, y m ) can be determined by the following formula:

F (();,, x y_{1} \dots y_{m}) = sup_{f ((),,,,, y_{1} \dots y_{m} x_{1} \dots x_{n}) = x} ((), min_{1 \leq i \leq k} ϒ_{i} ((), x_{i}) \land min_{k + 1 \leq i \leq n} ((), 1 - ϒ_{i} ((), x_{i})))

…”

Section: Preliminarymentioning

confidence: 99%

“…The results in this case are consistent with the reliability index proposed by Wang which is a new metric to deal with the epistemic uncertainty.…”

Section: The New Reliability Index Based On Chance Measurementioning

confidence: 99%

“…Bai and Jin introduced the uncertainty theory into structural reliability design as a new method of ocean structure analysis. By introducing the uncertainty theory, Wang adopted the uncertain measure to quantify the reliability of structures containing the subjective randomness and fuzziness, and a new definition of structural reliability index was developed. Zeng proposed a definition of belief reliability metric as the uncertain measure of a system to perform a given function within given time under the given operating conditions, and the belief reliability metric can overcome the drawback of the possibility theory‐based reliability metric.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Reliability analysis of structures based on a probability‐uncertainty hybrid model

Zhang

You

et al. 2018

Quality & Reliability Eng

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The traditional reliability analysis method based on probabilistic method requires probability distributions of all the uncertain parameters. However, in practical applications, the distributions of some parameters may not be precisely known due to the lack of sufficient sample data. The probabilistic theory cannot directly measure the reliability of structures with epistemic uncertainty, ie, subjective randomness and fuzziness. Hence, a hybrid reliability analysis (HRA) problem will be caused when the aleatory and epistemic uncertainties coexist in a structure. In this paper, by combining the probability theory and the uncertainty theory into a chance theory, a probability‐uncertainty hybrid model is established, and a new quantification method based on the uncertain random variables for the structural reliability is presented in order to simultaneously satisfy the duality of random variables and the subadditivity of uncertain variables; then, a reliability index is explored based on the chance expected value and variance. Besides, the formulas of the chance theory‐based reliability and reliability index are derived to uniformly assess the reliability of structures under the hybrid aleatory and epistemic uncertainties. The numerical experiments illustrate the validity of the proposed method, and the results of the proposed method can provide a more accurate assessment of the structural system under the mixed uncertainties than the ones obtained separately from the probability theory and the uncertainty theory.

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\begin{matrix} lefttrue \\ γ_{U} = \frac{E (f (τ_{1} \dots τ_{n}))}{\sqrt{V ((), f ((),, τ_{1} \dots τ_{n}))}} \\ \begin{array}{c}  \end{array} = \frac{\int_{0}^{1} f ({normalϒ}_{1}^{- 1} (α) {normalϒ}_{2}^{- 1} (α) \dots {normalϒ}_{k}^{- 1} (α) {normalϒ}_{k + 1}^{- 1} (1 - α) {normalϒ}_{k + 2}^{- 1} (1 - α) \dots {normalϒ}_{n}^{- 1} (1 - α)) dα}{\sqrt{\int_{0}^{1} {(f ({normalϒ}_{1}^{- 1} (α) {normalϒ}_{2}^{- 1} (α) \dots {normalϒ}_{k}^{- 1} (α) {normalϒ}_{k + 1}^{- 1} (1 - α) {normalϒ}_{k + 2}^{- 1} (1 - α) \dots {normalϒ}_{n}^{- 1} (1 - α)) - e)}^{2} italicdα}} \end{matrix}

…”

Section: The New Reliability Index Based On Chance Measuresupporting

confidence: 89%

normalϒ ((), x) = ({, \begin{array}{l} 0, \begin{matrix} center \\ if & x \leq a \end{matrix} \\ \begin{matrix} center \\ (x - a) / (b - a), & \begin{array}{c} italicif & a \leq x \leq b \end{array} \end{matrix} \\ \begin{matrix} center \\ 1, & \begin{array}{c} italicif & x \geq b \end{array} \end{matrix} \end{array})

where a and b are real numbers with a < b , as shown in Figure . Besides, the zigzag uncertain variable τ ~ Z ( a , b , c ) has an uncertainty distribution