Moment-based evaluation of structural reliability

Accepted Manuscript

Moment-based evaluation of structural reliability Cao Wang, Hao Zhang, Quanwang Li PII: DOI: Reference:

S0951-8320(17)30505-7 https://doi.org/10.1016/j.ress.2018.09.006 RESS 6257

To appear in:

Reliability Engineering and System Safety

Received date: Revised date: Accepted date:

27 April 2017 27 August 2018 6 September 2018

Please cite this article as: Cao Wang, uation of structural reliability, Reliability https://doi.org/10.1016/j.ress.2018.09.006

Hao Zhang, Quanwang Li, Moment-based evalEngineering and System Safety (2018), doi:

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Highlights • A moment-based reliability analysis method is proposed.

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• It provides a novel perspective for evaluating structural reliability.

• The method is efficient in solving multi-dimensional reliability prob-

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Moment-based evaluation of structural reliability Cao Wanga,b , Hao Zhangb , Quanwang Lia,∗ a

Department of Civil Engineering, Tsinghua University, Beijing 100084, China School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australia

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b

Abstract

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Reliability analysis is a widely-used tool to measure a structure’s ability of fulfilling safety and serviceability requirements. Existing methods for reliability assessment have, for the most part, been developed based on using the probability distribution functions of input random variables accounting for the uncertainties arising from both structural properties (e.g., material

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strength, geometry) and external loads. This paper develops a moment-based method for reliability assessment, which relies on the moment information

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of input variables rather than the probability distribution functions. It is shown that the proposed method is useful in solving multi-dimensional reli-

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ability assessment problems with improved efficiency compared with Monte Carlo simulation. The implementation, validity and efficiency of the pro-

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posed method are demonstrated through illustrative examples. Keywords: Structural reliability, Moment-based method, Probability

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distribution

∗

Corresponding author. Email addresses: [email protected] (C. Wang), [email protected] (H. Zhang), li [email protected] (Q. Li).

Preprint submitted to Reliability Engineering and System Safety

September 7, 2018

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1. Introduction The reliability theory has seen its rapid growth and widely-accepted im-

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portance of addressing safety issues of civil structures during the past four

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decades, motivated by the fundamental nature of uncertainties arising from

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both structural performance and external actions [1, 2]. Reliability analysis

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provides a measure of a structure’s capacity of fulfilling safety requirements

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within a considered reference period, and a quantitative link between the pro-

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fessional practice of structural engineering and its social consequence. This

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is particularly relevant as the structures are often subjected to natural or

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man-made hazards that may lead to catastrophic damages [3–6]. The design

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specifications in currently enforced standards and codes, which are for the

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most part reliability-based, also demonstrate the up-to-date applications of

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reliability theory in structural design and safety evaluation (e.g., [7]).

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exceeds its resistance, R. This criterion defines a limit state function (LSF)

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Conceptually, a structure is deemed to fail if the external load effect, S,

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Z = z(R, S) = R − S

(1)

where Z < 0 indicates that the structure fails. By noting that both R and

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S are practically random variables, the structural failure probability, Pf , is

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given by Pf = Pr(Z < 0) = Pr(R < S), where Pr( ) denotes the probability

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of the event in the bracket. With the assumption of statistically independent

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R and S, Pf is given by

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Pf =

Z

Z<0

fR,S (r, s)drds = 3

Z

0

∞

FR (s)fS (s)ds

(2)

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where fR,S ( ) is the joint probability density function (PDF) of R and S,

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FR ( ) is the cumulative density function(CDF) of R, and fS ( ) is the PDF

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of S. Eq. (2) is the basis of modern approaches of safety assessment and

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plays a key role in the implementation of probability-based limit state design

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standards [4, 7]. More generally, the LSF in Eq. (1), Z, is multi-dimensional.

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In such a case, Z can be expressed as Z = z(X), where X = (X1 , X2 , . . . Xn )

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is a random vector representing the uncertain quantities such as material

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property, structural geometry and external load magnitudes. Thus, Pf in

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Eq. (2) becomes Pf =

Z

fX (x)dx =

Z<0

Z

...

Z

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1 [z(x1 , x2 , . . . xn ) < 0]

n Y

fXj (xj )dx

(3)

j=1

where fXj ( ) is the PDF of Xj for j = 1, 2, . . . n, and 1[ ] is an indicator

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function, which returns 1 if [ ] is true and 0 otherwise. Due to the multi-fold

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integral involved in Eq. (3), the numerical calculation of Eq. (3) is practi-

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cally associated with great complexity and low efficiency. Alternatively, one

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may employ the Monte Carlo simulation (MCS) method to find an approx-

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imate solution. The convergence of MCS does not depend on the number

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of random variables of interest, making it a practical approach to deal with

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high-dimensional problems. The basic idea of Monte Carlo simulation is to

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repeat the experiment (it refers to whether a structure fails in this specific

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case) for many times and approximate the true solution by means of the law

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of large numbers or other methods of statistical inference [8–10]. For Eq. (3),

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one may sample a realization of X, (x1 , x2 , . . . xn ), and determine whether

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the structure fails according to the LSF in each simulation run. The failure

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probability can be approximated by Eq. (4) provided that the number of

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replications, N , is large enough. N 1 X 1 [z(x1 , x2 , . . . xn ) < 0] N j=1

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Pf ≈

(4)

Motivated by the fact that the brute MCS methods are often time-consuming

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(especially for the case of low failure probabilities), advanced simulation

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methods with variance-reduction techniques have been developed to improve

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the efficiency of MCS, such as importance sampling method [11–13], direc-

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tional simulation method [14], multiwavelet-based methods [15], among oth-

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ers. Nonetheless, simulation-based methods can only provide an input-output

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‘black box’ for the reliability problem rather than insights that analytical so-

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lutions may offer [16]. Thus, a practical question may be asked: ‘Can a

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multi-dimensional reliability problem be solved in an efficient way using an

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analytical approach rather than simulation-based methods?’.

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A promising answer to this question is that one can utilize the the mo-

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ment information of random variables instead of the probability distribution

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in reliability assessment. This idea is motivated by the two facts. Firstly,

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the PDF and the moments of a random variable can determine each other

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uniquely; they are linked via the moment generating function (MGF) or the

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characteristic function, see, e.g., [17, 18] for details. Both of them are repre-

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sentative of the probabilistic characteristics of a random variable. Secondly,

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the use of a random variable’s moments is powerful in dealing with many

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multi-dimensional problems (e.g., finding the probability distribution of the

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sum of several random variables). In fact, it has been partially adopted in

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the field of structural reliability analysis to use the moment information of

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a random variable. For instance, with the LSF in Eq. (1), provided the

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first- and second-order moments of R and S, the classical first-order secondp 2 moment (FOSM) method gives β = (µR − µS )/ σR + σS2 , where β is the

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reliability index, µ• and σ• denote the mean value and standard deviation

of • respectively (e.g., [19, 20]). Further, the failure probability is given by

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Pf = Φ(−β), in which Φ( ) is the CDF of the standard normal distribution.

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Der Kiureghian et al. [21] proposed a second-order reliability approxima-

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tion method. Zhao and Ang [22] developed a method for system reliability

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analysis based on up to fourth moments of system LSF obtained from point

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estimates. Zhao and Lu [23] studied the reliability assessment problem in the

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presence of random variables with unknown distributions, and proposed an

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explicit fourth-moment standardization method using the third-order poly-

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nomial normal transformation technique. Wang et al. [24] proposed an

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approach to estimate the averaged time-dependent reliability of aging struc-

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tures in the presence of incomplete deterioration information. However, these

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methods only make use of lower orders of moments of random variables.

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This paper proposes a moment-based approach for structural reliability

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analysis, which relies on the moment information of input variables rather

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than the probability distribution functions. This paper is organized as fol-

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lows. After the introduction part, the one-dimensional reliability problem is

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first discussed in Section 2, which forms a simple yet complete platform for

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the development of the moment-based approach. The method is further ex-

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tended to solve the reliability problem of a parallel system in Section 3, where

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the limit state function associated with each element is also one-dimensional

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as in Section 2. Section 4 addresses the generalized case of multi-dimensional

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problems using a moment-based approach, where more than one random vari-

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able may be involved in the limit state function. Illustrative examples are

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presented in Section 5 to demonstrate the implementation, validity and effi-

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ciency of the proposed method. Finally, conclusions are drawn in Section 6.

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2. Moment-based reliability analysis: One-dimensional problems

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Consider the LSF in Eq. (1). Let smax represent the maximum of S.

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Normalizing Eq. (1) with respect to smax gives

R S Ze = R − S = − smax smax

Similar to Eq. (2), the structural failure probability, Pf , can be obtained as1

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(5)

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Pf = where ξ(x) is the CDF of R =

Z

1

ξ(x)fS (x)dx

(6)

0

R , smax

and fS (x) is the PDF of S =

S . smax

It

is noticed that while two random variables are involved in Eq. (5), i.e., R

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and S, the estimate of failure probability2 is converted to a one-dimensional

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problem as in Eq. (6) by incorporating the CDF of R.

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Considering the Fourier expansion of ξ(x), ξ(x) is expanded by defining

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1

It is obvious that Eqs. (2) and (6) are mathematically equivalent. The normalization of Eq. (1) herein is motivated by the fact that the high-order moment of S is bounded since 0 < S < 1, which is for R Rthe sake of computation (c.f. Eq. (24) in the following). 2 With Eq. (3), Pf = 1 (r − s < 0) fR (r)fS (s)drds by definition, representing a two-dimensional reliability problem, where fR (r) is the PDF of R.

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ξ(−x) = ξ(x) for 0 < x ≤ 1, with which ∞

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where aj is a Fourier coefficient, aj = 2

Z

1

ξ(x) cos(jxπ)dx

0

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∞

(9)

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a0 1 X ξ(x) = + aj [exp(iπ · jx) + exp(−iπ · jx)] 2 2 j=1 108

(8)

eix + e−ix for j = 0, 1, 2, . . .. By noting that cos x = holds for an arbitrary 2 √ real number x with i = −1, Eq. (7) becomes

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(7)

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a0 X ξ(x) = + aj cos(jxπ) 2 j=1

Substituting Eq. (9) to Eq. (6) yields ∞

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a0 1 X + aj [φ(iπ · j) + φ(−iπ · j)] Pf = 2 2 j=1

(10)

where φ(t) is the MGF of S, i.e., φ(t) = E [exp(St)], in which E( ) represents

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the mean value of the random variable in the bracket. φ(t) can be expressed

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in a polynomial form of Eq. (11),

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109

112

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φ(t) =

∞ X

ck tk

(11)

k=0

where each ck is a constant. The moment generating function satisfies that φ(j) (0) = E(S j ) for j =

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0, 1, 2, ..., where φ(j) ( ) is the jth order derivative of φ. With this, according

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to Eq. (11), E(S j ) = φ(j) (0) = cj · j!,

(12)

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j = 0, 1, 2, . . .

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Consider the first m orders of moment of S. It can be proven that there ex-

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ist two coefficient sequences {αl , l = 1, 2, . . . m}, and {βl > 0, l = 1, 2, . . . m}

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such that

m X l=1

j = 0, 1, . . . m − 1

(13)

Eq. (13) can be rewritten in a matrix form as 

···

1

1

β1

β2

β12 .. .

β22 .. .

β1m−1 β2m−1







α 1    1       · · · βm   α2   E(S)           2   ·  α3  =  E(S 2 )  · · · βm       ..   ..   .. .. .  .   .   .      m−1 m−1 αm E(S ) · · · βm 1

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         

 

(14)

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αl · βlj ,

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E(S j ) =

or β · α = E in short. Note that β is a Vandermonde matrix, whose deter-

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minant is given by

Y

det(β) =

1≤l
(βk − βl )

(15)

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which is non-zero if βk 6= βl for ∀k 6= l. In such a case, the inverse of β exists

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and Eq. (14) holds for properly selected {αl } and {βl }. Substituting Eq. (13) into Eq. (12), it follows

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c2k

E(S 2k ) = = (2k)!

Pm

αl · βl2k , (2k)!

l=1

k = 0, 1, 2, . . .

(16)

With Eqs. (10), (11) and (16), as m is sufficiently large, the failure probability 9

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is given by3

(17)

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# " ∞ P ∞ X m αl · β 2k a0 X l l=1 + · (jπ)2k (−1)k Pf = aj 2 (2k)! j=1 k=0 127

By noting that for an arbitrary real number x, the Taylor expansion of func-

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tion cos x gives cos x = one has

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∞ X x2k (−1)2k (2k)! k=0

∞ Pm m 2k X X l=1 αl · βl · (jπ)2k (−1)k = αl cos(βl · jπ) (2k)! k=0 l=1 130

Thus, Eq. (17) becomes

(19)

(20)

Recall Eq. (7), assigning x = βl gives

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"∞ # m X a0 X Pf = + αl aj cos(βl · jπ) 2 j=1 l=1

(18)

∞

a0 X ξ(βl ) = + aj cos(βl · jπ) 2 j=1

3

(21)

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The failure probability associated with the first m orders of moment of S converges to the true value when m is sufficiently large. In practical application, the choice of m may depend on the accuracy requirement of the failure probability. For instance, provided that the maximum error is pf , m shall satisfy |Pf (m) − Pf (m − 1)| ≤ pf , where Pf (m) and Pf (m − 1) are the failure probabilities associated with m and m − 1 terms of summation respectively.

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Substituting Eq. (21) into Eq. (20) yields a0 X h a0 i Pf = + αl ξ(βl ) − 2 2 l=1 m

By noting that

Pm

l=1

αl = 1, Eq. (22) can be further written as Pf =

m X

αl ξ(βl )

(23)

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l=1

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(22)

Equivalently, Eq. (23) can be written in a form of matrix as follows, h i h i Pf = ξ(β1 ), ξ(β2 ), . . . ξ(βm ) · α = ξ(β1 ), ξ(β2 ), . . . ξ(βm ) · (β −1 E) (24)

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where α, β and E are defined in Eq. (14).

Eq. (24) is the proposed method for estimating structural failure proba-

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bility. It can be seen that only the moments of S is incorporated in Eq. (24),

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while the distribution function of S is not involved. The implementation and

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validity of Eq. (24) will be demonstrated in Section 5.

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Let

h i h i −1 γ1 , γ2 , . . . γm = ξ(β1 ), ξ(β2 ), . . . ξ(βm ) · β

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Pm

(25)

γi E (S i−1 ), indicating that Pf can be expressed

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Eq. (24) becomes Pf =

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as the weighted sum of the moments of S. As an application in practical

i=1

engineering practice, this fact can be applied to data analysis-based reliabil-

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ity analysis problems. Consider the case that in Eq. (6), the probabilistic

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information of S is unknown but a set of observed samples of S are available.

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Using the traditional approach, the failure probability can be estimated by

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substituting the fitted probability distribution type of S, making the confi-

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dence level of the estimated Pf difficult to determine (i.e., what is the variance

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of Pf provided n samples of S?). Alternatively, we reconsider this problem

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with Eq. (24). With sufficient large m and n, Pf is estimated by

Pf =

m X

γi E S

i−1

i=1




=

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m X

n γi X i−1 S n j=1 j

i=1

!

(26)

where each S j is statistically independent and identically distributed with S.

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Treating Pf as a random variable, it follows

=

i=1 m m XX i=1 j=1

n γi X i−1 S n j=1 j

γi γj n2

n X n X

!!

=

m X m X γi γj i=1 j=1

p=1 q=1

n2


j−1 C S i−1 = p , Sq

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V(Pf ) = V

m X

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C

n X

S pi−1 ,

n X

p=1 q=1 m m n X X γi γj X C n2 p=1 i=1 j=1

S j−1 q

!

S pi−1 , S j−1 p




(27)

where V( ) and C(·, ·) represent the variance and covariance of the random

154

variable(s) in the bracket, respectively. Due to the identical distribution and

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statistical independence of each S j ,

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n X

Thus,

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p=1



j−1 C S i−1 = n E(S i+j−2 ) − E(S i−1 )E(S j−1 ) p , Sp

m m   1 XX γi γj E(S i+j−2 ) − E(S i−1 )E(S j−1 ) V(Pf ) = n i=1 j=1

(28)

(29)

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Eq. (29) clearly implies that when the moments of S are estimated by n

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realizations of S, and are further applied to the failure probability assessment

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as in Eq. (6), the variance of the estimated Pf decreases with n and is in

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proportion to 1/n.4 Furthermore, in the presence of the samples of S, the

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probability distribution fitting is often based on the assignment of a specific

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distribution type for S according to the fitting goodness, and thus may result

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in epistemic error, especially for cases where the “realistic” distribution type

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is singular. Instead, the proposed method (Eq. (26)) first estimates the

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moments of S with the samples uniquely and thus offers a straightforward

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approach for the reliability assessment avoiding the potential epistemic error

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that could be induced by the traditional distribution function fitting.

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3. Moment-based reliability analysis: parallel systems

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Recall that Eq. (24) links the failure probability and the moments of S.

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We further discuss the generalized form of Eq. (24) in this section. First, we

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define a matrix

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1 1    ξ(β1 ) ξ(β2 )   2 ξ =  ξ (β1 ) ξ 2 (β2 )   .. ..  . .  ξ m−1 (β1 ) ξ m−1 (β2 )

4

···

1



  ··· ξ(βm )    2 ··· ξ (βm )    .. .. .  .  m−1 ··· ξ (βm )

(30)

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Recall the classical problem in statistics that to estimate the mean value of a random variable from n observed samples. The variance of the estimated mean value also decreases with n in proportion to 1/n. This conclusion is consistent with that obtained from Eq. (29).

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and a column matrix P which satisfies ξ·α=P

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(31)

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Obviously, according to Eq. (24), the failure probability Pf equals the second

174

element in P .

Now consider a parallel system with k components, where each component

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has a statistically independent and identically distributed resistance of R, and

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a common (normalized) load effect S ∈ [0, 1]. The system is deemed to fail

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if the load effect exceeds all the resistances associated with each component

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(i.e., S > maxkj=1 Rj ). An illustrative example for such a parallel system

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is a parallel electricity transmission system subjected to wind actions (or

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a parallel bridge network subjected to earthquakes). By noting that the

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hazardous events usually have a large footprint and are spatially correlated

183

(e.g.,[25, 26]), the load effects at different locations can be approximately

184

treated as fully correlated when the considered scale is small (i.e., the load

185

effects applied to different components are identical).

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∗ The failure probability of the system, Pf,k is given by

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∗ Pf,k

=

Z

ξ k (x)fS (s)ds =

k X

αl ξ k (βl )

(32)

l=1

∗ With Eqs. (31) and (32), it is seen that Pf,k equals the (k + 1)th element in

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P , i.e.,

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where the superscript

h iT ∗ ∗ ∗ P = Pf,0 Pf,1 . . . Pf,m−1 T

(33)

denotes the transpose of the matrix. Comparing 14

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Eqs. (31) and (33), it follows P = ξβ −1 E

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(34)

Eq. (34) establishes a relationship between the moments of S and the parallel

192

system failure probability. It provides a new perspective for moment-based

193

reliability assessment of parallel systems. Furthermore, if ξ(x) is a continuous

194

monotonous function, and all values of βl are distinct, ξ is a Vandermonde

195

matrix with a non-zero determinant (the inverse of ξ exists). With this, a

196

symmetric relationship is given by

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ξ −1 P = β −1 E

4. Moment-based reliability analysis: general multi-dimensional

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(35)

problems

199

In this section, we generalize Eq. (24) to the case of multi-dimensional

200

LSF. This is motivated by the fact that many of the commonly-used LSFs in

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practical engineering involve multiple random variables and sometimes are

202

nonlinear. A simple example is that in the load and resistance factor design

203

(LRFD), for a structure that is subjected to both live load and dead load,

204

the LSF may take a linear form of G = 0.9Rn − 1.2Dn − 1.6Ln , where Rn , Dn

205

and Ln are the nominal resistance, dead load and live load, respectively [4,

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27]. Further, if we consider the resistance as a function of several structural

207

parameters (e.g., geometry size, material strength, and others), then the LSF

208

becomes nonlinear. In this section, we discuss both the linear and nonlinear

209

forms of LSF in the presence of multiple random variables.

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4.1. Linear LSF First, we consider the case of a linear LSF, Z, which takes the form of Z = z(X1 , X2 , . . . Xn ) =

n X

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δj X j

j=1

where δj is the coefficient of Xj for j = 1, 2, . . . n. Rearranging Eq. (36) gives

δ1 X1 − min

Z=

215

216

217

where min

220

222

is the minimum of

is given by

δj Xj − min

n X

j=2

δj X j

Pn

j=2 δj Xj

j=2

!

(37)

(a deterministic value),

δ1 X1 − min max

n P

n P

δj Xj

j=2

δj Xj − min

n P

δj X j

j=2

!−

j=2

max

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j=2

n P

δj Xj − min

Pn

j=2 δj Xj

n P

δj Xj

j=2

− min

n P

j=2

δj Xj

!

(38)

e1 − X e ∗ in short. Obviously, X e ∗ ∈ [0, 1]. The structure fails if or Ze = X e1 < X e ∗ and survives otherwise. It is seen from Eq. (38) that the multiX

e1 dimensional reliability problem can be simplified as one-dimensional if X

e ∗ are treated as generalized resistance and load effect, respectively. and X

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−

n X

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219

j=2 δj Xj

!

which is introduced so that the second term in Eq. (37) is definitely positive.  P Pn n δ X δ X − min The normalized form of Z with respect to max j=2 j j j=2 j j

Ze = 218

Pn

δj X j

j=2

ED

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n X

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(36)

Similar to Eq. (23), one has

Pf =

m X

αl∗ ξ(βl∗ )

l=1

16

(39)

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225

226

e ∗j−1 ). βj∗i−1 and (E ∗ )j = E(X

Note that with the multinomial theorem, Eq. (40) holds, where k1 , k2 , . . . kp P are non-negative integers satisfying pj=1 kj = q. p X

xj

j=1

227

228

!q

=

p Y q! k xj k1 !k2 ! . . . kp ! j=1 j

(40)

With Eq. (40), by noting that E(A1 A2 ) = E(A1 )E(A2 ) for two independent e ∗ is estimated by variables A1 and A2 , the jth order moment of X n X 1 Yj E(X ) = j E ma j=1

e ∗j

P

!

n j=2 δj Xj

p   Y 1 X q! kj = j E Yj k1 !k2 ! . . . kp ! j=1 ma

− min

where ma = max

230

and Yj = δj Xj for j = 2, 3, . . . n.

Pn

M

229

j=2 δj Xj



, Y1 = − min

(41)

Pn

j=2 δj Xj ,

The variables in Eq. (38), Xi for i = 2, 3, . . . n, can be generalized to take

ED

231

X

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224

where the sequences {αl∗ } and {βl∗ } satisfy β ∗ · α∗ = E ∗ , in which (β ∗ )ij =

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223

the form of

233

(1) Xi is a function of another random variable, e.g., Xi = h(Yi );

234

(2) Xi is the product of several random variables.

235

In such cases, the LSF can also be treated as linear, and Eq. (39) applies.

236

For the case of Xi = h(Yi ), the jth order moment of Xi can be estimated by

AC

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PT

232

237

E(Xij )

= E[h (Yi )] = j

where fYi is the PDF of Yi .

17

Z

hj (y)fYi (y)dy

(42)

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239

240

4.2. Nonlinear LSF More generally, we consider the case of a nonlinear LSF. The failure probability, Pf , is estimated by Pf =

Z

Z

n Y

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. . . ϑ(x1 , x2 , . . . xn ) fXj (xj )dx j=1 | {z } n−fold

(43)

where ϑ(x1 , x2 , . . . xn ) is the structural failure probability conditional on x =

242

(x1 , x2 , . . . xn ), and fXj ( ) is the PDF of Xj , j = 1, 2, . . . n. Assume that

243

each element in X, Xi , is statistically independent mutually. To consider the

244

normalized form of each Xj , let

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241

Xj − X j

Xj − Xj

(44)

M

Yj =

where X j and X j denote the maximum and minimum values of Xj , respec-

246

tively. Obviously, Yj ∈ [0, 1]. Substituting Eq. (44) into Eq. (43) gives

ED

245

|

0

1

... {z

Z

n−fold

0

1

φ(y1 , y2 , . . . yn )

}

n Y

fYj (yj )dy

(45)

j=1

where fYj ( ) is the PDF of Yj , j = 1, 2, . . . n, and φ(y1 , y2 , . . . yn ) = ϑ((X 1 − X 1 )y1 + X 1 , . . . (X n − X n )yn + X n )

AC

247

CE

PT

Pf =

Z

18

(46)

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We define a function φ1 (y1 ) =

Z

Z

n Y

. . . φ(y1 , y2 , . . . yn ) fYj (yj )dy j=2 | {z }

(n−1)−fold

249

with which Eq. (45) becomes

φ1 (y1 )fY1 (y1 )dy1

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Pf =

Z

(47)

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248

(48)

250

With Eqs. (14) and (24), there exists a sequence of {1 βl > 0, l = 1, 2, . . . m}

251

such that

h i Pf = φ1 (1 β1 ), φ1 (1 β2 ), . . . φ1 (1 βm ) · (1 β −1 E1 )

(49)

where (1 β)ij = 1 βji−1 and (E1 )j = E(Y1j−1 ). Next, to find the terms φ1 (1 βj )

253

in Eq. (49), we define a function

Z

PT

(n−2)−fold

254

with which

CE

φ1 (y1 ) =

256

257

Z

φ2 (y1 , y2 )fY2 (y2 )dy2

(50)

(51)

Again, with Eq. (14), there exists a sequence of {2 βl > 0} such that

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255

n Y

fYj (yj )dy . . . φ(y1 , y2 , . . . yn ) j=3 | {z }

ED

φ2 (y1 , y2 ) =

Z

M

252

h i φ1 (y1 ) = φ2 (x1 , 2 β1 ), φ2 (x1 , 2 β2 ), . . . φ2 (x1 , 2 βm ) · (2 β −1 E2 )

(52)

in which (2 β)ij = 2 βji−1 and (E2 )j = E(Y2j−1 ). Similar to Eqs. (47) and (50), we can define the function sequence φ3 , φ4 19

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258

through φn , and calculate φj with φj+1 for j = n − 1, n − 2, . . . 2, 1 (note that

259

φn = φ). The calculation of Pf is finalized once φ1 is obtained. In summary, Eqs. (49) through (52) build a recursion to calculate the

261

sequence φn , φn−1 ,. . . φ2 , φ1 and finally completes the calculation of Pf with

262

Eq. (49). The efficiency of the method is due to the straightforward calcu-

263

lation procedure without replications of simulation. However, it is noticed

264

that the proposed approach is based on the closed-form expression of the

265

LSF and thus its applicability may be haltered when the explicit LSF is not

266

available.

268

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Specifically, if n = 2, with Eqs. (49) and (52), the failure probability is given by 

φ1 (1 β1 )



. . . φ(1 β1 , 2 βm )



  . . . φ(1 β2 , 2 βm )   (2 β −1 E2 )  .. .. .  .  . . . φ(1 βm , 2 βm ) (53)

Note that each element in X, Xi , has been assumed statistically indepen-

AC

269

CE

PT

ED

M

      φ ( β ) T −1 T  1 1 2  Pf = (E1 ) (1 β ) ·   ..   .   φ1 (1 βm )  φ( β , β ) φ(1 β1 , 2 β2 )  1 1 2 1   φ(1 β2 , 2 β1 ) φ(1 β2 , 2 β2 ) = (E1 )T (1 β −1 )T   .. ..  . .  φ(1 βm , 2 β1 ) φ(1 βm , 2 β2 )

270

dent in the aforementioned derivation. There are also some practical cases

271

where the random variables are mutually correlated. For instance, the re-

272

sistance deterioration process of an aging structure may highly depend on

273

the loading process [28]. Mathematically, consider the case where the ele20

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ments X1 through Xk in X are statistically correlated while the remaining

275

elements in X are independent. We use the Gaussian copula function to

276

model the joint distribution function of X1 , X2 , . . . Xk , with which we can

277

further employ the Nataf transformation approach to convert {X1 , . . . Xk }

278

into an uncorrelated sequence [3, 29, 30]. Suppose that the correlation ma-

279

trix of {X1 , . . . Xk } is ρ = [ρij ]k×k (i.e., the correlation coefficient between

280

Xi and Xj is ρij for 1 ≤ i, j ≤ k), with which one has Φ−1 [FX1 (X1 )]





U1





 

a11

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

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274

U1



            −1      U2   Φ [FX2 (X2 )]  U2  a21 a22 ·    = Ach   =    ..   ..   ..   .. . . .. .   .    .   .  . .         −1 Uk ak1 ak2 . . . akk Uk Φ [FXk (Xk )]

where {U1 , . . . Uk } is a sequence of uncorrelated standard normal variables,

M

281

(54)

FXi ( ) is the CDF of Xi , and Ach is a lower triangle matrix satisfying Ach ATch =

283

ρ (Ach can be obtained by the Cholesky decomposition of ρ). Thus,

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282

"

aij Uj

j=1

!#

,

∀i = 1, 2, . . . k

(55)

With this, the probability of failure in Eq. (43) can be estimated by

CE

284

PT

Xi = FX−1i Φ

i X

AC

Pf =

·

Z

"

Z

. . . ϑ FX−11 [Φ (a11 u1 )] , . . . FX−1k Φ | {z } n−fold

k Y j=1

ϕ(uj )

n Y

k X j=1

akj uj

!#

, xk+1 , . . . xn

!

fXj (xj )dxdu

j=k+1

(56)

21

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where ϕ( ) is the PDF of a standard normal variable. As such, the reliability

286

problem that involves correlated random variables can be transferred into

287

that with independent variables only.

288

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For visualization purpose, a flowchart is given in Fig. 1 to demonstrate the calculation procedure of the proposed moment-based approach.

Linear LSF

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Start

Nonlinear LSF

Normalize the LSF m=1

Normalize each random variable m=1

Find the sequence ϕn, ϕn−1,...ϕ2, ϕ1

M

Assign β1, β1,...βm, and calculate E(S i), ξ(βi) for i =1,2,… m

m=m+1 No

ED

Calculate the failure probability with Eq.(24)

m=m+1 No

|Pf (m) − Pf (m-1)| ≤ ϵpf

PT

Calculate the failure probability with Eq.(49)

|Pf (m) − Pf (m-1)| ≤ ϵpf

AC

CE

Yes

Yes Failure probability = Pf (m)

End

Figure 1: Flowchart of the proposed moment-based approach.

289

22

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290

5. Implementation and validation of the proposed method Illustrative examples are presented to demonstrate the implementation

292

and validity of the proposed method when applied to one-dimensional and

293

multi-dimensional reliability problems.

294

5.1. Example 1: One-dimensional reliability problem

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Consider the LSF in Eq. (1), where R is assumed to have a normal dis-

296

tribution with a mean value of 60 and a standard deviation of 10. The load

297

effect, S, is Gamma distributed with µS = 30 and σS = 8. The exact failure

298

probability Pf is 0.012263 according to Eq. (2).

299

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Now we consider the calculation of Pf with Eq. (24) using the moment information of S. First, assign smax = µS + 10σS . The jth moment of

301

is given by S

smax

j #

j bjS Y = j (aS + j − k), smax k=1

ED

E

"

S smax

M

300

j = 1, 2, . . .

(57)

where aS and bS are the shape and scale parameters of S, respectively. Let

303

βl =

304

m are obtained according to Eq. (24) and are listed in Table 1. It is seen

305

that the failure probability converges gradually with increasing m. With

306

a permissible error of 10−6 , the failure probability is obtained as 0.012263

l = 1, 2, . . . m, the failure probabilities associated with different

CE

l , m

PT

302

when m = 21. This result is consistent with that obtained from Eq. (2), and

308

suggests that the proposed method uses the sum of only 21 terms to solve

309

the integral in Eq. (2) with sufficient accuracy. Theoretically, the assignment

310

of the values βl does not affect the failure probability according to Eq. (24).

AC 307

23

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l 2 m

311

For example, we recalculate the failure probability by letting βl =

312

l = 1, 2, . . . m and can obtain the same solution as above (Pf = 0.012263).

314

However, from a view of numerical calculation, each βl is suggested to vary p within [0, E(R)+5 V(R)] since the function ξ is the CDF of R (c.f. Eq. (6)).

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313

for

Table 1: Convergence of Pf obtained from Eq. (24) with increasing m, Example 1.

m

Failure probability Lognormal Adaptive (Eq. (59)) 0.014513 0.013706 0.011477 0.010679 0.017494 0.031436 0.023587 −0.057347 −0.148467 0.182730 1.354148 0.983493 −9.160334

M

0.013500 0.011974 0.012547 0.012420 0.012085 0.012233 0.012320 0.012266 0.012249 0.012264 0.012267 0.012263 0.012263

0.013757 0.013767 0.013766 0.013759 0.013753 0.013756 0.013766 0.013771 0.013765 0.013754 0.013750 0.013762 0.013777

PT

ED

10 11 12 13 14 15 16 17 18 19 20 21 22

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Gamma

315

Note that the MGF of S was employed in the derivation of Eq. (24);

317

however, for some distribution types, the MGF does not exist (e.g., lognor-

318

mal, [31]). To demonstrate the applicability of Eq. (24) in the presence of

319

the random variables that do not have an MGF, we assume that S follows a

320

lognormal distribution instead. With this, the failure probability is obtained

321

as 0.013761 according to Eq. (2). Using Eq. (24) to calculate the failure

AC

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316

24

ACCEPTED MANUSCRIPT

probability, similar to Eq. (57), the jth moment of

E

"

S

smax

j #

=

1 sjmax

S smax

  1 2 2 exp λ0 j + 0 j , 2

is estimated by

j = 1, 2, . . .

(58)

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322

where λ0 and 0 are the location and the scale parameters of S, respectively

324

(i.e., the mean value and standard deviation of ln S). The failure probabilities

325

associated with different values of m are presented in Table 1. It is seen that

326

Pf fluctuates with an increasing m but does not converge, indicating that the

327

moment-based method is not applicable when the random variable does not

328

have an MGF, because the moment information cannot determine the PDF

329

uniquely in such a case.

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323

To overcome this difficulty, this paper develops an adaptive method to

331

handle the random variables with no MGF. We reconsider Eq. (6) as follows,

M

330

332

ED

Pf =

Z

0

1

ξ(x)fS (x) fQ (x)dx fQ (x)

(59)

where fQ (x) is the PDF of an arbitrary random variable, Q, defined in [0, 1],

334

which has an MGF. With this, the failure probability is estimated according

335

to Eq. (24), with the exception that ξ(x) is replaced by

336

that Q follows a Gamma distribution with a mean value of

337

dard deviation of

338

probabilities obtained from the adaptive method (Eq. (59)) with different m

4 55

(the same distribution as

AC

CE

PT

333

S smax

ξ(x)fS (x) . fQ (x) 3 11

Assuming

and a stan-

in Eq. (57)), the failure

339

are also presented in Table 1. It is seen that with Eq. (59), Pf converges

340

gradually with an increasing value of m, indicating the applicability of the

341

adaptive method to the case where a random variable does not have an MGF.

25

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342

5.2. Example 2: Moment-based estimate of parallel system reliability The aim of this section is to verify the accuracy of Eq. (34). Illustratively,

344

consider a parallel system consisting of k components subjected to a common

345

load effect S. The resistance of each component is assumed to be statically

346

independent and identically distributed, denoted by R. Similar to Example

347

1, R is assumed to have a normal distribution with µR = 60 and σR = 10, and

348

S, is Gamma distributed with µS = 30 and σS = 8. In Table 2, the system

349

failure probabilities obtained from both the proposed method (Eq. (34)) and

350

Eq. (32) are presented for comparison purpose. The consistency of both

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343

results indicates the validity of Eq. (34).

Table 2: Failure probabilities of a parallel system with k components, Example 2.

System failure probability Proposed method (Eq. (34)) Eq. (32)

Error (%)

1 2 3 4

0.012263 0.001937 0.000676 0.000360

0.00 −0.18 −3.01 1.98

ED

5.3. Example 3: Multi-dimensional reliability problem Consider the LSF

CE

353

0.012263 0.001941 0.000697 0.000353

PT

351

352

M

k

Z = R − S1 − S2 S3

(60)

where the probabilistic models of R, S1 , S2 and S3 are summarized in Table 3.

355

The failure probability is obtained as 0.003718 via 1,000,000 replications of

356

MCS.

AC

354

357

Now we consider the calculation of Pf using the method developed in

358

Section 4.1. The LSF in Eq. (1) can be treated as linear, and is rewritten 26

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Table 3: Statistics of the variables in Example 3.

361

362

Distribution type

R S1 S2 S3

60 15 1/2 20

10 √5 3/6 4

Normal Gamma Uniform Weibull

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Standard deviation

as Z = R − S ∗ , where S ∗ = S1 + S2 S3 . Let s∗max and s∗min denote the maximum and minimum values of S ∗ , respectively. Assign s∗min = 0 and

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360

Mean

s∗max = µS1 + 3σS1 + µS3 + 3σS3 . Normalizing Z with respect to s∗max yields Ze =

R smax

E

−

"

S∗ . s∗max

S∗ s∗max

With Eq. (41), the jth moment of

j # l , m

S∗ s∗max

is given by

(   ) k  j   S j−k X j S1 3 j−k E S2 E ∗ = E ∗ smax smax k k=0

(61)

M

359

Variable

l = 1, 2, . . . m, results show that the failure probability is

Letting βl =

364

0.003710 when m = 12, which agrees well with the Monte Carlo simulation

365

solution.

ED

363

Furthermore, for illustration purpose, we re-calculate the failure proba-

367

bility using the method developed in Section 4.2 (i.e., the LSF in Eq. (60)

368

is treated as non-linear ). By definition, the failure probability can be esti-

369

mated by Eq. (43), with ϑ(s1 , s2 , s3 ) = FR (s1 + s2 s3 ), where FR is the CDF

AC

CE

PT

366

27

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370

of R. Similar to Eq. (53), the failure probability is 

φ2 (1 β1 , 2 β1 )

φ2 (1 β1 , 2 β2 ) . . . φ2 (1 β1 , 2 βm )



where φ2 (s1 , s2 ) =



Z

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371

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     φ ( β , β ) φ2 (1 β2 , 2 β2 ) . . . φ2 (1 β2 , 2 βm )  T −1 T  2 1 2 2 1  (2 β −1 E2 ) Pf = (E1 ) (1 β )   .. .. .. ...   . . .   φ2 (1 βm , 2 β1 ) φ2 (1 βm , 2 β2 ) . . . φ2 (1 βm , 2 βm ) (62)

φ(s1 , s2 , s)fS3 (s)ds

h i = φ(s1 , s2 , 1 β1 ), . . . φ(s1 , s2 , 1 βm ) · (3 β −1 E3 ) Sk −S k S k −S k

j−1 

(63)

372

and (Ek )j = E

373

l = 1, 2, . . . m, with which the failure probability is obtained as 0.003710 with

374

m = 10 according to Eq. (62), consistent with the result treating the LSF

375

in Eq. (60) as linear. Using the commercial software Matlab R2016a on an

376

R Intel CORETM i7-6700 CPU @3.40GHz computer, the CPU time is 14s for

377

106 Monte Carlo simulations and 0.003s for Eq. (62), indicating that the cal-

378

culation of failure probability using the proposed method is straightforward

379

and is more efficient than MCS.

PT

ED

M

for

Another approach to estimate the failure probability in the presence of

nonlinear LSF is the classical first order reliability method (FORM), which

AC

381

l m

CE

380

for k = 1, 2, 3. Let 1 βl = 2 βl = 3 βl =

382

is realized based on an iterative algorithm. The detailed procedure of FORM

383

has been well discussed in the literature (e.g., [3]). The basic idea is to find

384

the optimized design point by transforming the non-normal random vari-

385

ables into equivalent normal distributed variables. Herein, for the LSF in 28

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Eq. (60), the FORM yields a reliability index of 2.60 and correspondingly

387

a failure probability of 0.0047. The difference between the failure proba-

388

bilities associated with the proposed method and FORM shows the better

389

accuracy of the proposed moment-based reliability approach, by noting that

390

the FORM is based on the linearization of the nonlinear LSF. Moreover, us-

391

ing the same calculating platform, the CPU time is 0.06s for FORM, which

392

is slightly slower than the proposed approach but is more efficient than the

393

MCS method. This observation again indicates the efficiency of the proposed

394

method as it is based on a straightforward calculation procedure.

395

6. Conclusions

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386

A moment-based reliability analysis method has been proposed in this

397

paper, which takes advantage of the moment information of random variables

398

rather than the probability distribution. The following conclusions can be

399

made from this paper.

400

(1) The estimate of structural failure probability can be performed based on

401

the moments of random variable(s) instead of the probability distribu-

402

tion, due to the fact that the moments and probability distribution of

403

a random variable can determine each other uniquely. In the case when

ED

PT

the random variable does not have an MGF, an adaptive method (c.f. Eq. (59)) can be used.

AC

405

CE

404

M

396

406

(2) The moment-based reliability assessment method provides a different

407

perspective for evaluating structural reliability compared with the con-

408

ventional simulation-based approaches. The method works for both one-

409

dimensional and multi-dimensional reliability problems. The implemen29

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410

tation and accuracy of the method are verified through numerical exam-

411

ples. (3) For the data analysis-based reliability assessment problem in which the

413

probabilistic information of a random variable is obtained from observed

414

samples only, the proposed method shows that the variance of the esti-

415

mated failure probability decreases with the number of samples, n, and

416

is proportional to 1/n.

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412

(4) For a parallel system where each component is subjected to common

418

loads, a moment-based method is developed to estimate the system fail-

419

ure probability, which provides a new perspective for understanding the

420

linkage between system failure and the moment information of the com-

421

mon load effect. Acknowledgements

M

422

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417

This research has been jointly sponsored by the National Natural Sci-

424

ence Foundation of China (Grant No. 51578315), the National Key Research

425

and Development Program of China (Grant No. 2016YFC0701404) and the

426

Faculty of Engineering and IT PhD Research Scholarship (SC1911) from the

427

University of Sydney. The first author would like to thank the Department of

428

Civil Engineering, Tsinghua University, for hosting him as a visiting scholar

429

there. The thoughtful suggestions from two anonymous reviewers are grate-

AC

CE

PT

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423

430

fully acknowledged, which substantially improved the present paper.

30

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431

References

432

References

434

[1] B. R. Ellingwood, Probability-based codified design: past accomplish-

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[2] M. Ghosn, D. Frangopol, T. McAllister, M. Shah, S. Diniz, B. R. Elling-

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wood, L. Manuel, F. Biondini, N. Catbas, A. Strauss, et al., Reliability-

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based performance indicators for structural members, Journal of Struc-

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[3] R. E. Melchers, Structural reliability analysis and prediction, John Wiley & Son Ltd, 1999.

[4] B. R. Ellingwood, LRFD: implementing structural reliability in profes-

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[6] Q. Li, C. Wang, B. R. Ellingwood, Time-dependent reliability of ag-

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ing structures in the presence of non-stationary loads and degradation, Structural Safety 52 (2015) 132–141.

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[7] ASCE-7, Minimum design loads for buildings and other structures, American Society of Civil Engineers, 2012.

[8] W. L. Dunn, J. K. Shultis, Exploring Monte Carlo Methods, Elsevier, 2012. 31

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453

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[9] G. Fishman, Monte Carlo: concepts, algorithms, and applications, Springer Science & Business Media, 2013. [10] D. P. Kroese, T. Brereton, T. Taimre, Z. I. Botev, Why the monte

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