Reliability analysis based on a novel density estimation method for structures with correlations

CJA 835 11 May 2017 Chinese Journal of Aeronautics, (2017), xxx(xx): xxx–xxx

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Chinese Society of Aeronautics and Astronautics & Beihang University

Chinese Journal of Aeronautics [email protected] www.sciencedirect.com

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Reliability analysis based on a novel density estimation method for structures with correlations

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Baoyu Li a,b, Leigang Zhang b,*, Xuejun Zhu b, Xiongqing Yu a, Xiaodong Ma b

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a b

College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China China Academy of Launch Vehicle Technology, Beijing 100076, China

Received 8 April 2016; revised 19 September 2016; accepted 2 March 2017

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KEYWORDS

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Fractional moment; Maximum entropy; Probability density function; Reliability analysis; Unscented transformation

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Ó 2017 Published by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

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Abstract Estimating the Probability Density Function (PDF) of the performance function is a direct way for structural reliability analysis, and the failure probability can be easily obtained by integration in the failure domain. However, efficiently estimating the PDF is still an urgent problem to be solved. The existing fractional moment based maximum entropy has provided a very advanced method for the PDF estimation, whereas the main shortcoming is that it limits the application of the reliability analysis method only to structures with independent inputs. While in fact, structures with correlated inputs always exist in engineering, thus this paper improves the maximum entropy method, and applies the Unscented Transformation (UT) technique to compute the fractional moments of the performance function for structures with correlations, which is a very efficient moment estimation method for models with any inputs. The proposed method can precisely estimate the probability distributions of performance functions for structures with correlations. Besides, the number of function evaluations of the proposed method in reliability analysis, which is determined by UT, is really small. Several examples are employed to illustrate the accuracy and advantages of the proposed method.

1. Introduction Reliability analysis is used to assess the safety of an engineering system or structure, and reliability is defined as the proba* Corresponding author. E-mail address: [email protected] (L. Zhang). Peer review under responsibility of Editorial Committee of CJA.

bility that a system, subsystem, or device will perform adequately for a specified period of time under specific operating conditions.1 Commonly, the performance function of a system or structure is characterized by a function GðXÞ, which is a function of physical random variables X ¼ ½X1 X2 . . . Xn  (n is the dimensionality of the input vector). The definition equation of the failure probability, Pf , is given as Z Pf ¼ fX ðxÞdx ð1Þ F

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where fX ðxÞ is the joint Probability Density Function (PDF) of the input variables X, and F ¼ fX : GðXÞ 6 0g is the failure

http://dx.doi.org/10.1016/j.cja.2017.04.005 1000-9361 Ó 2017 Published by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Li B et al. Reliability analysis based on a novel density estimation method for structures with correlations, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.04.005

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region. During the past several decades, various reliability analysis methods have been proposed and developed with the difficulty to compute the failure probability, which can be classified into three groups generally, i.e., moment based analytical methods, sampling based numerical simulation methods, and surrogate model based methods. Moment based analytical methods have been firstly developed, and they assess system safety by the reliability index, which is a function of a few integral moments of the performance function. Among them, the First-Order Reliability Method (FORM) and Second-Order Reliability Method (SORM)2–4 are most classical and widely used. Both the FORM and SORM aim at searching for a design point, which locates on the limit state surface GðXÞ ¼ 0 and is closest to the origin of coordinate in a standard normal space. Then the original performance function is approximated by a low-order function at the obtained design point. Obviously, the accuracy of these two methods lies heavily on the search of the design point and the approximation of the performance function, and it would spend high costs to obtain the derivative information of the performance function with respect to input variables when searching for the design point using iteration algorithms for complicated engineering problems. Consequently, moment based analytical methods may have large errors when dealing with highly dimensional or implicit problems. Sampling based numerical simulation methods can artfully avoid the above shortcomings, and they are suitable for most of the reliability problems, with no constraint on the type of performance function. Doubtlessly, the Monte Carlo Simulation (MCS) method is a representative numerical simulation method, and it is easy to implement and probably the most widely used method for reliability analysis. However, it becomes quite inefficient when dealing with those rare events. In order to obtain convergent results, the simulation sampling size should be large enough, generally, ð102  104 Þ=Pf points should be sampled. Especially, for engineering problems with implicit performance functions, large numbers of mode simulations are unpractical and the computational cost is really unaffordable. The Importance Sampling (IS) method5 is a popular variance reduction technique, and its computational efficiency has been improved compared with MCS. Meanwhile, for engineering problems with implicit performance functions and small failure probabilities, IS is also incompetent and the accuracy relies on the estimated design point. In conclusion, sampling based numerical simulation methods show low efficiency, but it is worth emphasizing that this family of methods is always employed to test the accuracy and efficiency of newly developed approaches. Considering the computational burden of numerical simulation methods, surrogate model based methods have been widely researched. They aim at utilizing surrogate models to substitute the real performance functions, and thus the computational burden of evaluating the implicit performance functions can be reduced obviously. Many surrogate techniques have been developed up to now, and commonly used surrogate models include the response surface model,6,7 the artificial neural networks,8,9 the support vector machine,10,11 and the Kriging model.12–14 Generally, the accuracy of results obtained by these surrogate model based methods relies on the accuracy of the surrogate models, and research on the balance between

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B. Li et al. efficiency and accuracy when using this type of methods has attracted increasing attention. In recent years, a major breakthrough has been achieved in estimating the PDF of the response function of a system or structure using the concept of entropy, a measure of uncertainty. Under some given moment constraints of the response function, the Principle of Maximum Entropy (P-ME) proposed by Jaynes15 can estimate the PDF of the response function by a way of maximizing the entropy. Meanwhile, in order to model the distribution tail of the PDF of the response function accurately, a larger number of moments are required. Zhang and Pandey pointed out in Ref. 16 that the entropy maximization algorithm shows the numerical instability well as the number of moment constraints increases and the tail of the obtained PDF may become an oscillatory function. Significantly, fractional moments have promoted the development of the P-ME.16–20 Zhang and Pandey16 proved that a fractional moment contains information of a large number of integral moments. The concepts of the P-ME, fractional moment, and dimensional reduction method were used to accurately estimate the PDF of the structural response function, and the failure probability with high precision could be easily calculated based on the available PDF. There is no doubt that the method proposed by Zhang and Pandey16 is really efficient. A Multiplicative Dimensional Reduction Method (M-DRM) based on the concept of high-dimensional model representation was proposed by Zhang and Pandey16 to transform the original response function into the form of a product of univariate functions, and thus the fractional moments could be easily computed by the integrations of one-dimensional functions using the Gaussian integration scheme. Consequently, only a few functional evaluations are essential for structural reliability analysis. However, some shortcomings and limitations have been detected with the method proposed by Zhang and Pandey16 through deep research, which come from the MDRM actually, as shown in Section 2.2. The major limitation is that it can only deal with reliability problems with mutually independent input variables. While in many cases, dependent input variables exist in structural systems,21–25 thus it is very necessary to extend the fractional moment based P-ME method to the correlated field. Note that a new technique, called Unscented Transformation (UT), was applied by Julier and Uhlmann26 to propagate mean and covariance information through nonlinear transformations. A set of weighted sigma points are chosen deterministically, considering the mean and covariance of the sigma points must match those of the prior distribution to be transformed. Researchers have applied UT in many fields, such as Kalman filter,26 statistical robust design,27 wind production system,28 and sensitivity analysis,29. Among them, UT was used to compute the mean and covariance of outputs, namely lower integral moments. In this paper, we apply UT to calculate the fractional moments of the performance function, so as to extend the fractional moment based P-ME to the correlated field, and widen its engineering applicability. The remainder of the paper is organized as follows. Section 2 briefly reviews the fractional moment based P-ME and presents some discussions about this method. Section 3 describes the UT method and gives the computational effort of the proposed method. In Section 4, numerical and engineering examples are introduced and analyzed. Finally, Section 5 draws conclusions.

Please cite this article in press as: Li B et al. Reliability analysis based on a novel density estimation method for structures with correlations, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.04.005

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Reliability analysis based on a novel density estimation method 157 158

2. Brief review of fractional moments based maximum entropy analysis

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In Ref. 16, Zhang and Pandey16 implemented structural reliability analysis based on the concepts of the P-ME, fractional moment, and dimensional reduction method. It is really a useful method, but some limitations still exist.

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2.1. Maximum entropy analysis using fractional moments

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The P-ME with constraints of fractional moment is shown as 8 > < find fY ðyÞ max H½fY ðyÞ ð2Þ > R a : k f ðyÞdy ¼ Mak ðk ¼ 1; 2; . . . ; mÞ s: t: y Y Y Y

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Y

By constructing the Lagrange function associated in Eq. (2) and deriving the optimal solution, we can easily obtain the generic form of the estimated PDF of fY ðyÞ as ! m X ak b f Y ðyÞ ¼ exp  kk y ð4Þ k¼0

where k ¼ ½k0 k1    km T is the Lagrange multiplier vector, and R  P  ak . a0 ¼ 0; k0 ¼ lg Y exp  m k¼1 kk y Then the main task is to compute k and a. Zhang and Pandey16 introduced the Kullback-Leibler entropy between the true PDF, fY ðyÞ, and the estimator, fbY ðyÞ, as R b ¼ f ðyÞlg½f ðyÞ= fbY ðyÞdy Kðf; fÞ Y Y Y ð5Þ R ¼ H½f ðyÞ  f ðyÞlg½ fbY ðyÞdy Y Y

b will It can be seen that if fbY ðyÞ is very close to fY ðyÞ, Kðf; fÞ be close to zero. Therefore, fbY ðyÞ can be obtained by minimizab Since H½f ðyÞ is independent of k and a, minition of Kðf; fÞ. Y mization of Eq. (5) means minimization of the following function: m X b þ H½f ðyÞ ¼ k0 þ Iðk; aÞ ¼ Kðf; fÞ kk MaYk Y

ð6Þ

k¼1

Consequently, the P-ME problem for estimating fbY ðyÞ is transformed into the following optimization problem16: 8 > an T & k ¼ ½k1 k2   kn T > < find a ¼ ½a1 a2   " ! # m m X X R ð7Þ a k > kk MaYk > : min Iðk;aÞ ¼ lg Y exp  kk y dy þ k¼1

2.2. Introduction and discussions of M-DRM

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The M-DRM is proposed based on the concept of HighDimensional Model Representation (HDMR),30,31 which decomposes a multivariate function into orthogonal component functions involving low-dimensional vectors only. Firstly, we assume Y ¼ gðXÞ is the response function, and the HDMR expansion of gðXÞ can be expressed as

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n X X gðXÞ ¼ g0 þ gi ðXi Þ þ gij ðXi ; Xj Þ þ    :

where fY ðyÞ is the PDF of the response function Y, and MaYk is the akth-order fractional moment of Y. H½fY ðyÞ is the information-theoretic entropy of Y, and it is defined as Z H½fY ðyÞ ¼  fY ðyÞlg½fY ðyÞdy ð3Þ

Y

3

k¼1

The optimization has been transformed and becomes clear now. While the next work is to efficiently compute the fractional moments nested in the optimization problem, Zhang and Pandey16 introduced an M-DRM to compute these MaYk ðk ¼ 1; 2; . . . ; mÞ. In next subsection, the M-DRM is presented, and some discussions are described after deep research on this method.

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ð8Þ 217

16i
i¼1

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In Ref. 16, the basic idea of the Cut-HDMR method is applied, and the component functions are evaluated at the mean values of the input variables, i.e., l ¼ ½l1 l2 . . . ln T : 8 g0 ¼ gðlÞ > > > > > g > i ðXi Þ ¼ gðl1 ; l2 ; . . . ; li1 ; xi ; liþ1 ; . . . ; ln Þ  g0 > < gij ðXi ; Xj Þ ¼ gðl1 ; l2 ; . . . ; li1 ; xi ; liþ1 ; . . . ; lj1 ; xj ; ljþ1 ; . . . ; ln Þ > > gi ðXi Þ  gj ðXj Þ  g0 > > > > > : .. . ð9Þ

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Then, a simple representation of the response function gðXÞ can be obtained by retaining only up to the first-order components of Eq. (8), i.e.,

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n X gðXÞ  gðl1 ; l2 ; . . . ; li1 ; xi ; liþ1 ; . . . ; ln Þ  ðn  1ÞgðlÞ

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i¼1 n X ¼ gðxi ; lÞ  ðn  1ÞgðlÞ i¼1

ð10Þ 16

While Zhang and Pandey made an improvement based on the Cut-HDMR, they applied the Cut-HDMR to the logarithmic transformation of the response function, i.e., lg½absðgðXÞÞ. Substituting gðÞ with lg½absðgðÞÞ in Eqs. (9) and (10), and then inverting the transformation, the original response function can be expressed as16 gðXÞ ¼ exp½gðXÞ  ½gðlÞ

n Y 1n

gðxi ; lÞ

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ð11Þ

i¼1

Then the fractional moment, which is defined as a multidimensional integration, can be decomposed as16 R MaY ¼ X ½gðXÞa fX ðxÞdx " #a " # n n Y Y R 1n  X ½gðlÞ gðxi ; lÞ fXi ðxi Þ dx ð12Þ i¼1 i¼1 n n o Y R ¼ ½gðlÞaan ½gðxi ; lÞa fXi ðxi Þdxi Xi i¼1

where fXi ðxi Þ is the marginal PDF of variable Xi . The integration of a one-dimensional function can be computed by the Gaussian integration scheme. Actually, the M-DRM described above is really an efficient and useful method to compute fractional moments. However, there are still some limitations with this technique. Some discussions are presented as follows. Firstly, the M-DRM was proposed based on the CutHDMR, and the mean values of input variables are chosen

Please cite this article in press as: Li B et al. Reliability analysis based on a novel density estimation method for structures with correlations, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.04.005

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as a good choice for the coordinates of the cut-point. Therefore, it can be seen from Eq. (11), the multiplicative approximation of the response function, that it requires the response value of the mean point gðlÞ be larger than zero. That is to say, for computing the integral moments of the response function with multi-dimensional inputs, the M-DRM cannot be competent if gðlÞ 6 0, because of the term ½gðlÞ1n in Eq. (11). For example, consider a very simple bivariate function gðXÞ ¼ X1  X2 , where X1 and X2 are mutually independent input variables, and X1  Nð1; 0:12 Þ, X2  Uð0; 2Þ. The M-DRM cannot be applied on this function to compute the mean, standard deviation, skewness, and kurtosis of the response function owing to gðlÞ ¼ 0. However, the conclusion is different for computing fractional moments. As we all know that fractional moments only work with positive variables, before computing the fractional moments of a response function, we need to make sure that the values of the response function should be positive no matter which moment estimation method is chosen, and gðlÞ > 0 in this context. Therefore, the M-DRM is a useful method for computing moments with gðlÞ > 0 and Zhang and Pandey16 have demonstrated its precision. However, what we mainly consider is another issue about the M-DRM, and it is just the limitation that we intend to improve in this paper. Reconsider Eq. (11), it can be seen that the original response function is approximated by a multiplicative form. There is no doubt that this new expression is propitious to compute fractional moments. Because the multivariate function is expressed as a product of univariate functions, an ath-order fractional moment can be approximated by a product of ath-order moments of the univariate functions, which can be clearly seen from Eq. (12). Meanwhile, if more attention is paid to Eq. (12), we will find that during the derivation of MaY , the joint PDF Q of the input variables, fX ðxÞ, is substituted by ni¼1 fXi ðxi Þ. Qn While fX ðxÞ ¼ i¼1 fXi ðxi Þ only exists when all the input variables are mutually independent with each other, thus it can be concluded that the M-DRM is only competent for response functions with mutually independent inputs, which determines that the reliability analysis method proposed by Zhang and Pandey16 is valid only if structures with independent variables further. However, functions with correlated variables exist generally in engineering, and researchers have studied a lot on them, such as uncertainty propagation and sensitivity analysis.21–25 Consequently, introducing a very efficient method for structural reliability analysis with correlations is very necessary. Considering that the method in Ref. 16 is really outstanding, and its accuracy and efficiency have already been illustrated, we intend to make an improvement on this method based on the above discussions. Actually, improving the technique of computing the fractional moments is about to implement in the paper, and finally, the paper aims at widening the application of the fractional moments based P-ME method to the correlated field, and making this accurate and efficient method more popular for structural reliability analysis in engineering. In this paper, a recently developed method called Unscented Transformation (UT) is introduced to calculate the fractional moments involved in the optimization problem presented in Eq. (7), and Section 3 gives a detailed description of UT.

B. Li et al. 3. Computation of fractional moments using UT

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The UT method, which was originally introduced in Ref. 32, is based on the idea that it is ”easier to approximate a probability distribution than to approximate an arbitrary nonlinear function or transformation”.26 UT can be used to easily estimate the mean and covariance of response functions. In this paper, we extend the application of UT and apply it to compute the fractional moments involved in Eq. (7).

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3.1. Basic technique of UT method

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The UT method is based on selecting a set of weighted points called sigma points, so that their mean and covariance match the mean and covariance of a selected distribution. The basic procedure of UT can be summarized through the following steps:

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(1) Select the sigma points s and weights W. There are several methods for selecting the sigma points,26,32–34 and the number of sigma points to be selected is linearly proportional to the number of input variables. A basic sampling scheme called the standard or symmetric UT32 selects a set of p ¼ 2n þ 1 sigma points, and the points s and corresponding weights W are shown as

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For i ¼ 0,

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s0 ¼ lX

ð13Þ

W0 ¼ w0 For i ¼ 1; 2; . . . ; n, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 n > si ¼ lX þ ð 1w PXX Þ > 0 > i > > > > 0 < Wi ¼ 1w 2n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > n > s ¼ l  P > iþn X 1w0 XX > > i > > : 0 Wiþn ¼ 1w 2n

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ð14Þ

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where lX and PXX are the mean vector and covariance of the pffi input variables X, respectively.  denotes a matrix square root, which can be implement by the Cholesky decomposition, and ðÞi means the ith column or row. Generally, zero is selected as the value of w0 .26 Note that the weighs Wi can be positive or negative, but in order to provide an unbiased estiP mate, they must obey the condition that pi¼1 Wi ¼ 1. (2) Compute the corresponding response values of the sigma points. The one-dimensional output response function Y ¼ gðXÞ is considered in this paper, so Y i ¼ gðsi Þ. We can see that the number of function evaluations is 2n þ 1. (3) Calculate the mean and variance of Y . The low-order integral moments are calculated by

lY ¼

2nþ1 X

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ð15Þ

Wi Yi

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i¼1

VY ¼

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2nþ1 X

Wi ðYi  lY Þ

363 2

ð16Þ

i¼1

Please cite this article in press as: Li B et al. Reliability analysis based on a novel density estimation method for structures with correlations, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.04.005

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where lY and VY are the mean and variance of Y, respectively. While, in this paper, we use UT to calculate the fractional moments of Y, according to Eq. (16), the ath-order fractional moment of Y can be derived as MaY ¼

2nþ1 X

Wi Yai

ð17Þ

i¼1

It has been demonstrated that UT can effectively estimate the low-order moments of an output function,26–29 so for fractional orders, UT can also estimate the corresponding moments with high precision, which can be seen from test example 4.1 in Section 4. Reconsider the simple example in Section 2.2, the estimates of the first four integral moments obtained by UT are 0, 0.5860, 0, and 1.8869, respectively, which are very close to the reference results obtained by the MCS method with 105 samples, i.e., 0, 0.5864, 0, and 1.8675, respectively. It can be found that UT is very easy to implement, while most importantly, the correlation coefficients of inputs are considered in the covariance PXX in Eq. (14), which determines that UT is competent for moment estimation of models with correlations, and correspondingly, the proposed UT based PME method can deal with reliability analysis problems with dependent input variables. Besides, UT is a derivative-free technique, and it can also be used if the response function is non-smooth. Therefore, UT is a very useful method for the computation of fractional moments in the P-ME. The standard or symmetric UT is used to generate sigma points and compute fractional moments, so actually, this sampling scheme is a very efficient and useful technique for general models. Besides, the P-ME method generally needs some low-order fractional moments, and the standard or symmetric UT is competent for computing these moments. While the computational precision may decrease with very highly nonlinear models, now a high-order UT technique can be used, and correspondingly the computational cost will increase.32–34

5 Z Pf ¼

fbT ðtÞdt ¼

Z

1

fbT ðtÞdt

ð19Þ

0

Fratio

Note that during the whole implementation process of the P-ME, only the step of computing fractional moments calls for the performance function. In addition, we can see that UT only selects 2n þ 1 sigma points and compute the corresponding response values, and thus it can be concluded that the method proposed in this paper only needs 2n þ 1 evaluations of the performance function, where n is the dimensionality of the input vector. Because in each iteration loop of the optimization problem in Eq. (7), the values of Ti ¼ Gðsi Þ ði ¼ 1; 2; . . . ; 2n þ 1Þ can be reused, no extra function evaluations are needed. Let NUT be the number of function evaluations of the proposed method in this paper, therefore it is given as NUT ¼ 2n þ 1

ð20Þ

The number of function evaluations of the method proposed by Ref. 16 is given as NMDRM ¼ 1 þ

k X

n X

i¼1

i¼kþ1

ðNi  1Þ þ

Ni

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ð21Þ 447

where k is the number of input variables with the symmetric distribution (e.g., normal distribution, uniform distribution). Ni is the adopted number of Gaussian nodes for variable Xi , and Ni ¼ 5 in this paper. It can be seen that the proposed UT based P-ME method and the M-DRM based one in Ref. 16 are both efficient enough, and they both decrease the cost of function evaluations to a great extent. Considering an example of 5 independent normal variables, it can be derived that NUT ¼ 11, and NMDRM ¼ 21. In a word, the proposed method in this paper can greatly decrease the computation burden with assurance of reasonable precision.

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4. Examples

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In this section, three examples involving correlated input variables are introduced to illustrate the advantages of the proposed UT based P-ME method. Firstly, a numerical response function is adopted to demonstrate the accuracy of the moments computed by UT and the PDF estimated by the proposed UT based P-ME method. In fact, there are a few available PDF estimation methods for comparison. Ref. 35 introduced a Kernel Density Estimation (KDE) method to estimate the PDF of the output response, while it needs thousands of function evaluations to get a reasonable PDF generally,35 and thus, compared with the KDE method, the proposed UT based P-ME method is more efficient, and the KDE method is not introduced to illustrate the efficiency of the proposed method in this paper. The proposed method is applied on the reliability analysis of two engineering structures, i.e., a cantilever beam and a roof truss structure, to further demonstrate its accuracy and efficiency. Besides, the engineering applicability for reliability analysis of the proposed method can also be seen from these examples.

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4.1. Test example: A numerical example

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Consider a response function

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3.2. Computational effort In this paper, a novel density estimation method for structural reliability analysis with correlations is proposed, based on the method introduced by Zhang and Pandey.16 We make an improvement on the computation of fractional moments, in order to make the P-ME competently deal with problems involving correlated inputs. Here, the P-ME method is applied to directly estimate the PDF of the performance function of a structure. For a response function gðXÞ of a structure, of which the critical threshold is assumed as YCT , the performance function is constructed considering the fact that the fractional moment only works with a positive variable as YCT T ¼ GðXÞ ¼ gðXÞ

ð18Þ

so that the values of GðXÞ are positive and the P-ME can be applied on it to estimate the probabilistic distribution. Note that the corresponding failure domain is Fratio ¼ fX : GðXÞ 6 1g, so when the estimated PDF fbT ðtÞ of the performance function is available, the failure probability can be easily computed by

Please cite this article in press as: Li B et al. Reliability analysis based on a novel density estimation method for structures with correlations, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.04.005

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B. Li et al.

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Y ¼ X1 þ ð0:2X3 Þ2 þ

1:5X2 X24 X1

ð22Þ

where X1 and X3 are two uniformly distributed variables, and X2 and X4 are normally distributed variables. Their distribution parameters are given as: X1  Uð3:7; 4:9Þ, X2  Nð1; 0:12 Þ, X3  Uð2:65; 4:5Þ, and X4  Nð2; 0:22 Þ. X2 and X4 are correlated and the Pearson correlation coefficient q ¼ 0:4. This example is only used to illustrate the accuracy of the estimated moments and PDF of the response function. Table 1 gives the results of fractional and integral moments computed by the UT method, which are compared to the results of the MCS method. Besides, the estimated PDF of the response function obtained by the proposed method (denoted as UT_P-ME) is shown in Fig. 1, and it is also compared to the reference one estimated by MCS. 106 points are sampled in the MCS method, so each order moment computed by MCS can be set as a reference. Obviously, it can be seen that the moments computed by UT are very close to those computed by MCS, especially for the low fractional orders. In order to demonstrate the applicability and precision of the proposed method for correlation problems, the correlation coefficient is changed from 0.4 to 0.2, and then using the proposed method, the results are listed in Table 1, and the results match well with the MCS results. Therefore, it can be found that the UT method is an available and useful method for computation of the low real order

Table 1 Comparison of estimated fractional and integral moments of test example. a 0.3 0.05 0.23 0.6 1 2 3

q ¼ 0:4

q ¼ 0:2

UT

MCS

UT

MCS

0.57761 0.91256 1.52365 3.0016 6.25239 39.2981 248.2797

0.57762 0.91256 1.52362 3.0014 6.25213 39.2990 248.2912

0.57775 0.91259 1.52334 2.99989 6.24615 32.20988 247.3607

0.57772 0.91258 1.52340 3.00013 6.24681 39.21251 247.3259

Fig. 1

moments, which is very important for the fractional moments based P-ME method, because accurate constraints in the PME can improve the accuracy of the estimated PDF. However, it needs to point out that the UT method only selects 9 sigma points to calculate all the moments, and only 9 evaluations of the response functions are needed, so it really greatly decreases function evaluations compared with a sampling based numerical method. Fig. 1 shows the estimated PDFs obtained by the proposed UT_P-ME and MCS methods, respectively, and we can see that the PDF estimated by the UT_P-ME method matches very well with the simulation result obtained by MCS with 106 samples, especially for the tail of the PDF curve, which closely relates to the failure probability. Besides, 9 evaluations of the response function are the cost during the whole process, so it is efficient enough. According to the comparison, we can find that with only a few function evaluations, the accuracy of the proposed UT_P-ME method has been well demonstrated.

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4.2. Engineering example 1: A cantilever beam

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In this subsection, we introduce a cantilever beam structure,25 which is shown in Fig. 2. Dimensional parameters of the beam, such as width, height, and length, are denoted as w, t, and L, respectively. E is the elastic modulus. Two forces FX and FY are random forces exerted on the tip section. According to the mechanical analysis, it can be obtained that the tip section produces the maximum displacement in the vertical direction, and this maximum displacement can be expressed as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2  2ffi 4L3 FX FY DðFX ; FY ; E; w; t; LÞ ¼ þ 2 ð23Þ 2 Ewt w t The distribution information of the six input variables, i.e., L, FX , FY , E, w, and t is listed in Table 2. Besides, the Pearson correlation coefficients of the input variables are set to qwt ¼ 0:55 and qwL ¼ qtL ¼ 0:45. Assume the critical threshold of the maximum displacement of the tip section is 0.066 m, so the performance function of the cantilever beam is built as T ¼ GðXÞ ¼ 0:066=DðFX ; FY ; E; w; t; LÞ according to Eq. (18). Here, we directly apply the proposed UT_P-ME method on the performance function. Six input variables determine that

Probability density function of response function of test example.

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Reliability analysis based on a novel density estimation method

Fig. 2

Sketch of a cantilever beam.

Table 2 Distribution information of input variables of a cantilever beam.

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Input variables

Distribution

Mean

Variation coefficient

L FX FY w t E

Normal Lognorm Lognorm Normal Normal Lognorm

4.29 m 556.8 N 453.6 N 0.062 m 0.0987 m 200 GPa

0.10 0.08 0.08 0.10 0.10 0.06

there are 13 sigma points to be selected, and the PDF estimated by the UT_P-ME is given in Fig. 3. Only taking 13 evaluations of the performance function, the UT_P-ME method estimates a PDF of the performance function which is very close to the one obtained by MCS with 106 samples. Again, the precision of the proposed method in this paper is illustrated.

Fig. 3

Table 3 Method

MCS UT_P-ME AFOSM IS

7 Based on the available estimated PDF fbT ðtÞ of the performance function, reliability analysis can be implemented easily. Referring to Eq. (19), the failure probability can be easily computed by integrating fbT ðtÞ on [0, 1], and we get Pf ¼ 0:0111. For comparison, the reference result obtained by MCS is computed by Eq. (24) as PfMCS

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N 1X ¼ IF ðXi Þ N i¼1 ratio

ð24Þ 563

where N ¼ 106 is the number of random samples obtained by the simple sampling method using the joint PDF of input variables. Xi ði ¼ 1; 2; . . . ; NÞ are the random samples and IFratio ðXi Þ is the indicator function of the failure region Fratio , i.e., 8 < 1; Xi 2 Fratio IFratio ðXi Þ ¼ 0; Xi R Fratio . By 106 evaluations of the perfor: mance function, the failure probability estimated by Eq. (24) is 0.0131 with a variation coefficient of 0.0058. Applying the Advanced First Order and Second Moment (AFOSM) method36 with the finite difference approximation technique on this cantilever beam, the result is 0.0155 with 117 function evaluations. Besides, the failure probability estimated by an improved IS method36 is 0.0145 with a variation coefficient of 0.0153, taking 5117 evaluations of the performance function. Compared with results computed by these existing meth-

PDF of performance function of a cantilever beam.

Failure probabilities of a cantilever beam for different correlation coefficients. Pf qwt ¼ 0:55; qwL ¼ 0:45 qtL ¼ 0:45

qwt ¼ 0:4; qwL ¼ 0:2 qtL ¼ 0:3

qwt ¼ 0:1; qwL ¼ 0:6 qtL ¼ 0:3

0.0131 (0.0058) 0.0111 0.0155 0.0145 (0.0153)

0.0655 (0.0064) 0.0638 0.0710 0.0691 (0.0147)

0.0620 (0.0061) 0.0610 0.0694 0.0657 (0.0139)

Note: The numbers in the brackets denote the variation coefficient.

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B. Li et al. see that the precision of the proposed UT_P-ME method has been demonstrated, and the proposed method is a useful method for correlation problems.

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4.3. Engineering example 2: A roof truss

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A roof truss25 is shown in Fig. 4, in which the top boom and compression bars are reinforced by concrete, and the bottom boom and tension bars are steel. Assume that a uniformly distributed load q is applied on the roof truss, and this load can be transformed into a nodal load P ¼ ql=4. According to the mechanical analysis, the perpendicular defection of Node C

 2 , which is a function can be obtained as DC ¼ ql2 A3:81 þ A1:13 c Ec s Es

Fig. 4

Table 4 truss.

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Schematic diagram of a roof truss.

Distribution information of input variables of a roof

Input variables

Distribution

Mean

Variation coefficient

q l Ac As Ec Es

Normal Normal Normal Normal Normal Normal

20,000 N/m 12 m 0.04 m2 9.82  104 m2 20 GPa 100 GPa

0.07 0.01 0.12 0.06 0.06 0.06

ods, we can see that the proposed UT_P-ME method only needs to run a very small number of evaluations of the performance function to get a reasonable reliability result. Similarly, in order to demonstrate the advantages of the proposed method, the results of failure probabilities for different correlation coefficients are given in Table 3, from which we can

Fig. 5

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of random variables. Ac , As , Ec , Es , and l are the sectional area, elastic modulus, length of the concrete and steel bars, respectively. DC not exceeding 3 cm is set as the constraint condition to define the performance function, and referring to Eq. (18), the performance function is constructed as follows:  2  ql 3:81 1:13 ð25Þ T ¼ GðXÞ ¼ 0:03 þ 2 Ac Ec As Es

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The distribution information of these independent normal basic random variables is listed in Table 4, and the Pearson correlation coefficients of the input variables are given as qqAc ¼ qqAs ¼ 0:5 and qAc As ¼ 0:6. The reliability analysis of this roof truss structure is similar to that of the cantilever beam structure. Using the proposed UT_P-ME method, the estimated PDF is presented in Fig. 5. Similarly, the reference PDF obtained by MCS with 106 random samples is also shown in Fig. 5 for comparison. We can also see that the two results match very well, especially for the tail of the PDF. With the available estimated PDF, reliability analysis of the roof truss structure can be implemented. According to the integration of the PDF obtained by UT_P-ME, we obtain that the probability of the perpendicular defection of Node C exceeding 3 cm is 2.371  103. Similarly, the failure probability obtained by MCS using Eq. (24) is 2.211  103, with 106 evaluations of the performance function, and the variation coefficient of the result is 0.0086. The result obtained by the

603

PDF of performance function of a roof truss.

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Reliability analysis based on a novel density estimation method Table 5

9

Failure probabilities of a roof truss for different correlation coefficients.

Method

MCS UT_P-ME AFOSM IS

Pf qqAc ¼ 0:5; qqAs ¼ 0:5 qAc As ¼ 0:6

qqAc ¼ 0:3; qqAs ¼ 0:6 qAc As ¼ 0:4

qqAc ¼ 0:1; qqAs ¼ 0:2 qAc As ¼ 0:9

2.211  103 (0.0086) 2.371  103 1.872  103 2.168  103 (0.0133)

2.015  103 (0.0099) 2.179  103 1.552  103 1.896  103 (0.0125)

1.3521  102 (0.0085) 1.3156  102 9.9841  103 1.2015  102 (0.0144)

Note: The numbers in the brackets denote the variation coefficient.

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AFOSM method36 is 1.872  103 with 104 evaluations of the performance function. Moreover, taking 5104 function evaluations, the IS method36 gives the result 2.168  103 with a variation coefficient of 0.0133. We can see that the result of our proposed UT_P-ME method is with high precision compared to the MCS reference result. Note that the number of function evaluations of UT_P-ME is just 13, so the advantage is extraordinarily obvious and it is really a very useful method for structural reliability analysis in engineering. Similarly, the results of failure probabilities for different correlation coefficients are given in Table 5, and the results of the proposed method match well with the MCS reference results.

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5. Conclusions

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In this paper, a new fractional moment based P-ME method is proposed to provide an available approach for structural reliability analysis. In the maximum entropy method, a new moment calculation method called UT is applied to calculate the fractional moments involved in the P-ME method. The standard UT method only needs to select 2n þ 1 weighted sigma points, and output moments can be computed for either independent or dependent inputs. Therefore, the proposed UT_P-ME method can be competent for reliability analysis of structures with correlations, and it only needs a low computation cost. Consequently, for structures with mutual independent input variables or structures with correlations, UT_P-ME is a very useful and available reliability analysis method.

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648

Acknowledgements

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This study was supported by the Equipment Development Department ‘‘13th Five-year” Equipment Research Field Foundation of China Central Military Commission (No. 6140244010216HT15001).

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