Efficient reliability analysis of structures with the rotational quasi-symmetric point- and the maximum entropy methods

Mechanical Systems and Signal Processing 95 (2017) 58–76

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Efficient reliability analysis of structures with the rotational quasi-symmetric point- and the maximum entropy methods Jun Xu a,⇑, Chao Dang b, Fan Kong c a Department of Structural Engineering, College of Civil Engineering & Hunan Provincial Key Lab on Damage Diagnosis for Engineering Structures, Hunan University, Changsha 410082, PR China b College of Civil Engineering, Hunan University, Changsha 410082, PR China c School of Civil Engineering and Architecture, Wuhan University of Technology, 122 Luoshi Road, Wuhan 430070, PR China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 15 August 2016 Received in revised form 9 March 2017 Accepted 12 March 2017

Keywords: Structural reliability Rotational quasi-symmetric point method Fractional moment Maximum entropy Limit state function

This paper presents a new method for efficient structural reliability analysis. In this method, a rotational quasi-symmetric point method (RQ-SPM) is proposed for evaluating the fractional moments of the performance function. Then, the derivation of the performance function’s probability density function (PDF) is carried out based on the maximum entropy method in which constraints are specified in terms of fractional moments. In this regard, the probability of failure can be obtained by a simple integral over the performance function’s PDF. Six examples, including a finite element-based reliability analysis and a dynamic system with strong nonlinearity, are used to illustrate the efficacy of the proposed method. All the computed results are compared with those by Monte Carlo simulation (MCS). It is found that the proposed method can provide very accurate results with low computational effort. Ó 2017 Published by Elsevier Ltd.

1. Introduction A fundamental problem in reliability analysis of structures is the determination of the probability of failure pf [1], which is defined by

Z pf ¼ Pr½GðZÞ 6 0 ¼

GðZÞ60

f Z ðzÞdz

ð1Þ

where Pr stands for probability, GðÞ is the explicit or implicit limit state function or the performance function, Z ¼ fZ 1 ; Z 2 ; . . . ; Z d gT is the random vector consisting of d basic random variables involved in both structural parameters and loads, and f Z ðzÞ denotes the joint probability density function (PDF) of Z. Despite the simplicity of the formulation of the problem, the exact solution of Eq. (1) is difficult since the explicit expressions of limit state functions, in general, are not available for realistic engineering problems, particularly when nonlinear behavior needs to be considered [2]. Difficulty in computing this probability has led to the development of various approximation techniques, among which three kinds of methods are usually adopted. The first kind is the first- or the second-order reliability method (FORM or SORM) [3]; those methods were considered to be the most popular for reliability analysis in the ⇑ Corresponding author. E-mail addresses: [email protected] (J. Xu), [email protected] (C. Dang), [email protected] (F. Kong). http://dx.doi.org/10.1016/j.ymssp.2017.03.019 0888-3270/Ó 2017 Published by Elsevier Ltd.

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59

past decades. However, in the reliability evaluation of nonlinear structures using direct finite element formulations, the derivatives of limit state functions with respect to basic random variables, required for FORM/SORM, may be not readily available [4]. On the other hand, the most widely adopted methods to determine the probability of failure are the Monte Carlo simulation (MCS) and its variants, which are grouped under the second kind. The MCS in its crude form may require excessive computational efforts in evaluating problems of small failure probability or in problems involving a large number of costly finite element analysis. Therefore, in the past decades, some remarkable improvements on the efficiency of MCS have been made, which could be called the advanced MCS. Representative contributions include subset simulation [5], asymptotic sampling [6] and line sampling [7], etc. The third is the response surface methodology [8], which builds a surrogate model for the target limit state function, defined in a simple and explicit mathematical form for reliability analysis of linear and nonlinear structures [9,10]. Response surface models can be built to find the design point with a much reduced computational cost [11], where FORM or MCS can be applied for reliability analysis with high efficiency. Here it is convenient to explicitly mention neural networks [12,13], support vector machines [9,14,15] and Gaussian processes (kriging) [16] as representative methodologies of response surface models. Extensive developments have been studied in the aforementioned methods. Nevertheless, other efficient reliability methods still attract interest in the structural reliability community. Another route for obtaining structural reliability or probability of failure is directly utilizing the probability density function (PDF) of the performance function. This subject could be classified into two categories. The first is the probability density evolution method (PDEM) [2,17–21], which straightforwardly derives the PDF from a partial differential equation. The second is the method of moments [22], which approximates the distribution of a random variable using its moments of finite orders. Specifically, in the present paper, the method of moments is of interest; this method requires the computation of moments of a multivariate function involving multi-dimensional integrals and therefore numerical integration methods, which can keep the tradeoff between efficiency and accuracy, are preferred. Among the numerical methods for multi-dimensional integration, the univariate dimension reduction method (UDRM) [23], the high dimensional model representation method (HDMRM) [24,25] and the quasi-symmetric point method (Q-SPM) [26,27] might be the best choices. The virtue of these methods is that they need considerably fewer response function evaluations in comparison to other simulation based methods. The UDRM decomposes a multi-dimensional integral into several one-dimensional integrals; it has been used to capture the moments information for reliability analysis [1,28]. However, this method may be not adequate for a system with a large number of random variables or strong nonlinearity due to the high-dimensional integrals retained in the residue error of this method [29]. On the other hand, since the higher order terms in HDMRM are usually negligible, then the first-order and the second-order HDMRM are mostly adopted. In fact, the first-order HDMRM is equivalent to the UDRM while the secondorder HDMRM might be called the bivariate dimension reduction method (BDRM). Although the second-order HDMRM provides more accurate results for general nonlinear structures, the computational efforts may significantly increase compared with that of UDRM [30]. The quasi-symmetric point method (Q-SPM), which is established based on the invariant theory and orthogonal arrays with a 5th degree of algebraic accuracy for numerical integration, is applicable to linear/nonlinear systems with multiple random parameters. The number of deterministic analysis required in Q-SPM is quite close to that of UDRM, which indicates the high efficiency of Q-SPM. In a recent implementation of this method [31], the Q-SPM was found to be not accurate enough for stochastic dynamic response analysis of structures. An improvement of Q-SPM for the computation of moments is necessary. Besides, the adaptability of this method for structural reliability analysis has not been explored yet. The objective of the present paper is to develop an efficient method based on the rotational quasi-symmetric point method (RQ-SPM) and the maximum entropy method (MEM) for structural reliability analysis. The paper is arranged as follows. Section 2 devotes to introducing the fundamentals of the quasi-symmetric point method. In Section 3, a new rotational quasi-symmetric point method is proposed for calculation of the fractional moments of the performance function’s probability density function (PDF). Determination of the PDF of the performance function by the maximum entropy method in such a manner that its fractional moments are used as constraints is conducted in Section 4. In Section 5, integration of the performance function’s PDF over the failure domain gives the probability of failure. Numerical examples are investigated in Section 6 to verify the proposed method. Concluding remarks and the problems to be further studied are included in the final section. 2. Fundamentals of the quasi-symmetric point method The quasi-symmetric point method (Q-SPM) was proposed by Victoir [26] for the purpose of Gaussian weighted multidimensional numerical integration. This method has also been applied to stochastic dynamic response analysis of structures in Ref. [27]. Brief basics of the Q-SPM are given in this section. Let us define the l-th Z moment of the performance function GðZÞ as

ml ¼ E½Gl ðZÞ ¼

XZ

Gl ðZÞf Z ðzÞdz

ð2Þ

where l is a natural number, XZ is the distribution domain of Z. It is well known that non-normal random variables might be transformed into standard normally distributed random variables by some specific transformation, e.g. Rosenblatt transformation [32]. In consequence, Eq. (2) can be evaluated by

Z

ml ¼

XX

Gl ðR1 ½XÞf X ðxÞdx 

N X k¼1

ak Gl ðR1 ½xk Þ

ð3Þ

60

J. Xu et al. / Mechanical Systems and Signal Processing 95 (2017) 58–76

where X is the standard normal random vector, whose density function is

f X ðxÞ ¼

1 d=2

ð2pÞ

ex

Tx

ð4Þ

R1 ½ denotes the transformation, xk is the quasi-symmetric point (integration point) and ak is the corresponding weight for the point xk . Two classes of SPM are provided for numerical integration. The first class of SPM employs fixed coordinates to construct the symmetric point sets [26], whereas the coordinates in the second class vary with the dimension. Because relatively fewer points are involved in the second class of SPM, herein we will deal exclusively with the second class of SPM. The interested readers can refer to Refs. [26,27] for the first class. The two symmetric point sets, denoted as xx0 and xx1 , involved in the second class of SPM are given by

xx0 ¼ ðhr; 0; . . . ; 0Þ; r¼

rffiffiffiffiffiffiffiffiffiffiffiffi dþ2 ; 2

g¼

xx1 ¼ ðhg; . . . ; hgÞ

ð5Þ

rffiffiffiffiffiffiffiffiffiffiffiffi dþ2 d2

ð6Þ

where h is the permutation of 1. It is noted that 2d and 2d points are involved in these fully symmetric point sets when all the points are considered. Note that the number of points in xx0 grows linearly with dimension, whereas the number of points in xx1 increases exponentially with dimension. As is seen, the total number of points in the fully symmetric point set xx1 is often quite large. For example, if the number of independent random variables d ¼ 15 is considered, the total number of points in the fully symmetric point set xx1 is 215 = 32,768, which may result in an intractable computational effort if applied for numerical integration. On the contrary, in Q-SPM, only a subset of points of this symmetric point set are used for numerical integration without losing accuracy. This means that the total integration points for numerical integration (Eq. (3)) are contributed by all the points in xx0 , where the number of points is n1 ¼ 2d and a subset of the points in xx1 with n2 points, where n2 is much smaller than 2d [26]. Since only a subset of points in the symmetric point set xx1 is employed, we will call this method the quasisymmetric point method (Q-SPM) and we will refer to that subset of points of xx1 as the quasi-symmetric points. The invariant theory and orthogonal arrays can be applied to determine the exact value of n2 . Due to its complexity, the total number of points N ¼ n1 þ n2 required in Q-SPM for the dimensions from 3 to 20 has been calculated in Ref. [26] and is listed in Table 1. In this regard, we can specify n1 ¼ 2d and n2 ¼ N  n1 . It is seen that a small number of points are involved in Q-SPM. This feature is extremely exciting for the numerical integration with high efficiency. Thus, the integration points in Q-SPM can be expressed as



xk ¼

ðhr; 0; . . . ; 0Þ k ¼ 1; 2; . . . ; n1

ð7Þ

ðhg; . . . ; hgÞ k ¼ n1 þ 1; n1 þ 2; . . . ; n2

qffiffi qffiffi qffiffi pffiffiffi pffiffiffi pffiffiffi 3 ; 32; . . . ; 32 , For example, if d ¼ 10, we have x1 ¼ ð 6; 0; . . . ; 0Þ, . . ., x10 ¼ ð0; 0; . . . ; 6Þ, x20 ¼ ð0; 0; . . . ;  6Þ, x21 ¼ 2 qffiffi qffiffi  qffiffi qffiffi  qffiffi qffiffi 3 x22 ¼ ;  32; . . . ; 32 , . . ., and x276 ¼  32;  32; . . . ;  32; , etc. 2 The sum of weights for the integration points in xx0 is given by

A0 ¼

8d ðd þ 2Þ

ð8Þ

2

Each point in xx0 has the same weight, which is ak ¼ A0 =n1 . On the other hand, the sum of weights for the integration points in xx1 is

A1 ¼

ðd  2Þ

2

ðd þ 2Þ

2

ð9Þ

For each point in xx1 , the weight is ak ¼ A1 =n2 . P Clearly, A0 þ A1 ¼ Nk¼1 ak ¼ 1. Remark. The main advantages of using the Q-SPM to calculate the moment ml of the performance function are twofold: the replacement of a complicated multi-dimensional integral with a weighted summation on the basis of integration points in the two symmetric point sets above and the moment evaluation for its good accuracy and efficiency compared with other Table 1 The number of points in Q-SPM. Dimension d Number N Dimension d Number N

3 14 12 280

4 24 13 282

5 42 14 284

6 44 15 286

7 78 16 288

8 144 17 546

9 146 18 548

10 276 19 550

11 278 20 552

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61

numerical integration methods, such as trapezoidal integration, Simpson integration and Sparse grid. Besides, like UDRM, the Q-SPM does not require the calculation of any partial sensitivities of response or the inversion of random matrices, as compared with commonly used Taylor expansion/perturbation methods and Neumann expansion methods, respectively [23,28].

3. Proposed rotational quasi-symmetric point method for fractional moment assessment of the performance function 3.1. Definition of fractional moment If Z is a positive random variable, the fractional moment of Z is defined as

E½Z a  ¼

Z

za f ðzÞdz

ð10Þ

Z

where E½ is the expectation operator and a is a real number, called the fractional order. Mathematical properties of the fractional moment can be found in Refs. [33–35]. The Taylor series expansion of Z a about its mean Z 0 is [36] 0

Za ¼

00

ðiÞ

a ðZ a ÞjZ¼Z0 ðZ a ÞjZ¼Z0 Z a0 ðZ ÞjZ¼Z0 Za þ ðZ  Z 0 Þ þ ðZ  Z 0 Þ2 þ    þ ðZ  Z 0 Þi þ    ¼ 0 0! 1! 2! i! 0! a a1 a  ða  1Þ a2 a  ða  1Þ      ða  i þ 1Þ ai 2 Z 0 ðZ  Z 0 Þ þ    þ Z 0 ðZ  Z 0 Þi þ Z 0 ðZ  Z 0 Þ þ 2! i! 1! 1   X a ai þ  ¼ Z 0 ðZ  Z 0 Þi i i¼0

ð11Þ

where the binomial coefficient is

 

a i

¼

a! a  ða  1Þ      ða  i þ 1Þ ¼ i!ða  iÞ! i!

ð12Þ

and thus, we have

E½Z a  ¼

1   X a i¼0

i

Z 0ai E½ðZ  Z 0 Þi 

ð13Þ

where E½ðZ  Z 0 Þi  denotes the i-th central moment of the random variable Z. Actually, the infinite series in Eq. (13) turns to be finite if a is adopted as an integer number. This is because the binomial coefficient turns to be zero when i > a, i.e.

 

a i

¼

a  ða  1Þ      ða  aÞ      ða  i þ 1Þ

¼0 i>a

ð14Þ

2  ð2  1Þ      ð2  2Þ      ð2  i þ 1Þ ¼0 i>2 i!

ð15Þ

i!

For example, if a ¼ 2 is considered, we have

  2 i

¼

and

E½Z 2  ¼ ¼ ¼

2   X 2 2i Z 0 E½ðZ  Z 0 Þ2  i i¼0 Z 20 2ð21Þ 22 þ 1!2 Z 21 Z 0 E½ðZ 0 E½ðZ  Z 0 Þ þ 0! 2! 2 Z 0 þ 2Z 0 E½ðZ  Z 0 Þ þ E½ðZ  Z 0 Þ2 

 Z 0 Þ2 

ð16Þ

Therefore, the i-th integer moment only contains the information of a small number of central moments. On the other hand, when a real number is assigned to a, the binomial coefficient will never become zero, i.e.

 

a i

¼

a  ða  1Þ      ða  i þ 1Þ i!

– 0;

a2RnN

ð17Þ

Obviously, it is seen from Eqs. (13) and (17) that a single fractional moment embodies the information about a large number of central moments. In this regard, a few of fractional moments enable to characterize the true PDF more clearly than a finite number of integer moments does. Also, high-order integer moments might be required to achieve a reasonable accuracy for modeling the tail distribution. However, it might be difficult to compute the high-order integer moments because the integrands could be strongly oscillatory functions. The use of fractional moments may circumvent this limitation to some degree since low-order fractional moments might be adequate to remodel the entire range of distribution [36]. A method, which

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J. Xu et al. / Mechanical Systems and Signal Processing 95 (2017) 58–76

evaluates the fractional moment of the performance function with a tradeoff between accuracy and efficiency, is of paramount importance. For this purpose, a new rotational quasi-symmetric point method will be developed. 3.2. Formulation of the rotational quasi-symmetric point method Although the Q-SPM in Section 2 behaves quite well in the application to stochastic dynamic response analysis of nonlinear structures [27], there are still some deficiencies in this method. The problem of accuracy deterioration for the Q-SPM could be encountered even when evaluating the second-order moment, that is, the standard deviation, of nonlinear structures [31]. The reason could be attributed to the quasi-symmetry and sparseness of this method, where the marginal probability density function or the marginal tail distribution of basic random variables is not captured sufficiently [31]. To overcome such a problem, a new rotational quasi-symmetric point method (RQ-SPM) will be proposed. The basic idea is to rotate these quasi-symmetric points by the Givens transform [37] in multi-dimensional spaces. For example, rotating a point xk ¼ ðx1;k ; x2;k ; . . . ; xd;k Þ in the (i; j) plane by an angle h counterclockwise gives [37]

~k ðhÞ ¼ Mij ðhÞxk x

ð18Þ

~ k is the point after rotation and where x

is the rotational matrix in the (i; j) plane. Thus, the rotation of the point in the space can be represented as

~k ðhÞ ¼ x

d Y d Y

Mij ðhij Þxk ¼ RðhÞxk

ð20Þ

i¼1 j¼iþ1

Q Q where RðhÞ ¼ di¼1 dj¼iþ1 Mij ðhij Þ is the rotational matrix in the space.It is seen that the rotation angles h ¼ fh1;2 ; h1;3 ; . . . ; hd1;d g determine the RQ-SPM based on the integration points in Q-SPM. To ensure the performance of the RQ-SPM, h needs to be optimally specified. To this end, an index needs to be specified as the objective function. Besides, the computation of such an index in multi-dimensional cases should be efficient. Based on this thought, a new index will be proposed to determine the optimal angles to formulate the new RQ-SPM. It should be pointed out that such an index is established based on the marginal probabilistic information of the input random vector. 3.3. Determination of optimal angles in RQ-SPM As mentioned earlier, fractional moments contain the information about a large number of central moments. In this regard, the information of a marginal PDF of the input random vector, may be equivalent to the information included in its fractional moments. In addition, it is quite easy to determine the fractional moments of a marginal PDF both analytically and numerically, when only one-dimensional integrations are involved. Thus, the fractional moments of marginal PDFs of the input random vector can be adopted to formulate the required index as the objective function to determine the optimal angles in the proposed RQ-SPM. Besides, the fractional moments related index can be computed very efficiently in multidimensional cases. In order to avoid the fractional order operation of a negative random variable, a translation is firstly implemented such that

U¼XþN

ð21Þ T

where N ¼ fjcj; jcj; . . . ; jcjg is a constant vector that makes U a positive random vector and c can be adopted as the truncated boundary of a standard normal random variable. In this paper, the truncated boundary jcj ¼ 5 is considered because U i ¼ X i þ jcj P 0; i ¼ 1; 2; . . . ; d. According to the probability theory, the translation transformation only changes the position and will not change the distribution properties of the original random vector. The basic idea is minimizing the differences between the estimated marginal moments from the integration points after the rotation together with their weights and the exact ones. Therefore, an objective function is proposed such that

J. Xu et al. / Mechanical Systems and Signal Processing 95 (2017) 58–76

 aj aj !) l  ~ Ui ðhÞ  lUi  JðhÞ ¼ max max    16i6d 16j6N f  laj

63

(

ð22Þ

Ui

which is the maximum relative error between the estimated fractional moments and the exact ones of U. In Eq. (22), aj s, j = 1, 2, . . . , Nf are the fractional orders, which are adopted such that aj ¼ 4 : 0:1 : 4 because fractional moments of low orders are usually adequate to rebuild the normal PDF, JðhÞ denotes the objective function of the rotational point set ~ ¼ fx ~k ; k ¼ 1; 2; . . . ; Ng, laUj denotes the exact fractional moment of random variable U i of order aj , i.e. P

laUji ¼ E½U ai j  ¼

i

Z

Ui

a

ui j pUi ðui Þdui

ð23Þ

~ aUj ðhÞ is the estimated fractional moment of order aj , defined as and l i

l~ aUji ðhÞ ¼

N N X X a ~ i;k ðhÞÞaj ¼ ak ðu ak ð~xi;k ðhÞ þ jcjÞ j k¼1

ð24Þ

k¼1

~k . where ~ xi;k represents the i-th coordinate of the point x We believe that the smaller the objective function of Eq. (22) is, the better performance of the rotational point set will exhibit in the numerical integration (Eq. (3)). Therefore, the task then changes to find the optimal angles, which vary between [0, 2p] that minimizes the objective function. In this regard, an optimization problem can be formulated such that

find h

objectiv e

ð25Þ

minðJðhÞÞ

s:t: 0 6 hi;j 6 2p;

8hi;j 2 h

Clearly, this is a problem involving multi-variable optimization, which could be solved by employing a global optimization method (like for example a genetic algorithm or a particle swarm optimization). When the optimization is performed based on the integration points in Q-SPM, the resulting points belong to the new RQ-SPM. It is assumed that the rotation transformation will not reduce the algebraic accuracy compared with that of the original Q-SPM. Besides, by performing such a rotation, the marginal probability density could be better reproduced and the accuracy of numerical integration could be improved considerably. 3.4. Fractional moments assessment of the performance function based on the RQ-SPM The moment in Eq. (3) can be alternatively evaluated by

Z ml ¼

XX

Gl ðR1 ½XÞf X ðxÞdx 

N X

~k Þ ak Gl ðR1 ½x

ð26Þ

k¼1

It is seen that the proposed RQ-SPM does not increase any computational effort, however, it may significantly improve the accuracy of numerical integration of a highly nonlinear response surface. As a matter of fact, the numerical evaluation of the l-th order central moment jl is the rank-l integral [27]. In order to understand the rank-l integral concept, let us consider for example that the mean is a rank-1 integral, i.e.

Z

j1 ¼ m1 ¼

XX

G1 ðR1 ½XÞf X ðxÞdx 

N X ~k Þ ak G1 ðR1 ½x

ð27Þ

k¼1

where ak ’s occur linearly. The second order central moment, which is the variance, is a rank-2 integral, i.e.

j2 ¼ r2 ¼ m2  m21 

N X k¼1

~k Þ  ak G2 ðR1 ½x

" #2 N X ~k Þ ak G1 ðR1 ½x

ð28Þ

k¼1

where ak ’s occur in a quadratic form. Likewise, the skewness j3 and the kurtosis j4 are the rank-3 and the rank-4 integrals respectively. In general, to evaluate the l-th order central moment jl , a rank-l integral is involved. Therefore, the structural reliability analysis, which is based on driving the PDF of the performance function, is a rank-1 integral because the PDF contains the information of an infinite number of central moments j1 , j2 , . . ., j1 . Moment method based on a finite number of central moments, for example, the fourth moment method [38], may be not adequate for accurate reliability assessment [36]. On the other hand, a fractional moment of the performance function contains the information about numerous central moments (see Eq. (13)), where j1 ; j2 ; . . . ; j1 are all involved. In this regard, a couple of fractional moments could be regarded to be equivalent to a rank-1 integral. In this regard, recovering the PDF of the performance function with high accuracy based on fractional

64

J. Xu et al. / Mechanical Systems and Signal Processing 95 (2017) 58–76

moments is possible for structural reliability analysis. Due to the better performance in marginal regions, the proposed RQSPM can be employed to efficiently evaluate the fractional moment of the performance function such that

ma ¼

R

XZ

Ga ðZÞf Z ðzÞdz

XX

Ga ðR1 ½XÞf X ðxÞdx

R

¼

ð29Þ

N X ~k Þ ak Ga ðR1 ½x



k¼1

The accuracy of such implementation of the proposed RQ-SPM for the fractional moment evaluation will be explored and verified in the following numerical examples. Then, the method which can derive the PDF from a couple of fractional moments is of great interest. In the following section, a new procedure will be developed in which the proposed RQ-SPM is considered to be combined with the maximum entropy method (MEM) to derive the PDF of the performance function. In this regard, the efficiency of the proposed method is ensured for structural reliability analysis. 4. Determination of the performance function’s PDF by the maximum entropy method The MEM is regarded as the most unbiased estimation of the PDF, which means the most probable PDF from all the PDF under the moments constraint since ‘‘it is maximally noncommittal with regard to missing information” [39]. Let Y ¼ GðZÞ be a positive random variable with density f Y ðyÞ; its differential entropy H½f Y ðyÞ is given by

Z

H½f Y ðyÞ ¼ 

f Y ðyÞ ln½f Y ðyÞdy

ð30Þ

In general, the maximum entropy density estimation is obtained by maximizing the differential entropy in Eq. (30) subject to a finite number of integer moment constraints

Z

ml ¼

yl f Y ðyÞdy;

l ¼ 1; 2; . . . ; L

ð31Þ

where ml is the l-th raw moment of the performance function, L stands for the number of the given moment constraints. Therefore, the f Y ðyÞ can be obtained in the following way

8 > < find f Y ðyÞ R Maximize H½f Y ðyÞ ¼  f Y ðyÞ ln½f Y ðyÞdy > R : Subject to ml ¼ yl f Y ðyÞdy; l ¼ 1; 2; . . . ; L

ð32Þ

The Lagrangian function corresponding to Eq. (32) is given as [36,40]

Z L¼

Z f Y ðyÞ ln½f Y ðyÞdy  ðk0  1Þ½

f Y ðyÞdy  1 

Z L X kl ½ yl f Y ðyÞdy  ml 

ð33Þ

l¼1

For optimal solution, following condition must be fulfilled, i.e. @L @f Y ðyÞ

¼ dL

R R ¼ d f Y ðyÞ ln½f Y ðyÞdy  ðk0  1Þd½ f Y ðyÞdy  1 L X R  kl d½ yl f Y ðyÞdy  ml  Rl¼1 ¼  fdf Y ðyÞ ln½f Y ðyÞ þ f Y ðyÞdfln½f Y ðyÞggdy L X R R kl ½ yl df Y ðyÞdy ðk0  1Þ½ df Y ðyÞdy  ¼

Rn

l¼1

o ln½f Y ðyÞ þ f Y ðyÞ f Y1ðyÞ df Y ðyÞdy

L RX R kl yl df Y ðyÞdy  ðk0  1Þdf Y ðyÞdy 

¼

R

(

l¼1

 ln½f Y ðyÞ  1  ðk0  1Þ 

) L X kl yl df Y ðyÞdy l¼1

¼0 so that

ð34Þ

J. Xu et al. / Mechanical Systems and Signal Processing 95 (2017) 58–76

 ln½f Y ðyÞ  1  ðk0  1Þ 

L X kl yl ¼ 0

65

ð35Þ

l¼1

which leads to the analytical expression of f Y ðyÞ i.e.

"

L X kl yl

f Y ðyÞ ¼ exp k0 

#

ð36Þ

l¼1

The normalization condition is given as

Z

Z f Y ðyÞdy ¼ which yields

"Z

k0 ¼ ln

h P i h P i R R # L exp  Ll¼1 kl yl dy exp  Ll¼1 kl yl dy X l R ¼ exp k0  kl y dy ¼ ¼1 exp½k0  exp½k0 dy l¼1 "

" # # L X exp  kl yl dy

Y

ð37Þ

ð38Þ

l¼1

It is seen that once the integer moment constraints in Eq. (31) are determined, the optimization problem in Eq. (32) can be solved accordingly. In fact, the integer moments defined in Eq. (31) are actually equivalent to those in Eq. (26), which could be straightforwardly evaluated by the RQ-SPM. However, a relatively large number of moments, say L P 6, are required to achieve a reasonable accuracy in the modeling of the distribution tail, which is of paramount importance to reliability assessment [33]. The entropy maximization process may encounter numerical instability when the number of integer moment constraints becomes large. Alternatively, fractional moments, which embody the information of a large number of central moments, are suggested as constraints. Therefore, a small number of fractional moments, say L < 6, may be adequate enough to model the distribution tail with fair accuracy. The MEM with fractional moments provides a powerful tool for estimating an unknown PDF. Actually, such a strategy of defining PDF can be found in Refs. [36,40], in which a modified UDRM for the evaluation of fractional moments of structural response is adopted. The fractional moment constraints in the MEM can be expressed as

Z

ma~l ¼

ya~l f Y ðyÞdy;

l ¼ 1; 2; . . . ; L

ð39Þ

Y

~ l th order fractional moment of Y. To distinguish from the fractional orders used in the RQ-SPM, the fracwhere ma~ l is an a ~ l ; l ¼ 1; 2; . . . ; L. tional orders in the MEM are denoted as a Likewise, the fractional moment constraints can be evaluated by the proposed RQ-SPM such that

Z ma~l ¼

Y

ya~l f Y ðyÞdy ¼

N X ~ ~k Þ; ak Gal ðR1 ½x

l ¼ 1; 2; . . . ; L

ð40Þ

k¼1

Thus, Eq. (40) will be employed instead of Eq. (26) in the MEM for deriving the PDF. Herein, if we assume that the fractional moments can be accurately evaluated by the proposed RQ-SPM, the entropy optimization problem in Eq. (32) changes to

8 > < find f Y ðyÞ R Maximize H½f Y ðyÞ ¼  f Y ðyÞ ln½f Y ðyÞdy > R : Subject to ma~ l ¼ Y ya~l f Y ðyÞdy; l ¼ 1; 2; . . . ; L

whose solution is

"

L X f Y ðyÞ ¼ exp k0  kl ya~ l

ð41Þ

# ð42Þ

l¼1

It is seen that using the fractional moments instead of integer moments as constraints poses no theoretical or numerical difficulties. It is demonstrated [30,36] that the maximization of the entropy (Eq. (41)) is equivalent to the minimization of the difference between the true PDF, denoted as ~f Y ðyÞ, and its estimation f ðyÞ, which can be characterized by the relative entropy Y

(also known as Kullback-Leibler divergence). The relative entropy between ~f Y ðyÞ and f Y ðyÞ is defined as

R K½~f ; f  ¼ ~f Y ðyÞ ln½~f Y ðyÞ=f Y ðyÞdy R R ¼ ~f Y ðyÞ ln½~f Y ðyÞdy  ~f Y ðyÞ ln½f Y ðyÞdy

Substituting Eqs. (30) and (42) into Eq. (43) further gives

ð43Þ

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J. Xu et al. / Mechanical Systems and Signal Processing 95 (2017) 58–76

R K½~f ; f  ¼ H½~f Y ðyÞ 

k0 

! L X kl ya~l ~f Y ðyÞdy l¼1

L X R kl ya~ l ~f Y ðyÞdy ¼ H½~f Y ðyÞ þ k0 þ

ð44Þ

l¼1

¼ H½~f Y ðyÞ þ k0 þ

L X kl ma~ l l¼1

~ ¼ ½a ~1; a ~2; . . . ; a ~ L T and where H½~f Y ðyÞ is the differential entropy of the true PDF, which does not depend on a k ¼ ½k1 ; k2 ; . . . ; kL T . Therefore, we can define 





Wða; kÞ ¼ K½f ; f  þ H½f Y ðyÞ L P ¼ k0 þ kl ma ¼ ln

R

ð45Þ

l

l¼1

L



L P P  exp  kl yal dy þ kl ma l¼1

l¼1

l

In this regard, one can transform the constrained optimization problem (Eq. (41)) to be an un-constrained form [36] such that

8 > <

find









T

a ¼ ½a1 ; a2 ; . . . ; aL  and k ¼ ½k1 ; k2 ; . . . ; kL T

L



L R P P   > : Minimize Wða; kÞ ¼ ln exp  kl yal dy þ kl ma l¼1

l¼1

ð46Þ l

~1 ; a ~2 ; . . . ; a ~L This optimization can be carried out with the simplex search method in Matlab [36]. Then, the fractional orders a and the Lagrange multipliers k1 ; k2 ; . . . ; kL can be determined accordingly for obtaining the PDF of the performance function. 5. Probability of failure evaluation When f Y ðyÞ is completely determined by the fractional orders a0 ; a1 ; . . . ; aL and the Lagrange multipliers k0 ; k1 ; . . . ; kL , the approximated value of the probability of failure can be obtained by integrating f Y ðyÞ over the failure domain, i.e.

Z pf ¼ Pr½GðZÞ 6 0 ¼

0

1

f Y ðyÞdy

ð47Þ

By using the proposed method, the probability of failure can be determined with high efficiency. This is because the RQSPM embedded in the methodology requires a relatively small number of response evaluations for fractional moments, which is incorporated into the MEM based derivation of the PDF f Y ðyÞ. It is noted that the proposed reliability analysis method is established based on the RQ-SPM and the maximum entropy method. It is actually relatively simple and easy to be implemented. Since a small number of performance function evaluations are involved, it is very convenient for implementation into the existing design procedures [41]. 6. Numerical examples The proposed method, in which the RQ-SPM are combined with the MEM, is applied to six numerical examples to investigate its accuracy and efficiency for structural reliability assessment. Each example contains the comparison of the results by the proposed method and MCS. 6.1. Example 1 A plastic collapse mechanism of a one-bay frame [42] (Fig. 1), which involves a linear performance function is firstly considered

Y ¼ GðZÞ ¼ Z 1 þ 2Z 2 þ 2Z 3 þ Z 4  5Z 5  5Z 6

ð48Þ

where Z i , i ¼ 1; 2; . . . ; 6 are statistically independent lognormal random variables and their statistical characteristics (the mean and the coefficient of variation (C.O.V.)) are given in Table 2. In order to avoid computing the fractional moment of a random variable with negative realizations, let Y 0 ¼ Y þ Q ¼ GðZÞ þ Q , where Q is a large positive value. From the probability theory, we know that the distribution of Y 0 is the same with that of Y. Performing a simple translation of pY 0 ðyÞ will get pY ðyÞ. In this example, Q ¼ 200 is adopted.

J. Xu et al. / Mechanical Systems and Signal Processing 95 (2017) 58–76

67

Z6 Z3

Z5

Z2

Z1

Z4

Fig. 1. A plastic collapse mechanism of one-bay frame.

Table 2 Statistical characteristics of the random variables for the Example 1. Designation

Distribution

Mean

Coefficient of variation

Z1 Z2 Z3 Z4 Z5 Z6

Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal

120 120 120 120 50 40

0.10 0.10 0.10 0.10 0.20 0.20

In the present paper, we assume that four fractional moments constraints are enough for deriving the distribution. Then, ~ 2 ¼ 2:0579, a ~ 3 ¼ 3:2028, a ~ 4 ¼ 4:0532. The frac~ 1 ¼ 1:3005, a the fractional orders can be determined by Eq. (46), which are a tional moments of Y 0 , say ma~1  ma~ 4 , are computed by the proposed RQ-SPM and MCS (107 runs), respectively. The results are listed in Table 3, which show that the proposed RQ-SPM estimates have quite small relative errors for all the four fractional moments. It should be noted that the calculation of fractional moments and the following PDF evaluation are only based on 44 deterministic performance function evaluations. Further, the PDF of Y obtained by the proposed method is shown in Fig. 2; this PDF is in close agreement with the results obtained by MCS, demonstrating the accuracy of the proposed method for the PDF evaluation of the performance function. In Fig. 2(b) shown is the cumulative distribution function (CDF) of the PDF in logarithmic scale. Likewise, the result is compared with that given by MCS. Again, the CDF in logarithmic scale by the proposed method still accords quite well with that by MCS, further indicating the accuracy of the proposed method for structural reliability analysis. Actually, the value of CDF at the abscissa of zero gives the probability of failure. The probabilities of failure by the proposed method and MCS are 7.14 104 and 8.33 104 , respectively, which again manifests the high efficiency and accuracy of the proposed method for structural reliability analysis.

6.2. Example 2 A simply-supported I-beam [39], which is shown in Fig. 3, is adopted to demonstrate the effectiveness of the proposed method. The beam is acted on a concentrated force F with a distance b away from the fixed end [28]. The maximum stress due to bending can be expressed as

rmax ¼

FbðL  bÞd 2LI

3

I¼

bf d  ðbf  t w Þðd  2t f Þ 12

3

! ð49Þ

The performance function can be defined as

Y ¼ GðZÞ ¼ S  rmax

ð50Þ

The uncertainty properties of eight random variables involved in this example are shown in Table 4. Likewise, let ~ 1 ¼ 1:3979, a ~ 2 ¼ 2:1315, Y 0 ¼ Y þ Q ¼ GðZÞ þ Q , where Q ¼ 1  105 . The fractional orders specified by Eq. (46) are a a~ 3 ¼ 2:9713, a~ 4 ¼ 4:4551. The four fractional moments of Y 0 when are compared in Table 5. Again, it is seen that the proposed RQ-SPM provides very accurate results, which are very close to those by MCS (107 runs), with only 144 performance

68

J. Xu et al. / Mechanical Systems and Signal Processing 95 (2017) 58–76 Table 3 Fractional moments of Y 0 comparison for the Example 1. Fractional order

RQ-SPM

Relative error

MCS

1.3005 2.0579 3.2028 4.0532

3000.01 323979.9 3.94  108 7.83  1010

0.02% 0.02% 0.09% 0.30%

3000.567 324052.7 3.93  108 7.81  1010

-3

6

x 10

5

0

10

MCS The proposed method

-2

CDF

PDF

4 3

10

2 -4

10

1 0 -400

-200

0

y

200

400

600

The proposed method MCS

-200

(a) PDF

0

200

y

400

600

800

(b) CDF in logarithmic scale

F d

b

tw

tf

Fig. 2. Comparisons of PDF and CDF in logarithmic scale by the proposed method and MCS for the Example 1.

bf

L Fig. 3. Simply-supported I beam.

Table 4 Statistical characteristics of the random variables for the Example 2. Designation

Distribution

Mean

Standard deviation

F L b S d bf tw tf

Normal Normal Normal Normal Normal Normal Normal Normal

7010 120 72 210000 2.3 2.3 0.16 0.26

200 6 6 4760 1/24 1/24 1/48 1/48

Table 5 Fractional moments of Y 0 comparison for the Example 2. Fractional order

RQ-SPM

Relative error

MCS

1.3979 2.1315 2.9713 4.4551

15,184,069 9.07  1010 1.93  1015 8.90  1022

0.03% 0.04% 0.07% 0.10%

15,180,001 9.06  1010 1.93  1015 8.89  1022

function evaluations. The comparisons of the PDF, the CDF in logarithmic scale are shown in Fig. 4. A similar conclusion in terms of the accuracy of the proposed method for structural reliability analysis could be drawn. The probabilities of failure by the proposed method and MCS are 0.03301 and 0.03315, respectively.

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J. Xu et al. / Mechanical Systems and Signal Processing 95 (2017) 58–76 -5

2.5

x 10

10

MCS The proposed method

2

10

CDF

PDF

1.5 1

10

0.5 0

10

10

-4

-2

0

2

4

6

8

10

12 4 x 10

y

0

-1

-2

-3

-4

The proposed method MCS -1

-0.5

0

0.5

1

1.5

y

(b) CDF in logarithmic scale

(a) PDF

2 5

x 10

Fig. 4. Comparisons of PDF and CDF in logarithmic scale by the proposed method and MCS for the Example 2.

6.3. Example 3 A nine-bar truss structure is shown in Fig. 5. Random variables associated with this problem are listed in Table 6. Note that a closed form expression for the deflection at each node may be not available and it has to be computed through finite element analysis. Linear structural analysis is employed to calculate the responses. The implicit limit state function is defined such that [36]

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U 2Xi þ U 2Y i Y ¼ GðZÞ ¼ U b  max

ð51Þ

16i66

where U X i and U Y i are the horizontal and the vertical displacement at the i th node and U b ¼ 0:20 m is the deterministic allowable value of displacement. nqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio If we define Y 0 ¼ max16i66 U 2X i þ U 2Y i , the probability of failure in Eq. (47) is equivalent to

Z pf ¼ Pr½GðZÞ 6 0 ¼

Z

0

1

f Y ðyÞdy ¼

þ1

Ub

f Y 0 ðyÞdy

ð52Þ

The four fractional orders are determined as a1 ¼ 0:9037, a2 ¼ 0:8557, a3 ¼ 0:7800, a4 ¼ 1:4376 and the corresponding fractional moments of Y 0 evaluated by the proposed RQ-SPM are listed in Table 7. Likewise, all the results are compared with those given by MCS. It is seen that the proposed RQ-SPM is of accuracy and efficiency for the fractional moments evaluation. It should be noted that only 42 deterministic performance function evaluations are carried out to achieve quite accurate fractional moments by RQ-SPM. The PDF and the CDF in logarithmic scale are shown in Fig. 6. It is seen that the PDF and the CDF in logarithmic scale evaluated by the proposed method accord quite well with those by MCS (107 runs), which further demonstrates the accuracy of the proposed method for reliability analysis. Furthermore, in this example, if the first four integer moments are applied as the constraints in the MEM to derive f Y 0 ðyÞ, the PDF and the CDF in logarithmic scale are compared with those by MCS in Fig. 7. Obviously, the PDF accords not quite well with the histogram and the left tail of the CDF in logarithmic scale severely deviates from the result by MCS. This indicates the integer moments based MEM is unable to capture the reliability with accuracy. In contrast, the fractional moments always results in a highly accurate PDF and the tail distribution for reliability analysis.

3

3m

5

1

4

2 4m

F1

4m

F2

Fig. 5. Planar nine bar truss structure.

6 4m

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J. Xu et al. / Mechanical Systems and Signal Processing 95 (2017) 58–76

Table 6 Statistical characteristics of the random variables for the Example 3. Variable

Description

Distribution

Units

Mean

Coefficient of Variation

E A

Young’s Modulus Cross-sectional Area Mass density External load External load

Normal Lognormal Lognormal Lognormal Lognormal

Pa m2 kg/m3 N N

2.0  1011 2.5  103 7860 544,822 444,766

0.2 0.1 0.15 0.2 0.2

q F1 F2

Table 7 Fractional moments of Y 0 comparison for the Example 3. Fractional order

RQ-SPM

Relative error

MCS

0.9037 0.8557 0.7800 1.4376

0.0901 10.3463 0.1248 0.0225

0.10% 0.11% 0.09% 0.09%

0.0902 10.3345 0.1249 0.0225

0

25

10 MCS The proposed method

-1

10

-2

15

CDF

PDF

20

10

-3

10

10

5

10

-4

0 0

0.05

0.1

0.15

0.2

The proposed method MCS

0.25

0

0.05

0.1

0.15

0.2

0.25

y'(m)

y'(m)

(b) CDF in logarithmic scale

(a) PDF

Fig. 6. Comparisons of PDF and CDF in logarithmic scale by the proposed method and MCS for the Example 3.

0

25

10 MCS The proposed method

-1

10

-2

15

CDF

PDF

20

10

-3

10

10

5

10

0

-4

0

0.05

0.1

0.15

y'(m)

(a) PDF

0.2

0.25

The proposed method MCS 0

0.05

0.1

0.15

0.2

0.25

y'(m)

(b) CDF in logarithmic scale

Fig. 7. Comparisons of PDF and CDF in logarithmic scale by the MEM with integer moments as constraints and MCS.

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J. Xu et al. / Mechanical Systems and Signal Processing 95 (2017) 58–76

6.4. Example 4 A nonlinear performance function is considered [43]

Y ¼ GðZÞ ¼ 1000Z 1 Z 2  7:51Z 1 Z 3 þ Z 1 Z 4 þ 40Z 5  0:5Z 23

ð53Þ

where the independent random variables are shown in Table 8. Like the situation in Example 2, we define Y 0 ¼ Y þ Q ¼ GðZÞ þ Q . In this example, Q ¼ 4000 is specifically adopted and the resulting fractional orders are a1 ¼ 0:1345, a2 ¼ 1:2128, a3 ¼ 2:1929, a4 ¼ 4:5560. Again, only 42 deterministic evaluations need to be implemented. The fractional moments of Y 0 , the PDF and the CDF in logarithmic scale of Y together with the results by MCS (107 runs) are shown in Table 9 and Fig. 8. It is seen that the proposed method can keep the balance between accuracy and efficiency. The probabilities of failure evaluated by the proposed method and MCS are 0.01842 and 0.01843, respectively. Again, all the results validate the effectiveness of the proposed method for structural reliability assessment. 6.5. Example 5 A nonlinear un-damped single-degree-of-freedom system is considered [44], which is shown in Fig. 9. The statistics of the basic random variables are given in Table 10. The limit state function is given as

  2   2F 1 x0 t1  Y ¼ GðZÞ ¼ 3r  jZ max j ¼ 3r   sin 2  mx20

ð54Þ

where r is the displacement at which one of the spring yields, Z max is the maximum displacement of the system and

x20 ¼ ðc1 þ c2 Þ=m. Similarly, we define Y 0 ¼ Y þ Q , where Q ¼ 1:8 is considered. The fractional orders are specified as a1 ¼ 0:7335, a2 ¼ 1:0345, a3 ¼ 2:4762, a4 ¼ 1:6978. Table 11 and Fig. 10 compare the results by the proposed method and MCS (107 runs). Only 44 deterministic evaluations need to be carried out. The probability of failure calculated by the proposed method is 0.03904, which is fairly close to that of MCS (0.03850). Again, it can be seen that the proposed method is of high accuracy. 6.6. Example 6 A single-degree-of-freedom (SDOF) structural dynamic system with hysteretic property (Fig. 11) is considered

_ ¼ FðtÞ m€x þ cx_ þ f ðx; xÞ

ð55Þ

_ is the restoring where m, c and FðtÞ are the mass, the damping coefficient and the applied external load, respectively, f ðx; xÞ force which is described by the Bouc-Wen model [45]

_ ¼ aB kx þ ð1  aB Þkz f ðx; xÞ

ð56Þ

where k is the initial stiffness, aB is the post- to pre-yield stiffness ratio, z is the hysteretic component, which is governed by [45]

z_ ¼ hðzÞ

_ n1 z þ cxjzj _ nÞ Ax_  mðbjxjjzj

ð57Þ

g

where _ u =f2 hðzÞ ¼ 1:0  f1 e½zsgnðxÞqz 2

ð58Þ

in which sgnðÞ is the signum function and

f1 ðeÞ ¼ fS ð1  epe Þ;

f2 ðeÞ ¼ ðw þ dw eÞðk þ f1 ðeÞÞ

ð59Þ

A typical sample of the displacement x v.s. the hysteretic displacement z is shown in Fig. 12, where strong nonlinearity is clearly observed. The external excitation is adopted as

Table 8 Statistical characteristics of the random variables for the Example 4. Designation

Distribution

Mean

Standard deviation

Z1 Z2 Z3 Z4 Z5

Normal Lognormal Lognormal Normal Weibull

1.20 2.40 50 25 10

0.36 0.072 3 7.5 3

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J. Xu et al. / Mechanical Systems and Signal Processing 95 (2017) 58–76

Table 9 Fractional moments of Y 0 comparison for the Example 4. Fractional order

RQ-SPM

Relative error

MCS

0.1345 1.2128 2.1929 4.5560

3.1881 35267.9435 170287928.8461 1.11  1017

0.002% 0.01% 0.02% 0.65%

3.1880 35263.7241 170257296.8288 1.11  1017

-4

6

x 10

5

0

10

MCS The proposed method

-2

CDF

PDF

4 3

10

2 -4

10

1 0 -4000

-2000

0

y

2000

4000

6000

The proposed method MCS

-2000

(a) PDF

0

2000

y

4000

6000

(b) CDF in logarithmic scale

Fig. 8. Comparisons of PDF and CDF in logarithmic scale by the proposed method and MCS for the Example 4.

Zt

c1

F t F t

F1

c2 t1 Fig. 9. A nonlinear oscillator.

Table 10 Statistical characteristics of the random variables for the Example 5. Designation

Distribution

Mean

Coefficient of Variation

m c1 c2 r F1 t1

Normal Normal Normal Normal Normal Normal

1.0 1.0 0.1 0.5 1 1

0.05 0.1 0.1 0.1 0.2 0.2

Table 11 Fractional moments of Y 0 comparison for the Example 5. Fractional order

RQ-SPM

Relative error

MCS

0.7335 1.0345 2.4762 1.6978

1.8708 2.4254 8.5782 4.3201

0.01% 0.01% 0.02% 0.02%

1.8707 2.4252 8.5762 4.3194

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J. Xu et al. / Mechanical Systems and Signal Processing 95 (2017) 58–76 0

1.4

10

MCS The proposed method

1.2 1

-2

CDF

PDF

10

0.8 0.6

-4

10

0.4 0.2 0

The proposed method MCS

-6

-1

-0.5

0

0.5

1

1.5

2

10

-2

-1

0

y

1

2

y

(b) CDF in logarithmic scale

(a) PDF

Fig. 10. Comparisons of PDF and CDF in logarithmic scale by the proposed method and MCS for the Example 5.

x t

c k F t

Fig. 11. A hysteretic oscillator.

0.02

Hystetrtic Displacement (m)

0.015 0.01 0.005 0 -0.005 -0.01 -0.015 -0.02 -0.03

-0.02

-0.01

0

0.01

0.02

0.03

Displacement (m) Fig. 12. Hysteretic behavior.

h i FðtÞ ¼ m n1 €xNS ðtÞ þ n2 €xEW ðtÞ g g

ð60Þ

where € xNS ðtÞ and € xgEW ðtÞ are the El Centro accelerogram in the N-S and W-E direction, respectively, n1 and n2 denote the g amplitudes. The statistic properties of the input random variables are given in Table 12 and the other parameters in the Bouc-Wen model are listed in Table 13. The implicit limit state function is defined as

Y ¼ GðZÞ ¼ xb  jxmax j

ð61Þ

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J. Xu et al. / Mechanical Systems and Signal Processing 95 (2017) 58–76

Table 12 Statistical characteristics of the random variables for the Example 6. Designation

Distribution

Mean

Coefficient of variation

m c k n1 n2 b

Normal Normal Normal Normal Normal Normal Normal

3.798  105 kg 7.9897  105 N m/s 0.858  108 N/m 0.8 0.5 30 10

0.1 0.1 0.1 0.2 0.2 0.05 0.05

c

Table 13 Parameters in Bouc-Wen model. Parameter

aB

A

n

q

p

dw

k

w

dm

dg

fS

Value

0.01

1

1

0.25

1000

5

0.5

0.05

2000

2000

0.99

Table 14 Fractional moments of Y 0 comparison for the Example 6 (1). Fractional order

RQ-SPM

Relative error

MCS

1.5745 0.1187 1.7022 2.1099

564.8753 0.6247 0.0013 2.69  104

1.58% 0.15% 2.00% 2.25%

573.9287 0.6238 0.0013 2.63  104

0

100 80

-2

60

CDF

PDF

10

MCS The proposed method

10

40 -4

20 0 0

10

0.01

0.02

0.03

y'(m)

0.04

0.05

The proposed method MCS 0

0.01

0.02

0.03

y'(m)

0.04

0.05

(b) CDF in logarithmic scale

(a) PDF

Fig. 13. Comparisons of PDF and CDF in logarithmic scale by the proposed method and MCS for the Example 6.

where xmax is the maximum displacement of the oscillator, xb ¼ 0:045 m is the deterministic threshold. Like the situation in Example 3, if we adopt Y 0 ¼ jxmax j, the probability of failure reads

Z pf ¼ Pr½GðZÞ 6 0 ¼

Z

0 1

f Y ðyÞdy ¼

þ1 xb

f Y 0 ðyÞdy

ð62Þ

The fractional moments and the distribution of Y 0 are of great interest. In this example, only 78 times of deterministic dynamic response analyses need to be carried out. The results of fractional moments of Y 0 are shown in Table 14. The computational results imply that the proposed RQ-SPM is of indeed efficiency and accuracy for fractional moments evaluation in the proposed method. In Fig. 13, it is seen that the proposed method still has very good accuracy even for the system exhibiting strong nonlinearity compared with MCS (106 runs). The proposed method and MCS produce the probabilities of failure of 2.3840 104 and 4.3333 104 , respectively. If only the Q-SPM without rotation is employed to assess the fractional moments of Y 0 , the results are shown in Table 15. It is noted that the relative errors of fractional moments are quite large by using the Q-SPM, which can further creep into the MEM and result in the inaccurate PDF and tail distribution, which are depicted in Fig. 14.

75

J. Xu et al. / Mechanical Systems and Signal Processing 95 (2017) 58–76 Table 15 Fractional moments of Y 0 comparison for the Example 6 (2). Fractional order

Q-SPM

Relative error

MCS

0.4950 2.9368 2.7666 1.4687

0.1382 9.83  106 1.89  105 0.0029

1.78% 13.70% 12.37% 5.46%

0.1407 1.14  105 2.16  105 0.0031

0

100 80

-2

60

CDF

PDF

10

MCS MEM+Q-SPM

10

40 -4

20 0 0

10

0.01

0.02

y'(m)

(a) PDF

0.03

0.04

MCS MEM+Q-SPM 0

0.01

0.02

0.03

0.04

0.05

y'(m)

(b) CDF in logarithmic scale

Fig. 14. Comparisons of PDF and CDF in logarithmic scale by the MEM with Q-SPM and MCS.

7. Concluding remarks This paper proposes an efficient method for structural reliability analysis based on a new rotational quasi-symmetric point method (RQ-SPM) and the maximum entropy method (MEM). In this method, the reliability is assessed based on the determination of the performance function’s PDF. Firstly, the fractional moments are evaluated by using the proposed RQ-SPM with efficiency and accuracy. Then the MEM is carried out to derive the performance function’s PDF, where the fractional moments are specified as the moment constraints. Several examples are presented to elucidate the numerical efficiency and accuracy of the proposed method in comparison with the Monte Carlo method. In these examples, linear and nonlinear, explicit and implicit performance functions are all involved. It is found that the proposed method is very efficient, where accurate results are obtained based on a few of deterministic model evaluations. Furthermore, it could be concluded that the proposed method can be successfully applied to solve the reliability–related problems, e.g. reliability based optimal control, of large-scale engineering structures due to its high efficiency and accuracy and its adaptability to almost arbitrary performance function. It is worth pointing out that the number of fractional moment constraints required in the proposed method is assumed according to computational experiences. A rational choice of the number of fractional moment constraints will be of great necessity. On the other hand, the proposed method is only applicable to structural reliability analysis where the number of random variables should be no larger than 20. Further investigations will concern efficient structural reliability analysis involving a high dimensional random vector, e.g. d > 100. Acknowledgement The support of the National Natural Science Foundation of China (Grant No. 51608186) and the Fundamental Research Funds for the Central Universities (No. 531107040890) is highly appreciated. The anonymous reviewers are greatly acknowledged for their constructive criticisms to the original version of the paper. References [1] K. Piric, Reliability analysis method based on determination of the performance function’s PDF using the univariate dimension reduction method, Struct. Saf. 57 (2015) 18–25. [2] J. Li, J.B. Chen, Stochastic Dynamics of Structures, John Wiley & Sons, 2009. [3] H.O. Madsen, S. Krenk, N.C. Lind, Methods of Structural Safety, Courier Corporation, 2006. [4] J. Huh, Reliability analysis of nonlinear structural systems using response surface method, KSCE J. Civil Eng. 4 (3) (2000) 135–143. [5] S.-K. Au, J.L. Beck, Estimation of small failure probabilities in high dimensions by subset simulation, Probab. Eng. Mech. 16 (4) (2001) 263–277.

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