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

Boyu LI,Leigng ZHANG,Xuejun ZHU,Xiongqing YU,Xiodong MA

aCollege of Aerospace Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

bChina Academy of Launch Vehicle Technology,Beijing 100076,China

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

Baoyu LIa,b,Leigang ZHANGb,*,Xuejun ZHUb,Xiongqing YUa,Xiaodong MAb

aCollege of Aerospace Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

bChina Academy of Launch Vehicle Technology,Beijing 100076,China

Available online 8 May 2017

*Corresponding author.

E-mail address:leigang_zhang@163.com(L.ZHANG).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2017.04.005

1000-9361©2017 Production and hosting 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/).

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 eff icient 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.

©2017 Production and hosting 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/).

Fractional moment;

Maximum entropy;

Probability density function;

Reliability analysis;

Unscented transformation

1.Introduction

Reliability analysis is used to assess the safety of an engineering system or structure,and reliability is defined as the probability that a system,subsystem,or device will perform adequately for a specif i ed period of time under specific operating conditions.1Commonly,the performance function of a system or structure is characterized by a functionG（X）,which is a function of physical random variablesX= ［X1X2...Xn］(nis the dimensionality of the input vector).The definition equation of the failure probability,Pf,is given as

wherefX（x）is the joint Probability Density Function(PDF)of the input variablesX,andF= ｛X:G（X）≤ 0｝is the failure 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 beenfirstly 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)2–4are most classical and widely used.Both the FORM and SORM aim at searching for a design point,which locates on the limit state surfaceG（X）=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, （102-104）/Pfpoints 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)method5is 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,6,7the artificial neural networks,8,9the support vector machine,10,11and the Kriging model.12–14Generally,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 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 Jaynes15can 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.16that 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.16–20Zhang and Pandey16proved 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 Pandey16is really efficient.A Multiplicative Dimensional Reduction Method(M-DRM)based on the concept of highdimensional model representation was proposed by Zhang and Pandey16to 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 Pandey16through deep research,which come from the M-DRM 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,21–25thusit 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 Uhlmann26to 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,26statistical robust design,27wind production system,28and 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 brief l y 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.

2.Brief review of fractional moments based maximum entropy analysis

In Ref.16,Zhang and Pandey 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.

2.1.Maximum entropy analysis using fractional moments

The P-ME with constraints of fractional moment is shown as

wherefY（y）is the PDF of the response functionY,andis the αkth-order fractional moment ofY.H［fY（y）］is the information-theoretic entropy ofY,and it is de fi ned as

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 offY（y）as

where λ= ［λ0λ1···λm］Tis the Lagrange multiplier vector,and

Then the main task is to compute λ and α.Zhang and Pandey16introduced the Kullback-Leibler entropy between the true PDF,fY（y）,and the estimator,（y）,as

Consequently,the P-ME problem for estimatingis transformed into the following optimization problem16:

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 Pandey16introduced an M-DRM to compute these（k=1，2，...，m）.In next subsection,the M-DRM is presented,and some discussions are described after deep research on this method.

2.2.Introduction and discussions of M-DRM

The M-DRM is proposed based on the concept of High-DimensionalModelRepresentation(HDMR),30,31which decomposes a multivariate function into orthogonal component functions involving low-dimensional vectors only.

Firstly,we assumeY=g（X）is the response function,and the HDMR expansion ofg（X）can be expressed as

In Ref.16,the basic idea of the Cut-HDMR method14is applied,and the component functions are evaluated at the mean values of the input variables,i.e.,μ = ［μ1μ2...μn］T:

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

While Zhang and Pandey16made an improvement based on the Cut-HDMR,they applied the Cut-HDMR to the logarithmic transformation of the response function, i.e.,lg［abs（g（X））］.Substitutingg（·）with lg［abs（g（·））］in Eqs.(9)and(10),and then inverting the transformation,the original response function can be expressed as16

Then the fractional moment,which is defined as a multidimensional integration,can be decomposed as16

wherefXi（xi）is the marginal PDF of variableXi.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 Cut-HDMR,and the mean values of input variables are chosen 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 pointg（μ）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 ifg（μ）≤ 0,because of the term ［g（μ）］1-nin Eq.(11).For example,consider a very simple bivariate functiong（X）=X1-X2,whereX1andX2are mutually independent input variables,andX1～N（1，0.12）,X2～U（0，2）.The M-DRM cannot be applied on this function to compute the mean,standard deviation,skewness,and kurtosis of the response function owing tog（μ）=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,andg（μ）＞ 0 in this context.Therefore,the M-DRM is a useful method for computing moments withg（μ）＞ 0 and Zhang and Pandey16have 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 αth-order fractional moment can be approximated by a product of αth-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 ofthe 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 Pandey16is 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.21–25Consequently,introducing a very eff icient method for structural reliability analysis with correlations is very necessary.

Considering that the method in Ref.16is 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 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.

3.Computation of fractional moments using UT

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”.26UT 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).

3.1.Basic technique of UT method

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:

(1)Select the sigma pointssand weightsW.There are several methods for selecting the sigma points,26,32–34and 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 UT32selects a set of p=2n+1 sigma points,and the pointssand corresponding weightsWare shown as

where μXandPXXare the mean vector and covariance of the input variablesX,respectively.enotes a matrix square root,which can be implement by the Cholesky decomposition,and （·）imeans theith column or row.Generally,zero is selected as the value ofw0.26Note that the weighsWican be positive or negative,but in order to provide an unbiased estimate,they must obey the condition that∑

(2)Compute the corresponding response values of the sigma points.The one-dimensional output response function Y=g（X）is considered in this paper,so Yi=g（si）.We can see that the number of function evaluations is 2n+1.

(3)Calculate the mean and variance of Y.The low-order integral moments are calculated by

where μYandVYare the mean and variance ofY,respectively.While,in this paper,we use UT to calculate the fractional moments ofY,according to Eq.(16),the αth-order fractional moment ofYcan be derived as

It has been demonstrated that UT can effectively estimate the low-order moments of an output function,26–29so 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 105samples,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 covariancePXXin 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

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.16We 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 functiong（X）of a structure,of which the critical threshold is assumed asYCT,the performance function is constructed considering the fact that the fractional moment only works with a positive variable as

so that the values ofG（X）are positive and the P-ME can be applied on it to estimate the probabilistic distribution.Note that the corresponding failure domain isFratio= ｛X:G（X）≤ 1｝,so when the estimated PDF^fT（t）of the performance function is available,the failure probability can be easily computed by

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+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+1 evaluations of the performance function,wherenis the dimensionality of the input vector.Because in each iteration loop of the optimization problem in Eq.(7),the values ofTi=G（si）（i=1，2，...，2n+1）can be reused,no extra function evaluations are needed.LetNUTbe the number of function evaluations of the proposed method in this paper,therefore it is given as

The number of function evaluations of the method proposed by Ref.16is given as

wherekis the number of input variables with the symmetric distribution(e.g.,normal distribution,uniform distribution).Niis the adopted number of Gaussian nodes for variableXi,andNi=5 in this paper.

It can be seen that the proposed UT based P-ME method and the M-DRM based one in Ref.16are 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 thatNUT=11,andNM-DRM=21.In a word,the proposed method in this paper can greatly decrease the computation burden with assurance of reasonable precision.

4.Examples

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.35introduced 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,35and 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.

4.1.Test example:A numerical example

Consider a response function

whereX1andX3are two uniformly distributed variables,andX2andX4are normally distributed variables.Their distribution parameters are given as:X1～U（3.7，4.9）,X2～N（1，0.12）,X3～U（2.65，4.5）,andX4～N（2，0.22）.X2andX4are correlated and the Pearson correlation coefficients ρ=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.

106points 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 arevery 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 coefficients 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 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.

Table 1 Comparison of estimated fractional and integral moments of test example.

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 106samples,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.

4.2.Engineering example 1:A cantilever beam

In this subsection,we introduce a cantilever beam structure,25which is shown in Fig.2.Dimensional parameters of the beam,such as width,height,and length,are denoted asw,t,andL,respectively.Eis the elastic modulus.Two forcesFXandFYare 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

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

The distribution information of the six input variables,i.e.,FX,FY,E,w,t,andLis listed in Table 2.Besides,the Pearson correlation coefficients of the input variables are set to ρwt=-0.55 and ρwL= ρtL=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 asT=G（X）=0.066/D（FX，FY，E，w，t，L）according to Eq.(18).

Here,we directly apply the proposed UT_P-ME method on the performance function.Six input variables determine that 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 106samples.Again,the precision of the proposed method in this paper is illustrated.

Based on the available estimated PDF^fT（t）of the performance function,reliability analysis can be implemented easily.Referring to Eq.(19),the failure probability can be easily computed by integrating^fT（t）on[0,1],and we getPf=0.0111.For comparison,the reference result obtained by MCS is computed by Eq.(24)as

whereN=106is the number of random samples obtained by the simple sampling method using the joint PDF of input variables.Xi（i=1，2，...，N）are the random samples andIFratio（Xi）is the indicator function of the failure regionFratio,i.e.,By 106evaluations of the performance function,the failure probability estimated by Eq.(24)is 0.0131 with a variation coefficients of 0.0058.Applying the Advanced First Orderand Second Moment(AFOSM)method36with 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 method36is 0.0145 with a variation coefficients of 0.0153,taking 5117 evaluations of the performance function.Compared with results computed by these existing methods,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 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.

4.3.Engineering example 2:A roof truss

Table 3 Failure probabilities of a cantilever beam for different correlation coefficients.

A roof truss25is 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 loadqis applied on the roof truss,and this load can be transformed into a nodal loadP=ql/4.According to the mechanical analysis,the perpendicular defection of NodeCcan be obtained as ΔC=which is a function of random variables.Ac,As,Ec,Es,andlare the sectional area,elastic modulus,length of the concrete and steel bars,respectively.ΔCnot 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:

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 ρqAc= ρqAs=0.5 and ρAcAs=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 106random 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.

Table 4 Distribution information of input variables of a roof truss.

Table 5 Failure probabilities of a roof truss for different correlation coefficients.

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 NodeCexceeding 3 cm is 2.371×10-3.Similarly,the failure probability obtained by MCS using Eq.(24)is 2.211×10-3,with 106evaluations of the performance function,and the variation coefficient of the result is 0.0086.The result obtained by the AFOSM method36is 1.872×10-3with 104 evaluations of the performance function.Moreover,taking 5104 function evaluations,the IS method36gives the result 2.168×10-3with a variation coefficients 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.

5.Conclusions

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+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.

Acknowledgements

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