孟廣偉，馮昕宇，周立明，李鋒

基于降維算法的結(jié)構(gòu)混合可靠性分析方法

孟廣偉，馮昕宇，周立明，李鋒

(吉林大學(xué) 機械與航空航天工程學(xué)院，吉林 長春，130025)

針對工程實際中同時帶有模糊變量與隨機變量的結(jié)構(gòu)混合不確定性問題，提出1種基于降維算法的混合不確定性變量的可靠性分析模型。首先，利用模糊數(shù)學(xué)中的?截集概念，將模糊變量轉(zhuǎn)變?yōu)樗浇丶孪鄳?yīng)的區(qū)間變量，再借助于降維算法，將含有個隨機變量的結(jié)構(gòu)功能函數(shù)展開為個一維隨機變量函數(shù)；將所得到的降維表達式進行泰勒展開，得到結(jié)構(gòu)功能函數(shù)的上、下界表達式；運用變量轉(zhuǎn)換方法將其中的隨機變量轉(zhuǎn)換為均值為0，方差為0.5的正態(tài)分布變量，并結(jié)合二項式展開定理、Gauss?Hermite積分方法與變量轉(zhuǎn)換方法計算出結(jié)構(gòu)功能函數(shù)上下界的統(tǒng)計矩；將所得的矩信息應(yīng)用到Edgeworth級數(shù)展開式中，計算得到對應(yīng)于?截集的失效概率區(qū)間，從而獲得失效概率的隸屬度函數(shù)。研究結(jié)果表明：本文提出的方法不僅計算精度高，而且計算量小。

結(jié)構(gòu)可靠性分析；混合可靠性；降維算法；模糊變量；Edgeworth級數(shù)

結(jié)構(gòu)可靠度分析是工程實踐中的重點研究內(nèi)容之一[1?6]。帶有隨機變量的失效概率計算問題是概率可靠性設(shè)計中的重點研究內(nèi)容之一，帶有模糊變量的失效概率計算問題同樣也是模糊可靠性設(shè)計中的重點問題。然而在工程實踐中，往往僅帶有單一類型變量的結(jié)構(gòu)比較少，混合類型變量共同存在的情況較多。目前，已有很多方法用于求解帶有隨機變量和模糊變量的結(jié)構(gòu)失效概率f，例如，CHAKRABORTY等[7?8]基于信息熵方法，將模糊隸屬函數(shù)轉(zhuǎn)換為等效概率密度函數(shù)進行可靠性分析；SMITH等[9]基于尺度變換方法，將傳統(tǒng)概率可靠性分析理論應(yīng)用于混合可靠性分析；ADDURI等[10?12]基于概率理論利用一次二階矩法和一次二階矩法的改進方法求解混合失效概率等。在利用上述方法計算時，信息熵法在理論上與模糊可靠度的結(jié)果存在一定偏差，且喪失了模糊性。另外，尺度變換法對模糊數(shù)類型的要求較為嚴格；一次二階矩法在非線性情況下，存在迭代收斂慢甚至不收斂的情況。因此，本文作者結(jié)合降維算法和Edgeworth級數(shù)展開法[13]，針對同時含有隨機變量和模糊變量的混合結(jié)構(gòu)，提出1種簡便的計算失效概率方法：通過?截集概念將模糊變量轉(zhuǎn)化為水平截集下相應(yīng)的區(qū)間變量；利用降維算法，將功能函數(shù)分解為個一維隨機變量函數(shù)疊加的形式；借助于泰勒展開，得到功能函數(shù)的上、下界；再結(jié)合變量轉(zhuǎn)換法、Gauss-Hermite數(shù)值積分計算獲得功能函數(shù)上、下界的矩信息，并將其應(yīng)用到Edgeworth級數(shù)展開式中，最終計算得到結(jié)構(gòu)功能函數(shù)的失效概率隸屬度。

1 模糊變量λ?截集

對于同時帶有隨機變量和模糊變量且兩者相互獨立的結(jié)構(gòu)，其結(jié)構(gòu)功能函數(shù)可表示為

圖1 模糊變量與λ?截集關(guān)系

2 降維算法

h與0的上、下界分別為

式中：(x)為隨機變量x的概率密度函數(shù)。

3 變量轉(zhuǎn)換

針對式(19)，可運用傳統(tǒng)的數(shù)值積分公式進行計算。計算積分的方法有很多種，其中值得提出的有快速傅里葉變換法和蒙特卡羅方法(MCS)。前者的前提條件較為嚴格，要求存在解析解且要求功能函數(shù)相互獨立；后者是較為普遍的方法，雖然簡單易行但計算量相當(dāng)大，耗時較長，其工程應(yīng)用范圍受到了一定限制。因此，本文作者引入變量轉(zhuǎn)換的方法，將相互獨立的各隨機變量均轉(zhuǎn)換為正態(tài)分布變量(即均值為0，方差為0.5的正態(tài)分布變量)。為確保準確性，假設(shè)Z(=1，2，…，)是相互獨立的，則變量轉(zhuǎn)換的表達式可寫為

4 Edgeworth級數(shù)

計算結(jié)構(gòu)失效概率的前提是具備功能函數(shù)的概率密度函數(shù)或者其聯(lián)合概率密度函數(shù)表達式，但是在工程實踐中，很難通過大量的數(shù)據(jù)得出精確的概率密度函數(shù)表達式。因而，可選用擬合功能函數(shù)的概率密度函數(shù)或累積分布函數(shù)的方式近似求解復(fù)雜結(jié)構(gòu)的失效概率。本文選用Edgeworth級數(shù)方法進行擬合，將式(17)與式(18)計算得到的功能函數(shù)原點矩信息代入式(28)與式(29)，最終將所得中心矩信息作為Edgeworth級數(shù)展開式系數(shù)，擬合結(jié)構(gòu)功能函數(shù)相應(yīng)的累積分布函數(shù)表達式。

5 數(shù)值算例

十桿桁架結(jié)構(gòu)如圖2所示。桿件長為=3.6 m；于節(jié)點2與節(jié)點4處施加豎直向下載荷(為外載荷，其服從均值為710 kN，變異系數(shù)為0.03的正態(tài)分布)；水平方向的桿①~④面積為1m2，豎直方向的桿⑤~桿⑥面積為2m2，斜向的桿⑦~桿⑩面積為3m2(1，2，3和均為模糊變量)；為材料的彈性模量，其服從均值為2.1×1011Pa，變異系數(shù)為0.01的正態(tài)分布。當(dāng)在節(jié)點2處的垂直位移max不大于允許位移allow(allow=0.004 2 m)時，結(jié)構(gòu)功能函數(shù)可表示為：

MCS方法在工程實踐中被廣泛認可為精確解。表1所示為本文方法與MCS方法的十桿桁架失效概率對比。圖3所示為本文方法與MCS方法的失效概率?隸屬度的關(guān)系。由表1和圖3可知：本文方法與MCS方法計算結(jié)果非常接近。但在每個?截集處，本文方法的樣本點數(shù)僅為6個，遠低于MCS方法的106個，體現(xiàn)了本文方法的簡單性和有效性。

圖2 平面十桿桁架結(jié)構(gòu)

表1 2種方法十桿桁架失效概率對比

1—MCS；2—本文方法(即Edgeworth)。

6 結(jié)論

1) 提出1種針對同時帶有模糊變量和隨機變量不確定性問題的結(jié)構(gòu)失效概率計算方法。

2) 借助于?截集概念，將模糊變量轉(zhuǎn)變?yōu)樗浇丶孪鄳?yīng)的區(qū)間變量，將隨機模糊問題有效地轉(zhuǎn)變?yōu)殡S機區(qū)間問題。

3) 利用降維算法建立結(jié)構(gòu)功能函數(shù)的一維隨機變量降維表達式，無需借助于多重積分求解結(jié)構(gòu)功能函數(shù)的統(tǒng)計矩，無需求解結(jié)構(gòu)功能函數(shù)矩陣的逆。

4) 高效且穩(wěn)定地降低計算成本，有效避免了由于迭代而存在的收斂慢甚至不收斂情況的問題。

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(編輯 伍錦花)

Hybrid structural reliability analysis method based on dimension reduction algorithm

MENG Guangwei, FENG Xinyu, ZHOU Liming, LI Feng

(School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130025, China)

A new hybrid reliability analysis method based on dimension reduction algorithm was proposed, aiming at solving the hybrid uncertain problems of structures with both fuzzy variables and random variables in engineering practice. Firstly, fuzzy variables were transformed into corresponding interval variables within the level set using the concept of?cut set in fuzzy mathematics. Taking advantage of dimension reduction method, structural performance functionwithrandom variables was expanded tofunctions, each with one random variable. This dimension reduction formula is then expanded by Taylor method to get the expression of the upper and lower limits of the performance function. By using the variable transformation method, the random variables could be changed to mutual independent normal variables, with mean value zero and the variance 0.5. Combining the binomial theorem, Gauss-Hermite integration method and variable transformation method, the statistical moments of the upper and lower limits of the structural performance function were computed. After substituting the moment information into the Edgeworth series expansion formula, the intervals of structural failure probability corresponding to the λ?cut set were obtained. Therefore, the membership function of the failure probability was calculated. The results show that the proposed method has high precision with low computational cost.

structural reliability; hybrid reliability; dimension reduction method; fuzzy variable; Edgeworth series

TB114.3

A

1672?7207(2018)08?1944?06

10.11817/j.issn.1672?7207.2018.08.015

2017?08?26；

2017?10?26

國家自然科學(xué)基金資助項目(51305157)；吉林省科技廳基金資助項目(20160520064JH)(Project(51305157) supported by the National Natural Science Foundation of China; Project(20160520064JH) supported by the Science and Technology Department Fund of Jilin Province)

李鋒，博士，副教授，從事結(jié)構(gòu)可靠性研究：E-mail：fengli@jlu.edu.cn