﻿ 一个与多个失效单元间相关系数的计算方法
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 应用科技  2017, Vol. 44 Issue (6): 41-47  DOI: 10.11991/yykj.201609017 0

### 引用本文

LI Jiancao, YU Yanchun, CHEN Yun, et al. The computing method of correlation coefficient between one element end multi-elements[J]. Applied Science and Technology, 2017, 44(6), 41-47. DOI: 10.11991/yykj.201609017.

### 文章历史

1. 滁州学院 地理信息与旅游学院，安徽 滁州 239000;
2. 东北农业大学 水利与建筑学院，黑龙江 哈尔滨 150030

The computing method of correlation coefficient between one element end multi-elements
LI Jiancao1, YU Yanchun2, CHEN Yun1, YANG Qing1
1. College of Geography Information and Tourism, Chuzhou University, Chuzhou 239000, China;
2. College of Water Conservancy and Architecture, Northeast Agricultural University, Harbin 150001, China
Abstract: Based on error analysis of dimensionality reduction method in structural system reliability, the problem of inaccurate correlation coefficient between one element and multi-elements in dimensionality reduction method was studied. First, the states of structural elements were mapped to standard normal space, thereby establishing the element failure model based on state vector. Based on structural reliability and the basic principle of probability theory, the change rule of state vector was analyzed when multi-elements were joint failure. Then, the change rule that multi-elements are joint failure was described by equivalent state vector. The correlation analysis between one element and multi-elements was converted into the computing problem of included angle cosine between equivalent state vector of one element and equivalent state vector of multi-elements; On account of the difference between reliability index and correlation coefficient between one element and multi-elements, the calculation method of correlation coefficient between one element and multi-elements was discussed in three different cases. Finally, the precision and rationality of this method was discussed when compared with Monte Carlo Method. It is proved that the method is rational and it is better than multiple correlation theory; this method is applicable to reliability analysis of structural system, and a better result can be obtained by this method.
Key words: structural reliability    reliability    dimensionality reduction method    multiple correlation    correlation analysis    dependent failure    failure mode    performance function

1 单元状态在标准正态空间的映射

 $\left\{ \!\!\!{\begin{array}{*{20}{c}}{Z(\mathit{\boldsymbol{X}}) > 0,}&{ \text{单元安全}}\\{Z(\mathit{\boldsymbol{X}}) \leqslant 0,}&{ \text{单元失效}}\end{array}} \right.$

 $Z(\mathit{\boldsymbol{X}}) = {\mathit{\boldsymbol{a}}^{\rm{T}}}\mathit{\boldsymbol{X}} + \beta$

 $\lambda = \mathit{\boldsymbol{a}} \cdot \mathit{\boldsymbol{X}}$

$\lambda$ 服从标准正态分布，结构单元的失效可通过状态值 $\lambda \leqslant - \beta$ 描述。

 $P(\lambda \leqslant - \beta ) = \varPhi ( - \beta ) = 1 - \varPhi (\beta ) = P(\lambda \geqslant \beta )$

 ${\mathit{\boldsymbol{M}}_i} = {\lambda _i}{\mathit{\boldsymbol{a}}_i}\;\;\;\;\left( {i = 1,2, \cdots ,m} \right)$

 $\cos {\theta _{ij}} = \frac{{{\mathit{\boldsymbol{M}}_i} \cdot {\mathit{\boldsymbol{M}}_j}}}{{\left| {{\mathit{\boldsymbol{M}}_i}} \right|\left| {{\mathit{\boldsymbol{M}}_j}} \right|}} = {\mathit{\boldsymbol{a}}_i} \cdot {\mathit{\boldsymbol{a}}_j} = {\rho _{ij}}$

 $\left| {{\mathit{\boldsymbol{M}}_i}} \right| = {\lambda _i}$

i单元失效事件对应失效域：

 \left\{ \begin{aligned}& \left| {{\mathit{\boldsymbol{M}}_i}} \right| \geqslant \left| {{\mathit{\boldsymbol{M}}_i}^{\rm{0}}} \right|\\& {\mathit{\boldsymbol{M}}_i}^{\rm{0}} = {\beta _i}{\mathit{\boldsymbol{a}}_i}\end{aligned} \right.

X构成多维空间中点的角度描述，i单元状态可行域为αi轴上所有点的集合；失效域为αi轴正向上至原点距离大于 ${\beta _i}$ 的点集合 ${\varOmega _i}^0$ 。单元状态矢量具有相同的自变量，且在同一坐标系下。当某一单元状态矢量为其可行域内指定值时，其余单元状态矢量必有一指定值与其对应，如图1(b)。这正揭示了单元间的相关程度，相关系数也正是描述该变化规律间相互关系的量。

 图 1 状态矢量示意图

 图 2 等效状态矢量示意图

 $\mathit{\boldsymbol{M}}_R^0 \cdot {\mathit{\boldsymbol{a}}_i} = \left| {\mathit{\boldsymbol{M}}_R^0} \right|\cos \, {\theta _{Ri}} = \left| {\mathit{\boldsymbol{M}}_i^0} \right| \,\left( {i = 1,2, \cdots, m} \right)$

$\left| {{\mathit{\boldsymbol{M}}_R}} \right| \geqslant \left| {\mathit{\boldsymbol{M}}_R^0} \right|$ 时，有

 $\left| {\mathit{\boldsymbol{M}}_i^{}} \right|{\rm{ = }}\left| {{\mathit{\boldsymbol{M}}_R}} \right|\cos \, {\theta _{Ri}} \geqslant \left| {\mathit{\boldsymbol{M}}_R^0} \right|\cos \, {\theta _{Ri}} = \left| {\mathit{\boldsymbol{M}}_i^0} \right|\;\;\left( {i = 1,2, \cdots, m} \right)$ (1)

 $\left| {\mathit{\boldsymbol{M}}_i^{}} \right| < \left| {\mathit{\boldsymbol{M}}_i^0} \right|\;\;\;\;\;\left( {i = 1,2, \cdots, m} \right)$

${\mathit{\boldsymbol{M}}_R}$ 属于失效域时，多个单元均处于失效域，反之亦然。这样多个结构单元联合失效问题，就转化为等效状态矢量属于失效域的问题。需要注意的是，等效状态矢量属于失效域的概率，并不等于多个单元的联合失效概率。因为空间轴上的一点实际代表着一个平面(或超平面)。 ${\mathit{\boldsymbol{M}}_R}$ 属于失效域，代表着能使多个单元同时失效的空间点一定在这些面上，但每个面上的所有点未必都能导致多个单元同时失效。所以，等效状态矢量只是描述了各单元状态值同时属于失效域这一现象及内在规律，而与各单元状态值大于某一指定阈值的概率不存在必然联系。

2 等效状态矢量建立

 \begin{aligned}& {P_f} = P\left( {{Z_1} \leqslant 0 \cap {Z_2} \leqslant 0 \cap \cdots } \right) = \\& P\left( {\left( {{Z_1} \leqslant 0 \cap {Z_2} \leqslant 0} \right) \cap {Z_3} \leqslant 0 \cap \cdots } \right)\\& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \vdots \end{aligned}

 图 3 不同条件下的等效状态矢量

 $\frac{{\left| {\mathit{\boldsymbol{M}}_i^0} \right|}}{{\left| {\mathit{\boldsymbol{M}}_j^0} \right|}} = {\rho _{ij}}$

 $\left| {{\mathit{\boldsymbol{M}}_i}} \right| = {\mathit{\boldsymbol{M}}_j} \cdot {\mathit{\boldsymbol{a}}_i} = \left| {{\mathit{\boldsymbol{M}}_j}} \right|{\rho _{ij}} \geqslant \left| {\mathit{\boldsymbol{M}}_j^0} \right|{\rho _{ij}} = \left| {\mathit{\boldsymbol{M}}_i^0} \right|$

 ${\rho _{ij}} < \frac{{\left| {\mathit{\boldsymbol{M}}_i^0} \right|}}{{\left| {\mathit{\boldsymbol{M}}_j^0} \right|}} < \frac{1}{{{\rho _{ij}}}}$

 $\frac{{\left| {\mathit{\boldsymbol{M}}_i^0} \right|}}{{\left| {\mathit{\boldsymbol{M}}_j^0} \right|}} < {\rho _{ij}}\text{或}\frac{{\left| {\mathit{\boldsymbol{M}}_i^0} \right|}}{{\left| {\mathit{\boldsymbol{M}}_j^0} \right|}} > \frac{1}{{{\rho _{ij}}}}$

 $\left| {{\mathit{\boldsymbol{M}}_i}} \right| \geqslant \left| {\mathit{\boldsymbol{M}}_i^0} \right|$

 $\left| {\mathit{\boldsymbol{M}}_*^0} \right| = {\rho _{ij}}\left| {\mathit{\boldsymbol{M}}_i^0} \right|$

3 相关系数计算

 $\mathit{\boldsymbol{\rho }} = \left[ {\begin{array}{*{20}{c}}1&{{\rho _{12}}}&{{\rho _{13}}}\\{{\rho _{12}}}&1&{{\rho _{23}}}\\{{\rho _{13}}}&{{\rho _{23}}}&1\end{array}} \right]$ (2)

 ${\rho _{3 \cdot 12}} = \left\{ {\begin{array}{*{20}{c}}{{\rho _{13}},}&{{\beta _1} > {\beta _2}}\\{{\rho _{23}},}&{{\beta _1} < {\beta _2}}\end{array}} \right.$

 \left\{ {\begin{aligned}& {{\mathit{\boldsymbol{a}}_1} = \left[ {\begin{array}{*{20}{c}}1& \, 0& \, 0\end{array}} \right]}\\& {{\mathit{\boldsymbol{a}}_2} = \left[ {\begin{array}{*{20}{c}}{{b_1}}&{{b_2}}&0\end{array}} \right]}\\& {{\mathit{\boldsymbol{a}}_3} = \left[ {\begin{array}{*{20}{c}}{{c_1}}&{{c_2}}&{{c_3}}\end{array}} \right]}\end{aligned}} \right.

 \left\{ \begin{aligned}& {\rho _{12}} = {\mathit{\boldsymbol{a}}_1} \cdot {\mathit{\boldsymbol{a}}_2} = {b_1}\\& {\rho _{13}} = {\mathit{\boldsymbol{a}}_1} \cdot {\mathit{\boldsymbol{a}}_3} = {c_1}\\& {\rho _{23}} = {\mathit{\boldsymbol{a}}_2} \cdot {\mathit{\boldsymbol{a}}_3} = {b_1}{c_1} + {b_2}{c_2}\end{aligned} \right.

 \left\{ \begin{aligned}& b_1^2 + b_2^2 = 1\\& c_1^2 + c_2^2 + c_3^2 = 1\end{aligned} \right.

 \left\{ \begin{aligned}& {\mathit{\boldsymbol{a}}_2} = \left[ {\begin{array}{*{20}{c}}{{\rho _{12}}}&{\sqrt {1 - \rho _{12}^2} }&0\end{array}} \right]\\& {\mathit{\boldsymbol{a}}_3} = \left[ {\begin{array}{*{20}{c}}{{\rho _{13}}}&{\displaystyle\frac{{{\rho _{23}} - {\rho _{13}}{\rho _{12}}}}{{\sqrt {1 - \rho _{12}^2} }}}&{{c_3}}\end{array}} \right]\end{aligned} \right.

 $\mathit{\boldsymbol{M}}_R^0 = \left[ {\begin{array}{*{20}{c}}{{d_1}}&{{d_2}}&0\end{array}} \right]$

 \left\{ \begin{aligned}\mathit{\boldsymbol{M}}_R^0 \cdot {\mathit{\boldsymbol{a}}_1} = {\beta _1}\\\mathit{\boldsymbol{M}}_R^0 \cdot {\mathit{\boldsymbol{a}}_2} = {\beta _2}\end{aligned} \right.

 \left\{ \begin{aligned}& {d_1} = {\beta _1}\\& {d_2} = \frac{{{\beta _2} - {\beta _1}{\rho _{12}}}}{{\sqrt {1 - \rho _{12}^2} }}\end{aligned} \right.

 $\begin{split}& {\rho _{3 \cdot 12}} = \frac{{\mathit{\boldsymbol{M}}_R^0 \cdot {\mathit{\boldsymbol{a}}_3}}}{{\left| {\mathit{\boldsymbol{M}}_R^0} \right|}} = \displaystyle\frac{{{\beta _1}{\rho _{13}} + \displaystyle\frac{{{\rho _{23}} - {\rho _{13}}{\rho _{12}}}}{{\sqrt {1 - \rho _{12}^2} }} \cdot \displaystyle\frac{{{\beta _2} - {\beta _1}{\rho _{12}}}}{{\sqrt {1 - \rho _{12}^2} }}}}{{\sqrt {\beta _1^2 + {{\left( {\displaystyle\frac{{{\beta _2} - {\beta _1}{\rho _{12}}}}{{\sqrt {1 - \rho _{12}^2} }}} \right)}^2}} }} = \\& \quad \quad \frac{{{\beta _1}{\rho _{13}}\left( {1 - \rho _{12}^2} \right) + \left( {{\rho _{23}} - {\rho _{13}}{\rho _{12}}} \right)\left( {{\beta _2} - {\beta _1}{\rho _{12}}} \right)}}{{\sqrt {{{\left[ {{\beta _1}\left( {1 - \rho _{12}^2} \right)} \right]}^2} + \left( {1 - \rho _{12}^2} \right){{\left( {{\beta _2} - {\beta _1}{\rho _{12}}} \right)}^2}} }}\end{split}$ (3)

 $\frac{{{\beta _1}{\rho _{13}}\left( {1 - \rho _{12}^2} \right) + \left( {{\rho _{23}} - {\rho _{13}}{\rho _{12}}} \right)\left( {{\beta _2} - {\beta _1}{\rho _{12}}} \right)}}{{\sqrt {{{\left[ {{\beta _1}\left( {1 - \rho _{12}^2} \right)} \right]}^2} + \left( {1 - \rho _{12}^2} \right){{\left( {{\beta _2} - {\beta _1}{\rho _{12}}} \right)}^2}} }} = {\rho _{23}}$

1)判断是否满足条件 ${\rho _{12}} \leqslant {\beta _1}/{\beta _2} \leqslant 1/{\rho _{12}}$

2)分别计算1、2单元与其余m-2个单元的相关系数，满足条件采用式(3)，1、2单元联合失效概率 ${P_r} = {\varPhi _2}\left( { - {\beta _1}, - {\beta _2};{\rho _{12}}} \right)$ ；若条件不满足，当 ${\beta _1} > {\beta _2}$ 时，相关系数取 ${\rho _{13}}$ ，失效概率

 ${P_r} = \min \left[ {{\varPhi _2}\left( { - {\beta _1}, - {\beta _2};{\rho _{12}}} \right),{\varPhi _2}\left( { - \frac{{{\beta _2}}}{{{\rho _{12}}}}, - {\beta _2};{\rho _{12}}} \right)} \right]$

${\beta ^{}}_1 < {\beta ^{}}_2$ 时，相关系数取 ${\rho _{23}}$ ，失效概率

 ${P_r} = \min \left[ {{\varPhi _2}\left( { - {\beta _1}, - {\beta _2};{\rho _{12}}} \right),{\varPhi _2}\left( { - {\beta _1}, - \frac{{{\beta _1}}}{{{\rho _{12}}}};{\rho _{12}}} \right)} \right]$

 $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

4 精度分析及讨论

 ${g_i} = \sqrt \rho {x_0} + {x_i} + {\beta _i}\;\;\;\;\;(i = 1,2, \cdots ,n)$

n=3时，相关系数ρ=0～1.0，单元可靠性指标分别为β1= β2= β3= 2.0。当β1=2.0、β2=2.5、β3=3.5时，并联体系失效概率Pf计算结果见图4

 图 4 3个单元并联

 图 5 10个单元并联

 图 6 单元数量变化的并联体系

 \begin{aligned}{g_1} = & \sqrt {\displaystyle\frac{{{\rho _{12}}{\rho _{13}}}}{{{\rho _{23}}}}} {x_0} + \sqrt {1 - \frac{{{\rho _{12}}{\rho _{13}}}}{{{\rho _{23}}}}} {x_1} + {\beta _1}\\{g_2} = & \sqrt {\displaystyle\frac{{{\rho _{12}}{\rho _{23}}}}{{{\rho _{13}}}}} {x_0} + \sqrt {1 - \frac{{{\rho _{12}}{\rho _{23}}}}{{{\rho _{13}}}}} {x_2} + {\beta _2}\\{g_3} = & \sqrt {\displaystyle\frac{{{\rho _{13}}{\rho _{23}}}}{{{\rho _{12}}}}} {x_0} + \sqrt {1 - \frac{{{\rho _{13}}{\rho _{23}}}}{{{\rho _{12}}}}} {x_3} + {\beta _3}\end{aligned}

β1=2.1、β2=1.8、β3=2.5，ρ12=0.3~0.65、ρ13=0.4、ρ23=0.6时，计算结果见图7

 图 7 ${\rho _{12}}$ 变化的三单元并联

β1=2.1、β2=2.3、β3=2.7，ρ23=0.3~0.65，ρ12=0.6、ρ13=0.4时，计算结果见图8

 图 8 ${\rho _{23}}$ 变化的三单元并联

5 结论

1）本文方法是在线性功能函数基础上建立的，虽然非线性功能函数可线性化处理，但要考虑线性化带来的误差，不代表一定能获得文中的计算精度。非线性函数可近似为多个线性函数，所以文中算法在功能函数线性化方面也能发挥作用。

2）文中分析过程中，相当于建立了一个状态矢量模型，状态矢量对结构系统失效描述较为直观，便于相关程度分析，值得作为一种新的结构系统可靠性分析模型开展深入研究。

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