Spherical Layer Sampling Method for Probability Evaluation on Structural Failure
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摘要:
传统Monte Carlo抽样方法应用于小失效概率等复杂可靠度问题时，存在效率低下、精度有限等不足，针对这一问题提出一种球层抽样分析方法. 首先，通过分离距离与方向参量，对标准正态随机向量进行重构，并验证其标准正态性与相互独立性；然后，采用分层抽样策略，将半径大于一阶可靠度指标外的正态空间划分为多个球层，进而利用所重构的向量逐层进行抽样，并结合全概率公式，形成估计结构失效概率的球层抽样算法；最后，通过对3个典型算例进行对比分析，验证算法性能. 分析表明：所提出算法具有较高的抽样效率与收敛性能，算例计算结果误差在3%以内；与其他算法相比，其估计方差更小，且可有效解决多设计验算点等复杂可靠度问题；算法在抽样效率、适用范围以及稳定性等方面具有优势，更适用于实际复杂结构可靠度的求解分析.
Abstract:When the traditional Monte Carlo sampling method is applied to complex reliability problems such as small failure probability, there are some shortcomings such as low efficiency and limited accuracy. To solve this problem, a spherical layer sampling analysis method is developed. Firstly, by dividing the distance and direction parameters, the standard normal random vector is reconstructed, and its standard normality and mutual independence are verified. Thereafter, based on a layered sampling strategy, the standard normal space with the radius beyond first order reliability index is divided into multiple spherical layers, which are then sampled by the reconstructed vector layer by layer. Combined with the full probability formula, a spherical layer sampling algorithm is developed to estimate the structural failure probability. Finally, three typical examples are taken as objects of interest, and the performance of the algorithm is verified through comparative analysis. The results show that, the proposed algorithm has high sampling efficiency and convergence performance, and the error of calculation results is within 3%. Compared with other algorithms, its estimation variance is smaller, and it can effectively solve complex reliability problems such as multiple design check points. The algorithm has advantages in sampling efficiency, scope of application, and stability, and is more suitable for solving and analyzing the reliability of actual complex structures.
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图 2 Zi抽样频率密度与标准正态分布概率密度对比
Figure 2. Fig. 2 Comparison of probability density between Zi sampling and standard normal distribution
图 3 n维标准正态空间中n−1维球面均匀分布
Figure 3. Spherical uniform distribution of n−1 dimensions in standard normal space
图 4 不同维度下Zi前四阶矩抽样结果
Figure 4. Sampling results of first four-order moments of Zi in different dimensions
图 8 算例1极限状态曲面及球层划分
Figure 8. Limit state surface and spherical layer division in standard normal space for case 1
图 9 算例1球层抽样失效概率变异系数随抽样次数变化
Figure 9. Coefficient of variation in failure probability of spherical layer sampling with sampling times for case 1
图 11 算例3钢斜拉桥最大单悬臂施工工况示意（单位：m）
Figure 11. Illustration of case 3 steel cable-stayed bridge in longest single cantilever construction phase （unit: m）
表 1 算例1不同方法计算结果对比
Table 1. Comparison of calculation results with different sampling methods for case 1
计算方法 抽样
数/次Pf/×10−3 βform 相对误差/% 变异
系数HL-RF 1.832 2.9057 38.3 重要抽样 1000 2.090 2.8642 29.6 0.360 线抽样 1000 1.957 2.8850 34.1 0.004 球层抽样 1000 2.998 2.7480 0.9 0.062 MCS 1×106 2.970 2.7511 注：各算法变异系数结果为重复100次计算得到，后续算例同. 
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表 2 算例2随机变量的分布参数
Table 2. Distribution parameters of random variables for case 2
项目 q/
（kN•m−1）ls/m As/m2 Ac/m2 Es/
GPaEc/
GPa均值 20 12 9.82 × 10-4 0.04 100 20 变异系数 0.07 0.01 0.06 0.12 0.06 0.06
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表 3 算例2不同方法计算结果对比
Table 3. Comparison of calculation results with different sampling methods for case 2
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表 4 算例3抗力不确定性变量分布参数
Table 4. Distribution parameters of resistance random variables for case 3
项目 km kf kp t≤16 mm 16 mm<t≤25 mm 25 mm<t≤38 mm 38 mm<t≤50 mm 偏差系数 1.1511 1.1228 1.0991 1.2074 1.0000 1.0890 变异系数 0.0656 0.0670 0.0843 0.0917 0.0200 0.1080
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表 5 算例3荷载变量分布参数
Table 5. Distribution parameters of temporary load random variables during construction for case 3
项目 桥面吊机和焊接设备 焊接平台 张拉平台 均值/kN 2402 150 399 变异系数 0.0500 0.0500 0.0500
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