Failure probability estimation of dynamic systems employing relaxed power spectral density functions with dependent frequency modeling and sampling基于松弛功率谱密度函数相依频率建模与采样的动力系统失效概率估计
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Behrendt M, Lyu MZ, Luo Y, Chen JB, Beer M, 2024. Failure probability estimation of dynamic systems employing relaxed power spectral density functions with dependent frequency modeling and sampling. Probabilistic Engineering Mechanics, 75: 103592.DOI: 10.1016/j.probengmech.2024.103592
摘要 | Abstract
本文采用具有相似特征数据集的概率荷载模型，即松弛功率谱密度 (power spectral density, PSD) 函数，解决了动力系统中有效估计失效概率的关键问题。松弛功率谱密度函数的主要缺点是频率间缺少相依性，这导致采样会得到不真实的功率谱密度函数，对失效概率估计产生不利影响。本文通过多种相依性建模方法解决这一限制，考虑数据集中存在的相依性统计量。具体地，提出了一种新技术，结合不同频率间概率相依性进行功率谱密度函数的代表性采样，从而提高荷载表示的真实性。通过考虑频率间相依性，松弛功率谱密度函数提高了失效概率估计的精度，为在不确定性下进行更鲁棒和准确的可靠性评估提供了可能。通过数值算例演示了所提方法的有效性和精度，展示了进行动力系统失效概率估计的能力。关键词: 功率谱密度函数, 随机过程, 随机动力学, 不确定性量化, 概率相依性This work addresses the critical task of accurately estimating failure probabilities in dynamic systems by utilizing a probabilistic load model based on a set of data with similar characteristics, namely the relaxed power spectral density (PSD) function. A major drawback of the relaxed PSD function is the lack of dependency between frequencies, which leads to unrealistic PSD functions being sampled, resulting in an unfavorable effect on the failure probability estimation. In this work, this limitation is addressed by various methods of modeling the dependency, including the incorporation of statistical quantities such as the correlation present in the data set. Specifically, a novel technique is proposed, incorporating probabilistic dependencies between different frequencies for sampling representative PSD functions, thereby enhancing the realism of load representation. By accounting for the dependencies between frequencies, the relaxed PSD function enhances the precision of failure probability estimates, opening the opportunity for a more robust and accurate reliability assessment under uncertainty. The effectiveness and accuracy of the proposed approach is demonstrated through numerical examples, showcasing its ability to provide reliable failure probability estimates in dynamic systems.Keywords: Power spectral density function; Stochastic processes; Stochastic dynamics; Uncertainty quantification; Probabilistic dependency创新点 | Highlights
- Improved failure probability estimation utilizing the relaxed PSD function
- Addressing the lack of frequency dependencies in the relaxed PSD function
- Probabilistic dependency modeling in adjacent frequencies of the relaxed PSD function
- Enhancing the realism of the probabilistic load representation for dynamic systems
- Numerical examples demonstrate the effectiveness and accuracy
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Fig. 1. Generated samples without dependency modeling or consideration of correlations
Fig. 2. Generated sample PSD functions for the one RV model
图 3: 多变量 Gauss 模型的功率谱密度函数生成样本Fig. 3. Generated sample PSD functions for the MVG model
Fig. 4. Generated sample PSD functions for the proposed sampling approach
Fig. 5. Ensemble of PSD functions utilized in this work
图 6: 由各功率谱密度函数集合生成的松弛功率谱密度函数Fig. 6. Relaxed PSD function generated from the ensemble of individual PSD functions
Fig. 7. Schematic representation of the SDOF system
Tab. 1. Failure probabilities of the SDOF system for the different dependency models方法 | 蒙特卡罗模拟
| 子集模拟
|
基准模型
| 0.002746 | --
|
均值模型 | 0.000594 | 0.000539
|
不相关模型 | 0.001616 | 0.001425
|
单随机变量模型 | 0.003573
| 0.003584
|
多变量 Gauss 模型 | 0.002593
| 0.002699 |
所提方法 | 0.002763 | 0.002662 |
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Fig. 8. Schematic representation of the nine storey shear-frame structure
Tab. 2. Mass and lateral stiffness of the shear-frame structure model for the individual storeys层号 | 质量 (1e6 kg)
| 刚度 (1e8 N/m)
|
1 | 3.5
| 1.47 |
2 | 3.3 | 1.63 |
3 | 3.0
| 1.62
|
4 | 3.0
| 1.60
|
5 | 3.0
| 1.60 |
6 | 3.0
| 1.92
|
7 | 3.0
| 1.85 |
8 | 2.7
| 0.96
|
9 | 2.7
| 0.89
|
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图 9: (a) 所提方法采样的功率谱密度函数; (b) 谱表达方法生成的平稳地震动加速度样本; (c) 剪切型框架结构的层间位移与回复力结果Fig. 9. (a) A PSD function sampled by the proposed method; (b) A sample of a stationary ground motion acceleration generated with SRM; (c) Resulting inter-storey drift vs. restoring force of the shear-frame structure
表 3: 不同相依模型下剪切型框架结构模型的失效概率Tab. 3. Failure probabilities of the shear-frame structure model for the different dependency models方法 | 蒙特卡罗模拟
| 子集模拟
|
基准模型
| 0.014642 | --
|
均值模型 | 0.005426 | 0.00591
|
不相关模型 | 0.012289 | 0.01242
|
单随机变量模型 | 0.017010
| 0.01706
|
多变量 Gauss 模型 | 0.014473
| 0.01481 |
所提方法 | 0.015085 | 0.01472 |
表 4: 不同相依模型下采用低相关性数据集的单自由度系统失效概率
Tab. 4. Failure probabilities of the SDOF system for the different dependency models by utilizing a data set with low correlation方法 | 蒙特卡罗模拟
|
基准模型
| 0.043357 |
均值模型 | 0.043022 |
不相关模型 | 0.043595 |
单随机变量模型 | 0.045229
|
多变量 Gauss 模型 | 0.043279
|
所提方法 | 0.042622 |
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图 10: 由低相关性数据集松弛功率谱密度采样的功率谱密度函数Fig. 10. Sampled PSD functions from the relaxed PSD of a data set with low correlation
作者信息 | Authors
德国莱布尼茨汉诺威大学 (Leibniz Universität Hannover) 风险与可靠性研究所
律梦泽 Meng-Ze Lyu, 通讯作者 (Corresp.) 同济大学 (Tongji University) 土木工程学院Email: lyumz@tongji.edu.cn
德国莱布尼茨汉诺威大学 (Leibniz Universität Hannover) 风险与可靠性研究所
同济大学 (Tongji University) 土木工程学院
德国莱布尼茨汉诺威大学 (Leibniz Universität Hannover) 风险与可靠性研究所
律梦泽 M.Z. Lyu | 编辑 (Ed)
P.D. Spanos | 审校 (Rev)
陈建兵 J.B. Chen | 审校 (Rev)
彭勇波 Y.B. Peng | 审校 (Rev)
