Метод вероятностного анализа надежности элементов конструкций на основе граничных функций распределения

С. А. Соловьев, А. А. Соловьева
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Abstract

Introduction. The research is aimed at the development of methods for assessing and analyzing the structural reliability of elements of building structures in practical tasks when statistical data about random variables may be incomplete or limited. In such cases, it is difficult to identify the specific type of the exact cumulative distribution function of a random variable or to give an accurate estimate of the distribution parameter, because there is a need to take into account epistemic uncertainty in addition to aleatory uncertainty. Materials and methods. For effective modelling of two types of uncertainties, it is proposed to use boundary distribution functions of a random variable that form a p-box (probability box). P-boxes take into account both types of uncertainty: caused by natural variability of random parameters and uncertainty caused by lack of knowledge about the random variable (number of control samples, accuracy of measuring instruments, etc.). Results. The paper proposes a new type of p-box based on the Dvoretzky – Kiefer – Wolfowitz inequality and Chebyshev’s inequality, which form two boundary distribution functions based on the sample population data. The numerical example shows the approach to arithmetic operations with p-boxes, which make it possible to bring complex mathematical models to simpler ones and estimate the failure probability in an interval form. The difference between the analytical and numerical solution for the example is 0.9 %. Conclusions. Boundary distribution functions form a more cautious and reliable approach to the structural reliability analysis. The result of reliability assessment using p-boxes is presented in an interval form. If the interval turns out to be too wide and uninformative, then it is necessary to increase the quantity or quality of statistical data or to increase the cross-sections of structural elements to achieve the lower limit of the interval of the required reliability level.
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基于边界分布函数的概率结构分析方法
介绍。本研究旨在发展在实际任务中,当随机变量的统计数据可能不完整或有限时,评估和分析建筑结构构件结构可靠度的方法。在这种情况下,很难确定随机变量的确切累积分布函数的具体类型或给出分布参数的准确估计,因为除了选择性不确定性之外,还需要考虑认知不确定性。材料和方法。为了对两类不确定性进行有效的建模,建议使用随机变量的边界分布函数,形成一个p盒(概率盒)。p -box考虑了两种不确定性:一种是由随机参数的自然变异性引起的,另一种是由于缺乏对随机变量的了解(控制样本的数量、测量仪器的精度等)引起的。结果。本文提出了一种基于Dvoretzky - Kiefer - Wolfowitz不等式和Chebyshev不等式的新型p-box,它们构成了两个基于样本总体数据的边界分布函数。数值算例显示了用p-box进行算术运算的方法,它可以将复杂的数学模型转化为简单的数学模型,并以区间形式估计失效概率。解析解与数值解的差异为0.9%。结论。边界分布函数为结构可靠度分析提供了一种更为谨慎可靠的方法。以区间形式给出了用p-box进行可靠性评估的结果。如果区间太宽且信息量不足,则需要增加统计数据的数量或质量,或者增加结构单元的截面,以达到所需可靠度水平区间的下限。
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审稿时长
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