【作者】 徐瑞；

【导师】 苏成；

【作者基本信息】 华南理工大学 ， 结构工程， 2010， 博士

【摘要】 现实世界充满偶然性,大到星体的产生、形成至毁灭的整个过程,小到生命个体的孕育、成长至死亡的整个过程。在自然科学技术领域,用随机性的观点对偶然性加以描述是大势所趋。对数学、物理或力学中的变量、场或过程用随机模型加以反映,就表述为随机变量、随机场或随机过程。把随机变量、随机场或随机过程与描述物理和工程振动问题的控制微分方程相结合就成了一个新的研究领域:随机动力微分方程。本文致力于大型复杂结构离散系统随机动力微分方程的求解。对于包含随机性的结构动力系统,其安全性能需要在概率意义上加以评估,结构动力可靠度分析是本文致力于研究的另外一个主要方面。系数项为确定值、输入项为随机过程的随机动力微分方程所描述的物理问题就是通常所谓的经典随机振动问题,即确定性结构系统受随机激励作用的问题。功率谱法已经可以有效解决经典随机振动问题中的平稳随机振动问题。而对于非平稳随机振动问题,大型复杂结构随机振动的精确高效求解方法是目前的研究热点之一。根据线性结构输入与输出之间具有线性关系这一特点,在激励于离散时间间隔内随时间线性变化的假定下,推导了结构动力响应的显式表达式。依据动力响应的显式表达式,一方面可以根据矩的运算规律,直接计算结构响应的数字统计特征,提出了非平稳随机振动分析的时域显式直接法;另一方面可以实施蒙特卡罗模拟,利用数理统计方法计算结构响应的概率密度演化函数,提出了非平稳随机振动分析的时域显式随机模拟法。对于系数项和输入项同时具有随机性的随机动力微分方程所描述的物理问题,通常称之为复合随机振动问题,即随机结构系统受随机激励作用的问题。已有的针对随机结构的随机振动分析方法各有优缺点,然而在工程应用中都未获推广。利用随机激励与结构随机参数之间具有相互独立性这一特点,采用分两步走的策略先后考虑激励随机性和结构参数随机性对随机响应的影响。以确定性结构随机振动分析的时域显式直接法为基础,以概率论中的条件数学期望为纽带,提出了随机结构非平稳随机振动分析的全数学期望法。结构动力可靠度研究分为构件动力可靠度研究和体系动力可靠度研究两个层次。由于受到随机振动分析方法的限制,无论是确定性结构动力可靠度问题,还是随机结构动力可靠度问题,它们的研究仍停留在利用结构随机动力响应分析结果来计算结构动力可靠度的层次上,主要的研究基础并没有突破按照特殊跨越假定进行分析的格局。为克服已有动力可靠度计算方法的缺点,对于确定性结构,在动力响应显式表达式的基础上进行蒙特卡罗模拟以求解结构动力可靠度,提出了结构构件动力可靠度和体系动力可靠度分析的时域显式随机模拟法;对于随机结构,同样采用随机结构随机振动分析分两步走的策略,以确定性结构动力可靠度分析的时域显式随机模拟法为基础,以概率论中的条件概率为纽带,提出了随机结构构件动力可靠度和体系动力可靠度分析的全概率法。前述随机振动分析方法和动力可靠度分析方法的计算效率取决于激励离散后时间截口随机变量的个数。为了进一步提高所提出方法的计算效率,把随机过程的KL分解(Karhunen-Loeve expansion)理论应用于随机激励的分解,提出了基于KL分解的确定性结构和随机结构的随机振动分析方法和动力可靠度分析方法。理论方法的研究目的在于指导工程实践,并在工程实践中检验理论方法的有效性。新光大桥作为一种新型的、复杂的大跨度拱桥,由于结构的重要性、几何形状的特殊性、结构组成的多样性以及动力响应的复杂性,其抗震安全性能研究显得十分重要。利用本文提出的随机振动分析方法和动力可靠度分析方法,对新光大桥在非平稳地震作用下的随机响应和动力可靠度进行了分析。数值算例和工程算例都表明本文所提出的随机振动分析方法和动力可靠度分析方法的精确性和高效性,它们为解决大型复杂结构的随机振动分析和动力可靠度分析提供了一条有效途径。更多还原

【Abstract】 It is well acknowledged that the real world is full of contingency. This means that the lifetime of an astral body or a life being is a random process in nature. Therefore it is a general trend to describe the contingency from the randomness viewpoint in the field of natural science and technology. The variables, fields and processes associated with random models in mathematics, physics or mechanics could be represented by random variables, random fields and random processes, respectively. A new research field, the stochastic dynamic differential equations, emerges as vibration problems in physics and engineering are described by random variables, random fields and random processes. How to solve such equations of the discrete system of a large and complex structure is the focal point of this thesis. Besides, the research of this thesis also concentrates on the analysis of structural dynamic reliability, because the security of a dynamic system with random parameters has to be assessed in the sense of probability.A problem described by a stochastic dynamic differential equation with deterministic coefficients and stochastic inputs of random processes, is the so-called classical random vibration problem, i.e. the problem of deterministic structures subjected to random excitations. Such problems have been studied intensively and solved efficiently by the power spectral method. However, for the problems of non-stationary random vibration analyses, especially for large and complex structures, seeking for an accurate and efficient method is still a research hotspot. In this thesis, explicit expressions in the time domain for dynamic responses of a structure are derived based on the linear relationship between the input and output of the linear structure and the assumption that the excitations vary linearly with time within a discrete time interval. Applying such explicit expressions, a direct formulation method in the time domain is proposed by which mean the values and variances of structure responses could be obtained directly according to the moment operation rules. Furthermore, a stochastic simulation method, termed as the explicit Monte-Carlo simulation method, is proposed by incorporating the derived explicit expressions to obtain the evolutionary probability functions of non-stationary responses.The problems involving stochastic dynamic differential equations with stochastic coefficients and stochastic inputs are the so-called composite random vibration problems, in which the structure is stochastic and subjected to random excitations. The existing analysis methods for such problems have not yet been well accepted in engineering. In consideration of the mutual independence of randomness between structural parameters and excitation parameters, a strategy is adopted to reflect the influences of excitation parameters and structural parameters in two steps. By using the concept of conditional mathematical expectation in probability theory, a total mathematical expectation method for non-stationary random responses analysis of stochastic structures is proposed based on the direct formulation method for deterministic structures.Dynamic reliability of a structure consists of two levels, the component dynamic reliability and the system dynamic reliability. Due to the limitation of existing random vibration analysis methods,the studies of dynamic reliability, no matter for deterministic structures or for stochastic structures, are still restricted by using particular excursion assumptions. In order to overcome these shortcomings, an explicit Monte-Carlo simulation method is proposed to analyze the component dynamic reliability and the system dynamic reliability of a deterministic structure based on explicit expressions of dynamic responses. Moreover, as to stochastic structures, the two-step strategy is also adopted and a total probability method is proposed based on the dynamic reliability analysis of deterministic structures and the concept of conditional probability in probability theory.The computational efficiency of the proposed random vibration analysis methods and the dynamic reliability analysis methods depends on the number of random variables of excitations at discrete instants. Therefore, Karhunen-Loeve expansion theory is used to represent the random excitations in order to get higher computational efficiency. New random response analysis methods and dynamic reliability methods are proposed based on Karhunen-Loeve expansion theory.To demonstrate the effectiveness of the proposed methods, the application of the present methods to the Xinguang Bridge is introduced as an example. A strict seismic safety evaluation is required for this new-style and complex large-span arch bridge due to the importance of the structure, the particular form of the geometry, the diversity of the structural composition and the complexity of the dynamic response. Random vibration analysis and dynamic reliability analysis of the Xinguang Bridge under non-stationary seismic excitations are conducted in this thesis.Both numerical examples and engineering examples show good accuracy and efficiency of the proposed methods, indicating that such methods offer a novel and effective way to analyze the random vibration responses and dynamic reliabilities for large and complex structures.更多还原