A Study On Non-stationary Random Vibration Analysis For Large And Complex Structures

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.