The Interval Finite Element Method For The Dynamic And Static Structures Analysis

In the past, it is dominative for the traditional definite mathematical models to be used in most engineering problems, whatever it is static problem or an dynamic one, i.e. the structural parameters are certain in the calculation. However, there are always uncertain factors in the structural engineering practices, such as the inaccuracy of the measurement, etc. In the case where the structures are simple in topology or small in size, the uncertainty in the structures can be neglected without any more bad effect. However, it is infeasible for the large complex structures.There are several mathematical models in present researches or applications about uncertain structural analysis, such as convex model, fuzzy model, stochastic model and interval analysis. The most common approach to uncertain problem is stochastic one. In this method, all information about the structural parameters is provided by the joint probability density function of them. Unfortunately, the probabilistic modeling is not the only way we can use to describe the uncertainty, and uncertainty is not tantamount to randomness. In many cases, the uncertainty phenomena do not have a stochastic nature. The reason why many researchers studying uncertain problems utilize stochastic modeling is that this randomization is the result of an established scientific stereotype. Indeed, probabilistic approaches are not able to deliver reliable results without sufficient experiment date. For the fuzzy and convex models, there are certain subjective information limited.More attention has been given to the interval model since the interval mathematics appears, because it needs only the bounds of the structural parameters, which can be always obtained in the engineering. The workload is very large to use directly the interval calculation in the structural analysis, and the result is not very ideal. Based on the previous researches on interval structural analysis, the interval FEM for structural analysis is set up. The interval characterized matrices have been acquired from the interval structural parameters and then the mathematical descriptions about the interval static problem and the interval response problem are presented; with the perturbation methods and the interval extension theory, the interval displacement, and response are given.There are five chapters in the thesis, and the main contents are as follows:In the first chapter, we introduce firstly the background of the uncertain structural analysis with its great value. Then the uncertainty in the engineering are classified and the mathematical models are given. The present research works on uncertain analysis, both here and abroad, are summarized; especially the interval model.In the second chapter, some basic background on interval mathematics is introduced, including the operations of the real intervals, and the interval extension theorem.In the third chapter, the interval static FEM is set up. The interval stiffness matrix and interval load vector are obtained from the interval parameters and then the interval static problems are described. Based on the perturbation method on displacement and the interval extension theory, the interval displacements are obtained for the two kinds of interval parameters, the element interval parameters and the element group ones.In the fourth chapter, the interval dynamic response problems are proposed. The interval characterized matrices are obtained from the interval parameters and then interval dynamic response problems are described. Based on the matrix perturbation method of the dynamic response and the interval extension theory, the interval dynamic response is obtained.In the 5th chapter, the conclusions of the thesis are given and some potential fields on the interval structural analysis are suggested.