The Static Optimization For Uncertain Structures With Interval Parameters

The analysis and the design methods of practical engineering structure are often based on deterministic parameters and deterministic model. However, there are errors or uncertainty related to the material properties, the geometrical character, the loading, the initial condition and the boundary condition, etc. are existed. The errors or uncertainty may be very small, but the combination of these factors may cause large deviations of the responses of the structure,especially in multi- component system. Therefore, the concept of uncertainty plays an important role in the investigation of various engineering problems.The most common approach to uncertain problems is to model the structural parameters as random variables or fields. In this case, all information about the structural parameters is provided by the joint probability density function(or distribution 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 data. Recently, Chen Suhuan, Qiu Zhiping,etc. have used interval method in the study of the static response and eigenvalue problems of structures with bounded uncertain parameters. In the finite element equation Ku = f,where K , f are functions of the structural parameters, andΔK ,Δf are functions of the structural parameters too, so they must can be calculated according to the uncertainties of the structural parameters. Chen Suhuan,Yang Xiaowei and Lian Huadong have presented some effective interval methods for structures with interval parameters.Since the mid-1960's, a new method called the interval analysis has appeared. Moore and his co-workers, Alefeld and Herzberger have done the pioneering work. Hansen in his book discussed the global optimization using interval analysis. Because of the complexity of the interval algorithm, it is difficult to deal with practical engineering problems. However, few papers can be found about the optimization of structures with interval parameters in engineering. Hence, it is necessary to develop an effective method to solve the optimal problems of structures with interval parameters. Recently, Chen Suhuan and Wu Jie have discussed the optimization of dynamic problems for structures with interval parameters. This paper presents an interval optimization method based on the interval static finite element analysis. The idea of this method can be considered as follows: first, transform the interval problem into approximate deterministic one, then optimize the upper bound of the response by using the standard optimal calculate method in order to make them in a narrowest interval.The contents of the paper are as follows:1. In chapter 4, the interval finite element methods of the structural static analysis based on 1st-order perturbation and 2nd-order perturbation are presented.2. In chapter 5, a static interval optimization method for structures with deterministic parameters and interval loading is presented. In this method, using 1st-order Taylor expantion to expand the loading about its mid-vector of the interval loading vector, then transform the interval problem into approximate deterministic one to solve.3. In chapter 6, a static interval optimization method for structures with interval parameters and deterministic loading is presented. In this method, using 1st-order Taylor expantion to expand the stiffness matrix about the mid-vector of the interval parameter vector, then transform the interval problem into approximate deterministic one to solve.4. In chapter 7, a static interval optimization method for structures with interval parameters and interval loading is presented. In this method, using 1st-order Taylor expantion to expand the stiffness matrix about the mid-vector of the interval parameter vector, and using 1st-order Taylor expantion to expand the loading about its mid-vector of the interval loading vector at the same time. Then transform the interval problem into approximate deterministic one to solve.5. In chapter 8, a static interval optimization method for structures based on 2nd-order Taylor expantion is presented. In this method, using 2st-order Taylor expantion to expand the stiffness matrix about the mid-vector of the interval parameter vector, and using 2st-order Taylor expantion to expand the loading about its mid-vector of the interval loading vector at the same time. Then transform the interval problem into approximate deterministic one to solve.