Analysis of structures with uncertain parameters using interval method.
Date of Issue2012
School of Civil and Environmental Engineering
The uncertainties of structural parameters may lead to large and unexpected excursion of responses that may result in drastic reduction in accuracy and precision of the operation. Therefore, the concept of uncertainty plays an important role in the investigation of various engineering problems. In this thesis, the effect of uncertainty in structural parameters and applied forces on the linear structural systems is studied. Uncertain parameters are introduced in the form of intervals. The structural stiffness matrix, mass matrix and loading vectors are described as the sum of two parts corresponding to the deterministic matrix and the uncertainty matrix of the interval parameters. Interval analysis is conducted with the finite element (IFE) method to analyze the system response. The static analysis, the eigenvalue analysis, the frequency response function (FRF) analysis and the dynamic response analysis of structures are carried out. Some general interval techniques are used to solve equations with intervals, such as the element-by-element (EBE) technique, a special matrix treatment, the perturbation technique and so on.In the static analysis, the element-by-element (EBE) technique is implemented to avoid the element coupling. A special matrix treatment to handle the interval matrix multiplication is used to reduce the overestimation due to multi-occurrence of interval variables. Then, the Brouwer’s fixed point theorem and Krawczyk’s operator are applied to obtain the displacements of the structure. Illustrative examples show that the proposed IFE method can achieve the sharpest possible results.In the eigenvalue analysis, the perturbation technique is applied to remove the higher order terms of uncertainties in the eigenvalue equation, and a special treatment is used to reduce the overestimation of interval calculation. In the proposed method, it is only required to solve a system of linear equations for each eigenvalue and its corresponding eigenvector. Neither iteration nor calculation of inverse of interval matrices is needed, which saves a lot of computational efforts. The effectiveness of the proposed method is demonstrated by numerical results which are compared with the vertex solutions and other researchers’ work.