Buckling Analysis of Structures with Uncertain Properties and Loads Using an Interval Finite Element Method

In order to ensure the safety of a structure, one must provide for adequate strength of structural elements. In addition, one must prevent large unstable deformations such as buckling. In most analyses of buckling, structural properties and applied loads are considered certain. This approach ignores the fact that imperfections and unknown changes in properties, albeit small, are required for onset of buckling. In this paper, we extend the interval finite element methods developed by the authors to solve for the possible values of loads that will result in a structural stability failure. The analysis requires that interval axial element forces in each frame element in a structure be calculated. These values are calculated from a linear system of interval equations resulting from the static structural analysis. Using the calculated axial loads, a subsequent interval eigenvalue problem is solved for the buckling loads. For both solutions of the linear system of equations and the eigenvalue problem, the unique properties of the finite element methods result in sharp solutions. Several structural problems are presented as exemplars. The sharpness of the solution is demonstrated by comparing to combinatorial solutions.