In order to ensure the safety of a structure, adequate strength for structural elements must be provided. In addition, the catastrophic deformations such as buckling must be prevented. In most buckling analyses, structural properties and applied loads are considered certain. Using the linear finite element method, the deterministic buckling analysis is done in two main steps. First, a static analysis is performed using an arbitrary ordinate of applied load. Using the obtained element axial forces, the geometric stiffness of the structure is assembled. Second, performing an eigenvalue problem between the structure's elastic and geometric stiffness matrices yields the structure's critical buckling loads. However, these deterministic approaches disregard uncertainty in the structure's material and geometric properties.
In this work, an interval formulation is used to represent the uncertainty in the structure's parameters such as material characteristics. Then, the developed interval finite element methods are extended to solve for the bounds of possible values of loads that will result in a structural instability. Analogous to deterministic problem, the analysis requires that bounds on element axial forces in each frame element in a structure be calculated. These axial forces 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 critical buckling loads of the structure. The unique properties of finite element methods result in sharp solutions for both the interval linear system of equations and the interval eigenvalue problem. A structural problem is presented as an exemplar. The sharpness of the solution is demonstrated by comparing to exact solutions.