Wang, W., Gu, R., Li, C., and Yang, L., "Buckling of Structures Subject to Multiple Forces," SAE Technical Paper 2013-01-1370, 2013, doi:10.4271/2013-01-1370.
Frames are important structures found in many transportation applications such as automotive bodies and train cars. They are also widely employed in buildings, bridges, and other load bearing designs. When a frame is carrying multiple loads, it can potentially risk a catastrophic buckling failure. The loads on the frame may be non-proportional in that one force stays constant while the other is increased until buckling occurs. In this study the buckling problem is formulated as a constrained eigenvalue problem (CEVP). As opposed to other CEVP in which the eigenvectors are forced to comply with a number of the constraints, the eigenvalues in the current CEVP are subject to some equality constraints. A numerical algorithm for solving the constrained eigenvalue problem is presented. The algorithm is a simple trapping scheme in which the computation starts with an initial guess and a window containing the potential target for the eigenvalue is identified. Using a quadratic interpolation scheme the eigenvalue satisfying the constraints is further located. Several examples are presented to show the accuracy and effectiveness of the proposed numerical algorithm, including the buckling of a two-dimensional truss structure, a plane frame, a three-dimensional frame, and a thin-walled structural component.