Generalized Computation of Buckling Loads via Curvature-Based Displacement Interpolation

The calculation of flexural buckling loads for structural members remains a relevant topic of practical and theoretical interest. In addition to well-known solutions for prismatic members, solutions are available for the buckling loads of nonprismatic and weakened members; however, these solutions are often specialized and difficult to implement, as are finite element analyses that require a refined mesh in order to achieve accurate solutions. A straightforward approach for computing buckling loads based on eigenvalue analysis of the curvature-based displacement interpolation (CBDI) influence matrix is developed. The CBDI influence matrix, which is a byproduct of a force-based frame element formulation of geometric nonlinearity, simplifies the calculation of flexural buckling loads for nonprismatic and weakened members while also providing accurate results for prismatic members. Comparisons with previously published solutions show the CBDI approach gives accurate first-mode buckling loads for prismatic and nonprismatic columns when the CBDI influence matrix is formed using at least three interpolation points. More interpolation points are required for the critical buckling loads of columns where the change in flexural stiffness is more abrupt. The CBDI approach is easy to implement and provides engineers and researchers a means of calculating flexural buckling loads for members with arbitrary changes in flexural stiffness.

Critical buckling load calculations are required for a variety of methods to determine column strength and to assess frame stability. The proposed approach using curvature-based displacement interpolation (CBDI) can be applied directly in these methods and is easily implemented in the Python programming language or any software capable of matrix algebra. As presented, the proposed approach is applicable to critical load calculations for flexural buckling of pin-ended initially straight members, inclusive of nonprismatic, e.g., tapered, stepped, or weakened, members. The approach does not apply to torsional and flexural-torsional buckling, and further research is required to extend the approach to members with other boundary conditions, curved members, and members with prestressing. Although not shown in this work, the proposed approach can be extended to columns with distributed and intermediate axial loads. In addition, the method can be extended to three dimensions for columns subjected to combined axial load and biaxial bending as well as shear deformable columns.