Buckling in columns. Solution of the indeterminations of Euler's theory and derivation of an equation for inelastic buckling

Abstract

Buckling in columns under central load has always been a complex problem and until now remains unclear. This work consists of two parts. In the first, a theory is presented to solve the indeterminations of Euler's theory, such as: Moment, slope, deflection, moment and shear reactions. In this sense, the column is analyzed as if it were a prismatic member under pure bending, due to the application of a bending moment and consequently the elastic curve of the column is calculated by the traditional method. Using a second hypothesis, it is established that elastic buckling occurs when the compressive and bending energies become equal; and solving it the Euler's Formula for the critical load is obtained exactly. In the second part, and with the indeterminations already resolved, the maximum-strain-energy failure criterion is raised, in the critic section, where the maximum moment occurs, due to the compressive load and subsequent bending. This allows us to obtain an extraordinary equation for inelastic flexural buckling that predicts highly accurate results consistent with laboratory tests. In the stress diagram versus slenderness, we can see the flat zone of the short columns, and the critical slenderness ratio, that in the case of ASTM-A36 structural steel, points directly to one hundred and twenty (120). And, of course, we can calculate the critical stress according to the type of cross-section and buckling axis and direction. An ideal column is considered, i.e., without initial deflection, with the application point of load on the centroid, and without residual stresses. A lineal stress-strain diagram is used until the yield point.