On the applicability of Southwell’s method to the determination of the critical force of elastic columns of variable cross sections

Abstract

AbstractBrilliant in its simplicity, Southwell’s method of experimentally determining the critical force on the basis of the so-called Southwell diagram has been of interest to structural stability specialists since its presentation in 1932. In its original version, the correctness of the Southwell method was proved for compressed bars with a constant cross section and showing the loss of stability only within the elastic range while maintaining the constant value of Young's modulus. In numerous works on the method, proposals for its modification have emerged ensuring the extension of the scope of applications with problems other than the classic buckling of a compressed bar. There has also been an attempt to justify the correctness of Southwell’s method in the case of a bar with a variable cross section. This paper presents a strict solution of the differential equation of bar stability with a parabolic variable moment of inertia of the cross section and a parabolic form of initial geometric imperfections. The solution arrived at with the use of the Mathematica™ system was the basis for constructing the Southwell diagram and justifying the correctness of the method for this particular case of a bar. To confirm the correctness of Southwell’s method also the numerical approach exploiting FEM was adopted. Two different cases of column’s shape variabilities and different boundary conditions were considered.