Buckling load of heavy columns with variable cross section and flexible end restraints

Effective and economical design of heavy columns, taking into account their own self weight is a common problem in engineering. The degree of restraint provided at the ends of the column should also be taken into account and it is addressed by adding rotational and/or translational springs to simulate several levels of fixity. In this work results are given for five different combinations of boundary conditions at the two ends, and for several restraint rigidities at the supports. An approximate solution using Rayleigh's method is presented and compared to numerical finite element results and to the exact solution previously derived for general cases of variable cross section columns with variable axial force.