Stability of bars with variable rigidity compressed by a distributed force

A variable cross-section bar is considered. The bar is not uniform in length. The bar is compressed by a variable longitudinal force distributed along its axis. The stability loss of the straight-line shape of the bar's equilibrium is discussed when a curved shape is also possible. The critical combination between rigidity and the longitudinal force is a result of using an integral representation for the solution to the original stability equation with variable coefficients by the aid of the solution to a similar equation with constant coefficients. The integral representation contains the Green function of the original equation specified by the method of perturbations. The numerical results obtained by the derived formulas are compared with the known exact solutions to the stability equations for various particular cases.