Shape optimization of beam due to lateral buckling problem

Based on a non-linear mathematical model of lateral buckling of a slender beam with a narrow rectangular cross section, the variational formulation of the two-parametric optimization problem is given in the dimensionless form. An optimal shape is obtained by solving the variational problem using the Rayleigh–Ritz method with the orthogonal system of trigonometric functions. By a partial solution of the Euler–Lagrange differential equation of the variational problem, a proof is given that in the case of the optimal shape, a maximal reference stress according to the total strain energy theory is constant along the beam. An example of extrapolation of the two-parametric optimization problem solution is represented.