The energy method in problems of buckling of bars with quantifier elimination

The classical energy method for the approximate determination of critical buckling loads of bars is revisited. This method is based on the stability condition of the bar and on the appropriate selection of an approximation to the deflection of the bar. Moreover, it is frequently related to the Rayleigh quotient or to the Timoshenko quotient for the determination of the critical buckling load. Here we will use again the energy method for the determination of critical buckling loads of bars but now on the basis of a new computational approach. This new approach consists of using the modern computational method of quantifier elimination efficiently implemented in the computer algebra system Mathematica instead of partial differentiations when we use the stability condition of the bar or essentially equivalently when we minimize the Rayleigh quotient or the Timoshenko quotient. This approach, which avoids partial differentiations, is also more rigorous than the classical approach based on partial derivatives because it does not require the use of the conditions for a minimum based on second partial derivatives, which are generally ignored in practice. Moreover, it is very simple to use inside the powerful computational environment offered by Mathematica. The present approach is illustrated in several buckling problems of bars including parametric buckling problems. Buckling problems of bars with two internal unilateral constraints, where the classical energy method is difficult to apply, are also studied. Even in this rather difficult application the critical buckling load is directly determined with a sufficient accuracy.