On the characteristics of optimum beams with optimum length surrounded by a Winkler’s medium

In this paper an analytical approach is used to optimize a beam surrounded by a Winkler’s medium and laterally loaded by a force at the top. The optimization regards both the distribution of the mass along the beam and the beam length in order to minimize the top displacement. Therefore, after having defined a transversality condition, we implement an algorithm to optimize the length of optimum beams. After having achieved the dimensionless extremals of optimum beams with optimum length, we show that the found solutions describe a central field of moment extremals defined along the beam and with the origin at the top. Having achieved the Jacobi’s condition and the strengthened Legendre’s condition for an extremal of optimum beam length, a sufficient condition for a weak minimum along the beam is also achieved. Besides, a given set of cross-sections whose moment of inertia divided by a dimensioned constant is obtained by raising their area to the same exponent. The optimized distribution of the dimensionless cross-sectional area as well as the dimensionless rigidity of the Winkler’s medium are intrinsic properties of all optimum beams with optimum length whose cross-sections belong to the same set.