Design problems in structural engineering are often modeled as differential equations. These problems are posed as initial or boundary value problems with several possible variations in structural designs. In this paper, we have derived a mathematical model that represents different structures of beam-columns by varying axial load with or without internal forces including bending rigidity. We have also developed a novel solver, the LeNN-NM algorithm, which consists of weighted Legendre polynomials, and a single path following optimizer, the Nelder–Mead algorithm. To (...) evaluate the performance of our solver, we have considered three design problems representing beam-columns. The values of performance indicators, MAD, TIC, NSE, and ENSE, are calculated for a hundred simulations. The outcome of our statistical analysis points to the superiority of the LeNN-NM algorithm. Graphical illustrations are presented to further elaborate on our claims. (shrink)