Axially-loaded structures play an integral role in engineering design. Some of these structures exhibit a nonlinear response behavior under harmonic loading. Methods aimed at eliminating these behaviors are often employed in design of such structures. Our hypothesis is that the nonlinear dynamics can be used to optimize desired system response characteristics. In this paper, the dynamic response of a buckled beams under harmonic excitation is considered. An optimization approach is formulated that achieves maximum amplitude, periodic, and stable responses of the beam systems. Case studies are presented that demonstrate the utility of this optimization approach to exploit the nonlinear dynamics to achieve desired responses.