Based on numerical solutions that minimize the total potential energy of trusses subjected to static loads, with specified displacements at selected support nodes and simultaneous fulfillment of the isoperimetric condition on the structure’s volume, several properties of optimally designed structures are revealed. The most significant finding is the relatively frequent occurrence of non-unique global solutions, represented as vectors of cross-sectional areas of members in the stress-based version of topology optimization problem. A key aspect of the presented method is the objective function, derived from Castigliano’s theorem, which minimizes the total potential energy. Unlike traditional approaches that optimize transverse areas of the members, this method uses statically admissible forces in the truss members as design variables. This formulation allows for a free search of solutions, including cases where certain members can disappear in the optimal design. Numerous tests have revealed an interesting property of the objective function, indicating that global solutions are located at the bottom of a long valley in its graph within the space R^r +1, where r denotes the size of the kernel of the equilibrium matrix governing the force balance equations of the nodes of the truss structure.

Computer Assisted Methods in Engineering and Science, 32(1): 77-115, 2025