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The structural optimization problem of choosing the profile of each member belonging to a framed structure in order to minimize its weight while satisfying stress, displacement, stability, and other applicable constraints is often complicated by the requirement of considering non-linear structural behavior. The problem is further complicated if the members are to be chosen from a discrete set of commercially available sizes, which is frequently the case. The solution of the commonly occurring case where the cardinality of the set of distinct values of the design variables (for instance, cross-sectional areas) should be smaller than a given value is still an open area for investigation. In this paper a genetic algorithm encoding, previously proposed in the literature, is used to directly enforce such cardinality constraint for design optimization of geometrically nonlinear truss structures. The impact of performing a more rigorous (geometrically nonlinear) structural analysis, on both safety and cost of the optimized structure is also pointed out.