The increasingly common design of reinforced concrete (RC) structures with soft stories in seismically active areas requires a more detailed calculation approach and the introduction of special seismic solutions. A soft story refers to an open ground floor space, while the upper floors are masonry. Regarding earthquake effects, the most unfavorable and at the same time the most common variant is the construction of "soft stories" on the ground floor of buildings. The paper presents the principle of nonlinear static analysis, based on the application of the N2 method as a representative nonlinear static method supported by Eurocode 8. A nonlinear static analysis model of two RC structures with a soft story is shown, where the impact of masonry infill is considered. In one model, metal-yielding dampers, specifically the DC90 SYSTEM, are used on the ground floor as a special type of passive seismic protection. The results of the nonlinear static analysis are then presented in the form of a pushover curve, displacement diagrams, relative story displacements, and the distribution scheme of plastic hinges, all with the aim of evaluating the efficiency of metal-yielding dampers. The general conclusion that emerges from the presented results of the numerical investigation is that the DC90 SYSTEM has a significant impact on improving the seismic performance of RC frame structures with a soft story. By applying this passive seismic protection system, inter-story drifts on the ground floor are reduced, the overall displacement of the building is decreased, and the soft story mechanism transforms into a more favorable beam failure mechanism. Considering the market price of this type of damper, it can be concluded that the initial investment in seismic protection is significantly lower than the cost of repairing a damaged building after a moderate earthquake, while the contribution to preventing the collapse of the structure during strong earthquakes and preserving human lives is invaluable.