The present article addresses the study of an adaptive-passive beam structure with a shape-memory alloy based actuator. In order to mitigate adverse dynamic effects resulting from externally induced vibrations, the structure is able to automatically tune its natural frequency to avoid resonance. The adaptive-passive beam configuration is based on an underslung cable-stayed girder concept. Its frequency tuning is achieved by temperature modulation of the shape-memory alloy elements through a closed-loop control process based on a proportional-integral-derivative algorithm. The effectiveness of the proposed control solution is substantiated by numerical simulations and experimental tests on a small-scale prototype. The validated numerical model enables the simulation of the proposed control approach in a real-scale footbridge, subjected to a prescribed pedestrian loading. The results are very encouraging and show that, by activating the shape-memory alloy elements, the system is able to successfully shift its natural frequency and to mitigate the effects of induced vibrations.