[en] This paper is concerned with the derivation of exact solutions for the static responses of simply supported nanobeams integrated with a flexoelectric layer acting as the distributed nano actuator. Considering both the direct and the converse flexoelectric effects, the governing equations and the associated boundary conditions of the overall beams are derived to obtain exact solutions for the displacements and the electric potential in the flexoelectric layer and the substrate beam. Due to the converse flexoelectric effect, the active flexoelectric layer significantly counteracts the deformations of the substrate beams caused by the applied mechanical load, resulting in the coupling of bending and stretching deformations in the substrate beams. For particular values of the length and thickness of the substrate beam and the applied voltage in the flexoelectric layer, the deflection of the substrate beam due to the converse flexoelectric effect increases with increasing thickness of the flexoelectric layer up to a certain value of the latter. Beyond this value, any further increase in the thickness of the flexoelectric layer causes a decrease in its ability to actuate the substrate beam. The electric potential varies linearly across the thickness of the flexoelectric layer. The benchmark results presented here may be useful for verifying further research and the present study suggests that the flexoelectric layer may be effectively exploited for advanced applications as a smart nano actuator. (paper)