In this work, the dependence of the bending rigidity and the free vibrations in graphene nano-structures on the shape geometry and the boundary conditions has been investigated. For graphene-based square, rectangular, parallelogram and circular nanostructures, finite elements analysis allows the estimation of the frequency of the first vibration mode. Therefore, it was possible to evaluate the change of the bending rigidity upon varying boundary conditions. In addition, we calculate by finite element methods the first mode of vibration of state-of-the-art nanoelectromechanical devices, whose outer contour cannot be reduced to elementary geometrical shapes.