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The theory of vibration of engineering structures belongs to a known chapter of mechanics. Nevertheless, new emerging technologies operating at the smallest scales open a completely new landscape in the field of nanodevices, nanostructured materials as well as biological systems. Nanoscale structures are always involved and characterized by extremely high-frequency vibrations and very small energy, implying quantum effects. This approach, even for simple oscillators, implies both conceptual and mathematical difficulties. It is a matter of fact that the analysis of complex structures, as for example beams and shells, has never been considered in the physical and engineering recent literature in a complete quantum-mechanical context using directly the Schrodinger's equation. The present paper proposes an attempt in this direction, introducing a quantum Euler beam, the QUEB model at the ground state energy. The idea is to shape a new model for flexural structures that is mathematically similar to those used in classical mechanics, but mimicking the peculiarity of the quantum motion. [ABSTRACT FROM AUTHOR]