A nonlocal beam theory for bending, buckling, and vibration of nanobeams

A nonlocal shear deformation beam theory is proposed for bending, buckling, and vibration of nanobeams using the nonlocal differential constitutive relations of Eringen. The theory, which does not require shear correction factor, accounts for both small scale effects and quadratic variation of shear strains and consequently shear stresses through the thickness of the beam. In addition, it has strong similarities with nonlocal Euler–Bernoulli beam theory in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The equations of motion are derived from Hamilton’s principle. Analytical solutions of deflection, buckling load, and natural frequency are presented for a simply supported beam, and the obtained results compare well with those predicted by the nonlocal Timoshenko and Reddy beam theories.