A homogenized theory for functionally graded Euler–Bernoulli and Timoshenko beams

A new theoretical approach to resolve functionally graded beams is the subject of the present work. In particular, it is shown how the definition of some particular generalized quantities allows to simplify the form of the differential equations governing the response of both Euler–Bernoulli and Timoshenko functionally graded beams. Indeed, they take the same form as the differential equations governing the axial and the bending equilibrium in the Euler–Bernoulli theory. This result is obtained in both the cases of material variation in transversal and axial direction.