A strain gradient functionally graded Euler–Bernoulli beam formulation

A size-dependent functionally graded Euler–Bernoulli beam model is developed based on the strain gradient theory, a non-classical theory capable of capturing the size-effect in micro-scaled structures. The governing equation and both classical and non-classical boundary conditions are obtained using variational approach. To develop the new model, the previously used simplifying assumption which considered the length scale parameter to be constant through the thickness is avoided in this work. As a consequence, equivalent length scale parameters are introduced for functionally graded microbeams as functions of the constituents’ length scale parameters. Moreover, a generally valid closed-form solution is derived for static deflection of the new model. As case studies, the static and free-vibration of the new model are investigated for FG simply supported microbeams in which the properties are varying through the thickness according to a power law and the results of the new model are compared to those of the modified couple stress and the classical continuum theories, noted that the two latter theories are special cases of the strain gradient theory utilized in this paper.