Exact solution of Eringen's nonlocal integral model for bending of Euler–Bernoulli and Timoshenko beams

Despite its popularity, differential form of Eringen nonlocal model leads to some inconsistencies that have been demonstrated recently for the cantilever beams by showing the differences between the integral and differential forms of the nonlocal equation, which indicates the importance and necessity of using the original integral model. With this motivation, this paper aims to derive the closed-form analytical solutions of original integral model for static bending of Euler Bernoulli and Timoshenko beams, in a simple manner, for different loading and boundary conditions. For this purpose, the Fredholm type integral governing equations are transformed to Volterra integral equations of the second kind, and Laplace transformation is applied to the corresponding equations. The analytical expressions of the beam deflections which are obtained through the utilization of the proposed solution technique are validated against to those of other studies existing in literature. Furthermore, for all boundary and loading conditions, in contrast to the differential form, it is clearly established that the integral model predicts the softening effect of the nonlocal parameter as expected. In case of Timoshenko beam theory, an additional term that includes the nonlocal parameter is introduced. This extra term is related to the shear rigidity of the beam indicating that the nonlocal effect manifests itself via not only bending, but also shear deformation.