  We present the analytical formulation and the finite element solution of a
fractional-order nonlocal continuum model of a Euler-Bernoulli beam. Employing
consistent definitions for the fractional-order kinematic relations, the
governing equations and the associated boundary conditions are derived based on
variational principles. Remarkably, the fractional-order nonlocal model gives
rise to a self-adjoint and positive-definite system accepting a unique
solution. Further, owing to the difficulty in obtaining analytical solutions to
this boundary value problem, a finite element model for the fractional-order
governing equations is presented. Following a thorough validation with
benchmark problems, the fractional finite element model (f-FEM) is used to
study the nonlocal response of a Euler-Bernoulli beam subjected to various
loading and boundary conditions. The fractional-order positive definite system
will be used here to address some paradoxical results obtained for nonlocal
beams through classical integral approaches to nonlocal elasticity. Although
presented in the context of a 1D Euler-Bernoulli beam, the f-FEM formulation is
very general and could be extended to the solution of any general
fractional-order boundary value problem.
