On the stability response of a restrained beam using nonlocal elasticity and nonlocal foundation

Abstract This study considers the buckling problem of beams resting on a nonlocal foundation. A nonlocal foundation is modeled with two different spring structures extending under and along the length of the beam and a shear layer between these springs. This study investigates the buckling of a beam on the nonlocal foundation using a method based on the Fourier sine series and Stokes’ transformation. Thus, an analytical solution of a beam on a nonlocal foundation modeled by transverse springs at both boundaries is presented. This solution can solve arbitrary, i.e., rigid and non-rigid, boundaries since it includes the stiffnesses of the transverse springs positioned as bearings at both ends of the beam resting on the nonlocal foundation. The stiffnesses of the springs at the ends of the beam can be set arbitrarily and the effects of numerous boundary conditions, whether rigid or deformable, can be studied. After the derivation of the solution formulation, the effects of different parameters on the critical buckling loads of beams resting on nonlocal foundations are investigated through a series of numerical examples.