Postbuckling and free vibration of multilayer imperfect nanobeams under a pre-stress load

This paper investigates the postbuckling and free vibration response of geometrically imperfect multilayer nanobeams. The beam is assumed to be subjected to a pre-stress compressive load due to the manufacturing and its ends are kept at a fixed distance in space. The small-size effect is modeled according to the nonlocal elasticity differential model of Eringen within the nonlinear Bernoulli-Euler beam theory. The constitutive equations relating the stress resultants to the crosssection stiffness constants for a nonlocal multilayer beam are developed. The governing nonlinear equation of motion is derived and then manipulated to be given in terms of only the lateral displacement. The static problem is solved for the buckling load and the postbuckling deflection in terms of three parameters: Imperfection amplitude, size, and lamination. A closed-form solution for the buckling load in terms of all of the beam parameters is developed. With the presence of imperfection and size effects, it has been shown that the buckling load can be either less or greater than the Euler buckling load. Moreover, the free vibration in the pre and postbuckling domains are investigated for the first five modes. Numerical results show that the effects of imperfection, the nonlocal parameter, and layup on buckling loads and natural frequencies of the nanobeams are significant.