Buckling of Composite Nonlocal Or Gradient Connected Beams

The buckling of an axially loaded elastic composite beam with a nonlocal core or a nonlocal connection system is studied in this paper. The composite beam or the sandwich beam is composed of two Euler–Bernoulli beams with a nonlocal elastic interaction. This nonlocal interaction is physically based on the Reissner's model based on three-parameters' interaction function. The energy equations are first presented, and the differential equations are rigorously obtained from a variational principle. We show that the connection model can be expressed in an integral format, therefore, inducing the nonlocal character of this beam elastic interaction model. Furthermore, the variational format of this nonlocal composite model is given, leading to meaningful natural and higher-order boundary conditions. The system of these differential equations can be reduced to a single 10th-order differential equation. We present an exact method to solve this stability problem, based on Ferrari or Cardano's method. The solution can be fully simplified in case of specific boundary conditions with symmetrical considerations. The stability domain is analytically characterized in the loading space for the pinned–pinned boundary conditions. The correspondence between the buckling of the nonlocal composite column and the shear composite column is discussed. Finally, it is shown that the Timoshenko beam model is a nonlocal integral model.