Conjugate-beam analogy for inflexed laminates

A conjugate beam analogy is here proposed to evaluate the shear coupling in three-layered laminates formed by two external elastic plies bonded by a compliant interlayer. Based on the formal analogy of the governing equations, the method allows the evaluation of the shear coupling in the external plies by studying the response of a monolithic conjugate beam under a tensile constant axial force and a transversal load with the same shape of that acting on the parent laminate. Remarkably, the boundary conditions for the conjugate beam depend not only upon the external constraints of the laminate, but also upon possible internal constraints that limit the relative sliding of the plies at the end points. The axial force at a point of each ply corresponds to the displacement of the elastic curve of the conjugate beam at the same point, while the flux of shear stress from the interlayer is related to the corresponding slope. The static state of the laminate can finally be determined by analyzing a simple structure under an effective transversal load, which accounts for the contribution of the shear coupling. Since, in most cases, the deformation of the conjugate beam can be intuitively guessed, the proposed analogy permits a synthetic view of the interaction between the various layers of the laminate. This can be very useful especially in preliminary design.