MACRO-ELEMENT MODELLING

The macro-element (ME) technique is the methodology for the simplified stress analysis of bonded, bolted and hybrid (bonded/bolted) joints. Based on simplified hypotheses, the actual joint is meshed in particular elements. The fasteners are modelled by 2-DDL or 6-DDL fastener elements, while the bonded overlap are modelled by dedicated 4-nodes Bonded-bars (BBa) and Bonded-beams (BBe) espacially formulated. Only one BBa or BBe, depending on the chosen kinematics, is sufficient to be representative for an entire bonded overlap in the frame of a linear elastic analysis. When the geometrical or material properties of the adherends or the adhesive layer vary along the overlap a mesh is necessary along the overlap length direction only. The ME technique is inspired by the FE method and differs in the sense that the interpolation functions are not assumed. Indeed, they take the shape of solutions of the governing ordinary differential equations (ODEs) system, coming from the constitutive equations of the adhesive and adherends and from the local equilibrium equations, related to the simplifying hypotheses. The main work is thus the formulation of the elementary stiffness matrix of the ME. Once the stiffness matrix of the complete structure is assembled from the elementary matrices and the boundary conditions are applied, the minimization of the potential energy provides the solution, in terms of adhesive stress distributions along the overlap, internal forces and displacements in the adherends. The ME technique can be regarded as mathematical procedure allowing for the resolution of the system of ODE, under a less restricted application field of simplifying hypotheses, in terms of geometry, material behaviours, kinematics, boundary conditions and loads.

Presentation of ME technique

Presentation of ME technique