4. Multiscale Nonlinear Analysis of Beam Structures by Means of the Carrera Unified Formulation

This chapter addresses a multi-scale analysis of beam structures using the Carrera Unified Formulation (CUF). Under the framework of the \(\text {FE}^{2}\) method, the analysis is divided into a macroscopic/structural problem and a microscopic/material problem. At the macroscopic level, several higher-order refined beam elements can be easily implemented via CUF by deriving a fundamental nucleus that is independent of the approximation order over the thickness and the number of nodes per element (they are free parameters of the formulation). The unknown macroscopic constitutive law is obtained by numerical homogenization of a Representative Volume Element (RVE) at the microscopic level. Vice versa, the microscopic deformation gradient is calculated from the macroscopic model. Information is passed between the two scales in a \(\text {FE}^{2}\) sense. The resulting nonlinear problem is solved through the Asymptotic Numerical Method (ANM) that is more reliable and less time consuming when compared to classical iterative methods. The developed models are used as a first attempt to investigate the microstructure effect on the macrostructure geometrically nonlinear response. Results are compared regarding accuracy and computational costs towards full FEM solutions demonstrating the robustness and efficiency of the proposed approach.