Geometrically nonlinear transient analysis through direct and reduced order methods

New challenges demanded by aerospace, automotive, and other engineering fields require the design of sophisticated and lightweight components. These highly flexible structures generally exhibit large displacements/rotations without showing plastic deformations. Thus, dynamic and static analyses that consider geometrical nonlinearities become essential to investigate their structural responses accurately. In particular, the nonlinear transient response of isotropic and composite structures can be analyzed using time integration schemes and reduced-order methods. Undoubtedly, the latter techniques are preferred because direct algorithms become time-consuming, even for simple configurations. Indeed, the stiffness matrix must be computed at each time step since it changes with the deformation. The mode superposition approach represents a practical reduced-order method. According to this technique, some mode shapes are used to transform the nodal dynamic equilibrium equations into a system of equations expressed in generalized (modal) coordinates. Since a limited number of modes is usually considered, the number of modal equations is reduced with respect to nodal coordinates. Then, the nodal response can be calculated through an inverse transformation with a significant decrease in the computational effort compared to the direct integration algorithms. However, the method’s accuracy depends on the number of modes retained in the analysis and, in addition, mode aberrations phenomena, such as crossing, veering, and mode jumping, which may occur for various loadings and material configurations can significantly affect the response. For these reasons, this work aims at evaluating the range of applicability of the mode superposition method in geometrical nonlinear problems. To this end, transient analyses will be performed by either updating or not the mode shapes used to reduce the system of equations. The updating procedure will be carried out by considering various equilibrium states of the structure. The related results will be compared with those obtained by direct simulations to evaluate the effectiveness of the updating strategy. The comparisons will be presented for compact and thin-walled structures of metallic or composite materials subjected to various loading conditions. Low and high-fidelity finite elements developed with the Carrera Unified Formulation will be used to derive the equations of motion; therefore, the effect of the kinematic models will also be investigated.