Geometrically Nonlinear Dynamic Analysis of Notched Beams Using the Positional Finite Element Method

Resumo

The accurate assessment of structural elements with local geometric discontinuities, such as notches, remains a recurrent challenge in civil engineering, especially under dynamic loading conditions. Notches introduced due to architectural, constructional, or functional requirements act as critical regions of stress concentration, potentially compromising structural integrity and triggering instabilities even in otherwise robust systems. In this context, the use of models capable of accurately capturing geometric nonlinear effects becomes essential, particularly in scenarios involving large displacements.This study presents a numerical analysis of the dynamic response of two-dimensional notched beams using the Positional Finite Element Method (FEM-P), in which the equilibrium equations are formulated in the deformed configuration. Unlike conventional formulations based on nodal displacements, FEM-P adopts nodal positions as primary variables, allowing a more direct treatment of geometric nonlinearity, especially in simulations involving large displacements. Time integration is performed using the implicit Newmark-beta scheme, ensuring numerical stability and good accuracy. Additionally, the adopted constitutive model is the Saint Venant–Kirchhoff hyperelastic formulation, suitable for large elastic deformation regimes.The computational code developed was initially verified through classical benchmark examples widely referenced in the literature on solid dynamics, aiming to ensure the fidelity of the numerical implementation against established solutions. After this verification stage, the model was applied to the analysis of notched beams with various geometric configurations, including variations in depth, shape, and location of the notches, in order to investigate their effects on internal stress redistribution and overall dynamic response.The results indicate that the presence of notches significantly contributes to displacement amplification, redistribution of internal forces, and early onset of geometric instabilities. The proposed approach highlights the importance of advanced modeling techniques for structures with geometric discontinuities, especially in scenarios where the interaction between nonlinear kinematics and dynamic excitations may compromise structural integrity.