Element free Galerkin method of composite beams with partial interaction

Abstract

Composite beam with partial interaction behaviour has ignited many studies, not just on its mechanics but also on solutions of its one-dimensional partial differential equation. Inadequate solution by available analytical methods for this high order differential equation has demanded for numerical approach and therefore Element Free Galerkin (EFG) method is applied for the first time in this present work. The work consists of three parts; first is the formulation of Galerkin weak form and assemblage of the EFG discrete equilibrium equation. One-dimensional formulation of the weak form is performed by adopting the variational approach and the discrete equation, which is in matrix form and written using the Matlab programming code. Subsequently in second part, the EFG formulation is developed for both the slip and uplift models, where the former adopted equal curvature deflection assumption while the latter considered the unequal curvature. The proposed EFG formulation gives comparable results in both models, after been validated by established analytical solutions, thus signify its application in partial interaction problems. The third part provides numerical tests result on EFG numerical parameters such as size of support domain, polynomial basis and quadrature points with seven different types of weight functions for this composite beams behaviour. Conclusively, Cubic Spline and Quartic Spline weight functions yield better accuracy for the EFG formulation results, compares to other weight functions. The capability of the EFG formulation was also studied in terms of its application on free vibration problem and various composite beam cross-sections. Results from the numerical tests deduced the demand for optimised parameters value as the parameters are highly reliant on user-defined value. Additionally, the research supports the need for more efficient EFG code’s algorithm, stiffness matrix, shape function formulation and background integration methods, in approximating the higher order differential equation which refers to dynamics analysis.