Closed form solutions of bending Bi-directional functionally graded tapered beams using Euler and Timoshenko theories
Omar K. Omar, Mohammad Alshawabkeh, H Alkhatib
Abstract
• Closed-form solutions for graded tapered beams using euler and timoshenko theories. • Examined how material gradation and tapering affect beam stiffness. • Higher gradient indices reduce deflection by up to 39%, and taper ratios by 58%. • Material grading shifts the neutral axis, enhancing stiffness. • Provides a practical framework for understanding advanced beam mechanics. This study focuses on developing closed-form analytical solutions for the bending analysis of bi-directional functionally graded tapered beams, a technically challenging area with limited existing solutions. Employing both Euler-Bernoulli and Timoshenko beam theories, the research provides a step-by-step derivation methodology, validated through detailed formulations and computational tools such as Maple software. The analysis examines the effects of material gradation, geometric tapering, and shear deformation on beam behavior, offering comprehensive displacement and stress evaluations. Key findings reveal that Timoshenko theory yields more accurate results for shorter beams due to shear deformation, while variations in gradient indices, taper ratios, and axial parameters significantly influence deflection. For instance, increasing the material gradient index reduces deflection by up to 39%, and higher taper ratios decrease deflection by 58% in both theories. This work bridges theoretical and practical understanding, providing a valuable framework for educators and students to explore advanced beam mechanics and apply closed-form solutions in engineering education.