A Hybrid Approach to Bending Analysis of Axially Graded Tapered Timoshenko Beams Using Analytical and FEM with a Wind Turbine Blade as a Case Study

This study investigates the bending behavior of axially functionally graded material (AFGM) tapered beams using a combination of analytical and numerical methods based on Timoshenko beam theory. The analytical approach accounts for both bending and shear deformations, providing a closed-form solution for deflection. To validate the analytical results, finite element method (FEM) simulations are conducted in MATLAB using Timoshenko beam elements. Additionally, FEM is performed in ANSYS Workbench, where a layer-by-layer material gradation technique is applied to accurately represent the axial variation in material properties. The study explores the effects of key parameters, including the grading parameter, taper ratio, and aspect ratio, on the bending response of AFGM beams. The findings indicate that increasing the grading parameter significantly enhances beam stiffness, reducing deflection by approximately 39.35% for both tapered and prismatic beams. Likewise, increasing the taper ratio leads to a notable decrease in deflection, with a reduction of 57.97% observed for tapered beams. In contrast, variations in the aspect ratio have a relatively minor effect on stiffness compared to material gradation and tapering. A case study is conducted on a wind turbine blade modeled with a NACA 4418 airfoil to demonstrate the practical application of AFGMs in structural optimization. The results show that material grading significantly reduces deflection under bending loads, with a 30.3% reduction observed when the grading parameter is set to 0.5 compared to the non-graded case. The numerical results from FEM simulations closely match the analytical solution, with maximum deviations of 6.67% in MATLAB and 5.09% in ANSYS, confirming the accuracy and reliability of the proposed approach.