Free vibration analysis of Bi-Directional Functionally Graded Beams using a simple and efficient finite element model

This research explores a new finite element model for the free vibration analysis of bi-directional functionally graded (BDFG) beams. The model is based on an efficient higher -order shear deformation beam theory that incorporates a trigonometric warping function for both transverse shear deformation and stress to guarantee traction -free boundary conditions without the necessity of shear correction factors. The proposed two -node beam element has three degrees of freedom per node, and the inter -element continuity is retained using both C1 and C0 continuities for kinematics variables. In addition, the mechanical properties of the (BDFG) beam vary gradually and smoothly in both the in -plane and out -of -plane beam's directions according to an exponential power -law distribution. The highly elevated performance of the developed model is shown by comparing it to conceptual frameworks and solution procedures. Detailed numerical investigations are also conducted to examine the impact of boundary conditions, the bi-directional gradient indices, and the slenderness ratio on the free vibration response of BDFG beams. The suggested finite element beam model is an excellent potential tool for the design and the mechanical behavior estimation of BDFG structures.