Vibration of four-phase bidirectional functionally graded beams based on trigonometric enriched beam element

Improving the convergence rate in finite element formulation plays an important role in studying the behavior of structures. This paper presents an efficient beam element to investigate the free vibration of bidirectional functionally graded beam. The beam is composed of four materials whose properties vary along both the length and thickness directions according to the power function, and these properties are evaluated by Voigt model. The equations of motion are derived using Hamilton’s principle within the framework of the higher-order shear deformation beam theory. A two-node beam element is formulated by enriching the conventional Lagrange and Hermite interpolations with trigonometric functions, leading to rapid convergence. The finite element formulation has been validated through comparison with previously published results, and showing good agreement. The enriched beam element is employed to compute the natural frequencies of bidirectional functionally graded (BFG) beams under different boundary conditions. The influence of the grading indices, slenderness ratio and boundary conditions on the natural frequency is examined in detail and highlighted