A refined finite element method for bending of smart functionally graded plates

This study presents a finite element (FE) formulation based on four-variable refined plate theory for a bending analysis of functionally graded (FG) plates integrated with a piezoelectric fiber reinforced composite (PFRC) actuator under electrical and mechanical loadings. The four-variable refined plate theory accounts for a parabolic variation of the transverse shear stresses across the plate thickness, which satisfies zero traction conditions on the plate free surfaces. The principle of minimum potential energy is used to derive the weak form of governing equations, and a 4-node nonconforming rectangular plate element with eight degrees of freedom (DoFs) per node is introduced for discretizing the domain of the bending variables. Some benchmark problems are also solved using the developed MATLAB code. A comparison of the results of the obtained displacements and stresses with the exact and other numerical solutions shows good agreement, thereby proving the simplicity and efficiency of the present finite element (FE) solutions. In addition, the effects of several parameters on the results, including the thickness ratio, Young's modulus ratio, the types of boundary conditions, and the distribution and amount of loadings, are investigated.