Warping-included mixed FE approach of beating characteristics in functionally graded graphene platelet-reinforced composite spatially curved beams under harmonic excitation force

This study is the first in the literature to investigate the beating characteristics of functionally graded graphene platelet reinforced composite (FG-GPLRC) spatially curved (helical) beams using a warping-included mixed finite element (W-MFE) formulation. Integrating GPLs into the matrix material significantly increases the strength and load-carrying capacity of GPLRC structures. This addition also allows the tailoring of properties such as stiffness and tensile strength within the composite structure through the FG dispersion of GPLs. In this study, the GPLRC helical beams are modeled with uniform and nonuniform FG gradation patterns through the thickness. The beam is subjected to a uniformly distributed dynamic load characterized by a half-wave rectified sine function. The forced vibration analysis is carried out using the Newmark time integration scheme. A two-noded curved element is utilized with twelve field variables at each node, three displacements, three cross-sectional rotations, three forces, and three moments expressed in the Frenet coordinate frame. Satisfactory results are obtained for the warping-included natural frequency, normal/shear stresses, displacements, and reactional forces of an FG-GPLRC helical beam with lesser degrees of freedom compared to the three-dimensional behavior of brick finite elements. Through the examples, the effect of the distribution patterns and weight fractions of GPL nanofillers, and the pitch angle of the helical beam on the dynamic behavior of the FG-GPLRC semi-circular helical beam under half-rectified sinusoidal dynamic load are studied in detail. By increasing the pitch angle, the oscillation magnitude of displacements and normal stress distributions of the helical beam decreases for non-uniform distribution patterns. The distribution pattern with the GPL-rich mid-part of the cross-section is more affected by the variation of the pitch angle compared to the case where the top and bottom of the cross-section are GPL-rich.