Natural frequency of Sandwich Beam Structures with Two Dimensional Functionally Graded Porous Layers Based on Novel Formulations

This study presents an analytical solution for free vibration analysis of two-dimensional functionally graded (2D-FG) porous sandwich beams. The equations of motion for the beam were derived using Hamilton's principle, and then the Galerkin method was employed to solve the equations. The material properties of the sandwich beams vary with the thickness and length of each layer according to the power-law function. The mechanical properties gradually changed from aluminum to alumina as the metal and ceramic, respectively. The vibration analysis was investigated based on two new higher-order shear deformation beam theories (NHSDBTs). These two new theories do not need any shear correction factor and have fewer unknown variables than other higher order shear beam theories.  The obtained natural frequencies for the three types of beams were compared with the results of the Timoshenko, first-order , and parabolic shear deformation beam theories. In addition, the effects of porosity, L/h, and FG power indexes along the thickness and length on the non-dimensional frequency of three special types of beams are presented and discussed. Furthermore, the mode shapes of the beam are depicted for various FG power indexes based on these new theories. By comparing the results of the two proposed theories with those of existing studies, the accuracy of the proposed theories was validated. Power-law indexes shifted the node point to the left and resonance will be accrued sooner than the non-FGM beam.