Free vibrational characteristics of various imperfect FG beam via a novel integral Timoshenko’s theory

In the current paper, a new first-order Timoshenko’s theory (N-FSDBT) based on the indefinite integral is proposed to examine the vibrational behavior of power-law (P-FG), sigmoid (S-FG), and exponential (E-FG) functionally graded beams. The developed formulation takes into consideration the effect of the transverse shear deformation. During the manufacturing process, faults, such as the presence of micro voids or porosities in FGMs, structural imperfections may arise in the material. For this purpose, this research has focused on the modification of the mixture rules including the porosity volume fraction. To illustrate the porosity variation throughout the body of the structures, the uniform, nonuniform and mass-density porosity distribution are considered in the present examination. Mathematical approach is used to simplify the FG beam governing equation based on classical beam theory (CBT) and new-FSDBT frame works for free vibration. The resultant equations of motion are solved using the Navier’s technique, for simply supported FG beams. To validate the results, a parameter study was presented to analyze the impact of the types of volume fraction (FG beam-type), the effects of porosity distribution, material exponent parameter, slenderness ratio, and porosity index on the dynamic behavior of imperfect FG beams.