Frequency Analysis of Bi-directional Porous Functionally Graded Beams with Variable cross section on Elastic Foundation using Reddy Third Order Shear Deformation Theory

In this paper, frequency analysis of Bi-directional Porous functionally graded beams with variable cross section which are resting on elastic foundation based on Reddy third order shear deformation theory is studied. Mechanical property gradients defined in accordance with two models of exponential and volume fraction power law. Governing equation which is obtained with the aid of third order shear deformation theory and by considering elastic foundation effect in conjunction with Hamilton’s principle. Due to intrinsic closed form solution of equations, differential equations solved with using Generalized Differential Quadrature Method by considering various end conditions. In order to validate the results comparisons are made with solutions which are available for other papers. This study reveals that the difference between the results of this paper and the results of others is negligible. Eventually the effects of geometrical parameters, power and exponential law indexes and elastic foundation coefficients on natural frequencies of Bi-directional FGM beams is studied. The results reveal that non dimensional frequencies increase with the rise of elastic foundation coefficients and the soar of porosities and material gradients in two directions causes a sharp decrease in non dimensional frequencies.The results of this study can be used in optimal design, vibration control and detection of failure structure of functional graded structures