Analysis Of Static And Dynamic Mechanical Behavior Of Functionally Graded Beams By Differential Quadrature Method

To study the structural safety and stability of functionally graded material beams,this paper investigates the static and dynamic mechanical problems of functionally graded beams by differential quadrature method using two layouts: isometric and isometric Chebyshev.(1)The free vibration of functionally graded material beams along the axial direction is studied.Based on the Euler-Bernoulli beam theory,the mathematical model of the axial functional gradient beam is established,the differential control equations of the functional gradient beam are derived,and two types of distributions,the isometric series,and Chebyshev are used.The numerical calculation results show that the calculated values of the differential product method in this paper are consistent with those of the existing literature,proving the feasibility and accuracy of the differential product method for analyzing the free vibration of functional gradient beams are demonstrated.Then,the influence of the gradient parameters on the first three orders of inherent frequencies of the EulerBernoulli simply supported beam with functional gradient is investigated.(2)The free vibration problem of a bi-directional functionally graded rectangular section beam is investigated.Based on the Euler-Bernoulli beam theory,a mathematical model of a two-way functionally graded rectangular section beam is established,and the controlling equations of the two-way functionally graded rectangular section beam are derived and solved by the differential product method using both the isometric and Chebyshev distribution methods.The results of the calculations show that the values calculated by the differential product method are in full agreement with the existing literature results,which verifies the feasibility and accuracy of the differential product method for the study of free vibration problems of bi-directional functional gradient beams.Then,the effect of material gradient parameters on the first three orders of inherent frequencies of Euler-Bernoulli simply supported beams with bi-directional functionally graded is analyzed.(3)The static and dynamic mechanical problems of functionally graded beams along the thickness direction under different boundary conditions are investigated.A mathematical model of a functional gradient rectangular section beam is established based on the theory of EulerBernoulli beams,and the differential control equations for a functionally graded rectangular section beam with material properties varying along the thickness are derived.Both isometric and Chebyshev layouts are used,and the differential product method is applied to solve the equations.The results show that the calculated values of this method are in full agreement with the results of existing literature and the analytical solutions,which verifies the feasibility and accuracy of the differential product method for studying the static and dynamic mechanics of functionally graded beams.The effect of material gradient parameters on Euler-Bernoulli beams’ deflection,buckling load,and inherent frequency with functionally graded is analyzed.(4)Based on Levinson’s higher-order shear deformation theory,the free vibration problem of a beam with a functional gradient rectangular section along the thickness direction under different boundary conditions is investigated.According to the basic principle of the differential product method,the eigenvalue problem of the differential equation of motion of the beam is transformed into a common generalized algebraic eigenvalue problem,which is then solved by the orthogonal triangular decomposition(QR)method to obtain the inherent frequency of the beam and its corresponding vibration pattern in one go.The results show that the calculated values of the differential product method agree with the literature data.At the same time,as the length-to-height ratio of the beam increases,the error between the numerical solution and the literature solution gradually decreases.The differential quadrature method can efficiently and accurately obtain the buckling load,and vibration frequency of functionally graded beams,and play a certain theoretical guidance and reference role for engineering practice by analyzing and studying the harmonic vibration frequency and material gradient parameters.