Research On Mechanical Problems Of Functionally Graded Beams Resting On The Elastic Foundation

With the rapid development of economic construction,different styles of different types of structures appear.This makes the application of medium thick beams in engineering more and more widely.However,the traditional beam theory can not be applied to the research and application of medium thick beams.Therefore,the research and analysis of medium thick beams become a hot issue.For the solution of the basic equations of beam structure mechanics problems,the analytical solution is only limited to the condition of simply supported boundary conditions,while other supporting conditions need to be solved by numerical method.At present,numerical methods for higher-order shear deformation beams are rare.In this paper,the differential quadrature method is used to solve the mechanical problems of functionally graded beams on elastic foundations.The influences of different boundary conditions,higher-order beam theory and Winkler foundation parameters on the mechanical behavior of beams are emphatically analyzed.It is assumed that the functionally graded material changes along the thickness direction in a power function,and the statics and dynamics characteristics of the high order beam on the elastic foundation are analyzed.Firstly,based on the theory of highorder shear deformed beams and the principle of virtual work,the equilibrium equation of bending of functionally graded beams is derived.Differential quadrature method is used to discretize the differential equation of beam bending problem.Then the influence of different span-height ratio,material gradient factor,elastic foundation parameters and different boundary conditions on the displacement and stress of functionally graded beams subjected to uniform load is analyzed by MATLAB programming.Then,the buckling problem of high-order shear beams is studied.The analytical solution of the buckling equation with simply supported boundary conditions is solved.In addition,the numerical solution obtained by differential quadrature method is compared with the analytical solution of the literature,which verifies the accuracy and applicability of differential quadrature method for solving functionally graded high order shear beams,and discusses the effects of various parameters on the buckling problem of beams.Finally,the dynamic problems of functionally graded beams on elastic foundations are studied.Based on the theory of high-order shear deformation and Hamilton's principle,the dynamic equations of beams are established.The derived differential equations are discretized into linear algebraic equations by DQM method.The numerical solutions of beam vibration problems with various parameters are obtained.The effects of different elastic foundation conditions,different slenderness ratios and functional gradient factors on the vibration characteristics of beams are summarized.