Dynamic Buckling Of Functionally Graded Material Beam Under Axial Load

Functionally graded materials have broad application prospects in aerospace,shipbuilding,automobile,energy and biomedical industries.Many domestic and foreign scholars have conducted in-depth analysis and exploration of functionally graded materials.As one of the most common structural forms,beam has attracted much attention for the study of dynamic buckling of beams.However,the results are different because researchers have different methods and perspectives to deal with problems.Based on this,in this paper,the dynamic buckling of functionally graded material beam is studied by using the Ritz method and de Moivre's formula,several jobs are investigated in the following :1.The research status of the dynamic buckling of functionally graded materials at home and abroad and the research status of dynamic buckling of beams are introduced,and dynamic buckling criterion are summarized.2.Based on the plane section assumption,the governing equation of the Euler-Bernoulli beam is deduced by using Hamilton principle.From the first order shear theory,the governing equation of the Timoshenko beam is deduced.3.The study of dynamic buckling of functionally graded material Euler-Bernoulli beam and Timoshenko beam under axial load by using Ritz method with de Moivre's formula.The trial function of the beam is set up in the form of a trigonometric function multiplied by a time term.In combination with the boundary conditions,the exact form of the trial function that satisfies the boundary condition is judged by the parity of function.Based on the de Moivre's formula,the governing equations of functionally graded material beams are simplified,and the expressions of the critical loads of Euler-Bernoulli beam and Timoshenko beam are obtained.At the same time,the dynamic buckling critical load of functionally graded material Euler-Bernoulli beam is solved by using the method of separation of variables.The critical load-critical length curve obtained by the separation variable method and the Ritz method is compared by MATLAB,the result shows that the Ritz method and de Moivre's formula are effectively for the study of dynamic buckling.4.The critical load formula of Euler-Bernoulli beam and Timoshenko beam are programmed by MATLAB,and the critical load-critical length curve are drawn,the influence of geometric size,gradient index,modal number,material composition,Poisson 's ratio and elastic modulus on the critical load are discussed.The critical load-critical length curve of Euler-Bernoulli beam and Timoshenko beam is compared,and it shows that the effect of shearing is very small for slender beam with small height-to-span ratio.