Post-Buckling Analysis Of Slender Structure Considered Geometric Nonlinearity

In recent years,slender structure has been widely used in engineering.In the classical linear buckling theory,it is believed that after the critical load is reached,the axial compression structure will not be able to continue bearing and instability occurs,but in fact,the slender axial compression structure will appear the post-buckling behavior after the critical load,and still hold a capacity to bear load.In this paper,Euler-Bernoulli beam model is established for slender structure.Based on the geometrically nonlinear theory,the geometrically nonlinear equilibrium differential equation of post-buckling behavior of slender structure under axial load is established.Three different boundary conditions are considered: hinged-hinged,fixed-hinged,fixed-fixed.According to classical material mechanics,there is no shear force in the structure with hinged-hinged ends and fixed-fixed ends,but there is shear force in the structure with fixed-hinged ends.Firstly,the geometric nonlinear differential equations are reduced to linear equation,and the bifurcation solutions of the slender structure are obtained.Then,the exact solutions of nonlinear equilibrium differential equations are obtained by using elliptic integral method and shooting method respectively.Finally,the harmonic balance method and Galerkin method are used to obtain the analytical approximate solutions of the post-buckling problems of the slender structure.The comparisons of the analytical approximate solutions and the exact solutions are graphically shown,which included that the dimensionless axial force changing with the amplitude of rotation angle,dimensionless shear force changing with the amplitude of rotation angle,rotation angle changing with the dimensionless length of rod arc under different amplitudes,dimensionless axial displacement and dimensionless lateral displacement.By comparing with the exact solutions,the analytical approximate solutions show excellent accuracy.The nonlinear buckling behaviors of several groups of aluminum alloy slender columns with H-section and rectangular section are analyzed by using the finite element software ANSYS.The BEAM4 element and BEAM189 element are employed for modeling respectively,and the buckling and post-buckling behaviors of aluminum alloy axial compression columns with different length under different support conditions are analyzed.In the process of finite element analysis,geometric initial defects with different values are introduced to analyze the influence ofgeometric initial defects on post-buckling behavior of structure.By comparing the theoretical results of H-section aluminum alloy columns with those of BEAM 4element and BEAM 189 element,the results are very consistent.Comparing the numerical simulation results of BEAM4 element and BEAM189 element of rectangular cross-section columns,it is found that the smaller slenderness ratio,the greater influence of shear deformation on rectangular section column.Applying Timoshenko beam theory for calculation is more accurate.By comparing the load-displacement curves of different geometrical initial defects,it is found that the geometrical initial defects have little influence on the axial displacement,and have great influence on the lateral displacement in the initial post-buckling stage.By comparing load-displacement curves under different slenderness ratios,it is found that the greater slenderness ratio,the greater sensitivity of axial and lateral displacements to loads.