Generalized Fourier Series And Stability Of Euler Column With Elastic Supports At Critical Point

Columns with flexible restraints are widely used in engineering.This type of structure is easily destabilized under the pressure.The stability of columns under the Euler critical load and the post-buckling characteristics of this kind of rod should be fully understood,which can reduce accidents caused by instability,and also improve the utilization of materials.At present,there have been a lot of achievements in the research on the buckling and post-buckling of the struts,however,most of the researches are conducted on the rigid struts.Therefore,the post-buckling of the column with flexible restrains still needs to be further studied.In this paper,based on the Koiter stability theory,the following three types of slender columns are studied respectively:column with one end fixed and the other end clamped in rotation with a flexible support by a spring,column with one end fixed and the other end constrained by a spring,column with one end fixed and the other end constrained with a torsion spring and column with one end pinned,the other end constrained by a torsion spring.The stability under Euler critical load of the columns and the bifurcation in their initial post-buckling equilibrium range are analyzed.The potential energy of the system is expressed as the functional of the rotation angle,and the second-order variation and higher-order variations are obtained according to the increasement of the potential energy.For the columns with extension springs,the disturbance quantity is expanded into the form of ordinary Fourier series,for the columns with torsion spring,the disturbance quantity is expanded into the form of generalized Fourier series.The quadratic form of the second-order variation of the system potential is obtained,and all order principal minor determinants can be changed into an elementary expression.The semi-positive definiteness of the second variation of the potential energy is determined by the sign of the elementary expression.Further,the Euler critical load can be obtained when the second-order variation is semi-positive,and the buckling modes of the compression rod can be obtained.According to the positive definiteness of the fourth-order and sixth-order variations of the potential energy,the stability of the critical point can be judged.Then Koiter initial post-buckling theory is used to analyze the characteristics of the post-buckling equilibrium path.The results show that the stability of the critical state of the columns with extension springs is related to the relative stiffness of the flexible restraints.The potential energy may take the minimum or not,which means that the critical point can be either stable or unstable.The corresponding post-buckling equilibrium paths are the positive bifurcation and negative bifurcation respectively,where the positive bifurcation is a stable equilibrium path;the negative one is an unstable equilibrium path.The range of relative stiffness of elastic restraints corresponding to stable and unstable post-buckling is given.It is worth mentioning that there is a double-bifurcation in the anlysis of the column with one end fixed and the other end clamped in rotation with a flexible support by a spring.The critical point and the initial post-buckling of the columns with torsion springs are stable in large.The corresponding post-buckling equilibrium path is the positive bifurcation.The main innovations in this paper are:applying the generalized Fourier series analysis to analyze the stability of struts with flexible restraints;providing a methord to determining the positive definiteness of an infinite-order matrix;and presenting the unstable critical point of struts with flexible restraints.