A Theoretical Proof Of Stability Of Critical Equilibrium For Columns With Flexible Constraints And For Ring Under Pressure

With the application of high strength alloy materials and composite materials,people's understanding of system stability is further developed in modern structural design.On one hand,the results caused by instability are usually serious;On the other hand,the instability cases are also used in engineering practice.The columns with flexible constraints and the ring under external pressure are common structures.At present,the method of stability analysis is mainly based on the principle of minimum potential energy.According to Koiter's initial post buckling theory,it is necessary to strictly prove the positive definiteness of the second-order variation of potential energy.And the stability of the critical state is decided from the information of the higher-order variation corresponding to the critical point.Nonlinear finite element method is a numerical method for solving structural stability problems.However,there are two shortcomings: First,the nonlinear finite element method can only solve the post buckling equilibrium path under the condition of given specific stiffness,and can only solve this kind of forward problem.It can not directly and accurately give the range of the flexible constraint stiffness for the stable and unstable critical states.And it can not solve such inverse problems.Second,the nonlinear finite element method does not contain the information of higher-order variation,so it can not be directly used to decide the stability of the critical equilibrium state.Based on the above background,a theoretical proof of stability of critical equilibrium for columns with flexible constraints and for ring under pressure is studied in this paper.In this paper,a theoretical proof of stability of critical equilibrium for columns with flexible constraints is presented.Based on the principle of minimum potential energy,the properties of the second-order and higher order variation of potential energy are analyzed by analytic method.The stability of critical state is determined by the sign of high order energy variational potential energy.And the range is given of the flexible constraint stiffness for the stable and unstable critical states for the column.The main work is as follows:Firstly,the stability of a column with one end fixed and the other end constrained by a spring under Euler critical load is proved.The potential energy of the system is expressed as a functional of rotation;the perturbation is expanded into Fourier series;and the second-order variation of potential energy is expressed as an off diagonal quadratic form.The condition is obtained that the second-order variation of potential energy is positive in critical state.And the critical load and buckling mode are obtained.It is proved that the stability of the critical state is related to the relative stiffness of the constraint,which can be divided into stability and instability.This is different from the case of rigid constraints.The range of the stiffness of the flexible constraint is given when the critical state is stable or unstable.Research on an elastic foundation was published in 2017(Applied Mathematics and Mechanics,38(008): 877-887).Two years later,Batista,a Slovenian scholar,got the same result as this pape(INT J SOLIDS STRUCT,2019,169(9),72-80).The other research result of another kind of elastic foundation was published on February 20,2019(Arch Appl Mech,2019,89(8):1579-1587).Batista applied different method to the completely consistent model and got the same conclusion on February 22,2019(Int J Mech SCI,2019,155: 1-8).Secondly,by expanding the perturbation into Fourier series,the second-order variation of potential energy can be expressed as a quadratic form.But the coefficient matrix of the quadratic form is not diagonal in general,and the order is infinite.It is necessary to deduce the recurrence formula of finity and infinity order sequential principal minor to decide whether the quadratic form is positive or not.Thus,the stability of the critical equilibrium state of the compressed column with flexible constraints is proved.This leads to the fact that the proof of the critical state stability of the compressed column is very cumbersome with one end fixed and the other constrait by spring.For the pressure column with torsion spring constraint,this method can't get the recurrence formula of finite order and infinite order,and can't carry out the next step analysis.For the compressed column with torsion spring constraint,a generalized Fourier series is used to expand the perturbation.The second-order variation of potential energy can be expressed as a diagonal and infinite quadratic form of form.This makes it possible to decide whether the quadratic form is positive definite or not,and the proof process is very concise.According to the data obtained by the author,this is the first time that generalized Fourier series is applied to analyze the stability of the critical state of the compressed column with flexible constrained(Math Mech Solids,2020,25(4):961-967).In addition,the theoretical analysis method of stability of critical equilibrium for the ring under external pressure is discussed.The first,second and third order variational expressions of the potential energy functional are obtained,and an approximate solution of the critical load is obtained.The work of this paper lays a foundation for the further study of the stability of the critical equilibrium state for the structure.