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The stability of structures is important for dynamics in modern solid mechanics and it is noticed in engineering problem.Since the beam is a basic sort of structures,its buckling problem is researched all the while.Many methods are studied and applied to many engineering problems.For some non-linear problems of beam,the method is necessary to be researched specially,for example,for the thermal buckling and the rules of post-buckling.So far,the methods,which are analyzed and solved large deformation and post-buckling of Euler-Bernoulli beam,mainly include energy method,perturbation method,elliptic integral method.Nevertheless,these belong to inverse method or semi-inverse method and the results are not complete.In this paper,an effective method is presented to research the rule of dynamic and thermal buckling of beam.In the present research,dynamic buckling of an elastic beam under thermal shock is discussed in the Hamiltonian System.Firstly,the heat transfer process in the elastic beam is analyzed.The temperature distributing in the beam is given by the method of separation of variables.Next,the Hamiltonian system for the problem is established by introducing a Hamiltonian function.Thus,the problem is replaced by the basic problem of the Hamiltonian system.In the symplectic space,the critical buckling loads and buckling modes of the problem are reduced to symplectic eigenvalues and symplectic eigenfunctions.The zero-eigenvalue solutions and non-zero eigenvalue solutions are corresponding to problems of pre-buckling and post-bucking of the beam,respectively.Last,based on the completeness of the symplectic eigensolutions,the buckling mode of the post-buckling problems is obtained by using a symplectic eigenfunction expansion method and the pre-buckling mode,which is as the initial mode.In this way,the problem of post-buckling is correlated with the pre-buckling and the solution reveals the whole development process from the pre-buckling to post-buckling.A new method for solving the nonlinear problem is presented.Numerical results show the whole process of the post-buckling of a beam and the post-buckling behaviors,which are related with the impacted intensity,the thermal diffusion coefficients and the parameters of the beam.With the temperature increasing,the beam can buckle and vibrate mainly in the first buckling mode and the amplitude of the mode increases. Especially,if the temperature of thermal resource keeps a constant,the vibration of the beam is nearly equal amplitudes after enough time.
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