Thermal Buckling Of Elastic Beams In Hamilton System

Since beam is basic structure, its buckling problem is playing an important role in the theories of structural stability. The research of stability has achieved a lot of results and solved many practical problems. But it is necessary to do some research on some non-linear problems of beam which are arose by the temperature and the rules of post-buckling.In this paper, thermal pre-buckling and post-buckling, which are arose by the temperature, of elastic beams are investigated in Hamilton System. On the basis of linear theory for axially non-extensible beam, Hamilton system is established for thermal pre-buckling of elastic beam with fixed-fixed ends. The critical buckling loading and buckling modes are described by eigenvalues and eigensolutions in symplectic space. The bifurcation condition of pre-buckling is obtained from the boundary conditions, and differential equation is converted to algebra equation. The eigenvalues and eigensolutions are given out to describe the critical buckling loads and buckling modes. In the completed space of symplectic einensolutions, any buckling mode can be expressed by a combination of eigensolutions. For the thermal post-buckling in the nonlinear geometric large deflecton, the pre-buckling mode is as initial mode and the post-buckling mode is described by combined eigensolutions. Therefore, the problem is reduced to solving nonlinear algebra equations by using the Ritz method. The numerical results reveal the whole process from pre-buckling to post-buckling and give some rules.Numerical results show that developing with time, post-buckling mode presented as first order critical mode. So, the critical buckling loads and critical buckling mode can be described by eigenvalues and eigensolutions in symplectic space. The mathematical mode of thermal pre-buckling can be described by geometrical linear theory. For the thermal post-buckling of the beam, geometrical large deflection theory must be applied. The symplectic method and its results are useful for post-buckling questions that are non-linear and large-deformation. This method can also be extended to other fields.