A Hamiltonian System Method For Buckling Problem Of Thin Circular Plates

The thin plate structure is one of the most typical structures in engineering. The stability of this structure has been noticed all the way. On the one hand, the buckling of plate structures can bring security risks; On the other hand, the buckling of thin plates can absorb a lot of energy, so that the plate is often used for auxiliary structure in the energy absorbing device. Therefore, researching further the buckling problem of sheet structures is necessary and its application is valuable in engineering. In this paper, a new symplectic eigensolutions method and numerical simulation method are presented by constructing a Hamiltonian system. Pre-buckling and post-buckling problems of circular plates, which are impacted by mechanical and thermal coupling loads, are analyzed and some rules about them are obtained. It provides a new method and idea to solve the similar problems. The innovative research results are as follows:A Hamiltonian system method is presented for determining critical loads and buckling modes of thin circular plates. Based on the strain energy of thin circular plates, the Hamiltonian system is constructed for pre-buckling problems of the plates. In the Hamiltonian system, the critical loads and their buckling modes of plate are reduced to problems of the generalized symplectic eigenvalues and symplectic eigensolutions. This symplectic method is extended to the thermal buckling problem for circular plates and annular circular plates. Further, the critical temperatures and the corresponding buckling modes can be obtained directly. Thus, a completed space of symplectic eigensolutions is found. In this space, there are a symplectic relationship of adjoint orthonormality between buckling modes, so that the buckling mode can be obtained by expanding of symplectic eigensolutions. The numerical results reveal the characteristics of the critical loads and buckling modes.Considering the problem of geometric large deformation of elastic circular plates, a nonlinear Hamiltonian system is constructed for the post-buckling of circular plates. In this system, a new symplectic eigensolutions method is presented for the post-buckling of thin circular plates. Since pre-buckling modes of all orders generate a completed space of symplectic eigensolutions, any post-buckling mode can be expressed by expanding of symplectic eigensolutions. On the basis of the principle, three kinds of numerical simulation models are shown as:1) a direct model of symplectic eigensolutions for regular boundary condition;2) a numerical method to solve symplectic eigen values and an expansion model for irregular boundary conditions;3) a numerical simulation model of symplectic finite element. Thus, the development process of the post-buckling of circular plates is reduced to the evolution of different buckling modes. The pre-buckling problem and post-buckling problem of circular plates are Combined together in this method.For the post-buckling problem of elasto-plastic plates under nonlinear large deformation, a symplectic finite element numerical model is presented. The governing equations and finite element calculation formulas for elastic loading, plastic proportional loading and elastic unloading problems are unified by a form. The numerical results are found that the development paths of post-buckling of circular plates can be transited from the high order buckling mode to the low order mode under impact load. The results also show that plate vibration mode can appear on the new equilibrium position when the impact is unloaded. The equilibrium position relates to the residual strain, and the region and distribution of residual strain can be determined by the method. Using this method, the characteristics of plate buckling is analyzed further under the mechanical and thermal coupling loads, and some laws are revealed. The research results provide the basis for the control of buckling modes of structures.