A Symplectic Method In Buckling Of Donnell’s Cylindrical Shells

With the development of high-technology and innovation of manufacturing processing, various high-performance composite materials and light-weight structures are applied widely in the engineering. Cylindrical shells made of the innovative materials belong to the classical and important structures that fulfill the tendency of scientific and technological development. Stability for these structures is the main consideration in design and analysis. The traditional solving methods are all employed in Lagrangian system, such as semi-inverse method, expansion of modal functions and numerical methods and so on. These method all have limits more or less in application. To chase the progress of science and technology, it is necessary to develop a new research approach and solving system. In this context, this paper tries to present an analytical method for buckling of cylindrical shells.Based on the Donnell’s theory of shell expressed by stress function and radial displacement, a new energy conservative symplectic method is developed for the elastic-plastic buckling of isotropic and FGM cylindrical shell under the actions of single or coupled static, impact and thermal loads. With the purpose to innovate solving concept, the original problem in Lagrange system is converted into the Hamilton system. The high-order differential equations in Lagrangian system exist some difficulties in solution and proving process that can be overcome in symplectic system. In this paper, radial displacement, radial angle, stress function and its gradient are chosen as the original variables. According to the balance between the strain energy and external work and by employing the Hamilton variational principle, the high-order differential governing equations can be transformed into the lower-order Hamilton canonical equations through the Legendre transformation. The fundamental problems can then be solved completely due to the advantages of Hamilton system. In this way, the buckling characteristics are converted into the solving for symplectic eigenvalues and eigenvectors, respectively, and the complete solving space can also been obtained. Due to the dual relations in Hamilton system, one of the main advantages for Hamilton system is the performance in dealing with the problems under complex and mixed boundary conditionsBuckling of cylindrical shells subjected to the static loads is a classical mechanic problem. Most of the previous studies only concerned with the buckling behaviors of the shell under symmetric boundary conditions. By employing the symplectic solving method, this article not only obtains these existing results but also investigates this problem under the non-symmetric boundary conditions. The loading conditions have also been extended to the combined actions of three fundamental loads, e.g., axial compression, torque and external pressure. Especially for the buckling problems including the torsional feature, some complete analytical results can be obtained through this method. The completeness of the solving space is discussed and proven in detail. In the study of dynamic buckling, the effect of stress wave propagation is considered and dynamic stability problem is reduced into the bifurcation buckling. Some unmentioned dynamic buckling patterns are described in symplectic system. For the plastic problem, buckling behavior of cylindrical shell under axial compression is first investigated according to the J2deformation theory, and the geometric borders for the occurrences of elastic-plastic buckling are given in the article. With regard to the FGM cylindrical shells, the temperature-dependent material properties are considered and various material compositions and temperature field distributions are discussed. For the thermal buckling of FMG cylindrical shells, there may exist a critical material distribution exponent that can be used to determine whether thermal buckling occurs or not.