Symplectic Methods In Dynamic And Thermal Buckling Of Elastic Cylindrical Shells

The stability of structure is an important research subject in modern solid mechanics and much attention has been focused on the problem in industry equipments.Since cylindrical shell is a basic structure,its buckling problems play an important role in the theories of structural stability.Many researches have been carried out and great progress has been made in solving many practical problems.Some phenomena which deal with local buckling of a cylindrical shell in experiments are reported.To explain these,it is necessary to consider basic theories and a new method is needed.It is well known that an object can be impacted or heated.The buckling phenomena of a cylindrical shell subjected to an impact load or a thermal load are not the same because of their different characters.When a shell is impacted,the internal forces caused by the impact will be propagated and reflected in a stress wave form.Becuase stress waves are affected locally,the shell will be overall bucked and accompanied with local wrinkles,if both loads are applied. High-order partial differential equations often occur when one solves problems which deal with pre-buckling of a cylindrical shell.Sometimes traditional separation of variables is not work anymore if the problem is presented in a Lagrangian system.When this happens,it is necessary to introduce a new system.This paper deals with pre-buckling of a shell subjected to thermal load,impact load or coupling load.By introducing a Hamiltonian function,the Hamiltonian system for the problem is established.This means that the transformation from a Lagrangian system to a Hamiltonian system is finished.In a symplectic space,eigenvalues and eigenfunctions take the place of the critical buckling loads and buckling modes of the problem.The zero-eigenvalues describe axisymmetric buckling and non-zero eigenvalues mean non-axisymmetric bucking.In pre-buckling problems,small deformation theories are used,while lager deformation theories are applied in solving post-buckling problems.Based on the complete solutions obtained in the pre-buckling problems,a symplectic eigenfunction expansion method is developed and the whole progress from prebuckling to postbuckling are described.A new method for solving such nonlinear problems is presented.Stress waves play an important role in the local buckling of a shell when the waves propogate,reflect and transmit in the shell.After studying these effects,critical loads and buckling modes of a shell subjected to impact loads,thermal loads and coupling loads with different boundary condtions are presented.Numerical results show that when a shell is impacted,local buckling is unavoidable due to the local effects of the stress waves.There will be overall buckling if a shell is heated.When a shell is impacted,a big open is tend to occur at the impact end in the beginning and when the stress waves are reflected,the big open may appear at the reflect end.If a pulse is imposed,a bamboo-node type will come up in the middle of the shell.The pulse usually is pruduced by the propagation,reflection and transmission of the stress waves in the bullet.In fact,local buckling of a shell is a type of energy concentration in the shell.It can be concluded that local buckling in a shell are mainly caused by the propagation, reflection and transmission of stress waves.Local buckling have something to do with not only the phisical properties of the shell but also its geometric feature parameters.The length of a pulse and the wave impedence are of major signifance on a bamboo node-type.In post-buckling problems,an intial mode which is in the form of pre-buckling solutions is adopted,and the effects of the initial mode to the post-buckling are discussed in three ways,that is,its amplitudes,circular orders and axial branches.It is found that the shell is not sensitive to the intial mode.In post-buckling process,a thinner shell is inclined to a higher circular order and a longer shell tends to a higher axial order.These provide important rules for structure stability design in engineering.