Research On Buckling Behavior Of Functionally Graded Material Cylindrical Shells Based On The Symplectic Method

Cylindrical shell is a common basic structure in engineering,which has many advantages,such as simple structure,light weight,high strength,strong bearing capacity,etc.Cylindrical shells are widely used in engineering fields such as aerospace,ship,machine,civil engineering.With the development of technology,the service environment of engineering equipment is increasing significantly,and the requirements for carrying capacity are constantly improving.Therefore,functionally graded materials with excellent mechanical properties begin to replace traditional metal materials for the design and manufacture of cylindrical shell structures gradually.Buckling instability is one of the main failure forms of functionally graded material cylindrical shells.The sudden failure of the structure will cause the failure of the whole instrument or equipment,and even result in casualties and economic losses.Therefore,it is of great practical significance to study the buckling instability of functionally graded material cylindrical shells.The research results can provide theoretical basis and analysis methods for the engineering design and safety assessment of such structures.There are three main limitations in the current research work on buckling of functionally graded material cylindrical shells:(1)The existing research mainly focuses on the role of single load,while the research on buckling behavior under combined loads of axial compression,torsion and external pressure is relatively limited;(2)The existing research mainly focuses on numerical solution and analytical solution.The numerical solution is usually time-consuming,and repeated modeling is required for different parameters,which is not convenient for parameter analysis.The analytical solution is limited by the solution system and mainly focuses on the inverse method and semi-inverse method.It is necessary to assume trial functions meeting the boundary conditions in advance.For the buckling problem under the combined loads with complex deformation modes,it is difficult to accurately construct the trial function that can describe the buckling mode;(3)The existing buckling studies are mainly based on the linear eigenvalue buckling theory which neglecting the bending deformation before the buckling of the shell.In fact,the two ends of the shell have produced large bending deformation before the buckling,which makes the critical load of the linear eigenvalue buckling greater than the actual results and is not reliable as a design criterion in engineering.In order to solve the above problems,this dissertation conducts research on buckling problem of functionally graded material cylindrical shells under axial pressure,torsion,external pressure and their combined loads,proposes a new Hamilton system solution method and obtains the analytical solution of the problem,analyzes the influence of influencing parameters on the buckling behavior,and reveals the instability mechanism of this kind of cylindrical shells.The research work is as follows:(1)The unified Hamiltonian solution model of functionally graded material cylindrical shells under axial compression,torsion,external pressure and their combined loads is established.By introducing the dual variables(internal force,shear force and bending moment)of the original variables(displacement and rotation),the full state vector composed of the original variables and dual variables is constructed as the basic unknown in the Hamiltonian system.By using the Hamilton variational principle,the basic equation in the traditional Lagrangian system is transformed into a unified form of Hamilton canonical equation in symplectic space.The Hamilton operator matrix only contains the derivative to one coordinate,thus realizing the transformation from high-order partial differential equations to low-order ordinary differential equations.(2)The analytical solutions of functionally graded material cylindrical shells under different loads are solved,and the corresponding critical loads and buckling modes are obtained.In the Hamiltonian system,the original buckling problem is transformed into an eigenproblem in the symplectic space,and the critical loads and buckling modes can be directly represented by the symplectic eigenvalue and symplectic eigensolution.Therefore,the symplectic eigenfunction is brought into the corresponding boundary conditions by the separation of variables method to obtain the analytical critical loads and buckling modes.The solution process is rational and rigorous,which is not necessary to assume any approximate displacement function meeting the boundary conditions in advance,nor to conduct the approximate integration processing in the energy method.(3)The nonlinear buckling governing equations considering the pre-buckling deformation is established to obtain more accurate critical loads.By introducing axisymmetric pre-buckling deformation into the buckling governing equation,pre-buckling static equation and nonlinear buckling governing equation are obtained.Based on the symplectic eigenfunction,pre-buckling deformation function and buckling displacement function are constructed,and the pre-buckling deformation of the shell is determined by solving the pre-buckling static equation.Then,the pre-buckling deformation and buckling displacement function are introduced into the nonlinear buckling governing equation to obtain the critical load.Results show that the critical load considering the pre-buckling deformation is closer to the experimental results than the linear eigenvalue buckling.(4)Through the analysis of the key calculation parameters,the influence and action rules of circumferential wave number,boundary conditions,size parameters,material parameters,load forms,pre-buckling deformation on the buckling behavior are revealed.The results show that the theoretical solution is consistent with the existing literature results and finite element results,indicating that the theoretical model and solution method proposed can accurately analyze the buckling behavior of functionally graded material cylindrical shells under different load forms.Through parameter analysis,it is found that the size parameters of functionally graded material cylindrical shells have significant influence on the critical loads and buckling modes,and the size parameters have greater influence on the critical loads and buckling modes of torsion and external pressure than axial compression.For combined loads,the proportion and value of various loads will affect the critical loads and buckling modes.After considering the pre-buckling deformation,the critical load will be significantly reduced and closer to the experimental results,and the pre-buckling deformation exhibits more obvious effect on the thicker or shorter functionally graded material cylindrical shells.