Hamiltonian System Method For Post-Buckling Of Elastic Cylindrical Shells With Defect

The cylindrical shell structure has important applications in aerospace,marine engineering,nuclear industry,vehicle engineering and civil engineering due to their regular geometry,light weight,easy processing,and excellent bearing capacity.The main form of failure of cylindrical shells in extreme working conditions is instability,that is,when the external load exceeds a certain critical value,the shell will suddenly undergo buckling failure.Therefore,the evaluation for the stability behavior of cylindrical shells is the basis for the design of various engineering structures.In order to explain the huge discrepancy between the experimental results of cylindrical shells and the values predicted by classical linear theory,as well as the discreteness of experimental data,the stability theory of cylindrical shell has gradually developed from the initial small deflection linear theory to the nonlinear large deflection theory,initial postbuckling theory and nonlinear pre-buckling consistent theory.At present,it is generally believed that the critical axial compression of cylindrical shells is significantly affected by geometrical and material defects,load eccentricity,load uneven distribution and imprecise boundary conditions.The most important reason for the reduction of the bearing capacity of cylindrical shell is the defect produced in the process of manufacture or use.The small defects can also lead to large changes in the critical load and buckling deformation of the cylindrical shell.This defect-sensitive buckling characteristic leads to the experimental test results of the critical load of the cylindrical shell are much lower than the classical theoretical value,and the corresponding buckling deformation also shows very irregular.The assumed form of shell defects in previous studies is often relatively simple,and it is difficult to be used as a reasonable description of complex defect shapes in practice.The governing equations of post-buckling of cylindrical shells with defects are nonlinear differential equations,which can only be solved by numerical methods,such as finite element method and energy method.If the cylindrical shell contains local defects or complex external loads,it is more difficult to assume the deflection function when applying the energy method,and the accuracy of the calculation results is difficult to determine.Therefore,establishing a theoretical model of the stability of cylindrical shells with complex defects and providing an effective solution and analysis method will help to deeply understand the influence of various defects on the buckling behavior of cylindrical shells,and deepen the understanding of the complex buckling phenomenon of cylindrical shells.Under the above-mentioned difficulties and demands,this paper attempts to propose a new solution system for the post-buckling problem of cylindrical shells with defects.Based on the Reissner shell theory,considering the thickness defects and large geometric deformation of the shell,a nonlinear Hamiltonian system for post-buckling problems of cylindrical shells with defects is established,and the problem is reduced to a series of series expansion problems consisting of symplectic eigensolutions.By equating the angle coordinates in the cylindrical coordinate system to the time coordinates in traditional analytical mechanics,and using Legendre transformation and Hamilton’s variation principle,the traditional Lagrangian solution system for the post-buckling problem of cylindrical shells is converted into the Hamiltonian solution system.The variables to be determined are changed from three displacement variables under the original classical system to eight variables including generalized displacement and generalized force under the symplectic system.After the transformation of the solution system,the original high-order differential governing equations are transformed into low-order differential equations.After separating the related terms of external load,defects and large deflection separately,the general matrix equation covers four kinds of stability problems: small deflection buckling of non-defect cylindrical shell,large deflection post-buckling of non-defect cylindrical shell,small deflection buckling of cylindrical shell with defects and large deflection post-buckling of cylindrical shells with defects.The homotopy analysis method and the symplectic eigensolution expansion method are combined to form a post-buckling solution scheme for cylindrical shells with defects to obtain the critical load and post-buckling equilibrium path.In the homotopy analysis,the nonlinear system of the post-buckling problem of cylindrical shells is decomposed into several linear systems,and the relationship and recurrence relationship between the subsystems are established.The homotopy equations of each order are solved and analyzed by using the symplectic conjugate orthogonal relation.The zero-order homotopy deformation equation is constructed as the small-deflection buckling equation of the cylindrical shell.The post-buckling equilibrium path and deformation can be obtained asymptotically from the small deflection solution of the defective cylindrical shell.A direct solution method for eigenvalue buckling and post-buckling analysis under complex external loads is formed by using the symplectic conjugate orthogonal relationship between the symplectic eigensolutions containing externally loaded information and the symplectic eigensolutions.The deflection function in the symplectic eigensolution is determined by the external load and the ratio between the loads,which can naturally construct the buckling deformation under the action of the complex external load,which overcomes the difficulty of constructing the deflection function in the previous buckling analysis.The effects of complex forms of local defects including rectangle,cross and ellipse on the buckling of cylindrical shells are analyzed by using the above model and method.The results show that the symplectic method can accurately obtain the critical load and post-buckling equilibrium paths of the cylindrical shell with defects,and give the local buckling phenomenon of the shell which is difficult to describe by the previous analytical methods.The defect is most likely to induce the local buckling,when the width of the defect is close to the width of the corrugations that may buckling.The local defects significantly change the buckling mode of the cylindrical shell and reduce the critical load and post-buckling equilibrium path,especially for the cylindrical shell under axial compression.The local small ripples of small deflection buckling will gradually transform into local diamond-shaped depressions unique to the postbuckling stage with the fully developed post-buckling of the cylindrical shell.Local defects reduce the circumferential torsional buckling wave number of the cylindrical shell and concentrate the torsional buckling deformation on the defective side of the shell.The most unfavorable defect location for torsional buckling is only related to the width of the defect,not to the depth of the defect.The torsional buckling symmetry of the cylindrical shell is broken when the defect has no axial symmetry.The buckling analysis under complex external loads shows that a small lateral external pressure also greatly reduces the critical axial compression and post-buckling equilibrium path of the cylindrical shell.The local small ripple buckling mode in the defect area will gradually disappear,and the sensitivity of the cylindrical shell to the defect will also decrease with the increase of the external pressure.The effect of the end torque on the critical axial compression and post-buckling equilibrium path is not as significant as that of the external pressure.The small corrugated deformation in the defect area gradually slopes with the increase of the end torque,and a unique local compression-torsional coupling deformation mode appears.The validity and accuracy of the above model and solution system provide the basis for the post-buckling for other complex cylindrical shells.