An Extension Of Karman-Donnell’s Theory For Non-Shallow Cylindrical Shells Undergoing Large Deflection

The buckling of structures is always one of the most concerned problems among scientists and engineers from the establishment of stability theories to finding the big difference between the buckling experimental results and theoretical prediction. Many buckling theories are proposed to deal with the problems arising from the development of practice. Karman-Donnell governing equations are always the most widely used for its simplicity and precise prediction for short cylindrical thin-walled shells. However, because of its simplicity, this equations is not appropriate when used for long cylindrical shells, especially for infinite long cylindrical shells. This paper is aimed to extend the classical Karman-Donnell governing equations to enable it available for long cylindrical shells. The contents involved in this paper are as follow:First, the influence of initial curvature on the bending moments is studied based on the classical Karman-Donnell governing equations and the extended Karman-Donnell governing equations are deduced. The similarity and difference of these two governing equations are presented. Second, in order to certify the rationality of the extended Karman-Donnell governing equations, the derivation of the governing equations for the plane strain case of both the two theories are given. Last but not the least, discussion about the conformity to the classical result given by Timoshenko of the above two governing equations is also conducted.Secondly, the extended Karman-Donnell governing equations and the local buckling mechanismare used for long cylindrical shells. The governing equations are solved by assuming the buckling model of every cross section is elliptical shape and Ritz are used. Subsequently, the post buckling equilibrium path and the length of buckling transition are acquired. Several cases’ numerical analysis results are compared with previous results. Discussion about the conformity of the post buckling equilibrium path with the observation of buckling experiments and classical result given by Timoshenko are presented. In addition, the influential factors to the length of buckling transition are also given.Thirdly, the extended Karman-Donnell governing equations is used to analyse the buckling of long cylindrical clamped shells subjected to uniform pressure on the outer surface. The extended Karman-Donnell governing equations particular for this problem is established considering the influence of initial curvate on the bending moments. The boundary conditions is also built by taking into account the continuity of displacement. The governing equations are solved by introducing separation of variables method. The MATLAB program is developed to calculate the buckling pressure of the long cylindrical shells and analyse the influence of radius-thickness ratio and radius-length to the buckling pressure. Besides, the results of this question and the long cylindrical shells as well as present work are compared. What’more, the accordance of the shortest length given by this question and the length given by the long cylindrical shells based on the extended Karman-Donnell governing equations is discussed.