George J. Simitses

Buckling and Postbuckling of Imperfect Cylindrical Shells: A Review

Thin-walled cylinders of various constructions are widely used in simple or complex structural configurations. The round cylinder is commonly found in tubing and piping, and in offshore platforms. Depending on their use, these cylinders are subjected (in service) to individual and combined application of external loads. In resisting these loads the system is subject to buckling, a failure mode which is closely associated with the establishment of its load-carrying capacity. Therefore, the system buckling and postbuckling behavior have been the subject of many researchers and investigators both analytical and experimental. The paper is a state-of-the-art survey of the general area of buckling and postbuckling of thin-walled, geometrically imperfect, cylinders of various constructions, when subjected to destabilizing loads. The survey includes discussion of imperfection sensitivity and of the effect of various defects on the critical conditions.

Delamination buckling and growth of flat, cross-ply laminates

Abstract A one-dimensional model is presented in order to predict (a) delamination buckling loads of an across the width delaminated and axially loaded laminated plate and (b) the ultimate load-carrying capability of the above geometry, when delamination growth can take place. In order to study the possibility of spreading of the damaged (delaminated) area to the undamaged area, the post-buckling solution is employed. The energy release rate has been used to determine whether the delamination growth is stable or unstable. The results reveal that for a relatively small delamination length the buckling loads serve as a measure of the load-carrying capacity of the damaged plate, while for a relatively large delamination length, the plate could carry larger loads depending on the fracture toughness of the plate material. Moreover, the present model can be used to study the effect of the presence of coupling between bending and stretching on delamination growth. Note that for such geometries the possibility of bifurcational buckling does not exist, regardless of the level of the applied load.

Nonlinear analysis of axially-loaded laminated cylindrical shells

Abstract The nonlinear analysis of geometrically imperfect, thin, laminated, circular, cylindrical shells subjected to uniform axial compression, and for various in-plane and transverse supports, is presented. Moreover, the solution methodology is described, and results are generated for imperfect laminated (Boron/Epoxy) cylinders with symmetric, antisymmetric and asymmetric stacking of lamina. The applications deal with imperfection sensitivity studies and investigation of the effect of lamina stacking on the critical conditions. Finally, for some of the chosen geometries, experimental data is available and therefore these geometries serve as (partial) benchmarks for the developed solution scheme.

Buckling of long sandwich cylindrical shells under external pressure

The paper deals with the theoretical prediction of buckling loads for sandwich long cylindrical shells with laminated facings and foam core. The loading is a uniform hydrostatic pressure, which means that the loading remains normal to the deflected surface during the buckling process. Several fiber materials are used in the laminated facings. The materials are: Boron/Epoxy, Graphite/Epoxy and Kevlar/Epoxy laminates with 0° orientation with respect to the hoop direction. These various materials are employed to provide comparative data that can be used in design. Shell theory results are generated with and without accounting for the transverse shear effect. Moreover, results based on three-dimensional elasticity are also generated for comparison purposes. The effect of the ratio of radius to thickness is assessed.

Delamination buckling of cylindrical shells under axial compression

Abstract A model is developed for the study of delamination buckling of axially loaded cylindrical shells. Delamination is assumed to exist before load application, it spans the entire circumference, and it lies on the contact surface of neighboring laminae. The mathematical model employs Donnell-type kinematic relations and linearly elastic material behavior. Furthermore, each lamina is assumed isotropic, and the emphasis is on studying the effect of delamination size and position on the critical load. Two sets of boundary conditions are used with the model: simply supported and clamped. The study reveals several important conclusions. Among them, one may list the following: (a) the critical load is primarily controlled by the position of the delamination from the reference surface, provided that the delamination is not very close to the boundaries; and (b) for long delaminations (relative to the cylinder length), the critical load is not appreciably affected by the boundary conditions.

Nonlinear Response of a Shallow Sandwich Shell With Compressible Core to Blast Loading

This paper investigates the nonlinear dynamic response of a shallow sandwich shell subject to blast loading with consideration of core compressibility. The shallow shell consists of two laminated composite or metallic face sheets and an orthotropic compressible core. Experimental results and finite element simulations in literature have shown that the core exhibits considerable compressibility when a sandwich panel is subjected to impulse loading. To address this issue properly in the analysis, a new nonlinear compressible core model is proposed in the current work. The system of governing equations is derived by means of Hamiltons principle in combination with the Reissner-Hellingers variational principle. The analytical solution for the simply supported shallow shell is formulated using an extended Galerkin procedure combined with the Laplace transform. Numerical results are presented. These results demonstrate that this advanced sandwich model can capture the transient responses such as the stress shock wave effect and the differences in the transient behaviors of the face sheets and the core when a sandwich shadow shell is subjected to a blast loading. However, in the steady state dynamic stage, all the displacements of the face sheets and the core tend to be identical. This model can be further used to study the energy absorption ability of the core and the effects of different material and geometrical parameters on the behaviors of sandwich structures subject to blast loading.

Spring simulated delamination of axially-loaded flat laminates

Abstract A general one-dimensional model of a delaminated composite structural element under axial loading is used for predicting the classical buckling load, with the bending-stretching coupling effect. Two cases are examined. In the first case the delamination is such that it does not cause any resistance in the motion of the two detached laminae, due to the applied axial loads. In the second case, the delamination is such that both laminae feel the existence of a spring distribution along the length of the delamination.

Optimal beam frequencies by the finite element displacement method

Abstract A finite element displacement formulation is used to maximize the first mode natural frequency of a vibrating beam of specified volume with elastically restrained ends and resting on a continuous elastic foundatior subject to a constraint of minimum allowable moment of inertia. For cross-sections with moment of inertia and area related by I = pA n (p and n are positive constants), the optimality condition is reduced to a relation between the strain energy and kinetic energy densities. Beginning with a uniform beam, an iterative procedure is used to converge to the optimum material distribution and maximum first mode frequency. Results are presented for various boundary conditions, with and without the effect of any given arbitrarily varying axial load distribution and/or dead mass distribution for n = 1, 2 and 3.

Inperfection sensitivity of laminated cylindrical shells in torsion and axial compression

Abstract The imperfection sensitivity of thin cylindrical shells, made out of fiberreinforced composite material and subjected to either uniform axial compression or torsion, and the effects upon it of certain parameters, are investigated. The sensitivity is established through plots of critical loads (limit point loads) versus imperfection amplitude. The larger the drop in critical load value with increasing amplitude, the greater the sensitivity. Results are presented for four- and six-ply laminates with simply supported boundaries and various stacking sequences. These sequences lead to symmetric, antisymmetric and asymmetric configurations with respect to the laminate midsurface. The material for all configurations is boron/epoxy. The parametric studies include primarily the effect of lamina stacking and length-to-radius ratio on the critical loads. Among the important findings are that (a) laminated cylindrical shells are more imperfection sensitive under axial compression than under torsion, (b) the imperfection sensitivity decreases as the length-to-radius ratio increases and (c) lamina stacking has a pronounced effect on the imperfection sensitivity of the laminated shell.

Buckling of delaminated, long, cylindrical panels under pressure

Abstract Delamination is one of the basic defects inherent to laminar materials. The investigation of the buckling characteristics of delaminated cylindrical shells or panels, when subjected to external pressure, is presented herein. The geometry is such that it covers a wide range of length to radius ratios as well as panels of different widths. Results are presented only for very long cylinders and panels. The boundaries are either simply supported or clamped. Furthermore, the material is such that it leads to (quasi) isotropic laminates for all sections involved, the overall as well as the ones separated by the delamination. Finally, the geometry is free of initial geometric imperfections. Because of the last two assumptions, a primary membrane state exists and bifurcational buckling is possible. Buckling loads are calculated for a wide range of parameters. The width and the through-the-thickness position of delamination greatly affect the bifurcation load.