Buckling Of Orthotropic Cylindrical Shells Research Articles

BUCKLING OF THIN LAMINATED ORTHOTROPIC COMPOSITE RINGs/LONG CYLINDERS UNDER EXTERNAL PRESSURE

The buckling of isotropic rings under external pressure has attracted the interest of researchers since late 1950s. The formula for critical fluid buckling pressure of thin rings is very well known. This formula was directly extended to account for homogeneous orthotropic rings as well. The buckling of orthotropic cylindrical shells was also a subject of interest since the 1960s. However, the formulations developed, to date, require numerical solutions to obtain the critical pressure. In this work, a generalized closed form analytical formula for the buckling of thin orthotropic multi-angle laminated rings/long cylinders is developed. Standard energy based formulation is used to express the kinematics and equilibrium equations. Classical lamination theory is implemented to introduce the constitutive equations of thin shells. These equations are statically condensed, in terms of the ring's boundary conditions, to produce effective axial, coupling and flexural rigidities for the cases of rings and long cylinders. The critical buckling pressure may be calculated by hand using the derived equation in terms of these effective elastic rigidities. Comparisons are made with some existing results. Parametric studies are conducted to compare the present results with those of the buckling equations implemented by design standards. Various fiber orientations and stacking sequences are considered.

Effect of Normal Strains in Buckling of Thick Orthotropic Shells

An improved elasticity solution to the problem of buckling of orthotropic cylindrical shells sub- jected to external pressure is presented. The 2D axisymmetric cylindrical shell is studied (ring approximation). Specifically, in the development of the governing equations and boundary conditions for the buckling state, the solution includes the terms with the prebuckling normal strains and stresses as coefficients (i.e., the terms and , which were neglected in the earlier work as being too small compared to the terms and 00 e s9 s e9 s9 kk ij kk ij ij , respectively). The formulation results in a two-point boundary eigenvalue problem for ordinary differential 0 sv 9 kk j equations in r, with the external pressure p as the parameter. The results show that the effect of including the normal strains and stresses is to further decrease the critical load. This decrease (versus the earlier elasticity solution without these terms) depends on the shell thickness and is generally moderate, and in no event com- parable with the (quite large) decrease of the elasticity versus the shell theory prediction. This decrease depends also on the degree of orthotropy, and it is smaller for the isotropic case. Finally, a formula is derived for the critical pressure based on a first-order shear deformation formulation, and the comparison shows an improvement versus the classical shell for thick shells, but still the elasticity solution is noticeably lower than the first-order shear deformation prediction.

Nonlinear dynamic buckling of orthotropic cylindrical shells subjected to rapidly applied loads

Dynamic buckling of an orthotropic cylindrical shell which is subjected to rapidly applied compression is considered. A nonlinear differential equation of Donnell–Karman type is derived with the initial imperfection taken into account. An energy method is used to obtain the equation of motion which is then solved numerically by means of a Runge-Kutta method. These numerical results show that the critical load is increased over the corresponding static case. An analytical solution is also obtained for the problem of hydrostatic pressure.

Dynamic Buckling and Post-buckling of Imperfect Orthotropic Cylindrical Shells Under Mechanical and Thermal Loads, Based on the Three-Dimensional Theory of Elasticity

The three-dimensional theory of elasticity in curvilinear coordinates is employed to investigate the dynamic buckling of an imperfect orthotropic circular cylindrical shell under mechanical and thermal loads. Accurate form of the strain expressions of imperfect cylindrical shells is established through employing the general Green's strain tensor for large deformations and the equations of motion are derived in terms of the second Piola-Kirchhoff stress tensor. Then, the governing equations are properly formulated and solved by means of an efficient and relatively accurate solution procedure proposed to solve the highly nonlinear equations resulting from the above analysis. The proposed formulation is very general as it can include the influence of the initial imperfections, temperature distribution, and temperature dependency of the mechanical properties of materials, effect of various end conditions, possibility of large-deformation occurrence and application of any combination of mechanical and thermal loadings. No simplifications are done when solving the resulting equations. Furthermore, in contrast to the displacement-based layer-wise theories and the three-dimensional approaches proposed so far, the stress, force and moment boundary conditions as well as the displacement type ones, can be incorporated accurately in these formulations. Finally, a few examples of mechanical and thermal buckling of some orthotropic cylindrical shells are considered and results of the present three-dimensional elasticity approach are compared with the buckling loads predicated by the Donnell's equations, some single-layer theories, some available results of the layer-wise theory and the recently published three-dimensional approaches and the accuracy of the later methods are discussed based on the exact method presented in this paper.

Buckling of Thick Orthotropic Cylindrical Shells Under Torsion

A three-dimensional elasticity solution to the problem of buckling of orthotropic cylindrical shells under torsion is presented. A mixed form of the Galerkin method with a series of Legendre polynomials in the thickness coordinate has been applied to solve the governing differential equations. The accuracy of existing shell theory solutions has been assessed through a comparison study for both isotropic and orthotropic cylinders. For isotropic cylinders the solutions based on the Donnell shell theory were found to predict nonconservative values for the critical loads. As the circumferential wave numbers increase, shell theory solutions provide more accurate values. For orthotropic cylinders, the classical shell theory predicts much higher critical loads for a relatively short and thick cylinder, while the shear deformation theories provide results reasonably close to the elasticity solutions. Detailed data are also presented for the critical torsional loads over a wide range of length ratios and radius ratios for isotropic, glass/epoxy, and graphite/epoxy cylinders.

Benchmark three-dimensional elasticity solutions for the buckling of thick orthotropic cylindrical shells

Abstract Benchmark solutions to the problem of buckling of orthotropic cylindrical shells, which are based on the three-dimensional theory of elasticity, are presented in this review article. It is assumed that the shell is under external pressure or axial compression or a combination of these loadings. These solutions provide a means of accurately assessing the limitations of the various shell theories in predicting critical loads. A comparison with some classical shell theories shows that the classical shell theories may produce, in general, highly non-conservative results on the critical load of composite shells with thick construction. One noteworthy exception: the Timoshenko shell buckling equations produce conservative results under pure axial compression.

Buckling of Thick Orthotropic Cylindrical Shells Under External Pressure

An elasticity solution to the problem of buckling of orthotropic cylindrical shells subjected to external pressure is presented. In this context, the structure is considered a three-dimensional body. The results show that the shell theory predictions can produce nonconservative results on the critical load of composite shells with moderately thick construction. The solution provides a means of accurately assessing the limitations of shell theories in predicting stability loss.