Mechanical and thermal postbuckling of FGM thick circular cylindrical shells reinforced by FGM stiffener system using higher-order shear deformation theory

The postbuckling of the eccentrically stiffened circular cylindrical shells made of functionally graded materials(FGMs),subjected to the axial compressive load and external uniform pressure and filled inside by the elastic foundations in the thermal environments,is investigated with an analytical method.The shells are reinforced by FGM stringers and rings.The thermal elements of the shells and stiffeners in the fundamental equations are considered.The equilibrium and nonlinear stability equations in terms of the displacement components for the stiffened shells are derived with the third-order shear deformation theory and Leckhniskii smeared stiffener technique.The closed-form expressions for determining the buckling load and postbuckling load-deflection curves are obtained with the Galerkin method.The effects of the stiffeners,the foundations,the material and dimensional parameters,and the pre-existent axial compressive and thermal load are considered.     

Creep buckling and postbuckling of circular cylindrical shells under axial compression

An infinitely long, axially compressed, circular cylindrical shell with an imperfection in the shape of the axisymmetric classical buckling mode, undergoing steady or non-steady creep, is analyzed. The axisymmetric problem is solved incrementally using nonlinear shell equations The ratio of the applied stress to the classical buckling stress determines if the shell will collapse axisymmetrically or if it will bifurcate into a nonaxisymmetric mode, and whether or not bifurcation will result in instantaneous collapse. The bifurcation problem is formulated exactly and the initial postbuckling behavior is investigated via an asymptotic elastic analysis, based on Koiter's general theory Numerical results are compared with available experimental data.     

Postbuckling of FGM cylindrical shells under combined axial and radial mechanical loads in thermal environments

A postbuckling analysis is presented for a shear deformable functionally graded cylindrical shell of finite length subjected to combined axial and radial loads in thermal environments. Heat conduction and temperature-dependent material properties are both taken into account. The temperature field considered is assumed to be a uniform distribution over the shell surface and varied in the thickness direction only. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The formulations are based on a higher order shear deformation shell theory with von Kármán–Donnell-type of kinematic nonlinearity. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of functionally graded cylindrical shells. A singular perturbation technique is employed to determine the interactive buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect cylindrical shells with two constituent materials subjected to combined axial and radial mechanical loads and under different sets of thermal environments. The results reveal that the temperature field and volume fraction distribution have a significant effect on the postbuckling behavior, but they have a small effect on the imperfection sensitivity of the functionally graded shell.     

Dynamic thermal buckling of suddenly heated temperature-dependent FGM cylindrical shells,under combined axial compression and external pressure

FGM components are constructed to sustain high temperature gradients. There are many applications where the FGM components are vulnerable to transient thermal shocks. If a component is already under compressive external loads (e.g. under a combination of axial compression and external pressure), the mentioned thermal shocks will cause the component to exhibit dynamic behavior and in some cases may lead to buckling. On the other hand, a preheated FGM component may undergo dynamic mechanical loads. Only static thermal buckling investigations were developed so far for the FGM shells. In the present paper, dynamic buckling of a pre-stressed, suddenly heated imperfect FGM cylindrical shell and dynamic buckling of a mechanically loaded imperfect FGM cylindrical shell in thermal environment, with temperature-dependent properties are presented. The general form of Green’s strain tensor in curvilinear coordinates and a high order shell theory proposed already by the author are used. Instead of using semi-analytical solutions that rely on the validity of the separation of variables concept, the complicated nonlinear governing equations are solved using the finite element method. Buckling load is detected by a modified Budiansky criterion proposed by the author. The effects of temperature-dependency of the material properties, volume fraction index, load combination, and initial geometric imperfections on the thermo-mechanical post-buckling behavior of a shell with two constituent materials are evaluated. The results reveal that the volume fraction index and especially, the differences between the thermal stresses created in the outer and the inner surfaces may change the buckling behavior. Furthermore, temperature gradient and initial imperfections have less effect on buckling of a shell subjected to a pure external pressure.     

Thermal buckling and postbuckling behavior of FG-GRC laminated cylindrical shells with temperature-dependent material properties

Thermal postbuckling analysis is presented for graphene-reinforced composite (GRC) laminated cylindrical shells under a uniform temperature field. The GRC layers are arranged in a functionally graded (FG) graphene reinforcement pattern by varying the graphene volume fraction in each GRC layer. The GRCs possess temperature dependent and anisotropic material properties and the extended Halpin–Tsai model is employed to evaluate the GRC material properties. The governing equations are based on a higher order shear deformation shell theory and include the von Kármán-type kinematic nonlinearity and the thermal effects. A singular perturbation method in conjunction with a two-step perturbation approach is applied to determine the thermal postbuckling equilibrium path for a GRC shell with or without geometric imperfection. An iterative scheme is developed to obtain numerical thermal buckling temperatures and thermal postbuckling load–deflection curves for the shells. The results reveal that the FG-X piece-wise FG graphene distribution can enhance the thermal postbuckling capacity of the shells when the shells are subjected to a uniform temperature loading.

     

Torsional buckling of cross-ply laminated orthotropic composite cylindrical shells subject to dynamic loading

This study considers torsional buckling of cross-ply laminated orthotropic composite cylindrical thin shells under loads, which is a power function of time. The modified Donnell type dynamic stability and compatibility equations are obtained first. These equations are subsequently reduced to a time dependent differential equation with variable coefficients by using Galerkin's method. The critical parameters are found analytically by applying the Ritz type variational method. According to theoretical solutions, numerical analyses are done.     

Nonlinear buckling of torsion-loaded functionally graded cylindrical shells in thermal environment

Based on the nonlinear large deflection theory of cylindrical shells, this paper deals with the nonlinear buckling problem of functionally graded cylindrical shells under torsion load by using the energy method and the nonlinear strain–displacement relations of large deformation. The material properties of the functionally graded shells vary smoothly through the shell thickness according to a power law distribution of the volume fraction of the constituent materials. Meanwhile, on the base of taking the temperature-dependent material properties into account, various effects of external thermal environment on the critical state of the shell are also investigated. Numerical results show various effects of the inhomogeneous parameter, the dimensional parameters and external thermal environment on nonlinear buckling of functionally graded cylindrical shells under torsion. The present theoretical results are verified by those in literature.     

Nonlocal shear deformable shell model for torsional buckling and postbuckling of microtubules in thermal environments

Buckling and postbuckling analysis is presented for microtubules subjected to torsion in thermal environments. The microtubule is modeled as a nonlocal shear deformable cylindrical shell which contains small scale effects. The governing equations are based on a higher order shear deformation theory. The thermal effects are included and the material properties are assumed to be temperature-dependent. The small scale parameter e₀a is estimated by matching the buckling twist angle of microtubules obtained from the nonlocal shear deformable shell model with the existing result. The results show that the small scale effect plays an important role in the postbuckling of microtubules.     

Nonlinear buckling and postbuckling of heated functionally graded cylindrical shells under combined axial compression and radial pressure

The nonlinear large deflection theory of cylindrical shells is extended to discuss nonlinear buckling and postbuckling behaviors of functionally graded (FG) cylindrical shells which are synchronously subjected to axial compression and lateral loads. In this analysis, the non-linear strain-displacement relations of large deformation and the Ritz energy method are used. The material properties of the shells vary smoothly through the shell thickness according to a power law distribution of the volume fraction of the constituent materials. Meanwhile, by taking the temperature-dependent material properties into account, various effects of external thermal environment are also investigated. The non-linear critical condition is found by defining the possible lowest point of external force. Numerical results show various effects of the inhomogeneous parameter, dimensional parameters and external thermal environments on non-linear buckling behaviors of combine-loaded FG cylindrical shells. In addition, the postbuckling equilibrium paths are also plotted for axially loaded pre-pressured FG cylindrical shells and there is an interesting mode jump exhibited.     

Vibration analysis of bi-layered FGM cylindrical shells

In the present study, a vibration frequency analysis of a bi-layered cylindrical shell composed of two independent functionally graded layers is presented. The thickness of the shell layers is assumed to be equal and constant. Material properties of the constituents of bi-layered functionally graded cylindrical shell are assumed to vary smoothly and continuously through the thickness of the layers of the shell and are controlled by volume fraction power law distribution. The expressions for strain–displacement and curvature–displacement relationships are utilized from Love’s first approximation linear thin shell theory. The versatile Rayleigh–Ritz approach is employed to formulate the frequency equations in the form of eigenvalue problem. Influence of material distribution in the two functionally graded layers of the cylindrical shells is investigated on shell natural frequencies for various shell parameters with simply supported end conditions. To check the validity, accuracy and efficiency of the present methodology, results obtained are compared with those available in the literature.     

Thermal buckling and postbuckling of FGM circular plates with in-plane elastic restraints

Based on von Karman's plate theory, the axisymmetric thermal buckling and post-buckling of the functionally graded material(FGM) circular plates with inplane elastic restraints under transversely non-uniform temperature rise are studied. The properties of the FGM media are varied through the thickness based on a simple power law. The governing equations are numerically solved by a shooting method. The results of the critical buckling temperature, post-buckling equilibrium paths, and configurations for the in-plane elastically restrained plates are presented. The effects of the in-plane elastic restraints, material property gradient, and temperature variation on the responses of thermal buckling and post-buckling are examined in detail.     

Experimental buckling of GRP cylindrical shells

During the setup of an experiment, errors may occur. Sources of such errors may be due to several factors which sometimes accumulate and then cause erroneous results. An experimental investigation on buckling of GRP (glass-reinforced-plastic) cylindrical shells, subject to axial compression and/or external pressure loading, has been carried out. At the beginning of the experiment, the initial geometrical imperfections were measured. Because of the small size of these quantities and the great effect these imperfections had on buckling loads, any small errors in the measurement procedure may lead to unreasonable results. Attempts have been made to detect these errors, and to identify and minimize their effect on experimental results. Tables are provided to show a comparison between the final experimental results and the corresponding theoretical ones.     

Nonlinear stability of functionally graded material (FGM) sandwich cylindrical shells reinforced by FGM stiffeners in thermal environment

In this paper, Donnell's shell theory and smeared stiffeners technique are improved to analyze the postbuckling and buckling behaviors of circular cylindrical shells of stiffened thin functionally graded material(FGM) sandwich under an axial loading on elastic foundations, and the shells are considered in a thermal environment. The shells are stiffened by FGM rings and stringers. A general sigmoid law and a general power law are proposed. Thermal elements of the shells and reinforcement stiffeners are considered. Explicit expressions to find critical loads and postbuckling load-deflection curves are obtained by applying the Galerkin method and choosing the three-term approximate solution of deflection. Numerical results show various effects of temperature, elastic foundation, stiffeners, material and geometrical properties, and the ratio between face sheet thickness and total thickness on the nonlinear behavior of shells.     

Dynamic buckling analysis of functionally graded material cylindrical shells under thermal shock

Continuum Mechanics and Thermodynamics - This study focuses on dynamic buckling of functionally graded material (FGM) cylindrical shells under thermal shock. The transient non-uniform temperature...     

Buckling and postbuckling of stiffened cylindrical shells under axial compression

Buckling and postbuckling behaviors of perfect and imperfect,stringer andorthotropically stiffened cylindrical shells have been studied under axial compression.Based on the boundary layer theory for the buckling of thin elastic shells suggested in ref.[1],a theoretical analysis is presented.The effects of material properties of stiffeners andskin,which are made of different materials,on the buckling load and postbuckling behaviorof stiffened cylindrical shells have also been discussed.     

Axial compression buckling of conical and cylindrical shells

Experiments on the axial compression buckling of high-quality epoxy cylindrical shells with imposed dimpletype defects are described. Additionally, a technique for the manufacture of high-quality epoxy conical shells which buckle at loads approaching the classical critical load is presented. For both types of shells, prebuckling deformations have been monitored optically. The sizes of defects determined from the optical examination when applied in the space-frame approach to shell buckling have led to predicted knock-down factors which are remarkably consistent with measured knock-down factors (i.e., the ratio of actual collapse to classical critical load).     

Torsional buckling of spherical shells under circumferential shear loads

ＩｎｔｒｏｄｕｃｔｉｏｎＩｍｐｏｒｔａｎｔａｐｌｉｃａｔｉｏｎｓｏｆｔｈｅｓｔａｂｉｌｉｔｙａｎａｌｙｓｉｓｏｆｓｈｅｌｓｃａｎｂｅｆｏｕｎｄｉｎｔｈｅｍｏｄｅｒｎｅｎｇｉｎｅｒｉｎｇｒａｎｇｉｎｇｏｖｅｒｔｈｅａｅｒｏｓｐａｃｅ,ｍａｒｉｎｅ,ａｒ...     

Initial postbuckling behavior of cylindrical composite shells under axisymmetric deformation

A solution is obtained that describes the postbuckling behavior of cylindrical shells in the case of axisymmetric buckling. The basis for this solution is Koiter’s asymptotic method and the nonlinear equations of the third-order Timoshenko theory of shells. It is shown that the bifurcation point in this case is a symmetrically unstable one. The effect of the initial axisymmetric deflections on the buckling loads is weaker when buckling is axisymmetric. The results obtained by Koiter’s special theory evidence this __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 4, pp. 108–118, April 2006.     

Plastic buckling of cylindrical shells under biaxial loading

The predictions for plastic buckling of shells are significantly affected by the plasticity model employed, in particular in the case of nonproportional loading. A series of experiments on plastic buckling of cylindrical aluminum alloy shells under biaxial loading (external pressure and axial tension), with well-defined loading and boundary conditions, was therefore carried out to provide experimental data for evaluation of the suitability of different, plasticity models. In the experiments, initial imperfections and their growth under load were measured and special attention was paid to buckling detection and load path control. The Southwell plot was applied with success to smooth the results. The results show that axial tension decreases resistance to buckling under external pressure in the plastic region due to softening of the material behavior. Comparison with numerical calculations usingJ ₂ deformation and incremental theories indicate that both theories do not predict correctly plastic buckling under nonproportional loading.Babcock (SEM Member), deceased, was Professor of Aeronautics and Applied Mechanics, California Institute of Technology, Pasadena, CA 91125.