Nonlinear dynamic buckling of functionally graded cylindrical shells subjected to time-dependent axial load

This paper is presented to solve the nonlinear dynamic buckling problem of a new type of composite cylindrical shells, made of ceram/metal functionally graded materials. The material properties vary smoothly through the shell thickness according to a power law distribution of the volume fraction of the constituent materials. The dynamic axial load is set in a linear increase form with regard to time. By taking the temperature-dependent material properties into account, the effect of environmental temperature rise is included. The nonlinear dynamic equilibrium equation of the shell was obtained by applying an energy method, and was then solved using the four-order Runge–Kutta method. The critical condition was eventually determined using B-R dynamic buckling criterion. Numerical results show the dynamic buckling load is higher than its static counterpart. Meanwhile, various effects of the inhomogeneous parameter, loading speed, dimension parameter, environmental temperature rise and initial geometrical imperfection on nonlinear dynamic buckling are discussed.     

Research on nonlinear postbuckling of functionally graded cylindrical shells under radial loads

The nonlinear postbuckling behaviors of functionally graded cylindrical shells (FGCSs) under uniform radial pressure are investigated by using the nonlinear large deflection theory of cylindrical shells. According to an inhomogeneous parameter, the material properties of functionally graded materials vary smoothly through the thickness. With the temperature-dependent material properties taken into account, various effects of thermal environment are compared. In addition, the effects of the inhomogeneous parameter and the dimensional parameters are also investigated. The present theoretical results are verified by the experimental results of homogeneous cylindrical shells.     

Buckling characteristics and layup optimization of long laminated composite cylindrical shells subjected to combined loads using lamination parameters

The buckling characteristics and layup optimization of long laminated composite cylindrical shells subjected to combined loads of axial compression and torsion are examined on the basis of Flügge’s theory. In the buckling analysis of long laminated composite cylindrical shells, 12 lamination parameters are introduced and used as design variables for layup optimization. Applying a variational approach, the feasible region in the design space of the 12 lamination parameters is numerically obtained. The buckling characteristics are discussed in the design space of the 12 lamination parameters. In the layup optimization, the optimum lamination parameters for maximizing the buckling loads and the laminate configurations for realizing the optimum lamination parameters are determined by mathematical programming methods. It is found that in case of combined loads of axial compression and torsion, the optimum laminate configurations are unsymmetric.     

The buckling of FGM truncated conical shells subjected to combined axial tension and hydrostatic pressure

The purpose of this paper is to investigate the elastic buckling of FGM truncated thin conical shells under combined axial tension and hydrostatic pressure. Here axial tensions are separately applied to small and large bases of the truncated conical shell, respectively. It is assumed that the cone is a mixture of metal and ceramic, and that its properties changes as the power and exponential functions of the shell thickness. After giving the fundamental relations, the stability and compatibility equations of an FGM truncated conical shell, subject to combined axial tension and hydrostatic pressure, have been derived. Applying Galerkin’s method general formulas have been obtained for the critical combined and separate loads of FGM conical shells. The appropriate formulas for homogenous and FGM cylindrical shells are found as a special case. Effects of changing shell characteristics, material composition and volume fraction of constituent materials on the critical combined and separate loads of FGM shells with simply supported edges are also investigated. The results obtained for homogeneous cases are compared with their counterparts in the literature.     

Dynamic buckling of FGM truncated conical shells subjected to non-uniform normal impact load

Dynamic buckling of functionally graded materials truncated conical shells subjected to normal impact loads is discussed in this paper. In the analysis, the material properties of functionally graded materials shells are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Geometrically nonlinear large deformation and the initial imperfections are taken into account. Galerkin procedure and Runge–Kutta integration scheme are used to solve nonlinear governing equations numerically. From the characteristics of dynamic response obtain critical loads of the shell according to B-R criterion. From the research results it can be found that gradient properties of the materials have significant effects on the critical buckling loads of FGM shells.     

Buckling of conical composite shells

A semi-analytical approach is proposed to obtain the linear buckling response of conical composite shells under axial compression load. A first order shear deformation shell theory along with linear strain-displacement relations is assumed. Using the principle of minimum total potential energy, the governing equilibrium equations are found and Ritz method is applied to solve them. Parametric study is performed by finding the effect of cone angle and fiber orientation on the critical buckling load of the conical composite shells.     

Free vibration analysis of rotating functionally graded cylindrical shells in thermal environment

The free vibration analysis of rotating functionally graded (FG) cylindrical shells subjected to thermal environment is investigated based on the first order shear deformation theory (FSDT) of shells. The formulation includes the centrifugal and Coriolis forces due to rotation of the shell. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The initial thermo-mechanical stresses are obtained by solving the thermoelastic equilibrium equations. The equations of motion and the related boundary conditions are derived using Hamilton’s principle. The differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to discretize the thermoelastic equilibrium equations and the equations of motion. The convergence behavior of the method is demonstrated and comparison studies with the available solutions in the literature are performed. Finally, the effects of angular velocity, Coriolis acceleration, temperature dependence of material properties, material property graded index and geometrical parameters on the frequency parameters of the FG cylindrical shells with different boundary conditions are investigated.     

Finite-element modelling and buckling analysis of anisogrid composite lattice cylindrical shells

The paper investigates the buckling behaviour of anisogrid composite lattice cylindrical shells under axial compression, transverse bending, pure bending, and torsion. The lattice shells are modelled as three-dimensional frame structures composed of curvilinear ribs subjected to the tension/compression, bending in two planes and torsion. The specialised finite-element model generation procedure (model generator/design modeller) is developed to control the orientation of the beam elements allowing the original twisted geometry of the curvilinear ribs to be closely approximated. The effects of varying the length of the shells, the number of helical ribs and the angles of their orientation on the buckling behaviour of lattice structures are examined using parametric analyses. Buckling of the lattice shells with cutouts is also analysed. The results of these studies indicate that the modelling approach presented in this work can be successfully applied to the solution of design problems.     

A limit-based approach to the stress analysis of cylindrically orthotropic composite cylinders (0/90) subjected to pure bending

Composite cylinders are used in many engineering applications. Solution for a single orthotropic cylinder subjected to bending load was provided by Lekhnitskii [1]. Solution for composite cylinders made of many layers subjected to pure bending has been provided by Jolicoeur and Cardou [4]. However these methods do not provide systematic solution for the cases where the layers are oriented at 0° or 90° with respect to the cylinder axis. The reason is that the number of available equations is fewer than the unknowns due to the fact that some of the equations simplify down to identities. Thus the unique solution cannot be obtained. In order to overcome this problem, a limit-based approach is presented in this paper, where the limit to the cylindrical orthotropy is considered, and by using the Taylor series of expansion, the equations replacing the identically satisfied ones are derived. It is also found that this method is a generalization of the L’Hopital’s Rule to the case of multi-unknowns with multi-equations, some of which are identically satisfied. In the numerical examples, the results by the proposed approach are compared with those obtained from NASTRAN. It is found that they are in good agreement to each other.     

On the vibration and stability of shear deformable FGM truncated conical shells subjected to an axial load

The aim of present study is to investigate the vibration and stability of functionally graded (FG) conical shells under a compressive axial load using the shear deformation theory (SDT). The basic equations of shear deformable FG conical shells are derived using Donnell shell theory and solved using Galerkin's method. The novelty of this study is to achieve closed-form solutions for the dimensionless frequencies and critical axial loads for freely-supported FG truncated conical shells on the basis of the SDT. Parametric studies are made to investigate effects of shear stresses, compositional profiles and conical shell characteristics on the critical parameters. Some comparisons with the various studies have been performed in order to show the accuracy of the present study.     

Analytical study on the buckling of cylindrical shells with stepwise variable thickness subjected to uniform external pressure

This article focuses on the buckling of cylindrical shells with stepwise variable thickness subjected to uniform external pressure. First, combining the method of separation of variables, perturbation method, and Fourier series expansion, an analytical method for the buckling analysis of cylindrical shells with axisymmetric thickness variation subjected to external pressure is established. The method is verified by comparing with the previous results. Then, the stepwise variable thickness of cylindrical shells is described exactly by the arc tangent function. Finally, using the presented method, a general formula for the critical buckling load of cylindrical shells with stepwise variable thickness subjected to uniform external pressure is derived. This general formula is compared and discussed with some empirical formulae in the current design standards. This study lays a theoretical foundation for the calculation of the buckling load of cylindrical shells with stepwise variable thickness subjected to uniform external pressure. Moreover, it provides a reference and guidance for the further revision of related standards.     

Nonlinear bending and postbuckling of FGM cylindrical panels subjected to combined loadings and resting on elastic foundations in thermal environments

A nonlinear analysis is presented for FGM cylindrical panels resting on elastic foundations subjected to the combined actions of uniform lateral pressure and compressive edge loads in thermal environments. The two cases of postbuckling of initially pressurized FGM cylindrical panels and of nonlinear bending of initially compressed cylindrical panels are considered. Heat conduction and temperature-dependent material properties are both taken into account. Material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction based on Mori-Tanaka micromechanics model. The formulations are based on a higher order shear deformation theory and von Kármán strain displacement relationships. The panel-foundation interaction and thermal effects are also included. The governing equations are solved by a singular perturbation technique along with a two-step perturbation approach. The numerical illustrations concern the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels with two constituent materials resting on Pasternak elastic foundations. The effects of volume fraction index, temperature variation, foundation stiffness as well as initial stress on the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels are discussed in detail.     

Buckling of moderately thick laminated cylindrical shells: a review

The present paper is a review article on the problem of buckling of moderately thick, laminated, composite shells subjected to destabilizing loads. The loads consist of uniform axial compression, uniform lateral pressure and torsion applied individually or in combination. In all the works reported in the literature, the analysis is based on higher-order shear deformation (HOSD) shell theory and/or first-order shear deformation (FOSD) shell theory with or without a shear correction factor. Results obtained by these two shell theories and by employing classical thin shell theory are compared to determine the range of applicability of each in predicting critical conditions. The effect of stacking sequence, radius-to-thickness ratio and length-to-radius ratio is assessed. Typical numerical results are presented in tabular form. Moreover, some limited results, which are based on limit point analysis are also presented (imperfection sensitivity studies).     

Buckling and collapse of open and closed cylindrical shells

The buckling of a finite section of a cylindrical shell resembling a two-dimensional contact lens, and the collapse of a tubular shell of infinite extent are considered. The deformation is due, respectively, to the application of an edge force or to a negative transmural pressure. In both cases, the shell develops elastic bending moments due to the deformation from a specified resting shape according to a linear constitutive equation, accompanied by in-plane and transverse shear tensions. In the case of a section of a shell with a flat resting shape, classical results due to Euler and Love show that, as the applied edge force is increased beyond a sequence of thresholds, an infinite family of deformed shapes becomes possible corresponding to buckled states that bifurcate from the zero-curvature resting configuration. It is shown here that a corresponding infinite family of shapes is also possible for a finite shell whose resting shape is a section of circle. These shapes, however, no longer arise from bifurcations, but rather constitute disconnected solution branches of a nonlinear boundary-value problem. A closed cylindrical shell whose cross-section has a circular resting shape exhibits similar bifurcations when the difference between the exterior and interior pressure exceeds a sequence of thresholds, but a shell with a non-circular resting shape deforms into a multitude of shapes described by isolated solution branches. The computed two-dimensional buckled shapes are used to reconstruct the three-dimensional shape of a slowly collapsing fluid-conveying vessel. The reconstruction procedure involves stacking together cross-sections at axial positions that are found by integrating the differential equation determining the axial pressure distribution in unidirectional pressure-driven flow, subject to a constant flow rate. The dimensionless coefficient relating the local pressure gradient to the flow rate is computed by solving the Poisson equation governing unidirectional viscous flow using a boundary-element method, and expressing the flow rate as a boundary integral involving the shear stress which is available from the solution of the boundary-integral equation. In an appendix, the energy of the bending state is discussed with reference to specific choices made by previous authors in various branches of science and engineering.     

Hybrid layerwise-differential quadrature transient dynamic analysis of functionally graded axisymmetric cylindrical shells subjected to dynamic pressure

In this paper, two computationally efficient and accurate solution methods for transient dynamic analysis of functionally graded (FG) cylindrical shells subjected to internal dynamic pressure are presented. In order to accurately account for the thickness effects, the layerwise theory is employed to approximate the displacement components in the radial direction. In the first solution method, differential quadrature method (DQM) is implemented to discretize the resulting equations in the both spatial and time domains. In the second approach, DQM is applied to discretize equations in the axial direction while Newmark’s time integration scheme is used to solve the problem in the time domain. The fast convergence rate of the methods is demonstrated and their accuracy is verified by comparing the results with those obtained using ANSYS and also with available exact solution of a particular problem. Considerable less computational efforts of the proposed approaches with respect to the finite element method is observed. Furthermore, comparative studies are performed between two approaches in different cases and it is found that the two techniques give very close results. The effects of geometrical parameters and boundary conditions on the transient behavior of shells are also investigated.     

Analysis of various thick-walled cross-ply composite cylindrical shells subjected to lateral pressures

A simple and accurate finite cylindrical element method was formulated by the authors to determine stresses and deformations in laminated thick-walled cylindrical composite shells. This method reduces the overall problem to a one-dimensional one. In the previous study it was found that significant tensile radial strains occur near the inner shell surface, when the shell is subjected to external pressures. These radial tensile strains are of the same magnitude as the failure strains for the composite shells for shell dimensions and pressures of practical use. In the present study various stacking sequences are investigated to determine the one resulting in the lowest radial tensile strains. Using this optimum stacking sequence, results are obtained for shells composed of E-glass/epoxy, T300/SP-286 graphite/epoxy, and Kevlar 49/epoxy. In addition, two hybrid composite systems consisting of graphite/epoxy and glass/epoxy are studied.     

轴压作用下矩形开口圆柱壳的稳定性

利用数值方法系统研究带矩形开口的薄壁圆柱壳的稳定性能．对矩形开口的周向角度、高度、轴向位置、开口数量等几何参数进行比较分析,并考察两类几何初始缺陷对结构稳定性的影响．分析表明,影响矩形开口圆柱壳轴压下稳定性的主要因素是周向开口角度,屈曲荷载随开口角度的增大而线性下降,而开口高度及开口的轴向位置对屈曲荷载的影响很小．此外,矩形开口圆柱壳对以特征值屈曲模态分布的初始缺陷并不敏感,但对周向轴对称凹陷十分敏感．     

Crushing behaviour of composite hemispherical shells subjected to quasi-static axial compressive load

This paper examines the effects of composite constituents and geometry on the energy absorption capability of composite hemispherical shells. To examine the effects of matrix types on their energy absorption capability, glass fibre/epoxy and glass fibre/polyester hemispherical shells were fabricated. While glass fibre/epoxy and carbon fibre/epoxy hemispherical shells were fabricated to investigate the effect of fibre reinforcements. Effect of aspect ratio (R/t) was also examined and the results were presented. The results obtained showed that the energy absorption capability of the hemispherical shells significantly affected by the composite constituents as well as R/t ratio.     

Post-buckling analysis of stiffened braided cylindrical shells under combined external pressure and axial compression

A post-buckling analysis is presented for a stiffened braided thin shell subjected to combined loading of external pressure and axial compression. The effects of the nonlinear large deflection and the initial geometrical imperfection are considered in the formulations. The analysis uses perturbation method to determine the interactive buckling loads and the post-buckling equilibrium paths and a three-cell model are used to obtain the elastic constants of braided shell. Some effects such as imperfection parameter, stiffened and Braiding parameters on the post-buckling path are discussed in the article.