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.     

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 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.     

Dynamic pulse buckling of composite shells subjected to external blast

Dynamic pulse buckling of woven E-Glass/Vinyl Ester and laminated E-Glass/Epoxy cylindrical shells subjected to uniform overpressure and asymmetric pressure pulse (side-on explosion) were examined. The solutions for the radial shell deformations were represented by Mathieu differential equations. The dynamic instability of the shells was determined from a Mathieu stability diagram. It was found that the stability of the shells depended on lay-up, aspect ratio as well as impulse distribution. The stable vibration response of the shells with side-on explosion compared well with finite element solutions using a Dynamic, Implicit analysis in ABAQUS Standard. First-ply failure of the woven E-Glass/Vinyl Ester shell with side-on explosion was predicted using a modified Hashin–Rotem failure criterion. It was shown that the thinner woven E-Glass/Vinyl Ester shells were more likely to fail by dynamic instability, whereas the thicker woven E-Glass/Vinyl Ester shells were more likely to fail by first-ply failure.     

Delamination buckling and postbuckling in composite cylindrical shells under combined axial compression and external pressure

A series of finite element analyses on the delaminated composite cylindrical shells subject to combined axial compression and pressure are carried out varying the delamination thickness and length, material properties and stacking sequence. Based on the FE results, the characteristics of the buckling and postbuckling behaviour of delaminated composite cylindrical shells are investigated. The combined double-layer and single-layer of shell elements are employed which in comparison with the three-dimensional finite elements requires less computing time and space for the same level of accuracy. The effect of contact in the buckling mode has been considered, by employing contact elements between the delaminated layers. The interactive buckling curves and postbuckling response of delaminated cylindrical shells have been obtained. In the analysis of post-buckled delaminations, a study using the virtual crack closure technique has been performed to find the distribution of the local strain energy release rate along the delamination front. The results are compared with the previous results obtained by the author on the buckling and postbuckling of delaminated composite cylindrical shells under the axial compression and external pressure, applied individually.     

Non-linear buckling and postbuckling of shear deformable anisotropic laminated cylindrical shell subjected to varying external pressure loads

Non-linear buckling and postbuckling of a moderately thick anisotropic laminated cylindrical shell of finite length subjected to lateral pressure, hydrostatic pressure and external liquid pressure has been presented in the paper. The material of each layer of the shell is assumed to be linearly elastic, anisotropic and fiber-reinforced. The governing equations are based on a higher order shear deformation shell theory with von Kármán–Donnell-type of kinematic non-linearity and including the extension/twist, extension/flexural and flexural/twist couplings. The non-linear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, moderately thick, anisotropic laminated cylindrical shells with different values of shell parameters and stacking sequence. The results confirm that there exists a circumferential stress along with an associate shear stress when the shell is subjected to external pressure.     

Dynamic short-wave buckling of thin-walled cylindrical shells upon local action of an external pressure pulse

We present the results of experimental investigations of dynamic instability (buckling) of thin-walled cylindrical shells upon local action of an external pressure impulse. In order to create the local short-term loading, we used pulsed radiation from a CO ₂ laser. The test pieces were smooth and reinforced shells made from aluminum alloys. The nature of the buckled wave in the treated region and the magnitude of the critical impulse correspond to the short-wave type of buckling in the shell and are determined by the amplitude of the pressure pulse. Axial static compression of the shell leads to a decrease in the critical impulse and an increase in the number of waves in the meridional direction within the treated region.Translated from Problemy Prochnosti, No. 4, pp. 36–43, April, 1995.     

充液及内空圆柱壳在爆炸荷载下动力屈曲特性研究

采用实验与数值模拟相结合方法,对充液及内空圆柱壳在爆炸载荷下动力屈曲响应特性进行对比研究。将壁厚δ=2.0 mm、外径Φ=100 mm钢圆柱壳（内空及充水）置于75gTNT药柱、200gTNT药柱产生的爆炸场中进行冲击实验,获得不同工况下圆柱壳变形破坏模式。利用动力有限元程序LS-DYNA及Lagrangian-Eulerian流固耦合方法进行数值计算,分析壳壁屈曲变形过程及壳壁关键点速度、水介质内压力等动态参数。计算结果与实验结果一致性较好。研究表明,由于内充水介质的近似不可压缩性,承受冲击荷载时内压增大,因而参与对外界爆炸冲击载荷抗力作用,圆柱壳抗爆能力显著提高。     

Nonlinear buckling and postbuckling behavior of cylindrical nanoshells subjected to combined axial and radial compressions incorporating surface stress effects

In the present study, the Gurtin-Murdoch elasticity theory, as a theory capable of capturing size effects, is implemented to predict the nonlinear buckling and postbuckling response of cylindrical nanoshells under combined axial and radial compressive loads in the presence of surface stress effects. For this purpose, a size-dependent shell mode containing geometric nonlinearity is proposed within the framework of the classical shell theory. Because it is necessary to satisfy balance conditions on the surfaces of nanoshell, it is assumed that the normal stress component of the bulk varies linearly through the shell thickness. On the basis of a variational formulation using the principle of virtual work, the non-classical governing differential equations are derived. Subsequently, a boundary layer theory is employed including the nonlinear prebuckling deformations and the large deflections in the postbuckling regime. Then a two-stepped perturbation methodology is utilized to obtain the size-dependent critical buckling loads and the postbuckling equilibrium paths of nanoshells corresponding to the axial dominated and radial dominated loading cases. It is revealed that in the radial dominated loading case, a positive value of surface elastic constants leads to increase the critical buckling load but decrease the critical end-shortening of nanoshell. However, in the axial dominated loading case, surface elastic constants with positive sign causes to increase the both critical buckling load and critical end-shortening of nanoshell.     

复合材料圆柱壳的轴压屈曲失效试验

为了研究高径比大于1的复合材料圆柱壳的轴压屈曲性能及其失效模式,对2组单向纤维圆柱壳和3组外侧环裹环向纤维圆柱壳进行了轴压试验,观察了试件的受力过程和破坏形态,获得了荷载-位移曲线和荷载-应变曲线,利用有限元模型分析了单向纤维圆柱壳两种屈曲形式的破坏机制,对比分析了两种铺层试件的轴压性能。结果表明:单向纤维复合材料圆柱壳出现先纵向劈裂后板壳屈曲和先柱壳屈曲后纵向劈裂的两种破坏模式;外侧环向纤维可改善圆柱壳的轴压性能,屈曲发展有一定的阶段性并表现出延性特征,破坏形式和承载力均较为稳定。     

On the influence of laminate stacking on buckling of composite cylindrical shells subjected to axial compression

The buckling loads of laminated cylinders can strongly depend on the position of the differently oriented layers within the shell. This paper deals with two different laminated orthotropic cylinders with opposite stacking sequence of the laminate layers. Cylinders of this construction had been thoroughly tested within a BRITE EURAM project. Analytical and semi-analytical methods have been used to predict the buckling loads, and the results are reported in this paper as well as test results for comparison. An explanation of the striking influence of stacking sequence is given. With some more examples the findings are verified. It is suggested that the presented results can be used for benchmarking purpose.     

Elasticity, shell theory and finite element results for the buckling of long sandwich cylindrical shells under external pressure

The buckling of a sandwich cylindrical shell under uniform external hydrostatic pressure is studied in three ways. The simplifying assumption of a long shell is made (or, equivalently, ‘ring’ assumption), in which the buckling modes are assumed to be two-dimensional, i.e. no axial component of the displacement field, and no axial dependence of the radial and hoop displacement components. All constituent phases of the sandwich structure, i.e. the facings and the core, are assumed to be orthotropic. First, the structure is considered a three-dimensional (3D) elastic body, the corresponding problem is formulated and the solution is derived by solving a set of two linear homogeneous ordinary differential equations of the second-order in r (the radial coordinate), i.e. an eigenvalue problem for differential equations, with the external pressure, p the parameter/eigenvalue. A complication in the sandwich construction is due to the fact that the displacement field is continuous but has a slope discontinuity at the face-sheet/core interfaces, which necessitates imposing ‘internal’ boundary conditions at the face-sheet/core interfaces, as opposed to the traditional two-end-point boundary value problems. Second, the structure is considered a shell and shell theory results are generated with and without accounting for the transverse shear effect. Two transverse shear correction approaches are employed, one based only on the core, and the other based on an effective shear modulus that includes the face-sheets. Third, finite element results are generated by use of the ABAQUS finite element code. In this part, two types of elements are used: a shear deformable shell element and a solid 3D (brick) element. The results from all these three different approaches are compared.     

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.     

Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite cylindrical shells

Thermal postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to a uniform temperature rise. The SWCNTs are assumed to be aligned and straight with a uniform layout. Two kinds of carbon nanotube-reinforced composite (CNTRC) shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The governing equations are based on a higher order shear deformation theory with a von Kármán-type of kinematic nonlinearity. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. Based on the multi-scale approach, numerical illustrations are carried out for perfect and imperfect, FG- and UD-CNTRC shells under different values of the nanotube volume fractions. The results show that the buckling temperature as well as thermal postbuckling strength of the shell can be increased as a result of a functionally graded reinforcement. It is found that in most cases the CNTRC shell with intermediate nanotube volume fraction does not have intermediate buckling temperature and initial thermal postbuckling strength.     

Nonlinear torsional buckling and postbuckling of eccentrically stiffened FGM cylindrical shells in thermal environment

The main aim of this paper is to investigate the nonlinear buckling and post-buckling of functionally graded stiffened thin circular cylindrical shells surrounded by elastic foundations in thermal environments and under torsional load by analytical approach. Shells are reinforced by closely spaced rings and stringers in which material properties of shell and the stiffeners are assumed to be continuously graded in the thickness direction. The elastic medium is assumed as two-parameter elastic foundation model proposed by Pasternak. Based on the classical shell theory with von Karman geometrical nonlinearity and smeared stiffeners technique, the governing equations are derived. Using Galerkin method with three-term solution of deflection, the closed form to find critical torsional load and post-buckling load–deflection curves are obtained. The effects of temperature, stiffener, foundation, material and dimensional parameters are analyzed.     

Tests on ring–stiffened cylindrical shells with external pressure

External pressure tests were conducted on thin multi–bay cylindrical shells to investigate the significance of out-of-roundness in relation to elastic and collapse behaviour.
Conclusions are drawn about the developing elastic deflection profiles of the shells during loading and suggestions are proposed for the prediction of the collapse mode and the collapse pressure.     

基于波动法的静水压力下环肋圆柱壳耦合振动特性研究

研究了静水压力下环肋圆柱壳的振动特性。在Flügge理论的基础上,考虑流体的影响,通过变换轴向波数,先后采用波动法和牛顿迭代法得到了不同边界条件下环肋圆柱壳的固有频率值。经过与已有文献数据进行对比,验证了研究的有效性和正确性。通过算例,分析了静水压力、肋条截面尺寸和数目、边界条件等因素对水下环肋圆柱壳固有频率的影响。     

Buckling optimization of hybrid-fiber multilayer-sandwich cylindrical shells under external lateral pressure

This paper describes a method of analyzing the maximum buckling strength of hybrid-fiber multilayer-sandwich cylindrical shells under external lateral pressure with respect to fiber orientations and weighting factors. Formulae for optimizing these parameters are presented and conditions for the application of these formulae are discussed. A discriminant is derived for assessing whether the hybrid sandwich cylindrical shells for given different unidirectional composites should reach the extreme value of critical pressure.     

Delamination buckling of cylindrical shells under axial compression

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.