Experimental investigation of the buckling of shallow spherical shells

In the present paper, the buckling behavior of clamped thin shallow spherical shells under external pressure is studied. Seventy-nine plastic shells formed by thermovacuum process were tested. The distributions of initial geometrical imperfections and vertical displacements were minutely measured with a differential transformer. It was possible to control the symmetrical initial geometrical imperfection of each specimen.Results indicate that the buckling phenomena of shallow spherical shells vary greatly with the symmetrical initial imperfection parameter η. In the case of the geometrical parameter λ larger than 5.5, the amplitude of the asymmetrical displacement component with the bifurcation buckling wave calculated by Huang becomes large immediately before buckling. The validity of Huang's theory for an initially perfect shell is experimentally demonstrated.     

Non-linear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects

This paper presents an analytical approach to investigate the non-linear axisymmetric response of functionally graded shallow spherical shells subjected to uniform external pressure incorporating the effects of temperature. Material properties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Equilibrium and compatibility equations for shallow spherical shells are derived by using the classical shell theory and specialized for axisymmetric deformation with both geometrical non-linearity and initial geometrical imperfection are taken into consideration. One-term deflection mode is assumed and explicit expressions of buckling loads and load-deflection curves are determined due to Galerkin method. Stability analysis for a clamped spherical shell shows the effects of material and geometric parameters, edge restraint and temperature conditions, and imperfection on the behavior of the shells.     

Buckling and postbuckling behavior of half-loaded shallow spherical shells

Applying the Galerkin procedure to Marguerre's shallow shell equations, reasonably accurate solutions are obtained for the buckling and postbuckling behavior of clamped shallow spherical shells subjected to external pressure distributed over half the surface.For the geometrical parameter λ smaller than 4.5, the half-loaded shells do not buckle. The buckling loads for 6 ≦ λ ≦ 15 lie about 17–30% below those of uniformly loaded shells. However, the lowest loads after buckling are higher than those for uniformly loading. As λ keeps increasing, the buckling mode shows more and more waves along the circumferential direction. The amplitude of the asymmetrical deflection component Δ₁ becomes remarkably large immediately after buckling. After passing the postbuckling lowest equilibrium point, the sinusoidal deflection disappears and the deflection is changed such that only one location on the shell surface, including the non-loaded area, has been deflected almost symmetrically. The effects of initial imperfections are not so great as for the uniformly loaded case.     

夹层扁球壳的非线性稳定性

基于Reissner假设和变分原理,给出夹层扁球壳在均布压力作用下的大挠度方程,采用修正迭代法求得了夹层扁球壳非线性稳定问题的解析解,得到两类边界条件下临界屈曲载荷的表达式,讨论了几何参数和物理参数对临界屈曲载荷的影响     

Buckling of spherical sandwich shells

An experimental investigation was carried out to determine the critical buckling loads of several shallow spherical sandwich shells. A cold-forming process simultaneously using pressure and vacuum was employed to manufacture the nearly perfect spherical facing layers from 5052 aluminum-alloy sheets of 0.006 and of 0.008-in. thicknesses. Eight shallow spherical-shell specimens of 20-in. base diameter and of 20 and 30-in. radii with 1/8 and 1/4-in. thickness of “Flexcore” have been tested in a 300-psi autoclave specifically designed for these experiments. The pressure on shells was developed by the differential pressure between the inner and the outer chambers separated by the shell being tested. When the inner chamber was maintained at atmospheric pressure and gas pressure was applied in the outer chamber, the testing procedure was termed “soft.” Alternatively, the inner chamber would be filled with fluid with the outer chamber remaining filled with gas. By initially pressurizing both chambers equally, a load on the shell could be developed by the differential pressure due to controlled bleeding of the fluid inside the inner chamber, while the gas in the outer chamber was maintained at the initial pressure. This is an accurate volume-control experiment and this testing procedure was termed “hard.” In the latter case, it was possible to monitor the displacements of the shell for each load increment with a nest of clip gages of an unique design. It was found that there is no substantial difference in the buckling loads between the hard and “soft” systems. All shells buckled in the plastic range. A reasonably good correlation is obtained with a linear theory using the double modulus for the sandwich segments.     

On the nonlinear stability of symmetrically laminated cylindrically orthotropic shallow spherical shells

In this paper,the large deflection theory of symmetrically laminated cylindricallyorthotropic shallow spherical shells is established.Based on this theory,applying themodified iteration method,the analytic solution for critical buckling loads of the shells withrigidly clamped edges under actions of uniform pressure has been obtained.     

大跨度双层网壳的非线性动态响应

网壳结构在大跨度结构中得到广泛应用．在建立了双层网格扁球壳的非线性强迫振动微分方程的基础上,研究了在边缘滑动固定的边界条件下,双层网格扁球壳的非线性动态响应问题．用突变理论建立了该网壳的尖点突变模型,得出了突变的临界方程,并阐述了网壳参数对该结构动态屈曲的影响．     

Effects of uni-directional geometric imperfections on vibrations of pressurized shallow spherical shells

This paper deals with the effects of initial geometric uni-directional imperfections on vibrations of a pressurized spherical shell or spherical cap. The analysis is based upon shallow shell theory. Frequency vs applied pressure interaction curves are plotted for various values of the imperfection amplitude. Imperfections are shown to have a severe effect in reducing the natural frequencies similar to that demonstrated in the buckling behavior of spherical shells.     

Torsional buckling of spherical shells under circumferential shear loads

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A rigorous solution for the axisymmetrical buckling of simply supported cylindrical sandwich shells under axial load

In this paper,based on ref[1],the axisymmetrical buckling of simply supportedcylindrical sandwich shells under the action of uniform axial load is solved by a rigorousmethod.The classical theory of shells is used for the two face sheets and the core isconsidered as a three-dimensional elastic body.A series of transcendental equations areobtained,from which the critical loads can be calculated by numerical methods.Numericalexamples are given to compare with the solutions of sandwich shell theories.     

Nonlinear stability of advanced sandwich cylindrical shells comprising porous functionally graded material and carbon nanotube reinforced composite layers under elevated temperature

The nonlinear stability of sandwich cylindrical shells comprising porous functionally graded material(FGM) and carbon nanotube reinforced composite(CNTRC)layers subjected to uniform temperature rise is investigated. Two sandwich models corresponding to CNTRC and FGM face sheets are proposed. Carbon nanotubes(CNTs) in the CNTRC layer are embedded into a matrix according to functionally graded distributions. The effects of porosity in the FGM and the temperature dependence of properties of all constituent materials are considered. The effective properties of the porous FGM and CNTRC are determined by using the modified and extended versions of a linear mixture rule, respectively. The basic equations governing the stability problem of thin sandwich cylindrical shells are established within the framework of the Donnell shell theory including the von K'arm'an-Donnell nonlinearity. These equations are solved by using the multi-term analytical solutions and the Galerkin method for simply supported shells.The critical buckling temperatures and postbuckling paths are determined through an iteration procedure. The study reveals that the sandwich shell model with a CNTRC core layer and relatively thin porous FGM face sheets can have the best capacity of thermal load carrying. In addition, unlike the cases of mechanical loads, porosities have beneficial effects on the nonlinear stability of sandwich shells under the thermal load. It is suggested that an appropriate combination of advantages of FGM and CNTRC can result in optimal efficiency for advanced sandwich structures.     

Elasto-plastic buckling and post-buckling analysis of sandwich plates with functionally graded metal-metal face sheets and interfacial damage简

Based on the elasto-plastic theory, considering the effect of spherical stress tensor on the elasto-plastic deformation and using the slicing treatment to deal with the plasticity of functionally graded coatings, the elasto-plastic increment constitutive equations of the sandwich plates with functionally graded metal-metal face sheets can be derived. Applying the weak bonded theory to the interfacial constitutive relation and taking into account the geometric nonlinearity, the nonlinear increment differential equilibrium equations of the sandwich plates with functionally graded metal-metal face sheets are obtained by the minimum potential energy principle. The finite difference method and the iterative method are used to obtain the post-buckling path. When the effect of geometrical nonlinearity of the plate is ignored, the elasto-plastic critical buckling load of the sandwich plates with functionally graded metal-metal face sheets can be solved by the Galerkin method and the iterative method. In the numerical examples, the effects of the interface damages, the induced load ratio, the functionally graded index, and the geometry parameters on the elasto-plastic post-buckling path and the elasto-plastic critical buckling load are investigated.     

Thermoelastic stability of bimetallic shallow shells of revolution

This article considers the thermoelastic stability of bimetallic shallow shells of revolution. Basic equations are derived from Reissner’s non-linear theory of shells by assuming that deformations and rotations are small and that materials are linear elastic. The equations are further specialized for the case of a closed spherical cup. For this case the perturbated initial state is considered and it is shown that only in the cases when the cup edge is free or simply supported buckling under heating is possible. Further the perturbated flat state is considered and the critical temperature for buckling is calculated for the case of free and simply supported edges. The temperature–deflection diagrams are calculated by the use of the collocation method for shallow spherical, conical and cubic shells.     

A boundary layer theory for the buckling of thin cylindrical shells under external pressure

Based on the boundary layer theory for the buckling of thin elastic shells suggested in ref. [14]. the buckling and postbuckling behavior of clamped circular cylindrical shells under lateral or hydrostatic pressure is studied applying singular perturbation method by taking deflection as perturbation parameter. The effects of initial geometric imperfection are also considered. Some numerical results for perfect and imperfect cylindrical shells are given. The analytical results obtained are compared with some experimental data in detail, which shows that both are rather coincident.     

爆炸冲击下复合材料层合扁球壳的动力屈曲

研究计及横向剪切的复合材料层合扁球壳在爆炸冲击载荷作用下的非线性轴对称动力屈曲问题。通过在复合材料层合扁球壳非线性稳定性的基本方程中增加横向转动惯量项并引入R.H.Cole理论的爆炸冲击力,得到爆炸冲击下复合材料层合扁球壳的动力控制方程,应用Galerkin方法得到用顶点挠度表达的爆炸冲击动力响应方程,并采用Runge-Kutta方法进行数值求解,采用Budiansky-Roth准则确定冲击屈曲的临界载荷,讨论了壳体几何尺寸对复合材料层合扁球壳冲击屈曲的影响;数值算例表明,此方法是可行的。     

Experimental investigation of the buckling instability of monocoque shells

Buckling of a series of thin-metal, shallow spherical shells under a uniform hydrostatic pressure has been investigated. Stress and deformation histories, as well as the critical buckling pressure and the post-buckling behavior, have been determined. Comparisons with theoretical analyses for buckling of spherical caps are given. Results are presented for an initial phase of a stability study of truncated conical sections which have been subjected to combinations of axial load and lateral pressure. A series of roll-formed and butt-welded, truncated aluminum conical shells with a 75-deg base angle have been tested. Buckle modes for axial-load condition alone, laterial-pressure load alone and combinations of these loading conditions are described. Interaction curves for the conditions investigated are given.     

Non-linear studies of orthotropic shallow spherical shells on elastic foundation

Governing non-linear integro-differential equations for cylindrically orthotropic shallow spherical shells resting on linear Winkler-Pasternak elastic foundations, undergoing moderately large deformations are presented. Three problems, namely, non-linear static deflection response, non-linear dynamic deflection response and dynamic snap-through buckling of orthotropic shells have been investigated. The influences of material orthotropy, foundation parameters and shell-material damping on the deflection response are determined for the clamped and the simply- supported immovable edge conditions accurately. Orthotropy, foundation interaction and material damping play significant roles in improving the load carrying capacity of the shell structures.     

矩形脉冲作用下复合材料层合扁球壳的冲击屈曲分析

研究了计及横向剪切的复合材料层合扁球壳在矩形脉冲载荷作用下的非线性动力屈曲问题;采用Galerkin方法得到以顶点挠度表达的动力响应方程,并用Runge-Kutta方法进行数值求解,应用Budiansky-Roth准则(简称B-R准则)确定冲击屈曲的临界荷载;讨论了壳体几何尺寸和物理参数对复合材料层合扁球壳冲击屈曲的影响;数值算例表明,该方法是可行的.     

Analysis of nonlinear dynamic response and dynamic buckling for laminated shallow spherical thick shells with damage

Based on the nonlinear theory of shallow spherical thick shells and the damage mechanics, a set of nonlinear equations of motion for the laminated shallow spherical thick shells with damage subjected to a normal concentrated load on the top are established. According to Hertz law, the contact force acted upon the shells is determined due to the impact of a mass, and it is related to the mass and initial velocity of the striking object, the geometrical and physical character of the shell. By using the finite difference method and the time increment procedure, the nonlinear equations are resolved. In the numerical examples, the effects of the damage, the initial velocity, and mass of the striking object, the shells’ geometrical parameters on the dynamic responses and dynamic buckling of the laminated shallow spherical thick shells are discussed. Research of Y. Fu, Z. Gao and F. Zhu was supported by National Natural Science Foundation of China (No. 10572049).     

安全壳中钢衬壳热屈曲问题理论与实验研究(I)———理论分析部分

钢衬壳热屈曲问题是核工程安全壳设计中的主要问题把铆固之间的钢衬壳视为钢衬板的特殊缺陷形式，利用Ｋｏｉｔｅｒ初始后屈曲理论分析了完善和具有初始缺陷钢衬壳的弹性热后屈曲性态给出了用挠度－温度载荷表示的钢衬壳的后屈曲平衡路径表达式和屈曲临界载荷表达式具体分析了三种钢衬壳模型：四点铆固钢衬壳、四边固支钢衬壳和五点铆固钢衬壳给出了钢衬的初始缺陷、锚钉间距、钢衬厚度等参数对钢衬热屈曲载荷的影响结果对安全壳中钢衬壳的设计有很好的参考价值