Classical and nonlinear buckling analyses of spherical sandwich shells

The buckling and initial postbuckling behavior of clamped shallow spherical sandwich shells with dissimilar face sheets under a uniform pressure is studied. The numerical results show that the buckling and initial post-buckling behavior of clamped shallow spherical sandwich shells with dissimilar face sheets is similar to that of the corresponding homogeneous shell.     

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     

Torsional buckling of spherical shells under circumferential shear loads

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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.     

Nonlinear static and dynamic analysis for laminated,annular, spherical caps of moderate thickness

Based on Timoshenko-Mindlin kinematic hypothesis, the shallow shell theory is extended to include the transverse shear deformation for the nonlinear axisymmetric dynamic analysis of the symmetric cross-ply shallow spherical shell. Using the orthogonal point collocation method and the Newmark scheme, an iterative solution is formulated. The numerical results for the nonlinear static and dynamic responses and dynamic buckling of these shallow spherical shells with circular holes under uniformly distributed static or dynamic normal impact loads are presented and compared with available data.     

A numerical calculation of dynamic buckling of a thin shallow spherical shell under impact

Assuming the deformation of the shell has an axial symmetrical form, we transform Marguerre's equations into difference equations, and use these equations to discuss the buckling of an elastic thin shallow spherical shell subjected to impact loads. The result shows when impact load acts on the shells, a jump of the shell takes place dependent on the values λ and the critical buckling load increases with the enlargement of the loading area.     

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.     

Nonlinear stability of double-deck reticulated circular shallow spherical shell

Based on the variational equation of the nonlinear bending theory of doubledeck reticulated shallow shells, equations of large deflection and boundary conditions for a double-deck reticulated circular shallow spherical shell under a uniformly distributed pressure are derived by using coordinate transformation means and the principle of stationary complementary energy. The characteristic relationship and critical buckling pressure for the shell with two types of boundary conditions are obtained by taking the modified iteration method. Effects of geometrical parameters on the buckling behavior are also discussed.     

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.     

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.     

S型功能梯度扁球壳非线性屈曲的渐近分析

本文利用渐近迭代法获得了边界弹性支撑的功能梯度扁球壳的非线性屈曲问题的理论解．假设材料组分体积分数沿壳体厚度方向呈sigmoid幂函数变化,边界上考虑一般的弹性支撑约束．基于经典的薄壳理论和几何非线性关系,导出了S型功能梯度扁球壳的非线性屈曲问题的控制方程．采用渐近迭代法通过两次迭代得到了无量纲挠度和均布荷载之间的非线性特征关系．通过与已有有限元方法和其他数值方法的结果对比,验证了本文解的有效性和高精度．同时,通过计算阐述了缺陷因子、材料参数、边界约束系数及特征几何参数对壳体临界屈曲荷载的影响．     

An approximation to red blood cells with a model of three-center-combined shells

Red blood cells present a biconcave shape and bear an inner pressure (osmotic pressure) when they are in the static state. In this paper, a model of “three-center-combined shells”, which consists of two spherical shells and a toroidal shell, is employed to describe the geometric shape of red blood cells. Surface area and volume of the combined shells model are very close to those measured from experiment. The stress distribution in the cell membrane is formulized as a closed form according to the Novozhilov's theory of the three-center-combined shells. Calculating results in terms of Novozhilov's formula give a good agreement with the numerical results given by ABAQUS when using actual measurements. It is concluded that the combined shells model can well approximate to the biconcave structure of red blood cells. In addition, stress calculation shows that the membrane of biconcave red blood cells can carry bending moments, and the moments reach a maximum value in the vicinity of joint line of the spherical shell and the toroidal shell in the combined shells model.     

Thermal buckling of axisymmetrically laminated cylindrically orthotropic shallow spherical shells including transverse shear

The nonlinear thermal buckling of symmetrically laminated cylindrically orthotropic shallow spherical shell under temperature field and uniform pressure including transverse shear is studied. Also the analytic formulas for determining the critical buckling loads under different temperature fields are obtained by using the modified iteration method. The effect of transverse shear deformation and different temperature fields on critical buckling load is discussed.     

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．     

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.     

Experiments on the stability of spherical and paraboloidal shells

This paper presents the experimental part of a study of the elastic stability of clamped and hinged spherical and paraboloidal shell caps subjected to external uniform pressure loading. Ninety-six poly-vinyl-chloride (PVC) shells of 15-in. base diameter and formed by thermovacuum process, were tested and their critical loads recorded. The effects of geometry, clamped and hinged boundary conditions, height/span and radius of curvature/thickness ratios on the critical pressure were investigated. The results are compared, where possible, with experimental data obtained by previous investigators.     

扁球壳轴对称非线性弯曲和稳定的权余解

本文从扁球壳的积分方程组出发,通过新定义的残差表达式,用权余法详细地研究了扁球壳轴对称非线性弯曲和稳定问题.通过数值计算可以看出,本方法应用方便,精确可靠.     

Repeated buckling and its influence on the geometrical imperfections of stiffened cylindrical shells under combined loading

The present experimental study aims at providing better inputs for improvement of the buckling load predictions of stiffened cylindrical shells subjected to combined loading. The work focuses on two main factors which considerably affect the combined buckling load of stiffened shells, namely geometric imperfections and boundary conditions. Six shells with nominal simple supports were tested under various combinations of axial compression and external pressure. The vibration correlation technique is employed to define the real boundary conditions. The geometric imperfections of the integrally stiffened shells are measured in the present experiments in situ and are used as inputs to a multimode analysis which yields the corresponding “knockdown” factor for various combinations of loading. Thus, when employing the repeated buckling procedure for obtaining interaction curves, each point on the curve is adjusted (using the multimode analysis) for the measured “new” surface of the shell and this results in more realistic interaction curves. The geometrical imperfections of the preloaded shells can also serve as an input to the International Imperfection Data Bank for future studies on the correlation between the manufacturing method of the shell and their geometric imperfections.     

NON-SYMMETRICAL LARGE DEFORMATION OF A SHALLOW THIN SPHERICAL SHELL

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