The possibility of a theoretical confirmation of the experimental values of the external critical pressure of thin-walled cylindrical shells

The problem of the buckling of elastic, isotropic, thin-walled cylindrical shells with small initial shape defects that are under the action of an external pressure is solved in a geometrically non-linear formulation. Equations that are identical to Marguerre's equations for a shallow cylindrical shell are used in formulating the problem. The solution is constructed by the Rayleigh–Ritz method with the points of the middle surface of the shell approximated by double functional sums over trigonometric and beam functions. The system of non-linear equations obtained is solved by arc-length methods. Cases of the clamped and supported shells when loading with a lateral and uniform hydrostatic pressure are considered. Its deflections from the limit points of the postbuckling branches of its loading trajectory are used as the initial imperfections. An inspection of the different forms of the initial imperfections when they have maximum values of up to 30% of the shell thickness made it possible to obtain practically the whole range of experimentally found critical pressures.     

复合材料层合剪切圆柱曲板在侧压作用下的后屈曲

基于Reddy高阶剪切变形理论的Kármám-Donnell型非线性壳体方程,给出复合材料层合剪切圆柱曲板在侧压作用下的后屈曲分析。将壳体屈曲的边界层理论推广到复合材料层合剪切圆柱曲板受侧压作用的情况。相应的奇异摄动法,用于确定圆柱曲板的屈曲荷载和后屈曲平衡路径。分析中同时考虑非线性前屈曲变形和初始几何缺陷的影响。数值算例给出完善和非完善,中等厚度正交铺设层合圆柱曲板的后屈曲荷载-挠度曲线。讨论了横向剪切变形,曲板几何参数,铺层数,铺展方式和初始几何缺陷等各种参数变化的影响。     

Creep buckling of glass-reinforced plastic cyclindrical shells

The buckling modes of cylindrical shells made of a polymeric composite with creep properties are considered. The initial imperfections of the shell are characterized by a Fourier series. The time dependence of a large number of harmonics of the Fourier series is investigated by means of nonlinear equations of the Timoshenko type. It is established that some are damped, while others grow at an ever-increasing rate. It is shown that in the course of time the effective shape of the deflections is transformed and at the critical moment the shell buckles in a mode that differs from the shape of the initial deflections.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 697–703, July–August, 1971.     

热-压电效应对轴压混合层合圆柱曲板后屈曲的影响

基于Karman-Donnell型非线性壳体方程，给出带压电作动器混合层合圆柱曲板在机械荷载、电荷载和热荷载作用下的后屈曲分析．假定温度场为均匀分布，电场仅有沿板厚方向的分量Ez，且假定材料性能常数与温度和电场的变化无关。将壳体屈曲的边界层理论推广到混合层合圆柱曲板受复合荷载作用的情况.相应的奇异摄动法用于确定圆柱曲板的屈曲荷载和后屈曲平衡路径.分析中同时考虑非线性前屈曲变形和初始几何缺陷的影响．数值算例给出完善和非完善，含整体覆盖或内埋压电作动器正交铺设层合圆柱曲板的后屈曲平衡路径。讨论了温度变化、控制电压、铺层方式、面内边界条件和初始几何缺陷等各种参数变化的影响。     

Limit equilibrium of a closed transversally isotropic cylindrical shell with a nonthrough longitudinal crack

In the context of an analog of the Leonov-Panasyuk-Dagdeil model we consider the problem of limit equilibrium of a nonshallow transversally isotropic cylindrical shell weakened by a nonthrough surface longitudinal crack. Based on the equations that take account of the initial stresses, we reduce the problem to a system of two singular integral equations with unknown limits of integration. We carry out a numerical analysis of the dependence of the opening of the edges of the crack on the load and the geometric and physico-mechanical parameters of the shell. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 31–36.     

Nonaxisymmetric bulging of cylindrical shells made from polymeric material on prolonged loading

Forms of bulging of cylindrical shells prepared from polymeric materials caused by the action of a constant compressing load in the direction of the generatrix are discussed. The functions of the initial irregularities and the additional radial deflection and stresses are characterized by the Fourier series in a complex representation. When the problem in a geometrically and physically nonlinear formulation was solved, it was found that the form of the deformed surface is transformed, and the bulging of the shell at the critical instant takes place asymmetrically. This agrees with the experimental results.     

Buckling Instability of Sectional Noncircular Cylindrical Composite Shells under Axial Compression

The problem of buckling instability of cylindrical shells under axial compression is considered. The shells consist of cylindrical sections of smaller radius. The geometrical parameters of the shells are approximated by Fourier series on a discrete point set. A Timoshenko-type shell theory is used. The solution is obtained in the form of trigonometric series. It is shown that shells consisting of cylindrical sections have considerable advantages over circular ones. At a constant shell weight, the choice of suitable parameters of shell sections leads to a significant increase in the critical load. The composite shells considered possess higher efficiency indices in comparison with isotropic ones.     

Formulation of the problem of the flutter of a shell of revolution and a shallow shell in a high-velocity supersonic gas flow

Aero-elastic vibration is investigated in the case of a shallow shell of revolution or a cylindrical panel, which respectively occupy a part of a thin cylindrical body or a thin profile, in a high-velocity supersonic gas flow at zero angle of attack. Particular attention is paid to finding the pressure interaction and this problem is solved within the framework of the law of plane sections in boundary-layer theory. An expression is obtained which refines and supplements the well-known formula of “piston” theory. A linearized formulation of the problem of the panel flutter of a shallow shell is presented. Using the example of a plate located on one of the sides of a wedge, it is shown that the formula of “piston” theory is complemented with a term which has the meaning of a compressive force in the plane of the plate. It is shown that, when account is taken of this term, there is a reduction in the critical flow velocity.     

Postbuckling of a shear-deformable anisotropic laminated cylindrical shell under external pressure in thermal environments

A postbuckling analysis is presented for a shear-deformable anisotropic laminated cylindrical shell of finite length subjected to external pressure in thermal environments. The material properties are expressed as linear functions of temperature. The governing equations are based on Reddy’s higher-order shear-deformation shell theory with the von Karman-Donnell-type kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. The boundary-layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling region, and the initial geometric imperfections of the shell, is extended to the case of shear-deformable anisotropic laminated cylindrical shells under lateral or hydrostatic pressure in thermal environments. The singular perturbation technique is employed to determine the interactive buckling loads and postbuckling equilibrium paths. The results obtained show that the variation in temperature, layer setting, and the geometric parameters of such shells have a significant influence on their buckling load and postbuckling behavior. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 789–822, November–December, 2007.     

On a representation of the Green's functions for shallow panels supported on the boundary of a rectangular base

To enhance the convergence of double series in the case of a shallow panel supported on a rectangular base and a long closed cylindrical shell we propose an approximate representation of the Green's function as a combination of rapidly convergent expansions and a closed-form analytic component obtained by approximate summation of one particular part of series using the two-dimensional integral Fourier transform and reduction to the Kelvin functions.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 78–83.     

Investigation of the initial imperfections and buckling modes of glass-reinforced plastic shells under hydrostatic pressure

The results of testing the stability of glass-reinforced plastic cylindrical shells under hydrostatic pressure are examined. The initial imperfections and deflections of the shells were directly measured. It is found that during loading or in the course of time (under load) the shape of the initial imperfections is transformed. An algorithm (see [11]) based on the approximation of the experimental deflection function by a Fourier series is used to establish the characteristic coefficients of the series important in connection with the elastic loss of stability and progressive buckling of the shell.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 6, pp. 1057–1063, November–December, 1971.     

Contact of shells with a sharp linear stamp of finite length

The action of a plane, absolutely rigid stamp on a transversely isotropic shell is investigated. The use of the equations of shells with finite shear stiffness enables the correct formulation of the problem of the action on a shell by a stamp of fixed length. The problem is reduced to an integral equation. Applying the Fourier transform, the kernel of the integral equation is represented in the form of an expansion with respect to Chebyshev polynomials. By the representation of the solution of the integral equation in the form of a product, of a series of Chebyshev polynomials and a function that takes into account the singularities of the solution at the boundary of the contact zone, the considered problem is reduced to the solving of an infinite system of linear algebraic equations, whose coefficients have been determined by the methods of numerical integration. As an example a problem for a cylindrical shell has been solved.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 59–63, 1989.     

The effect of the batdorf parameter on the post-critical behaviour of an axially compressed imperfect cylindrical shell

The loss of stability and post-critical behaviour of a geometrically imperfect elastic cylindrical shell subjected to axial compression at moveable hinged endfaces are asymptotically analysed in the limit as Z → ∞ (where Z is the Batdorf parameter). The asymptotic behaviour of the eigenvalues and associated vectorial eigen-functions, linearized about a torqueless solution of the boundary-value problem are constructed when Z → ∞. The Lyapunov-Schmidt method is applied in the neighbourhood of each eigenvalue for which the asymptotic behaviour has been determined. For Z → ∞ equilibrium eigenshapes that are odd with respect to the axial coordinate are shown to be unstable (the Koiter parameter b < 0), and the even ones (b 0) are shown to be stable. It is shown that by an appropriate choice of initial imperfection the upper critical load for shell loss of stability (the limiting point) can be made to correspond to any of the close to (Z → ∞) critical loads for loss of stability of an ideal shell.     

Investigation of frequencies of natural vibrations of a transversally isotropic cylindrical panel with a circular hole

We consider the problem of natural vibrations of a hingedly supported transversally isotropic cylindrical panel with a circular hole. The deformation of the shell is described by modified equations of the Timoshenko theory of shells. The numerical solution of the problem is constructed by the indirect method of boundary integral equations based on the sequential representation of Green functions.     

A study of the deflection of an isotropic shell under the action of an abruptly applied concentrated normal force

We consider the perturbed deformed state of a thin shallow isotropic shell of nonnegative Gaussian curvature under the action of an abruptly applied concentrated normal force. Using the integral Fourier transform method with respect to the geometric coordinates and the Laplace transform with respect to time we obtain a solution of this problem in the form of double series in generalized Fresnel sine and cosine integrals. We give numerical results for the initial reaction of the shell at the point of application of the load. We give a comparison with known results. Two figures. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 58–63, 1991.     

湿热环境中复合材料层合圆柱薄壳屈曲和后屈曲

在宏-细观力学模型框架下,讨论湿热环境对复合材料层合圆柱薄壳在轴向压缩作用下屈曲和后屈曲行为的影响。基于细观力学模型复合材料性能与湿度和温度变化有关。壳体控制方程基于经典层合壳理论,并包括湿热效应。壳体屈曲的边界层理论被推广用于湿热环境的情况,相应的奇异摄动法用于确定层合圆柱薄壳的屈曲荷载和后屈曲平衡路径。分析中同时计及壳体非线性前屈曲变形和初始几何缺陷的影响。数值算例给出完善和非完善正交铺设层合圆柱薄壳在不同湿热环境中的后屈曲行为。讨论了温度和湿度,纤维体积比率,壳体几何参数,铺层数,铺层方式和初始几何缺陷等各种参数变化的影响。     

湿热环境中复合材料层合圆柱薄壳的屈曲和后屈曲

在宏-细观力学模型框架下,讨论湿热环境对复合材料层合圆柱薄壳在轴向压缩作用下屈曲和后屈曲行为的影响.基于细观力学模型复合材料性能与湿度和温度变化有关.壳体控制方程基于经典层合壳理论,并包括湿热效应.壳体屈曲的边界层理论被推广用于湿热环境的情况,相应的奇异摄动法用于确定层合圆柱薄壳的屈曲荷载和后屈曲平衡路径.分析中同时计及壳体非线性前屈曲变形和初始几何缺陷的影响.数值算例给出完善和非完善正交铺设层合圆柱薄壳在不同湿热环境中的后屈曲行为.讨论了温度和湿度,纤维体积比率,壳体几何参数,铺层数,铺层方式和初始几何缺陷等各种参数变化的影响.     

The non-axisymmetric postbuckling behaviour of shallow spherical domes

The non-axisymmetric postbuckling behaviour of an elastic, shallow spherical dome, which is rigidly clamped along its contour and loaded with a uniform transverse pressure, is considered. The solution of the problem is constructed using the Rayleigh-Ritz method based on the Marguerre equations in which the displacements in the circumferential direction are approximated by a Fourier series, and the radial displacements by Bessel functions. The resulting system of non-linear algebraic equations is solved by prolongation methods. It is shown for the first time that a shell has postbuckling, non-axisymmetric equilibrium states with loads which are significantly less than the upper critical load as well as the loads corresponding to bifurcation points. It is suggested that, taking into account the forms of these equilibrium states as the initial inaccuracies of a spherical dome should enable one to model the spread in its experimentally obtained critical loads.     

The Effect of Inhomogeneity of Elastic Properties on the Stress State in Composite Cylindrical Panels

The effect of inhomogeneity of elastic properties in the circumferential direction on the distribution of stress and displacement fields in orthotropic cylindrical panels is studied. The mechanical properties of the panels and the load acting on them are constant in the axial direction, which makes it possible to neglect the influence of the curvilinear ends. From the initial relations of a three-dimensional problem of the elasticity theory of inhomogeneous anisotropic bodies, a resolving system of partial differential equations is obtained, whose solution is presented in the form of truncated Fourier series, so that the conditions of free support of the rectilinear ends are satisfied. This allows us to separate the variables and to get a system of ordinary high-order differential equations, which is integrated by a stable numerical method. The problem on the stress-strain state of an orthotropic composite panel with a varying relative volume content of reinforcing elements in the circumferential direction is solved. The effect of the change in the reinforcement density on the stresses and displacements of the panel is studied.     

Korn inequalities for shells with zero Gaussian curvature

We consider shells with zero Gaussian curvature, namely shells with one principal curvature zero and the other one having a constant sign. Our particular interests are shells that are diffeomorphic to a circular cylindrical shell with zero principal longitudinal curvature and positive circumferential curvature, including, for example, cylindrical and conical shells with arbitrary convex cross sections. We prove that the best constant in the first Korn inequality scales like thickness to the power 3/2 for a wide range of boundary conditions at the thin edges of the shell. Our methodology is to prove, for each of the three mutually orthogonal two-dimensional cross-sections of the shell, a “first-and-a-half Korn inequality”—a hybrid between the classical first and second Korn inequalities. These three two-dimensional inequalities assemble into a three-dimensional one, which, in turn, implies the asymptotically sharp first Korn inequality for the shell. This work is a part of mathematically rigorous analysis of extreme sensitivity of the buckling load of axially compressed cylindrical shells to shape imperfections.