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.     

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。     

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.     

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.     

Some features of the natural flexural vibrations of circular rings with initial imperfections

The effect of small initial deviations from an ideal circular shape on the frequencies and modes of the natural oscillations of a thin ring under conditions of plane deformation is investigated. The analysis is based on the equations of motion of the shallow shell theory when the length of the shell tends to infinity. A new approach is proposed to the construction of a finite-dimensional model of a ring in which it is assumed that initial imperfections lead to the interaction of flexural and radial vibrations. A number of assertions are formulated and proved regarding the specific features which accompany the vibrations of an imperfect ring that apparently also have to be taken into account when investigating the dynamic characteristics of thin circular cylindrical shells with initial imperfections.     

Initial imperfections and buckling modes of polymeric cylindrical shells under long-term loading

The problem of buckling of polymeric cylindrical shells under long-term loading is considered. The results of compressive and external-pressure tests on shells [10, 11, 12, 17] are discussed and their carrying capacity is estimated [16]. The buckling problem is solved theoretically on the basis of an investigation of the development of axisymmetric modes and their conversion to nonaxisymmetric modes [15]. The equations of the nonlinear theory of viscoelasticity are used as the physical relations. The theoretical model gives qualitative agreement with the experimental behavior of the shell under load.     

非完善加筋圆柱壳在外压和热荷载共同作用下的后屈曲

基于壳体屈曲的边界层理论,本文给出有限长加筋圆柱壳在侧向外压和均布热荷载共同作用下的后屈曲分析。分析中同时考虑壳体非线性前屈曲变形,大挠度和初始几何缺陷的影响。肋条的处理采用“平均刚度”法。采用奇异摄动方法导得壳体屈曲载荷关系曲线和后屈曲平衡路径,并给出完善和非完善,纵向加筋或环向加筋圆柱壳数值算例。     

Deformation of cylindrical composite shells under combined dynamic loading

Conclusions A procedure has been shown for calculating the stress-strain state of cylindrical multilayer shells made from composite materials under the combined action of dynamic axial compression and dynamic external pressure, as well as with different variants of combined loading with static and dynamic forces. An investigation has been made of the effect on the mode of the buckled shell surface of the ratio of the application rate of dynamic loads; ranges of loading rates have been established in which stresses predominate caused either by axial compression or external pressure. It has been shown that, as a result of preliminary static loading, a marked change occurs in the initial imperfections of the shell mode which affects subsequent dynamic buckling. To calculate the time when the first defect occurs and its location in the shell body, a procedure has been devised for layer-by-layer strength analysis employing a tensor-polynomial criterion. It was demonstrated that the level of preliminary static loading noticeably affects the time until the first failure of the layer, not only a reduction of this time being possible with an increase in the static loads, but also an increase in it.We should also point out the work in [10] where it is shown that it is possible to weaken the susceptibility of the shell to initial imperfections when internal pressure is applied.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 461–473, May–June, 1981.     

Optimization of cylindrical composite shells with regard to their critical mode of imperfections

This study deals with the optimum design of composite shells under external pressure with material strength and loss of stability according to the critical mode of imperfections taken as the failure criterion. The problem of optimum design is solved and the critical mode is obtained by nonlinear optimum programming for which the geometric and initial imperfection parameters are treated as variables. Numerical results are obtained for a cylindrical composite shell supported freely at its ends. The effect of shear forces between layers on the load-carrying capacity of the shell is also investigated.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 5, pp. 613–620, September–October, 1998.     

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。     

环肋加劲圆柱壳在静水压力作用下的初始后屈曲分析

本文用Koiter理论分析环肋加劲圆柱壳在静水压力作用下的后屈曲性能.前屈曲状态采用与边界条件一致的非线性有矩方程,本征值问题的解用伽辽金方法求出,得到的临界载荷与经典线性解作了比较.具体计算了三种不同环肋参数的外肋加劲圆柱壳.结果表明,肋的强弱不仅显著影响临界载荷值,同时也改变了柱壳的缺陷敏感度.     

Propagation of Transient Hydroelastic Waves in a Semiinfinite, Piecewise-Constant Cylindrical Shell Containing a Fluid

A solution is formulated for a new problem of wave propagation in a semiinfinite cylindrical shell with a junction connecting two shells of different radii. The material of the shell is assumed to be viscoelastic, and the fluid is assumed to be viscous. The motion of the shell is described by Kirchhoff–Love theory, and the motions of the fluid are described by equations averaged over the cross section. The problem is solved by means of the time Laplace transform and subsequent numerical inversion. The numerical results for the pressure and radial displacement of the shell are analyzed for various values of the parameters.     

Non-classical forms of loss stability of cylindrical shells joined by a stiffening ring for certain forms of loading

A structure in the form of two coaxial cylindrical shells with different radii, joined by a stiffening ring either rigidly or by hinges, is considered. Starting out from improved equations of general form constructed earlier, a linearized contact problem is formulated that enables all possible classical and non-classical forms of loss of stability to be investigated in the case of axisymmetric forms of loading of the structure. The initial relations of the problem are transformed to an equivalent system of integro-algebraic equations containing integral Volterra-type operators by integrating along the longitudinal coordinate and representing the two-dimensional and one-dimensional required unknowns introduced into the treatment in the form of the sum of trigonometric functions in the circumferential coordinate that, in changing into a perturbed state, allows the possibility of the shell deforming in antiphase forms. A numerical algorithm for constructing solutions of the resulting equations is proposed, based on the method of finite sums, that enables all the boundary conditions of the problem and the conditions for the joining of the shells with the stiffening ring to be satisfied exactly. Retaining and discarding parametric terms in the relations for the shells, the stability of a structure of the class considered is investigated in the case when an external pressure acts on the stiffening ring and, also, in the case of its axial tension during which the stiffening ring is found to be under wrench deformation conditions and, in a shell of larger diameter, subcritical circumferential compressive stresses are formed.     

Problems of geometric non-linearity and stability in the mechanics of thin shells and rectilinear columns

An analysis of the current state of the geometrically non-linear theory of elasticity and of thin shells is presented in the case of small deformations but large displacements and rotations, the ratios of which are known as the ratios of the non-linear theory in the quadratic approximation. It is shown that they required specific revision and correction by virtue of the fact that, when they are used in the solution of problems, spurious bifurcation points appear. In view of this, consistent geometrically non-linear equations of the theory of thin shells of the Timoshenko type are constructed in the quadratic approximation which enable one to investigate in a correct formulation both flexural as well as previously unknown non-classical forms of loss of stability (FLS) of thin plates and shells, many of which are encountered in practice, primarily in structures made of composite materials with a low shear stiffness. In the case of rectilinear elastic whereas, which are subjected to the action of conservative external forces and are made of an orthotropic material, the three-dimensional equations of the theory of elasticity are reduced to one-dimensional equations by using the Timoshenko model. Two versions of the latter equations are derived. The first of these corresponds to the use of the consistent version of the three-dimensional, geometrically non-linear relations in an incomplete quadratic approximation and the Timoshenko model without taking account of the transverse stretching deformations, and the second corresponds to the use of the three- dimensional relations in the complete quadratic approximation and the Timoshenko model taking account of the transverse stretching deformations. A series of new non-classical problems of the stability of columns is formulated and their analytical solutions are found using the equations which have been derived with the aim of analyzing their richness of content. Among these are problems concerning the torsional, flexural and shear FLS of a column in the case of a longitudinal axial, bilateral transverse and trilateral compression, a flexural-torsional FLS in the case of pure bending and axial compression together with pure bending and, also, a flexural FLS of a column in the case of torsion and a flexural-torsional FLS under conditions of pure shear. Five FLS of a cylindrical shell under torsion are investigated using the linearized neutral equilibrium equations which have been constructed: 1) a torsional FLS where the solution corresponding to it has a zero variability of the functions in the peripheral direction, 2) a purely beam bending FLS that is possible in the case of long shells and is accompanied by the formation of a single half-wave along the length of the shell while preserving the initial circular form of the cross-section, 3) a classical bending FLS, which is accompanied by the formation of a small number of half-waves in the axial direction and a large number of half-waves in a peripheral direction which is true in the case of long shells, 4) a classical bending FLS which holds in the case of short and medium length shells (the third and fourth types of FLS have already been thoroughly studied in the case of isotropic cylindrical shells), 5) a non-classical FLS characterized by the formation of a large number of shallow depressions in the axial as well as in the peripheral directions; the critical value of the torsional moment corresponding to this FLS is practically independent of the relative thickness of the shell. It is established that the well-known equations of the geometrically non-linear theory of shells, which were formulated for the case of the mean flexure of a shell, do not enable one to reveal the first, second and fifth non-classical FLS.     

Computation of worst geometric imperfection profiles of composite cylindrical shell panels by minimizing the non-linear buckling load

In this article the “most unfavorable” shape of initial geometric imperfection profile for laminated cylindrical shell panel is obtained analytically by minimizing the limit point load. The partial differential equations governing the shell stability problem are reduced to a set of non-linear algebraic equations using Galerkin's technique. The non-linear equilibrium path is traced by employing Newton–Raphson method in conjunction with the Riks approach. A double Fourier series is used to represent the initial geometric imperfection profile for the cylindrical shell panel. The optimum values of these Fourier coefficients are determined by minimizing the limit point load using genetic algorithm. The results are determined for simply supported composite cylindrical shell panel. Numerical results show that more number of terms is needed in Fourier series representation to obtain the “worst” geometric imperfection profile which gives lower limit load compared to single term representation of imperfection. We have incorporated constraints on the shape of imperfection to avoid unrealistic limit point loads (due to imperfection shape) as we have assumed that the imperfection is due to machining/manufactuting.     

Geometrically non-linear equations in the theory of momentless shells with applications to problems on the non-classical forms of loss of stability of a cylinder

Geometrically non-linear and linearized equations in the theory of momentless shells are set up based on the kinematic relations in [Paimushin VN, Shalashilin VI. Relations of the theory of deformations in the quadratic approximation and problems of constructing refined versions of the geometrically non-linear theory of laminar structural components. Prikl Mat Mekh 2005; 69(5): 861–81]. The use of these equations, unlike in the case of the well-known equations, enables one to avoid the occurrence of spurious bifurcation points in solving real problems. Non-classical problems of the stability of cylindrical shells under an external pressure, axial compression and torsion are considered, which can be formulated on the basis of the derived equations of the theory of momentless shells. Their exact analytical solutions are found and enable one to estimate the quality of the previously obtained relations [Paimushin VN, Shalashilin VI. Relations of the theory of deformations in the quadratic approximation and problems of constructing refined versions of the geometrically non-linear theory of laminar structural components. Prikl Mat Mekh 2005; 69(5): 861–81] and the richness of content of the equations which have been constructed compared with well-known equations in the mechanics of thin shells. It is established that the majority of the new forms of loss of stability of cylindrical shells which are revealed relate to a number of shear forms, the onset of which is possible before the flexural forms which have been well studied up to now, in the case of small values of the shear modulus of a shell material with a very highly pronounced anisotropy in its properties.     

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。     

The vibrations of a thin elastic orthotropic circular cylindrical shell with free and hinged edges

The problem of the existence of natural oscillations of a thin elastic orthotropic circular closed cylindrical shell with free and hinge-mounted ends and of an open cylindrical shell with free and hinge-mounted edges, when the two boundary generatrices are hinge-mounted is investigated. Dispersion equations and asymptotic formulae for finding the natural frequencies of possible vibration modes are obtained using the system of equations corresponding to the classical theory of orthotropic cylindrical shells. A mechanism is proposed by means of which the vibrations can be separated into possible types. Approximate values of the dimensionless characteristic of the natural frequency and the attenuation characteristic of the corresponding vibration modes are obtained using the examples of closed and open orthotropic cylindrical shells of different lengths.     

The equations of the geometrically non-linear theory of elasticity and momentless shells for arbitrary displacements

To validate earlier results for the case of arbitrary deformations and displacements in orthogonal curvilinear coordinates, kinematic and static relations of the non-linear theory of elasticity are set up which, in the limit of small deformations, lead, unlike the known relations, to correct and consistent relations. The same relations are also constructed for momentless shells of general form for the case of arbitrary displacements and deformations on the basis of which the problem of the static instability of a cylindrical shell with closed ends, made of a linearly elastic material and under conditions of an internal pressure (the problem of the inflation of a cylinder), is considered. It is shown that, in the case of momentless shells, the components of the true sheat stresses are symmetrical, unlike the three-dimensional case. All the above-mentioned relations are constructed for the loading of deformable bodies both by conservative external forces of constant directions and, also, by two types of “following” forces.     

On the theory of layered anisotropic cylindrical shells stiffened with ribs

The deformation, stability and vibration equations for anisotropic cylindrical shells stiffened with individual longitudinal and circumferential ribs are derived without introducing the hypothesis of nondeformable normals. The more general assumption adopted for layered materials (for example, glass-reinforced plastics) of a linear variation of the displacements over the thickness of the shell and the height of the ribs is used; in this case for the points of contact of the shell and the ribs after deformation the common normals form broken lines. The solution of the problem of the stability of a cylindrical shell stiffened with circumferential ribs is examined. For a shell with different, arbitrarily located ribs the problem is reduced to a homogeneous algebraic system of equations equal in number to three times the number of ribs.Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 647–654, July–August, 1974.