Minimization of the weight of ribbed cylindrical shells made of a viscoelastic composite

The minimization of the weight of ribbed viscoelastic composite cylindrical shells under a long-term external pressure is considered. The shells are strengthened with six inner stiffening rings with identical geometric parameters and a square cross section. It is assumed that the shell material obeys the linear law of hereditary creep and the displacements across the shell wall are distributed according to the Timoshenko hypothesis. The shell must withstand an external pressure of –0.5 MPa without the loss of stability for an unlimited time. The parameters of optimization are the intensity of reinforcement and thickness of its covering and the height and width of the stiffening rings. It is found that the weight of an optimum ribbed shell is 24% lower than that of an optimum cylindrical shell without ribs.     

An isoperimetric inequality in the optimization of circular rings under normal pressure

The problem of determining the optimal cross section of a circular ring so as to maximize the buckling pressure under a given total volume is formulated and solved. An isoperimetric inequality is proved: Among all the circular rings of given mass and radius, the ring with constant bending rigidity along the arc length has the largest critical buckling pressure.     

Critical time of a long multilayer viscoelastic shell

The buckling in stability of a long multilayer linearly viscoelastic shell, composed of different materials and loaded with a uniformly distributed external pressure of given intensity, is investigated. By neglecting the influence of fastening of its end faces, the initial problem is reduced to an analysis of the loss of load-carrying capacity of a ring of unit width separated from the shell. The new problem is solved by using a mixed-type variational method, allowing for the geometric nonlinearity, together with the Rayleigh-Ritz method. The creep kernels are taken exponential with equal indices of creep. As an example, a three-layer ring with a structure symmetric about its midsurface is considered, and the effect of its physicomechanical and geometrical parameters, as well as of wave formation, on the critical time of buckling in stability of the ring is determined. It is found that, by selecting appropriate materials, more efficient multilayer shell-type structural members can be created. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 5, pp. 617–628, September–October, 2007.     

A cylindrical plate with a weakly fixed curvilinear edge made of a transversally isotropic material

Free oscillations and stability under an axial compression of a thin cylindrical plate with a weakly fixed rectilinear edge made of a transversally isotropic material with low stiffness with respect to transverse displacements are considered. The curvilinear edges of the plate are assumed to be hingedly supported. The oscillation frequencies and the critical load for a plate with a free or weakly fixed edge are smaller than those for a shell closed in the circumferential direction. The shapes of oscillations and the forms of stability loss localized near the weakly fixed edge and damped at a distance from it are considered. The Timoshenko-Reissner model is used. Localized forms are analyzed by using a system of equations for Timoshenko-Reissner shallow shells, which is derived for this purpose. The main special feature of this system is that it contains a separate equation describing a solution with large variability. For the example of the stability problem under consideration, the error involved in the system of equations for Timoshenko-Reissner shallow shells is studied. The critical load values obtained with the use of the Kirchhoff-Love and Timoshenko-Reissner models are compared.     

Mechanical stability of functionally graded stiffened cylindrical shells

This paper is concerned with the elastic buckling of stiffened cylindrical shells by rings and stringers made of functionally graded materials subjected to axial compression loading. The shell properties are assumed to vary continuously through the thickness direction. Fundamental relations, the equilibrium and stability equations are derived using the Sander’s assumption. Resulting equations are employed to obtain the closed-form solution for the critical buckling loads. The results show that the inhomogeneity parameter and geometry of shell significantly affect the critical buckling loads. The analytical results are compared and validated using the finite element method.     

Dynamic stability of a porous cylindrical shell

This paper is devoted to a closed cylindrical shell made of a porous-cellular material. The mechanical properties vary continuously on the thickness of a shell. The mechanical model of porosity is as described as presented by Magnucki, Stasiewicz. A shell is simply supported on edges. On the ground of assumed displacement functions the deformation of shell is defined. The displacement field of any cross section and linear geometrical and physical relationships are assumed in cylindrical coordinate system. The components of deformation and stress state were found. Using the Hamilton's principle the system of differential equations of dynamic stability is obtained. The forms of unknown functions are assumed and the system of a differential equations is reduced to a simple ordinary equation of dynamic stability of shell (Mathieu's equation). The derived equation are used for solving a problem of dynamic stability of porous-cellular shell with intensity of load directed in generators of shell. The critical loads are derived for a family of porous shells. The unstable space of family porous shells is found. The influence a coefficient of porosity on the stability regions in Figures is presented. The results obtained for porous shell are compared to a homogeneous isotropic cylindrical shell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A coupled two-scale shell model for comb-like sandwich structures

In this work a coupled two-scale sandwich shell model is proposed, where 4-node quadrilaterals are employed both on the global and the local scale. The coupled global-local boundary value problem is derived by means of a variational formulation and ensuing linearization. A numerical simulation is carried out for linear elastic and elasto-plastic material behavior with small strains. The resulting coupled nonlinear boundary value problem is solved simultaneously in a Newton iteration with incremental load steps. Various types of sandwich models are investigated in the form of uni- and bidirectionally stiffened structures. For the unidirectionally stiffened beam, an analytical reference solution is present by means of classical beam theory. In addition, the numerical results of all coupled calculations are compared to full scale shell models, showing very good agreement while significantly reducing the size of occurring system matrices. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Flattening of a long multilayered viscoelastic cylindrical shell with a varying wall thickness

The stability of a multilayered linearly viscoelastic cylindrical shell with a varying wall thickness under the action of uniformly distributed lateral pressure is investigated. Assuming that the shell is sufficiently long and neglecting the influence of its fastening, the problem posed is reduced to the examination of stability of a compressed ring. The urgency and importance of such problems are connected with the search for reserves of saving of materials with simultaneously increasing the bearing capacity of the structure. In solving problems of this class, the geometric nonlinearity must also be taken into account. The acquisition of efficient analytical solutions here is a rather difficult and sometimes impossible task. This is connected with the integration of nonlinear boundary-value problems with discontinuous coefficients. Therefore, to avoid the existing mathematical obstacles, the problem is solved by using a variational method of mixed type in combination with the Rayleigh–Ritz method. For a three-layer ring with a symmetric structure relative to its median surface, the effect of various geometrically nonlinear theories and of the varying wall thickness on the critical time of stability is revealed numerically.     

Action of a concentrated load on an elastic ring pressed into a circular hole in an anisotropic plate

A solution is obtained for the elastic equilibrium problem for an anisotropic plate with a circular hole into which is pressed a ring of arbitrary cross section symmetrical about the middle surface of the plate. Before deformation the outside radius of the ring differs from the radius of the hole in the plate by the amount of the permissible elastic displacements. A normal concentrated load is applied to the ring. The ring is analyzed in accordance with the theory of thin curved bars. A numerical example is given.I. Franko L'vov State University. Translated from Mekhanika Polimerov, No. 6, pp. 1054–1059, November–December, 1973.     

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.     

Stability of orthotropic cylindrical shells under variable pressure

A method of determining the critical stresses is developed for elastic orthotropic cylindrical shells subjected to nonuniform pressure. It is assumed that the external pressure varies over the cross section and is constant along the length of the cylinder. A shell stability analysis is given for the case of a weakly varying load.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 5, pp. 897–902, September–October, 1970.     

Limiting state of a multilayered nonlinearly elastic long cylindrical shell under the action of nonuniform external pressure

The buckling of a long multilayered nonlinearly elastic shell made of different materials and subject to the action of external pressure is investigated. The load is not hydrostatic and greatly varies in value and direction. Neglecting the effect of end fastening of the shell, the problem is reduced to an analysis of the loss of load-carrying ability of a ring of unit width separated from the shell. The solution is based on a variational method of mixed type formulated for heterogeneous nonlinearly elastic bodies, taking into account the geometrical nonlinearity, in a combination with the Rayleigh–Ritz method. The initial analysis is reduced to solving the Cauchy problem for a nonlinear ordinary differential equation resolved for the derivative. Numerically, using the Runge–Kutta method, the effect of the number of layers and of the parameter of nonuniformity of the external pressure on the critical buckling force is revealed. The urgency and importance of the problem are connected with the research of reserves in the saving of materials with a simultaneous possibility of increasing the load-carrying ability of a structure.     

复合材料三角形网格加筋圆锥壳体位移型稳定性方程及其总体稳定性分析

本文采用Donnell型扁壳理论,首先利用最小势能原理和广义平均筋条刚度法推导出用位移分量表示的复合材料三角形网格加筋叠层圆锥壳体的稳定性方程,考虑了蒙皮最一般的拉弯与拉扭耦合关系和加筋筋条的偏心效应,并讨论了该方程的基本性质.根据外压实验观察结果,通过选取适当的位移分量表达式,并运用Galerkin法分析了在均布外压作用下复合材料三角形网格加筋叠层圆锥壳体总体稳定性,得到了临界载荷的解析表达式,并对某一类C/E复合材料三角形内网格加筋圆锥壳体的临界外压进行了计算,所得理论值与实验结果很好地吻合.最后,讨论了有关参数对临界载荷的影响.本文所建立的新方程和所得结果对于航空航天结构非常有用.     

Dynamic stability of a porous rectangular plate

The study is devoted to a axial compressed porous-cellular rectangular plate. Mechanical properties of the plate vary across is its thickness which is defined by the non-linear function with dimensionless variable and coefficient of porosity. The material model used in the current paper is as described by Magnucki, Stasiewicz papers. The middle plane of the plate is the symmetry plane. First of all, a displacement field of any cross section of the plane was defined. The geometric and physical (according to Hook's law) relationships are linear. Afterwards, the components of strain and stress states in the plate were found. The Hamilton's principle to the problem of dynamic stability is used. This principle was allowed to formulate a system of five differential equations of dynamic stability of the plate satisfying boundary conditions. This basic system of differential equations was approximately solved with the use of Galerkin's method. The forms of unknown functions were assumed and the system of equations was reduced to a single ordinary differential equation of motion. The critical load determined used numerically processed was solved. Results of solution shown in the Figures for a family of isotropic porous-cellular plates. The effect of porosity on the critical loads is presented. In the particular case of a rectangular plate made of an isotropic homogeneous material, the elasticity coefficients do not depend on the coordinate (thickness direction), giving a classical plate. The results obtained for porous plates are compared to a homogeneous isotropic rectangular plate. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Excitation of a Fluid-loaded Plate Stiffened by a Semi-infinite Array of Beams

The excitation of an infinite, fluid-loaded plate with parallel,equally spaced stiffening beams reinforcing one half of it isstudied. The problem is formulated in terms of a discrete convolutionequation for the displacement at the beam positions and is solvedby discrete Fourier transforms coupled with the Wiener-Hopftechnique. The basic ideas are introduced through a reconsiderationof the excitation of an infinite, fluid-loaded plate, stiffenedby a periodic array of beams. As an example, asymptotic expressionsare derived for the reflected, transmitted and scattered fieldsgenerated when a free wave in the unstiffened half of the plateimpinges upon the semi-infinite array. It is shown, in particular,that the far-field motion in the stiffened half of the platehas the form of a Floquet wave. Numerical results for reflectionand transmission coefficients, and for the pressure field radiatedinto the fluid, are presented graphically. Finally, a briefoutline is given of a number of related problems that are solubleby similar techniques.     

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.     

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。     

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.     

Static and dynamic beam forms of the loss of stability of a long orthotropic cylindrical shell under external pressure

A cylindrical shell with end sections which are closed and supported by hinges, in accordance with the concepts of the rod theory, is considered to be under the action of an omnidirectional external pressure which remains normal to the lateral surface during the deformation process. It is shown that, for such shells, the previously constructed consistent equations of the momentless theory, reduced using the Timoshenko shear model to the one-dimensional equations of the rod theory, describe three forms of loss of stability: (1) static loss of stability, which occures through a bending mode from the action of the total end axial compression force since, under the clamping conditions considered, its non-conservative part cannot perform work on deflections of the axial line; (2) also a static loss of stability but one which occurs through a purely shear mode with the conversion of a cylinder with normal sections into a cylinder with parallel sloping sections and a corresponding critical load which is independent of the length of the shell; (3) dynamic loss of stability which occurs through a bending-shear form and can only be revealed by a dynamic method using an improved shear model.