Dynamics and optimization of cylindrical composite shells

The stability of cylindrical composite shells under dynamic external pressure is discussed. A criterion for determining the load-carrying capacity based on Malmeister's equation with respect to bending parts of deformation is proposed. Optimization of the shell mass relative to various structural parameters has been carried out as a nonlinear programming problem. Numerical results are given.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 1, pp. 81–87, January–February, 1995.     

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.     

Formulation and evaluation of a finite element model for the linear analysis of stiffened composite cylindrical panels

A finite element model for linear static and free vibration analysis of composite cylindrical panels with composite stiffeners is presented. The proposed model is based on a cylindrical shell finite element, which uses a first-roder shear deformation theory. The stiffeners are curved beam elements based on Timoshenko and Saint-Venant assumptions for bending and torsion respectively. The two elements are developed in a cylindrical coordinate system and their stiffness matrices result from a hybrid-mixed formulation where the element assumed stress field is such that exact equilibrium equations are satisfied. The elements are free of membrane and shear locking with correct satisfaction of rigid body motions. Several examples dealing with stiffened isotropic and laminated plates and shells with eccentric as well as concentric stiffeners are analyzed showing the validity of the models.     

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 two-dimensional model of the dynamics of sharp bending of a non-linearly elastic plate

To describe the dynamics of the bending of a thin non-linearly elastic plate, a version of perturbation theory is proposed which correctly takes into account the non-linearity of the medium, the non-uniformity of the deformations along the plate thickness and the boundary conditions on its surface. An effective (2 + 1)-dimensional model is constructed which generalizes the static non-linearly geometrical Föppl-Karman equations. Two-dimensional solitons of the longitudinal deformation are obtained. The conditions for their existence and stability are investigated.     

弹性基上锥壳一般弯曲问题的精确解

本文应用Donnell的简化假定,从弹性基上锥壳位移型微分方程组出发,通过引入一个位移函数U(s,θ)(在极限情况下就退化成V.S.Vlasov对于圆柱壳所引的位移函数[5]),将基本微分方程组化成为一个八阶可解偏微分方程.这个方程的一般解用级数形式给出.对于在实际中有广泛应用价值的Winkler弹性基上锥壳的轴对称弯曲问题,本文给出了详细的数值结果,并求出了边缘荷载作用下的影响系数,这对计算弹性基上锥壳组合结构有着重要的意义.     

Stability and free vibration analyses of double-bonded micro composite sandwich cylindrical shells conveying fluid flow

In this study, based on Reddy cylindrical double-shell theory, the free vibration and stability analyses of double-bonded micro composite sandwich cylindrical shells reinforced by carbon nanotubes conveying fluid flow under magneto-thermo-mechanical loadings using modified couple stress theory are investigated. It is assumed that the cylindrical shells with foam core rested in an orthotropic elastic medium and the face sheets are made of composites with temperature-dependent material properties. Also, the Lorentz functions are applied to simulation of magnetic field in the thickness direction of each face sheets. Then, the governing equations of motions are obtained using Hamilton's principle. Moreover, the generalized differential quadrature method is used to discretize the equations of motions and solve them. There are a good agreement between the obtained results from this method and the previous studies. Numerical results are presented to predict the effects of size-dependent length scale parameter, third order shear deformation theory, magnetic intensity, length-to-radius and thickness ratios, Knudsen number, orthotropic foundation, temperature changes and carbon nanotubes volume fraction on the natural frequencies and critical flow velocity of cylindrical shells. Also, it is demonstrated that the magnetic intensity, temperature changes and carbon nanotubes volume fraction have important effects on the behavior of micro composite sandwich cylindrical shells. So that, increasing the magnetic intensity, volume fraction and Winkler spring constant lead to increase the dimensionless natural frequency and stability of micro shells, while this parameter reduce by increasing the temperature changes. It is noted that sandwich structures conveying fluid flow are used as sensors and actuators in smart devices and aerospace industries. Moreover, carotid arteries play an important role to high blood rate control that they have a similar structure with flow conveying cylindrical shells. In fact, the present study can be provided a valuable background for more research and further experimental investigation.     

Free vibrations of thick layered cylindrical shells

Two approaches to the calculation of closed thick layered cylindrical shells are developed. They are based on division of the cylindrical shell across its thickness by concentric circumferential surfaces into a series of constituent cylindrical shells. Satisfying the contact conditions on the surfaces between constituent shells, it is possible to determine the frequency of free bending vibrations of the initial shell with a sufficient accuracy. In the first approach, the distribution of unknown functions across the shell thickness is sought on the basis of an analytical solution to the corresponding system of differential equations; in the second one, the distribution is assigned by polynomial approximation functions.     

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.     

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.     

Application of the method of partitioning the state of stress to the analysis of thermoelastic shells : PMM vol. 40, n 6, 1976, pp. 1085–1092

By using an asymptotic approach [1], the method of partitioning the state of stress is extended to thermoelastic shells. It is examined in detail in [2] forun-heated shells subjected to the effect of external forces, and consists of representing the total state of stress of the shell as the sum of those simpler states of stress for each of which the simplest methods for their construction can be given.Partitioning of the state of stress was performed in [3] for shells with a constant temperature over the thickness. It was noted in [4] in an analysis of a circular cylindrical shell by bending theory that integrals extended over the whole middle surface, which describe the fundamental state of stress, and integrals which damp out with distance from the edges and represent edge effects are contained in the general solution. In a number of papers, [5] for example, partitioning is performed on the basis of graphic physical representations for simple examples of analyzing circular cylindrical shells.A general approach to the analysis of rigid thermoelastic shells by the partitioning method is described below.     

The dynamic stability and nonlinear vibration analysis of stiffened functionally graded cylindrical shells

A theoretical model is developed to study the dynamic stability and nonlinear vibrations of the stiffened functionally graded (FG) cylindrical shell in thermal environment. Von Kármán nonlinear theory, first-order shear deformation theory, smearing stiffener approach and Bolotin method are used to model stiffened FG cylindrical shells. Galerkin method and modal analysis technique is utilized to obtain the discrete nonlinear ordinary differential equations. Based on the static condensation method, a reduction model is presented. The effects of thermal environment, stiffeners number, material characteristics on the dynamic stability, transient responses and primary resonance responses are examined.     

周向加肋非圆柱壳谐振分析的一个新矩阵方法

基于一类柱壳谐振控制方程呈一阶常微分矩阵方程形式以及傅立叶级数展开,提出了一种新矩阵方法,求解两端简支具有环肋加强非圆柱壳在谐外压作用下的稳态响应．该方法和以往同类方法相比,有两个突出的优点:1) 矩阵微分方程的解采用齐次扩容精细积分法替代龙格-库塔法,提高了精度;其中传递矩阵能实现计算机精确计算．2) 环肋作用力借助Dirac-δ函数和三角级数逼近可以解析求出;除法向作用力外,还考虑了切向作用力．通过数值计算,还研究了外激励频率对壳体位移和应力的影响规律．对比有限元分析与其它方法的计算结果,表明了该方法的正确性和有效性．     

On exact expressions of the bending tensor in the nonlinear theory of thin shells

Some equivalent exact expressions of the bending tensor in the nonlinear theory of thin shells are reviewed. It is noted that the bending tensor, proposed by Shen et al. (2010) [X.Q. Shen, K.T. Li, Y. Ming “The modified model of Koiter’s type for the nonlinearly elastic shells”, Appl. Math. Model. 34 (2010) 3527-3535] as a third-degree polynomial of displacements, is an approximate expression, not the exact one. Then integrability of the fourth kinematic boundary condition, associated with two different but equivalent exact expressions of the bending tensor, is briefly discussed. Finally, a few modified definitions of the bending tensor proposed in the literature are recalled. Within the first-approximation theory they all lead to energetically equivalent models of elastic shells.     

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.     

旋转壳的抗扭刚度

本文列出了旋转壳在包括扭转在内的轴对称变形下的一般平衡方程,并证明了旋转对称壳内的剪应力独立于壳内其它薄膜和弯曲应力.本文求解了只考虑薄膜应力的扭转问题,也求解了考虑弯曲扭应力在内的扭转问题,并指出了在薄壳中,抗扭刚度的主要部份来源于薄膜应力.     

非均匀圆柱壳非线性轴对称变形一般解

本文利用阶梯折算法[1],得到了非均匀圆柱壳非线性轴对称变形的一般解.文中导出了在任意轴对称载荷下求解非均匀圆柱壳非线性弯曲的位移和内力的一般公式,并给出一致收敛于精确解的证明.问题最后归结为求解二元一次代数方程组,文末给出算例.算例表明,无论内力和位移都可得到满意的结果,并收敛于精确解.     

Identification of the elastic modulus of polymeric materials by using thin-walled cylindrical specimens

A method for determining the elastic modulus of polymeric materials from deformation diagrams of thin-walled circular cylindrical shells in compression and tension in the region of geometrical nonlinearity has been elaborated. A numerical solution is found by the finite-element method (ANSYS.) The existence of a unified deformation diagram in generalized coordinates is established, from which the elastic modulus is determined. To validate the method, the eigenfrequencies of cylindrical specimens were found experimentally. The results obtained are compared with FEM calculations.