Natural Vibrations of a Cylindrical Shell Reinforced with Two-Layer Rings

The vibrations of a cylindrical shell reinforced with ring ribs attached to the shell by means of elastic elements are studied. The problem is solved by the finite-element method. The shell and ribs are modelled by a plane four-node finite element, which is a combination of a four-node plane stress element and a four-node flexural element. The effect of the stiffness of the elastic elements on the natural frequencies and modes is examined __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 8, pp. 105–110, August 2005.     

具有复合正交异性铺层和偏心加筋的圆柱壳体的屈曲问题

本文导出了具有正交异性复合铺层和偏心加筋的圆柱壳体在轴压、横向压力或它们的任意组合作用下的屈曲问题的近似解,文中提出的方法使壳中不同铺层及偏心加筋引起的弯曲与拉伸间的耦合研究成为可能。以前的研究方法表明由于忽略了弯一拉耦合效应,所予测的屈曲结果是不完全正确的。     

Study of nonlinear deformation and stability of reinforced shells under combined loading by a bending moment and a transverse boundary force

We present a finite-element statement for the solution of stability problems for reinforced elliptic cylindrical shells with moment properties and nonlinearity in their precritical stressstrain state taken into account. Integrating the equations obtained by equating the linear strain components with zero, we find explicit expressions for the displacements of elements of noncircular cylindrical shells as rigid bodies. Using these expressions, we construct the shape functions of a fourangle finite element of natural curvature and develop an effective algorithm for studying nonlinear deformation and stability of shells. We study the stability of reinforced elliptic cylindrical shells under combined loading by a transverse boundary force and a bending moment and investigate how the ellipticity of the shells and the nonlinearity of deformation at the precritical stage affect the shell stability.     

Natural Vibrations of a Cylindrical Shell with Circular Ribs Attached by Means of Discrete Elastic Elements

The vibrations of a cylindrical shell reinforced with circular ribs attached to it by means of discrete elastic elements are studied. The problem is solved by the finite-element method. The shell and ribs are modeled by a plane four-node finite element, which is a combination of a four-node plane stress element and a four-node flexural element. The effect of the stiffness of the elastic elements on the natural frequencies and modes is examined __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 10, pp. 108–113, October 2005.     

Some problems of optimizing shell shape and thickness distribution on the basis of a genetic algorithm

We consider problems related to designing axisymmetric shells of minimal weight (mass) and the development of efficient nonlocal optimization methods. The optimization problems under study consist in simultaneous search for the optimal geometry and the shell thickness optimal distribution from the minimal weight condition under strength constraints and additional geometric constraints imposed on the thickness function, the transverse cross-section radii distribution, and the volume enclosed by the shell. Using the method of penalty functions, we reduce the above optimal design problem to a nonconvex minimization problem for the extended Lagrange functional. To find the global optimum, we apply an efficient genetic algorithm. We present the results of numerical solution of the optimal design problem for dome-like shells of revolution under the action of gravity forces. We present some data characterizing the convergence of the method developed here.     

Analysis of a strengthened weakly conical shell using a complete system of orthogonal functions on an arbitrary contour of its transverse cross-section

Thin-walled weakly conical and cylindrical shells with arbitrary open, simply or multiply closed contour of transverse cross-sections strengthened by longitudinal elements (such as stringers and longerons) are used in the design of wings, fuselages, and ship hulls. To avoid significant deformations of the contour, such structures are stiffened by transverse elements (such as ribs and frames). Various computational models and methods are used to analyze the stress-strain states of such compound structures. In particular, the ground stress-strain states in bending, transverse shear, and twisting of elongated structures are often analyzed with the use of the theory of thin-walled beams [1] based on the hypothesis of free (unconstrained) warping and bending of the contour of transverse cross-sections. In general, the computations with the contour warping and bending constraints caused by the variable load distribution, transverse stiffening elements, and the difference in the geometric and rigidity parameters of the shell cells are usually performed by the finite element method or the superelement (substructure) method [2, 3]. In several special cases (mainly for separate cells of cylindrical and weakly conical shells located between transverse stiffening elements, with the use of some additional simplifying assumptions), efficient variation methods for computations in displacements [4–8] and in stresses [9] were developed, so that they reduce the problem to a system of ordinary differential equations. In the one-and two-term approximations, these methods permit obtaining analytic solutions, which are convenient in practical computations. But for shells with multiply closed contours of transverse cross-sections and in the case of exact computations by using the Vlasov variational method [4], difficulties are encountered in choosing the functions representing the warping and bending of the contour of transverse cross-sections. In [10], in computations of a cylindrical shell with simply closed undeformed contour of the transverse section, warping was represented in the form of expansions in the eigenfunctions orthogonal on the contour, which were determined by the method of separation of variables from a special integro-differential equation. In [11], a second-order ordinary differential equation of Sturm-Liouville type was obtained; its solutions form a complete system of orthogonal functions with orthogonal derivatives on an arbitrary open simply or multiply closed contour of a membrane cylindrical shell stiffened by longitudinal elements. The analysis of such a shell with expansion of the displacements in these functions leads to ordinary differential equations that are not coupled with each other. In the present paper, by using the method of separation of variables, we obtain differential and the corresponding variational equations for numerically determining complete systems of eigenfunctions on an arbitrary contour of a discretely stiffened membrane weakly conical shell and a weakly conical shell with undeformed contour. The obtained systems of eigenfunctions are used to reduce the problem of deformation of shells of these two types to uncoupled differential equations, which can be solved exactly.     

任意边界条件下环肋圆柱壳振动特性的建模与求解

边界条件对环肋圆柱壳的振动特性有重要影响.基于能量法,把环肋看作离散模型,构建了任意边界条件下加环肋圆柱壳的动力学模型.采用一种改进的傅里叶级数作为位移容许函数,通过瑞利里兹程序求解结构的拉格朗日方程,得到环肋圆柱壳的振动模态和频响特性.通过与实验和有限元(FEM)方法的计算结果进行对比,验证了论文方法的准确性,在此基础上分析了环肋偏心方式、截面尺寸、位置分布和边界弹簧刚度等参数对环肋圆柱壳振动特性的影响.     

Effect of the Number of Ribs on a Transient in a Cylindrical Shell under a Disturbing Load*

International Applied Mechanics - A technique to study the transient process of the forced vibrations of a cylindrical shell stiffened with longitudinal ribs is developed. The shell is under an...     

求解水下纵向加肋无限长非圆柱壳声辐射问题的一种新的半解析方法

基于齐次扩容精细积分法和复数矢径虚拟边界谱方法,利用Fourier积分变换和稳相法,提出了一种具有较高效率和精度的新的求解水下纵向加肋无限长非圆柱壳声辐射问题的半解析方法.考虑了非圆柱壳和肋骨之间同时存在多种相互作用力和力偶矩,较已住很多学者仅计及法向相互作用力更加符合实际.不仅比较了该文方法和精确计算纵向加肋圆柱壳在集中点力激励下的声辐射计算结果,同时还研究了肋骨数量、大小以及椭圆柱壳横截面椭圆度对声辐射特性的影响.数值计算结果表明该文方法较已有的混合FE-BE法更为有效.     

Propagation of harmonic waves along open circular cylindrical shells reinforced with a quasiregular set of discrete longitudinal ribs

We obtain the exact solution describing the propagation of harmonic waves along an open cylindrical shell reinforced with a quasiregular set of discrete longitudinal ribs. Numerical examples are used to examine the effect of discrete ribs on the number and shape of dispersion curves and the effect of the stiffness and inertial characteristics of the ribs on the excitation frequency for given wave parameters     

Free vibrations of rotating composite conical shells with stringer and ring stiffeners

In this paper, an analytical solution for the free vibration of rotating composite conical shells with axial stiffeners (stringers) and circumferential stiffener (rings), is presented using an energy-based approach. Ritz method is applied while stiffeners are treated as discrete elements. The conical shells are stiffened with uniform interval and it is assumed that the stiffeners have the same material and geometric properties. The study includes the effects of the coriolis and centrifugal accelerations, and the initial hoop tension. The results obtained include the relationship between frequency parameter and circumferential wave number as well as rotating speed at various angles. Influences of geometric properties on the frequency parameter are also discussed. In order to validate the present analysis, it is compared with other published works for a non-stiffened conical shell; other comparison is made in the special case where the angle of the stiffened conical shell goes to zero, i.e., stiffened cylindrical shell. Good agreement is observed and a new range of results is presented for rotating stiffened conical shells which can be used as a benchmark to approximate solutions.     

Nonlinear Deformation and Stability of a Noncircular Cylindrical Shell Under Combined Loading with Bending and Twisting Moments

A previously developed technique is used to solve problems of strength and stability of discretely reinforced noncircular cylindrical shells made of a composite material with allowance for the moments and nonlinearity of their subcritical stress–strain state. Stability of a reinforced bay of the aircraft fuselage made of a composite material under combined loading with bending and twisting moments is studied. The effects of straining nonlinearity, stiffness of longitudinal ribs, and shell thickness on the critical loads that induce shell buckling are analyzed.     

The Nested Substructures Method for Solving Large Finite-Element Systems as Applied to Thin-Walled Shells with High Ribs

A direct method is proposed to solve large systems of linear algebraic equations that arise in using the finite-element method. The method suggests to subdivide the initial structure into nonoverlapping, deeply nested substructures. The subdivision is performed automatically by the nested dissection method. Initially, the structure is subdivided into isolated finite elements. The solution process is step-by-step assembly of subsystems with simultaneous elimination of unknowns for completely assembled ones. The efficiency of the method is demonstrated by comparing it with the incomplete Cholesky conjugate-gradient method and the traditional envelope method as applied to a circular cylindrical shell with high ribs     

Dispersion Curves for Harmonic Waves in Longitudinally Reinforced Cylindrical Shells

The paper studies the effect of the discrete arrangement of ribs and their number on the number of dispersion curves for harmonic waves that have a deflection node on a rib's axis and propagate along a cylindrical shell. It is found out that these curves split and the distance between the split curves depends strongly on the torsional stiffness of the ribs     

Propagation of harmonic waves in longitudinally reinforced cylindrical shells with low shear stiffness

The exact solutions of the equations of motion derived under refined theories of shells and ribs based on the Timoshenko model are used to plot dispersion curves for harmonic waves propagating along a cylindrical shell reinforced with discrete longitudinal ribs __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 37–41, May 2006.     

On the stability of nearly cylindrical long shells of revolution

In contrast to [1–4], where the stability problem was studied for shells of medium length, in the present paper we study the stability problem for nearly cylindrical long shells under the action of meridian forces uniformly distributed over their ends and under the action of the normal pressure distributed over the entire lateral surface of the shell. We consider the shells whose generating midsurface shape is determined by a parabolic function. The study is performed for nonaxisymmetric buckling modes by using an equation refined as compared with the equation given in [1]. We consider shells of both positive and negative Gaussian curvature. We assume that the shell ends are freely supported and obtain formulas for the critical load under both separate and joint action of the meridian forces and the pressure. In the specific case of a cylindrical shell under the action of longitudinal compression, the formulas thus obtained imply both the Euler formula and the Southwell-Timoshenko formula [5]. When solving the problem, we use the Bubnov-Galerkin method combined with the optimal approximation method [6].     

A NOVEL SEMI-ANALYTICAL METHOD FOR SOLVING ACOUSTIC RADIATION FROM LONGITUDINALLY STIFFENED INFINITE NON-CIRCULAR CYLINDRICAL SHELLS IN WATER

Based on the extended homogeneous capacity high precision integration method and the spectrum method of virtual boundary with a complex radius vector, a novel semi-analytical method, which has satisfactory computation effectiveness and precision, is presented for solving the acoustic radiation from a submerged infinite non-circular cylindrical shell stiffened by longitudinal ribs by means of the Fourier integral transformation and stationary phase method. In this work,besides the normal interacting force, which is commonly adopted by some researchers, the other interacting forces and moments between the longitudinal ribs and the non-circular cylindrical shell are considered at the same time. The effects of the number and the size of the cross-section of longitudinal ribs on the characteristics of acoustic radiation are investigated. Numerical results show that the method proposed is more efficient than the existing mixed FE-BE method.     

变厚度圆柱壳的强度优化设计

对在任意轴对称分布荷载作用下体积保持常数的变厚度圆柱壳的强度优化设计问题进行了研究。当中面形状固定时 ,采用阶梯折算法 ,用传递矩阵导出了变厚度圆柱壳的初参数解的显式表达式。根据Huber-Mises-Hencky强度准则 ,将变厚度圆柱壳的强度优化转化为极小化当量应力的非线性规划问题 ,并采用投影梯度法建立了问题的优化方法。文中对几个典型问题进行了计算。与等厚度圆柱壳相比较 ,优化圆柱壳的最大当量应力得到了显著降低。本文的研究方法和结果可以用于指导大型圆柱壳体的加肋设计     

Wrinkling of tubes in bending from finite strain three-dimensional continuum theory

The problem of a tube under pure bending is first solved as a generalised plane strain problem. This then provides the prebifurcation solution, which is uniform along the length of the tube. The onset of wrinkling is then predicted by introducing buckling modes involving a sinusoidal variation of the displacements along the length of the tube. Both the prebuckling analysis and the bifurcation check require only a two-dimensional finite element discretisation of the cross-section with special elements. The formulation does not rely on any of the approximations of a shell theory, or small strains. The same elements can be used for pure bending and local buckling a prismatic beam of arbitrary cross-section. Here the flow theory of plasticity with isotropic hardening is used for the prebuckling solution, but the bifurcation check is based on the incremental moduli of a finite strain deformation theory of plasticity.For tubes under pure bending, the results for limit point collapse (due to ovalisation) and bifurcation buckling (wrinkling) are compared to existing analysis and test results, to see whether removing the approximations of a shell theory and small strains (used in the existing analyses) leads to a better prediction of the experimental results. The small strain analysis results depend on whether the true or nominal stress–strain curve is used. By comparing small and finite strain analysis results it is found that the small strain approximation is good if one uses (a) the nominal stress–strain curve in compression to predict bifurcation buckling (wrinkling), and (b) the true stress–strain curve to calculate the limit point collapse curvature.In regard to the shell theory approximations, it is found that the three-dimensional continuum theory predicts slightly shorter critical wrinkling wavelengths, especially for lower diameter-to-wall-thickness (D/t) ratios. However this difference is not sufficient to account for the significantly lower wavelengths observed in the tests.     

A genetic algorithm optimization of ring-stiffened cylindrical shells for axial and radial buckling loads

In this research, the general axial and radial buckling optimization of ring-stiffened cylindrical shells is implemented by the genetic algorithm (GA). The stiffened shell is subjected to four constraints including the fundamental frequency, the structural weight, the axial buckling load, and the radial buckling load. In addition, six design variables including shell thickness, number of stiffeners, stiffeners width and height, stiffeners eccentricity distribution order, and stiffeners spacing distribution order are considered. In analytical solution, the Ritz method is applied and stiffeners are treated as discrete elements. The effect of the weighting coefficients of the objective functions on the optimum solution is studied. The results show that optimized stiffening a cylindrical shell leads to a lower structural weight, higher natural frequencies, and larger axial and radial buckling loads, simultaneously. In addition, the upper and lower bounds of the design variables influence the optimum results considerably. It is also found that the distributions of eccentricity and spacing of the stiffeners influence the magnitudes of the axial and radial buckling loads considerably.