Noncircular Cylindrical Shells Research Articles

Localized families of bending waves in a non-circular cylindrical shell with sloping edges

The initial-boundary-value problem for the equations of shallow shells describing the motion of a non-circular cylindrical shell is considered. The shell edges are given by not necessarily plane curves. The conditions of a joint support or a rigid clamp are considered as boundary conditions. It is assumed that the initial displacements and velocities of the points of the median surface of the shell are functions which decrease rapidly away from some generatrix. In the case when the shell edges lie in planes perpendicular to the generatrix, the solution of the problem can be constructed as an expansion in beam functions along the generatrix. The expansion enables the original initial boundary-value problem to be reduced to an initial problem, the solution of which can be constructed [1] by Maslov's method [2]. A complex WKB procedure is proposed, which is suitable for non-circular cylindrical shells with sloping edges. An asymptotic solution of the equations of motion is constructed by superimposing localized families (wave packets) of flexural waves travelling in a circular direction. A qualitative analysis of the solutions is carried out. As an example wave forms of motion of a cylindrical shell of oblique section are considered.

Vibrations of Composite Noncircular Cylindrical Shells

The free vibrational characteristics of composite noncircular cylindrical shells are investigated in this paper. The shells are composed of layered media of different material properties. The thickness of each layer is considered to be constant. First order composite shell theory, which includes the effects of shear deformation and rotary inertia, is used in the formulation. A combination of Bezier functions and beam functions is used to describe the displacement fields along the circumference and longitudinal directions, respectively, of the shell surface. The shell is modelled using a number of curved cylindrical panels. Displacement (C0), slope (C1) and curvature (C2) continuities between the panels are enforced by proper blending of the Bezier curves. Numerical results are included for a circular sandwich shell and a two-layer cross-ply oval cylinder that provide excellent agreement with those from the literature. The natural frequencies of clamped oval sandwich cylinders made of stiff outer layers and light middle layer are also presented.

A refined problem on the strain of orthotropic noncircular cylindrical shells

Based on the refined Timoshenko theory, a semi-analytic method for solving problems of statics of orthotropic noncircular cylindrical shells is developed. The essence of this method consists in the spline-approximation of a solution in one coordinate direction and utilization of the collocation method and numerical solution of a high-order one-dimensional boundary-value problem by the discrete orthogonalization method in the second direction. The state of stress and strain of an open elliptic cylindrical shell under external load is investigated in the case where three contours rest upon supports and the fourth contour is rigidly fixed. Bibliography: 4 titles.

Nonlinear analysis of laminated non-circular cylindrical shells

The nonlinear analysis of laminated initially imperfect non-circular cylindrical shells is presented. The analytical model is based on Donnell's nonlinear kinematic relations. The equations are derived via the Hu-Washizu mixed formulation, and are expressed in terms of the transverse displacement and the Airy stress function. The curvature of the non-circular cross-section is expanded into a Fourier series, allowing for representation of arbitrary closed cross-sections. The solution procedure is based on expansion of the variables into truncated trigonometric series in the circumferential direction and a finite difference scheme in the longitudinal one. Errors introduced by the truncated series are minimized by the Galerkin procedure and the equations are linearized by the Newton-Raphson method. Solutions beyond the limit point are obtained by Riks' constant arc-length algorithm. Results of both isotropic and laminated, axially loaded oval and elliptic shells are presented. The non-circular configurations are found to be less imperfection sensitive than the circular ones, and for largely eccentric cross-sections the shells are insensitive to initial imperfections.

Steady-State Responses of Laminated Composite Noncircular Cylindrical Shells.

his paper deals with the vibration and steady-state response problems for laminated composite noncircular cylindrical shells. Based on the classical lamination shell theory, equations of motion and boundary conditions are obtained from the stationary condition of the Lagrangian. The equations of motion are solved exactly by means of a power series expansion for symmetrically laminated, cross-ply shells having both ends freely supported and a load distributed uniformly in the circumferential direction and sinusoidally in the axial direction. The effects of the number of laminae and stacking sequences upon the steady-state responses of displacement and bending moment are discussed.

Free Vibrations of Laminated Composite Noncircular Thin Cylindrical Shells

An exact solution procedure is presented for solving free vibration problems for laminated composite noncircular cylindrical shells. Based on the classical lamination theory, strain energy and kinetic energy functional are first derived for shells having arbitrary layer stacking sequences. These functional are useful for a general analysis based upon energy principles. However, in the present work equations of motion and boundary conditions are obtained from the minimum conditions of the Lagrangian (Hamilton’s principle). The equations of motion are solved exactly by using a power series expansion for symmetrically laminated, cross-ply shells having both ends freely supported. Frequencies are presented for a set of elliptical cylindrical shells, and the effects of various parameters upon them are discussed.

Deformation of orthotropic noncircular cylindrical shells in an elastic bed

Deformation of orthotropic noncircular cylindrical shells in an elastic bed

Numerical solution of multipoint boundary-value problems in improved shell theory

For some classes of stratified shells subject to transverse shear (shells of revolution with parameters varying along the generatrix, noncircular cylindrical shells with characteristics varying along the directrix, rectangular shallow shells), we propose an approach that reduces the solution of multipoint boundary-value problems to a number of two-point problems. As an example, we consider the stressed state of an open noncircular cylindrical shell supported in some section along the generatrix.

Vibration Analysis Of Non-circular Cylindrical Shells Using Bezier Functions

The free vibration of non-circular cylindrical shells, having a circumferentially varying thickness, is studied in this paper. The variations of curvature and thickness along the circumference are expressed in terms of two independent parameters. Kirchhoff-Love assumptions of the classical theory of thin shells are used in arriving at the final equations. The analysis is based on the Ritz method in which a combination of eigenfunctions for beams and quintic Bezier functions are used to represent the displacement fields along the longitudinal and circumferential directions, respectively. The shell model comprises a number of panels in the circumferential direction. The C (0) , C (1) and C (2) continuities are enforced between the two adjoining panels. Doubly symmetric oval cylinders having constant thickness and second degree thickness variation in each quadrant are investigated in this paper. The results are compared with those which are available from the literature, and were obtained using conventional closed form and finite strip solutions. A study of the changes in mode shapes with the variation of curvature eccentricity and thickness parameters is also presented.

Large amplitude free vibrations of cylindrical shell on Pasternak foundations

A large amplitude free vibration analysis of non-circular cylindrical shells on Pasternak foundations has been carried out using a new approach due to Sinharay and Banerjee. A modified energy expression for deriving the differential equations for a cylindrical shell on Pasternak foundations is used. Investigations are made for both the movably clamped as well as the immovably clamped edge conditions and it is noticed that even large changes in the values of nondimensional foundation parameters do not have a significant effect on the frequency-amplitude relationship.

Nonlinear Analysis of Transverse Shear Deformable Laminated Composite Cylindrical Shells—Part I: Derivation of Governing Equations

Based on the concept of an “intermediate” class of deformations, a theory suitable for the nonlinear static and dynamic analysis of transverse shear deformable circular and noncircular cylindrical shells, composed of an arbitrary number of linearly elastic monoclinic layers, is developed. The theory is capable of satisfying zero shear traction boundary conditions at the inner and outer shell surfaces. Upon assuming that the shell is subjected to a certain initial stress state and applying the highly nonlinear governing equations derived to the adjacent equilibrium criterion, a set of Love-type linearized equations is further derived. These latter equations are suitable for buckling and/or vibration analyses; in a companion paper, they are solved and used for the study of the influence of transverse shear deformation on the buckling loads of axially compressed cross-ply laminated circular and oval cylindrical shells.

Nonlinear Analysis of Transverse Shear Deformable Laminated Composite Cylindrical Shells—Part II: Buckling of Axially Compressed Cross-Ply Circular and Oval Cylinders

A linearized transverse shear deformable shell theory presented in a companion paper is confined to consideration with the buckling problem of axially compressed, cross-ply laminated noncircular cylindrical shells. Based on a solution of its governing differential equations, obtained for simply supported shells by means of Galerkin’s method, a study of the buckling problem of axially compressed circular and oval cylindrical shells, of a regular antisymmetric cross-ply laminated arrangement, is presented. Moreover, by comparing the numerical results obtained with corresponding results based on a classical Love-type shell theory, the combined influence of both the transverse shear deformation and the shell eccentricity on the buckling loads of such laminated composite shells is examined.

Improved calculation of the stress-strain state of orthotropic noncircular cylindrical shells

Improved calculation of the stress-strain state of orthotropic noncircular cylindrical shells

Analysis of closed cross-section thin-walled structures

A new analytical method is presented for the analysis of the stresses and displacements in a finitely long, thin-walled cylindrical shell structure of arbitrary cross-section whose median surface cross-section curvature is described approximately by a special coordinate transformation function that can map the non-circular cross-section curvature into a circular one. The cylindrical structure is acted upon by an external load and is allowed to have various boundary conditions at its edges. The displacement functions are first of all assumed for a circular cylindrical shell to be respectively a Fourier series in the circumferential direction and the characteristic beam functions which satisfy the various end conditions in the longitudinal direction and then mapped into the arbitrary cross-sectional cylindrical shell. The principle of minimum total potential energy is employed to obtain stresses and displacements for the non-circular cylindrical shell structures. Problems involving oval, square, and U-shaped cross-sectional cylindrical shell structures are studied.

Two-dimensional statics problems of noncircular cylindrical shells

Two-dimensional statics problems of noncircular cylindrical shells

Transient response of cylindrical shells with arbitrary shaped sections

A new coordinate transformation function is presented, with the help of which a cylindrical shell of arbitrary cross-section can be mapped approximately into a circular cylindrical shell. Using this method, the transient response of a non-circular cylindrical shell subjected to a shock wave is obtained by the finite strip method. The versatility and accuracy of the method are amply demonstrated through a number of numerical examples.

Vibrations of circular and noncircular cylindrical shells

In order to better understand the vibrational modes of Chinese two-tone bells [Rossing et al., J. Acoust. Soc. Am. 83, 369–373 (1988)], the vibrational modes of capped and uncapped cylinders with circular and nearly elliptical cross sections have been studied. Mode doublets in the noncircular cylinders replace single modes in the circular cylinders. The higher frequency mode in each doublet, in both the capped and uncapped cylinders, generally belongs to the mode that has a node where the curvature is greatest. The frequency ratio of the doublet pair appears to decrease with increasing m (number of wavelengths around the circumference) but to increase with n (number of nodal circles). The splitting increases with eccentricity, as expected.

Optimal design of a cylindrical shell under overall bending and axial force with respect to creep stability

Optimal structural design of a noncircular cylindrical shell under overall bending and axial force is considered. The material is assumed to be governed by the Norton nonlinear steady creep law. Minimal cross-sectional area is the design objective, the middle line of the profile and the wall-thickness are the design variables and the constraints refer to local stability of the wall according to Gerard's criterion and to brittle creep rupture as described by the Kachanov-Robinson hypothesis. In view of bending, optimal design requires some concentrated areas (longitudinal ribs) located at the outer fibres of the cross-section.

Numerical analysis of the stress state of open noncircular cylindrical shells

Numerical analysis of the stress state of open noncircular cylindrical shells

Exact solutions for the free vibrations of open cylindrical shells with circumferentially varying curvature and thickness

An exact solution procedure is developed for determining the free vibration frequencies and mode shapes of open non-circular cylindrical shells having circumferentially varying thickness and the two opposite, curved edges supported by shear diaphragms. The remaining two edges, which are straight line segments parallel to the shell generators, may have arbitrary boundary conditions. The method is demonstrated for shells having elliptical cylindrical curvature and a thickness which varies quadratically in the circumferential direction, and straight edges which are clamped. For this symmetric configuration, vibration modes separate into symmetric and antisymmetric classes, and the exact frequencies are the roots of fourth order determinants. Numerical results are given showing the variations of frequencies and mode shapes of both symmetry classes with the shell length.