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Buckling behavior of a radially loaded corrugated orthotropic thin-elliptic cylindrical shell on an elastic foundation
Abstract
This paper is aimed to investigate how the corrugation parameters and the Winkler foundation affect the buckling behavior of isotropic and orthotropic thin-elliptic cylindrical shells with cosine-shaped meridian subjected to radial loads. The buckling three-dimensional equations of the shell are amended by including the Winkler foundation modulus based on the Flügge thin shell theory and Fourier's approach is used to deform the displacement fields as trigonometric functions in the longitudinal direction of shell. Using the transfer matrix of the shell, the governing equations of buckling can be written in a matrix differential equation of variable coefficients as a one-dimensional boundary-value problem that is solved numerically as an initial-value problem by the Romberg integration approach. The proposed model is adopted to determine the basic loads and the corresponding buckling deformations for the symmetrical and antisymmetrical modes of buckling. The sensitivity of the buckling behavior and bending deformations to the corrugation parameters, Winkler foundation moduli, ellipticity and orthotropy of the shell structure is studied for different type-modes of buckling. The obtained results indicate that this design model allows us to find out how the local properties of the shell and its stiffness in existence of an elastic foundation are related and would serve as benchmarks for future works in this important area.
... In order to elaborate a quantitative description of the effective shape transformations of the corrugated shells, such parameters related to the mechanical properties of the folded sheets as shear or normal stresses and strains are examined [42,43]. The effect of these transformations is to be such a state of equilibrium in which the effort of each shell fold is going to be the smallest possible and the work of the internal forces is balanced out [44]. ...
... The above dependencies were obtained by means of the experimental tests on the experimental stand [14,20,42] (see Figure 6). They made it possible to configurate the elaborated thin-walled FEM computer models [23] (Figures 14 and 15) used in computer simulations whose results are presented in the next section. ...
... Based on the experimental studies carried out in the Rzeszow University of Technology hall [19,42], there was elaborated a diagram composed of curves corresponding to various profiles, representing relations between the normal stresses acting orthogonally to the directions of the longitudinal axes and α j unit twist angle of a shell fold ( Figure 21). The curves presented in Figure 22 and corresponding to various profiles express the relations between the σ xx normal stresses acting along the longitudinal axes of the shell folds and α j unit angle of these folds. ...
This article provides a novel insight into specific properties of flat folded sheets transformed elastically into building roof shells. Elastic twist transformations of the sheets resulting from the arrangement of the sheets on two skew roof directrices cause changes in the geometric and mechanical sheet properties of the roof shell sheeting composed of these sheets. Regular smooth-ruled surfaces and their characteristic lines are used in the analysis of changes in the geometric properties. In the analysis of the mechanical changes, the constitutive relations and complex state of stresses are considered. The analysis is carried out on the basis of the results of the experimental tests and FEM computer simulations. They have led to the development of such a method of shaping of the effectively transformed folded covers that ensures the initial effort of each shell fold to be the smallest possible.
... Ghazijahani et al. [28] presented experimental results on the buckling behavior of corrugated cylindrical shells and cylindrical shells stiffened by corrugated rings under a uniform peripheral pressure. Ahmed [29] developed a simply analytical approach for investigating the linear buckling behavior of a cosine corrugated elliptic cylindrical shell subjected to external pressure and surrounded by an elastic foundation. Additionally, Xiong et al. [30] presented a fabrication process and experimental results of mechanical responses of carbon fiber composite core-corrugated sandwich cylindrical shells. ...
This paper presents a semi-analytical approach for investigating the nonlinear buckling and postbuckling of spiral corrugated sandwich functionally graded (FGM) cylindrical shells under external pressure and surrounded by a two-parameter elastic foundation based on Donnell shell theory. The improved homogenization theory for the spiral corrugated FGM structure is applied and the geometrical nonlinearity in a von Karman sense is taken into account. The nonlinear equilibrium equation system can be solved by using the Galerkin method with the three-term solution form of deflection. An explicit solution form for the nonlinear buckling behavior of shells is obtained. The critical buckling pressure and the postbuckling strength of shells are numerically investigated. Additionally, the effects of spiral corrugation in enhancing the nonlinear buckling behavior of spiral corrugated sandwich FGM cylindrical shells are validated and discussed.
... Duc et al. (2016) analyzed nonlinear thermal stability of FGM elliptical cylindrical shells reinforced by eccentrically stiffened. Ahmed (2016) considered the effects of the crease parameters and the elastic foundation on the buckling behavior of isotropic and orthotropic elliptic cylindrical shells with cosineshaped meridian under radial loads. Recently, Xu et al. (2017) introduced a recent computational study to obtain the nonlinear buckling resistance of perfect elastic circular cylindrical shells subjected to uniform bending. ...
This paper investigated the nonlinear vibration and dynamic response of the carbon nanotube polymer composite elliptical cylindrical shells on elastic foundations in thermal environment. The material properties of the nanocomposite elliptical cylindrical shells are assumed to depend on temperature and graded in the thickness direction according to various linear functions. The shell is subjected to the combination of the uniformly distributed transverse load in harmonic form and the uniform temperature rise. The motion and geometrical compatibility equations are derived based on the Reddy’s higher order shear deformation shell theory. The natural frequencies and the deflection amplitude–time curves of the shell are determined by using the Galerkin method and fourth-order Runge–Kutta method. The numerical results show not only the positive influences of carbon nanotube volume fraction and elastic foundations but also the negative influences of initial imperfection and temperature increment on the nonlinear vibration and dynamic response of the carbon nanotube polymer composite elliptical cylindrical shells. The reliability of the present results is verified by comparing with other publications.
... Zhang et al. [19] proposed an analytical method to investigate free and forced vibration behavior of submerged finite elliptic cylindrical shells. Ahmed [20] investigated the buckling behavior of corrugated orthotropic thin elliptic cylindrical shells subjected to radial loads. Silvestre and Gardner [21] studied elastic local post-buckling behavior of elliptical tubes under compression. ...
In this study, geometrically nonlinear dynamic behavior of laminated composite super-elliptic shells is investigated using generalized differential quadrature method. Super-elliptic shell can represent cylindrical, elliptical or quasi-rectangular shell by adjusting parameters in super-ellipse formulation (also known as Lamé curve formulation). Geometric nonlinearity is taken into account using Green–Lagrange nonlinear strain–displacement relations that are derived using differential geometry and theory of surfaces. Transverse shear effect is considered through the first-order shear deformation theory. Equation of motion is obtained using virtual work principle. Spatial derivatives in equation of motion is expressed with generalized differential quadrature method and time integration is carried out using Newmark average acceleration method. Several super-elliptic shell problems under uniform distributed load are solved with the proposed method. Effects of layer orientations, boundary conditions, ovality and ellipticity on dynamic behavior are investigated. Transient responses are compared with finite element solutions.
... Hosseini-Hashemi et al. [5] concentrated on presenting a reliable and accurate method for free vibration of functionally graded viscoelastic cylindrical panel under various boundary conditions. Based on 3D elasticity theory, Norouzi and Alibeigloo [6] carried out thermoviscoelastic analysis of FGM cylindrical panel under thermal and/or mechanical load; whilst Ahmed [7] investigated how the corrugation parameters and the Winkler foundation affect the buckling behavior of isotropic and orthotropic thin-elliptic cylindrical shells with cosine-shaped meridian subjected to radial loads. Quan et al. [8] focused on the nonlinear dynamic analysis and vibration of shear deformable imperfect eccentrically stiffened functionally graded thick cylindrical panels, taking into account the damping when subjected to mechanical loads. ...
This study follows an analytical approach to investigate the nonlinear dynamic response and vibration of eccentrically stiffened sandwich functionally graded material (FGM) cylindrical panels with metal–ceramic layers on elastic foundations in thermal environments. It is assumed that the FGM cylindrical panel is reinforced by the eccentrically longitudinal and transversal stiffeners and subjected to mechanical and thermal loads. The material properties are assumed to be temperature dependent and graded in the thickness direction according to a simple power law distribution. Based on the Reddy’s third-order shear deformation shell theory, the motion and compatibility equations are derived taking into account geometrical nonlinearity and Pasternak-type elastic foundations. The outstanding feature of this study is that both FGM cylindrical panel and stiffeners are assumed to be deformed in the presence of temperature. Explicit relation of deflection–time curves and frequencies of FGM cylindrical panel are determined by applying stress function, Galerkin method and fourth-order Runge-Kutta method. The influences of material and geometrical parameters, elastic foundations and stiffeners on the nonlinear dynamic and vibration of the sandwich FGM panels are discussed in detail. The obtained results are validated by comparing with other results in the literature.
... Shariati and Rokhi [28] proposed the numerical and experimental investigations on buckling of steel cylindrical shells with elliptical cutout subject to axial compression. Ahmed [29] investigated the buckling behavior of a radially loaded corrugated orthotropic thin-elliptic cylindrical shell on an elastic foundation. ...
Elliptical cylindrical shell is one of shells with special shape. Up to date, there is no publication on vibration and dynamic of functionally graded elliptical cylindrical shells. Therefore, the purpose of the present study is to investigate the nonlinear dynamic response and vibration of imperfect eccentrically stiffness functionally graded elliptical cylindrical shells on elastic foundations using both the classical shell theory (CST) and Airy stress functions method with motion equations using Volmir's assumption. The material properties are assumed to be temperature - dependent and graded in the thickness direction according to a Sigmoid power law distribution (S-FGM). The S-FGM elliptical cylindrical shell with metal-ceramic-metal layers are reinforced by outside metal stiffeners. Both the S-FGM elliptical shell and metal stiffeners are assumed to be in thermal environment and both of them are deformed under temperature simultaneously. Two cases of thermal loading (uniform temperature rise and temperature variation through thickness) are considered. The nonlinear motion equations are solved by Galerkin method and Runge-Kutta method (nonlinear dynamic response, natural frequencies). The effects of geometrical parameters, material properties, elastic foundations Winkler and Pasternak, the nonlinear dynamic analysis and nonlinear vibration of the elliptical cylindrical shells are studied. The some obtained results are validated by comparing with those in the literature.
The protective shield of PMT (photon multiplier tubes) is a typical spherical-cylindrical thin-walled shell structure which works in the environment of deep water pressure with the material of plexiglass and character of low Young's Modulus. Therefore, it is significant to analyze the stability of the structure. The validity of eigenvalue buckling prediction and arc length method was verified by classic buckling theory of rotation shell firstly. Then the stability of plexiglass shield was researched by these two methods and a general calculation formula of shallow shell was fitted. The results show that the critical stress of plexiglass shield decreases with the reduction of section roundness; if the height of cylindrical shell is less than 0.3R, there is no influence on sphere-cylindrical shell structure, but when the height is more than 0.3R, the structure stability decreases with the increasing of cylindrical shell height. If the influence of exhaust orifice is considered, the geometry of shell is the main factor for the stability of shell when the orifice radius is less than 0.3 times of orifice spacing. Otherwise, orifice is the main factor for the stability of shell. There is a significant influence of initial imperfection on the critical stress of the protective shield and the critical stress decreases with the increasing of initial deformation. The research of this paper can provide useful guidance for the design of PMT plexiglass shield.
Based on the framework of Flügge's shell theory, transfer matrix approach and Romberg integration method, we investigated how the thermal gradient affects the vibration behavior of rotating isotropic and orthotropic oval cylindrical shells. The governing equations of orthotropic oval cylindrical shells, under parabolically varying thermal gradient around its circumference, with consideration of the effects of initial hoop tension and centrifugal forces due to the rotation are derived, and they are put in a matrix differential equation as a boundary-value problem. As a semianalytic solution, the trigonometric functions are used with Fourier's approach to approximate the solution in the longitudinal direction, and also to reduce the two-dimensional problem in to an one-dimensional one. Using the transfer matrix approach, the equations can be written in a matrix differential equation of first-order and solved numerically as an initial-value problem. The proposed model is applied to get the natural frequencies and vibratory displacement of the symmetrical and antisymmetrical vibration modes. The sensitivity of the vibration behavior to the rotational speed, the thermal gradient, the ovality and orthotropy of the shell is studied for different type-modes of vibration. The present method is found to be accurate when compared with the results available in the literature.
In this paper, the framework of the Flügge's shell theory, the transfer matrix approach and the Romberg integration method has been employed to investigate the buckling analysis of radial loaded oval cylindrical shell with parabolically varying thickness along of its circumference. Trigonometric functions are used to form the modal displacements of the shell and Fourier's approach is used to separate the variables. The mathematical analysis is formulated to overcome the difficulties related to mode coupling of variable curvature and thickness of the shell. Using the transfer matrix of the shell, the buckling equations of the shell are reduced to eight first-order differential equations in the circumferential coordinate and rewritten in a matrix form and solved numerically. The proposed model is adopted to get the critical buckling loads and the corresponding buckling deformations for the symmetrical and antisymmetrical modes of buckling. The influences of the shell geometry, orthotropic parameters, ovality parameter, and thickness ratio on the buckling parameters and the bending deformations are presented for different type-modes of buckling.
The structural behaviour of elliptical hollow sections has been examined in previous studies under several loading conditions, including pure compression, pure bending and combined uniaxial bending and compression. This paper examines the elastic buckling response of elliptical hollow sections under any linearly varying in-plane loading conditions, including the most general case of combined compression and biaxial bending. An analytical method to predict the elastic buckling stress has been derived and validated against finite element results. The predictive model first identifies the location of the initiation of local buckling based on the applied stress distribution and the section geometry. The critical radius of curvature corresponding to this point is then introduced into the classical formula for predicting the elastic local buckling stress of a circular shell. The obtained analytical results are compared with results generated by means of finite element analysis. The comparisons between the analytical and numerical predictions of elastic buckling stress reveal disparities of less than 2.5% for thin shells and, following an approximate allowance for the influence of shear, less than 7.5% for thick shells. (c) 2012 Elsevier Ltd. All rights reserved.
Many elliptical shells are used in structural applications in which the dominant loading condition is axial compression. Due to the fact that the radius varies along the cross-section midline, the buckling behavior is more difficult to identify than those of cylindrical shells. The general concerned aspects in cylindrical shell buckling analyses such as the buckling mode, the pre-buckling deformation and post-buckling deformation are all quite different related to specific elliptical shell geometry. The buckling behavior of elliptical cylindrical shells with uniform thickness has been widely studied by many researchers. However, the thickness around the circumference may change for some specific structural forms, the femoral neck for example, which makes the buckling behavior more complex. It is known that the buckling strength of thin cylindrical shells is quite sensitive to imperfections, so it is natural to explore the imperfection sensitivity of elliptical shells. This paper explores the buckling behavior of imperfect elliptical shells under axial compression. It is hoped that the results will make a useful contribution in this field.
Flügge's shell theory and solution for the vibration analysis of a non-homogeneous orthotropic elliptical cylindrical shell resting on a non-uniform Winkler foundation are presented. The theoretical analysis of the governing equations of the shell is formulated to overcome the mathematical difficulties of mode coupling of variable curvature and homogeneity of shell. Using the transfer matrix of the shell, the vibration equations based on the variable Winkler foundation are written in a matrix differential equation of first order in the circumferential coordinate and solved numerically. The proposed model is applied to get the vibration frequencies and the corresponding mode shapes of the symmetrical and antisymmetrical vibration modes. The sensitivity of the vibration behavior and bending deformations to the non-uniform Winkler foundation moduli, homogeneity variation, elliptical and orthotropy of the shell is studied for different type-modes of vibrations.
A new vibration behavior is presented for an elastic oval cylindrical shell having circumferentially variable thickness with complex radius of curvature of an isotropic and orthotropic material. Based on the framework of the Flügge’s shell theory, the transfer matrix approach and the Romberg integration method, the governing equations of motion that have variable coefficients are formulated and solved. The analysis is formulated to overcome the mathematical difficulties related to mode coupling which comes from variable curvature and thickness of shell. The vibration equations of the shell are reduced to eight first-order differential equations in the circumferential coordinate, and by using the transfer matrix of the shell, these equations can be written in a matrix differential equation. The proposed model is adopted to get the vibration frequencies and the corresponding mode shapes for symmetric and anti-symmetric modes of vibration. The sensitivity of the frequency parameters and the bending deformations to the shell geometry, ovality parameter, thickness ratio, and orthotropic parameters corresponding to different types of vibration modes of shells is investigated.
This article is the result of an investigation on the effect of thermal load on vibration, buckling and dynamic stability of functionally graded cylindrical shells embedded in an elastic medium, based on the first-order shear deformation theory (FSDT) considering rotary inertia and the transverse shear strains. A heat conduction equation along the width of the shell is applied to determine the temperature distribution. Material properties are assumed to be graded with distribution along the width according to a power-law in terms of the volume fractions of the constituents. Calculations, effects of material composition, thermal loading, static axial loading, medium stiffness and shell geometry parameters on vibration, buckling and the parametric resonance are described. The new features of thermal vibration, buckling and dynamic stability of functionally graded cylindrical shells embedded in an elastic medium and some meaningful and interesting results in this paper are helpful for the application and design of functionally graded structures under thermal and mechanical loads.
Hot-rolled and cold-formed structural steel tubular members of elliptical cross-section have recently been introduced into the construction sector. However, there is currently limited knowledge of their structural behaviour and stability, and comprehensive design guidance is not yet available. This paper examines the elastic buckling response of elliptical hollow sections in compression, which has been shown to be intermediate between that of circular hollow sections and flat plates. The transition between these two boundaries is dependant upon both the aspect ratio and relative thickness of the section. Based on the results of numerical and analytical studies, formulae to accurately predict the elastic buckling stress of elliptical tubes have been proposed, and shortcomings of existing expressions have been highlighted. Length effects have also been investigated. The findings have been employed to derive slenderness parameters in a system of cross-section classification for elliptical hollow sections, and form the basis for the development of effective section properties for slender elliptical tubes.
The present paper contains a critical study of a number of foundation models as well as a further development of some of the ideas involved. Among others it is shown that the Pasternak foundation is a mechanical model for the so-called “generalized” foundation. It is also demonstrated that the kernel for the Pasternak foundation in plane stress or plane strain is identical with Wieghardt’s exponential kernel, and that for the three-dimensional case the kernel is a modified Bessel function. It is also shown that the “non solvability” of the problem of a finite beam or plate resting on a continuous foundation as posed by Wieghardt and further elaborated by Pflanz is not correct, and that problems of this type are solvable for any load distribution permissible in classical plate theory. Throughout the paper, emphasis is placed on the proper mathematical formulation of the physical problems in question.
The article discusses the buckling analysis of composite orthotropic truncated conical shells under a combined axial compression and external pressure and resting on a Pasternak foundation. The governing equations have been obtained for the orthotropic truncated conical shell resting on a Pasternak foundation and solved by applying the Superposition and Galerkin methods. The novelty of present work is to achieve closed-form solutions for critical combined loads. Finally, the effects of the Winkler and Pasternak elastic foundations, shell characteristics and material orthotropy on the values of critical combined loads have been studied. Comparison of results with those available in the literature has been presented.
The paper presents an approach based on three-dimensional elastic equations to solve boundary-value stress problems for hollow cylinders with corrugated elliptical cross section. Discrete Fourier series are used to make the problem one-dimensional and then to solve it by the stable discrete orthogonalization method. Solutions for cylinders of different thicknesses are presented
In this study, the buckling analysis of the simply supported truncated conical shell made of functionally graded materials (FGMs) is presented. The FGM truncated conical shell subjected to an axial compressive load and resting on Winkler–Pasternak type elastic foundations. The material properties of functionally graded shells are assumed to vary continuously through the thickness. The modified Donnell type stability and compatibility equations are solved by Galerkin’s method and the critical axial load of FGM truncated conical shells with and without elastic foundations have been found analytically. The appropriate formulas for homogenous and FGM cylindrical shells with and without elastic foundations are found as a special case. Several examples are presented to show the accuracy and efficiency of the formulation. Finally, parametric studies on the buckling of FGM truncated conical and cylindrical shells on elastic foundations are being investigated. These parameters include; power-law and exponential distributions of FGM, Winkler foundation modulus, Pasternak foundation modulus and aspect ratios of shells.
The stress state of longitudinally corrugated hollow orthotropic elliptic cylinders is analyzed using discrete Fourier series
and discrete orthogonalization
By using the relations of the Kirchhoff–Love theory of shells and the method of distortion of the shape of the boundary in the process of axisymmetric heating through the thickness of shells according to a linear law, we construct a solution of the problem of statics for a corrugated cylindrical shell in an axially symmetric temperature field under the action of internal pressure.
In this study, the buckling analysis of functionally graded material (FGM) circular truncated conical and cylindrical shells subjected to combined axial extension loads and hydrostatic pressure and resting on a Pasternak type elastic foundation is investigated. The critical combined loads of FGM truncated conical shells with or without elastic foundations have been found analytically. The appropriate formulas for FGM cylindrical shells with and without elastic foundations are found as a special case. Several examples are presented to show the accuracy and efficiency of the formulation.
The approach to the solution of a three-dimensional boundary-value stress problem for elastic hollow inhomogeneous cylinders
of corrugated elliptic cross-section is proposed. The boundary conditions make it possible to separate variables along the
length at the cylinder ends. It is proposed to include additional functions into the resolving system of differential equations.
These functions enable the variables to be separated along a directrix using discrete Fourier series. The boundary-value problem
derived for the system of ordinary differential equations is solved by the stable numerical method of discrete orthogonalization
over the cylinder thickness. Results in the form of plots and tables are presented.
The combination of Flüggeʼs shell theory, the transfer matrix approach and the Romberg integration method are used to investigate the free vibration behaviour of stepped orthotropic cylindrical shells. The hoop step on the shell surface is described by a reduced thickness over part of its circumference. Modal displacements of the shell can be described by trigonometric functions and Fourierʼs approach is used to separate the variables. The vibration equations of the shell are reduced to eight first-order differential equations in the circumferential coordinate, and by using the transfer matrix of the shell, these equations can be written in a matrix differential equation. The transfer matrix is derived from the non-linear differential equations of the cylindrical shells by introducing the trigonometric functions in the longitudinal direction and applying the numerical integration in the circumferential direction. The proposed model is used to get the vibration frequencies and the corresponding mode shapes for symmetrical and antisymmetrical type-modes. Computed results indicate the sensitivity of the frequency parameters and the bending deformations to the geometry of stepped shell, and also to the axial and circumferential rigidities of the shell.
An analytical and experimental investigation was conducted to determine the buckling behavior of elliptic cylinders under normal pressure. Tests were performed on a total of 80 models made of poly vinyl-chloride sheet. The models were simply supported and the buckling mode was found as an oval dimple which appears around the surface with the least curvature. Theoretical buckling pressure was also obtained by using Goldenveizer’s linear, thinshell theory in conjunction with Galerkin’s method for solution. The test data and theory agree fairly well, provided that the shell is not too short or too thick. For design purposes, curves are provided in nondimensional form to show the interaction of the physical quantities of the cylinders and the buckling pressures obtained by theory as well as by experiment. © 1970, American Institute of Aeronautics and Astronautics, Inc., All rights reserved.
This paper presents several issues that characterize the buckling behaviour of elliptical cylindrical shells and tubes under compression. First, a formulation of Generalised Beam Theory (GBT) developed to analyse the elastic buckling behaviour of non-circular hollow section (NCHS) members is presented. Since the radius varies along the cross-section mid-line, the main concepts involved in the determination of the deformation modes are adapted to account for the specific aspects related to elliptical cross-section geometry. After that, two independent sets of fully orthogonal deformation modes are determined: (i) local-shell modes satisfying the null membrane shear strain but exhibiting transverse extension and (ii) shell-type modes satisfying both assumptions of null membrane shear strain and null transverse extension. In order to illustrate the application, capabilities and versatility of the formulation, the local and global buckling behaviour of elliptical hollow section (EHS) members subjected to compression is analysed. In particular, in-depth studies concerning the influence of member length on the variation of the critical load and corresponding buckling mode shape are presented. Moreover, the GBT results are compared with estimates obtained by means of shell finite element analyses and are thoroughly discussed. The results show that short to intermediate length cylinders buckle mostly in local-shell modes, exhibiting only transverse extension, while intermediate length to long cylinders buckle mostly in shell-type modes (distortional and global modes), which are characterized by transverse bending and primary warping displacements. It is also shown that the present formulation is very efficient from the computational point of view since only three deformation modes (one local-shell, one distortional and one global) are required to evaluate the buckling behaviour of EHS cylinders for a wide range of lengths.
In this paper, the Flügge's shell theory is modeled and the transfer matrix approach is used to investigate the elastic buckling behaviour of a cylindrical shell with a four lobed cross section with reduced thickness over part of its circumference under axial compressive loads. Modal displacements of the shell can be described by trigonometric functions and Fourier's approach is used to separate the variables. The buckling equations of the shell are reduced to eight first-order differential equations in the circumferential coordinate, and by using the transfer matrix of the shell, these equations can be written in a matrix differential equation. The transfer matrix is derived from the differential equations of the cylindrical shells by introducing the trigonometric functions in the longitudinal direction and applying a numerical integration in the circumferential direction. The method is used to get the critical buckling loads and the buckling deformations for symmetrical and antisymmetrical shells. Computed results indicate the sensitivity of the buckling loads and the corresponding buckling deformations to the geometry of the non-uniformity of the shell, and also to the radius of curvature at the lobed corners.
Following the Eurocode 3 philosophy, it is expected that the design of elliptical hollow section (EHS) tubes will be based on the slenderness concept, which requires the calculation of the EHS critical stress. The critical stress of an EHS tube under compression may be associated with local buckling, distortional buckling or flexural buckling. The complexity in deriving analytical expressions for distortional critical stress from classical shell theories, led us to apply Artificial Neural Networks (ANN). This paper presents closed-form expressions to calculate the distortional critical stress and half-wave length of EHS tubes under compression, using ANN. Almost 400 EHS geometries are used and based solely on three parameters: the outer EHS dimensions (A and B) and its thickness (t). Two architectures are shown to be successful. They are tested for several statistical parameters and proven to be very well behaved. Finally, some simple illustrative examples are shown and final remarks are drawn concerning the accuracy of the closed-formed formulas.Graphical abstractHighlights► EHS tubes under compression may buckle in either local or distortional modes. ► Analytical formulas exist for local buckling of EHS tubes but they are missing for distortional buckling of EHS tubes, due to inherent complexity of distortional mechanics. ► Artificial Neural Networks (ANN) are used to develop closed-form expressions for the calculation of the distortional critical stress and half-wave length of EHS tubes under compression. ► They are tested for several statistical parameters and proven to be very well behaved.
Corrugated shells of revolution that may be considered cylindrical when the corrugation amplitude is small are analyzed for
stability. The corrugations are transverse to the axis of revolution. Isotropic and orthotropic shells with sine-shaped meridian
under uniform external compression are analyzed for stability. It is shown that the stability of corrugated shells can be
significantly improved, compared with cylindrical shells, by selecting appropriate number and amplitude of half-waves. A relationship
between the buckling modes and the change in the critical loads is established
Keywordsstability–corrugated shell of revolution–sine-shaped meridian–uniform external compression–buckling modes
The problem of stability of orthotropic noncircular cylindrical shells whose cross section can be described by a function in the form of superposition of a constant and a multiperiodic cosinusoid is considered. The solution of this problem is based on a rigorous consideration of the shell geometry and on the representation of the resolving functions in terms of trigonometric series in the circumferential coordinate. The determination of the bifurcation load is reduced to finding the minimum eigenvalue of a sequence of infinite systems of homogeneous algebraic equations. The effect of corrugation on the critical load of thin glass- and boron-reinforced shells of arbitrary length is analyzed. The data on the efficiency of corrugated shells, in comparison with circular ones, in relation to the mechanical properties of materials are obtained. The accuracy of the calculation procedure is estimated in the case where the corrugated shell is simulated by an equivalent circular one.
A technique is proposed for stability analysis of longitudinally corrugated shells under axial compression. The technique employs the equations of the Timoshenko-type nonlinear theory of shells. The geometrical parameters of shells are specified on discrete set of points and are approximated by segments of Fourier series. Infinite systems of homogeneous algebraic equations are derived from a variational equation written in displacements to determine the critical loads and buckling modes. Specific types of corrugated isotropic metal and fiberglass shells are considered. The calculated results are compared with those obtained within the framework of the classical theory of shells. It is shown that the Timoshenko-type theory extends significantly the possibility of exact allowance for the geometrical parameters and material properties of corrugated shells compared with Kirchhoff–Love theory.
Cylindrical shells consisting of cylindrical panels of smaller radius and subjected to uniform external pressure are analyzed
for stability. The geometrical parameters of the shells are approximated by Fourier series on a discrete set of points. The
Timoshenko theory of shells is used. The solution is represented in the form of trigonometric series. It is shown that short-and
medium-length shells with cylindrical panels are advantageous over circular shells. By selecting appropriate parameters of
the panels, keeping the mass of the shell constant, it is possible to achieve a significant gain in critical loads. The shells
under consideration are less effective than isotropic shells. Shells with sinusoidal corrugation under external pressure are
of no practical interest
A technique for stability analysis of cylindrical shells with a corrugated midsurface is proposed. The wave crests are directed along the generatrix. The relations of shell theory include terms of higher order of smallness than those in the Mushtari–Donnell–Vlasov theory. The problem is solved using a variational equation. The Lam parameter and curvature radius are variable and approximated by a discrete Fourier transform. The critical load and buckling mode are determined in solving an infinite system of equations for the coefficients of expansion of the resolving functions into trigonometric series. The solution accuracy increases owing to the presence of an aggregate of independent subsystems. Singularities in the buckling modes of corrugated shells corresponding to the minimum critical loads are determined. The basic, practically important conclusion is that both isotropic and orthotropic shells with sinusoidal corrugation are efficient only when their length, which depends on the waveformation parameters and the geometric and mechanical characteristics, is small
The paper presents an approach to solve the boundary-value stress-strain problem for circumferentially corrugated elliptic
cylindrical shells. The approach employs splines to approximate the solution and the stable discrete-orthogonalization method
to solve the resulting one-dimensional problem. The results are presented as plots and a table
The buckling problem for longitudinally corrugated cylindrical shells under external pressure is solved. The solution makes
practically exact allowance for the geometry and buckling modes of the shell. The inaccuracy of the results is due to the
assumption that the subcritical state is momentless. Shells consisting of cylindrical panels of smaller radius and noncircular
shells with sinusoidal corrugations are analyzed for stability. The practical applicability of such shells is demonstrated
A method to solve stability problems for corrugated shells with dished ends subject to hydrostatic pressure is outlined. Critical
loads are determined by solving a boundary-value problem with unknown amplitude of nonaxisymmetric perturbation. It is shown
that the shells are highly sensitive to the axial component of hydrostatic pressure
Keywordsstability–corrugated shell–critical load–hydrostatic pressure–inhomogeneous boundary-value problem
Thin-walled, corrugated, circular cylindrical shells under external pressure have repeatedly buckled at much lower values
than has been predicted from simple orthotropic shell theory, where a flexural mode of buckling is assumed, in-plane to the
cross-section of the tube. In the present paper it is shown that there exists always a combined, torsional–flexural, out-of-plane
mode of buckling, frequently corresponding to an extremely low critical pressure. The need to design these tubes also with
respect to torsional stiffness is emphasized, and a formula for the buckling pressure is proposed.
This paper presents a study on the postbuckling response of a shear deformable functionally graded cylindrical shell of finite length embedded in a large outer elastic medium and subjected to axial compressive loads in thermal environments. The surrounding elastic medium is modeled as a tensionless Pasternak foundation that reacts in compression only. The postbuckling analysis is based on a higher order shear deformation shell theory with von Kármán–Donnell-type of kinematic nonlinearity. The thermal effects due to heat conduction are also included and the material properties of functionally graded materials (FGMs) are assumed to be temperature-dependent. The nonlinear prebuckling deformations and the initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the postbuckling response of the shells and an iterative scheme is developed to obtain numerical results without using any assumption on the shape of the contact region between the shell and the elastic medium. Numerical solutions are presented in tabular and graphical forms to study the postbuckling behavior of FGM shells surrounded by an elastic medium of tensionless Pasternak foundation, from which the postbuckling results for FGM shells with conventional elastic foundations are also obtained for comparison purposes. The results reveal that the unilateral constraint has a significant effect on the postbuckling responses of shells subjected to axial compression in thermal environments when the foundation stiffness is sufficiently large.
A postbuckling analysis is presented for a shear deformable cross-ply laminated cylindrical shell of finite length subjected to combined loading of external pressure and axial compression. The governing equations are based on Reddy's higher order shear deformation shell theory with von Kármán–Donnell type of kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of shear deformable laminated cylindrical shells under combined loading cases. A singular perturbation technique is employed to determine interactive buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, unstiffened or stiffened, moderately thick, antisymmetric and symmetric cross-ply laminated cylindrical shells for different values of load-proportional parameters.
Nonlinear theory of elastic shells
- . M Kh
- K Z Mushtari
- Galimov
Design of structures on elastic foundation
- M I Gorbunov-Possadov
- T A Malikova
- V I Solomin
The Bending of the cylindrical shells in an elastic medium
- V A Bajenov