Shells Of Revolution Research Articles

Buckling behaviour of double-layer and single-layer stainless steel cylinders under external pressure

This study focused on the buckling behaviour of double-layer (DL) and single-layer (SL) stainless steel cylinders under uniform external pressure. Such cylinders are short thin-walled cylindrical shells of revolution. Three nominally identical DL cylinders and three nominally identical SL cylinders were designed, fabricated, geometrically measured, and subjected to hydrostatic testing. The length-to-diameter ratio of all cylinders was 1.7, while the thickness-to-diameter ratio was 0.0125 for SL cylinders and 0.0261 for DL cylinders. Analytical and numerical approaches were used to evaluate the buckling behaviour of experimental cylinders. Furthermore, the effects of initial clearance and imperfection on the ultimate strength of DL cylinders were studied. The results revealed that DL cylinders are clearance-sensitive and imperfection-sensitive structures. The imperfection-sensitivity of DL cylinders were determined by sublayered cylinders because they had nearly the same imperfection-sensitivity. The average ultimate strength of DL cylinders was approximately 1.62-times higher than that of SL cylinders. The clearance sensitivity and imperfection sensitivity decreased with the increase of length-to-thickness ratio.

Free bulging of thin-walled cylinders closed by two heavy plates

This study focuses on the dieless free bulging of thin-walled stainless cylinders closed by two heavy plates. Using the plastic mechanics of shells of revolution and by making several basic assumptions, a family of analytical formulae are derived to evaluate the shell bulging properties such as surface area, internal capacity, thickness distribution, equivalent strain, and bulging pressure. The analytical and numerical data were found to be in good agreement with each other; the results were validated through a free bulging experiments carried out on three nominally identical thin-walled cylinders, which will provide a reference for the manufacture of underwater thin-walled pressure structures.

Application of the Ritz Method to the Analysis of Free Vibrations of the Conjugate Shells of Revolution

We propose a variational method for the construction of approximate solutions of the spectral problem of vibrations of two conjugate shells of revolution that are not axially symmetric. The solution of the problem is based on the decomposition of the domain of integration of equations of the theory of shells performed by using the variational method. We construct a generalized functional of displacements of the shell for which the conditions of conjugation of solutions on the common boundary of the introduced subdomains are natural boundary conditions. For a shell formed by a truncated cone and a cylinder, the efficiency of the proposed approach is analyzed and the results of numerical calculations are compared with the available data obtained by the other authors.

ВИМУШЕНІ КОЛИВАННЯ ПАЛИВНИХ БАКІВ ПРИ РІЗНИХ УМОВАХ ЕКСПЛУАТАЦІЇ

Дослідження коливань рідини у паливних баках космічних апаратів є необхідним для забезпечення їх стійкості під час польоту. Рідинні ракети-носії мають на борту значні запаси рідких компонентів. В паливних баках та баках окисників виникають коливання вільної поверхні рідини, на характер яких впливають також і зовнішні чинники, такі як гравітація, рівень заповнення баків, їх форма тощо. Виникає необхідність визначення форм та частот коливань вільної поверхні рідини для забезпечення стійкого стану літального апарату під час польоту. Для вирішення цієї задачі застосовано методи математичного моделювання. Розглянуто дослідження задачі малих коливань у резервуарах, частково заповнених рідиною, що стосуються космічної галузі. Також розглянуті роботи вітчизняних та закордонних авторів щодо розв’язання задач коливань рідини в нелінійній постановці. У даній роботі проведено аналіз форм та частот коливань рідини в умовах перевантаження при різних рівнях заповнення. Виконано дослідження малих коливань (лінійне формулювання). Прийнято, що рідина є нестисливою та однорідною, а її рух безвихровий. Існує потенціал швидкості руху, що задовольняє рівнянню Лапласа. Виконуються кінематична та динамічна граничні умови на вільній поверхні. Для врахування впливу низької гравітації виконано узагальнення граничних умов. В динамічній умові на вільній поверхні враховано поверхневий натяг, що моє визначальне значення в умовах мікрогравітації. Отримано співвідношення, що має виконуватись на вільній поверхні. Отримане рівняння дозволяє проводити дослідження вимушених коливань жорсткої оболонки обертання, що частково заповнена рідиною. Отримано числові результати для циліндричної оболонки обертання. Прийняті нульові початкові умови. Виконані розрахунки при різних рівнях заповнення для циліндричної оболонки. Отримані найнижчі частоти коливань. Також досліджені коливання рідини, що знаходиться під дією вертикального навантаження. Отримані фазові портрети для різних значень частот вертикальної сили збудження. Отримані результати дозволяють дослідити амплітуди коливань вільної поверхні.

Funicularity in elastic domes: Coupled effects of shape and thickness

Abstract An historical overview is presented concerning the theory of shell structures and thin domes. Early conjectures proposed, among others, by French, German, and Russian Authors are discussed. Static and kinematic matrix operator equations are formulated explicitly in the case of shells of revolution and thin domes. It is realized how the static and kinematic matrix operators are one the ad-joint of the other, and, on the other hand, it can be rigorously demonstrated through the definition of stiffness matrix and the application of virtual work principle. In this context, any possible omission present in the previous approaches becomes evident. As regards thin shells of revolution (thin domes), the elastic problem results to be internally statically-determinate, in analogy to the case of curved beams, being characterized by a system of two equilibrium equations in two unknowns. Thus, the elastic solution can be obtained just based on the equilibrium equations and independently of the shape of the membrane itself. The same cannot be affirmed for the unidimensional elements without ‚exural stiffness (ropes). Generally speaking, the static problem of elastic domes is governed by two parameters, the constraint reactions being assumed to be tangential to meridians at the dome edges: the shallowness ratio and the thickness of the dome. On the other hand, when the dome thickness tends to zero, the funicularity emerges and prevails, independently of the shallowness ratio or the shape of the dome. When the thickness is finite, an optimal shape is demonstrated to exist, which minimizes the flexural regime if compared to the membrane one.

Control and measuring robot for determining the geometric parameters of large shells of revolution: metrological analysis

Control and measuring robot for determining the geometric parameters of large shells of revolution: metrological analysis

Numerical analysis of stresses and strains of a shell of revolution made from a laminate composite material with considering structural defects in the form of delamination

Numerical analysis of stresses and strains of a shell of revolution made from a laminate composite material with considering structural defects in the form of delamination

Strength of Shell Structure Operating in Hydrogenous Media under Pressure

Estimation of strength and durability of any structure can be carried out after finding components of stress-strain state of all its parts. The present work shows that in order to calculate stresses in a certain element of a structure that works under thermo-mechanical loading and contacts hydrogenous media, one should trace distribution of heat and hydrogen concentration in this element. Based on the results obtained when having solved thermo-diffusion problem of heat and hydrogen distribution in a casing of a diffusion apparatus, its stress state was found. Casing of the diffusion apparatus is represented in a shape of a shell of revolution. It was determined that in a thin shell, homogenous fields of temperature and hydrogen concentration are being set for a period that is significantly less than the design operating life of the apparatus. To solve the problem of stress distribution in a shell, S.K. Godunov’s method of discrete orthogonalization was used. In the structure that does not have any manifested stress raisers, a critical point with maximum value of shear stress intensity was found. Consistent patterns of change of the intensity in the critical point were found under the emergency pressure boost in the apparatus at various hydrogen densities. The paper shows that emergency pressure boost in a shell may cause the appearance of zones of plastic yield; when this boost is accompanied by increase in hydrogen concentration, the fracture pattern changes from viscous to brittle.

Forced vibrations of multilayered cylindrical shells taking into account the discresibility of the ribs with non-steady loads

This work considers the problem of nonstationary behavior of multilayered discretely reinforced cylindrical shells.By the way the problem is very important. Multiplayed shells with allowance for discrete ribs are widely used in engineering, industrial and public building, aviation and space technology, shipbuilding. In the framework of the Timoshenko type non – linear theory of shells and ribs nonstationary vibrations multilayered shells of revolution with allowance for discrete ribs are investigated. Reissner’s variational principle for dynamical processes is used for deduction of the motion equations. An efficient numerical method with using Richardson type finite difference approximation for solution of problems on nonstationary behaviour of multiplayer shells of revolution with allowance for discrete ribs which permit to realize solution of the investigated wave problems with the use of personal computers, as well as bringing their solutions to receiving concrete numerical results in wide diapason of geometrical, physico–mechanical parameters of structures are elaborated. In particular three-layer discretely reinforced cylindrical shells were investigated.

Stress-strain state of shell of revolution analysis by using various formulations of three-dimensional finite elements

The aim of the work is to perform a comparative analysis of the results of analyzing arbitrarily loaded shells of revolution using finite element method in various formulations, namely, in the formulation of the displacement method and in the mixed formulation. Methods. To obtain the stiffness matrix of a finite element a functional based on the equality of the actual work of external and internal forces was applied. To obtain the deformation matrix in the mixed formulation the functional obtained from the previous one by replacing the actual work of internal forces in it with the difference of the total and additional work was used. Results. In the formulation of the displacement method for an eight-node hexahedral solid finite element, displacements and their first derivatives are taken as the nodal unknowns. Approximation of the displacements of the inner point of the finite element was carried out through the nodal unknowns on the basis of the Hermite polynomials of the third degree. For a finite element in the mixed formulation, displacements and stresses were taken as nodal unknowns. Approximation of the target finite element values through their nodal values in the mixed formulation was carried out on the basis of trilinear functions. It is shown on a test example that a finite element in the mixed formulation improves the accuracy of the strength parameters of the shell of revolution stress-strain state.

Analysis of the fracture toughness parameters at the free edge in layered composites

The problem of deformation and elastoplastic buckling of shells of revolution with a thick-walled elastic core under combined static and dynamic loading is formulated in a two-dimensional planar formulation based on two approaches: full-scale modeling within the framework of continuum mechanics and a simplified formulation based on the hypotheses of the theory of shells of the Timoshenko type and the Winkler foundation. Both approaches allow solving the problems of deformation and stability of non-shallow shells on the basis of Timoshenko's hypotheses, taking into account geometric nonlinearities. The statement from the perspective of continuum mechanics makes it possible to approximate the shell in thickness by a number of layers of finite elements. The constitutive relations are formulated in Lagrange variables using a fixed Cartesian coordinate system as a reference one. Kinematic relations are recorded in the metric of the current state. The elastic-plastic properties of shells are described by the theory of plastic flow with isotropic hardening. The equations of motion follow from the balance of the virtual powers of the work. In the first approach, the contact interaction of a shell and an elastic body is modeled by the conditions of nonpenetration along the normal and free slip along the tangent. The nonpenetration conditions are satisfied only in the active phase of the contact interaction; if the contact is broken, they are replaced by conditions on the free surface. In the second approach, the contact interaction of the elastic core with the shell is modeled by the Winkler foundation. Both approaches allow one to describe the nonlinear subcritical deformation of shells of revolution with an elastic core, to determine the limiting (critical) loads in a wide range of loading rates, taking into account the geometric imperfections of the shape. Using both approaches, a numerical simulation of contact interaction problem of an elastoplastic cylindrical shell with a thick-walled elastic core at a quasi-static uniform external pressure is carried out. The study of the influence of the thickness and initial deflection of the shell, as well as the stiffness and thickness of the core, on the value of the critical pressure and the form of buckling has been carried out. Based on these calculations, a conclusion was made about a wide range of applicability of the Winkler foundation model.

Investigation of the Winkler foundation model applicability for describing the contact interaction of elastoplastic shells with a core under external pressure

The problem of deformation and elastoplastic buckling of shells of revolution with a thick-walled elastic core under combined static and dynamic loading is formulated in a two-dimensional planar formulation based on two approaches: full-scale modeling within continuum mechanics and a simplified formulation based on the hypotheses of the theory of shells of the Timoshenko type and the Winkler foundation. Both approaches allow solving the problems of deformation and stability of non-shallow shells on the basis of Timoshenko's hypotheses, taking into account geometric nonlinearities. The statement from the perspective of continuum mechanics makes it possible to approximate the shell in thickness by a number of layers of finite elements. The constitutive relations are formulated in Lagrange variables using a fixed Cartesian coordinate system as a reference one. Kinematic relations are recorded in the metric of the current state. The elastic-plastic properties of shells are described by the theory of plastic flow with isotropic hardening. The equations of motion follow from the balance of the virtual powers of the work. In the first approach, the contact interaction of a shell and an elastic body is modeled by the conditions of nonpenetration along the normal and free slip along the tangent. The nonpenetration conditions are satisfied only in the active phase of the contact interaction; if the contact is broken, they are replaced by conditions on the free surface. In the second approach, the contact interaction of the elastic core with the shell is modeled by the Winkler foundation. Both approaches allow one to describe the nonlinear subcritical deformation of shells of revolution with an elastic core, to determine the limiting (critical) loads in a wide range of loading rates, taking into account the geometric imperfections of the shape. Using both approaches, a numerical simulation of epy contact interaction problem of an elastoplastic cylindrical shell with a thick-walled elastic core at a quasi-static uniform external pressure is carried out. The study of the influence of the thickness and initial deflection of the shell, as well as the stiffness and thickness of the core, on the value of the critical pressure and the form of buckling has been carried out. Based on these calculations, a conclusion was made about a wide range of applicability of the Winkler foundation model.

Transient analysis of thermo-mechanically shock loaded four-parameter power law functionally graded shells

The present work identifies the mechanical and thermal response of ceramic–metal composed functionally graded cylindrical and spherical shells of revolution exposed to rapidly applied mechanical and thermal loads on their inner surface. The gradation of the constituent materials is done in the thickness direction of the FG shells according to four-parameter power laws. The mechanical and thermal properties of the constituent materials are considered to be temperature-dependent. Mori-Tanaka model is employed to obtain effective material properties based on the homogenization method. A nonlinear 1-D heat conduction equation is solved in each time-step of unconditionally stable Crank-Nicolson time integration scheme. Thereafter, an uncoupled thermoelastic finite element code based on higher-order shear deformation theory for shells and Newmark time integration scheme is developed in MATLAB environment. Numerical results considering the effect of the geometry of the shell, thermal boundary conditions, temperature-dependent material properties, different parameters of FGM laws, and thermomechanical load on the response characteristic of shells are presented. It was found that the oscillations of a thermomechanical shocked shell initially deviated from its quasi-static nature of response but started to oscillate about the quasi-static curve as its mean position on removal of the mechanical load.

The influence of mechanical uncertainties on the free vibration of functionally graded graphene-reinforced porous nanocomposite shells of revolution

This paper is concerned with the analysis of uncertainty propagation in the free vibration response of functionally graded porous (FGP) nanocomposite shells of revolution reinforced with graphene platelet (GPL). In this regard, the effects of uncertainties in the material properties as well as the geometry of the reinforcements on the free vibration responses of the FGP-GPL shells with different types of Gaussian curvature are assessed. The propagation of uncertainty and frequency sensitivity of the shells of revolution are examined for various GPL dispersions and porosity distributions. For this purpose, the interval analysis method which is an appropriate technique for the uncertainty analysis of systems with bounded uncertainties is utilized. In order to solve the equations of motion based on the higher-order shear deformation theory and to obtain the free vibration responses of the shells of revolution, the Fourier Differential Quadrature (FDQ) technique is employed. According to the analysis results, the shells’ type of curvature has a significant effect on the trend of uncertainty propagation and the sensitivity of the vibration frequencies in terms of various sources of uncertainty. Moreover, the results show that the uncertainty propagation and sensitivity are different in the frequencies corresponding to different modes of vibration.

Theoretical and experimental study of the free hydroforming of egg-shaped shell

ABSTRACT This study investigated the free hydroforming of a stainless-steel egg-shaped shell. An egg-shaped shell is a promising geometry for deep pressure hulls. The length and width of the egg-shaped shell were 512.40 and 356.25 mm, respectively. The wall thickness of the egg-shaped shell were 0.844 mm. The equivalent stress, yielding load, internal capacity, and surface area of the egg-shaped shell and its inscribed segmented preform were investigated analytically and experimentally. The hydroforming performance of the egg-shaped shell was analysed using the nonlinear finite element method, and the results were validated by experimental model tests. The results indicated that free hydroforming can be an effective technique for fabricating egg-shaped pressure hulls and can yield highly symmetric shells of revolution regardless of the large geometric deviation of the segmented preform.

A semi-analytic method for vibro-acoustic analysis of coupled propeller-shaft-hull systems under propeller excitations

A semi-analytic dynamic model is developed for vibro-acoustic analysis of submerged coupled propeller-shaft-hull systems under propeller forces. The propeller, shaft and hull are idealized as a lumped mass, multi-span beam and ring-stiffened shell of revolution, respectively. Stern, middle and thrust bearings are simulated as spring-damping systems. The hull is firstly decomposed into many segments, which are treated as conical shells and analyzed through combining Flügge shell theory and power series method. The shaft is divided into three Timoshenko beams according to the bearings. By expanding acoustic pressure and velocity as Fourier series in circumferential direction, surface Helmholtz integral equation is reduced to line integral, and acoustic pressure is further expressed as displacements of segments after meshing the line to some 3-node isoparameter elements. Boundary conditions and continuity conditions modified by external acoustic pressure are orderly assembled to the dynamic model. As considering the hull as a ring-stiffened conical-cylindrical-spherical shell, vibro-acoustic results of present method are compared with ones of coupled finite element-boundary element method, and the accuracy and efficiency of developed dynamic model are demonstrated. Furthermore, parameter analysis reveals that breathing and beam modes are predominant for vibro-acoustic responses, and sound power is mainly dominated by the cylindrical compartment.

Asymptotic Methods for Studying an Elliptical Boundary Layer in Shells of Revolution Under Normal Type Shock End Impacts

In this paper, asymmetric equations for an elliptical boundary layer in the vicinity of the conditional front of Rayleigh surface waves, which occurs in shells of revolution under shock end impacts of normal type are constructed. The technique of asymptotic derivation of these equations, based on the use of the symbolic Lurie method and the introduction of special coordinates that distinguish a small frontal zone required to reduce the original problem to an equivalent problem for an infinite shell by isolating a particular solution. The considered boundary layer complements the full description of the considered type of stress-strain state (SSS) in all sections of the phase plane. It also uses a quasi-static boundary layer of the Saint-Venant type in a small vicinity of the butt end, a parabolic boundary layer according to the two-dimensional Kirchhoff–Love theory, a quasi-plane short-wave component, and a hyperbolic boundary layer in a small neighborhood of the shear wave front. In conclusion, an example of constructing an elliptical boundary layer under shock action on the butt end of a cylindrical shell is considered.

Выбор схемы подкрепления параболической антенны из композиционных материалов с эллиптической линией раскрыва

Being prepared foThe problems of designing and calculating reinforced shells of revolution made of composite materials as constituent elements of parabolic antennas with an elliptical aperture line having a focal parameter value of not more than 0.03 m are examined in this paper. The structural layout schemes of reflectors of the parabolic-type mirror spacecraft antennas are reviewed. A technique for designing a bearing frame for reinforcing the reflector shell using numerical analysis based on the finite element method is presented. The technique is based on the assessment of the stiffness parameters of the structural elements of the reflector located between the annular ribs. It utilizes optimization in order to achieve a constant gradient of changes in the stiffness parameters. The deformability of the reflector design that uses a modern high-modulus composite material is evaluated. A finished product is presented as the result of the application of the technique. An assessment of manufacturing and design errors is performed based on the experimental data. The proposed approach to the design of reflectors with an elliptical aperture line eliminates warping of the molded product and deformation during operation such as twisting in the form of a propeller. The results of the study can be used for designing reflectors of spacecraft mirror antennas with a narrow radiation pattern that meet strict requirements of the accuracy of the reflecting surface geometry as well as mass restrictions.

A unified semi-analytic method for vibration analysis of functionally graded shells of revolution

A unified semi-analytic method is developed for free vibration analysis of functionally graded material (FGM) shells of revolution subjected to arbitrary boundary conditions, and independent and coupled shells of revolution are specific cases. Material properties vary continuously in thickness direction according to the four-parameter power-low distributions in terms of volume fractions of two constituents. The shell is firstly divided to some narrow shell segments to be treated as conical shells. Differential equations of motion based on first-order shear deformation theory (FSDT) are solved by expanding displacements as power series and Fourier series in meridional and circumferential directions, so five displacements and five forces at cross-section are only expressed as 10 unknown coefficients. Continuity conditions and elastic boundary conditions are assembled to the final governing equation. Through comparing natural frequencies of present method with the ones of opened works for several typical FGM shells of revolution with uniform/stepped thicknesses and classic/elastic boundaries, rapid convergence, high accuracy and wide application of the developed method are demonstrated. Furthermore, influences of material and geometry parameters reveal that increasing the power-law exponent does not change mode shapes and circumferential mode numbers of fundamental frequencies, and mode shapes are slightly affected by the variation of thickness.

Thermomagnetoelastic Deformation of Flexible Orthotropic Shells of Revolution of Variable Stiffness with Joule Heat Taken into Account

The equations of thermomagnetoelasticity for flexible orthotropic shells of revolution are derived taking into account orthotropic electrical conductivity and Joule heat. The thermomagnetoelasticity of a truncated orthotropic conical shell is analyzed using the axisymmetric problem formulation and taking into account the orthotropy of electrical conductivity and Joule heating in comparison with a flexible isotropic shell.