The exact solution for the general bending problems of conical shells on the elastic foundation

The general bending problem of conical shells on the elastic foundation (Winkler Medium) is not solved. In this paper, the displacement solution method for this problem is presented. From the governing differential equations in displacement form of conical shell and by introducing a displacement function U(s,θ), the differential equations are changed into an eight-order soluble partial differential equation about the displacement function U(s,θ) in which the coefficients are variable. At the same time, the expressions of the displacement and internal force components of the shell are also given by the displacement function U(s θ). As special cases of this paper, the displacement function introduced by V.S. Vlasov in circular cylindrical shell^[5], the basic equation of the cylindrical shell on the elastic foundation and that of the circular plates on the elastic foundation are directly derived.Under the arbitrary loads and boundary conditions, the general bending problem of the conical shell on the elastic foundation is reduced to find the displacement function U(s,θ).The general solution of the eight-order differential equation is obtained in series form. For the symmetric bending deformation of the conical shell on the elastic foundation, which has been widely usedinpractice,the detailed numerical results and boundary influence coefficients for edge loads have been obtained. These results have important meaning in analysis of conical shell combination construction on the elastic foundation,and provide a valuable judgement for the numerical solution accuracy of some of the same type of the existing problem.     

Displacement type unified equation for bending, buckling and vibration of conical shells and its applications

By using Donnell's simplication and starting from the displacement type equations of conical shells, and introducing a displacement functionU(s,,) (In the limit case, it will be reduced to cylindrical shell displacement function introduced by V. S. Vlasov) and a generalized loadq,(s,,),the equations of conical shells are changed into an eighth—order solvable partial differential equation about the displacement functionU(s,,). As a special case, the general bending problem of conical shells on Winkler foundation has been studied. Detailed numerical results and boundary coefficients for edge unit loads are obtained.The project supported by the National Natural Science Foundation of China.     

Analysis of the stress state of a flattened conical shell of moderate thickness with a structurally weakening hole

Conclusions We determined the relationship between the nature of the stress distribution on the hole surface in a flexible plate as a function of thickness. We observed a great difference between the stress densities in flattened, thin and moderate-thickness conical shells and the stress concentrations near holes in thin cylindrical shells and thin, almost cylindrical, conical shells. The stress distribution near the hole in flattened conical shells of moderate thickness is similar to the stress distribution near the holes in flexible, thick plates. During loading of conical shells by an axial force, the lowest stress concentration factor near the holes is obtained when the axis of the hole is parallel to the shell axis. As the thickness of the shell is increased, the stress concentration factor near the holes increases.Kiev University. Ukrainian Institute of Water Management Engineers, Rovno. Translated from Prikladnaya Mekhanika, Vol. 24, No. 9, pp. 65–70, September, 1988.     

Forced axisymmetric motions of viscoelastic cylindrical shells

The isothermal response of a viscoelastic cylindrical shell, of finite length, to arbitary axisymmetric surface forces, initial conditions, and boundary conditions is considered within the linear theory of thin shells. The problem is formulated with the effects of shear deformation and rotatory inertia included; the viscoelastic properties are assumed to be isotropic and homogeneous. The response is first found formally in terms of a causal Green's function. It is then shown that when Poisson's ratio is constant, the causal Green's function can be expanded in a series of orthonormal spatial eigenfunctions of an associated elastic shell eigenvalue problem. The resulting solution for the general problem is an eigenfunction series with Laplace transformed time-dependent coefficients. The general solution is applied to predicting the motion of a uniform, simply-supported cylindrical shell, initially quiescent, which is subjected to a step pressure moving with constant velocity. For this example, the relaxation function of the shell material in uniaxial extension is taken to be that of a standard linear solid. The motions predicted by simpler shell models, namely, shells with bending only and without bending, are also considered for comparison. Here, the absolute values of the Fourier coefficients in the shell displacement series go to zero faster than the inverse of the first or second power of positive integers when bending is excluded or included, respectively. Numerical results are presented for a moderately long and relatively thick, nearly elastic, cylindrical shell.     

Nonlinear deformation and stability of noncircular cylindrical shells under internal pressure and axial compression

Stability analysis of noncircular shells is performed with allowance for nonlinear subcritical deformation. Explicit expressions for the rigid displacements of elements of noncircular cylindrical shells are obtained and used to construct shape functions of an effective quadrilateral finite element of natural curvature. A finiteelement algorithm for solving problems of nonlinear deformation and stability of shells is developed. Stability problem of an elliptic cylindrical shell is considered. The effect of the ellipticity and subcritical nonlinear deformation of the shell on the critical load is studied. Results obtained are compared with available experimental data.     

Forced Nonstationary Vibrations of a Sandwich Cylindrical Shell with Cross-Ribbed Core

The equations of nonaxisymmetric vibrations of sandwich cylindrical shells with discrete core under nonstationary loading are presented. The components of the elastic structure are analyzed using a refined Timoshenko theory of shells and rods. The numerical method used to solve the dynamic equations is based on the integro-interpolation method of constructing finite-difference schemes for equations with discontinuous coefficients. The dynamic problem for a sandwich cylindrical shell under distributed nonstationary loading is solved with regard for the discreteness of the core__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 2, pp. 60–67, February 2005.     

Solving Stress Problems for Elastic Systems of Anisotropic Shells of Revolution with Regard for Transverse Shear and Reduction

An approach is proposed for refined solution of stress problems for elastic systems consisting of coaxial shells of revolution. Transverse shear and reduction are taken into account. Multivariant calculations made for orthotropic cylindrical shells with elliptical end-plates allow us to analyze the influence of the semiaxis ratio and intermediate supports on the stress–strain state of the shell systems under consideration     

A unified Chebyshev–Ritz formulation for vibration analysis of composite laminated deep open shells with arbitrary boundary conditions

In this paper, a unified Chebyshev–Ritz formulation is presented to investigate the vibrations of composite laminated deep open shells with various shell curvatures and arbitrary restraints, including cylindrical, conical and spherical ones. The general first-order shear deformation shell theory is employed to include the effects of rotary inertias and shear deformation. Under the current framework, regardless of boundary conditions, each of displacements and rotations of the open shells is invariantly expressed as Chebyshev orthogonal polynomials of first kind in both directions. Then, the accurate solutions are obtained by using the Rayleigh–Ritz procedure based on the energy functional of the open shells. The convergence and accuracy of the present formulation are verified by a considerable number of convergence tests and comparisons. A variety of numerical examples are presented for the vibrations of the composite laminated deep shells with various geometric dimensions and lamination schemes. Different sets of classical constraints, elastic supports as well as their combinations are considered. These results may serve as reference data for future researches. Parametric studies are also undertaken, giving insight into the effects of elastic restraint parameters, fiber orientation, layer number, subtended angle as well as conical angle on the vibration frequencies of the composite open shells.     

Stability of cylindrical shells subjected to combined axial compression and internal pressure under creep conditions

The critical strain of a cylindrical shell subjected to combined axial compression and internal pressure is computed under creep conditions. A method is proposed to determine values of the initial deflections by means of elastic shell test data for a creep analysis of shells. Data of an experimental investigation of the creep stability of shells are presented, which are compared with the results of the computation.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 109–116, September–October, 1974.     

Effects of partial crack–face contact for the bending of thin shell structures

The physical occurrence that crack surfaces are in contact at the compressive edges when a flat or a shell is subjected to a bending load has been recognized. This article presents a theoretical analysis of crack–face contact effect on the stress intensity factor of various shell structures such as spherical shell, cylindrical shell containing an axial crack, cylindrical shell containing a circumferential crack and shell with two non-zero curvatures, under a bending load. The formulation of the problem is based on the shear deformation theory, incorporating crack–face contact by introducing distributed force at the compressive edge. Material orthotropy is concerned in this analysis. Three-dimensional finite element analysis (FEA) is conduced to compare with the theoretical solution. It is found that due to curvature effect crack–face contact behavior in shells differs from that in flat plates, in that partial contact of crack surfaces may occur in shells, depending on the shell curvature and the nature of the bending load. Crack–face contact has significant influence on the stress intensity factor and it increases the membrane component but decreases the bending component.     

Imperfection sensitivity of laminated conical shells

The sensitivity of laminated conical shells to imperfection is considered, via the initial post-buckling analysis, on the basis of three different shell theories: Donnell’s, Sanders’, and Timoshenko’s. Unlike isotropic conical shells or laminated cylindrical shells, in the case of laminated conical shells the thickness and the material properties vary with the shell coordinates, which complicates the problem considerably. The main objective of the study is to investigate the influence of the variation of the stiffness coefficients on the buckling behavior and on the imperfection sensitivity of laminated conical shells. It is felt that by finding the various parameters that influence the shell’s imperfection sensitivity, it is possible to improve the behavior of the whole structure.A special Level-1 computer code ISOLCS (Imperfection Sensitivity of Laminated Conical Shells) had been developed. ISOLCS calculates the classical buckling load and the imperfection sensitivity via Koiter’s theory of laminated conical shells with consideration to the variation of the material properties in the shell’s coordinates. The range of validity of the Level-1 predictions by ISOLCS is verified by the Level-3 code STAGS-A.     

Thermomechanical vibration analysis of a functionally graded shell with flowing fluid

This paper reports the results of an investigation into the vibration of functionally graded cylindrical shells with flowing fluid, embedded in an elastic medium, under mechanical and thermal loads. By considering rotary inertia, the first-order shear deformation theory (FSDT) and the fluid velocity potential, the dynamic equation of functionally graded cylindrical shells with flowing fluid is derived. Here, heat conduction equation along the thickness of the shell is applied to determine the temperature distribution and material properties are assumed to be graded distribution along the thickness direction according to a power-law in terms of the volume fractions of the constituents. The equations of eigenvalue problem are obtained by using a modal expansion method. In numerical examples, effects of material composition, thermal loading, static axial loading, flow velocity, medium stiffness and shell geometry parameters on the free vibration characteristics are described. The new features in this paper are helpful for the application and the design of functionally graded cylindrical shells containing fluid flow.     

Nonlinear buckling of torsion-loaded functionally graded cylindrical shells in thermal environment

Based on the nonlinear large deflection theory of cylindrical shells, this paper deals with the nonlinear buckling problem of functionally graded cylindrical shells under torsion load by using the energy method and the nonlinear strain–displacement relations of large deformation. The material properties of the functionally graded shells vary smoothly through the shell thickness according to a power law distribution of the volume fraction of the constituent materials. Meanwhile, on the base of taking the temperature-dependent material properties into account, various effects of external thermal environment on the critical state of the shell are also investigated. Numerical results show various effects of the inhomogeneous parameter, the dimensional parameters and external thermal environment on nonlinear buckling of functionally graded cylindrical shells under torsion. The present theoretical results are verified by those in literature.     

Three-dimensional free vibration analysis of rotating laminated conical shells: layerwise differential quadrature (LW-DQ) method

This paper focuses on the free vibration analysis of thick, rotating laminated composite conical shells with different boundary conditions based on the three-dimensional theory, using the layerwise differential quadrature method (LW-DQM). The equations of motion are derived applying the Hamilton’s principle. In order to accurately account for the thickness effects, the layerwise theory is used to discretize the equations of motion and the related boundary conditions through the thickness of the shells. Then, the equations of motion as well as the boundary condition equations are transformed into a set of algebraic equation applying the DQM in the meridional direction. This study demonstrates the applicability, accuracy, stability and the fast rate of convergence of the present method, for free vibration analyses of rotating thick laminated conical shells. The presented results are compared with those of other shell theories obtained using conventional methods and a special case where the angle of the conical shell approaches zero, that is, a cylindrical shell and excellent agreements are achieved.     

Instability analysis in pressurized transversely isotropic Saint-Venant–Kirchhoff and neo-Hookean cylindrical thick shells

Under the effect of an inner pressure, a thick hyperelastic shell cylinder is susceptible to developing mechanical instabilities, leading to a bifurcated shape that no longer has the initial cylindrical symmetry. The considered problem has a strong resonance in a biomedical context, considering pathologies encountered for arteries, such as aneurysm. In this contribution, the mechanical behavior of thick elastic shells has been analyzed, considering a thick-walled cylindrical hyperelastic model material obeying a transversely isotropic behavior, first in a large-displacement situation and then in a large-deformation case. The response of the material is assumed to be instantaneous, so that time-dependent effects shall not be considered in this paper. The case of a Saint-Venant–Kirchhoff material is considered with special emphasis to exemplify a large-displacement small-strain situation; the neo-Hookean behavior is next considered to enlarge the constitutive law toward consideration of large strains. The stability conditions of the shell are studied and bifurcation conditions formulated in terms of the applied pressure and of the geometrical and mechanical parameters that characterize the shell. Analytical solutions of some bifurcation points are evidenced and calculated when the direction of the fibers coincide with the cylinder axis.     

The displacement solution of conical shell and its application

In this paper,the displacement solution method of the conical shell is presented.Fromthe differential equations in displacement form of conical shell and by introducing adisplacement function,U(s,θ),the differential equations are changed into an eight-ordersoluble partial differential equation about the displacement function U(s,θ)in which thecoefficients are variable.At the same time,the expressions of the displacement and internalforce components of the shell are also given by the displacement function.As special casesof this paper,the displacement function introduced by V.Z.Vlasov in circular cylindricalshell,the basic equation of the cylindrical shell and that of the circular plate are directlyderived.Under the arbitrary loads and boundary conditions,the general bending problem of theconical shell is reduced to finding the displacement function U(s,θ),and the generalsolution of the governing equation is obtained in generalized hypergeometric function,Forthe axisymmetric bending deformation of the     

Methods for calculation of free vibrations of a cylindrical shell with attached rigid body

We consider a mechanical system consisting of a circular cylindrical shell and an absolutely rigid body attached to one of the ends of the shell. Using the principle of possible displacements, we construct a mathematical model for the equilibrium state of the considered system under loads of general form. Using this model, we formulate an eigenvalue boundary-value problem that describes the free vibrations of the shell-body system and propose its approximate solution. In the case where the shell is replaced by an equivalent Timoshenko beam, we construct an exact solution of the problem under consideration. We also give an estimate for the effect of the rigid body on the vibrations of the system and investigate the accuracy of the beam approximation of flexural vibrations of the shell.Translated from Neliniini Kolyvannya, Vol. 7, No. 2, pp. 263–285, April–June, 2004.     

Nonaxisymmetric Thermoelastoplastic Analysis of Branched Shells of Revolution by the Semianalytic Finite-Element Method

A technique is proposed to solve elastoplastic deformation problems for branched shells of revolution under the action of asymmetric forces and a temperature field. The kinematic equations are derived within the framework of the linear Kirchhoff–Love theory of shells and the thermoelastic relations within the framework of the theory of small elastoplastic strains. The problem is given a variational formulation based on the virtual-displacement principle and the Fourier-series expansion of the unknown functions and loads with respect to the circumferential coordinate. The additional-load method is used to solve a nonlinear problem and the finite-elements method is used to carry out a numerical analysis. As an example, an asymmetric stress–strain analysis is performed for a cylindrical shell reinforced by a ring plate.     

Numerical Solution of Static Problems for Multitiered Structures

A linear stress—strain analysis is made of a structure consisting of two discretely reinforced cylindrical shells and an intermediate conical shell with large openings, which is simulated by a three-dimensional framework. This three-tier system is under longitudinal and local flexural loads. The calculations are made by the finite-difference method using modified equations of the mixed method for shells and by the deformation method for the framework. Two forms of the structure differing by specified parameters are studied     

Dynamic modeling thin skeletonal shallow shells as 2D structures with nonuniform microstructures

The subject of this consideration is a thin skeletonal elastic shallow shell with an orthogonal beam-grid microstructures. The important feature of the considered shells is that a dimension of the microstructure is of an order of the shell thickness. The formulation of 2D-macroscopic mathematical model of these shells, based on a tolerance averaging approximation (Wo?niak et?al., 2008), is the aim of the paper. During the modeling procedure, the shell under consideration is treated as a structure with a nonuniform microstructure. The general results of the contribution will be illustrated by the analysis of natural vibrations of a cylindrical thin skeletonal shallow shell.