Thin Shells Of Revolution Research Articles

A fuzzy optimum design of axisymmetrically loaded thin shells of revolution

This paper presents a fuzzy optimum design of axisymmetrically loaded thin shells of revolution. This paper consists of two parts, namely: an elastic analysis using the new curved element for finite element analysis developed in this study for axisymmetrically loaded thin shells of revolution, and the volume optimization on the basis of results evaluated from the elastic analysis. The curved element to meridian direction is used to develop the computer program. The results obtained from the computer program are compared by exact solution of each analytic example. The fuzzy optimizations of thin shells of revolution are done using [Model 2] which is in the form of a conventional crisp objective function and constraints with non-membership function, and nonlinear optimum GINO (General Interactive Optimizer) programming. In this paper, design examples show that the fuzzy optimum designs of the steel water tank and the steel dome roof could provide significant cost savings.

General axisymmetry problems for shells of revolution and analyses of integral equation

In this paper, a simplified equation in complex form for axisymmetry elastic thin shells of revolution under arbitrary distributed loads is given. The equation is equivalent to the exact equations within the error range of the thin shell theory, with the singularities at the points of meridional extreme values eliminated. A Volterra integral equation of the problem and the numerical solutions are given.

Alternative lower bounds analysis of elastic thin shells of revolution

Presents an alternative lower bound to the elastic buckling collapse of thin shells of revolution, in comparison with results from geometrically non‐linear elastic analysis. The numerical finite element method is based on axisymmetric rotational shell elements whose strain‐displacement relations are described by Koiter’s small finite deflection theory, with displacements expanded circumferentially using a Fourier series. First, compares the reduced stiffness linear analysis, based on the buckling equation without incremental linear in‐plane energy components corresponding to the lowest eigenmode (for a particular cylindrical shell under external pressure), with the results obtained by Batista and Croll. Second, the non‐linear astatic (quasi‐static) elastic analysis to clamped spherical caps under uniform external pressure is carried out in order to compare the results from a reduced stiffness analysis from viewpoints of not only buckling loads, but also total potential energy. Argues that the astatic buckling loads may relate to reductions due to a specific imperfection effect on elastic buckling collapses.

A formula for the frequency counting function of a thin shell of revolution for the case of a simple turning point

A formula for the frequency counting function of a thin shell of revolution for the case of a simple turning point

Solving some problems in the theory of thin shells of revolution with complex boundary conditions

An approach is proposed to solving linear boundary-value problems for shells of revolution that are closed in the circumferential direction, with complex boundary conditions in which the coefficients of the solving functions depend on the circumferential coordinate. The approach relies on reduction of the boundary-value problem to a number of boundary-value problems for systems of ordinary differential equations and systems of algebraic equations. We solve a specific problem for the stressed state of a conical shell with one of its ends supported by an elastic foundation with a variable modulus.

Un nouvel élément fini du type B-spline pour l'analyse dynamique des coques de révolution

ABSTRACT A new curved axisymmetric B-spline finite element for the free vibrations of thin shells of revolution is presented. The B-spline element, which is based on classical thin shell theory uses the uniform cubic B-spline functions for the interpolation of geometry as well as local displacements. It introduces three new features over the other known B-spline elements: i) instead of the usual control (or spline) variables, only nodal displacements at the internal nodes of a superelement (a group of elements) and their first derivatives with respect to the meridional length at the end nodes are used as degrees of freedom of a superelement; This choice reduces to the minimum the total number of degrees of freedom compared to standard elements, ii) a new approach for the discretization of the meridian, which dissociates the geometry and the behavior is used; thus, according to the problem in hand either the geometry or the behavior can be favoured by increasing its corresponding number of nodes. iii) all ...

Wrinkle-free solutions in the theory of curved circular membranes

Rotationally symmetric deformations of a curved circular elastic membrane under a vertical surface load are studied, with prescribed radial stresses or radial displacements at the edge. Considering Reissner's theory of thin shells of revolution suffering small strains but arbitrarily large deflections and rotations, the determination of the principal stresses in the membrane is shown to be equivalent to the solution of a single, second-order ODE, expressed in terms of a geodesic variable. Analytical techniques are applied in order to determine a limit curve of those boundary data, which subdivide the parameter range into complementary domains of existence and non-existence of tensile solutions. Finally, the more restricted subdomain of those boundary parameters is determined which admit wrinkle-free solutions, i.e. solutions governed by a nonnegative radial and circumferential stress component.

Free vibration analysis of thick spherical shells

The axisymmetric and non-symmetric vibrations of spherical shells are analysed using the thick shell theory. The semi-analytical method is used to reduce the dimension of the problem. A thick shell finite element is developed for the analysis of general shells of revolution and used for this purpose. The element can be used for the analysis of thick and thin shells of revolution. Effects of transverse shear, normal strain and rotary inertia are included in the formulation. Both shallow and deep shells are considered for the analysis. Frequencies are obtained for spherical caps with and without centre cutout.

Existence theorems for some boundary value problems in the nonlinear theory of annular elastic membranes

A simplified version of Reissner's theory of thin shells of revolution suffering small strains but arbitrarily large deflections and rotations reduces, when specialized to axi-symmetric deformations of annular membranes under a vertical surface load, to a nonlinear ordinary differential equation which is free of Poisson'ratio. Within this framework the questions of existence and non-existence of non-negative solutions of the associated stress and displacement boundary value problem are brought to a final answer. Progress in this direction was made in an earlier study (Grabmüller & Pirner 1987). In this paper a continuous monotone curve is constructed which effects a subdivision of the respective ranges of boundary data into complementary domains of existence and non-existence of strictly positive solutions

Wrinkle-free solutions of circular membrane problems

Axisymmetric deformations of a plane circular membrane subjected to an axial surface load are studied, with prescribed radial stresses or radial displacements at the edge. Considering the small-finite-deflection theory of Foppl-Hencky as well as a simplified version of Reissner's nonlinear theory of thin shells of revolution, the determination of the principal stresses in the membrane is shown to reduce to the solution of a nonlinear second-order ODE. In the Foppl case, the basic equation becomes singular when the membrane edge is free of traction. Nevertheless, existence and uniqueness of nonnegative solutions for zero edge traction is proved in both Foppl and Reissner models. Thereby, a limit curve for the radial displacement at the boundary is induced in the Reissner case, which subdivides the actual parameter range into complementary domains of existence and nonexistence of tensile solutions. The limit curve is studied, analytically and numerically. Finally, a maximum principle is established in order to determine the more restricted subdomain of those parameters which admit wrinkle-free solutions, i.e. solutions governed by a nonnegative radial and circumferential stress component.

A finite element scheme based on the simplified Reissner equations for shells of revolution

Abstract A C0 mixed finite element scheme is presented for the solution of the simplified Reissner equations for thin shells of revolution. The two governing equations, which were proposed by Simmonds and Libai, assume small normal strains but allow large rotations. They are nonlinear in one of the variables only, a fact which is exploited in the numerical scheme. The finite element formulation leads to a system of nonlinear algebraic equations in the rotation angle. The implementation of the scheme and some computational aspects are discussed. Numerical results are presented for a clamped cylindrical shell, and compared with those obtained by using moderate-rotation theory and by using the linear theory.

Wrinkle-free solutions in the theory of annular elastic membranes

Rotationally symmetric deformations of a flat annular elastic membrane under a gravitational force are studied, with prescribed radial stresses or horizontal displacements at the edges. The small-finitedeflection theory of Foppl-Hencky as well as a simplified version of Reissner's static first approximation theory of thin shells of revolution are applied which lead to consider a single, second-order, ordinary differential equation for the derivation of the principal stresses in the membrane. Using analytical methods, the range of those boundary data is determined for which the solutions of the differential equation are wrinkle free in the sense that both the radial and the circumferential stress components are nonnegative everywhere.

Finite Axisymmetric Deformation of Rubber-Like Shells of Revolution

The finite axisymmetric deformation of a thin shell of revolution is treated in this analysis. The governing differential equations are given for a hyperelastic shell material with the Mooney-Rivlin strain-energy-density function. These equations are solved numerically using a 4th-order Runge-Kutta integration method. A generalized Newton-Raphson iteration procedure is used to systematically improve trial solutions of the differential equations. The governing differential equations are differentiated with respect to a set of generalized coordinates to derive associated rate equations. The rate equations are solved numerically to generate the tangent stiffness matrix which is used to determine the load deformation history of the shell with incremental loading. Numerical examples are presented to illustrate the major characteristics of nonlinear shell behavior.

On Uniqueness of Axisymmetric Deformations of Elastic Plates and Shells

Finite axisymmetric deformations of thin shells of revolution are considered for problems where the radial membrane stress is nonnegative. It is rigorously proved that the solutions of the relevant boundary value problems for both closed and open (doubly connected) shallow shells subjected to arbitrary normal surface load and various edge conditions are unique. This result is shown to hold also for a restricted class of boundary value problems for nonshallow shells.

A simplified version of Reissner's non-linear equations for a first-approximation theory of shells of revolution

The static equations of the first-approximation theory of thin shells of revolution suffering small strains but arbitrarily large rotations are reduced, to within errors inherent in the assumption of a quadratic strain-energy density without transverse shear strains, to a coupled pair of equations for the meridional angle of rotation and a stress function. These equations are simpler than both Reissner's equations as well as a simpler version of Reissner's equations proposed by Koiter whose equations, like ours, are virtually free of Poisson's ratio.

Axisymmetrical elements of thin shell of revolution corresponding to different types of variational principles

The purpose of this paper is to investigate, to some extent, the influnce of variational constraints on the finite element properties, which are based on different types of variational principles. Taking axisymmetrical elements of thin shell of revolution (abbreviated as TSR element) as comparative elements, and with the same geometrical description, we derive seven kinds of TSR hybrid elements and two kinds of TSR conforming elements corresponding to three types of hybrid variational principles and potential energy principle respectively. By analysing the element stiffness formulations and comparing the numerical calculations, such as corrugated shell, we discuss the differences in properties of different models, and the adaptability, limitation as well as relationship between two types of models. We also point out a devergence case of TSR hybrid displacement element, and suggest two kinds of more acceptable TSR elements.

Effect of geometric nonlinearity on the stress-strain state of compensating members operating in the elastoplastic range

In some branches of modern engineering, such as in power and nuclear instrument construction, compensating members (CM) operating under cyclic loading are widely used. This article deals with an algorithm for calculating the state of stress and strain (SSS) with geometric nonlinearity taken into account and concerning CM operating in tension-compression in the elastroplastic range. The magnitudes of the obtained deformations are compared with the results of the geometrically linear problem. The SSS of a CM made in the form of a sectional thin shell of revolution is described by a system of nonlinear differential equations of first order. For the numerical realization of the described algorithm, a program in ALGOL was worked out, and with its aid the authors investigated the SSS of standard variants of CM. As an example, the single-lens CM Du1200 mm made of steel 1Kh18N9T was examined. An analysis of the results showed that in the elestoplastic range geometric nonlinearity has a slight effect on the SSS in the operation of CM in the case of tension, and that in the case of compression it has a considerable effect on its SSS.

On the general solution of stress equation for the membrane theory of shells of revolution

A new method is presented to solve the equation of Nemenyi-Truesdell, in the membrane theory of thin shells of revolution. It is an ordinary linear differential equation of the second order, for which resolution methods of Truesdell and Mitrinovitch are known. The proposed method is based on the transformation of the given equation into a linear integral equation of Volterra type. This allows to write the general solution of stress equation only in terms of equation of the meridian generating the shell.

Numerical asymptotic solution of strength and vibrations problems of thin shells of revolution

Numerical asymptotic solution of strength and vibrations problems of thin shells of revolution

Application of the method of removed elements to the investigation of residual stresses in thin shells of revolution made of composite materials

1. The distribution of residual stresses in the investigated shells is very nonaxisymmetric, whereas the elastic characteristics of the material of the shells are only slightly nonaxisymmetric. 2. Residual stresses attain their most substantial values at the bottoms of structures, and the most dangerous ones are circumferential residual stresses. 3. Residual stresses due to thermoelastic contraction of shells, determined theoretically, describe the distribution of residual stresses in shells only qualitatively.