Toroidal Shell Segments Research Articles

Post-buckling of sigmoid-functionally graded material toroidal shell segment surrounded by an elastic foundation under thermo-mechanical loads

The nonlinear buckling and post-buckling of ceramic–metal–ceramic layers (S-FGM) toroidal shell segment surrounded by elastic foundation under thermo-mechanical loads are investigated with an analytical approach in this paper. Based on the classical thin shell theory with geometrical nonlinearity in von Karman–Donnell sense, Stein and McElman assumption and Pasternak foundation model, the governing equations of nonlinear buckling of S-FGM toroidal shell segment are analyzed. The static critical buckling loads and the post-buckling analyses in two cases – movable and immovable boundary conditions including temperature effects are obtained. Furthermore, the effects of geometry ratios, characteristic of materials, elastic foundation and thermal environment on the nonlinear buckling of shells are presented.

Non-linear buckling analysis of FGM toroidal shell segments filled inside by an elastic medium under external pressure loads including temperature effects

The analytical approach is presented to investigate non-linear buckling analysis and post-buckling behavior of FGM toroidal shell segments filled inside by an elastic medium under external pressure loads including temperature effects. The governing equations of non-linear buckling of FGM toroidal shell segments are derived based on the classical thin shell theory with the geometrical nonlinearity in von Karman–Donnell sense, Stein and McElman assumption and elastic medium modeled by Pasternak's two-parameter elastic foundation. The static critical buckling loads and the post-buckling pressure–deflection curves in two cases: movable and immovable boundary conditions including temperature effects are obtained by using the Galerkin's method. In the paper, the results are also compared with the solutions published in the literature for the specific cases. Effects of geometrical and material parameters, elastic foundation and temperature on the nonlinear buckling behavior of shells are shown in obtained results.

Torsional buckling and post-buckling behavior of eccentrically stiffened functionally graded toroidal shell segments surrounded by an elastic medium

The nonlinear buckling and post-buckling problems of functionally graded stiffened toroidal shell segments surrounded by an elastic medium under torsion based on an analytical approach are investigated. The rings and stringers are attached to the shell, and material properties of the shell are assumed to be continuously graded in the thickness direction. The classical shell theory with the geometrical nonlinearity in von Karman sense and the smeared stiffeners technique are applied to establish theoretical formulations. The three-term approximate solution of deflection is chosen more correctly, and the explicit expression to find critical load and post-buckling torsional load-deflection curves is given. The effects of geometrical parameters and the effectiveness of stiffeners on the stability of the shell are investigated.

Buckling Analysis of Eccentrically Stiffened Functionally Graded Toroidal Shell Segments under Mechanical Load

AbstractAn analytical approach is presented to investigate the linear buckling of eccentrically stiffened functionally graded thin toroidal shell segments subjected to axial compression, lateral pressure, and hydrostatic pressure. On the basis of classical thin shell theory, the smeared stiffener technique and the adjacent equilibrium criterion, the governing equations of buckling of eccentrically stiffened functionally graded toroidal shell segments are derived. The functionally graded toroidal shell segments with simply supported edges are reinforced by a ring and stringer stiffener system on an external surface. The resulting equations in the case of compressive and pressure loads are solved directly. The obtained results show the effects of stiffeners and input factors on the buckling behavior of these structures. In this paper, the results are also compared with the solutions published in the literature for the specific cases of a toroidal shell.

Initial post buckling of toroidal shell segments pressurized by external load

Initial post buckling of toroidal shell segments pressurized by external load

Influence of extra terms on asymptotic analysis of imperfection sensitivity of toroidal shell segment with random imperfection

The bifurcation of a toroidal shell segment with initial imperfection which are subjected to lateral or hydrostatic pressure is studied under the assumption that the initial imperfection are Gaussian random stress-free displacement whose mean and autocorrelation function are given. We use a perturbation scheme developed by Amazigo [Amazigo, J.C., 1971. Buckling of stochastically imperfect columns on nonlinear elastic foundation. Quart. Appl. Math. 403–491]. A simple approximate asymptotic expression is obtained for the bifurcation load for small magnitudes of the imperfection. The result is compared with results obtained earlier under secondary bifurcation analysis for the imperfections in the shape of the buckling mode and the results in the literature, which shows some significant differences as a result of inclusion of extra terms in the buckling equation.

Disruption Induced Voltages and Loads on Torus Sectors

The rapid decay of magnetic flux during a plasma disruption induces voltages, currents, and Lorentz loadings in nearby electrically-conducting material. Present designs employ toroidal shells or shell segments near the plasma. These shells are divided into sectors for assembly and maintenance considerations, but may have toroidally-continuous conducting paths due to the need for vacuum boundaries. Voltages induced across sector gaps may initiate arcing and subsequent material damage. In addition, induced eddy currents in the shells can interact with the toroidal field and generate large net torques on a sector. A finite element model was used to estimate the induced sector gap voltages and net overturning moments following a 10 ms disruption. The number of shells, toroidal continuity, resistivity, and shell thicknesses were varied. Results are presented that show the effects of these changes on the sector gap voltages and induced loads.

Numerical improvement of asymptotic solutions for shells of revolution with application to toroidal shell segments

A simple technique for the analysis of shells of revolution is presented and its application to toroidal shells shown. The method employs the leading term of the formal asymptotic solution and the eigenvalue of the exact coefficient matrix. The one-term approximation is then used as a trial solution in an integral equation form of the problem. Through the method of successive substitutions, better approximations are obtained by numerical integration. The one-term solution provides the correct boundary-layer behavior for the steep shell and the successive integration converge to the exact solution in the shallow region. In contrast to finite element and finite difference methods, this technique becomes more efficient as the radius-to-thickness ratio of the shell increases.

Plastic Bifurcation Buckling of Toroidal Shells

The buckling characteristics of shells with double curvature is an important design consideration of such structures. Aircraft structural components are commonly of the double curvature type. Classical elastic buckling of toroidal shell segments with positive and negative Gaussian curvatures under pressure loading is given by Stein and McElmann. A further investigation of elastic toroidal shells, including bowed-out segments subjected to axial tension and a study of the initial post-buckling behavior, is performed by Hutchinson.

Solution of a homogeneous boundary value problem for the sector of a toroidal shell segment: PMM vol. 40, n≗ 4, 1976, pp. 755–759

Membrane forces in the segment of a thin toroidal shell loaded by an edge bending load are determined from the particular solution of the fundamental differential equation. Taking account of the asymptotic approximation of the special function in whose terms the particular solution is expressed, it is shown in [1] that the particular solution for a thin toroidal shell agrees with the membrane solution. In the general case, the tensile forces in a shell not closed in two coordinates are determined by membrane theory ; the membrane state of stress is determined taking the solution obtained into account. The method of characteristics, used in [2] to analyze membrane shells, is used to solve the problem. The fundamental differential equation is solved by means of a special integral transform using Fourier series. Results of a numerical computation on an electronic computer and of an experiment are presented.

Initial post-buckling behavior of toroidal shell segments

The initial post-buckling behavior of double curvature shell segments subject to several loading conditions is determined on the basis of Koiter's general theory of initial post-buckling behavior. Previously, the classical buckling loads associated with these shells were shown to be strongly dependent on the two radii of curvature and their relative magnitudes. Here, the initial post-buckling behavior and associated imperfectionsensitivity are also seen to be strongly dependent on the two curvatures.

Stiffness of thin pressurized shells of revolution

A quasi-linear approach is developed. The strains and rotations are assumed to be small, but the equilibrium conditions are fulfilled on the deformed shell. In the resulting differential equation, bending and appear side by side. The asymptotic form of this equation is discussed for the case of the circular torus. Considered separately, the two effects are fairly similar. Suitable superposition of the two limit solutions yields a close approximation to the composite solution. As a sample application, the transverse of torus shells is derived and is presented as a function of internal p and shell thickness h. It is found in particular that membrane theory remains correct for h small but finite. Linear shell theory, on the other hand, involves an error that is proportional to p if p is small. I. Introduction T HIS paper is concerned with the rotationally symmetric deformations that pressurized, thin elastic shells of revolution experience under axisymmetric external loads. The approach to this problem presented here is a quasilinear approach: the shell strains and rotations are assumed to be small, but, contrary to the approach of linear shell theory, the shell equilibrium conditions will be fulfilled on the deformed shell. Thereby, the stiffening that a shell experiences when it is pressurized is taken into account. This pressure stiffness is neglected in linear shell theory. This deficiency of linear theory has two aspects. Both are illustrated by the model, one half of which is shown in Fig. 1, a pressurized, complete circular torus shell $ subject to an axisymmetric system P of external loads. Consider the equilibrium of the vertical forces on either one of the two toroidal shell segments that connect the two crowns of S. If we assume that S is a membrane shell, then the only vertical forces are P and the vertical components of p. These are