Deflection Of Shell Research Articles

Large deflections of shells subjected to an external load and temperature changes

Shell stability in geometrically nonlinear range is considered. Influence of the temperature change on the value of the critical force is examined. The whole analysis is performed numerically using the Finite Element Method. To solve the problem, nonlinear equilibrium paths for given temperature change and various load values must be calculated. These are cross-sections of the nonlinear equilibrium surface. From such paths the critical value of the load for given temperature change can be determined and stability boundary can be found as the main purpose of the procedure. The detailed numerical analysis was performed in reference to the semi cylindrical shell segment subjected to temperature changes and loaded by the concentrated force.

Analysis of shell thickness irregularity in continuously cast middle carbon steel slabs using mold thermocouple data

Thermocouples buried in the mold wall of a continuous caster are used to investigate the nature and source of shell thickness irregularity in middle carbon steel slabs. Fourier analysis is used in conjunction with digital filters to determine the power spectra of time series mold temperature and mold level measurements. Direct evidence is obtained on the physical dimension of irregularity, as well as the phase relationships between neighboring thermocouples in both the transverse and longitudinal directions. In addition, mold thermocouple readings are used to set the boundary heat flux conditions for use in self-consistent mathematical modeling of mold thermal profiles. Temperature readings—average, minimum, and maximum—allow for the calculation of an envelope of shell thicknesses around the average distribution. These techniques are used to help explain a mechanism for the occurrence of shell thickness irregularity, in terms of both meniscus disturbances and shell deflections in response to such disturbances.

The Resisting Mechanism during the Large Normal Deflection of Thin Shells Subjected to Follower Forces

The load resistance mechanism of thin shells, such as the submerged shell structures in underwater marine constructions, are in principle of a nonconservative nature since the associated loads are the hydrostatic pressure and drag forces of the follower type.The general governing equations of equilibrium for thin shells are available from the literature in various details. In this research we consider shells defined in a monoclinically convected coordinate system subjected to follower loads and undergoing large deformations. The feasibility and significance of the theoretical formulations have already been substantiated through various numerical simulation results.In the simultaneous equations governing the equilibrium of shells, the presence of terms related to the shell curvature and other consequential terms such as the Christoffel symbols, clearly make them a substantially different class from the plate theory. This distinction, which may otherwise be denoted as the 'Form Effect' in more common terminology, renders the shell theory a lot more complicated and one that needs in-depth analytical effort to unravel the full significance of all the possible implications.The present paper elaborates on the mechanism of large deformation by studying the share of different stiffness factors on the total load resistance equilibrium picture of shells from the very shallow to the deep curvature range going through the subsequently increasing loading stages. This study concentrates mainly on the equilibrium in the normal direction. The resisting mechanism due to the extensional and the bending stiffness parts are considered as the two main entities and their subdivisions and other separable factors are studied in unison to draw out an overall picture. The partial cylindrical and spherical shells are considered and the results bring out the clear distinction between resistance mechanism of the two shells, especially in the deep curvature region.These results in their total details are believed to be helpful in understanding the deformation mechanism and formulate some design philosophy for shells and thereby serve as a guideline for appropriate formulations of the governing equations in particular cases.

Finite deflection and snapping of thin elastic flat panels

The problem of the non-linear. behaviour and stability loss of bars and shells under finite deflection is discussed. The onedimensional model, which plays a major part in the formulation of non-linear stability problems, is described in detail. In this connection, Ivan Grigor'yevich Bubnov's problem on the behaviour of a thin elastic cylindrical panel is analysed (the problem was solved by him in 192 and was the first study on finite deflections of thin shells). The significance of this result in the theory of shells in its own right and from the perspective of the subsequent development of the theory is discussed. The importance of Bubnov's results in shell theory is pointed out. This also includes his solution of the non-linear behaviour of circular plates and plane elastic panels. This is why he can be regarded as a forerunner in formulating the equations of finite deflections of elastic thin-walled Foeppl-Káimán-Marguerre surfaces.

Neoclassical Solution of Transient Interaction of Plane Acoustic Waves with a Spherical Elastic Shell

A detailed solution to the transient interaction of plane acoustic waves with a spherical elastic shell was obtained more than a quarter of a century ago based on the classical separation of variables, series expansion, and Laplace transform techniques. An eight­ term summation of the time history series was sufficient for the convergence of the shell deflection and strain, and to a lesser degree, the shell velocity. Since then, the results have been used routinely for validation of solution techniques and computer methods for the evaluation of underwater explosion response of submerged structures. By utilizing modern algorithms and exploiting recent advances of computer capacities and floating point mathematics, sufficient terms of the inverse Laplace transform series solution can now be accurately computed. Together with the application of the Cesaro summation using up to 70 terms of the series, two primary deficiencies of the previous solution are now remedied: meaningful time histories of higher time derivative data such as acceleration and pressure are now generated using a sufficient number of terms in the series; and uniform convergence around the discontinuous step wave front is now obtained, completely eradicating spurious oscillations due to the Gibbs' phenomenon. New results of time histories of response items of interest are pre­ sented. © 1996 John Wiley & Sons, Inc.

Neoclassical Solution of Transient Interaction of Plane Acoustic Waves with a Spherical Elastic Shell

A detailed solution to the transient interaction of plane acoustic waves with a spherical elastic shell was obtained more than a quarter of a century ago based on the classical separation of variables, series expansion, and Laplace transform techniques. An eight-term summation of the time history series was sufficient for the convergence of the shell deflection and strain, and to a lesser degree, the shell velocity. Since then, the results have been used routinely for validation of solution techniques and computer methods for the evaluation of underwater explosion response of submerged structures. By utilizing modern algorithms and exploiting recent advances of computer capacities and floating point mathematics, sufficient terms of the inverse Laplace transform series solution can now be accurately computed. Together with the application of the Cesaro summation using up to 70 terms of the series, two primary deficiencies of the previous solution are now remedied: meaningful time histories of higher time derivative data such as acceleration and pressure are now generated using a sufficient number of terms in the series; and uniform convergence around the discontinuous step wave front is now obtained, completely eradicating spurious oscillations due to the Gibbs' phenomenon. New results of time histories of response items of interest are presented.

Optimal design of axisymmetric plastic shallow shells of von Mises material

An optimal design technique developed earlier for axisymmetric plates and circular cylindrical shells is accommodated for shallow spherical shells subjected to uniform transverse pressure. Material of the shells is assumed to be rigid-plastic obeying the von Mises yield condition and the associated deformation law. The post-yield behaviour of the shells is taken into account. The weight minimization is performed under the condition that the maximal deflection of the shell of variable thickness coincides with the deflection of the reference shell of constant thickness. The problem is transformed into a non-linear boundary value problem which is solved numerically.

Experimental study of induced oscillations with large deflections of fiberglass shells of revolution

Experimental study of induced oscillations with large deflections of fiberglass shells of revolution

A study of the deflection of an isotropic shell under the action of an abruptly applied concentrated normal force

We consider the perturbed deformed state of a thin shallow isotropic shell of nonnegative Gaussian curvature under the action of an abruptly applied concentrated normal force. Using the integral Fourier transform method with respect to the geometric coordinates and the Laplace transform with respect to time we obtain a solution of this problem in the form of double series in generalized Fresnel sine and cosine integrals. We give numerical results for the initial reaction of the shell at the point of application of the load. We give a comparison with known results. Two figures. Bibliography: 4 titles.

Large deflection of an orthotropic shallow spherical shell on a Pasternak foundation

Non-linear static analysis of an orthotropic shallow spherical shell on a Pasternak type elastic foundation is carried out, using the Sinharay and Banerjee approach. The effect of the geometric parameter of the shell and the foundation and shear modulus of the foundation on load-deflection characteristics is investigated for movably and immovably clamped and for simply supported edge conditions.

The generation of non-linear stiffness matrix of triangle element when considering both the bending and in-plane membrane forces

Using Stricklin Method|5|, we have this paper has derived the formulas for the generation of non-linear element stiffness matrix of a triangle element when considering both the bending and the in-plane membrane forces. A computer programme for the calculation of large deflection and inner forces of shallow shells is designed on these formulas. The central deflection curve computed by this programme is compared with other pertaining results.

The large deflection of cylindrical shells on a Kerr foundation

The non-linear static behaviour of cylindrical shells resting on a Kerr type elastic foundation is investigated in detail. The influence of shell and foundation parameters on the load deflection relationship are also studied in detail. It is noticed that by assigning extreme values to certain foundation variables, the Kerr model can be reduced to correspond with the Winkler or Pasternak model. It is found that an increase in curvature, in Poisson's ratio, in shear layer or lower spring layer modulus results in reduced shell deflection. Further study reveals that shells with immovable edges are more sensitive to changes in geometry and to material and foundation variables than the ones with movable edges. This is true in the case of both static and dynamic analyses for clamped as well as for simply supported edges.

Plastic Deformation and Rupture of Ring-Stiffened Cylinders under Localized Pressure Pulse Loading

An analytical solution for the dynamic plastic deformation of a ring-stiffened cylindrical shell subject to high intensity pressure pulse loading is presented. By using an analogy between a cylindrical shell that undergoes large plastic deformation and a rigid-plastic string resting on a rigid-plastic foundation, one derives closed-form solutions for the transient and final deflection profiles and fracture initiation of the shell. Discrete masses' and springs are used to describe the ring stiffeners in the stiffened shell. The problem of finding the transient deflection profile of the central bay is reduced to solving an inhomogeneous wave equation with inhomogeneous boundary conditions using the method of eigenfunction expansion. The overall deflection profile consists of both global (stiffener) and local (bay) components. This division of the shell deflection profile reveals a complex interplay between the motions of the stiffener and the bay. Furthermore, a parametric study on a ring-stiffened shell damaged by a succession of underwater explosions shows that the string-on-foundation model with ring stiffeners described by lumped masses and springs is a promising method of analyzing the structure.

Plastic Deformation and Rupture of Ring-Stiffened Cylinders under Localized Pressure Pulse Loading

An analytical solution for the dynamic plastic deformation of a ring-stiffened cylindrical shell subject to high intensity pressure pulse loading is presented. By using an analogy between a cylindrical shell that undergoes large plastic deformation and a rigid-plastic string resting on a rigid-plastic foundation, one derives closed-form solutions for the transient and final deflection profiles and fracture initiation of the shell. Discrete masses' and springs are used to describe the ring stiffeners in the stiffened shell. The problem of finding the transient deflection profile of the central bay is reduced to solving an inhomogeneous wave equation with inhomogeneous boundary conditions using the method of eigenfunction expansion. The overall deflection profile consists of both global (stiffener) and local (bay) components. This division of the shell deflection profile reveals a complex interplay between the motions of the stiffener and the bay. Furthermore, a parametric study on a ring-stiffened shell damaged by a succession of underwater explosions shows that the string-on-foundation model with ring stiffeners described by lumped masses and springs is a promising method of analyzing the structure.

On the application of a limiting process to the dynamic relaxation analysis of circular membranes, circular plates and spherical shells

Two efficient techniques in the dynamic relaxation (DR) iteration procedure are described: (1) a limiting process technique for the governing equations expressed in polar or spherical coordinates and (2) a new DR-formulation for the time-step proposed by Mikami. The former technique overcomes the mathematical singularity at the coordinate origin without the use of interlacing finite difference meshes. The latter solution technique allows all variables to be evaluated at the same time without the use of fictitious densities. The validity of the use of these two techniques has been demonstrated through the numerical results for the problems of large deflection of axisymmetric circular membranes, large deflection of axisymmetric circular plates, and small deflection of axisymmetric spherical shells.

部分荷重を受ける球形シェルの非保存的安定性解析

The purpose of the-present study is to show theoretically that dynamic instability takes place in a clamped spherical shell under a partially distributed normal follower force. Large deflection and stability of shallow and deep spherical shells are studied. The axisymmetric equilibrium states are solved through a Newton-Raphson technique on discretized nonlinear shell equations by means of finite difference method. The stability of the respective equilibrium states on the axisymmetric fundamental equilibrium path is examined by the dynamic method. In this study, it is shown that transition from stability to instability takes place not only in the form of static instability, but also in the form of dynamic instability.

General Stability Analysis of Lattice Shells by Continuum Modeling

In this paper a linear elastic stability problem of lightweight lattice shells is studied. For this purpose a calculation on an equivalent rigidity continuum model is used. It is assumed that the rods of the analysed shells consist of a homogeneous, isotropic elastic material the mechanical properties of which are invariant in the time. The Author derived analytical expressions the point of departure of which is the theory of finite deflections of shallow shells. The use of the Bubnov-Galerkine method allows to define a relationships yielding the values of critical stress in closed analytical form. The effect of initial geometrical shell imperfections of the “medial” surface is taken into consideration, too. Sample numerical results are presented to demonstrate the effectiveness of the continuum modeling approach. The obtained theoretical results are documented some of the numerical examples.

The radiation of sound waves from a lightly loaded finite elastic shell, I: Duct and shell resonances for linear motions

This paper is the first half of a study into the scattering of acoustic waves by a finite circular cylindrical elastic shell. The flexible body is set into a rigid cylinder of infinite extent, and the shell motions are excited by waves which propagate from infinity inside the duct. In this paper we assume a linear response for the shell/fluid system. By employing the assumption that the fluid density is much smaller than the plate density, the scattered field both outside and interior to the cylinder is calculated from the shell deflections. At a forcing frequency near to that of the free vibration frequency of the shell, the asymptotic expansions must be modified to take into account the increased amplitude of the structural vibrations. We employ the method of multiple scales to calculate the solution near to these shell resonances, and the method is also applied near to the duct resonances. In a followup to this paper the effects of non-linearities in the shell equations will be investigated by the same method, and the modifications to the position and magnitudes of the resonant peaks in the scattered field will be discussed.

On equations for large deflections of elastic plates and shallow shells

On equations for large deflections of elastic plates and shallow shells

An assumed modes approach for higher-order effects in circular cylindrical shells.

Numerous investigators, notably Mindlin, have developed higher-order theories for beams, plates, and shells in which the variation of displacement through the thickness is represented in terms of series expansions. A major difficulty one encounters in those formulations is the derivation of hierarchal partial differential equations of motion accounting for the coupling between different orders of thickness dependence. The present work develops an alternative scheme that avoids the derivation of such equations by applying the assumed modes method in conjunction with Hamilton’s principle. An infinitely long cylinder provides the framework for the development. The basis functions representing the dependence on radial position consist of Legendre polynomials, which have been used in the past to analyze static deflections of thick cylindrical shells, while sinusoidal functions represent the variation along the circumference. The assumed mode representation leads to ordinary differential equations of motion governing the amplitudes of the basis functions. Because of the axisymmetric geometry, the modal equations decouple in the circumferential direction. When the shell is geometrically thin, h/a≪1, each radial mode couples to two adjacent modes above and below. The extent of this coupling increases with increasing h/a, but the mass and stiffness matrices for the modal amplitudes are banded in any case. [Work supported by ONR.]