Circumferential Displacement Research Articles

Buckling of eccentrically stringer-stiffened cylindrical panels under axial compression

This paper investigates the effects of boundary conditions and panel width on the axially compressive buckling behavior of eccentrically stringer-stiffened circular cylindrical panels. Numerical results are presented for eight different sets of boundary conditions along the straight edges of the panels. As the panel width is increased, the results show that the complete cylinder buckling load is reached only for one set of boundary conditions ( SS3, classical simple support conditions). However, for 180° and wider panels, the panel buckling loads are within ± 10% of the complete cylinder load for all cases except SS1 panels (free in-plane edge displacements) with outside stringers. Low buckling loads, as low as half the complete cylinder load, are found for some SS 1 panels. It is also observed that the prevention of circumferential edge displacement is the most important in-plane boundary condition from the point of view of increasing the buckling load, and that the prevention of edge rotation (i.e. clamping) in the circumferential direction is more effective in increasing the buckling loads of panels with free circumferential edge displacement υ that it is for panels with υ = 0. From stringer-eccentricity studies, it is shown that buckling loads are generally at least 40–50% higher for the case of outside stringers, and that eccentricity effects are generally similar for clamped and simply supported panels with the same in-plane boundary conditions.

Buckling of cylindrical panels under axial compression

This paper investigates the effects of boundary conditions and panel width on the axially compressive buckling behavior of unstiffened, isotropic, circular cylindrical panels. Numerical results are presented for eight different sets of boundary conditions along the straight edges of the panels. For all sets of boundary conditions except one (SSI), the results show that the panel buckling loads monotonically approach the complete cylinder buckling load from above as the panel width is increased. Low buckling loads, sometimes less than half the complete cylinder buckling load, are found for simply supported panels with free in-plane edge displacements (SSI). The SSI buckling loads are below the complete cylinder load even for ‘360° panels’. It is also observed that the prevention of circumferential edge displacement is the most important in-plane boundary condition from the point of view of increasing the buckling load, and that the prevention of edge rotation (i.e. clamping) in the circumferential direction also significantly increases the buckling load. Parametric studies are also performed to determine the effects of variations in panel length and thickness on the buckling loads.

The Axisymmetric Stress Distribution in an Infinite Elastic Solid Containing a Flat Annular Crack under Torsion

The authors analyze the axisymmetric stress distribution in an infinite elastic solid containing a flat annular crack under torsion on the basis of three dimensional theory of elasticity. Assuming that the circumferential displacement on the crack surface is continuous, we may represent it by Fourier series and reduce the problem to the solution of an infinite system of simultaneous equations. The radial distributions of the displacements and the stress components and the variations of the stress intensity factors at both crack tips with the ratio of the inner radius to the outer one are shown graphically.

Impact on a head-neck structure

An analytical and experimental study was executed involving non-destructive, axisymmetric impact on a fluid-filled shell constrained by a viscoelastic, artificial neck—a system intended to simulate a human head-neck structure. The shel material was considered to be homogeneous, isotropic and elastic, while the fluid was regarded as inviscid and compressible with its motion irrotational. The artificial neck was represented analytically by a linear viscoelastic cantilever beam that was rigidly connected to the shell. The physical model consisted of a water-filled uniform spherical thin-walled shell composed of lucite attached to an artificial neck developed by the Biomedical Laboratories of the General Motors Corporation. The system was loaded by the impact of a spherical aluminum shell with an o.d. of 7·4 in. that formed part of a simple pendulum. The impact force in both the analytical and experimental studies consisted of a half-sine pulse with a duration of 2·5 msec. The analytical solution first required the determination of the interaction forces and moments at the head-neck junction, which were ascertained from continuity conditions and an assumed rigid shell. These quantities, in turn, were applied simultaneously with the impact force to the deformable fluid-filled shell. yielding a boundary value problem that was solved on a digital computer. The resultant predictions of circumferential shell strain, fluid pressure, and displacement of the junction were found to be in good agreement with the experimental data. Implications of the results for head injury are discussed.

Finite element analysis of vibrations of initially stressed thin shells of revolution

A finite element is developed to analyse the vibration characteristics of initially stressed shells of revolution. The meridional, circumferential and normal displacements over the element are assumed to be cubic polynomials of the meridional co-ordinate. Numerical integration is used to generate the elastic stiffness, geometric stiffness and mass matrices. The element is used to analyse three typical problems pertaining to the vibrations of initially stressed thin shells of revolution.

Pipe Deformation During a Running Shear Fracture in Line Pipe

Under sponsorship of the American Iron and Steel Institute, U. S. Steel Research has been conducting full-scale burst tests of large-diameter submerged-arc-welded line pipe to determine the toughness required to arrest running shear fractures for different design conditions. As part of that program, the pipe were instrumented with crack detectors, strain gages, and pressure transducers to determine the crack velocities and the actual pipe deformation and strain fields associated with the shear fracture propagating along the top of the pipe. This paper summarizes the test data that document the manner in which the pipe deforms during this type of crack propagation. The data show that for a propagating shear fracture, each of four different locations along the pipe length (relative to the crack tip) has a distinctive type of pipe deformation. For a location many pipe diameters ahead of the crack tip, the circumferential strain first decreases because of flexural waves associated with the initiation process and then continues to decrease in proportion to the local gas decompression; however, the longitudinal strain continuously increases because of a longitudinal “tongue” of tensile straining on the top of the pipe caused by pressure-induced opening of the flaps of the pipe on both sides of the fracture behind the crack tip. At a distance about two diameters ahead of the crack tip, the pipe cross section becomes oval, and in the presence of this deformation the strain field is no longer determined by the local pressure; in fact, the circumferential strain is near zero at a distance two diameters ahead of the crack. The oval pipe shape ahead of the crack tip is caused by the venting of the gas behind the crack tip which creates a downward reactive force on the bottom portion of the pipe. Opening at the crack tip is the result of tensile straining caused by circumferential and radial displacement of the flaps behind the crack tip. Thus it is believed that the action of the pipe-wall flaps behind the crack tip provides the primary force driving the crack down the top of the pipe.

The bacterial flagellum as an imperfect cylindrical crystal: Flagellar geometry, movement and polymorphism, and the role of partial dislocations

A pairing attraction between helical turns of subunits in a cylindrical crystal, like that in the dahlemense strain of tobacco mosaic virus, can cause the axis of the rod or crystal to become helical. This is true only if the number of helices is odd. The shape of a bacterial flagellum can be accounted for then if, as Caspar & Holmes and Klug have suggested, rows of its subunits exhibit such a pairing interaction. Klug's thoughts on bacterial flagella are developed and extended into a model that accounts qualitatively for geometry, movement and polymorphism of flagella. If the number of helices between which there is a pairing interaction is odd, then the crystal is an imperfect cylindrical crystal. The geometry of such crystals is described. They contain a line defect, termed here an antiphase boundary, across which the pairing interaction is reversed. The boundary is a line of expansion on the convex side of a curved filament. Movement of flagella is explained by circumferential displacement of the antiphase boundary. One polymorphic form can convert to another if a dislocation passes along it. Straight flagella are perfect cylindrical crystals with no antiphase boundary.

The free vibrations of cylindrical shells with various end closures

The natural frequencies of vibration of cylindrical shells clamped at one end and closed at the other by different types of shells of revolution (cones, hemispheres, ellipsoids, etc.) were determined using programs written for the digital computer. The principal numerical methods investigated were variational finite-differences and finite elements. However, some results obtained by numerical integration of the differential equations of motion, together with some obtained from series solutions, are also given for purposes of comparison. For L D = 0.5 , h D = 0.01 the lowest four natural frequencies for all the cylinder/end closures investigated were concentrated in a band such that ω 4 ω 1 ⋍ 1.7 . Thus, changing the head shape does not cause radical changes in the natural frequencies for these geometric ratios. Results for other values of the geometric parameters are also given in the paper. Again for L D = 0.5 , h D = 0.01 , the effect of not equating to zero the in-plane circumferential displacement at the base of the cylinder caused a marked reduction in the n = 1 natural frequencies. In many of the cases studied, the n = 1 frequency was either the fundamental one or was close to it. Insofar as dynamic analyses are concerned, designers ought to be aware of the influence that the in-plane boundary conditions at the cylinder base can have on the natural frequencies. With regard to the main two numerical methods studied, the authors found both of them to be very effective for the natural frequency and mode shape predictions. For pointed shells (e.g. cones) it appears that for n = 0 and n = 1, the conditions employed at the apex in some of the programs need re-examination.

Line Load on Cylindrical Shell With End Plates

The elastic static stresses and displacements in a shell with end plates are determined when a circular cylinder is subjected to a uniform line load along a generator. The method of solution is the superposition of a particular solution in which the shell’s response is computed for simply supported end conditions and a complementary solution in which the true end plate reactions are imposed on the ends of the cylinder. Examination of the end conditions reveals that the assumptions Nx = w = 0 (no axial force and no radial displacement) are generally justified whereas the assumptions Mx = 0 and v = 0 (no axial moment and no circumferential displacement) will usually lead to considerable relative errors in the axial bending stresses and the circumferential membrane stresses, respectively.

The application of the simultaneous iteration method to flexural vibration problems

The simultaneous iteration method to determine the eigenfrequencies and eigenvectors is utilized for solving the flexural vibration problems of rotationally symmetric bodies. The finite element approach is used to predict the behaviour of the system. A suitable shape function for the circumferential displacement distribution has been proposed which reduces the three-dimensional problem to a two-dimensional one. The accuracy of the method has been checked by verifying the results of known cases. It has been applied to a complicated structure and results have been compared with experimental ones.

Wave transmission characteristics and anisotropy of canine carotid arteries

A method was developed to generate and record three types of small amplitude waves (pressure, torsion and axial) in the exposed carotid artery of anesthetized dogs. The pressure waves were studied with the aid of miniature pressure transducers; electro-optical tracking units monitored the axial and circumferential surface displacements. Results from 6 dogs are presented in the form of the phase velocities and attenuation of three types of waves. The data demonstrate incompatibility with an isotropic elastic model for the mechanical behavior of the artery. The measured damping appears to be primarily due to the viscoelastic properties of the vessel wall material.

Non-linear vibrations of orthotropic doubly-curved shallow shells

The purpose of this study is the evaluation of the effects of curvature, material orthotropy and internal pressure upon the non-linear vibrations of shallow shells. The shells considered here are complete in the circumferential coordinate. A mode shape that leads to a continuous circumferential displacement is used as the basic solution in a Galerkin type procedure. Among the results of interest are the effects of the aforementioned physical parameters on the relative hardness of the shell response, as well as some interesting new perspectives on jump phenomena in non-linear shell vibrations.

A Dynamic Shell Theory Coupling Thickness Stress Wave Effects With Gross Structural Response

A theory for thick cylindrical shells is presented that couples the early time thickness stress wave propagation with higher-order shell theory equations. The formulation completely describes the continuum response in the thickness direction for “early time” considerations while representing the circumferential response with a high-order circumferential displacement assumption. Late time (structural response) equations are developed to continue the analysis after thickness effects are no longer important. The finite-element method is used to obtain solutions of the theory. Calculations are presented which show that thickness stress need not be included for cylindrical shells with h/R ratios less than 0.2.

Solid flow pattern at the wall of a fluidization column induced by single bubbles

Particle flow induced by single bubbles at the wall of fluidized beds, 170–350-μm alumina catalyst, is observed by a photographic technique. The fluidization column is 350 mm i.d. Bubbles are injected by a vertical 6 mm i.d. nozzle placed at different distances between the column axis and the wall. Bubble diameters, ranging from 5 to 15 cm, are determined when emerging at the bed upper surface. The solid is traced with black particles, whose individual motion is filmed at the wall. Overall vertical and circumferential tangential particle displacements, upwards and downwards, are observed for different sizes of bubble, for different distances of the flow path of the bubble center from the wall and for circumferential distances from the point of the wall nearest to the bubble. Vertical particle displacement at this point is correlated to bubble diameter and position.

Further studies in the application of curved finite elements to circular arches

The work of Ashwell and Sabir,1 assessing the power of curved finite elements by applying them to circular arches, is extended. An element I based on the cylindrical shell element of Bogner et al.,3 converges satisfactorily for deflexions, for all arch types, but is inferior to a new element II based on simple strain functions. A natural (“exact”) shape function is also presented. When stress resultants are calculated, both I and II converge rapidly if the stress resultants are calculated from an element stiffness matrix, but are sometimes unsatisfactory (except for mid-element stress resultants) if a normal type of stress matrix is used.I becomes unsatisfactory if nodal continuity of the gradient of circumferential displacement is imposed.

Snapping of Imperfect Thin-Walled Circular Cylindrical Shells of Finite Length

The nonlinear partial differential equations of von Karman and Donnell governing the deformations of initially imperfect cylindrical shells are reduced to a consistent set of ordinary differential equations. A numerical procedure is then used to solve the equations together with the associated boundary conditions and to determine the number of waves at buckling as well as the load-carrying capacity of imperfect cylindrical shells of finite length subjected to uniform axial compression in the presence of a reduced restraint along the simply supported boundaries. It is found that details of the boundary conditions have little effect on the number of waves into which the shell buckles around the circumference. This number is determined essentially by the length-to-radius and radius-to-thickness ratios. The absence of an edge restraint to circumferential displacement reduces the classical value of the buckling load by a factor of about two. On the other hand, shells with these boundary conditions appear to be less sensitive to initial imperfections in the shape, and thus the maximal load supported in the presence of unavoidable initial deviations can be the same for shells with and without a restraint to circumferential displacements along the edges.

Limitations of certain curved finite elements when applied to arches

Several segmental arch problems were solved by finite elements using three shape functions, namely, I, cubic expression for radial, linear for circumferential displacements; II, a form of Cantin and Clough's shape function for cylindrical shells; III, a shallow element form of II. For a moderately deep arch with a thick rib all shape functions converged rapidly to correct solutions with increasing number of elements. For a deep arch with a thin rib, no shape function gave satisfactory solutions for a number of elements up to 34. For intermediate cases, II was superior to III which was superior to I. The relevance to shell studies is pointed out and several conclusions drawn.

Buckling of Open Cylindrical Shells with Imperfections

Buckling of open cylindrical shells with free longitudinal edges and simply supported ends, subjected to an axial compression, with imperfections due to end eccentricity or initial deflections, is examined. Field equations based on large deflection theory are reduced to a set of nonlinear finite difference equations and a computer technique for solving this set by the Newton-Raphson method is presented. Equilibrium paths and deflection patterns for examples with three types of imperfections (constant end moment, constant end eccentricity and specified initial deflection) are presented. The equilibrium paths exhibit an asymmetric type of postbuckling behavior; that is, negative imperfections which initially cause the shell to deflect outward would cause a buckling load lower than the linear buckling load without imperfections; whereas, positive imperfections would cause the opposite if there is no external disturbance. The behavior of a shell with Poisson's ratio μ ≠ 0 and circumferential displacement equal to zero at the ends, is similar to a shell with negative imperfections since the axial compression would induce an outward (negative) displacement.

On Gegenbauer transforms and forced torsional vibrations of thin spherical shells

The problem of determining the forced torsional deformation of thin, elastic, spherical and hemispherical shells subjected to either a free or a restrained edge is considered. It is shown that the Gegenbauer transform of order 3 2 defined in terms of the Gegenbauer polynomials C n ( 3 2 ) ( x) is appropriate to reduce the partial differential equation of torsional motion of a complete spherical shell to an elementary ordinary differential equation of second order in time. For a hemispherical shell with a restrained edge, it is shown that the odd Gegenbauer transform of order 3 2 is appropriate, and for a loaded edge the even Gegenbauer transform of order 3 2 may be used. Certain alternative representations of the circumferential displacement are presented, and the convergence of these representations is discussed. The response of a hemispherical nose radome in an aircraft subjected to circumferentially directed edge loads is determined, and the variation of the edge displacement as a function of time is demonstrated graphically.

薄肉円筒殻の非線形振動解析

The nonlinear flexural vibration property of thin circular cylindrical shells with simply supported ends is analytically studied. The deflection mode is assumed so as to satisfy the continuity of the circumferential displacement in the large deflection theory and consists of the driven mode and its companion mode. The governing equations are derived by using HAMILTON'S principle. For the approximate solution of the nonlinear equation and its stability analysis, the method of averaging is used. The steady solution of the driven mode has the resonance curve of the softening type, and becomes unstable with the appearance of the companion mode at the region of the resonance. The values of the structural damping and the exciting force are effective for the stability of thesteady solution. In this unstable region the vibration of circumferentially traveling wave mode is stable. The present analytical results offer a satisfactory explanation to the available experimental observation.