Circular Shells Research Articles

Non-linear general instability of ring-stiffened conical shells under external hydrostatic pressure

The paper presents the experimental results for 15 ring-stiffened circular steel conical shells, which failed by non-linear general instability. The results of these investigations were compared with various theoretical analyses, including an ANSYS eigen buckling analysis and another ANSYS analysis; which involved a step-by-step method until collapse; where both material & geometrical nonlinearity were considered. The investigation also involved an analysis using BS5500 (PD 5500), together with the method of Ross of the University of Portsmouth. The ANSYS eigen buckling analysis tended to overestimate the predicted buckling pressures; whereas the ANSYS nonlinear results compared favourably with the experimental results. The PD5500 analysis was very time consuming and tended to grossly underestimate the experimental buckling pressures and in some cases, overestimate them. In contrast to PD5500 & ANSYS, the design charts of Ross of the University of Portsmouth were the easiest of all these methods to use and generally only slightly underestimated the experimental collapse pressures. The ANSYS analyses gave some excellent graphical displays.

Displacement Fundamental Equations and Analysis of Governing Equations of the Circular Moderately Thick Shallow Spherical Shells

Displacement fundamental equations of moderately thick shallow spherical shells in polar coordinates concerning five independent variables, i.e. five middle surface displacements are established, based on the displacement fundamental equations of the moderately thick shells by transverse shearing deformation and basic hypothesis on shallow shells. Four assistant displacement functions are introduced to solve displacement fundamental equations of circular moderately thick shallow spherical shells , which are tenth-order differential equations with variable coefficient, then the decoupled governing differential equations are built up, and five displacement components through four assistant displacement functions are obtained.

The close circumstellar environment of Betelgeuse

Context: Mass-loss occurring in red supergiants (RSGs) is a major contributor to the enrichment of the interstellar medium in dust and molecules. The physical mechanism of this mass loss is however relatively poorly known. Betelgeuse is the nearest RSG, and as such a prime object for high angular resolution observations of its surface (by interferometry) and close circumstellar environment. Aims: The goal of our program is to understand how the material expelled from Betelgeuse is transported from its surface to the interstellar medium, and how it evolves chemically in this process. Methods: We obtained diffraction-limited images of Betelgeuse and a PSF calibrator (Aldebaran) in six filters in the N band (7.76 to 12.81 mic) and two filters in the Q band (17.65 and 19.50 mic), using the VLT/VISIR instrument. Results: Our images show a bright, extended and complex circumstellar envelope at all wavelengths. It is particularly prominent longwards of 9-10 mic, pointing at the presence of O-rich dust, such as silicates or alumina. A partial circular shell is observed between 0.5 and 1.0" from the star, and could correspond to the inner radius of the dust envelope. Several knots and filamentary structures are identified in the nebula, one of the knots, located at a distance of 0.9" west of the star, being particularly bright and compact. Conclusions: The circumstellar envelope around Betelgeuse extends at least up to several tens of stellar radii. Its relatively high degree of clumpiness indicates an inhomogeneous spatial distribution of the material lost by the star. Its extension corresponds to an important intermediate scale, where most of the dust is probably formed, between the hot and compact gaseous envelope observed previously in the near infrared and the interstellar medium.

Electron mean-free path in metal-coated nanowires

The mean-free path Leff is calculated for a circular shell geometry under the assumptions of (i) billiard scattering and (ii) diffusive scattering. Similarly to a spherical shell geometry, Leff displays qualitatively different behavior on the shell thickness D for the two models. Whereas, for the billiard model Leff=πD/2, the mean-free path in the diffusive scattering case is a complicated function involving complete and incomplete elliptic integrals of the first, second, and third kinds. Nevertheless, a linear combination of D and Dln (R/2D), where R is the outer circle radius, is shown to capture the functional dependence of Leff in the diffusive scattering case remarkably well over almost the entire parameter range. Hopefully, future experiments on single and well-controlled dielectric-core–metal-shell nanowires could differentiate between the two models.

Natural Frequencies Analysis of Moderately-Thick and Thick Toroidal Shells

Toroidal shells have traditionally found application in the pressure vessel and piping industry. Early theoretical work on these shells focused on their static properties, but with their consideration for newer applications more emphasis has been placed on their vibration properties. A major factor affecting shell vibrations is their wall thickness. The thin shell theory is firmly established for shell vibrations in the literature and is used extensively in analytical solutions and numerical analysis. With more applications of toroidal shells employed in engineering practice, such as space vehicle, liquid storage structures and water tanks, the thickness of these structures cannot be considered as thinwalled. For such applications, the corresponding wall thickness has to be moderately thick or thick and the background theory is required using the shear deformation theory or elasticity theory. The present study extends the vibration analysis of toroidal shells based on thin shell theory to the analysis for moderately-thick and thick toroidal shells. A method based on a shear deformation (Timoshenko-Mindlin) shell theory is investigated to determine the free vibration characteristics of a moderately-thick or thick toroidal shell. The analysis is developed for a shell of revolution of arbitrary meridian, and then applied to a complete circular toroidal shell. A modal approach in the circumferential direction is used in the analysis and the results are obtained using the differential quadrature method (DQM). The results are validated by comparing with previously published results and good agreement is observed in the comparison of different results. General equations are developed which enable the prediction of the vibration characteristics of moderately-thick and thick shells of revolution of arbitrary meridian. The results demonstrate the application of the shear deformation theory in determining natural frequencies for toroidal shells.

Free vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method

In this paper, the Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behaviour of laminated composite doubly-curved shells of revolution. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. The discretization of the system by means of the Differential Quadrature (DQ) technique leads to a standard linear eigenvalue problem, where two independent variables are involved. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Examples of hyperbolic, catenary, cycloid, parabolic, elliptic and circular shell and panel structures are presented to illustrate the validity and the accuracy of the GDQ method. Furthermore, GDQ results are compared with those presented in literature and the ones obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed.

Measures of Circularity for Shell Structures

Most design codes for circular shell structures include a limit on out-of-round imperfections. However, commonly used measures of out-of-roundness fail to identify shapes that have constant diameter but are not circular. It is shown that out-of-roundness imperfections that take these forms can significantly affect the strength of nominally circular shell structures. An alternative measure of out-of-roundness is proposed that captures all out-of-round shapes, gives the same measure of out-of-roundness for elliptical structures as currently used measures, and is easy to apply.

The peacock's train (Pavo cristatus and Pavo cristatus mut. alba) I. structure, mechanics, and chemistry of the tail feather coverts

The feathers in the train of the peacock serve not for flying but for sexual display. They are long, slender beams loaded in bending by their own weight. An outer circular conical shell, the cortex, is filled by a closed foam of 7.6% relative density, the medulla, both of feather keratin. Outer diameter and thickness of the cortex decrease linearly from the body toward the tip. This self-similar geometry leads to a division of labor. The cortex (longitudinal Young's modulus 3.3 GPa, transverse modulus 1 GPa) provides 96% of the longitudinal strength and bending rigidity of the feather. The medulla (Young's modulus 10 MPa) provides 96% of the transverse compressive rigidity. Fracture stress of the cortex, both longitudinal and transverse, is 120 MPa.

Buckling analysis of FGM circular shells under combined loads and resting on the Pasternak type elastic foundation

In this study, the buckling analysis of functionally graded material (FGM) circular truncated conical and cylindrical shells subjected to combined axial extension loads and hydrostatic pressure and resting on a Pasternak type elastic foundation is investigated. The critical combined loads of FGM truncated conical shells with or without elastic foundations have been found analytically. The appropriate formulas for FGM cylindrical shells with and without elastic foundations are found as a special case. Several examples are presented to show the accuracy and efficiency of the formulation.

An Analytical Method for Cylindrical Shells With Nozzles Due to Internal Pressure and External Loads—Part I: Theoretical Foundation

An improved version of the analytical solutions by Xue, Hwang and co-workers (1991, “Some Results on Analytical Solution of Cylindrical Shells With Large Opening,” ASME J. Pressure Vessel Technol., 113, 297–307; 1991, “The Stress Analysis of Cylindrical Shells With Rigid Inclusions Having a Large Ratio of Radii,” SMiRT 11 Transactions F, F05/2, 85–90; 1995, “The Thin Theoretical Solution for Cylindrical Shells With Large Openings,” Acta Mech. Sin., 27(4), pp. 482–488; 1995, “Stresses at the Intersection of Two Cylindrical Shells,” Nucl. Eng. Des., 154, 231–238; 1996, “A Reinforcement Design Method Based on Analysis of Large Openings in Cylindrical Pressure Vessels,” ASME J. Pressure Vessel Technol., 118, 502–506; 1999, “Analytical Solution for Cylindrical Thin Shells With Normally Intersecting Nozzles Due to External Moments on the Ends of Shells,” Sci. China, Ser. A: Math., Phys., Astron., 42(3), 293–304; 2000, “Stress Analysis of Cylindrical Shells With Nozzles Due to External Run Pipe Moments,” J. Strain Anal. Eng. Des., 35, 159–170; 2004, “Analytical Solution of Two Intersecting Cylindrical Shells Subjected to Transverse Moment on Nozzle,” Int. J. Solids Struct., 41(24–25), 6949–6962; 2005, “A Thin Shell Theoretical Solution for Two Intersecting Cylindrical Shells Due to External Branch Pipe Moments,” ASME J. Pressure Vessel Technol., 127(4), 357–368; 2005, “Theoretical Stress Analysis of Two Intersecting Cylindrical Shells Subjected to External Loads Transmitted Through Branch Pipes,” Int. J. Solids Struct., 42, 3299–3319) for two normally intersecting cylindrical shells is presented, and the applicable ranges of the theoretical solutions are successfully extended from d/D≤0.8 and λ=d/(DT)1/2≤8 to d/D≤0.9 and λ≤12. The thin shell theoretical solution is obtained by solving a complex boundary value problem for a pair of fourth-order complex-valued partial differential equations (exact Morley equations (Morley, 1959, “An Improvement on Donnell’s Approximation for Thin Walled Circular Cylinders,” Q. J. Mech. Appl. Math. 12, 89–91; Simmonds, 1966, “A Set of Simple, Accurate Equations for Circular Cylindrical Elastic Shells,” Int. J. Solids Struct., 2, 525–541)) for the shell and the nozzle. The accuracy of results is improved by some additional terms to the expressions for resultant forces and moments in terms of complex-valued displacement-stress function. The theoretical stress concentration factors due to internal pressure obtained by the improved expressions are in agreement with previously published test results. The theoretical results discussed and presented herein are in sufficient agreement with those obtained from three dimensional finite element analyses for all the seven load cases, i.e., internal pressure and six external branch pipe load components involving three orthogonal forces and the respective three orthogonal moments.

Percussion Synthesis Based on Models of Nonlinear Shell Vibration

The synthesis of sound based on physical models of 2-D percussion instruments is problematic and has been approached only infrequently in the literature. Beyond the computational expense inherent to the simulation of 2-D systems, a deeper difficulty is in dealing with the strong nonlinearity exhibited by thin structures when struck-this nonlinearity leads to phenomena which are not captured, even approximately, by a linear model, and nearly all synthesis work is based on the assumption that the distributed resonating component of a musical instrument is linear. Perceptually, the effects of the vibration of a thin structure at high amplitudes can be heard as crashes, pitch glides, and the slow buildup of high-frequency energy characteristic of gongs. A large family of instruments may be described, approximately, as circular thin shells, of approximately spherical geometry, in which case a tractable PDE description, described here, is available. Time-domain finite-difference schemes, in radial coordinates, are a suitable method for synthesis. Stability conditions, numerical boundary conditions both at the edge and center, and implementation details are discussed, and simulation results are presented, highlighting the various perceptual effects mentioned above.

A HIDDEN POPULATION OF MASSIVE STARS WITH CIRCUMSTELLAR SHELLS DISCOVERED WITH THESPITZER SPACE TELESCOPE

We have discovered a large number of circular and elliptical shells at 24 microns around luminous central sources with the MIPS instrument on-board the Spitzer Space Telescope. Our archival follow-up effort has revealed 90% of these circumstellar shells to be previously unknown. The majority of the shells is only visible at 24 microns, but many of the central stars are detected at multiple wavelengths from the mid- to the near-IR regime. The general lack of optical counterparts, however, indicates that these sources represent a population of highly obscured objects. We obtained optical and near-IR spectroscopic observations of the central stars and find most of these objects to be massive stars. In particular, we identify a large population of sources that we argue represents a narrow evolutionary phase, closely related or identical to the LBV stage of massive stellar evolution.

CIRCUMSTELLAR STRUCTURE AROUND EVOLVED STARS IN THE CYGNUS-X STAR FORMATION REGION

We present observations of newly discovered 24 micron circumstellar structures detected with the Multiband Imaging Photometer for Spitzer (MIPS) around three evolved stars in the Cygnus-X star forming region. One of the objects, BD+43 3710, has a bipolar nebula, possibly due to an outflow or a torus of material. A second, HBHA 4202-22, a Wolf-Rayet candidate, shows a circular shell of 24 micron emission suggestive of either a limb-brightened shell or disk seen face-on. No diffuse emission was detected around either of these two objects in the Spitzer 3.6-8 micron Infrared Array Camera (IRAC) bands. The third object is the luminous blue variable candidate G79.29+0.46. We resolved the previously known inner ring in all four IRAC bands. The 24 micron emission from the inner ring extends ~1.2 arcmin beyond the shorter wavelength emission, well beyond what can be attributed to the difference in resolutions between MIPS and IRAC. Additionally, we have discovered an outer ring of 24 micron emission, possibly due to an earlier episode of mass loss. For the two shell stars, we present the results of radiative transfer models, constraining the stellar and dust shell parameters. The shells are composed of amorphous carbon grains, plus polycyclic aromatic hydrocarbons in the case of G79.29+0.46. Both G79.29+0.46 and HBHA 4202-22 lie behind the main Cygnus-X cloud. Although G79.29+0.46 may simply be on the far side of the cloud, HBHA 4202-22 is unrelated to the Cygnus-X star formation region.

Nonlinear stability of double-deck reticulated circular shallow spherical shell

Based on the variational equation of the nonlinear bending theory of double-deck reticulated shallow shells, equations of large deflection and boundary conditions for a double-deck reticulated circular shallow spherical shell under a uniformly distributed pressure are derived by using coordinate transformation means and the principle of stationary complementary energy. The characteristic relationship and critical buckling pressure for the shell with two types of boundary conditions are obtained by taking the modified iteration method. Effects of geometrical parameters on the buckling behavior are also discussed.

Effect of Two Interacting Localized Defects on the Critical Load for Thin Cylindrical Shells Under Axial Compression

Problem statement: Many modern structures are made from thin shells. Design of these elements depends to a large extent on their bucklin g behavior which is hugely affected by the initial geometric imperfections. Approach: For axially compressed isotropic circular cylindric al shells, axisymmetric localized geometric imperfections were found to reduce severely the buckling strength. Among various axisymmetric shapes of localized defects that were investigated, the entering triangular form was recognized to yield the most adverse case. Since multiple localized defects may be present in the same shell structure and interact, studying the ir mutual effect on the buckling load is of great importance for shell design. Results: In this study, the effect of two interacting enteri ng triangular localized axisymmetric initial geometric imperfecti ons on shell buckling strength under uniform axial compression was modeled by means of the finite element method. A special software package which was dedicated to buckling analysis of quasi axisymmetric shells was used in order to compute the buckling load either via the linear Euler buckling analysis or through the full non linear iterative procedure. A set of five factors including shell a spect ratios, defect characteristics and the distan ce separating the localized initial geometric imperfec tions had been found to govern the buckling problem. A statistical approach based on the Taguch i method was used then to study their relative influence on the buckling load reduction. It was sh own by comparison with the single imperfection case that further diminution of the critical load w as obtained. Conclusion/Recommendations: In the range of investigated parameters, the distance sepa rating the localized geometric imperfections and imperfection wavelength were found to yield major i nfluences on the critical load. Further studies mus t be performed in order to assess shell buckling stre ngth in the presence of more than two defects and t o state the relative influence of the intervening fac tors.

Plane nonaxisymmetric deformation of a viscoelastic cylinder with consideration of its physical and geometric nonlinearity

The nonaxisymmetric plane problem of the nonlinear theory of viscoelasticity is solved for a cylinder reinforced by an elastic circular shell. The cylinder has an internal cut resembling a Maltese cross in shape. The identification of the nonlinear endochronous theory of aging viscoelastic materials is conducted by a genetic algorithm method on the basis a nonmonotonic experimental stress-strain dependence. Some numerical results obtained for the stress-strain state of this cylinder under the action of internal pressure are discussed with consideration of the above physical nonlinearity and the finite logarithmic strains.

On the solution of eigenvalue problems of laminated non-homogeneous orthotropic circular shells with clamped edges subjected to hydrostatic pressure

This article presents a method to study the free vibration and stability of laminated homogeneous and non-homogeneous orthotropic cylindrical, truncated and complete conical shells of general staking with clamped edges under a hydrostatic pressure. Based on the Love first approximation theory, the basic relations, the modified Donnell-type stability and compatibility equations have been obtained for laminated orthotropic truncated conical shells, the material properties of which vary piecewise continuously in the thickness direction. To solve this problem an unknown parameter λ was included in the approximation functions. Applying Galerkin methods, the buckling pressures and fundamental natural frequencies of laminated homogeneous and non-homogeneous orthotropic conical shells are obtained. The parameter λ which is included in the obtained formulas is obtained from the minimum conditions of critical stresses and frequencies. The different generalized values are obtained for the parameter λ for buckling pressures and frequencies of cylindrical shells, truncated and complete conical shells. The appropriate formulas for single-layer and laminated cylindrical shells made of homogeneous and non-homogeneous, orthotropic and isotropic materials are found as a special case. Finally, the influences of the degree of non-homogeneity, the number and ordering of layers and the variations of conical shell characteristics on the critical hydrostatic pressure and natural frequencies are investigated. The results obtained for homogeneous cases are compared with their counterparts in the literature.

Studies of nonlinear deformation and stability of oval and elliptic cylindrical shells under axial compression

The stability of noncircular shells, in contrast to that of circular ones, has not been studied sufficiently well yet. The publications about circular shells are counted by thousands, but there are only several dozens of papers dealing with noncircular shells. This can be explained on the one hand by the fact that such shells are less used in practice and on the other hand by the difficulties encountered when solving problems involving a nonconstant curvature radius, which results in the appearance of variable coefficients in the stability equations. The well-known solutions of stability problems were obtained by analytic methods and, as a rule, in the linear approximation without taking into account the moments and nonlinearity of the shell precritical state, i.e., in the classical approximation. Here we use the finite element method in displacements to solve the problem of geometrically nonlinear deformation and stability of cylindrical shells with noncircular contour of the transverse cross-section. We use quadrilateral finite elements of shells of natural curvature. In the approximations to the element displacements, we explicitly distinguish the displacements of elements as rigid bodies. We use the Lagrange variational principle to obtain a nonlinear system of algebraic equations for determining the unknown nodal finite elements. We solve the system by a step method with respect to the load using the Newton-Kantorovich linearization at each step. The linear systems are solved by the Kraut method. The critical loads are determined with the use of the Silvester stability criterion when solving the nonlinear problem. We develop an algorithm for solving the problem numerically on personal computers. We also study the nonlinear deformation and stability of shells with oval and elliptic transverse cross-section in a wide range of variations in the ovalization and ellipticity parameters. We find the critical loads and the shell buckling modes. We also examine how the critical loads are affected by the strain nonlinearity and the ovalization and ellipticity of shells.

OUT OF THE FRYING PAN: A YOUNG PULSAR WITH A LONG RADIO TRAIL EMERGING FROM SNR G315.9-0.0

The faint radio supernova remnant SNR G315.9-0.0 is notable for a long and thin trail that extends outward perpendicular from the edge of its approximately circular shell. In a search with the Parkes telescope we have found a young and energetic pulsar that is located at the tip of this collimated linear structure. PSR J1437-5959 has period P = 61 ms, characteristic age tau_c = 114 kyr, and spin-down luminosity dE/dt = 1.4e36 erg/s. It is very faint, with a flux density at 1.4 GHz of about 75 uJy. From its dispersion measure of 549 pc/cc, we infer d ~ 8 kpc. At this distance and for an age comparable to tau_c, the implied pulsar velocity in the plane of the sky is V_t = 300 km/s for a birth at the center of the SNR, although it is possible that the SNR/pulsar system is younger than tau_c and that V_t > 300 km/s. The highly collimated linear feature is evidently the pulsar wind trail left from the supersonic passage of PSR J1437-5959 through the interstellar medium surrounding SNR G315.9-0.0.

First record of the brachiopodLingulella waptaensiswith pedicle from the Middle Cambrian Burgess Shale

Pettersson Stolk, S., Holmer, L. E. and Caron, J -B. 2010. First record of the brachiopod Lingulella waptaensis with pedicle from the Middle Cambrian Burgess Shale. —Acta Zoologica (Stockholm) 91: 150–162 The organophosphatic shells of linguloid brachiopods are a common component of normal Cambrian–Ordovician shelly assemblages. Preservation of linguloid soft-part anatomy, however, is extremely rare, and restricted to a few species in Lower Cambrian Konservat Lagerstätten. Such remarkable occurrences provide unique insights into the biology and ecology of early linguloids that are not available from the study of shells alone. Based on its shells, Lingulella waptaensis Walcott, was originally described in 1924 from the Middle Cambrian Burgess Shale but despite the widespread occurrence of soft-part preservation associated with fossils from the same levels, no preserved soft parts have been reported. Lingulella waptaensis is restudied herein based on 396 specimens collected by Royal Ontario Museum field parties from the Greater Phyllopod Bed (Walcott Quarry Shale Member, British Columbia). The new specimens, including three with exceptional preservation of the pedicle, were collected in situ in discrete obrution beds. Census counts show that L. waptaensis is rare but recurrent in the Greater Phyllopod Bed, suggesting that this species might have been generalist. The wrinkled pedicle protruded posteriorly between the valves, was composed of a central coelomic space, and was slender and flexible enough to be tightly folded, suggesting a thin chitinous cuticle and underlying muscular layers. The nearly circular shell and the long, slender and highly flexible pedicle suggest that L. waptaensis lived epifaunally, probably attached to the substrate. Vertical cross-sections of the shells show that L. waptaensis possessed a virgose secondary layer, which has previously only been known from Devonian to Recent members of the Family Lingulidae.