Shell Junctions Research Articles

The Rayleigh and Courant variational principles in the six-parameter shell theory

Courant’s minimax variational principle is considered in application to the six-parameter theory of prestressed shells. The equations of a prestressed micropolar shell are deduced in detail. Courant’s principle is used to study the dependence of the least and higher eigenfrequencies on shell parameters and boundary conditions. Cases involving boundary reinforcements and shell junctions are also treated.

Non-Lithographic Growth of Core–Shell GaAs Nanowires on Si for Optoelectronic Applications

We demonstrated a nonlithographic method for integrating GaAs nanowire (NW) array-based light-emitting diodes (LEDs) on silicon (Si) substrates. A sub-100 nm hole array on a deposited SiO2 layer was patterned on an entire 2 in. Si wafer based on a sacrificed self-assembled InAs NW array. Then, a core–shell n–p junction GaAs NW array was grown on exposed Si windows via the selective-area growth method. The electrical properties of the core–shell n–p junction GaAs NW has been measured and compared to those of the core–shell junction NWs formed via the self-assembled growth method. Room temperature electroluminescence was successfully observed from the fabricated GaAs NW array-based LEDs. The core–shell junction III–V NW array epitaxially grown on a ubiquitous Si platform could be applied to future low-cost optical devices.

Recombination-tunneling conduction in Cu- and S-doped ZnO nanorods’ core–shell junction: dependence of diode parameters on thermal annealing temperature and role of interfacial defects

We have investigated the effect of thermal annealing on the diode characteristics of solution-processed novel core–shell coaxial n–p nanorods heterojunction consisting of n-type ZnO as a core and p-type CuS as a shell material. The values of turn-on-voltage, rectification ratio, reverse saturation current density, barrier height, and ideality factors have been improved as the as-prepared heterojunction is annealed at higher temperatures owing to the improvement in the interface between ZnO and CuS as evidenced from the high-resolution transmission electron microscopy. The X-ray diffraction results also confirm the improvement in the crystalline quality of ZnO and CuS through annealing. The experimental current–voltage data are consistent with the presence of dominating recombination-tunneling conduction occurring through the interface defect states between ZnO and CuS. The results demonstrate that an annealing process plays a dominant role in the interfacial defects which helps to modify the diode performance paving a way to cheaper electronic nanodevices.

Stress of Spherical Shell with Opening Nozzle

To study the stress condition at the junction of the spherical shell with opening nozzle, using the finite element analysis, a finite element model is built in view of the same spherical shell joining a flatting nozzle and inside-stretching nozzles with different inner lengths differently. The maximum stress and stress distribution are got. All kinds of stresses are obtained by the total stress which is carried on linear processing. The result shows the inside-stretching nozzle can reduce the maximum stress in comparison with the flatting nozzle, mainly reducing the local membrane stress, but not the peak stress. The maximum stress falls with increasing the inner length of the nozzle to some extent, and beyond the extent, the maximum stress tends to reach a stable value basically without changing the inner length. The stress variation can effectively provide a reference for improving the strength of the spherical shell.

Stress of Spherical Shell with Opening Nozzle

To study the stress condition at the junction of the spherical shell with opening nozzle, using the finite element analysis, a finite element model is built in view of the same spherical shell joining a flatting nozzle and inside-stretching nozzles with different inner lengths differently. The maximum stress and stress distribution are got. All kinds of stresses are obtained by the total stress which is carried on linear processing. The result shows the inside-stretching nozzle can reduce the maximum stress in comparison with the flatting nozzle, mainly reducing the local membrane stress, but not the peak stress. The maximum stress falls with increasing the inner length of the nozzle to some extent, and beyond the extent, the maximum stress tends to reach a stable value basically without changing the inner length. The stress variation can effectively provide a reference for improving the strength of the spherical shell.

Surface‐enhanced Raman microscopy of hemispherical shells stripped from templates of anodized aluminum

We report on the optical characterization of plasmonic metal nanostructures representing highly ordered interconnected hemispherical gold and silver shells that can be iteratively stripped from the same embossed templates (without template degradation) made from selectively etched anodized aluminum. By performing scanning high‐resolution confocal Raman microscopy of p‐aminothiophenol and Rhodamine 6G molecules homogeneously adsorbed to samples with different radii of shell curvature, we systematically investigate the applicability of the fabricated structures for surface‐enhanced Raman spectroscopy and correlate the results with linear reflection spectroscopy. We trace the origin of strong Raman signal enhancements (average relative enhancement of up to ~120) to electromagnetic hot‐spots located in sharp grooves and crevices at hemisphere shell junctions. Copyright © 2011 John Wiley & Sons, Ltd.

On modelling and non‐linear elasto‐plastic analysis of thin shells with deformable junctions

AbstractThe undeformed base surface of the irregular thin shell is modelled by the union of a finite number of regular smooth surface elements joined together along spatial curvilinear surface edges. The equilibrium conditions are formulated by postulating an appropriate form of the principle of virtual work, where also deformability of shell junctions is taken into account. The PVW is then discretised by C1 finite elements and the incremental‐iterative procedure is applied to solve the highly non‐linear BVP. As an example, the axisymmetric deformation state is calculated in the casing of devise measuring the external pressure and having two pairs of circular welded junctions. The problem is solved within the elasto‐plastic range of material behaviour with linear combination of isotropic and kinematic hardening, and deformability of the junctions is taken into account.

Asymptotic analysis of bending-dominated shell junctions

The subject of this paper is the asymptotic analysis of a multi-structure made of two thin linearly elastic shells. It is assumed that the middle surfaces of the two shells are linked together on a part of their boundary, that at each point of their intersection, the two middle surfaces have distinct tangent planes and that the order of the applied body force density is ɛ 2 , where ɛ is the half-thickness of each shell. We then identify the limit when ɛ tends to zero of the solution of the ‘scaled’ three-dimensional linearized elasticity problem. It is shown in particular that this limit is constituted of inextensional displacement fields for each shell and such that the junction is rigid (the angle between the middle surfaces of the two shells at the junction is unchanged by the deformation of the multi-structure). Moreover this limit is the same as the limit of the solution of the classical shell model (or Koiter model) for the multi-structure with the condition of rigid junction.

Discontinuity effects at cone-cone axisymmetric shell junctions

In this paper, the problem of a conical shell axisymmetrically intersecting another conical shell, such that the vertices of the cones lie on opposite sides of the plane of intersection, is considered, and associated discontinuity effects quantified for arbitrary loading and geometric parameters of the intersecting cones. The ensuing very practical closed-form results are based on the one-term asymptotic-series solution for the axisymmetric bending of a non-shallow conical shell, and are intended for use in the quick evaluation of stresses and deformations in double-cone pressure vessels, as well as liquid-retaining vessels with intersecting conical portions. As an example of the application of the developed formulation, the stress distribution in a large liquid-filled elevated rhombic tank is evaluated. The stresses obtained on the basis of the closed-form analytical approach developed in the paper are shown to be in good agreement with those obtained from a finite-element analysis, confirming the reliability of the presented formulation.

Unique design of the junction between a thick pressure vessel shell and a thinner hemispherical head

AbstractConsidering the fact that the required thickness of a hemispherical head for pressure loading is only half of that necessary for a cylindrical shell, the authors have developed a proprietary junction of shell to hemispherical head where the head is thinner than the shell. In the past 5 years, more than 100 high-pressure vessels with such junctions have been manufactured and safely used in the People's Republic of China. Theoretical investigations as well as engineering applications show that such junctions are rational in structure, convenient in fabrication and low in cost. For example, for a vessel with an inner diameter of 1000 mm and design pressure of 31.4 MPa, using the new junction, the head weight is reduced by 54.3 and 27.4 per cent respectively in comparison with a forged flat head and a hemispherical head with equal thickness of shell. The larger the vessel, the greater the economic benefits it achieves. In this paper, the authors describe its basic structure, stress characteristics and the engineering design method for the unique junction.

Unique design of the junction between a thick pressure vessel shell and a thinner hemispherical head

Considering the fact that the required thickness of a hemispherical head for pressure loading is only half of that necessary for a cylindrical shell, the authors have developed a proprietary junction of shell to hemispherical head where the head is thinner than the shell. In the past 5 years, more than 100 high-pressure vessels with such junctions have been manufactured and safely used in the People's Republic of China. Theoretical investigations as well as engineering applications show that such junctions are rational in structure, convenient in fabrication and low in cost. For example, for a vessel with an inner diameter of 1000 mm and design pressure of 31.4 MPa, using the new junction, the head weight is reduced by 54.3 and 27.4 per cent respectively in comparison with a forged flat head and a hemispherical head with equal thickness of shell. The larger the vessel, the greater the economic benefits it achieves. In this paper, the authors describe its basic structure, stress characteristics and the engineering design method for the unique junction.

Effect of internal inhomogeneities on acoustic scattering from finite cylindrical shell bounded by hemispherical endcaps

The acoustic scattering from an air-filled finite cylindrical shell with hemispherical endcaps is considered. In axial incidence, experimental studies have shown formation of compressional wave S0 resonances. These resonances are predicted and identified by applying the phase-matching condition over the meridian circumference. However, frequency irregularities observed experimentally on spectra have suggested that the assemblage quality during the manufacturing of the object may be not negligible. Indeed, internal nonuniformities of the shell at the junctions of cylindrical and hemispherical shells are present and modify certain wave propagation phenomena. The aim of this study is to discover these experimental conditions of the studied object through a numerical model with internal annular inhomogeneities localized at the junctions of the substructures and for two different b/a ratios (internal to external radius ratio) equal to 0.97 and 0.99. In the studied frequency range (50–300 kHz), in addition to the S0 wave, another peripheral wave A can be generated for the ratio b/a=0.97. The numerical approach is based on the discretization of the shell and determination of its eigenmodes (FEM). Acoustical pressure field is found by the boundary integral equation procedure (BEM). The comparison of numerical and experimental results is presented in frequency and time domains.

Stresses at the junctions of axisymmetric shells under axially varying load

This paper deals with the investigation of stresses and deformations in the neighborhood of junctions of axisymmetric shells, made of segments of different geometries and subjected to different edge restraints and axially varying internal pressure. The method of investigation involves solution of a set of six first order highly nonlinear differential equations seeking the state of deformation of the shell at which, for a given pressure, the potential energy in the deformed shell is a relative minimum. The basic concept of the multisegment integration, as developed by Kalnins and Lestingi (On nonlinear analysis of elastic shells of revolution, J. Appl Mech. 1967;34:59–64), has been utilized for obtaining the solutions of the governing equations. Solutions for various problems are obtained for increasing loading, starting from a low value. The soundness of the method and the correctness of the computer program used in the analysis are verified by comparing the results with that of the corresponding analytical results available in the literature. Results are presented in the form of curves for stresses and deformations for varying loads and varying shell parameters which can be used for locating the critical zones and determining suitable values of shell parameters. The results include both stresses and deformations in their nondimensional forms based on both the linear and nonlinear theories.

Numerical analysis of junctions between thin shells Part 1: Continuous problems

The junctions of beams, plates and shells are the basic components of any industrial structural construction. The numerical simulation of such junctions is a classical part of the commercial finite element codes. On the other hand, it seems that there are very few mathematical studies of such junctions. In this paper, we propose a variational formulation of junctions between thin shells when the junction can be considered as an elastic or a rigid hinge. Then, we study the mathematical properties of these equations.

Classical Versus Improved Thin Shell Theories: A Theoretical Argument or a Design Concern?

Differences in the response of thin nonshallow spherical shells resulting from the choice of the adopted shell theory (classical or improved) are addressed analytically and through a series of representative shell problems. The analytical approach utilized to study the variation between the two theoretical models is based on the response resulting from Singular loads. The differences are quantified in a set of problems that reflect on the assumptions used in formulating the analytical description of the two theories in question. The broad scope of this paper is to examine the impact of shear deformability, introduced by the improved theory on the stress field when amplified under specific loading and geometric conditions, when those are of primary concern to the engineers. Such cases associated with stress concentration around cutouts, interaction of shells with nozzles, stress field in the vicinity of concentrated surface loads, etc. The mathematical formulation is based on the derivation of appropriate Green functions and the computational scheme is formed upon a special type of boundary integral equation. Comparison solutions for stress concentration around circular cutouts of twisted and sheared shells, stress amplification in the junction of shell with nozzles, and local stress field induced by concentrated loads are presented for the two theories.

Scattering from fluid-loaded junctions of plates and shells

This paper will review recent results for several related problems involving plate and shell junctions in the presence of fluid loading. First, a general solution will be discussed which gives the acoustic and structural scattered response for two joined flat plates under unilateral fluid loading. By combining this with a related solution for the admittance matrix of the fluid-loaded plates, the behavior of a pair of semi-infinite plates in contact with fluid on one side and a mechanical structure on the other can be modeled. For simplicity, the internal frame is characterized by an impedance matrix. Based on these results, a perturbation solution can be developed for two joined curved shells under unilateral loading. The leading order term in the expansion is the previously solved case of two joined flat plates, and the next term gives an approximation of the diffracted longitudinal and shear wave fields originating from the junction of two shells. In each example considered, explicit formulas are obtained for the pressure transform, and corresponding explicit and relatively simple expressions are given for the various diffraction coefficients associated with the fluid/structure interaction. The general method of solution uses the Wiener-Hopf technique to solve dual integral equations for the acoustic pressure. The plates and shells are modeled by the classical theory of flexure and thin shell theory, respectively. Numerical results of the diffraction coefficients and the redistribution of energy at structural junctions will be presented. [Work supported by ONR.]

Waves in cylindrical shells with circumferential submembers: a matrix approach

A matrix approach is proposed to investigate waves in circular cylindrical thin shells jointed with circular plates. Both the general propagator matrix and S-matrix formalisms are presented, with emphasis on the latter. The loss of computational accuracy due to the inevitable exponentially growing terms in a propagator matrix is completely avoided by using the S-matrix. This paper demonstrates implementation of S-matrix methods for analyzing waves in complex shell structures with axial symmetry. The basic elements are laid out in detail, including the S-matrix for cylindrical shells, the propagator matrix and the asymptotic S-matrix for plates, and both the propagator matrix and the S-matrix for junctions of cylindrical thin shells with internal and/or external circular plates and for multi-channel elements. The general approach is demonstrated for several examples of cylindrical shells with periodic stiffeners and sub-elements. Dispersion curves are computed and compared with previous results.

Free vibration analysis of arbitrary thin shell structures by using spline finite element

This paper presents the free vibration analysis of arbitrary thin shell structures by using a newly developed spline finite element. The new element has three salient features in its formulation: (i) the use of B-spline shape functions for the interpolations of both in-plane and out-of-plane displacements of a general thin shell element; (ii) the use of “displacement constraints” and “parameters shifting” to construct a finite element model; (iii) that a proposed modified version of the Koiter’s thin shell theory is employed. The element is doubly curved, has nine primary nodes and eight auxiliary nodes, and has a total of 63 degrees of freedom. It is formulated through the conventional C 1displacement approach, and is capable of modelling sharp corners, arbitrary shapes and multiple junctions of thin shell structures. The numerical examples discussed include spherical panels, cylindrical panels and shells of revolution, as well as single- and double-cell boxes. These examples are typical shells of negative, zero and positive Gaussian curvatures. The effects of aspect ratios, distorted meshes and junctions of shells on the performance of the element are studied. It is shown that the new spline finite element is a reliable, versatile, accurate and efficient thin shell element suitable for the analysis of arbitrary thin shell structures.

Simplified Methods for Calculation of Redundant Moments and Forces at Discontinuities in Pressure Vessels—Part I: Thin Plate—Thin Shell

Junctions of circular flat plates and cylindrical shells in pressure vessels are often encountered, where redundant bending moments and redundant shearing forces are computed to satisfy the compatibility conditions for deformations of the joining members. The computation is based on elasticity theory of thin plates and shells, which can readily be found. The computation is tedious. However, the redundant moments and the redundant forces are unique functions of two nondimensionalized geometric parameters; thereby, the computation can be simplified as explained in the present paper.

Numerical analysis of junctions between plates

The junctions of bars, plates and shells are the basic ingredients of any industrial structural construction. The numerical simulation of such junctions is a classical part of the commercial finite element codes. On the other hand there seem to be very few mathematical studies of such junctions. In this paper, we propose a variational formulation of plate junctions when these junctions can be considered as elastic or rigid hinges. Then, we study the mathematical properties of these equations as their appoximation by finite element methods. We conclude by reporting some numerical experiments.