Axisymmetric Shell Structures Research Articles

Discrete differential geometry-based model for nonlinear analysis of axisymmetric shells

In this paper, we propose a novel one-dimensional (1D) discrete differential geometry (DDG)-based numerical method for geometrically nonlinear mechanics analysis (e.g., buckling and snapping) of axisymmetric shell structures. Our numerical model leverages differential geometry principles to accurately capture the complex nonlinear deformation patterns exhibited by axisymmetric shells. By discretizing the axisymmetric shell into interconnected 1D elements along the meridional direction, the in-plane stretching and out-of-bending potentials are formulated based on the geometric principles of 1D nodes and edges under the Kirchhoff–Love hypothesis, and elastic force vector and associated Hessian matrix required by equations of motion are later derived based on symbolic calculation. Through extensive validation with available theoretical solutions and finite element method (FEM) simulations in literature, our model demonstrates high accuracy in predicting the nonlinear behavior of axisymmetric shells. Importantly, compared to the classical theoretical model and three-dimensional (3D) FEM simulation, our model is highly computationally efficient, making it suitable for large-scale real-time simulations of nonlinear problems of shell structures such as instability and snap-through phenomena. Moreover, our framework can easily incorporate complex loading conditions, e.g., boundary nonlinear contact and multi-physics actuation, which play an essential role in the use of engineering applications, such as soft robots and flexible devices. This study demonstrates that the simplicity and effectiveness of the 1D discrete differential geometry-based approach render it a powerful tool for engineers and researchers interested in nonlinear mechanics analysis of axisymmetric shells, with potential applications in various engineering fields.

A MATLAB finite element toolbox for the efficient nonlinear analysis of axisymmetric shells

AbstractShells of revolution under axisymmetric conditions exhibit a circumferentially uniform pre‐buckling stress state and are important fundamental systems which often serve as reference systems for those under more complex conditions. Given this status, work is continuing on a careful and complete characterization of their buckling response with the aid of the Reference Resistance Design (RRD) framework for the ultimate benefit of the EN 1993‐1‐6 Eurocode on the strength of stability of metal shells. The situation is greatly complicated by the fact that while modern finite element software packages offer axisymmetric shell elements in an efficient 2D modelling plane, these are not capable of detecting bifurcation buckling into non‐axisymmetric modes which are often critical for slender systems. Reverting to a full 3D plane is possible, but grossly inefficient and the explicitly modelled circumferential direction is parasitic and detrimental to the overall solution quality. AQUINAS is an accessible and intuitive toolbox developed by the Authors in MATLAB for the efficient analysis of axisymmetric shell structures, aiming to reintroduce a modelling capability that was once standard in the field. Data input is entirely object‐oriented and matrix assembly is parallelized with pre‐compiled C++ routines, with users being able to take direct advantage of MATLAB's visualization properties. The software natively supports the LA, LBA, MNA, GMNIA etc. Eurocode analysis taxonomy. This paper demonstrates the current capabilities of the toolbox, describes the extensive programme of verification against existing established solutions that has been performed, and illustrates its ability to efficiently compute very detailed capacity curves using the EN 1993‐1‐6 capacity curve framework.

A frequency split trimming technique for axisymmetric shell structures

Axisymmetric shell structures are characterized by the presence of flexural modes which occur in degenerate pairs. However, since no practically realized structure is perfectly axisymmetric, the frequencies constituting a mode pair become unequal and causes what is commonly termed as “frequency split” between the modes. In applications such as vibratory gyroscopes, it is desired that the frequencies of the working pair of modes be equal. This work proposes an easily implementable (particularly in the industry) technique which reduces frequency split of a target mode pair. To address the frequency split trimming problem, this work investigates a model that describes the planar dynamics of a vibrating ring (which is prototypical to axisymmetric shell structures), with added mass segments. The effects of mass, as well as span, of an added segment on the ring’s natural frequencies and mode shapes are brought out. Novelties of the proposed trimming technique are that it employs added mass in the form of segments and that it has a negligible effect on the axes of the structure’s mode shapes. The technique is extended to general axisymmetric bodies, and it is validated experimentally for the case of a hemispherical resonator.

Free vibrations analysis of composite and hybrid axisymmetric shells

The free vibration of laminated composite (C) and hybrid axisymmetric shell structures, consisting of a composite laminated material sandwiched between two functionally graded material laminas (F1/C/F2), is analysed in the present work. The numerical solutions are obtained by expanding the variables in Fourier series in the circumferential direction and using conical frustum finite elements in the meridional direction. The implemented finite element is a simple conical frustum with two nodal circles, with ten degrees of freedom per nodal circle. This model requires only a reduced number of finite elements to model the geometry of axisymmetric structures, the integration procedures use one Gauss point, and the through the thickness properties variation in FGM laminas is modelled by a small number of virtual layers, resulting a very high computational efficiency. The in-house developed code presents very good solutions when compared with results obtained by alternative available models.

Hybrid FEM–SBM solver for structural vibration induced underwater acoustic radiation in shallow marine environment

This study focuses on the efficient computational model for predicting the structural vibration induced underwater acoustic radiation in shallow marine environment. In the proposed model, the finite element method (FEM) is used to calculate the vibration response of the shell structures and the singular boundary method (SBM) with near-field and far-field Green’s functions is used to simulate the underwater acoustic radiation excited by shell structural vibration in shallow water marine environment, namely, the hybrid FEM–SBM solver. For axisymmetric shell structures, the axisymmetric numerical formulation of the FEM–SBM solver is derived to save the computational cost. For general shell structures, the fast direct solver based on the recursive skeletonization factorization is introduced into the FEM–SBM solver to accelerate the computation of the underwater acoustic field. Numerical results demonstrate that the proposed FEM–SBM solver provides more efficient and reliable solutions in comparison with the coupled FEM/FEM solver (COMSOL software). The effect of the sound velocity profiles on the underwater acoustic radiation field in shallow water marine environment is numerically investigated.

Natural Vibrations Of Structurally Inhomogeneous Multi-Connected Shell Structures With Viscoelastic Elements

The methods for solving the problems of natural vibrations of structurally inhomogeneous, multi-connected, axisymmetric shell structures based on the hereditary theory of viscoelasticity are given in the paper. Using the basic relations of the hereditary theory of viscoelasticity and asymptotic methods, the problem of natural vibrations of structurally inhomogeneous, multi-connected, axisymmetric shell structures is reduced to the effectively solvable mathematical problem of complex eigenvalues, in which approximate engineering methods are proposed.

Temporally and Spatially Resolved Emission Spectroscopy of Hydrogen, Cyanide and Carbon in Laser-Induced Plasma

In this study, we examine the atomic and molecular signatures in laser-induced plasma. Abel inversions of measured line-of-sight data reveal insight into the radial plasma distribution. Laser-plasma is generated with 6 ns, Q-switched Nd:YAG radiation with energies in the range of 100 to 800 mJ. Temporally- and spatially-resolved emission spectroscopy investigates expansion dynamics. Specific interests include atomic hydrogen (H) and cyanide (CN). Atomic hydrogen spectra indicate axisymmetric shell structures and isentropic expansion of the plasma kernel. The recombination radiation of CN emanates within the first 100 nanoseconds for laser-induced breakdown in a 1:1 mole ratio CO2:N2 gas mixture. CN excitation temperatures are determined from fitting recorded and computed spectra. Chemical equilibrium mole fractions of CN are computed for air and the CO2:N2 gas mixture. Measurements utilize a 0.64-m Czerny–Turner type spectrometer and an intensified charge-coupled device.

Elastoplastic and nonlinear analysis of functionally graded axisymmetric shell structures under thermal environment, using a conical frustum finite element model

This work presents the formulation for static bending analysis of functionally graded axisymmetric plate/shell type structures under mechanical loading, and considering different structural behaviours: linear, geometric nonlinear and material nonlinear. The implemented model is based on a simple conical frustum finite element with 2 nodes, and 3 degrees of freedom per node, which includes shear deformation effects, and it shows to be extremely efficient in the analysis of axisymmetric shells subjected to axisymmetric loading. The used of reduced numerical integration procedure is essential for its success when applied to thin shells. The formulation accounts for the calculation of displacements and through-thickness stress distribution. The solutions for some illustrative examples involving variation of volume fractions are obtained, and the results are presented and compared with numerical alternative models when available, and discussed.

Calculation of structural-inhomogeneous multiply connected shell structures with viscoelastic elements

Using the basic relationships of the hereditary theory of viscoelasticity and asymptotic methods, the problem of natural oscillations of structural-inhomogeneous, multiply connected, axisymmetric shell structures is reduced to an effectively solvable mathematical problem of complex eigenvalues, in which approximate engineering methods are proposed.

Generalized Taylor polynomials for axisymmetric plates and shells

This work proposes the use of a mesh-free technique, derived from the generalized Taylor polynomials, for the analysis of axisymmetric plates and shells. The primary solution variable(s) is/are assumed to take the form of a truncated Taylor series around a point c, and the unknown coefficients of the expansion are determined using the governing differential equation(s) and boundary conditions. The method is free of shape-parameter calibration needed in some other famous mesh-free techniques such as the RBF, and is quite easy to formulate and program. Successful application of the method to several benchmark problems of axisymmetric plate and shell structures proves its robustness. The results have been verified using the existing rigorous analytical solutions that are in most cases not suited to practical engineering calculations.

Analysis of large sheet metal tailored tubes

The present study was triggered by the need to verify and optimize the primary constructive solution, for custom large tubes (section lengths larger than 1000 mm), under the gravity and pressure loads. The cases presented needed to be checked for the reinforcement design. Given the complex tridimensional geometry of the axisymmetric shell structures, the basic shape of the tubes was modelled, with its actual thickness. FEA was used to check the model under static loads and buckling. In order to optimize the weight, an alternative welded reinforcement's grid design was developed and checked for stability. Optimal welding sections along ribs in longitudinal and transversal directions were identified for easier design and further costs reduction.

Frequency split elimination method for a solid-state vibratory angular rate gyro with an imperfect axisymmetric-shell resonator.

The resonator of a solid-state vibratory gyro is responsible for sensing angular motion. Frequency splitting of an axisymmetric-shell resonator is a common problem caused by manufacturing defects. The defect causes a frequency difference between two working modes which consist of two nodes and two antinodes. The difference leads to the loss of gyroscopic effect, and thus the resonator cannot sense angular motion. In this paper, the resonator based on an axisymmetric multi-curved surface shell structure is investigated and an approach to eliminate frequency splits is proposed. Since axisymmetric multi-curved surface shell resonators are too complex to be modeled, this paper proposes a simplified model by focusing on a common property of the axisymmetric shell. The resonator with stochastic imperfections is made equivalent to a perfect shell with an imperfect mass point. Rayleigh's energy method is used in the theoretical analysis. Finite element modeling is used to demonstrate the effectiveness of the elimination approach. In real cases, a resonator's frequency split is eliminated by the proposed approach. In this paper, errors in the theoretical analysis are discussed and steps to be taken when the deviation between assumptions and the real situation is large are figured out. The resonator has good performance after processing. The elimination approach can be applied to any kind of solid-state vibratory gyro resonators with an axisymmetric shell structure.

On the fabrication of carbon fabric reinforced epoxy composite shell without joints and wrinkling

This article describes a simple and cost effective fabrication procedure by using hand lay-up technique that is employed for the manufacturing of thin-walled axi-symmetric composite shell structures with carbon, glass and hybrid woven fabric composite materials. The hand lay-up technique is very commonly used in aerospace and marine industries for making the complicated shell structures. A generic fabrication procedure is presented in this paper aimed at manufacture of plain Carbon Fabric Reinforced Plastic (CFRP) and Glass Fabric Reinforced Plastic (GFRP) shells using hand lay-up process. This paper delivers a technical breakthrough in fabrication of composite shell structures without any joints and wrinkling. The manufacture of stiffened CFRP shells, laminated CFRP shells and hybrid (carbon/glass/epoxy) composite shells which are valued by the aerospace industry for their high strength-to-weight ratio under axial loading have also been addressed in this paper. A fabrication process document which describes the major processing steps of the composite shell manufacturing process has been presented in this paper. A study of microstructure of the glass fabric/epoxy composite, carbon fabric/epoxy composite and hybrid carbon/glass/fabric epoxy composites using Scanning Electron Microscope (SEM) has been also carried out in this paper.

Analysis of axisymmetric shell structures under axisymmetric loading by the flexibility method

A method is developed for the static stress and deformation analysis of axisymmetric shells under axisymmetric loading by reduction of the shell to ring sections. In particular, the wall thickness of the shell may vary and the method is applicable to the analysis of shells with irregular meridional geometry. Explicit expressions for the influence coefficients for each ring element are derived. In the development of these expressions, exact evaluation of stresses in the circumferential direction of the ring is used. The distribution of stresses in the meridional direction of the ring element is assumed to be linear with each element. By using the derived influence coefficients, the unknown forces at the junctures of the ring elements are found by the standard flexibility method of indeterminate structural analysis. Subsequently, the displacements and internal stresses are determined. Example solutions for a flat circular plate under transverse loading and for a cylindrical shell under a boundary edge loading show excellent agreement with solutions found by solving the governing differential equations.

Analysis and Optimization of Prismatic and Axisymmetric Shell Structures: Theory, Practice, and Software

Analysis and Optimization of Prismatic and Axisymmetric Shell Structures: Theory, Practice, and Software

Review of Analysis and Optimization of Prismatic and Axisymmetric Shell Structures: Theory, Practice and Software

Review of Analysis and Optimization of Prismatic and Axisymmetric Shell Structures: Theory, Practice and Software

Mechanics of tether formation in liposomes.

It is well-known that a "tether" may be drawn out from a pressurized liposome by means of a suitably applied radial-outward force applied locally to the lipid bilayer. The tether is a narrow, uniform cylindrical tube, which joins the main vesicle in a short "transition region." A first-order energy analysis establishes the broad relationship between the force F needed to draw the tether, the radius R0 of the tether, the bending-stiffness constant B for the lipid bilayer and the membrane tension T in the pressurized liposome. The aim of the present paper is to study in detail the "transition region" between the tether and the main vesicle, by means of a careful application of the engineering theory of axisymmetric shell structures. It turns out that the well-known textbook "thin-shell" theory is inadequate for this purpose, because the tether is evidently an example of a thick-walled shell; and a novel ingredient of the present study is the introduction of elastic constitutive relations that are appropriate to the thick-shell situation. The governing equations are set up in dimensionless form, and are solved by means of a "shooting" technique, starting with a single disposable parameter at a point on the meridian in the tether, which can be adjusted until the boundary conditions at the far "equator" of the main vessel are satisfied. It turns out that the "transition region" between the tether and the main vessel is well characterized by only a few parameters, while the tether and main vessel themselves are described by very simple equations. Introduction of the thick-shell constitutive relation makes little difference to the conformation of and stress-resultants in, the main vessel; but it makes a great deal of difference in the tether itself Indeed, a kind of phase-change appears to take place in the "transition region" between these two zones of the liposome.

Dynamics of axisymmetric hyperbolic shell structures

Shell structures have many applications including cooling towers, liquid-retaining structures and roofs where large uninterrupted space is required. These structures are aesthetically pleasing, make efficient use of construction materials and can economically meet design criteria. A number of serious failures of shell structures during the last century has led to significant research into their behaviour, analysis, design and construction. Despite this, further research is still required to improve design, especially with respect to shell dynamics. While some research on free vibration has been carried out, very little or none has been documented to date on the response to earthquake loading. In earlier research carried out by the authors, it has been shown that the hyperbolic shape is very efficient for use as an axisymmetric shell. In view of this and other existing information, this paper treats the free vibration and seismic response of hyperbolic shells, and examines the influence of thickness, height and curvature on this response. It is found that the period of vibration decreases approximately linearly with increasing curvature, but at high curvatures this trend reverses. The early periods of the circumferential mode of vibration are also found to vary linearly with changes of height and thickness, with increasing thickness reducing the period and increasing height increasing the period. The response of the first lateral mode is also significantly affected by a change in the parameters. Transient earthquake analysis shows that height is the most important factor governing the dynamic response. However, significant changes in hoop and meridional stresses are also observed for structures with different shell thickness and curvature. Analysis has been undertaken using the finite element method, and results are presented in tabular and graphical format.

Ideal stretch forming for minimum weight axisymmetric shell structures

A companion paper “Minimum Weight Design of Axisymmetric Shell Structures” by Richmond and Azarkhin describes the design of axisymmetric thin-walled structures and parts that can support specified loads with minimum material. The result is an optimum shape and thickness distribution in the final part when the strength of the material is assumed to be uniform. Here, we demonstrate the application of ideal forming theory to design sheet stretching processes that can produce the optimum shapes and thickness distributions from flat sheets of uniform thickness. Specific designs are achieved for producing minimum weight shell structures that will support a specified uniform pressure assuming both the Mises and the Tresca yield criteria along with the rigid-perfectly plastic flow condition. In the case of the Tresca yield condition, the optimum structure is a spherical shell segment with uniform thickness, and an associated ideal stretching process is hydraulic bulging. Because the effects of strain hardening have been neglected in the structural optimization theory, it has been possible here to design the minimum weight structure and its forming process sequentially. In subsequent work, we plan to include the effects of strain hardening on the shell strength, which will then require coupled design of the structure and its forming process. Also extension of these methods to three-dimensional geometries will be considered.

Minimum weight axisymmetric shell structures

The membrane approximation was used to determine the shapes and thickness distributions for axisymmetric shell structures that can support specified loads with minimum material. A Lagrange multiplier technique was employed. It resulted in Euler field equations and associated boundary conditions for the optimum shells. It also generated expressions for the multipliers directly in terms of the shell geometry. Specific solutions were obtained for the cases where the prescribed forces were uniform pressure, uniform vertical load and the weight of the shell itself. Both the Mises and the Tresca yield condition were used, and the optimum shells were shown to be in a state of yielding everywhere. The optimum shells, however, differed in general from equal strength shells which also satisfied yielding everywhere.