Axisymmetric Shell Research Articles

Graphical Modelling of Hoop Force Distribution for Equilibrium Analysis of Masonry Domes

The behaviour of axisymmetric masonry shells can be simulated by a system of forces constituted by meridian forces acting in the vertical planes, and by hoop forces acting circumferentially. A crucial component for the assessment of these structures using the Modified Thrust Line Method (MTLM) is the determination of hoop forces, whose computation is strenuous, limiting the practical application of MTLM. Working around this limitation, the current research introduces a strategy to manipulate the hoop forces by graphically implementing a function describing their distribution. The adaptiveness of this distribution function not only allows the application of MTLM for the analysis of a range of geometries, but also enables the simulation of membrane behaviour, arch behaviour and their combination, for considering partially cracked structures. Taking this into account, the approach is applied in the case studies illustrated within the current research.

Natural frequency analysis of shells of revolution based on hybrid dual-mixed [formula omitted]-finite element formulation

A newly-developed, dimensionally reduced, hp-type axisymmetric shell finite element model is extended to linear elastodynamic problems of thin shells of revolution. The hp-shell finite element relies on the hybridized version of a three-field dual-mixed variational formulation, the application of which dictates the obligate usage of the inverse three-dimensional constitutive relation for homogeneous and isotropic materials, thereby ensuring the volumetric locking-free characteristic of the shell model at theory level. The fundamental fields are the a priori non-symmetric stress tensor, the displacement vector, the infinitesimal rotation vector and the hybrid variable defined on the element interfaces. Since the dimensional reduction process guided by this hybrid-mixed formulation does not necessitate the use of any classical kinematic assumptions appearing in the scientific literature, the inverse 3D Hooke’s law does not have to be modified. The numerical performance of the shell finite element is analyzed comprehensively for natural frequency computations of clamped-free and simply supported, silicone, conoid, spherical and hyperboloid shells of revolution. From their relative error convergence behaviors it follows that the extended hybrid-mixed shell finite element is not sensitive to the decrease of the slenderness ratio, namely providing reliable, uniformly stable numerical results for both h- and p-approximation. From theoretical point of view, the beneficial properties of the hybrid-mixed hp shell finite element model are as follows: (i) this is effectively applicable to modeling not only extremely thin but also moderately thick shell structures with transverse shear deformations, as well as (ii) both the through-the-thickness variation and the membrane stress normal to the shell mid-surface are retained as independent variables, making it much easier to upbuild shell model for contact problems. From numerical point of view, the nice feature of the hybrid-mixed hp shell finite element is that the global flexibility matrix of the system can be inverted block-wise at element level at cheap computational cost during the assembling procedure because of the hybridization technique.

Free Vibration Analysis of Multiphase Magneto-Electro-Elastic Composite Conical Shells

Objectives: In the present article, free vibration of multiphase Magneto-Electro-Elastic (MEE) conical shell having a uniform thickness is examined for Clamped-Free (C-F) boundary condition. Method: The study is carried out using a semi-analytical approach for different volume fractions (Vf ) 0, 0.2, 0.6 and 1.0 of BaTiO3 in BaTiO3-CoFe2O4 smart composite conical shell for three different semi-vertex angles 20o,35o and 50o. The piezoelectric (Pe) and piezomagnetic (Pm) phase on natural frequencies of MEE truncated conical shells are discussed for different circumferential modes. Findings: The parametric study indicates that natural frequency decrease with an increase in Vf of BaTiO3 in magnetoelectro-elastic truncated conical shells. Novelty: Studies on MEE constant thickness truncated conical shell using BaTiO3 and CoFe2O4 as (Pe) & (Pm) smart composite for clamped-free boundary condition to analyse the effect of the frequency with different semi-vertex angle and cone heights. Present commercial FEA software tools are limited to 2 coupling fields. In this research, coupling between 3 fields considered for MEE material. Hence, a computer code is developed to study the influence coupling between electric, elastic and magnetic fields, which can be used for any combinations of boundary conditions and volume fractions. Keywords: MagnetoElectroElastic (MEE); axisymmetric constant thickness conical shell; Smart Composites; volume fraction; free vibrations; finite element method

Numerical Modeling of Initial Stress Stiffening Effect on the Dynamic Behavior of Axisymmetric Shells

AbstractThis paper presents a numerical model to simulate the initial stress stiffening effect, induced by radial pressure and/or axial load on the dynamic behavior of axisymmetric shells. This effect is particularly important for thin shells since their bending stiffness is very small compared to membrane stiffness. The theoretical formulation is based on a combination of the finite element method and classical shell theory. For a perfect geometrical consistency, two semi-analytical finite elements, conical and cylindrical, are used to model axisymmetric shells. The displacement functions are derived from exact solutions of Sanders' shell equilibrium equations. The results obtained using this approach are remarkably accurate. The potential energy is calculated to estimate the initial stiffening effect using direct membrane forces per unit width and rotations about the orthogonal axes. The final stiffness matrix of each finite element is composed of the regular stiffness matrix and the added stiffness matrix generated by membrane loads. The frequencies of vibration are compared with those obtained in other experimental and theoretical research works and very good agreement is observed.

A new computational framework for the minimum thrust analysis of axisymmetric masonry domes

An original computational framework is presented for the minimum thrust analysis of axisymmetric masonry domes with arbitrary meridian curve, subject to their self-weight. The formulation is characterized by the ability to account for an arbitrary meridional stereotomy, which is reflected in the exact treatment of the dome self-weight and in the representation of the stress resultants. The classical equilibrium equations of axisymmetric shells are resorted to for formulating the equilibrium of the dome as that of its mid-surface, and the admissibility conditions on the stress resultants are consequently stated. The proposed formulation is proven to provide a unitary framework in which competing formulations for masonry domes available in the literature, classified as lunar-slices and membrane formulations, can be derived as particular cases. Furthermore, it fosters an extremely efficient and robust computational approach, which amounts to the solution of a straightforward Linear Programming problem. Numerical results are discussed, as parametric analyses of ogival domes with normal or vertical meridional stereotomies, and of domes with superellipse meridian curve, at varying of the rise-to-midspan ratio and of the shape exponent. Finally, the merit of the proposed procedure is shown in application to a real case.

A Morphoelastic Shell Model of the Eye

The eye grows during childhood to position the retina at the correct distance behind the lens to enable focused vision, a process called emmetropization. Animal studies have demonstrated that this growth process is dependent upon visual stimuli, while genetic and environmental factors that affect the likelihood of developing myopia have also been identified. The coupling between growth, remodeling and elastic response in the eye is particularly challenging to understand. To analyse this coupling, we develop a simple model of an eye growing under intraocular pressure in response to visual stimuli. Distinct to existing three-dimensional finite-element models of the eye, we treat the sclera as a thin axisymmetric hyperelastic shell which undergoes local growth in response to external stimulus. This simplified analytic model provides a tractable framework in which to evaluate various emmetropization hypotheses and understand different types of growth feedback, which we exemplify by demonstrating that local growth laws are sufficient to tune the global size and shape of the eye for focused vision across a range of parameter values.

Mechanical response of buried and covered pipes under water hammer

The classical water hammer model omits bending and inertial effects in the pipe wall. The validity of such assumptions for buried and covered pipes is assessed by developing a one-way coupled water hammer model based on an axisymmetric thin shell pipe finite element that captures bending and inertial effects. A finite/infinite pipe element formulation is developed to capture the stiffness contribution of the surrounding soil (or pipe cover) by adopting a decay function derived from plane strain analysis of the medium surrounding the pipe. Continuous and discrete elastic longitudinal restraint effects are also incorporated into the formulation. The model is then used to investigate the effects of viscous damping, elastic longitudinal restraints, presence of intermediate longitudinal anchors, and the elastic modulus of the surrounding soil. The pipe response is found insensitive to viscous damping. The present model shows that, while classical model predictions are valid for cases where the valve end of the pipe is fully (or nearly-fully) restrained in the longitudinal direction, this is not the case where the valve end is longitudinally unrestrained. Intermediate longitudinal restraints are found to divide the pipe into two segments with distinct responses. The ability of the proposed finite/infinite element to predict the response of buried pipe subjected to water hammer is illustrated through numerical examples.

Nonlinear dynamics of heterogeneous shells Part 1. Statics and dynamics of heterogeneous variable stiffness shells

The increasing complexity of the constructive forms and shell elements structure leads to the need to develop both the theory and methods for solving static and dynamic problems for non-homogeneous (heterogeneous) shells.By the shell heterogeneity, we mean heterogeneity in a broad sense: these are inclusions in the shell body of the different rigidity elements and, as a special case, these are holes; the material inhomogeneity caused by a change in the stress–strain state under the influence of both static and dynamic loads; stiffeners; and by taking into account the physical nonlinearity and different modulus of the shell material.This work is devoted to the mathematical model creation of the statics and dynamics for non-homogeneous shells in the above sense and consists of two parts.In the first part, a mathematical model of statics and dynamics for rectangular shells described by the Kirchhoff–Love kinematic model is constructed. Geometric nonlinearity is taken into account on the basis of T. von Kármán’s geometric model, physical nonlinearity — according to the deformation theory of plasticity, based on the elasticity variable parameters method. Stiffness heterogeneity is taken into account using the Heaviside function. The original equations were obtained from Hamilton’s variational principle. A numerical experiment is performed using the Faedo–Galerkin method in higher approximations. The convergence of this method is investigated.In the second part, the nonlinear dynamics of axisymmetric elastic variable thickness shells is investigated. In contrast to the first part of this work, where the Faedo–Galerkin method in higher approximations was used to reduce partial differential equations to the Cauchy problem, in this part the Ritz method in higher approximations is employed. For differential operators used in two parts of this work, according to the research of S.G. Mikhlin, the Ritz and Faedo–Galerkin methods are equivalent. The convergence of the applied numerical method is investigated. An approach is proposed for the zones study of oscillation types for variable thickness shells using dynamic modes maps.

Application of exponential functions in weighted residuals method in structural mechanics. Part 1: axisymmetrical shell problem.

Weighted residuals method gained a wide popularity during last years especially due to its application in finite element methods. Its goal is in approximate satisfaction of the governing differential equations while boundary conditions are to be fulfilled exactly. This goal is achieved by the proper choice of the sets of so-called trial (basic) functions which give the residuals. Residuals are multiplied by weight functions and minimized by integration over the whole area of task. In fact, they determine the peculiarity and advantages of each particular method. Most popular is the choice of trial and weight (test) function as the trigonometric and polynomial functions. In 2D applications so-called “beam functions” are often used, which are solutions of much simpler 1D problems for beam. In this methodological paper we explore the possibility of using the sets of functions constructed on the consequent exponential functions, which satisfy boundary conditions. The method is investigated on example of very simple 1D axisymmetrical task for shell, where exact solution exists for any loading. For several examples of distributed or concentrated loading the proposed method is compared with similar Navier’s method, which is the expansion on trigonometric functions. Also the proper choice of weight functions is carefully investigated. It is noted, that proposed sets (symmetrical or asymmetrical) of exponential functions has a good perspective in application for more complicated problems in structural mechanics.

Space-fixed formulation for the vibration of rotating, prestressed, axisymmetric bodies and shells

The vibration of high-speed, rotating, three-dimensional, flexible, axisymmetric bodies coupled to stationary structures motivates this work. Exploiting the axisymmetry of the body, a semi-analytical model is developed that only discretizes the two-dimensional radial cross-section; Fourier series are used in the circumferential direction. The corresponding formulation for thin-walled, axisymmetric shells is given. Even though the body is axisymmetric, its deflection as well as external forces, constraints, and supports acting on the body are allowed to be asymmetric. These asymmetric elements can be stationary or rotating. The model is developed in a stationary reference frame to avoid time-dependent coefficients in the equations of motion when coupled to stationary components. The model includes Coriolis and centripetal effects. The prestress (or geometric) stiffness matrix that arises from external forces and constant centripetal acceleration has additional terms compared to the literature, and these terms can significantly change the natural frequencies. The speed-dependent natural frequencies and complex-valued vibration modes, presence of multiple Fourier harmonics in each mode, changes to critical speeds, divergence and flutter instability phenomena, and eigenvalue veering are investigated for spinning systems with asymmetric features.

Nonlinear dynamics of heterogeneous shells. Part 2. Chaotic dynamics of variable thickness shells

In the second part of the article, the nonlinear dynamics and stability of nonhomogeneous variable thickness axisymmetric shells are analyzed. The mathematical model is based on the Kirchhoff–Love kinematic hypothesis. The resulting system of partial differential equations is reduced to an algebraic equations system by the Ritz method. The convergence of the applied numerical methods is investigated. The chaotic vibrations’ control of variable thickness shells using dynamic charts of vibration regimes is also developed.

Mechanical and thermal buckling of functionally graded axisymmetric shells

The buckling analysis of functionally graded materials (FGM) axisymmetric plate-shell type structures under mechanical and termal loading is presented in this work. A numerical solution is obtained by expanding the variables in Fourier series in the circumferential direction and using conical frustum finite elements in the meridional direction. The finite element model, having two nodal circles and ten degrees of freedom per node, is based in the Kirchhoff-Love theory that includes the transverse shear deformations by introducing a penalty function, which corresponds to the first order shear deformation theory (FSDT), is suitable for both thin and thick axisymmetric plate/shell structures. The reduced number of finite elements, which are required to model even complex structures, combined with the use of a small number of discrete layers to model the continuous variation of the mechanical properties through the thickness of the structure, results in an extremely low computational time required for FGM buckling applications. An in-house program has been developed, and applications in a variety of axisymmetric shells are solved, including circular plates. The solutions obtained in mechanical and thermal buckling are discussed and compared with alternative models..

Choice of materials properties and of shell molds structure by investment models

The paper presents a mathematical model optimizing the choice of material and morphological structure of the shell mold (SM), which has the highest resistance to cracking when pouring liquid metal into it. To solve this problem, the theory of small elastoplastic deformations and the heat equation, as well as proven numerical methods, were used. The objective function min – max was constructed from control variables characterizing the properties of the molding material of the shell. The process of heating an axisymmetric shell mold was considered when pouring liquid metal into it. The resistance of the shell form was estimated by the stresses arising in it. An algorithm for solving this problem was compiled. Using numerical schemes and program complexes developed in previous studies, an algorithm for solving the optimization problem was constructed and the values of control variables were found in which the shell mold does not break even in the presence of a rigid process – pouring steel into a cold shell mold. Analysis of the influence of weight of each of the found parameters on the value of the constructed objective function is given. Using a mathematical experiment, the morphological structure of the shell mold was studied. The shell mold of five layers is considered. The corrected system of equations makes it possible to take into account the properties of the layers made of different materials. Calculations were performed when the layer of the shell mold from material found by optimization occupies different positions in its cross section. In this case, the remaining layers of the mold are made of traditional ceramics. The optimal location of this layer was found. It is shown that the presence of several layers with the found properties does not affect the increase in crack resistance of the shell mold.

Two new nova shells associated with V4362 Sagittarii and DO Aquilae

ABSTRACT A classical nova is an eruption on the surface of a white dwarf in an accreting binary system. The material ejected from the white dwarf surface generally forms an axisymmetric shell. The shaping mechanisms of nova shells are probes of the processes that take place at energy scales between planetary nebulae and supernova remnants. We report on the discovery of nova shells surrounding the post-nova systems V4362 Sagittarii (1994) and more limited observations of DO Aquilae (1925). Distance measurements of $0.5\substack{+1.4 \\ -0.2}$ kpc for V4362 Sgr and 6.7 ± 3.5 kpc for DO Aql are found based on the expansion parallax method. The growth rates are measured to be 0.07 arcsec yr−1 for DO Aql and 0.32 arcsec yr−1 for V4362 Sgr. A preliminary investigation into the ionization structure of the nova shell associated with V4362 Sgr is presented. The observed ionization structure of nova shells depends strongly on their morphology and the orientation of the central component towards the observer. X-ray, IR, and UV observations as well as optical integral field unit spectroscopy are required to better understand these interesting objects.

Semi-analytical model development for preliminary study of 3D woven Composite/Metallic flange bolted assemblies

This research deals with the development of a new semi-analytical model for composite-metal eccentric tensile bolted joints of casing structures. The model is based on a direct stiffness method considering structural elements, where flanges are modeled with axisymmetric shell elements, the bolt with a beam element and contact behavior with a specific number of compressive linear springs, identified by means of convergence tests. A hybrid element is added in order to set the link between flange and bolt. Since the model is intended for the preliminary design of bolted joints including composite materials, we propose an extension of the stiffness matrix formulation of isotropic axisymmetric shell elements to orthotropic ones. This formulation is validated with respect to a 2D FE axisymmetric model. Then, a semi-analytical model is established for a casing assembly approximately corresponding to a real casing of an aircraft engine. Design criteria such as normal stress on the bolt, bending moment and axial load at a specific section of the composite flange, given by the semi-analytical model are compared to a 3D FE computation. Results are in very good agreement and the semi-analytical model offers considerable time saving where the computation time was 120 times faster than the 3D FE model. Moreover, relative error does not exceed 2% for normal stress of the bolt and 9% for the flange bending moment which is satisfactory in a preliminary design approach. However, due to the formulation of hybrid element, less accurate results were observed for axial displacement of the composite flange and it would be advisable to consider a new formulation. This will be detailed in future work. Finally, a parametric study showed the ability of the semi-analytical model to predict the physical behavior of joints and proved its usability for running optimization studies and its reliability as a preliminary design tool.

Influence of the Physical Properties of the Material on the Thermomagnetoelastic Behavior of a Flexible Conical Shell with Orthotropic Conductivity and Joule Heat

Differential equations of thermomagnetoelasticity for flexible axisymmetric conical shells made of isotropic and orthotropic materials taking into account Joule heating are obtained. A nonlinear analysis of the thermoelastic state of a truncated conical shell is carried out for different physical properties of its material.

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.

Vibrations of functionally graded material axisymmetric shells

The free-vibration analysis of functionally graded materials (FGM) axisymmetric plate-shell type structures are presented in this work. A numerical solution is obtained by expanding the variables in Fourier series in the circumferential direction and using conical frustum finite elements in the meridional direction. The finite element model, having two nodal circles and ten degrees of freedom per node, is based in the Kirchhoff-Love theory that include the transverse shear deformations by introducing a penalty function, and using one Gauss point integration scheme which gave excellent results for both thin and thick axisymmetric plate/shells structures. The reduced number of finite elements, which are required to model even complex structures, combined with the use of a small number of discrete layers to model the continuous variation of the mechanical properties through the thickness of the structure, result in an extremely low computational time required for FGM applications. An in-house program has been developed, and applications in a variety of axysimetric shells are solved, including circular plates. The solutions obtained are discussed and compared with solutions obtained by alternative models.

Dual-mixed hp-version axisymmetric shell finite element using NURBS mid-surface interpolation

A new, general hp-version axisymmetric finite element is derived for the boundary value problems of thin linearly elastic shells of revolution, applying a complementary strain energy-based three-field dual-mixed variational principle. For the interpolation of the mid-surface geometry, non-uniform rational B-splines—NURBS—is used. The independent field variables of the weak formulation are the a priori non-symmetric stress tensor, the displacement vector, and the infinitesimal skew-symmetric rotation tensor. The theoretical model of the shell formulation is based on a consistent dimensional reduction process and a systematic variable-number reduction procedure. The inverse of the unvaried three-dimensional constitutive equation is employed since neither the classical kinematical assumptions nor the stress hypotheses are built in the mathematical model; namely, both the through-the-thickness variation and the normal stress to the shell mid-surface are not excluded. The new hp axisymmetric shell finite element is tested by a representative model problem for extremely thin and moderately thick, singly and doubly curved shells of negative and positive Gaussian curvature. Following from the numerical experiments, the constructed hp-shell finite element gives locking-free results not only for the displacement but also for the stresses.

Prediction of the Boundary States for Thin-Walled Axisymmetric Shells Under Internal Pressure and Tension Loads

Abstract A method for calculating the ultimate true stresses arising in the walls of shells of revolution in the area of uniform plastic deformation is developed in the research. In order to derive the stability loss for the plastic deformation process the criterion of maximum load is taken as the basis, simple differential equations were solved. It has been shown analytically that the level of the boundary true stresses is much lower when the values of the principal stress ratios approach to 2 or 1/2 compared to the adjacent stress states.