Equilibrium Equations For Shells Research Articles

Numerical modeling and analysis of fluid-filled truncated conical shells with ring stiffeners

This study uses a hybrid finite element method to predict dynamic behavior of truncated conical shells with ring stiffeners under fluid loading. The proposed approach combines classical shell theory and the finite element method, making use of displacement functions derived from exact solutions of Sanders’ shell equilibrium equations for conical shells. The analysis of the shell-fluid interface involves leveraging the velocity potential, Bernoulli’s equation, and impermeability conditions to determine an explicit expression for fluid pressure. To the best of our knowledge, this paper is the first to compare the methods applied to ring-stiffened shells against other numerical and experimental findings. Our results on conical shells in various conditions, with and without ring stiffeners, are largely consistent with previous findings. This study also explores the influence of geometric parameters, stiffener quantity, cone angle, and applied boundary conditions on the natural frequency of fluid-loaded ring-stiffened conical shells. The paper concludes with a discussion of several useful implications for further research.

EQUILIBRIUM OF A NORMALLY LOADED ELASTIC SHALLOW CYLINDRICAL SHELL WITH DISLOCATIONS AND DISCLINATIONS

We consider the problem of the equilibrium of a normally loaded elastic shallow rectangular circular cylindrical shell, which contains sources of internal stress in the form of fields of continuously distributed edge dislocations and wedge disclinations or other sources, for example, thermal ones. Based on the modified system of Karman equations for the equilibrium of an elastic plate with dislocations and disclinations, a system of nonlinear equilibrium equations for the shell was constructed. The resulting system of equations differs from the Karman system of equilibrium equations for an elastic shallow cylindrical shell under the action of a normal load by the presence on the right side of the continuity equation of a nonzero function, called the incompatibility function, which is expressed through the densities of edge dislocations and wedge disclinations. The edges of the shell are freely clamped or hinged. In the absence of internal stress sources, this system transforms into a system of equilibrium equations for a shallow shell, and in the case of an infinitely large radius of curvature (and the presence of internal stress sources) into a modified system of Karman equilibrium equations for a flexible elastic plate with dislocations and disclinations. To solve the system of nonlinear shell equilibrium equations, the Newton – Kantorovich method in combination with the difference method is proposed. For the case of small values of the normal load and small values of the incompatibility function, linearization is carried out with respect to the deflection and stress function. As a result, a linear boundary value problem was obtained in the form of a system of differential equations for the deflection and the stress function, which is solved by the difference method. The solution of the linearized system is used as an initial approximation in the implementation of the Newton – Kantorovich method. Examples of numerical calculations of deflection and stress function for given values of normal load and incompatibility function are given, and corresponding graphs are constructed.

An analytical approach of nonlinear buckling behavior of torsionally loaded auxetic core toroidal shell segments with graphene reinforced polymer coatings

The buckling and nonlinear postbuckling analysis of toroidal shell segments with auxetic-core layer and Graphene-reinforced polymer coatings under torsional loads is reported in the present research. The functionally graded Graphene-reinforced polymer coatings are considered with three distribution laws, and the auxetic cores are designed in the honeycomb lattice forms. An auxetic homogenization technique is applied to establish the stiffness terms of the auxetic core. The combination of nonlinear von Karman Donnell shell theory and the Stain and McElman approximate is applied to formulate the nonlinear equilibrium equations of shells with the shallow longitudinal curvature considering the two-parameter foundation model. The deflection of shells is assumed to be a three-term form corresponding to the pre-buckling, linear and nonlinear postbuckling behaviors, and the Galerkin method is employed for three terms of defection. The torsion-deflection and torsion-twist angle postbuckling behaviors can be obtained in explicit forms. The numerical examinations validate the large effects of honeycomb lattice auxetic core, the functionally graded Graphene-reinforced polymer coatings, the parameters of shell’s geometric and foundation on the nonlinear buckling behaviors of shells.

A finite difference method for the static limit analysis of masonry domes under seismic loads

The static limit analysis of axially symmetric masonry domes subject to pseudo-static seismic forces is addressed. The stress state in the dome is represented by the shell stress resultants (normal-force tensor, bending-moment tensor, and shear-force vector) on the dome mid-surface. The classical differential equilibrium equations of shells are resorted to for imposing the equilibrium of the dome. Heyman’s assumptions of infinite compressive and vanishing tensile strength, alongside with cohesive-frictional shear response, are adopted for imposing the admissibility of the stress state. A finite difference method is proposed for the numerical discretization of the problem, based on the use of two staggered rectangular grids in the parameter space generating the dome mid-surface. The resulting discrete static limit analysis problem results to be a second-order cone programming problem, to be effectively solved by available convex optimization softwares. In addition to a convergence analysis, numerical simulations are presented, dealing with the parametric analysis of the collapse capacity under seismic forces of spherical and ogival domes with parameterized geometry. In particular, the influence that the shear response of masonry material and the distribution of horizontal forces along the height of the dome have on the collapse capacity is explored. The obtained results, that are new in the literature, show the computational merit of the proposed method, and quantitatively shed light on the seismic resistance of masonry domes.

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.

ПСЕВДОМИНИМАЛЬНОСТЬ И ЛИНЕЙЧАТЫЕ ПОВЕРХНОСТИ

This paper is a follow-up to the author's series of works about shape modeling for an orthotropic elastic material that takes an equilibrium form inside the area with the specified boundaries. V.M. Gryanik and V.I. Loman, based on thin shell equilibrium equations, solved about 30 years ago a similar problem for an isotropic mesh attached to rigid parabolic edges. With a view to extend modeling to orthotropic materials (and other boundary contours), the author in his publications of 2016–2017 proposed an approach to the problem based on the application of surfaces with a constant ratio of principal curvatures. These surfaces are called pseudo-minimal surfaces. A partial differential equation that defines (in the local sense) a class of pseudo-minimal surfaces is very complex for analysis. However, for some classes of surfaces, the analysis is greatly simplified, notably, the analysis can be performed without this inconvenient PDE, but with the method of moving frames. The author is referring to a class of ruled surfaces. This class is interesting not only due to the aforesaid but also due to an evident interest manifested by architects and builders. However, one should discuss not the pseudo-minimal ruled surfaces (they exist but are obviously trivial) but an invariant (principal curvatures ratio), which is not an identical constant on a given surface but its contour lines coincide with the lines of some invariant family. Roughly speaking, there are surfaces whose pseudo-minimal condition is satisfied identically, and surfaces that are pseudo-minimal "in a limited sense"—lengthways the lines of a certain family, internally connected with the surface. The article finds that the role of such a family can be obviously played by "equidistant" lines for the striction line of a skew ruled surface, and rays are the carriers of such a ruled surface, they form a regulus with constant Euclidean invariants.

A comparative study of composite structures reinforced with carbon, glass or natural fibers

PurposeThe purpose of this paper is to propose a comparative study between different structures composed of fiber-reinforced composite materials. Plates, cylinders and cylindrical and spherical shell panels in symmetric 0°/90°/0° and antisymmetric 0°/90°/0°/90° configurations are analyzed considering carbon fiber, glass fiber and linoleum fiber reinforcements.Design/methodology/approachA free vibration analysis is proposed for different materials, lamination sequences, vibration modes, half-wave numbers and thickness ratios. Such an analysis is conducted by means of an exact three-dimensional shell model which is valid for simply supported structures and cross-ply laminations. The employed model is based on a layer-wise approach and on three-dimensional shell equilibrium equations written in general orthogonal curvilinear coordinates.FindingsThe proposed study confirms the well-known superiority of the carbon fiber-reinforced composites. Linoleum fiber-reinforced composites prove to be comparable to glass fiber-reinforced composites in the case of free vibration analysis. Therefore, similar frequencies are obtained for all the geometries, thickness ratios, laminations sequences, vibration modes and a large spectrum of half-wave numbers. This partial conclusion needs further confirmations via static, buckling and fatigue analyses.Originality/valueAn exact three-dimensional shell model has been used to compare several geometries embedding carbon fiber composites and natural fiber composites.

Numerical model to analyze the aerodynamic behavior of a combined conical–cylindrical shell

A numerical model is presented in this paper to simulate the aerodynamic behavior of combined conical–cylindrical shells. This class of structures is of great interest due to its extensive use in aeronautical and aerospace applications. Two distinct semi-analytical finite elements are used to model a combined axisymmetric shell for better geometrical consistency. The structural formulation is a combination of the finite element method and classical shell theory. The displacement functions of each finite element are derived from exact solutions of Sanders' shell equilibrium equations. The linearized first-order piston theory formula is applied to take into account the aerodynamic interaction effect. For a liquid contained in the combined shell, the fluid pressure is derived from the velocity potential, Bernoulli equation and from the impermeability condition applied to ensure permanent coupling at the fluid–solid interface. Initial stress stiffening due to axial compression and/or radial pressure is accounted for by generating an additional stiffness matrix. The elementary matrices of the solid and fluid corresponding to each finite element are calculated using exact analytical integration. Results obtained using the present approach in various conditions such as under vacuum, filled with liquid and subjected to supersonic flow are compared to those in published experimental or numerical works. Good agreement is found.

Generalized equilibrium equations for shell derived from a generalized variational principle

In this paper, the general equations of equilibrium for axisymmetrical deformation including the torsional deformation of shells with general forms are derived as stationary conditions of a generalized variational principle, which is established by the semi-inverse method. It shows that they are different from the results obtained by Chien (1990) in case of revolutional shell, therefore Chien’s results can be further improved.

Stress and Strain Recovery of Laminated Composite Doubly-Curved Shells and Panels Using Higher-Order Formulations

The present paper investigates the static behaviour of doubly-curved laminated composite shells and panels. A two dimensional Higher-order Equivalent Single Layer approach, based on the Carrera Unified Formulation (CUF), is proposed. The differential geometry is used for the geometric description of shells and panels. The numerical solution is calculated using the generalized differential quadrature method. The through-the-thickness strains and stresses are computed using a three dimensional stress recovery procedure based on the shell equilibrium equations. Sandwich panels are considered with soft cores. The numerical results are compared with the ones obtained with a finite element code. The proposed higher-order formulations can be used for solving elastic problems involved in the first stage of any scientific procedure of analysis and design of masonry structures.

Static analysis of doubly-curved anisotropic shells and panels using CUF approach, differential geometry and differential quadrature method

The present paper investigates the static behavior of doubly-curved laminated composite shells and panels. A two dimensional General Higher-order Equivalent Single Layer (GHESL) approach, based on the Carrera Unified Formulation (CUF), is proposed. The geometry description of the middle surface of shells and panels is computed by means of differential geometry tools. All structures have been solved through the generalized differential quadrature numerical methodology. A three dimensional stress recovery procedure based on the shell equilibrium equations is used to calculate through-the-thickness quantities, such as displacements components and the strain and stress tensors. Several lamination schemes, loadings and boundary conditions are considered in the worked out applications. The numerical results are compared with the ones obtained with commercial finite element codes. New profiles, concerning displacements, strains and stresses, for doubly-curved multi-layered shell structures are presented for the first time by the authors.

Уравнения равновесия оболочек в координатах общего вида

Уравнения равновесия оболочек в координатах общего вида

Modelling apical constriction in epithelia using elastic shell theory

Apical constriction is one of the fundamental mechanisms by which embryonic tissue is deformed, giving rise to the shape and form of the fully-developed organism. The mechanism involves a contraction of fibres embedded in the apical side of epithelial tissues, leading to an invagination or folding of the cell sheet. In this article the phenomenon is modelled mechanically by describing the epithelial sheet as an elastic shell, which contains a surface representing the continuous mesh formed from the embedded fibres. Allowing this mesh to contract, an enhanced shell theory is developed in which the stiffness and bending tensors of the shell are modified to include the fibres' stiffness, and in which the active effects of the contraction appear as body forces in the shell equilibrium equations. Numerical examples are presented at the end, including the bending of a plate and a cylindrical shell (modelling neurulation) and the invagination of a spherical shell (modelling simple gastrulation).

Vibration analysis of truncated conical shells subjected to flowing fluid

A method to predict the dynamic behaviour of anisotropic truncated conical shells conveying fluid is presented in this paper. It is a combination of finite element method and classical shell theory. The displacement functions are derived from exact solutions of Sanders’ shell equilibrium equations of conical shells. The velocity potential, Bernoulli’s equation and impermeability condition have been applied to the shell–fluid interface to obtain an explicit expression for fluid pressure which yields three forces (inertial, centrifugal, Coriolis) of the moving fluid. To the best of the authors’ knowledge, this paper reports the first comparison made between two works which deal with conical shells subjected to internal flowing fluid effects. The results obtained by this method for conical shells with various boundary condition and geometries, in vacuum, fully-filled and when subjected to flowing fluid were compared with those of other experimental and numerical investigations and good agreement was obtained.

Least-squares finite element formulation for shear-deformable shells

We present a least-squares based finite element formulation for the numerical analysis of shear-deformable shell structures. The variational problem is obtained by minimizing the least-squares functional, defined as the sum of the squares of the shell equilibrium equations residuals measured in suitable norms of Hilbert spaces. The use of least-squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise, i.e. stability requirements such as inf–sup conditions never arise. The proposed formulation retains the generalized displacements and stress resultants as independent variables and, in view of the nature of the variational setting upon which the finite element model is built, allows for equal-order interpolation. A p-type hierarchical basis is used to construct the discrete finite element model based on the least-squares formulation. Exponentially fast decay of the least-squares functional is verified for increasing order of the modal expansions. Several well established benchmark problems are solved to demonstrate the predictive capability of the least-squares based shell elements. Shell elements based on this formulation are shown to be effective in both membrane- and bending-dominated states.

A GROUP THEORETIC APPROACH TO THE LINEAR FREE VIBRATION ANALYSIS OF SHELLS WITH DIHEDRAL SYMMETRY

This paper deals with a group theoretic approach to the finite element analysis of linear free vibrations of shells with dihedral symmetry. Examples of such shell structures are cylindrical shells, conical shells, shells with circumferential stiffeners, corrugated shells, spherical shells, etc. The group theoretic approach is used to exploit the inherent symmetry in the problem. For vibration analysis, the group theoretic results give the correct symmetry-adapted basis for the displacement field. The stiffness matrix K and the mass matrix M are identically block diagonalized in this basis. The generalized linear eigenvalue problem of free vibration gets split into independent subproblems due to this block diagonalization. The Simo element is used in the finite element formulation of the shell equilibrium equations. Numerical results for natural frequencies and natural modes of vibration of several dihedral shell structures are presented. The results are shown to be in very good agreement with those reported in the literature. The computational advantages and physical insights due to the group theoretic approach are also discussed.

On nonlinear asymptotic equilibrium equations for shells with geometric flaws

On nonlinear asymptotic equilibrium equations for shells with geometric flaws

Orthotropic behaviour of shells and stability of uniaxially compressed cylinders

AbstractShells, stability under various types of loadings and their combinations have not yet been completely investigated. This is due to incomplete knowledge of shell philosophy. The aim of this study is to provide a complete and integrated method for treating shell problems on a pure statical basis. This study introduces shell orthotropic behaviour, discusses hoop stress resultant actions, and applies the shell equilibrium equations to a uniaxially compressed cylindrical shell with finite length and fixed ends. Theoretical results are also compared with previously published experimental work. The behaviour of shells in elastic and plasto-elastic range is discussed for a wide range of shell geometries (dimensions), material properties, methods of manufacture, material imperfections, and methods of load application. From this study we conclude mainly that shells behave orthotropically. Their flexural rigidities are related to their radii of curvature and the resultant buckling mode. The buckling strength of a shell depends on shell geometry, material properties, buckling mode and shell coefficient H. This shell coefficient is affected by the method of manufacture, existing imperfections, degree of plasticity, and the method of load application. H can be evaluated by experiment.

Generic buckling of thick orthotropic cylindrical shells

AbstractA unified analytical method is presented for evaluating the generic buckling behaviours of simply-supported thick orthotropic cylindrical shells subjected to both axial compression and lateral pressure. The thick shell theory and Flugge type equations are employed. The equilibrium equations of shells become particularly simple in the chosen space by considering an appropriate transformation and utilizing some previous material constant definitions due to Brunelle. Hence, comprehensive solutions are found to this problem so that the curves presented in the text are generic rather than specific. The analysis is for the restricted case of a single homogeneous layer. The effects of various parameters on buckling loads are studied. The results indicate that the buckling coefficient increases with increasing generalised rigidity ratio D* but decreases with increasing generalised Poisson's ratio ε. The results of a reduced set of equations agree with those obtained by previous investigators.

A new approach to the asymptotic integration of the equations of shallow convex shell theory in the post-critical stage

A method is proposed for the asymptotic integration of the non-linear equations of shallow elastic shell theory on the basis of a new definition of the small parameter that is selected to be proportional to the ratio between the shell thickness and the amplitude of its deflection. This parameter is actually small if the shell is in the post-critical stage, i.e., its deflections are large. An asymptotic expansion of the solution of the shell equilibrium equations in the parameter mentioned is carried out. It is established that the first two approximations result in the geometric theory of shell stability formulated by Pogorelov /1/. By comparing the asymptotic and numerical solutions /2/ found for a spherical shell under axisymmetric deformation, satisfactory accuracy of the proposed method is obtained for fairly large deflection. The well-known Koiter approach is used in the small-deflection domain. The two asymptotic expansions, one of which is suitable for small deflections and the other for large, are merged using the Pade approximation.