Orthotropic Shells Research Articles

An analytical tire model using thin shell theory

Tire vibrations play an important role in determining vehicle ride characteristics. Tires are complex inflated structures making them difficult to model analytically. Due to this modeling complexity, a majority of researchers have resorted to numerical finite element (FE) based models. The aim of this work is to develop an analytical understanding of how tire constructional (aspect ratio, diameter, tire width, tread depth, stacking sequence) and operational properties (inflation pressure) affect its modal characteristics. The tire is represented as an inflated orthotropic toroidal shell with an elliptical cross-section. We begin with Sanders’ shell theory to formulate the governing equations as a set of partial differential equations. These are further reduced to an eigenvalue problem and solved using Galerkin projection. In our approach, tire properties are retained as free parameters facilitating quick parametric studies to estimate modal properties not possible with time-evasive FE models. The tire considered in this study is a 205/55 R16 aftermarket tire, a four-layered stacked composite. Tire material properties were obtained through a tensile test experiment. For model validation, a detailed FE model was prepared in ABAQUS and experimental modal analysis was also performed. The model shows a good agreement against both FE simulations and experiments.

GENERAL THEORY OF ORTHOTROPIC SHELLS. PART I

Modern mechanical engineering sets the tasks of calculating thin-walled structures that simultaneously combine sometimes mutually exclusive properties: lightness and economy on the one hand and high strength and reliability on the other. In this regard, the use of orthotropic materials and plastics seems quite justified.The article demonstrates the complex representation method of the equations of the orthotropic shells general theory, which allowed in a complex form to significantly reduce the number of unknowns and the order of the system of differential equations. A feature of the proposed technique for orthotropic shells is the appearance of complex conjugate unknown functions. Despite this, the proposed technique allows for a more compact representation of the equations, and in some cases it is even possible to calculate a complex conjugate function. In the case of axisymmetric deformation, this function vanishes, and in other cases the influence of the complex conjugate function can be neglected.
 Verification of the correctness of the proposed technique was demonstrated on a shallow orthotropic spherical shell of rotation under the action of a distributed load. In the limiting case, results were obtained for an isotropic shell as well.

REFINED DISCRETE METHOD FOR CALCULATING STIFFENED ORTHOTROPIC SHELLS

The author proposes a refined discrete method for taking into account stiffeners in the simulationof thin-walled shell structures. According to the method, it is necessary to add different reduction factors along different coordinate axes. For ribs directed perpendicular to the considered direction, a reduction factor is introduced equal to the ratio of the width of the ribs in this direction to the linear size of the shell in the considered direction. This method supplements the previously developed geometrically nonlinear mathematical model, which takes into account transverse shears and material orthotropy. The model is written as a functional of the total potential strain energy and can be used for different types of shells by specifying the Lame parameters and the radii of principal curvatures. The computational algorithm is based on the Ritz method and the method of continuation of the solution with respect to the best parameter. The software imple-mentation was carried out in the Maple software package. The applicability of the refined discrete method is shown by the example of orthotropic shallow shells of double curvature, simply supported along the contour and under the action of an external uniformly distributed transverse load. Material parameters were selected for T-10/UPE22-27 and 0/90 Woven Roving E-Glass/Vinyl Ester fiberglass. A comparison was made of the values of critical buckling loads for different stiffening options (a grid of ribs from 0 to 12 ribs in each direction) and a comparison of the values with the conventional discrete method, which showed that with the conventional discrete method, the values of buckling loads are significantly overestimated, especially with an increase in the number stiffening ribs. Comparison of the results of the test problem with the results of experiments obtained by other authors showed good agreement between the refined discrete method.

Simplification of the calculation scheme for determining the stress state of non-thin cylindrical shells with a complex cross-sectional shape ellipticity

An alternative to full-scale experiments is computer simulation, which allows studying a variety of states, phenomena, processes, etc. occurring in the environment.
 Conducting a computational experiment is an integral part of the design phase of new structures and their elements. One of the important issues is the choice of research model, feasible calculation scheme and possibility of its simplification.
 This research investigates orthotropic non-thin cylindrical shells with corrugated ellipses as cross sections, which has the two-parameter deviation of the cross-section shape from a circular one. Considered are shells, for which the cross-section curvature radius of the reference surface has the positive sign. The shells are subjected to internal pressure under conditions of simple support on the ends.The subject of the study is the stress state of shells and, as a consequence, the establishment of the relationship between the geometric parameters of the reference surface of the cross sections and the possibility of simplifying the calculation scheme (excluding from consideration the parameter characterizing ellipticity).The problem is solved using the spatial model of linear elasticity theory based on the method of approximation of functions by discrete Fourier series.For the class of considered shells, we find the limits of possible simplification of the calculation scheme during calculations on durability with use of the fourth theory of durability (the theory of the greatest specific potential energy), due to exclusion from consideration of the parameter, which characterizes ellipticity of corrugated cylindrical shells.The cross-sectional radius of curvature of the reference surface in the zone of greatest rigidity was chosen as a criterion for the possibility of using a simplified scheme. It is found that a simplified scheme can be used when the cross-section curvature radius of the reference surface of elliptical corrugated cylindrical shells in this section differs from that for circular corrugated shells by no more than 17%.

Identification of the patterns of influence the number of reinforcing elements and the inhomogeneity parameter of the shell material on frequencies of a reinforced inhomogeneous orthotropic spherical shell with a medium

Spherical shells are used in many areas of the national economy. Spherical domes are widely used in the construction of various structures (technoparks, testing laboratories, entertainment complexes, reservoirs, etc.). They are also used in aircraft, ship structures, radar antennas and other structures. It is known that coatings have sufficient strength and durability even with a small thickness. However, to increase the working life of coatings, to ensure their long-term operation, as well as to increase their hardness, it is necessary to strengthen them on the surface or inside with rods. Sometimes it is possible to reduce the weight of the structure and save material consumption by strengthening it with. One of the advantages of these structures is that they give the maximum useful volume, being both load-bearing and enclosing structures. Checking the shells for stability is a priority task, since it is known that the shells, even with an insignificant thickness, have great strength and therefore their insufficient stability can be a criterion determining the bearing capacity. This article is devoted to identifying the regularities of the influence of the number of reinforcing elements and the inhomogeneity parameter of the shell material on the frequencies supported by an inhomogeneous orthotropic spherical shell with a medium. To solve the problem under consideration, the Hamilton-Ostrogradsky variation principle is applied. The frequency equation is constructed and implemented numerically. Such studies have not been considered for a reinforced spherical shell with a no uniform filler in thickness

Free Oscillations of an Orthotropic Conical Shell

Free Oscillations of an Orthotropic Conical Shell

A comprehensive study on vibration characteristics of corrugated cylindrical shells with arbitrary boundary conditions

The vibration characteristics of corrugated shells under arbitrary boundary conditions are investigated by theoretical analysis, finite element simulation and experimental verification. Firstly, based on a homogenization analytical technique, the equivalent orthotropic shell model is set up. Then, the strain-displacement relationships are received by Love thin shell theory, where artificial springs are applied to denote the arbitrary boundary conditions. Afterwards, modified Fourier series are selected to represent the displacement fields, and the Rayleigh-Ritz method is employed to acquire natural frequencies of corrugated shells. The modal expansion approach is applied to gain forced vibration responses. Finally, the effects of unit cell half period, unit cell radius, unit cell height and different boundary conditions on vibration characteristics of corrugated shells are studied.

Анизотропные твердые цилиндрические волноводы

Object and purpose of research. The article studies the behavior of anisotropic elastic bodies of cylindrical shape (orthotropic shell and transversely isotropic rod). The intention is to find the phase velocities of elastic waves in these bodies using thin shell approximations and a rigorous approach based on the dynamic theory of elasticity using "Debye type" potentials. In previous studies of anisotropic structures, anisotropic media or anisotropic half-spaces were used. Materials and methods. Both an approximate thin shell method and a rigorous approach based on the dynamic theory of elasticity and "Debye type" potentials are used in the paper. Main results. Equations for finding the phase velocities of elastic waves in anisotropic cylindrical bodies are obtained. Phase velocities of longitudinal and bending waves in anisotropic cylindrical rod are calculated. Conclusion. As a result of the conducted research relations for calculating the phase velocities of elastic waves in an orthotropic shell and a transversely isotropic rod were found.

Efficient Model of the Interaction of Elastomeric Filler with an Open Shell and a Chrome-Plated Shaft in a Dry Friction Damper.

The results of a study of the contact interaction of an open shell and a chrome-plated shaft with elastomeric filler installed coaxially are presented. The considered contact system is a model of the original design of the shell damper of dry friction. The design feature is the following: the bearing link of the damper is a thin-walled cylindrical shell with a cut along the generatrix; the working body of the damper is elastomeric filler; a hollow chrome-plated shaft centers the damper elements and allows it to be used in technological processes with the presence of aggressive and abrasive-containing media. The mechanical-mathematical modeling of the behavior of the presented damper under the conditions of operational loads has been carried out. The idea of identifying the properties of a cut isotropic shell, which bends under the conditions of a nonaxisymmetric contact load, and a strongly orthotropic continuous shell is applied. As a result, dependences were obtained to determine the rigidity and the maximum allowable load of the damper. The effect of the coefficient of friction of the contact pairs elastomer-shell and elastomer-shaft on the damper performance properties has been studied. A technique for the quasi-static analysis of structural damping in non-mobile, non-conservative shell systems with deforming filler has been developed. The hysteresis loops of the damper under a nonmonotonic load are constructed, the dependence of the amount of dissipated energy on the cycle asymmetry coefficient is found. An analysis of the results obtained showed that the use of open shells in friction shock absorbers can significantly reduce their rigidity compared to solid shells and thereby reduce the resonant frequencies of the dynamic system. This circumstance makes such vibration isolators particularly attractive for use in superresonance vibrators as working modules of drilling shock absorbers and elastic hangers of sucker rods in oil and gas production.

Semi-analytical model of noise transmission through finite length cylinder with poroelastic core using Biot’s and Shell theories

In the open literature, sound transmission through cylindrical structures is often considered for infinitely long cylinders for which full analytical solutions can be derived. However, when dealing with finite cylinders subjected to arbitrary boundary conditions, establishing a general analytical solution is more complex to achieve and appropriate resolution techniques should be used. In the present study, a consistent semi-analytical model of sound transmission through such finite length and multilayered cylinders is presented. The orthotropic shell is modeled with 2D first order shear deformation theory and the poroelastic core is modeled with full 3D Biot’s theory in u-p format. Analytical expressions are used for the acoustic domains while closed form Rayleigh–Ritz expansion and 1D finite element method are used for the structural domains. Results are validated with those of literature and those given by full numerical simulations. Furthermore, the effect of structural and acoustical resonances on noise transmission are discussed through finite/infinite cylinders superposition and with respect to angle of incidence and porous layer thickness. The contribution of internal resonances demonstrates the importance of the cylinder finiteness for oblique incidences.

Geometrically nonlinear electro-thermo-static analysis of piezoelectric/CNT/GPL/fibre/polymer sandwich panels with double curvature resting on elastic foundation

Evaluation of the electro-thermo-static behaviour of the shell panels of double curvature using an analytical method is carried out in this paper. The proposed panels are made of a hybrid composite, and considered to be rested on the Winkler-Pasternak elastic foundation. The hybrid composite shell contains two thin piezoelectric sheets at its bottom and top surfaces, and some plies made of carbon fibres and polymeric resin which is reinforced by the randomly oriented nanocomposites. The employed nanocomposites for some layers are carbon nanotubes (CNTs) and for some other are graphene nanoplatelets (GPLs). The governing nonlinear system of differential equations of orthotropic doubly-curved shell panels are developed based on the first-order transverse shear deformation theory (FSDT) which incorporates moderately thick and also thin shells. The Gauss Quadrature method is applied to obtain the forces and moments due to the thermo-electrical loads, and the Galerkin method is employed to solve the developed system of equations which leads to a closed-form load-deformation relationship (equilibrium path of the shell). The novelty of the present study can be expressed in the solution and employed hybrid composite materials. First, by employing the proposed closed-form solution no time consuming computations are required. On the other hand, the effects of the piezoelectric and two different nanocomposite materials on the thermo-electro-mechanical nonlinear static behaviour of the hybrid composite shells can be evaluated. The accuracy of the present method is validated by comparing the achieved results for the some benchmark problems with those are existent in the literature. The impacts of various electro-thermo-mechanical and geometrical characteristics of the proposed shells on their equilibrium paths are also considered by carrying out an extended parametric study.

Theoretical and experimental investigation of circumferential guided waves in orthotropic annuli

The fiber composite materials fabricated by stacking the lamina structure obtain increasing attention in the industry, and at the same time, damage analysis is essential for safe operation. A proposed way to detect defects such as debonding and delamination is to induce circumferential guided waves. Accurate determination of the dispersion characteristics and wave fields is a prerequisite for using acoustic guided waves. The aim of this work is to provide such an accurate dispersion results in an arbitrarily thick cylindrically orthotropic homogeneous cylindrical shell of uniform thickness. Compared with the traditional SAFE method, the proposed method, which combines the two numerical methods, namely Floquet Boundary Condition (Floquet BC) Method and Sweeping Frequency Finite Element Modeling (SFFEM) method has obvious advantages. Through Floquet BC method, a theoretical result can be easily obtained without tedious code writing. Then SFFEM helps to verify the theoretical result in an experimental way. Besides, the slight gap between the theoretical and experimental results works as a basis for calibrating the elastic constants of composite materials provided in the manual. Simulations in COMSOL Multiphysics FE software supported by both methods and experiments by the latter have been carried out in this work. The calculated phase velocity dispersion curves demonstrated the efficiency of those two methods. The developed method can be adapted to other complex pipeline structures to extract circumferential guided wave dispersion characteristics by both simulation and experimental measurements.

Non-linear large amplitude vibration of orthotropic FGM convex and concave toroidal shell segments including the damping effect using the shear deformation theory

The aim of this paper is to investigate the vibration behavior of a functionally graded orthotropic toroidal shells segments (FGOTSSs), such as convex and concave shell segments, at large amplitudes. Material properties presumed to vary in thickness directions according to the exponential functions. The non-linear motion equations are conducted established on shear deformation theory in conjunction with Stein and McElman’s assumption and von-Karman type of nonlinearity. The technique of superposition and the Galerkin approach was used to transform the nonlinear partial differential equations of FGOTSSs to the nonlinear ordinary differential. Then, utilizing the harmonic equilibrium method, nonlinear equations are solved to extract the expression for frequency amplitude relationships and the non-linear frequency to linear-frequency ratio of FGOTSSs. The results are compared to those found in the literature with the expressions, and the precise of numerical calculations are performed after testing the proposed formulation’s reliability and accuracy. The homogeneous orthotropic and orthotropic FGM of the convex and concave shell segments are used for basic analyzes, and their vibrational behaviors with large amplitudes are examined in contrast to each other, with different examples revealing that heterogeneity has an effect.

Global dynamics of a structurally orthotropic stringer shell system

In this paper we are interested in a structurally orthotropic stringer shell system, which can be transformed into a planar reversible polynomial Hamiltonian system via the Galerkin method. Numerous articles have employed numerical methods to study its dynamics, but only partial dynamical behaviors are obtained due to the limitation of numerical methods. To capture all possible dynamical behaviors of this shell system, we resort to the Poincaré compactification method and the homogeneous directional blow up technique in the qualitative theory of dynamical systems. After reducing the original system to a normal form with less parameters, we prove that this shell system exhibits exactly 14 non-equivalent global dynamics. Moreover, the parameter range corresponding to each kind of global dynamics is accurately determined, and bifurcation diagrams are drawn.

Mathematical analogies in solving problems of strength, stability and vibrations of orthotropic plates and shells

Mathematical analogies in solving problems of strength, stability and vibrations of orthotropic plates and shells

Eigenfrequencies and eigenmodes of anisotropic thick cylindrical shell

Thick cylindrical shells are used in various applications, and there are many studies on strength and vibration analysis. In vibration analysis of isotropic and orthotropic thick cylindrical shells, it is commonly assumed that displacements are affected by rotational inertia and shear deformation and the shell thickness does not change during deformation. On the other hand, vibration analysis of orthotropic cylinders has been studied based on three-dimensional analysis theory without these assumptions. However, vibration analysis of thick cylindrical shells based on three-dimensional analysis theory has not been conducted. Previously, we studied an orthotropic parallelepiped as a single continuum and reported the relationships between dimensions and natural angular frequencies and vibration modes. In this report, a thick cylindrical shell with deformation in thickness direction and orthotropy is investigated. The natural angular frequencies and modal functions of the axisymmetric vibration are derived using three-dimensional analytical theory. In the calculation example, the relationship between dimensionless axial wavenumber and natural angular frequencies and modal functions of the model with varying thickness ratios are illustrated. The frequency veering of natural angular frequencies occurs, and the wavenumber where the frequency veering occurs depends on the thickness ratio. The number of radial nodes varies independently of the frequency veering, and the sign of the slope of radial node vs. dimensionless wavenumber changes around the wavenumber where veering occurs. The number of axial nodes and their positions vary only around the value of the wavenumber where veering occurs.

Determination of natural vibration frequencies of reinforced cylindrical shell

Free vibrations of a reinforced cylindrical shell filled with liquid are investigated. The case of an orthotropic shell is considered when the cord filament is placed symmetrically with respect to the meridian of the shell. The motion of a fluid is potential and is described by a wave equation. The fluid moves without separation from the walls of the cylinders. The fluid pressure is taken into account in the equations of motion of the shells, and the velocities of the fluid and the shell are equalized at the boundaries. Representing a solution in a harmonic form reduces to a system of transcendental equations. Comparison of the solutions of the problems without a liquid and with a liquid shows the dependence of the frequency of the system without a liquid at the frequency of the system with the liquid. An inverse method is proposed for solving the equation. The inverse method for solving the problem has made it possible to construct a more accurate frequency spectrum of free oscillations of the system. For some values of the system parameters, the natural frequencies of the cylinder are determined.

A symplectic analytical approach for free vibration of orthotropic cylindrical shells with stepped thickness under arbitrary boundary conditions

Exact analytical solutions for free vibration of isotropic and orthotropic cylindrical shells with uniform and stepped thickness subject to general boundary conditions are presented by means of a symplectic analytical approach. The Reissner shell theory is adopted to formulate a theoretical model. By introducing a Hamiltonian system, the governing higher order partial differential equation is reduced to a set of ordinary differential equations which can be analytically solved by separating the variables. Applying the end boundary and interface continuous conditions, a set of analytical characteristic frequency equations are obtained, and exact solutions can be determined. To ensure accuracy and validity of the symplectic method, the analytical solutions for uniform and stepped cylindrical shells with isotropic or orthotropic material properties and with arbitrary boundary conditions are compared with available published data and FEM solution. A set of comprehensive new result for orthotropic stepped cylindrical shells under classical and elastic restraints are presented. Some typical mode shapes for various examples are illustrated. In addition, the effects of boundary conditions and orthotropic properties on vibration frequency are analyzed. The result shows that the fundamental frequency is higher for a boundary with more constraints.

Deformation of Orthotropic Toroidal Shells of Superelliptic Cross-Section*

Deformation of Orthotropic Toroidal Shells of Superelliptic Cross-Section*

Axial tension of nonlinear elastic composite shells of zero Gaussian curvature with a rectangular hole

The formulation of physically nonlinear problems for composite shells of zero Gaussian curvature weakened by a rectangular hole under the action of axial loading is given. The initial equations are the equations of the theory of non-sloping shells, in which the Kirchhoff–Love hypotheses take place. Geometric relationships are written in vector form, and physical relationships are based on the deformation theory of plasticity for anisotropic materials. The system of resolving equations is obtained from the Lagrange variational principle. A technique has been developed for the numerical solution of two-dimensional physically nonlinear problems for orthotropic composite shells of this type, based on the use of the method of additional stresses and the method of finite elements. A variant of the finite element method is proposed, the peculiarity of which lies in the vector approximation of the sought values and the discrete execution of the geometric part of the Kirchhoff–Love hypotheses (at the nodes of finite elements). Using the developed technique, the nonlinear elastic state of an organoplastic conical shell with a rectangular hole, which at the ends is reinforced with frames and loaded with uniformly distributed tensile forces, has been investigated.