General Shell Theory Research Articles

Vibrational development of nanocomposite cylindrical shells by employing reduced graphene oxide (rGO) as a nanoscale strengthener

This study uses the harmonic differential quadrature technique to learn more about the free vibration of cylinder constructions made from reduced graphene oxide nanocomposite (rGON). To support, using the Halpin-Tsai homogenization form, the effective Young's modulus related to the rGON is considered here. In addition, five distinct functionally graded (FG) distribution types of reduced graphene oxide (rGO) are examined along shell thickness. The main equations of the cylinder structures are identified by combining the first-order shear deformation theory (FSDT), the general shell theory, and Hamilton's principle. A comprehensive analysis of the impact of changing factors, such as rGO weight percent and FG distribution types, is presented. With the significant results: Regarding the rGON cylinder structure reinforced by uniformly rGO, adding 0.1%, 0.5%, 1%, and 2% rGO to the polymer matrix can cause to improving the minimum natural frequency to 1.91%, 9.23%, 17.75%, and 33.17% improvements compare with a cylindrical shell made by the pure polymer.

General theory of orthotropic shells. Part II

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 orthotropic shells general theory, which allowed in a complex form to significantly reduce the number of unknowns and the order of the system of diferential 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.

ON THE QUESTION OF THE STABILITY OF SHELLS, PLATES AND RODS

Strict, often conflicting requirements are imposed on modern engineering structures. On the one hand, they must be light, and on the other hand, their load-bearing capacity must be high. The exhaustion of the bearing capacity often occurs due to the loss of stability of structural elements: shells, plates and rods. There are no reliable formulas for determining the critical loads of plates and, moreover, shells. The purpose of this paper is to find such formulas. A new approach to solving stability problems is used, based on the assumption that at the moment of stability loss, the axial compressive force decreases. Based on this approach, by the energy method using the relations of the general linear theory of thin-walled shells, formulas were obtained for calculating the axial critical forces of cylindrical shells with hinged and rigidly embedded ends, rectangular plates and rods. The results of critical load calculations using these formulas are in good agreement with the experimental data. The results of an experimental study of the stability of cylindrical shells reinforced with stringers under axial compression are presented. Conclusions are drawn and recommendations are given on the use of the proposed approach for further study of the stability of thin-walled structures.

Assessment of refined higher order theories for the static and vibration analysis of laminated composite cylindrical shells

In the present study, a generalized shell theory is presented and applied for the analysis of laminated composite cylindrical shells. A theoretical unification of the several refined shell theories is presented. The principle of work done is employed to derive five differential equations corresponding to five unknowns involved in the present generalized shell theory. Five differential equations are solved by an analytical procedure suggested by the Navier. The numerical results for simply supported laminated composite cylindrical shells are presented and compared with 3D elasticity solutions. Displacements, stresses and fundamental frequencies are obtained for isotropic, orthotropic, 00/900 and 00/900/00 laminated cylindrical shells. The numerical results are obtained for h/a=0.1, a/b=1 and different values of R/a ratio. Displacements and stresses of laminated cylindrical shells are estimated under sinusoidal transverse load. In the case of free vibration analysis, first five natural frequencies are presented. It is observed that refined theories predicts displacements and stresses in close agreement with 3D elasticity solutions whereas the FST and the CST underpredict the displacements and stresses. It is also observed that the CST overestimates the natural frequencies due to neglect of shear deformation effect.

Flexural mode, thickness-shear mode and thickness-twist mode frequencies of laminated composite shells of double curvature

The availability of limited literature on thickness-shear mode and thickness-twist mode frequencies of laminated composite plates and shells of double curvature motivated the authors to perform this research study. Cylindrical, spherical, hyperbolic paraboloid and elliptical paraboloid laminated composite shells are analyzed using a generalized higher-order shell theory. Governing equations of motion corresponding to the present theory are derived within the framework of Hamilton’s principle. Analytical solutions are obtained using Navier’s technique for the simply supported boundary conditions. Non-dimensional flexural mode, thickness-shear mode, and thickness-twist mode natural frequencies are presented. Natural frequencies obtained using the classical, first-order, and the present higher-order shell theories are compared with each other and previously published results wherever possible to verify the accuracy and efficiency of the present generalized shell theory. Thickness-shear mode and thickness-twist mode frequencies presented in this paper can be considered as one of the important contributions towards the research in dynamic analysis of laminated composite shells of double curvature.

Unified wavelet finite element formulation for static and vibration analysis of laminated composite shells

This study presents, for the first time, a unified wavelet finite element formulation for static and free vibration analysis of laminated composite shells, combining the wavelet finite element method and general shell theory. The governing equations of the proposed unified wavelet finite element formulation are derived from general shell theory and first-order shear deformation theory (FSDT) using the principle of minimum total potential energy. The formulated kinematics of laminated shells can be applied to flat plates, cylindrical shells, and doubly-curved shells with the selection of appropriate Lamé coefficients with surface metric tensor and radii of curvature parameters. With the excellent approximation of B-spline wavelets on the interval (BSWI) for structure analysis, BSWI scaling functions are used as interpolating functions to construct wavelet finite elements for laminated shells. The accuracy and efficiency of the proposed BSWI method are assessed through numerical comparisons with analytical 3-D elasticity solutions and other reference solutions in the literature for static and free vibration analysis of laminated composite plates, cylindrical shells, and doubly-curved shells.

A variational formulation for 2-D vibro-acoustic analysis of a circular ring in unbounded domain

This paper develops a variational formulation for two-dimensional (2-D) vibro-acoustic analysis of a circular ring in unbounded acoustic domain. The Reissner–Naghdi’s generalized shell theory is adopted to formulate the energy functional of the circular ring, while the Helmholtz equation is used to derive the stored energy in the acoustic domain. A modified variational principle combined with a domain partitioning technique is adopted to formulate the coupled vibro-acoustic model. The continuity constraints on the common interfaces of adjacent acoustic partitions are imposed by using a least square weighted residual method. To satisfy the Sommerfeld radiation condition, the local absorbing boundary condition is adopted at an artificial acoustic boundary. With the spectral polynomials or series employed as the admissible functions, the structural displacements and acoustic pressure are expanded by a spectral method. The convergence, accuracy and efficiency of the present method for predicting the vibro-acoustic responses are demonstrated by comparing the obtained results with the exact solutions or FEM results. The effects of the local absorbing boundary and the radius of the exterior acoustic boundary are also investigated in both the light and heavy acoustic mediums.

Static and free vibration analysis of doubly-curved functionally graded material shells

Static and free vibration analysis of doubly-curved (cylindrical, spherical, hyperbolic paraboloid, and elliptical paraboloid) functionally graded material (FGM) shells is presented here using various equivalent single-layer shell theories considering the shear deformation and rotary inertia effects. The kinematics of the various shell theories are presented using a generalized shell theory in which the displacement field is independent of the selection of transverse shear strain function which results in a theoretical unification of various equivalent single-layer shell theories. The present generalized formulation yields traction free boundary conditions on the top and bottom surfaces of the shell using constitutive relations. Equations of motion are derived by using Hamilton’s principle which are further solved analytically using the Navier’s technique of assuming unknown variables in the double trigonometric series. Displacements, stresses, and frequencies are presented for functionally graded cylindrical, spherical, hyperbolic paraboloid, and elliptical paraboloid shells. The present results are compared with previously published results wherever possible to verify the accuracy and efficiency of the present generalized shell theory. Also, this study contributes many new numerical results which can be useful for future researchers.

ON DIFFERENT DEFINITIONS OF STRAIN TENSORS IN GENERAL SHELL THEORIES OF VEKUA-AMOSOV TYPE

Several possible definiions of strains in a general shell theory of I.N. Vekua – A.A. Amosov type are considered. The higher-order shell model is definedon a two-dimensional manifold within a set of fieldvariables of the firstkind determined by the expansion factors of the spatial vector fieldof the translation. Two base vector systems are introduced, the firs one so-called concomitant corresponds to the cotangent fibrtion of the modelling surface while the other is defind on a surface equidistant to the modelling one. The distortion appears as a two-point tensor referred to both base systems after covariant differentiationof the translation vector feld. Thus, two main definition of the strain tensor become possible, the firstone referred to the main basis whereas the second to the concomitant one. Some possible simplificationsof these tensors are considered, and the interrelation between the general theory of A.A. Amosov type and the classical ones is shown.

A General Fourier Formulation for In‐Plane and Out‐of‐Plane Vibration Analysis of Curved Beams

Based on the principle of energy variation, an improved Fourier series is introduced as an allowable displacement function. This paper constructs a calculation model that can study the in‐plane and out‐of‐plane free and forced vibrations of curved beam structures under different boundary conditions. Firstly, based on the generalized shell theory, considering the shear and inertial effects of curved beam structures, as well as the coupling effects of displacement components, the kinetic energy and strain potential energy of the curved beam are obtained. Subsequently, an artificial spring system is introduced to satisfy the constraint condition of the displacement at the boundary of the curved beam, obtain its elastic potential energy, and add it to the system energy functional. Any concentrated mass point or concentrated external load can also be added to the energy function of the entire system with a corresponding energy term. In various situations including classical boundary conditions, the accuracy and efficiency of the method in this paper are proved by comparing with the calculation results of FEM. Besides, by accurately calculating the vibration characteristics of common engineering structures like slow curvature (whirl line), the wide application prospects of this method are shown.

Vibration analysis of coupled straight–curved beam systems with arbitrary discontinuities subjected to various harmonic forces

In this paper, a modified variational method is developed to study the free and forced vibration of coupled straight–curved beam systems with an arbitrary number of eccentric discontinuities (EDs). Based on the generalized shell theory, the kinetic and potential functional of the curved beam with arbitrary subtended angles is formulated. Since the shear and inertial (or radial–tangential–rotational coupling) effects are included for the curved beam, the longitudinal vibration is also introduced to the energy functional for a straight Timoshenko beam. Using corresponding coordinate transformations, the Lagrange multiplier method and least-square weighted residual method are employed to impose the continuity constraints on the internal interfaces and boundaries among the straight and curved beams. The proposed method allows a flexible choice of the admissible functions and can be used for various combinations of the straight and curved beams to model corresponding engineering structures. Concentrated forces, uniformly distributed loads and space-dependent loads are considered to demonstrate great efficiency and accuracy of the present approach for the forced as well as the free vibration of the coupled system. Most of the present results are compared with those from finite element program ANSYS, and good agreement is observed. Influences of the EDs on the dynamic responses of the coupled system are also examined.

Toward Numerical Solution of the Problem on Stressed State of Thermoelastic Physically Orthotropic Cylindrical Shells

To investigate the stressed-strained state of thermoelastic anisotropic cylindrical shells the original system of nineteen differential equations of eighth order in partial derivatives of the general theory of shells is transformed into a system of eight differential equations of first order. It is suggested to carry out numerical realization by the method of orthogonal marching. Numerical results are presented for the case of temperature drop over the shell thickness.

Transformable Steel Dome

The developed design of the lattice shell has the ability to fold by reducing its size in one direction and increasing it in another. The principal difference of this design lies in the feasibility of folding the dome as a wholly without the need for a complete disassembly of the shell into numerous elements. The positive side of this design is the possibility of factory production of elements, speeding up the installation process and increasing mobility – the dome can be easily folded and transported to a new place. The below step-by-step description of actions of the general method of constructing the surface of the shell allows you to simulate the frame of a folding reticulate dome. Recommendations for the construction calculation contain a description of the methods used for carrying out calculations manually, bringing the arguments in favor of a computer calculation, instructions on creating a model, setting rigidity and types of finite elements, as well as taking into account nonlinearity when designing the dome in the software package LIRA-CAD. The design of the calculation scheme and the calculation itself are carried out using the general theory of shells. The field of practical implementation of the folding net dome is quite wide – it is the construction industry, defense production, agricultural industry and industry as a whole.

In-plane vibration of a circular ring with arbitrary concentrated elements by an analytical method

This paper investigates the vibration characteristics of a circular ring with an arbitrary number of concentrated elements based on the Hamilton principle. The shear and inertia effects are introduced to the variational functional of system kinetic and potential energy by adopting the generalized shell theory. The concentrated elements are treated as the concentrated masses with elastic boundary condition. The system vibration displacements are analytically expanded in the form of Fourier series and substituted into the variational functional to obtain the equation of motion. Computed results are compared with those solutions obtained from the finite element program ANSYS to validate the accuracy of the present method. Effects of the asymmetrical concentrated elements on the convergence of circumferential wavenumbers are discussed. Moreover, the coupling characteristics of different circumferential wavenumbers caused by the asymmetrical or symmetrical concentrated elements and their influences on the vibration response characteristics of the system under simple harmonic excitations are studied.

Static and free vibration analysis of laminated composite and sandwich spherical shells using a generalized higher-order shell theory

In this article, higher-order closed-form solutions are obtained for static bending and free vibration analysis of laminated composite and sandwich spherical shells using a generalized higher-order shell theory. A theory is independent of the choice of shearing stress function (polynomial/non-polynomial) which eventually results in a theoretical unification of most of the classical and higher-order shear deformation theories. The present theory yields an accurate distribution of transverse shear stresses through the shell thickness; therefore, it does not require problem dependent shear correction factor. Governing equations and associated boundary conditions of the theory are derived by employing Hamilton’s principle. Navier type higher-order closed-form solutions are obtained for simply supported boundary conditions. Displacements, stresses and natural frequencies are presented for laminated composite and sandwich plates as well as shallow and deep spherical shells. The results of parabolic, trigonometric, hyperbolic, and exponential models are compared with each other and previously published results to verify the accuracy and efficiency of the present generalized shell theory.

Convertible mesh dome

The developed design of the mesh shell has the ability to fold by decreasing its dimension in one direction and increasing in the other. The fundamental difference of this design is the feasibility of folding the dome exactly wholly without the need for a complete separation of the shell into separate elements. The positive side of using such a design is expressed in the manufacture of elements, acceleration of installation and the mobility property, in which the dome can be folded and moved to a new location. The above described step-by-step description of actions in the general method of constructing the shell surface allows one to model the skeleton of a folded mesh dome. Recommendations for calculating design include a description of the methods used to perform calculations manually, the derivation of arguments in favor of computer calculation, the instructions for creating a model, the definition of rigidity and types of finite elements, as well as accounting for nonlinearity in the design of the dome in the software LIRA-CAD. The specification of the calculation scheme and the calculation itself are carried out using the general theory of shells.

Стійкість циліндричних оболонок

Мета: Отримати більш точні формули теоретичної осьової критичного навантаження шарнірно закріпленої циліндричної оболонки; знайти причину великих відмінностей між розрахунковими і експериментальними критичними навантаженнями. Метод: Використовується енергетичний критерій стійкості і співвідношення загальної лінійної теорії тонкостінних оболонок. Результати: Отримано нові формули залежності критичного навантаження від механічних і геометричних характеристик оболонки і параметрів хвилеутворення. Значення критичних навантажень, розраховані за цими формулами, близькі до експериментальних даних. Виявлено велика залежність критичних навантажень від: відносини радіуса до товщини оболонки; відношення довжини до радіусу; граничних умов. Обговорення: Циліндричні оболонки широко використовуються в інженерних конструкціях. Втрата стійкості оболонки може привести до руйнування конструкції. Для оцінки несучої здатності інженерних споруд, необхідні точні формули для обчислення критичних навантажень оболонок при осьовому стисненні. Такі формули поки не отримані. Причину великих розбіжностей між теоретичними і експериментальними значеннями осьових критичних навантажень циліндричних оболонок не знайдено. У даній роботі зроблена спроба вирішити цю проблему. На відміну від традиційного підходу, тут передбачається, що при випинанні оболонки, відстань між її кінцями не змінюється. Такий підхід дозволив отримати формули осьових критичних навантажень які більш точно описують процес втрати стійкості циліндричної оболонки при осьовому стисненні. Аналіз отриманих результатів дозволяє зробити висновок, що отримані формули можна використовувати для реальних розрахунків критичних навантажень циліндричних оболонок, а запропонований підхід може бути використаний для продовження досліджень стійкості тонкостінних конструкцій.

A variational formulation for vibration analysis of curved beams with arbitrary eccentric concentrated elements

In this paper, a modified variational method is developed to study the free and forced vibration of curved beams subjected to different boundary conditions. An arbitrary number of eccentric concentrated elements (ECEs) attached to the beams are taken into account. A modified variational principle and least-square weighted residual method are employed to impose the continuity constraints on the internal interfaces and the boundaries of the curved beam. The shear and inertial (or radial–tangential–rotational coupling) effects are incorporated into the system kinetic and potential functional using the generalized shell theory. To test the efficiency and accuracy of the present method, both free and force vibrations of the curved beams are examined under various boundary conditions including non-classical boundary conditions. Good agreement is observed between the results obtained by the present method and those from finite element program ANSYS. Corresponding curved beams with non-eccentric concentrated elements are also developed to investigate the effects of the ECEs on the vibration behaviors of the curved beams.

A MITC-based procedure for the numerical integration of a continuum elastic-plastic theory of through-the-thickness-jacketed shell structures

Through-the-Thickness Jacketing (TTJ) is a technique for repairing and retrofitting shell structures by inducing in the shell core a beneficial confining stress state created by a net of broadly distributed retrofitting links crossing the shell thickness and tying externally applied layers. The paper presents the derivation, the algorithmic implementation and the numerical assessment of a predictor-corrector computational strategy for the integration of a shell FE-model obtained by combining a discrete MITC quadrilateral element with a layered continuum-based generalized shell theory of TTJ-reinforced structures, essentially based upon a Winkler-like idealization of TTJ. This theory of Through-the-Thickness-Jacketed Shells (TTJS) captures the onset of complex triaxial stress states originated by the interaction between core and TTJ reinforcements.Results of benchmark numerical applications in OpenSees with flat and curved elastic-plastic shell structures are presented in order to assess and illustrate the consistency and the general modelling features of the proposed TTJS-MITC framework endowed with the Drucker-Prager elastic-perfectly-plastic idealization of the nonlinear behavior of the material composing the shell. Numerical results exhibit quadratic convergence and show that the model captures marked strength increments over the in-plane membrane response, albeit these are lower when the response is predominantly of out-of-plane flexural type.

A Geometric Theory of Nonlinear Morphoelastic Shells

Many thin three-dimensional elastic bodies can be reduced to elastic shells: two-dimensional elastic bodies whose reference shape is not necessarily flat. More generally, morphoelastic shells are elastic shells that can remodel and grow in time. These idealized objects are suitable models for many physical, engineering, and biological systems. Here, we formulate a general geometric theory of nonlinear morphoelastic shells that describes both the evolution of the body shape, viewed as an orientable surface, as well as its intrinsic material properties such as its reference curvatures. In this geometric theory, bulk growth is modeled using an evolving referential configuration for the shell, the so-called material manifold. Geometric quantities attached to the surface, such as the first and second fundamental forms, are obtained from the metric of the three-dimensional body and its evolution. The governing dynamical equations for the body are obtained from variational consideration by assuming that both fundamental forms on the material manifold are dynamical variables in a Lagrangian field theory. In the case where growth can be modeled by a Rayleigh potential, we also obtain the governing equations for growth in the form of kinetic equations coupling the evolution of the first and the second fundamental forms with the state of stress of the shell. We apply these ideas to obtain stress-free growth fields of a planar sheet, the time evolution of a morphoelastic circular cylindrical shell subject to time-dependent internal pressure, and the residual stress of a morphoelastic planar circular shell.