Classical Shell Research Articles

Vibration Analysis of Porous Functionally Graded Material Truncated Conical Shells in Axial Motion

In this study, a vibration equation for axially moving truncated conical thin shells made of functionally gradient materials with uniformly and non-uniformly distributed pores has been established based on classical thin shell theory. The free vibration and dynamic response solutions are obtained using the Galerkin method. The effects of axial velocity, half cone angle, ceramic material mass composition, material component index, and internal porosity on the free vibration and dynamic response of mentioned shells were analyzed and discussed. The results show that the increase of axial velocity, half cone angle, and material composition index all decrease the natural frequency of the truncated conical shell but amplify its dynamic response, and the rise of the mass fraction of the ceramic material increases the natural frequency of the truncated conical shell but reduces the dynamic response. The results also demonstrate that compared with non-uniformly distributed pores, the effects of uniformly distributed pores on the shells’ dynamic responses are more evident under axial motion.

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.

Free vibration analysis of functionally graded shallow shells with variable thickness and physical neutral surface effects

The study and manufacture of functionally graded (FG) shallow shells with the variable thickness are essential for the actual application of these types of structures. Previously, many researchers have developed different approaches to analyzing the mechanical behaviors of the FG structures, in which the thickness parameter has been assumed to be constant. Moreover, studies on these types of structures with the variable thickness are still quite limited. For this reason, this paper provides an effective and simple approach via mathematical formulations for analyzing the free vibrational responses of FG shallow shells with the variable thickness. Also, the presence of porosities that can occur during the fabrication of FG materials is considered. One of the highlights here is to consider the influence of the physical neutral surface. Before conducting the analysis of these FG structures, firstly, it is important to establish governing equations based on the classical shell theory. These essential equations are used for the FG shallow shells. Then, by setting the Navier solution with the help of simply supported boundary conditions, the natural frequencies are obtained. Finally, the numerical examples will be given to illustrate the effects of porosity coefficient, material and geometry parameters as well as the physical neutral surface on the natural frequencies. In addition, two numerical examples for verification will also be performed and thereby show the accuracy of the current method and model. It is hoped that the main results of this study can be useful not only to designers and researchers but also to be extended to more complex problems.

Analytical and numerical buckling of bi-material pipelines reinforced by novel composite polyhedral linings

This study concerns the mechanical stability performances of a composite lining constrained in a cracked pipeline. A polyhedral cross-section is used to improve the load capacity of the tightly constrained lining. The lining deflects inward due to the confinement of the pipeline. The contact surface is frictionless between the lining and the pipeline. A periodic variation is found for both the radius and bending rigidity of the cross-section of the polyhedral lining, which is approximated analytically to simplify the derivation. By adopting the classical shell criteria, and a trigonometric function to express the radial deflection, the potential energy function is expressed, yielding two formulae of equilibrium. Then, the path of equilibrium is illustrated analytically for a composite polyhedral short lining under a point load. The present analytical predictions are compared with the other two results, and good agreements are obtained. A profile factor is introduced by defining the buckling load of the polyhedral lining over the circular one. Finally, parametric evaluations indicate that the profile factor reduces with the rise of thickness, and side numbers, respectively. Therefore, a lining with a thin-walled polyhedral cross-section is recommended in engineering practices.

Analyzing vibrational characteristics of fluid-filled symmetric cross-ply delaminated composite cylindrical shells

This study investigates the vibration behavior of delaminated cylindrical shells composed of symmetric cross-ply composite material are partially or completely filled with fluid. An incompressible, non-viscous and irrotational internal fluid is considered. The classical shell theory and Helmholtz’s equation with the velocity potential function are employed to derive the governing equation for the vibration of delaminated symmetric cross-ply laminated cylindrical shells. The primary focus is to understand the impact of varying fluid heights, densities and velocities on the natural frequencies of the shells, as well as the effects of shell length, thickness-to-radius ratio, and delamination’s position in the thickness and circumferential directions. The study findings indicate that the natural frequencies decrease considerably as the height of the fluid increases and that the delamination position significantly influences the observed frequency variations. Moreover, the internal fluid density and shell length affected natural frequencies. This research contributes not only to our knowledge of the behavior of cylindrical shells with delamination and fluid filling but can also assist in designing structures that mitigate undesirable vibrations and promote the creation of more efficient and dependable structures.

Investigation of sound transmission through a finite length cylindrical shell in the presence of an exterior turbulent boundary layer

This paper studies sound transmission through a thin cylindrical shell of finite length. The structure is subjected to an exterior turbulent boundary layer (TBL) excitation and the boundary conditions of its two ends are considered as simply-supported. The classical shell theory (CST) is employed to derive wave propagation in the cylindrical shell, and the Corcos model is applied to describe pressure fluctuation due to TBL. According to the existence of two axial and radial modes, a complete convergence study is presented. To examine sound transmission through the structure, a comprehensive parameters study is provided on the resulting power spectral density (PSD) of the kinetic energy. To validate the results of the present method, the statistical energy analysis (SEA) method is used. For this purpose, modeling is performed by SEA in VA One software, which shows the high accuracy of the results of the present method. The analytical predictions suggest that the sound insulation performance of such a finite length structure is enhanced considerably by increasing the thickness of the shell in a broad-band frequency range. Similar effects on sound transmission are observed for Young’s modulus and density of the shell particularly in the low and high-frequency ranges, respectively. Whereas, enhancing the turbulent pressure and freestream velocity adversely affects sound radiation into the structure.

Vibration and thermomechanical transient response of doubly curved FGM panels with porosities and elastically restrained edges

The present paper aims to analyze the influences of porosities, elevated temperature and elasticity of in-plane boundary conditions on the free vibration and nonlinear transient response of functionally graded material (FGM) doubly curved panels subjected to uniform lateral pressure. Porosities are assumed to exist in the FGM according to even and uneven distributions. Effective properties of porous FGM are determined using a modified rule of mixture. Basic equations in terms of deflection and stress function are derived basing on the classical shell theory taking into account von Kármán–Donnell nonlinearity and initial geometrical imperfection. Analytical solutions for simply supported panels are assumed and Galerkin method along with Runge–Kutta scheme are applied to obtain the deflection–time response for nonlinear dynamic analysis. Numerical results demonstrate that tangential constraints of edges have extremely significant influences on the vibration and transient response of porous FGM panels exposed to an elevated temperature. The analysis also reveals that porosities have negative and positive effects on the dynamic response of FGM panels at reference and elevated temperatures, respectively. Furthermore, when the edges are restrained, initial imperfection importantly influence the behavior tendency of porous FGM panels exposed to preexist thermal environment.

Simple first-order shear deformation theory for free vibration of FGP-GPLRC spherical shell segments

This study presents a free vibration analysis of spherical shell segments using simple first-order shear deformation shell theory (S-FSDT) for the first time. The shell structure is made of functionally graded porous graphene platelet reinforced composite (FGP-GPLRC) – one kind of porous material strengthened by graphene platelets (GPLs). Effective material properties of the FGP-GPLRC are determined by the modified Halpin-Tsai micromechanical model and the mixture rule. Four types of porosity distributions and GPL dispersions are considered fully in this study. The governing equations of the shell are derived based on the S-FSDT and classical shell theory, then solved by the well-known Rayleigh-Ritz method and the artificial spring technique. The results show that the S-FSDT can capture well the behaviors of the spherical shell segments. Besides, the best profile of novel FGP-GPLRC is not fixed; the physical parameters, geometric parameters, and material characteristics of the shells need to be investigated fully.

Stability Analysis of Shear Deformable Inhomogeneous Nanocomposite Cylindrical Shells under Hydrostatic Pressure in Thermal Environment

In this study, the stability of inhomogeneous nanocomposite cylindrical shells (INCCSs) under hydrostatic pressure in a thermal environment is presented. The effective material properties of the inhomogeneous nanocomposite cylindrical shell are modeled on the basis of the extended mixture rule. Based on the effective material properties, the fundamental relations and stability equations are derived for thermal environments. In this process, the first-order shear deformation theory (FSDT) for the homogeneous orthotropic shell is generalized to the inhomogeneous shell theory. This is accomplished using the modified Donnell-type shell theory. The analytical expressions are obtained for hydrostatic buckling pressure of INCCSs in the framework of FSDT and classical shell theory (CST) by obtaining a solution based on Galerkin’s procedure. The numerical examples presented include both comparisons and original results. The last section shows the influences of carbon nanotube (CNT) models, volume fraction, and shell characteristics on the hydrostatic buckling pressure in the thermal environment.

A hybrid shell-beam element for straight thin-walled tubular structures

A novel beam model is proposed in order to consider deformations of its cross-section in the context of straight thin-walled tubes. For this purpose, the straight beam kinematics is enriched by addition of orthogonal shell-type displacement field of the tube section. Linear strain in terms of displacement is considered in conjunction with the non-linear coupling between local deformation of the section and its global rotation. Both Euler–Bernoulli and Love-Kirchhoff hypotheses for beam and shell kinematics, respectively, are adopted as well as the thin-walled assumption. First-order shear deformation for the radial variable and Fourier expansion in terms of the circumferential variable are also considered for the mid-surface displacement field. Then, the stress tensor is obtained under plane stress conditions. The virtual power principle is finally used to obtain the equations of motion satisfied by the corresponding generalized forces. Afterwards, an explicit updated Lagrangian Finite-Element approach using a lumped mass matrix is proposed for solving the tube governing equations and the stability condition of the time integration is given. Test-cases are then chosen to assess the present tube finite-element. Both static and dynamic problems are considered. First, the proposed model is compared to analytical solutions. Finally, a tube subjected to a distributed patch loading is studied. The influence of the number of Fourier modes, of warping and coupling terms is examined. The proposed model makes it possible to retrieve classical shell solution of the cross-section deformation with significant computational savings.

A Comparison of Shell Theories for Vibration Analysis of Single-Walled Carbon Nanotubes Based on an Anisotropic Elastic Shell Model.

In the present paper, a comparison is conducted between three classical shell theories as applied to the linear vibrations of single-walled carbon nanotubes (SWCNTs); specifically, the evaluation of the natural frequencies is conducted via Donnell, Sanders, and Flügge shell theories. The actual discrete SWCNT is modelled by means of a continuous homogeneous cylindrical shell considering equivalent thickness and surface density. In order to take into account the intrinsic chirality of carbon nanotubes (CNTs), a molecular based anisotropic elastic shell model is considered. Simply supported boundary conditions are imposed and a complex method is applied to solve the equations of motion and to obtain the natural frequencies. Comparisons with the results of molecular dynamics simulations available in literature are performed to check the accuracy of the three different shell theories, where the Flügge shell theory is found to be the most accurate. Then, a parametric analysis evaluating the effect of diameter, aspect ratio, and number of waves along the longitudinal and circumferential directions on the natural frequencies of SWCNTs is performed in the framework of the three different shell theories. Assuming the results of the Flügge shell theory as reference, it is obtained that the Donnell shell theory is not accurate for relatively low longitudinal and circumferential wavenumbers, for relatively low diameters, and for relatively high aspect ratios. On the other hand, it is found that the Sanders shell theory is very accurate for all the considered geometries and wavenumbers, and therefore, it can be correctly adopted instead of the more complex Flügge shell theory for the vibration modelling of SWCNTs.

Stability of submarine bi-material pipeline-liner system with novel polyhedral composites subjected to thermal and mechanical loading fields

This study is concerned with the thermal and mechanical instability behaviors of composite novel liners encased in deteriorated pipelines. A liner may contain many connected segments, and two adjacent segments may become disconnected after long years of service. In such a case, the single disconnected segment may reduce to a ring. An innovative polyhedral configuration is introduced to improve the bending stiffness of the composite ring that is confined by the pipeline. The radius and bending rigidity of the ring are simplified analytically to facilitate the derivation of the critical buckling load. By employing the classical shell criteria, and defining an admissible displacement function, the expression of the potential energy function is obtained with only two unknown parameters. Taking the first derivative of the energy function to the two unknowns respectively generates two equilibrium equations. By solving the two equations, the analytical buckling load is obtained for a composite polyhedral ring in a thermal variational field. Then, two comparisons are taken between the present analytical predictions and results from elsewhere, and good agreements are obtained. An amplification coefficient is defined as the ratio between the buckling load of the polyhedral and circular rings. Finally, parametric evaluations indicate the amplification coefficient reduces with the increase of thickness-to-radius ratio, the increase of the number of sides, and the increase of the temperature variation, respectively. Therefore, a polyhedral ring with a low thickness-to-radius ratio is recommended in engineering applications.

Buckling behavior analysis of hybrid-honeycomb sandwich cylindrical shells

In this research, the buckling behavior of hybrid-honeycomb sandwich (HHS) cylindrical shells under hydrostatic pressure is investigated. The HHS shells consist of two thin ceramic-metal functionally graded (FG) layers on the outer and inner surfaces, and a layer of lightweight, high-strength, low-density hybrid-honeycomb material in the center, with a thickness that is larger compared to the FG layers. The ceramic composition of the FG layers covers the entire outer surface of the cylindrical shells, which is intended to improve the performance of structures in highly corrosive and acidic environments, such as the ocean, and in other mining and metallurgical industries. The hybrid-honeycomb core has a negative Poisson's ratio, also called as auxetic materials, which have a unique microstructure that can affect the buckling behavior of cylindrical shells made from these materials, resulting in improved buckling resistance, stability, and enhanced energy absorption. Using classical shell theory, the governing equations for the buckling behavior problem are derived, and the critical buckling loads are then gained using the Galerkin solution. The results of the present study are validated through comparison with existing literature, and numerous numerical examples are conducted to showcase the interesting results of the study with various material and geometric parameters.

Vibration analysis of sandwich cylindrical shells made of graphene platelet polymer–viscoelastic–ceramic/metal FG layers

In this paper, natural frequencies and loss factors of cylindrical sandwich shells composed of the viscoelastic core layer, surrounded by functionally graded graphene-platelet reinforced polymer composite (FG-GPLRPC) and ceramic/metal (FG-ceramic/metal) are investigated. The viscoelastic layer is modeled via the fourth parameter fractional viscoelastic pattern, and the functionally graded ceramic/metal layer is theoretically modeled using a power-law function. The uniform, symmetric and un-symmetric patterns are considered for simulating the graphene platelet (GPL) nanofillers distributions along with the thickness direction. The classical shell theory is used for functionally graded layers and properties of the effective materials of GPLRPC multilayers are determined by using a modified Halpin–Tsai micromechanics model and the rule of mixture. The governing equations of motion are extracted by applying the Lagrange equation and the Rayleigh-Ritz method. The determinant of the coefficient matrix of the characteristic equation is calculated, and the natural frequencies and loss factors of the system are extracted. A study of the interactions of materials and geometrical factors such as the ratio of radius to length, the properties of functionally graded materials, and GPL weight fractions for patterns of proposed distributions are presented and some conclusions have been formed.

Postbuckling behavior of functionally graded porous toroidal shell segments with surrounding elastic media subjected to combined mechanical loads

AbstractBuckling and postbuckling behavior of toroidal shell segment (TSS) made of functionally graded porous material (FGPM), exposed to a thermal environment, surrounded by an elastic medium and subjected to combined action of axial compression and lateral pressure are investigated in this paper. Material constituents are varied across the thickness according to power‐law functions, while porosities are distributed in the TSS in the form of cosine functions of thickness variable. Governing equations in terms of deflection and stress function are derived based on classical shell theory taking into account geometrical nonlinearity and interactive pressure from surrounding elastic media. Multi‐term analytical solutions are assumed for simply supported TSSs and Galerkin method is used to obtain nonlinear load–deflection relations from which buckling loads can be derived. Parametric studies are carried out to access the influences of material and geometry properties, porosity coefficient and distributions, preexisting loading conditions, thermal environments and surrounding elastic media on the buckling resistance and postbuckling load carrying capabilities of FGPM TSSs.

Linear constrained Cosserat-shell models including terms up to $${O}(h^5)$$: conditional and unconditional existence and uniqueness

In this paper we linearise the recently introduced geometrically nonlinear constrained Cosserat-shell model. In the framework of the linear constrained Cosserat-shell model, we provide a comparison of our linear models with the classical linear Koiter shell model and the "best" first order shell model. For all proposed linear models we show existence and uniqueness based on a Korn's inequality for surfaces.

Nonlinear Electro-Thermo-Torsional Buckling Analysis of Stiffened Functionally Graded Graphene-Reinforced Composite Laminated Toroidal Shell Segments

A new design of functionally graded graphene-reinforced composite laminated (FG-GRCL) toroidal shell segments in the thermal environment with a piezoelectric layer attached to the surface of the shells is presented in this paper. The piezoelectric FG-GRCL toroidal shell segments are reinforced by a FG-GRCL stringer and/or ring stiffener system. The nonlinear electro-thermo-torsional buckling and postbuckling of the shells are analyzed considering the geometrical nonlinearities described by von Kármán in the framework of the classical Donnell thin shell theory. Airy’s stress function and Galerkin’s process are applied, in which the deflection solution of the shell is approximately presented. The effects of thermal load, volume fraction and distribution type of graphene, piezoelectric layer, geometric parameters of the toroidal shell segments, and stiffeners on the torsional buckling load and postbuckling curves are analyzed and investigated in the numerical investigations.

Torsional instability of functionally graded porous toroidal shell segments including elastic edge constraints and elevated temperature

This article aims to analyze the nonlinear instability of toroidal shell segment (TSS) made of functionally graded porous material (FGPM), exposed to elevated temperature and subjected to uniform torsion. The volume fraction of two material constituents is varied through the shell thickness according to a power-law function, while porous distribution into FGPM is modeled in forms of cosine functions. Governing equations in terms of deflection and stress function are established within the framework of classical shell theory incorporating geometrical nonlinearity. Two edges of the TSS are assumed to be simply supported and elastically restrained. Multi-term analytical solutions are assumed to satisfy boundary conditions and Galerkin method is applied to derive closed-form expressions of nonlinear load–deflection relations and buckling loads. Parametric studies are carried out to analyze the separate and combined influences of material and geometry properties, porous coefficient and distributions, degree of edge constraint and elevated temperature on the buckling resistance and postbuckling strength of torsionally loaded FGPM TSSs.

Kinematic modeling of transverse shear in textile composite reinforcements forming

During the draping of dry textile reinforcements, the high tensile stiffness of fibers and the possible slippage between them significantly modify the transverse shear deformation mechanism. In order to show the limits of classical shell elements (Kirchhoff and Mindlin shell) for textile reinforcements, four-point bending tests of multi-layer fibrous material were conducted and analyzed through experiment and simulation. In forming cases, the mid-surface deformation was obtained by a finite element stress resultant shell, which takes the in-plane shear and bending behavior into account. Based on the quasi-inextensibility of fibers, a kinematic modeling approach was then proposed to efficiently calculate the transverse shear strain. This approach has been implemented in Matlab software as a post-processing application, and can therefore be carried out by any user, together with any finite element software. Different bending tests and hemispherical forming experiments proved the effectiveness and correctness of the approach through comparisons between experimental and numerical results.

LAMINATED NANOCOMPOSITE STRUCTURAL ELEMENTS: MATHEMATICAL MODELING AND APPLICATIONS

Abstract. In this study, first general and basic information about carbon nanotubes (CNTs) are presented. Then, information about single-layer and laminated nanocomposites and their applications and micromechanical models of functionally graded nanocomposites are given. Studies on the buckling and vibration behaviors of single-layer and laminated nanocomposite cylindrical shells (LNCSs) are briefly evaluated and an example of the stability behaviors of LNCSs within shear deformation theory (SDT) and classical shell theory (CST) is presented.