Multilayered Cylindrical Shells Research Articles

Natural Vibrations and Stability of Composite Cylindrical Shells Containing a Quiescent Fluid

This paper presents the results of investigation of the natural vibrations and stability of circular vertical multilayered cylindrical shells, fully or partially filled with a quiescent compressible fluid subjected to hydrostatic and external static loads. The behavior of the elastic structure and fluid medium is described based on the classical shell theory and Euler’s equations. The linearized equations of motion of the shell, together with the corresponding geometrical and physical relations are reduced to a system of ordinary differential equations with respect to new unknowns. The acoustic wave equation is transformed to a system of differential equations using the method of generalized differential quadrature. The solution of the formulated boundary value problem is developed using Godunov’s orthogonal sweep method. The dependences of the lowest vibration frequencies and critical external pressure on the ply angle and the filling level of two-layer and three-layer cylindrical shells are analyzed in detail. It is demonstrated that, in contrast to the ply angle and a given combination of boundary conditions, the lay-up scheme of composite materials plays different parts in the problems of maximizing the fundamental frequency of vibrations and extending the stability boundaries.

Free vibration analysis of fiber-reinforced composite multilayer cylindrical shells under hydrostatic pressure

With the development and application of composite materials in deepwater submersibles, the influence of high external pressure from deepwater environments on the vibration characteristics of composite pressure-resistant cylindrical shells cannot be ignored. In this paper, the free vibration characteristics of prestressed composite multilayer shells are methodically examined by establishing partial differential equations based on the three-dimensional elasticity theory. To solve these equations, the state-space technique and double Fourier series expansion are employed to transform them into state-space ordinary differential equations. These equations are utilized to analyze vibration problems of simply supported composite multilayer cylindrical shells with arbitrary layered angles. The initial stress equilibrium equation is solved to determine the prestress induced by the hydrostatic pressure. This overcomes the limitation of membrane stress and considers non-uniform distributions of radial and interlaminar prestresses. The proposed theoretical model and solution strategies are validated by comparing with theoretical and experimental results from the existing literature and the finite element method. Additionally, the present investigation reveals that the natural frequencies of thick shells can be calculated more accurately using the prestress derived by the proposed approach, especially subjected to noticeable hydrostatic pressure. Finally, the performed investigations indicate that the circumferential prestressing has a more pronounced effect on the shell's natural frequencies than axial prestressing.

Numerical Study of the Buckling Response of Stiffened FG Graphene-Reinforced Multilayer Composite Cylindrical Panels

The present research aims at determining the axial buckling load of stiffened multilayer cylindrical shell panels made of functionally graded graphene-reinforced composites (FG-GPL RCs). Rings and stringers are applied as stiffening tools for shell panels, whose elastic properties are determined according to the Halpin–Tsai relations. The virtual work principle and finite element approach are implemented here, according to a first-order shear deformation theory (FSDT) and Lekhnitskii smeared stiffener approach, in order to determine the governing equations of the stability problem. Four different dispersions of nanofillers are assumed in the thickness direction, including the FG-X, FG-A, FG-O, and UD distributions. A large systematic investigation considers the effect of different geometric and material parameters on the buckling loads and mode shapes of the stiffened FG-GPL RC cylindrical shell panel, primarily the dispersion and weight fractions of the nanofiller, the number of rings and stringers, and the boundary conditions, with useful insights for design purposes.

Multi-objective optimization of multi-layered cylindrical shells with opening under axial load using the NSGA-II genetic algorithm

Composite cylindrical shells play a crucial role in aerospace and marine structures. This study investigates the optimal structure for cylindrical multilayer composite shells under the effect of axial pressure using the finite element method and NSGA-II genetic algorithm to determine the maximum buckling load capacity. The critical buckling load of multilayer composite shells depends on various parameters, such as fiber angle, the number of layers, the material of the layers, and their thickness. The objective functions are used to increase the structure load capacity and reduce its weight. ABAQUS software was used to perform finite element analysis on the composite cylindrical shell for determining the buckling load. Using the response surface model, the relationship between variables and objective functions has been determined. Results of the proposed response surface model for the training stages are evaluated using various statistical indices and the root mean square error for buckling load and shell weight variables is 0.065 and 0.140, respectively. In the next step, the NSGA-II genetic optimization algorithm was used to modify the layout and thickness of the composite layers to optimize the buckling strength and weight of the structure. A genetic algorithm based on NSGA-II was used to optimize the geometric characteristics.

Dynamic response and vibration modes of multi-layer wound cylindrical shell for hydrogen storage under external blast loading

Multi-layer wound cylindrical shells are the preferred structures for storage vessels working in high pressure and hydrogen atmosphere. However, they are susceptible to external blast loads from accidental hydrogen explosions. In order to study the response of such structures under external blast loading, a numerical model was developed, which combined a thermo-viscoplastic constitutive model and a fluid-structure coupling approach, and was subjected to external blast loads of different TNT equivalency weights. The strain responses and initial velocity fields of the structure were analyzed. A rigid-plastic theoretical model was also developed to calculate the maximum displacement of the inner shell and was compared with the simulation results. Moreover, the axial velocity vibration histories and dominant frequencies at different locations of the steel belts were numerically obtained. In addition, four vibration modes of the steel belts were presented in this paper. Furthermore, strain growth was observed in the high-frequency vibration area, and the possible reasons for the strain growth were analyzed and discussed in detail. The results will be useful for predicting the deformation modes of such multi-layer wound cylindrical shell structures under the risk of external hydrogen explosions.

A convergence criterion for sound transmission of single-, double-, and triple-walled cylindrical shell based on Love's, Donnell's, and Flügge's shell theories

Different sound transmission loss (STL) analytical methods based on Love's, Donnell's and Flügge's cylindrical thin-shell theories have been widely used. However, there are no available convergence criterion and formulas of sound transmission losses for multi-layer cylindrical shell structures. The current convergence determination approach is to select several typical frequencies (low, middle, and high frequencies) for convergence analysis, and then estimate the convergence of all other frequencies. This typically results in sound transmission loss jumps, leading to inaccurate sound transmission loss results in the high-frequency region. In this study, a novel convergence criterion and formulas are developed for calculating sound transmission loss mode numbers in the complete convergence zones of single-, double-, and triple-walled cylindrical shells based on Love's, Donnell's, and Flügge's thin shell theories, which can be used to directly determine the truncation iterations for sound transmission losses in all frequency ranges.

Buckling behavior of multilayer cylindrical shells composed of functionally graded nanocomposite layers under lateral pressure in thermal environments

In this study, the stability behavior of multilayer cylindrical shells made of functionally graded nanocomposite layers (FG-NCLs) subjected to the lateral pressure in thermal environments is investigated. It is postulated that nanocomposite layers forming layered cylindrical shells are made of single-walled carbon nanotube (SWCNT)-reinforced polymers that have four types of profiles based on the uniform and linear distributions of mechanical properties. The material properties of SWCNTs are assumed to be dependent on location as well as temperature and are obtained from molecular dynamics simulations. The governing equations are derived as partial differential equations within shear deformation theory (SDT) and solved in a closed form, using the Galerkin procedure, to determine the lateral critical pressure (LCP) in thermal environments. The numerical representations relate to the buckling behavior of multilayer cylindrical shells made of functionally graded nanocomposite layers under the uniform lateral pressure for different CNT patterns and temperatures within SDT and Kirchhoff-Love theory (KLT).

Seismic experiments on short coaxial multi-layered cylindrical shells with fluid–structure interaction in fast reactors

As the most important component of pool-type fast reactors, the main vessel contains most of the equipment for the primary circuit and thousands of tons of liquid sodium. The fluid–structure interaction effects of the coaxial multi-layer thin-walled cylindrical shells formed by the main vessel and its thermal shield cannot be neglected. Though the research on the fluid–structure interaction of the liquid in the pool is sufficient, the research on the short coaxial multi-layer cylindrical shells is not enough, especially the axial and circumferential mode shapes and the ratio of each mode under seismic conditions. So, it is necessary to study the fluid–structure interaction of short coaxial multilayer shells. This paper presents shaking table tests performed on double- and triple-layered cylindrical shells to investigate fluid–structure interaction and seismic response, respectively. The experiments reveal that when the aspect ratio is less than 2, Mode (1,6) exhibits the most significant response under sine wave excitation conditions, while Mode (1,1) demonstrates the highest response under seismic conditions. The first three modes are characterized as out-of-phase modes. Additionally, the pressure component of the first circumferential mode contributes 20% to the overall response of the first 12 modes under seismic conditions.

The modified CUF-EFG method for the dynamic analysis of GPLs-CNTs-reinforced FG multilayer thick cylindrical shells under shock loadings: A modified meshless implementation

The main scope of this paper is to develop the Carrera unified formulation in cylindrical coordinates for two-dimensional meshless modelling of graphene platelet and carbon nanotube reinforced functionally graded multilayer thick cylindrical shells with the same accuracy of three-dimensional modelling. The cylindrical shell is composed of several homogeneous layers with uniform distribution of both GPL and CNT contents in each layer. The effective macroscopic mechanical properties of each GPLs and CNTs reinforced layer are estimated using the modified Halpin-Tsai (H-T) micromechanical model and rule of mixture. The volume fraction of nanofillers gradually changes layer to layer by nonlinear continuous function which leads to functionally graded (FG) distribution of nanofillers. Five grading patterns of layer distribution along the thickness direction of cylindrical shell are considered to investigate the effect of layer arrangement on dynamic behavior of cylindrical shells. The FG multilayer hybrid nanocomposite cylindrical shell is assumed to be under mechanical shock loading. The numerical method for dynamic analysis of multilayer cylindrical shell is based on element free Galerkin (EFG) method with radial basis shape functions in the framework of displacement-based Carrera unified formulation (CUF). The present EFG-CUF method is compared with analytical method for isotropic homogeneous shells and results available in the literature for FG multilayer cylindrical shells. Considering the results related to various expansion orders, proved accuracy and efficiency of proposed method for dynamic analysis of thin to thick FG multilayer cylindrical shells at high expansion orders ‘n > 2’. After verifying the analytical method, the effects of several key factors such as layer distribution patterns, volume fraction index, number of layers, nanofiller weight fraction, aspect ratios of cylinder, and damping ratio on static, free vibration, time history and wave propagation of cylinder are studied in detail.

Free Vibration Analyses of Stiffened Functionally Graded Graphene-Reinforced Composite Multilayer Cylindrical Panel

In this paper, the free vibration response of a stiffened functionally graded graphene nanoplatelet (GPL)-reinforced composite multilayer cylindrical shell panel is studied for the first time. The shell is stiffened by both stringers and rings. Additionally, the effect of reinforcing the shell panel, ring and stinger with GPLs is independently studied. Halpin–Tsai relations are employed to evaluate the mechanical properties of the shell panel, rings and stringers. The first-order shear deformation shell theory, accompanied by the Lekhnitsky smeared stiffener model, using the numerical finite element method and Hamilton principle, is employed to develop the governing motion equations of the shell panel. Four different types of GPL patterns, including FG-A, FG-X, FG-O and UD, are assumed across the thickness of the shell panel, rings and stringers. The effects of different factors, including various weight fractions and patterns of GPLs nanofillers, the geometry of the shell panel and stiffeners and two displacement boundary conditions, on the natural frequencies of the shell panel, have been studied.

Deformation and Dynamic Response of Steel Belt Staggered Multilayer Cylindrical Shell Under External Blast Loading

Abstract The deformation and dynamic response of a multilayer cylindrical shell composed of an inner shell and fourteen outer layers under external blast loads of different trinitro-toluene equivalency weights were studied. A numerical model using the thermo-viscoplastic constitutive model and considering fluid–structure coupling between explosion wave and structure was developed. The displacement in axial direction and cross section, as well as the effective strain responses, were analyzed to demonstrate the potential deformation of the shell structure. Results demonstrate that different materials cause inconsistent displacement and separation to develop in the inner and outer shells. In order to address the problem that the displacement of the inner shell is hard to measure due to the shielding and covering of the outer shell, a theoretical formula for calculating the maximum displacement of the inner shell was developed. The deflection process and stress triaxiality histories of the inner shell were investigated, and the results showed that compressive stress is the primary cause of plastic deformation. Additionally, the delamination that appeared in the outer shell was discussed, and it was revealed that there are two factors of delamination: (1) Stress waves spread across adjacent layers in the opposite direction because steel belts were wound in the opposite direction between the two adjacent layers; (2) Outer layers experienced uneven compressive loads. The results will be helpful to provide a reference for the intrinsic safety design of such multilayer cylindrical structures for hydrogen storage, etc.

Generalized differential quadrature element solution, swarm, and GA optimization technique to obtain the optimum frequency of the laminated rotary nanostructure

Optimization of the fiber angle in the laminated composite material to achieve the highest strength or stability has great importance in structural design. In the present study, we aim to utilize genetic algorithm (GA) and particle swarm optimization (PSO) together to find the best angle of composite layers in a multilayer cylindrical shell structure. To this aim, a detailed formulation of the composite cylindrical structure is presented using nonlocal strain gradient theory (NSGT), zigzag theory, and Hamilton's principle for a shear deformable shell displacement field. The modal equations of motion are further solved using the general differential quadrature element method (GDQEM). Having the route of obtaining the resonance frequency of the cylindrical composite structure, GA-PSO is engaged to obtain the optimal angle of the laminate composite to achieve the highest frequency. It is revealed that angle θ = 37.5° provides the highest frequency for a single-layer cylindrical shell. Using this angle, stacking sequences for 3, 5, and 7 layers’ composite are optimized to have the highest frequency. In the end, a detailed parametric study is presented using optimum condition shell composite. The results show that after 30 composite layers, changing the number of layers seems to be not effective on the resonance frequency of the cylinder for both simply supported and clamped boundary conditions.

An analytical method for free vibration analysis of multi-layered transversely isotropic cylindrical shells

This work presents an effective analytical method based on displacement potential functions (DPF) for solving 3-D free vibration problem of thick and multi-layered transversely isotropic cylindrical shells with simply supported end boundary conditions. By using the DPF method, the three-dimensional elasticity equations are simplified and decoupled into two linear partial differential equations of fourth and second order as governing differential equations. These equations are analytically solved using the separation of variables method in terms of fields that exactly satisfy end boundary conditions and the continuity of a closed cylinder in the hoop direction. One of the advantages of the present work is the analysis of thick cylindrical shells using an exact method without any approximations and simplifying assumptions in the distribution of stress or strain along the thickness of the shells. The analysis is extended to multi-layered cylindrical shells by applying appropriate boundary conditions at the interfacial surfaces of the layers. Numerical comparisons to other analytical works shows the accuracy of the present work and also due to the lack of study for multi-layered transversely isotropic cylindrical shells, 3-D finite element analysis has been performed for comparison. In addition, various more involved problems are studied and solved analytically for single and multi-layered shells of transversely isotropic material and the effect of various parameters such as shell dimensions, mechanical properties, number of layers and other parameters on shell behavior are evaluated. The results of the present study can be used as benchmark solutions for other studies.

Natural vibrations of composite cylindrical shells partially filled with fluid

The article presents the results of studies of natural vibrations of circular vertical multilayer cylindrical shells completely or partially filled with a stationary compressible fluid and subjected to hydrostatic load. The behavior of the elastic structure and the fluid medium is described using the classical shell theory and the Euler equations. The effects of sloshing on the free surface of the fluid are not considered. The linearized equations of motion for shells together with the corresponding geometric and physical relations are reduced to a system of ordinary differential equations with respect to new unknowns. The acoustic wave equation is transformed to a system of differential equations using the generalized differential quadrature method. The formulated boundary value problem is solved by Godunov’s method of orthogonal sweep. The natural frequencies of vibrations are calculated based on the combination of a stepwise procedure and subsequent refinement by the method of dividing in half. The reliability of the obtained results is verified by comparison with the known numerical solutions. The dependence of low vibration frequencies on the reinforcing angle and the level of the structure filling with a fluid for simply supported rigidly clamped and cantilevered two-layer and three-layer cylindrical shells with a fluid are analyzed in detail. It is demonstrated that the possibility of changing the frequencies and vibration modes through a suitable choice of the layer arrangement scheme and the reinforcing angle of the composite material is notably determined by a prescribed combination of boundary conditions for an elastic body.

3-D analytical solution of non-homogeneous transversely isotropic thick closed cylindrical shells

This paper presents an effective analytical method based on displacement potential functions (DPF) for solving 3D static problem of thick and multilayer transversely isotropic cylindrical shells with simply supported end boundary conditions. By using the DPF method, the three-dimensional elasticity equations are simplified and decoupled into two linear partial differential equations of fourth and second order as governing differential equations. The governing equations are solved by the separation of variable method in terms of fields that exactly satisfy end boundary conditions and the continuity of a closed cylinder in the hoop direction. The analysis covers a straightforward solution process for handling problems on multilayered cylindrical shells of transversely isotropic material, adopting all boundary and continuity conditions. Extensive sets of general radial loads located on the inner and outer faces of the cylindrical shell may be stated and examined with in a systematic manner. Comparisons are performed to other existing analytical results for one and multilayered cylindrical shells, and show excellent agreement for different materials, thicknesses and aspect ratios of the shell. In addition, various more involved problems are studied and solved analytically for single and three-layered shells of transversely isotropic material with different sets of radial loading functions at the outer and inner shell surfaces. The results of the present study can be used as benchmark solutions for other studies.

3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions

This paper presents the steady-state stress analysis of single-layered and multilayered plates and shells embedding Functionally Graded Material (FGM) layers under moisture conditions. This solution relies on an exact layer-wise approach; the formulation is unique despite the geometry. It studies spherical and cylindrical shells, cylinders, and plates in an orthogonal mixed curvilinear coordinate system (α, β, z). The moisture conditions are defined at the external surfaces and evaluated in the thickness direction under steady-state conditions following three procedures. This solution handles the 3D Fick diffusion equation, the 1D Fick diffusion equation, and the a priori assumed linear profile. The paper discusses their assumptions and the different results they deliver. Once defined, the moisture content acts as an external load; this leads to a system of three non-homogeneous second-order differential equilibrium equations. The 3D problem is reduced to a system of partial differential equations in the thickness coordinate, solved via the exponential matrix method. It returns the displacements and their z-derivatives as a direct result. The paper validates the model by comparing the results with 3D analytical models proposed in the literature and numerical models. Then, new results are presented for one-layered and multilayered FGM plates, cylinders, and cylindrical and spherical shells, considering different moisture contents, thickness ratios, and material laws.

Calculation of multilayer cylindrical shells taking into account thermoelectric effect based on refined theory

Calculation of multilayer cylindrical shells taking into account thermoelectric effect based on refined theory

Free Vibration Analysis of Composite Grid Stiffened Cylindrical Shells Using A Generalized Higher Order Theory

The present study analyzes the free vibration of multi-layered composite cylindrical shells and perforated composite cylindrical shells via a modified version of Reddy’s third-order shear deformation theory (TSDT) under simple support conditions. An advantage of the proposed theory over other high-order theories is the inclusion of the shell section trapezoidal form coefficient term in the displacement field and strain equations to improve the accuracy of results. The non-uniform stiffness and mass distributions across reinforcement ribs and the empty or filled bays between the ribs in perforated shells were addressed via a proper distribution function. For integrated perforated cylindrical shells, the results were validated by comparison to other studies and the numerical results obtained via ABAQUS. The proposed theory was in good consistency with numerical results and the results of previous studies. It should be noted that the proposed theory was more accurate than TSDT.

MATHEMATICAL MODELING OF THE HEAT TRANSFER PROCESS IN THE SYSTEM OF MULTILAYER CYLINDRICAL SOLID BODIES CONSIDERING INTERNAL SOURCES OF HEAT

The article is devoted to the application of the direct method to the study of heat transfer processes in the "continuous cylinder inside a multilayer cylindrical shell" system. To solve the initial problem, an auxiliary problem is posed with a “remote” cylinder of sufficiently small radius. The solution is based on the reduction method, the concept of quasiderivatives, the Fourier scheme using the modified eigenfunctions method. The solution to the original problem was obtained by following the radius of the remote cylinder to zero.

Postbuckling of multilayer cylindrical and spherical shell panels reinforced with graphene platelet by isogeometric analysis

The present work fills a gap on the postbuckling behavior of multilayer functionally graded graphene platelet reinforced composite (FG-GPLRC) cylindrical and spherical shell panels resting on elastic foundations subjected to central pinching forces and pressure loadings. Based on a higher-order shear deformation theory and the von Karman’s nonlinear strain–displacement relations, the governing equations of the FG-GPLRC cylindrical and spherical shell panels are established by the principle of virtual work. The non-uniform rational B-spline (NURBS) based isogeometric analysis (IGA), the modified arc-length method and the Newton’s iteration method are employed synthetically to obtain nonlinear load–deflection curves for the panels numerically. Several comparative examples are performed to test reliability and accuracy of IGA and arc-length method in present formulation and programming implementation. Parametric investigations are carried out to illustrate the effects of dispersion type of the graphene platelet (GPL), weight fraction of the GPL, thickness of the panel, radius of the panel and parameters of elastic foundation on the load–deflection curves of the FG-GPLRC shell panels. Some complex load–deflection curves of the FG-GPLRC cylindrical and spherical shell panels resting on elastic foundations may be useful for future references.