Shell Theory Research Articles

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.

On the Static Instability of FG-GNP-Reinforced Composite Cylindrical Shells Under Thermo-Mechanical Loading

The nonlinear buckling response of laminated composite cylindrical shells reinforced with graphene nanoplatelets (GNPs) is studied in this paper. The functionally graded (FG) shell reinforced by GNPs is analytically studied under external pressure and uniform temperature rise loadings. It is also assumed that the GNP-reinforced laminated composite shell is in contact with an elastic foundation. Various types of profiles are employed for the GNP distribution patterns in the shell thickness including 10 nanocomposite layers. The nonlinear strain-displacement relations of the shallow cylindrical panel are established utilizing the third-order shear deformation shell theory. Governing equilibrium equations of the laminated GNP-reinforced composite shell are formulated employing the principle of virtual displacement. The coupled system of nonlinear differential equations is solved analytically for the hinged–hinged and fixed–fixed boundaries of the shell using a perturbation-based technique. Correctness of presented formulations and obtained solutions is proved by comparisons with results from previous studies for an isotropic cylindrical shell. Novel numerical results reveal that the material properties, geometrical characteristics and load parameters significantly affect on the buckling behavior of laminated composite cylindrical shells.

Theoretical modeling of the pre-cutting system performance for the tunnel face stability in very weak rock masses

By forming continuous pre-lining around the tunnel's perimeter, pre-cutting is a recognized tunneling technique that enables full-face excavation of a tunnel even in unfavorable ground conditions. In this study, a theoretical method is put out to simulate the performance of this kind of pre-support in squeezing conditions. The loading distribution on the pre-cutting shells is determined for this purpose using the LDP of an unsupported tunnel, and the non-linear stiffness of the pre-cutting shell is then determined using the shell theory. The convergence confinement method is used to calculate how much the pre-supported tunnel isdeformed. The non-uniform loading distribution also results in the tangential and longitudinal bending moments on the pre-cutting shell. To show the effectiveness of pre-cutting shells in extremesqueezingconditions, several examples are solved. As a result, a few graphs for the preliminarydesign of a tunnel using this technique are given. The graphs show the overall tunnel displacement for various pre-cutting shells under various geological conditions. The outcomes demonstrate good agreement between FALC 3D resultsand the ones by thesuggested analytical model. A preliminary design can be made using the assessment of the proposed methodonthe loads on the pre-cutting system.

Nonlinear dynamics of bolted joined conical-cylindrical shells considering displacement-dependent characteristics

This paper investigates the nonlinear dynamic characteristics of joined conical-cylindrical shells (JCCSs) with bolt connection for the first time. In the process of developing mechanical model of the bolt connection, the displacement-dependent stiffness and damping are taken into account, which could simulate the change of the contact state at the connection interface under different excitations. Both theoretical and experimental studies are performed to illustrate the nonlinear vibration behaviors of the bolted JCCSs. Successive uniform distributed artificial springs between the conical and cylindrical shells are adopted to simulate the rigid joint of the two shells. Donnell's shell theory is employed in theoretical modeling, and the governing equation is obtained by the Lagrange equation. The displacement admissible function applied in the research is Chebyshev polynomial. By comparison with existing literature and experimental data, the accuracy of present theoretical model is validated. It is found that both theory and experiment show a dynamic softening phenomenon with the increase of excitation level. Taking the displacement-dependent characteristics in bolt connection zone into consideration can effectively illustrate this soft phenomenon. The established model is capable of predicting the nonlinear vibration characteristics of JCCSs with bolt connection.

Buckling Analysis of Grid-stiffened Composite Cylindrical Shell

An energy based smeared stiffener model is developed to obtain equivalent stiffness coefficients of a general grid-stiffened composite cylindrical shell. These equivalent stiffness coefficients are used in the Ritz buckling analysis of the shell. Transverse shear properties are important in a stiffened shell and results are obtained for First order Shear Deformation Theory (FSDT) as well as for Classical Laminated Shell Theory (CLST). Parametric study is carried out to find out the effects of various design parameters on specific buckling load of the shell.

Structural optimization of an encased bi-material FGP-GNF pipeline-liner system with novel polyhedral configurations

This article proposes a novel thin-walled polyhedral functionally graded porous (FGP) liner consolidated by graphene nanofillers (GNF) to rehabilitate the cracked pipeline. A buckle propagation may occur since the liner is subjected to hydrostatic pressure and temperature variational field simultaneously. A cosine function is developed to describe the deformed shape of the liner. Both pores and the GNF are distributed symmetrically on the cross-section of the liner, which also has a polyhedral profile to improve its bending stiffness. The nonlinear equilibrium equations are accurately obtained by introducing the thin-walled shell theory and the principle of minimum potential energy. The critical buckling pressure is obtained by solving the nonlinear equilibrium equations. Analytical solutions are compared successfully with other closed-form results to accomplish the verification when the FGP-GNF polyhedral liner reduces to an isotropic circular one. Furthermore, one improvement factor is proposed to analyze the effect of different polyhedral shapes on the buckling pressure of the encased polyhedral FGP-GNF liner. Finally, parameters are evaluated to further understand the buckling behavior of the FGP-GNF polyhedral liner, including the number of polygons, porosity coefficient, mass fraction, geometric shape of the GNF, radius-to-thickness ratio, temperature field, etc.

Vibration Control of Composite Thick Shells Using Magnetostrictive Material and Higher Order Shear Deformation Theory

Analytical solutions of laminated composite shells with embedded actuating layers are presented in this study. The actuating layers are used to control natural vibration of laminated composite shell panels. The first order shear deformation theory (FSDT) and higher order shear deformation theory (HSDT) for shells are used to represent the shell kinematics and equations of motion. The exact solution for symmetric laminated shells with embedded actuating layers under simply supported boundary conditions is obtained using the Navier solution procedure. Negative velocity feedback control is used. The parametric effect of the position of the magnetostrictive layers, material properties and control parameters on the vibration suppression are investigated in detail. It is found that (i) the shortest vibration suppression time is achieved by placing the actuating layers farthest from the neutral axis (ii) using thinner smart material layers leads to better vibration attenuation characteristics and, (iii) the vibration suppression time is longer for a lower value of the feedback control coefficient. For thicker shells HSDT predicts larger amplitude of vibration and longer vibration suppression time as compared to FSDT predictions.

Research on prediction of critical pressure of FGM cylindrical shells under hydrostatic pressure based on homogenization transformation method

This study uses the classical Flügge shell theory and homogenization transformation method to predict the critical pressure of functionally graded material (FGM) cylindrical shells under hydrostatic pressure. The method considers the similarity in physical properties and mechanical behavior between FGM and uniform materials. The vibration equation of the coupled system of an underwater FGM cylindrical shells is derived using the wave method, taking into account the influence of fluids. The natural frequency of the FGM cylindrical shell under hydrostatic pressure is obtained using the Newton iteration method. Based on the linear correlation between critical load and load level with zero natural frequency, the fitting curve method and the formula method after homogenization transformation are used to predict and analyze the critical pressure of FGM cylindrical shells under hydrostatic pressure. The effects of various parameters on the critical pressure are discussed. The results show that the material elastic modulus E, geometric dimensions h/R and L/R of FGM cylindrical shells, and different boundary conditions have a significant impact on the critical pressure. Through multiple comparative analyses of examples, the correctness and effectiveness of the proposed method are verified. The method has high prediction accuracy and low computational cost, providing a new exploration path for the analysis of non-uniform structural mechanical behavior.

Nonlinear dynamic response and vibration of spiral stiffened FG toroidal shell segments with variable thickness

In this study, the nonlinear vibratory responses and primary resonance (PR) of the spiral stiffened functionally graded (SSFG)-toroidal shell segments (TSSs) are investigated. The shell segments considered are with variable thickness and subjected to external excitations. Both the concave and convex TSSs are investigated in this study. The TSSs considered are reinforced with internal spiral stiffeners and are assumed to be continuously graded along their thickness direction. To model the stiffeners, the smeared stiffeners technique is utilized. Also, the improved Donnell shell theory, von-Kármán nonlinear geometric assumptions, and Galerkin’s method are applied to derive the discretized nonlinear governing equation of SSFG-TSSs with variable thickness. To examine the PR responses, the multiple scales method is utilized. To investigate the nonlinear vibratory behaviors, the 4th order P-T method is used. Although the results of the present research are verified with previous research available in the literature, the response of nonlinear vibration is compared with the 4th order Runge–Kutta (R-K) method as well. The influences of geometrical and material parameters on the natural frequencies, nonlinear vibration, and PR responses of SSFG-TSSs are also studied.

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.

Lagrange multipliers at the boundary in the inextensional bending theory of thin elastic shells

The theory of inextensional deformations of shells is extended by introducing Lagrange multipliers at the boundary to supplement those operating in the interior. These are associated with a constraint on the metric on the closure of the shell midsurface. The additional multipliers allow for a richer set of external actions than those admitted by the conventional theory. In particular, they balance concentrated forces in the tangent plane of the surface applied at corners of the boundary.

Robust optimization of prismatic lithium-ion cells for reducing thermal performance fluctuations and manufacturing costs

The thermal performances of lithium-ion cells are significantly influenced by unavoidable uncertainties during manufacturing. However, there is a lack of uncertainty optimization methods to control the thermal performance fluctuations of lithium-ion cells. This paper proposes a robust optimization approach for prismatic lithium-ion cells, aiming to optimize the manufacturing cost and control the thermal performance fluctuations without removing the uncertainties. The cost objective function and thermal robustness constraints are established considering the uncertainties of the manufacturing parameters, thereby formulating the robust optimization model of the prismatic lithium-ion cell. The robust optimization performs a nested optimization by optimizing the parameter standard deviations for an optimal cost and evaluating the cell thermal robustness at each design point. An analytical expression for the cell expansion behaviour is created based on the elastic shell theory. An electrochemical-thermal-mechanics multidisciplinary analysis method with ordinary differential equations is given to evaluate the cell thermal performances such as temperature, pressure, and expansion under thermal abuse. It addresses the efficiency issue in the nested optimization. The robust optimization approach is applied to design problems for two-type lithium-ion cells packaged in pouch and hard-pack. Numerical and experimental results demonstrate the excellent potential of the approach due to its advantages in applicability, accuracy, and efficiency.

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.

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.

Critical velocities of a two-layer composite tube incorporating the effects of transverse shear, rotary inertia and material anisotropy

Critical velocities of a two-layer composite tube subjected to a uniform internal pressure moving at a constant velocity are analytically derived by using a first-order shear deformation shell theory incorporating the transverse shear, rotary inertia and material anisotropy. The composite tube consists of two perfectly bonded axisymmetric circular cylindrical layers of dissimilar materials, which can be orthotropic, transversely isotropic, cubic or isotropic. Closed-form expressions for four critical velocities are first derived for the general case by including the effects of transverse shear, rotary inertia, material orthotropy and radial stress. The formulas for composite tubes without the transverse shear, rotary inertia or radial stress effect and with simpler anisotropy are then obtained as special cases. In addition, it is shown that the model for a single-layer, homogeneous tube is included in the current model as a special case. To illustrate the newly derived closed-form formulas, a composite tube with an isotropic inner layer and an orthotropic outer layer is analyzed as an example. The numerical values of the lowest critical velocity of the two-layer composite tube predicted by the new formulas compare well with existing data.

Analytical Modeling of Buckling of Carbon Nanotubes Reinforced Sandwich-Structured Composite Shells Resting on Elastic Foundations

Sandwich-Structured Composites (SSCs) are widely used in lightweight construction, especially in the aerospace sector, due to their high specific stiffness and strength. Therefore, it is important to develop their quality by using new techniques. Today, nanotechnologies offer new perspectives for the reinforcement of construction materials. This paper assumes that the reinforcement of the sandwich shell is performed by CNT reinforced face sheets with a uniform or Functionally Graded (FG) distribution of CNTs. The effective properties of the Carbon Nanotubes Reinforced Sandwich-Structured Composite (CNT-RSSC) shells are calculated using the rule of mixture. This study presents novel exact analytical formulas to predict the critical buckling load of the CNT-RSSC shells resting on elastic foundations based on Donnell cylindrical shell theory. These analytical formulas provide the most meaningful answer because we get an equation showing us exactly what happens with each variable. The effects of various parameters on the buckling stability of the RSSC shells are examined.

Forced vibration analysis of thin cross-ply laminated circular cylindrical shells with arbitrary boundary conditions using the symplectic wave-based method

A symplectic wave-based method is proposed in this paper for the forced vibration analysis of thin cross-ply laminated cylindrical shells with arbitrary boundary conditions. Firstly, based on Kirchhoff–Love’s shell theory, the unified governing equation of thin cross-ply laminated cylindrical shells in the symplectic duality system is established by selecting an appropriate state vector, which leads to the transformation of the vibration analysis of cylindrical shells from two-dimensional to one-dimensional. Then, the response of laminated cylindrical shells can be described in terms of wave shapes in the symplectic duality system. The relationship between incident and reflected waves at a boundary is established to analytically describe the boundary condition, which allows the present method to deal with arbitrary boundary conditions and has high computational accuracy. Finally, the displacement and internal force responses can be simultaneously and explicitly calculated by using the adjoint symplectic orthogonal relation of wave shapes in the symplectic duality system, which ensures that the present method has high computational efficiency. The present method has truncation errors only in the circumferential direction, and is analytical in the axial direction, which can lead to a high accuracy and convergence rate. Several examples demonstrate the effectiveness, convergence and accuracy of the present method, as well as its ability to handle arbitrary boundary conditions.

Delamination Buckling of Composite Conical Shells Under External Pressure

Airframe construction in conical form is the most desired shape of flight hardware due to their low drag profileand are located at the fore-end region of flight vehicles encountering high drag loads. Owing to their tailoringcapability, materials with orthotropic mechanical properties are preferred choice. Delamination defects formed inthem while manufacturing or when subjected to loads would unfavorably influence the mechanical performanceof the orthotropic airframe. In the current work, FE simulation of delamination which is embedded in orthotropiccone shaped shells under external pressure load is performed as per the method cited in published literature. A layer wise element based on shell theory has been used and the effect of delamination size and its through the thickness position on the mechanical performance of the cone shaped shell is investigated. Circumferential and rectangular shapes of defects have been simulated. The investigation is performed for metal and composite materials with 3 types of stacking sequences generally used in practical designs. Verification of the procedure is carried out by equating with the procedure cited in published studies on shells of thin orthotropic cylinders. The eigen value of the first mode is taken as the critical buckling factor under external pressure. The buckling factor of the delaminated cone is normalized with the buckling factor of the ideal cone. The normalized buckling factor is showed graphically with the normalised defect size. Global, as well as local buckling and also symmetric as well as asymmetric buckling shapes, are observed in the results of the simulation. Shift from global mode to local mode of buckling is also observed in certain cases. Drastic reduction in buckling capability with the local mode is observed when the defect location is close to the surface and more prominent for an outer surface case.

Coupled vibration of an annular plate-cylindrical drum structure with material-nonlinearity damping patches considering position-mistuned and stiffness-mistuned connections

Considering position-mistuned and stiffness-mistuned connections, the coupled vibration of an annular plate-cylindrical drum structure with material-nonlinearity damping patches is studied. These two mistuned connections are inevitable, but they are seldom concerned in published works. Moreover, the mistuned connections can cause local excessive vibrations, so vibration suppression is in demand. Spraying hard coatings with material nonlinearity is a good damping method. Their damping-related and stiffness-related parameters will change with strain, which will improve the vibration-reduction ability but bring obstacles to modelling. This paper focuses on the consideration of both mistuned connections and material-nonlinearity damping patches (MDP) and presents a general modeling method of connection-mistuned annular plate-cylindrical drum structures with MDP. Position-mistuned and stiffness-mistuned connections are emulated by developing the spring approach, and dynamic equations are created by combining Sanders’ shell theory, Kirchhoff’s plate theory, and Lagrange equation. Furthermore, the region decomposition approach is developed for considering the material nonlinearity, and an efficient solution is provided to solve the corresponding nonlinear complex equations. The study is adequately validated. Finally, numerical results indicate that mistuned connections cause the local-vibration behavior, frequency separation, and coupled vibration between substructures and modes. Moreover, mistuned connections also seriously affect the response curves and response cloud charts.