Toroidal Shell Segments Research Articles

On the nonlinear buckling and postbuckling responses of sandwich FG‐GRC toroidal shell segments with corrugated core under axial tension and compression in the thermal environment

AbstractThis paper presents an analytical approach for buckling and postbuckling analysis for functionally graded graphene‐reinforced composite (FG‐GRC) toroidal shell segments with the trapezoidal or round corrugated core under the axial tension and compression. The considered shells are placed in a thermal environment and surrounded by an elastic foundation. Based on the von Kármán‐Donnell shell theory with geometrical nonlinearities, Stein and McElman approximation, and a homogenization technique for corrugated shells, the basic equations of shells are established. The previous homogenization technique is improved by adding the thermal forces in the internal force expressions. The shell‐foundation interaction is expressed using the model of the Pasternak assumption. The Ritz energy method is used to obtain the pre‐buckling and postbuckling behaviors of the shells, from which the critical buckling tensions and compressions can be investigated. The influences of FG‐GRC face sheets, corrugated core, and foundation on the buckling behavior of sandwich shells can be shown in the numerical analysis.Highlights Postbuckling of toroidal shell segments with the corrugated core is analyzed. A homogenization technique is improved for the thermal forces in the core. The shells are made of functionally graded graphene‐reinforced composite. The nonlinear Kármán‐Donnell shell theory and Ritz energy method are applied. Special effects of input parameters on the buckling behavior are investigated.

Free Vibration Analysis of Toroidal Shell Segments Reinforced by Graded Graphene in Elastic Medium

Free Vibration Analysis of Toroidal Shell Segments Reinforced by Graded Graphene in Elastic Medium

Nonlinear forced vibration of functionally graded hybrid three-phase nanocomposite toroidal shell segments reinforced by carbon nanotubes (CNTs) and graphene nanoplatelets (GPLs)

This study proposes a unique three-phase functionally graded (FG) hybrid nanocomposite material reinforcing the toroidal shell segments to investigate the nonlinear forced vibration using Reddy's higher-order shear deformation theory, von Karman geometrical nonlinearity and Stein-McElman's assumption along with the Hamilton principle. The examined toroidal shell's material consists of polymeric resin, carbon nanotubes (CNTs), and graphene platelets (GPLs) as nanocomposites. The material properties have been derived based on a modified Halpin-Tsai micromechanical model. Four distribution patterns have been studied: uniform (UD), FG-X, FG-O, and FG-V. The continuous distribution of GPLs and CNTs leads to inhomogeneous position-dependent properties throughout the shell thickness. The obtained differential equations of motion have been reduced to ordinary equations using Galerkin's technique. A multi-scales method (MSM) is used to estimate a closed-form solution representing the frequency-amplitude relation, and the state space representation is used along with numerical fourth-order Runge Kutta method (RK4) to obtain the nonlinear dynamic response of the toroidal shell. The accuracy of the current results obtained has been verified by comparing them with the relevant literature and numerical results using (RK4). In addition, the influence of both GPLs and CNTs weight fractions, nanofillers’ distribution types, longitudinal and circumferential wave numbers, elastic foundation parameters, static axial compression load, transverse excitation load, damping ratio, geometrical characteristics of the toroidal shell on the dimensionless natural frequency, nonlinear primary resonance (frequency-amplitude curve), and nonlinear dynamic response are carefully studied. The results found that combining (GPLs & CNTs) into the shell's matrix improves performance, especially in concave shells compared to convex ones. The FG-X pattern reduces peak resonance amplitude and improves natural frequency. A reduction of 12.3 % is observed for convex shells, and 21.5 % for concave shells compared to FG-O.

A New Analytical Approach for Nonlinear Global Buckling of Axially Compressed and Tensiled Sandwich Toroidal Shell Segments with CNTRC Coatings and Corrugated Core in Thermal Environment

In this paper, the nonlinear global buckling and postbuckling responses of axially compressed and tensiled sandwich toroidal shell segments with functionally graded carbon nanotube-reinforced composite (FG-CNTRC) coatings and corrugated core in the thermal environment are investigated. Three distributed types of FG-CNTRC coatings with two geometrical types of corrugated core, and three shell types including convex, concave, and cylindrical shells are considered. By adding the thermal forces to the governing equations, a homogenization model for corrugated structures is developed to simulate the thermal and mechanical behavior of the trapezoidal and round corrugated core. The potential energy expression of sandwich shells is established according to the Donnell shell theory with the nonlinearities of deflection, an algorithm to solve the governing equation is developed using the Ritz energy method, and three equilibrium equations are obtained respecting three amplitudes of deflection. The axial critical buckling loads and postbuckling curves of shells can be achieved after some calculations. The numerical examples prove the beneficial effects of corrugated core and CNTRC coatings on nonlinear global buckling responses of sandwich toroidal shell segments significantly.

Nonlinear torsional buckling of corrugated core sandwich toroidal shell segments with graphene-reinforced coatings in temperature change using the Ritz energy method

An investigation on the nonlinear torsional buckling of corrugated core sandwich toroidal shell segments with functionally graded graphene-reinforced composite (FG-GRC) laminated coatings in temperature change is mentioned in this paper using the Ritz energy method. The complex configuration of sandwich toroidal shell segments with double curvatures and the combination of the corrugated core and the FG-GRC coatings create the special behavior of structures. A homogenization model for corrugated structures and the extended Halpin-Tsai model can be applied to estimate the stiffnesses of shells. The effects of uniform temperature change are taken into account by supplementing the thermal forces into the total forces of shells. The Donnell shell theory and Pasternak's foundation model are used to establish the total potential energy of the structures. Ritz energy method is applied, and the closed condition and circumferential stress are taken into account, the postbuckling curve expressions of torsion-deflection and torsion-twist angle are obtained in explicit forms. The numerically obtained examples can show the significantly beneficial effects of FG-GRC laminated coatings and corrugated core on nonlinear buckling responses of structures.

Nonlinear thermo-elastic stability of corrugated-core toroidal shell segments with carbon nanotube-reinforced face sheets under radial pressure

An analytical approach for nonlinear buckling and postbuckling responses of corrugated-core toroidal shell segments with carbon nanotube (CNT) reinforced face sheets in thermal environment subjected to radial pressure is reported in this work. Three distribution laws of functionally graded CNT-reinforced layers and the trapezoidal and round forms of the corrugated core are investigated. An equivalent technique for corrugated panels is improved by adding thermal forces to model the behavior of the corrugated core. The nonlinear equations are formulated by using the geometrically nonlinear Donnell shell theory considering Pasternak’s foundation. An algorithm to solve a system of nonlinear equations is established using the Ritz energy method with three shell deflection amplitudes. The postbuckling expressions between the compressive and tensile loads with maximal deflection and the critical compressive and tensile loads are determined. The numerically investigated examples present the significantly beneficial effects of corrugated core and functionally graded CNT-reinforced face sheets on nonlinear buckling responses of structures.

Electro-Mechanical Buckling of Shallow Segments of Functionally Graded Porous Toroidal Shell Covered by Piezoelectric Actuators

The current work deals with the mechanical buckling behavior of saturated porous toroidal shell segments sandwiched by piezoelectric actuator layers. Material properties of the porous shells with nonuniform distributed porosities are assumed to vary as a specific function of the thickness and porosity parameters. The nonlinear equations of the sandwich toroidal shell having either positive or negative Gaussian curvature are obtained on the basis of Biot’s poroelasticity theory. The energy method as the minimum potential energy principle is applied to derive the governing equations of the sandwich shells. The buckling equations of the piezo-porous sandwich shells under various types of mechanical loading are established by employing the adjacent equilibrium criterion. The governing equations as a set of the coupled partial differential equations are solved using an analytical method including the Airy stress function. Closed-form solutions are derived for shallow segments of the sandwich toroidal shell with positive/negative Gaussian curvature under lateral pressure, axial compression, and hydrostatic pressure loadings. Novel numerical results are presented in this work to show the effects of important factors such as the piezoelectric layer thickness, applied voltage, shell geometrical ratio, and variation of the porosity coefficient on the buckling behavior of these structures.

A new analytical approach for nonlinear buckling and postbuckling of torsion-loaded FG-CNTRC sandwich toroidal shell segments with corrugated core in thermal environments

The design of the torsion-loaded toroidal shell segment with two functionally graded carbon nanotube-reinforced composite (FG-CNTRC) coatings and the corrugated core is presented. Round and trapezoidal shapes of the corrugated core are considered. A homogenization model for the corrugated core is applied to predict the stiffnesses of the corrugated core. The total potential energy equation, according to the Donnell shell theory, is established. The Ritz energy method is applied to determine the expressions of the buckling and postbuckling responses. The numerical investigations present the significantly beneficial effects of FG-CNTRC coatings and corrugated core on the buckling responses of the shells.

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.

Vibration analysis of variable thickness functionally graded toroidal shell segments

Vibration analysis of variable thickness functionally graded toroidal shell segments

Buckling and Postbuckling of Laterally Pressured Sandwich FG-GRC Laminated Toroidal Shell Segments with Corrugated Core in Thermal Environment

The nonlinear buckling behaviors of sandwich functionally graded graphene reinforced composite (FG-GRC) laminated toroidal shell segments with corrugated core under lateral pressure in thermal environment are reported in this study. Three distributed types of the FG-GRC face sheets and two forms of corrugation including the trapezoidal and round corrugations are considered. By adding the thermal forces into an existing equivalent technique, the behavior of the corrugated core is modelled. The nonlinear formulations can be obtained by using the nonlinear Donnell shell theory taking into account the shell-foundation interaction. The Ritz energy procedure is used for three states of deflection to obtain the postbuckling curve expressions of the load-maximal deflection. The bifurcation buckling type of critical buckling pressure is obtained in the explicit form. The numerical examinations present the significant effects of corrugated core and FG-GRC laminated face sheets on nonlinear buckling behaviors of shells.

Nonlinear buckling responses of radially pressured FG-GPLRC toroidal shell segments

An analytical approach for nonlinear buckling of functionally graded graphene platelet reinforced composite toroidal shell segments is presented in this paper. The Ritz energy procedure is executed, and radial pressure–deflection expression is constituted to obtain the postbuckling strength and critical buckling pressure of the shells. Significant influences on the buckling responses of shells with three different material distribution rules and mass fractions of graphene platelet, and geometrical dimensions are exemplified and in numerical examples.

Nonlinear Stability and Vibration Analyses of Functionally Graded Variable Thickness Toroidal Shell Segments Reinforced with Spiral Stiffeners

In this study, an analysis of nonlinear stability and vibration of functionally graded (FG) variable thickness toroidal shell segments (TSSs) reinforced with spiral stiffeners exposed to axial loading is presented using a combination of semi-analytical and analytical methods. Three types of variable thickness TSSs, including concave, convex, and cylindrical shells (CSs), are studied. Moreover, these structures are reinforced by external spiral stiffeners with various angles whose material properties are considered to be continuously graded along the thickness direction. In this regard, the smeared stiffeners technique is utilized to model the stiffeners, and the Donnell shell theory and the von Kármán equation are applied to derive the nonlinear governing equation for variable thickness TSSs reinforced with spiral stiffeners. Galerkin’s method is then used to obtain a discretized nonlinear governing equation to analyze the shells’ behavior. Also, the fourth-order P-T method is applied to analyze the nonlinear dynamic behaviors and the Budiansky–Roth criteria are used to examine the dynamic post-buckling (DPB) behavior. In this regard, it is noted that in terms of reliability and accuracy, the fourth-order P-T method has demonstrated advantages over the other numerical methods. Results are reported to evaluate the influences of stiffeners with different angles and input factors on the nonlinear vibration, dynamic and static post-buckling (SPB) behaviors of FG variable thickness TSSs reinforced with spiral stiffeners.

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.

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.

Free vibration analysis of rotating stiffened functionally graded graphene-platelet-reinforced composite toroidal shell segments with novel four-unknown refined theories

Free vibration analysis of rotating stiffened functionally graded graphene-platelet-reinforced composite toroidal shell segments with novel four-unknown refined theories

Torsional buckling and postbuckling behavior of stiffened FG-GRCL toroidal shell segments surrounded by elastic foundation

In this study, a new algorithm for the nonlinear buckling behavior of the stiffened functionally graded graphene-reinforced composite laminated (FG-GRCL) toroidal shell segments in the thermal environment surrounded by the elastic foundation and subjected to torsional load is introduced. The FG-GRCL shell segment is stiffened by the FG-GRCL ring and/or stringer stiffener system. Based on the Donnell shell theory with the Stein and McElman approximation, considering the von Karman geometrical nonlinearity and the anisotropic smeared stiffener technique for FG-GRCL stiffeners, the formulations to investigate the critical buckling load and postbuckling torsional load-deflection curves are determined. A new algorithm is fully developed by adding the effect of circumferential stress into the stress function. The significant effects of temperature, stiffener system, geometrical parameters, and elastic foundation are considered and studied in detail.

Analytical solution for nonlinear buckling of convex and concave auxetic-core toroidal shell segments with graphene-reinforced face sheets subjected to radial loads

Analytical solution for nonlinear buckling of convex and concave auxetic-core toroidal shell segments with graphene-reinforced face sheets subjected to radial loads

Reddy’s third-order shear deformation shell theory for free vibration analysis of rotating stiffened advanced nanocomposite toroidal shell segments in thermal environments

Reddy’s third-order shear deformation shell theory for free vibration analysis of rotating stiffened advanced nanocomposite toroidal shell segments in thermal environments