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The buckling and postbuckling behavior of thin toroidal shell segments composed of auxetic core and graphene-reinforced face sheets under radial loads is reported in the present research combining exiting analytical solutions with the new material designs. Three types of graphene distribution of laminated face sheets and the lattice auxetic core are considered for convex, concave toroidal shell segments and cylindrical shells. The honeycomb lattice auxetic core can be modeled applying a homogenization technique. The Stein and McElman approximation can be used for longitudinally shallow shells to establish the nonlinear equilibrium equations in the framework of the Donnell shell theory with geometrically nonlinearities taking into account the two-parameter foundation model. The expressions of the radial load-maximal deflection postbuckling curves are achieved using the Galerkin method. The numerical investigations indicate the remarkably positive effects of honeycomb auxetic core and graphene-reinforced face sheets on nonlinear buckling responses of shells.

In this work, novel four-unknown refined theories were used to evaluate the free vibration of rotating stiffened toroidal shell segments subjected to varying boundary conditions in thermal environments. The shell segments consist of a functionally graded graphene-platelet-reinforced composite (FG-GPLRC). The effective material properties of the composite were calculated using the modified Halpin–Tsai model and the mixture rule. The governing equations of motion for the shell were formulated within the novel four-unknown refined shell theory framework. The effects of centrifugal and Coriolis forces and the initial hoop tension resulting from rotation were all included. The Rayleigh–Ritz procedure and smeared stiffener technique were subsequently used to determine the natural frequencies of the shells with stiffeners. The advantages of the adopted shell theory result directly from the reduction of key unknowns without the need for the shear correction factor, and it can predict better results for FG-GPLRC structures. Finally, numerical examples were provided to validate the proposed solution and demonstrate the effects of four-unknown refined theories, material distribution patterns, boundary conditions, rotating speed, and temperature rise on the natural frequencies of toroidal shell segments.

In this paper, for the first time, the nonlinear vibration response of toroidal shell segments with varying thickness subjected to external pressure is investigated analytically using Reddy’s third-order shear deformation shell theory. The variable thickness shells are made of functionally graded material (FGM) that is created from ceramic and metal constituents. The material properties of FGM shells are assumed to be gradually graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of constituents. Equations of motion of variable thickness FGM toroidal shell segments are established based on Reddy’s third-order shear deformation shell theory with von Kármán nonlinearity. The Galerkin method and the Runge–Kutta method are used to solve the governing system of partial differential equations of motion, and then the nonlinear vibration response of variable thickness FGM toroidal shell segment is analyzed. A numerical analysis is also performed to show the effects of material and geometrical parameters on the nonlinear vibration response of variable thickness FGM toroidal shell segments.

This study determined the buckling characteristics of sphere-segmented toroidal shells subjected to external pressure. The proposed toroidal vessel comprises six spheres and six rings. Two laboratory models with the same nominal dimensions were manufactured, measured, tested, and evaluated. To investigate whether sphere-segmented toroidal shells are imperfection-sensitive structures with closely spaced eigenvalues, the subspace algorithm was applied to evaluate the first 50 eigenmodes, and the modified Riks algorithm was used to obtain post-buckling characteristics. The results indicated that the deviation between the results of the experimental and numerical analyses was within a reasonable range. The proposed sphere-segmented toroidal shells were highly imperfection-sensitive structures with closely spaced eigenvalues. Subsequently, imperfection sensitivity analysis confirmed this conclusion. In numerical analyses, the first eigenmode could be considered as the worst eigenmode of sphere-segmented toroidal shells. The trend of the equilibrium path of sphere-segmented toroidal shells was consistent with spherical shells, revealing instability. In addition, ellipticity and completeness exerted a negligible effect on the buckling load of sphere-segmented toroidal shells.

In this study, we investigated toroidal shell wrinkling in free hydroforming. We specifically focused on toroidal shells with a regular hexagonal cross-section. Membrane theory was used to examine the distribution of stress and yield load in both preform and toroidal shells. The wrinkling moment was then predicted using an empirical formula of shell buckling. In addition, the wrinkling state was investigated using a general statics method, and the free hydroforming of toroidal shells was simulated using the Riks method. Subsequently, nonlinear buckling and equilibrium paths were analyzed. A toroidal preform was manufactured, and free hydroforming experiments were conducted. Overall, the experimental results confirmed the accuracy of the theoretical predictions and numerical simulations. This indicates that the prediction method used in the study was effective. We also found that wrinkling occurs during hydroforming in the inner region of toroidal shells due to compressive stress. Consequently, we improved the structure of the toroidal shells and performed analytical calculations and numerical simulations for the analysis. Our results indicate that wrinkling can be eliminated by increasing the number of segments on the inner side of toroidal preforms, thereby improving the quality of toroidal shells.

The nonlinear stress–strain state of an open elliptical semi-toroidal thin shell is analyzed using the variational-difference method and mixed functionals. This approach allows algorithmizing the geometric part of the Kirchhoff–Love hypotheses and avoiding membrane locking. The stress–strain state of the shell under uniformly distributed internal pressure is calculated in the following three cases of fixation of edges: immovable hinged, clamped, and movable hinged. It is shown that the hinged edges considerably decrease the circumferential stresses near them. Due to significant meridional moments, the clamped edges cause compression on the outer shell surfaces.

Thin-walled toroidal shells are widely used in the construction During operation, various defects appear on the surface of the shells, in particular, local depressions on the outer and inner surfaces, causing stress concentration in the structure. A three-dimensional spline option of the finite element method was developed to determine the stress-strain state of a toroidal shell with a local deepening on the outer and inner surface. The numerical experiments were carried out. The regularities of the changes in a stress-strain state of the shell with the change in the geometric parameters of the deepening were noted.

This study investigates the free vibration of rotating stiffened toroidal shell segments in thermal environments. The shell segments are made of a functionally graded graphene platelet reinforced composite (FG-GPLRC)—advanced nanocomposite. Functionally graded (FG)-X, FG-O, and uniform distribution-type graphene platelet distribution patterns are considered. The equations of motion of rotating stiffened FG-GPLRC toroidal shell segments are derived based on variants of Reddy’s third-order shear deformation shell theory (TSDT) and the smeared-stiffener technique. Then, solutions are obtained using the Rayleigh–Ritz procedure. The effects of centrifugal and Coriolis forces and the initial hoop tension resulting from rotation are considered. Various numerical examples are produced to verify the implemented scheme and demonstrate the effects of material properties, rotating speed, temperature increment, boundary conditions, geometric parameters, and stiffeners on the natural frequencies of the shell. In addition, while an incomplete version of Reddy’s TSDT has been used widely in many recent studies, the present study points out significant differences between that version and the complete version of Reddy’s TSDT by numerical results and discussions.

Higher order equivalent layer and layer-wise models of elastic composite sandwich toroidal shells are developed using the Unified Carrera Formulation. Also, axisymmetric shell fixed at the ends is considered in detail. Numerical calculations are performed using the computer algebra software Mathematica 14.0. The results of calculation can be used as benchmark examples for finite element analysis of higher order composite sandwich toroidal shells.

The present work aims to quantify the influence of uncertainties in the ply orientation of multi-layered bio-inspired helicoidal laminated composite conical, hemispherical, and toroidal shells under thermal conditions. Any change in the ply orientation affects the free vibration behavior of the laminates. The present investigation focuses on the different levels of uncertainties in the ply orientation on the free vibration behavior of the shell. Moreover, the study also focuses on the sensitivity of the uncertainties in ply orientations on the free vibration behavior of the shells, which is also quantified. To quantify the stochastic free vibration behavior of the shells, the Gaussian process regression (GPR) machine learning algorithm-based surrogate model is developed to predict the frequencies of the shells. The surrogate is created in the framework of higher-order shear deformation theory. The uncertainties in the ply orientations are introduced using bootstrapping. The present results are compared with the stochastic frequencies obtained using Monte Carlo simulations (MCS) to determine the model’s accuracy. The study highlights the influence of the temperature, type of shell, and end conditions on the stochastic free vibration behavior of bio-inspired laminated shells.