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The nonlinear buckling analysis of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates subjected to axial compression load is analytically examined in this paper. Assuming that the FG-CNTRC plates are stiffened by an oblique FG-CNTRC stiffener system. Reddy’s higher-order shear deformation plate theory (HSDPT) with the geometrical nonlinearities of von Kármán is applied to establish the basic formulations. Moreover, the smeared stiffener technique is successfully improved for the higher-order shear deformable anisotropic oblique stiffener system by using a homogeneous model of the anisotropic beam. Galerkin’s method is used to achieve the expressions of critical buckling loads and postbuckling load-deflection curves in explicit form. The numerical values display the influences of FG-CNTRC stiffeners, material, and geometrical properties on the nonlinear buckling response of plates.

Auxetic materials are usually designed as cores for structures subject to high impulse loads. Furthermore, the lightweight and high load capacity of the auxetic core construction is also an important advantage even for structures subjected to static loads. The combination of auxetic core and face sheets made by the advanced composite materials is a solution to dramatically increase the load-carrying capacity of the structure. In this paper, a new design of auxetic-core cylindrical shells with carbon nanotube-reinforced coatings is presented. Additionally, the nonlinear buckling behaviors of auxetic-core cylindrical shells with carbon nanotube-reinforced coatings under axially compressive loads are investigated. Three distributed types of functionally graded carbon nanotube-reinforced coatings and the honeycomb lattice form of the auxetic core are investigated. The homogenization model for auxetic lattice structures is considered to constitute the formulations of stiffnesses of the core layer. The nonlinear basic formulations are formulated by using the geometrically nonlinear Donnell shell theory considering Pasternak’s foundation. The Galerkin procedure can be applied three times for three states of buckling behaviors, and the expressions of the compressive load-maximal deflection and compressive load-average end shortening postbuckling curves are achieved. The numerically obtained investigations present the significant effects of auxetic core, volume fraction, direction arrangement and distributed law of carbon nanotube, foundation stiffnesses, geometrical parameters of auxetic core and shell on the critical buckling load and postbuckling behavior of structures.

In this paper, a new improved smeared stiffener technique for the higher-order shear deformable anisotropic stiffeners is presented. The thermo-electro-mechanical nonlinear buckling response of functionally graded graphene reinforcement composite (FG-GRC) laminated plates subjected to the combination of external and axial compression loads in the thermal environment with the inverse piezoelectric effect of a piezoelectric layer is analytically considered. It is proposed that the FG-GRC laminated plates are stiffened by the FG-GRC laminated stiffener system in the longitudinal or transverse directions at the bottom surface and the piezoelectric layer are attached at the top surface. The distribution law of graphene is suitably designed to ensure the piecewise continuity of material. The nonlinear higher-order shear deformation plate theory (HSDPT) is employed to formulate the stability equation system. The stress function is introduced and Galerkin’s method is employed to achieve the algebraically nonlinear stability equation system. Then, the simple calculation procedure can be applied to solve the acquired equation systems, and the expressions of critical buckling loads and postbuckling load-deflection curves are presented in explicit forms. The numerical investigations validate the significant influences of temperature, material, laminated stiffeners, and geometrical features on the nonlinear thermo-electro-mechanical buckling behavior of FG-GRC laminated plates.

In this article, the nonlinear thermomechanical buckling behaviors of sandwich functionally graded plates subjected to an axial compression and external pressure are analytically analyzed resting on nonlinear elastic foundation. Assuming that the plates are reinforced by oblique stiffeners and rested on nonlinear elastic foundation. The formulations are established using the higher-order shear deformation theory taking into account the geometrical nonlinearity of von Kármán. The Lekhnitskii’s smeared stiffener technique is developed for shear deformable oblique stiffener system using the coordinate transformation technique with both mechanical and thermal terms. The Galerkin method is utilized to obtain the nonlinear algebraically equation system, then, solve it to determine the explicit expressions of critical buckling loads and postbuckling load–deflection curves. Numerical results show the effects of temperature, nonlinear elastic foundation, stiffeners, and material and geometrical properties on nonlinear behaviors of plates.

An analytical approximation for nonlinear stability analysis of core-corrugated sandwich graphene-reinforced cylindrical shells is presented in this paper. A homogenization model for corrugated structures is used for corrugated core layer and the governing equation system for cylindrical shells are formulated considering the von Karman-Donnell nonlinear theory. Three-term solution form of deflection is chosen, and the nonlinear postbuckling relation can be formulated by applying the Galerkin procedure. The result of examinations validates the effects of the corrugated core layer, graphene volume fraction and graphene distributions with different geometric ratios on the nonlinear stability behaviors of corrugated cylindrical shells.

This paper presents the axisymmetric vibration responses of sandwich Functionally graded material (FGM) shallow spherical caps with lightweight porous core subjected to uniformly external loads and shell-foundation Pasternak interaction. Based on the first-order shear deformation theory (FSDT) with von Karman geometrical nonlinearity, the governing equations are established. By using the Galerkin method and the Runge–Kutta method, the fundamental frequencies and the nonlinear dynamic responses of the shell are obtained. The effects of porosity coefficient, geometrical parameters, and foundation are considered.

In this paper, a new design of sandwich cylindrical panels is proposed with two functionally graded carbon nanotube reinforced composite (FG-CNTRC) face sheets and lattice core. The lattice core layer is modeled by improving the novel smeared technique for stiffener according to the first-order shear deformation theory (FSDT). Nonlinear vibration behavior of first-order shear deformable cylindrical panels with the geometric nonlinearities is analyzed in present paper. The stress function is considered and the Galerkin method is used to formulate the nonlinear motion equation system. Nonlinear dynamic responses of panels can be achieved by using the fourth order Runge-Kutta method. Numerical investigations can show the very large effects of lattice core layer, volume fraction of carbon nanotube, type of carbon nanotube distribution on the nonlinear vibration behavior of sandwich FG-CNTRC cylindrical panels.

Nonlinear buckling analysis for honeycomb auxetic-core sandwich toroidal shell segments with CNT-reinforced face sheets surrounded by elastic foundations under the radial pressure is presented in this study. The basic equation system of shells is established based on the von Kármán–Donnell nonlinear shell theory, combined with Stein and McElman approximation. Meanwhile, the foundation-shell elastic interaction is simulated by the foundation model based on the Pasternak assumption. The Galerkin procedure is utilized to achieve the pre-buckling and post-buckling responses for the shell, from which the radially critical buckling load is determined. Numerical analysis shows the various influences of auxetic-core layer, CNT-reinforced face sheets, and elastic foundation on the pre-buckling and postbuckling behavior of sandwich shells with CNT reinforced face sheets.

A semi-analytical approach of nonlinear vibration of functionally graded carbon nanotubes reinforced composite (FG-CNTRC) plates and cylindrical panels stiffened by longitudinal or transversal FG-CNTRC stiffeners by using the first-order shear deformation theory (FSDT) with the geometrical nonlinearities in von Karman sense is reported in this paper. The novel smeared stiffener technique is improved for FG-CNTRC stiffeners according to the FSDT. The stress function is considered and the Galerkin method is used to formulate the nonlinear motion equation system, and dynamic responses of panels are obtained by using the fourth order Runge-Kutta method. Numerical investigations can show the very large effects of FG-CNTRC stiffeners, volume fraction of carbon nanotube, type of carbon nanotube distribution of panels and stiffeners with different geometrical parameters on the nonlinear vibration behaviors of stiffened FG-CNTRC plates and cylindrical panels.

In this paper, an analytical approach for nonlinear buckling and post-buckling behavior of stiffened porous functionally graded plate rested on Pasternak's elastic foundation under mechanical load in thermal environment is presented. The orthogonal and/or oblique stiffeners are attached to the surface of plate and are included in the calculation by improving the Lekhnitskii's smeared stiffener technique in the framework of higher-order shear deformation plate theory. The complex equilibrium and stability equations are established based on the Reddy's higher-order shear deformation plate theory and taken into account the geometrical nonlinearity of von Kármán. The solution forms of displacements satisfying the different boundary conditions are chosen, the stress function method and the Galerkin procedure are used to solve the problem. The good agreements of the present analytical solution are validated by making the comparisons of the present results with other results. In addition, the effects of porosity distribution, stiffener, volume fraction index, thermal environment, elastic foundation… on the critical buckling load and post-buckling response of porous functionally graded material plates are numerically investigated.

This paper presents an analytical approach to investigate the nonlinear stability of multilayer-functionally graded plates stiffened by orthogonal and/or oblique functionally graded stiffeners subjected to axial compression and/or thermal load. The equilibrium equation system is established by using the first-order shear deformation plate theory taking into account the plate–foundation interaction, geometrical nonlinearity in von Kármán sense and initial geometrical imperfection. An improved Lekhnitskii’s smeared stiffener technique is applied for oblique stiffeners with thermal terms and shear deformation of stiffeners. The governing equations are solved by Galerkin procedure to obtain the explicit expressions of buckling loads and postbuckling load–deflection curves. Results show the effects of material and geometrical properties, boundary conditions, elastic foundation parameters and initial imperfection on the buckling and postbuckling load-carrying capacity of plates.

This paper presents a semi-analytical approach for investigating the nonlinear buckling and postbuckling of spiral corrugated sandwich functionally graded (FGM) cylindrical shells under external pressure and surrounded by a two-parameter elastic foundation based on Donnell shell theory. The improved homogenization theory for the spiral corrugated FGM structure is applied and the geometrical nonlinearity in a von Karman sense is taken into account. The nonlinear equilibrium equation system can be solved by using the Galerkin method with the three-term solution form of deflection. An explicit solution form for the nonlinear buckling behavior of shells is obtained. The critical buckling pressure and the postbuckling strength of shells are numerically investigated. Additionally, the effects of spiral corrugation in enhancing the nonlinear buckling behavior of spiral corrugated sandwich FGM cylindrical shells are validated and discussed.

A semi-analytical approach to investigate the nonlinear vibration axisymmetric analysis of functionally graded sandwich shallow spherical caps under external pressure resting on elastic foundation in thermal environment is presented. The governing equations are derived by using the first-order shear deformation theory taking into account von Karman geometrical nonlinearity and Pasternak’s two-parameter elastic foundation. The motion equations are determined by Galerkin method and the obtained equation is numerically solved by using Runge–Kutta method. Results of nonlinear dynamic responses show the effects of foundation, material, geometric parameters, and temperature change on the nonlinear vibration of shells.

In part 1, the governing nonlinear dynamic equations of FGM sandwich doubly curved shallow shells reinforced by FGM stiffeners on elastic foundation subjected to mechanical and thermal loading are established based on the first order shear deformation theory (FSDT) with von Kármán - type nonlinearity and smeared stiffener technique. In the present part, the fourth-order Runge-Kutta method is applied to investigate influences of models of the shells, FGM stiffeners, thermal environment, elastic foundation, and geometrical parameters on the natural frequencies and dynamic nonlinear responses of stiffened FGM sandwich doubly curved shallow shells.

Nonlinear vibration of FGM sandwich doubly curved shallow shells reinforced by FGM stiffeners subjected to mechanical and thermal loading are investigated based on the first-order shear deformation theory (FSDT) with von Karman type nonlinearity, taking into account initial geometrical imperfection and smeared stiffener technique. Four material models of the FGM sandwich shells are presented. Explicit expressions for natural frequencies, nonlinear frequency-amplitude relation, and time-deflection curves of the FGM sandwich shallow shells are derived using Galerkin method.

Based on the third order shear deformation shells theory (TSDT) with von Karman geometrical nonlinearity taking into account initial imperfection and smeared stiffener technique, this paper presents post-buckling analyses for imperfect functionally graded stiffened doubly curved shallow shells resting on elastic foundations under mechanical load, thermal load and thermo-mechanical load by an analytical method. Shells are reinforced by stiffeners system in which material properties of the stiffeners are assumed to be continuously graded in the thickness direction. The material properties are assumed to be temperature- dependent. Two cases of uniform rising temperature and nonlinearly changing of temperature across the thickness are considered. The expressions of thermal parameters are found in the analytical form. Applying Galerkin method, explicit expressions to determine the critical buckling load and the post-buckling mechanical and thermal load-deflection curves are obtained. In the case of temperature-dependent material properties, an iterative algorithm has been presented. The effects of thermal element, FGM stiffeners, geometrical and material parameters, initial imperfection, and foundation are investigated.