Post-buckling Curves Research Articles

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.

A new analytical approach for nonlinear thermo-mechanical postbuckling of FG-GPLRC circular plates and shallow spherical caps stiffened by spiderweb stiffeners

For the first time, the problem of nonlinear postbuckling of circular plates and shallow spherical caps reinforced by meridian, parallel stiffeners, and spiderweb stiffeners based on the Donnell shell theory (DST) and von Kármán geometric nonlinearities is presented. The caps/plates and stiffeners are made from functionally graded graphene platelet-reinforced composite (FG-GPLRC). These stiffened structures are subjected to uniformly distributed external pressure or/and uniformly distributed thermal loads and are rested on a nonlinear elastic foundation. By expanding Lekhnitskii's smeared stiffener technique and employing the Ritz method of energy, the formulas to determine the postbuckling curves of the external pressure–deflection and thermal load-deflection relations of stiffened plates/spherical caps are derived. Meaningful discussions of the various influences of FG-GPLRC stiffeners, material distributions of plate/cap and stiffeners, and geometrical, material, and foundation parameters are shown in the content of the numerical investigations.

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 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.

Buckling of Stiffened Cone–Cylinder Structures Under Axial Compression

This paper studies the instability behavior of stringer-stiffened cone–cylinder shells under axial compressive load via computational finite element (FE) code, ABAQUS. First, the FE model was validated using past experiments on four cone–cylinder specimens (two unstiffened and two stringer-stiffened) fabricated in pairs using mild steel. Results show that the comparison between numerical and experimental collapse loads was less than 10%. Second, parametric study was conducted using the validated FE model to investigate the effects of the location and number of stringers on the buckling behavior of axially compressed cone–cylinder structures. It has been found that reinforcing the cone–cylinder structures tends to produce higher buckling load when compared to the unstiffened ones. The presence of stringers changes the failure mode of the structure, and the initial post-buckling curves show a less steep negative slope when compared to that of the unstiffened counterpart. Externally stiffened structures were able to sustain more load as compared to the internally stiffened counterpart. Introducing longitudinal reinforcement on the conical section and ring reinforcement on the cone–cylinder junction produces a much higher buckling load compared to having stringer-stiffeners on the conical part only.

Analytical and mathematical simulation for nonlinear stability of scale-dependent magneto-electro-elastic system

By using differential quadrature method (DQM), a numerical investigation was provided for nonlinear stability behavior of magneto-electro-elastic (MEE) cylindrical shells at microscale. It is assumed that the cylindrical shell has been subjected to compressive loads leading to buckling phenomena in geometrically nonlinear regime. The non-uniformity of strain field has been inserted in the formulation for considering the microscale effects. The material properties of the shell are considered to be inhomogeneous with graded distribution. After solving the governing equations using DQM, it is realized that if the nanoscale shell is subjected to electrical and magnetic fields, the post-buckling path may be changed with the value of electrical voltage and magnetic potential. Also, strain gradient effects have remarkable influence on post-buckling curves and critical voltages.

An analytical approach to the nonlinear buckling behavior of axially compressed auxetic-core cylindrical shells with carbon nanotube-reinforced coatings

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.

Nonlinear buckling analysis of stiffened FG-GRC laminated cylindrical shells subjected to axial compressive load in thermal environment

The main aim of this paper is to investigate the postbuckling and buckling of functionally graded graphene-reinforced composite (FG-GRC) laminated cylindrical shells strengthened by stiffeners under axial compression with the uniform temperature change effect. The stiffener design option for FG-GRC cylindrical shells under axial compression is presented and a suitably smeared stiffener technique for GRC laminated stiffener system is developed. Basing on the von Kármán Donnell shell theory, the Galerkin’s procedure, and the three-term solution of deflection, the explicit forms of critical buckling and expression of load-deflection postbuckling curves relation are obtained. Effects of the GRC laminated stiffeners, the environment temperature, the graphene-reinforced parameters on the compressed buckling behavior are validated.

Nonlinear torsional buckling of functionally graded graphene‐reinforced composite (FG‐GRC) laminated cylindrical shells stiffened by FG‐GRC laminated stiffeners in thermal environment

AbstractThe main aim of this article is to establish an analytical approach on the torsional postbuckling for functionally graded graphene‐reinforced composite (FG‐GRC) laminated circular cylindrical shells stiffened by FG‐GRC laminated stiffener system in thermal environment and subjected to torsion loads. The polymer matrixes of shell skin and stiffeners are reinforced by graphene with two graphene reinforcement directions, namely, zigzag and armchair directions. The anisotropic smeared stiffener technique for FG‐GRC stiffener system is used, and the governing equations are established based on the Donnell shell theory with von Kármán–Donnell‐type geometrical nonlinearity and the Airy's stress function. A three‐state solution of deflection is chosen and Galerkin's method is applied, the explicit forms of critical buckling torsion and expression of load‐deflection postbuckling curves relation are obtained. The effects of graphene‐reinforced composite laminated stiffeners, uniformly distributed temperature, the volume fraction of graphene of stiffened shell on the nonlinear torsional buckling behavior are numerically investigated.

Thermal Postbuckling of Temperature-Dependent Functionally Graded Nanocomposite Annular Sector Plates Reinforced by Carbon Nanotubes

In this study, the thermal buckling and postbuckling of functionally graded (FG) nanocomposite annular sector plates reinforced by carbon nanotubes (CNTs) are numerically analyzed. The effective material properties of FG nanocomposite are temperature-dependent (TD) and evaluated via the modified micromechanical method and rule of mixture. Based on the higher-order shear deformation theory (HSDT) and using the principle of virtual work and variational differential quadrature (VDQ) approach, the unified weak form of discretized nonlinear governing equilibrium equations is derived. Then, by using the linear part of equations and solving the derived eigenvalue problem, the critical temperature rise and associated mode shapes are obtained, which are used as the initial guess in solving the nonlinear thermal postbuckling problem. The pseudo-arc-length method and an iterative solver are employed to obtain the nonlinear thermal postbuckling equilibrium path of the FG nanocomposite annular sector plates. The influences of geometrical parameters, boundary conditions (BCs), CNT volume fraction, and CNT distribution pattern on the critical temperature rise and thermal postbuckling behavior of the FG nanocomposite annular sector plates are evaluated and discussed. Also, comparisons are made between the results considering the TD and temperature-independent (TID) properties. It is demonstrated that for higher values of sector angle, the effect of sector angle on the critical temperature rise and thermal postbuckling path is negligible. Moreover, by increasing the sector angle, the effect of BCs of straight edges vanishes, and the critical temperature rise and thermal postbuckling curves of for BCs of CSCS and SCSC approach those for CCCC and SSSS ones.

Investigation of thermal preloading and porosity effects on the nonlocal nonlinear instability of FG nanobeams with geometrical imperfection

The aim of the present research is twofold. The first is to present a study on nonlinear thermo-mechanical bending and thermal postbuckling analysis of functionally graded (FG) porous perfect/imperfect nanobeam according to the nonlocal elasticity theory. The second, concurrent aim is to address the snap-through phenomenon in the thermally preloaded graded porous nanobeams due to lateral mechanical load. Two types of thermal loading including, uniform temperature rise and heat conduction through the thickness as well as two cases of porosity distribution, including uniform and uneven, are considered. Geometrically imperfect FG porous nanobeam with four different types of immovable boundary conditions is also taken into account. Thermo-mechanical material properties of the porous nanobeam are assumed to be position-dependent according to the modified rule of mixture. The nonlinear equilibrium equations are constructed using the Euler–Bernoulli beam model and von-Kármán type of geometrical nonlinearity. Chebyshev polynomial based Ritz method is utilized into the principle of virtual displacement to obtain the matrix representation of the nonlocal governing equations. Two different strategies, including the Newton–Raphson technique and direct displacement control scheme, are first proposed to extract the nonlinear bending and postbuckling curves of the graded nanobeam. Afterwards, to capture the snapping behavior and trace the path beyond the limit loads of the thermally preloaded nanobeam, the cylindrical arch-length technique is adopted. Numerical results are provided to explore the effect of different parameters such as the power-law index, nonlocal parameter, boundary conditions, porosity distribution, thermal loading, and imperfection amplitude on the nonlinear thermo-mechanical bending and instability of the FG nanobeam.

Nonlinear buckling and post-buckling behavior of shear deformable sandwich toroidal shell segments with functionally graded core subjected to axial compression and thermal loads

This paper presents an analytical investigation on buckling and post-buckling of shear deformable sandwich toroidal shell segments with functionally graded core and homogeneous face sheets. The effective material properties of core layer are graded in the thickness direction according to a simple power law distribution in terms of volume fraction index. The shells are surrounded by an elastic foundation and subjected to axial compressive or thermal loads. Reddy's third-order shear deformation shell theory (TSDT) is used to established formulations. A new solution using Galerkin method for solving a governing system of four-partial differential equations is carried out to obtain closed-form expressions of buckling stresses and post-buckling curves. Two cases of bowed-out and bowed-in sandwich toroidal shells are considered. Effects of material, geometric parameters and elastic foundation on stability behavior of the shells is analyzed in detail. The difference between bucking stresses basing on TSDT and those basing on classical shell theory (CST) is also clearly shown in this study.

Post-buckling behavior of functionally graded and carbon-nanotubes based structures with different mechanical loadings

This article aims to investigate the post-buckling behavior of plates and cylindrical panels made of functionally graded materials (FGM) and carbon-nanotubes reinforced composites (FG-CNTRC) under mechanical loads. Unlike the other high order shear deformation theories, the proposed formulation is elaborated within a double directors finite shell element which allows a parabolic distribution of the transverse shear strains, guarantee the zero condition of the transverse shear stresses on the extreme surfaces of the shell and introduces large deformations and finite rotations through a non-linear framework. Newton-Raphson with arc length control solution strategy is used to resolve the equilibrium paths. A power-law distribution and the extended rule of mixture are used to determine the effective material properties of functionally graded materials and carbon-nanotubes reinforced composites. The effectiveness and the accuracy of the formulation is checked via several comparisons with published results in the literature. Especially, some complex post-buckling curves of FGM and FG-CNTRC curved panels and plates with different mechanical loadings are provided that could be useful for future references. The effects of various parameters are also reported.

Post-buckling analysis of geometrically imperfect tapered curved micro-panels made of graphene oxide powder reinforced composite

The present research investigates post-buckling behavior of geometrically imperfect tapered curved micro-panels made of graphene oxide powder (GOP) reinforced composite. Micro-scale effects on the panel structure have been included based on strain gradient elasticity. Micro-panel is considered to be tapered based on thickness variation along longitudinal direction. Weight fractions of uniformly and linearly distributed GOPs are included in material properties based on Halpin-Tsai homogenization scheme considering. Post-buckling curves have been determined based on both perfect and imperfect micro-panel assumptions. It is found that post-buckling curves are varying with the changes of GOPs weight fraction, geometric imperfection, GOP distribution type, variable thickness parameters, panel curvature radius and strain gradient.

Post-buckling analysis of geometrically imperfect nanoparticle reinforced annular sector plates under radial compression

Buckling and post-buckling behaviors of geometrically imperfect annular sector plates made from nanoparticle reinforced composites have been investigated. Two types of nanoparticles are considered including graphene oxide powders (GOPs) and silicone oxide (SiO2). Nanoparticles are considered to have uniform and functionally graded distributions within the matrix and the material properties are derived using Halpin-Tsai procedure. Annular sector plate is formulated based upon thin shell theory considering geometric nonlinearity and imperfectness. After solving the governing equations via Galerkin\'s technique, it is showed that the post-buckling curves of annular sector plates rely on the geometric imperfection, nanoparticle type, amount of nanoparticles, sector inner/outer radius and sector open angle.

Evaluation of geometrically nonlinear terms in the large-deflection and post-buckling analysis of isotropic rectangular plates

The nonlinear mechanical response of highly flexible plates and shells has always been of primary importance due to the widespread applications of these structural elements in many advanced engineering fields. In this study, the Carrera Unified Formulation (CUF) is used in a total Lagrangian framework to analyze the large-deflection and post-buckling behavior of isotropic rectangular plates based on different nonlinear strain assumptions. The scalable nature of the CUF provides us with the ability to tune the structural theory approximation order and the strain–displacement assumptions opportunely. In this work, the Newton–Raphson linearization scheme with a path-following constraint is used in the framework of the 2-D CUF to solve the geometrically nonlinear problems to draw important conclusions about the consistency of many assumptions made in the literature on the kinematics of highly flexible plates. In this regard, the effectiveness of the well-known von Kármán theory for nonlinear deformations of plates is investigated with different modifications such as the thickness stretching and shear deformations due to transverse deflection. The post-buckling curves and the related stress distributions for each case are presented and discussed. According to the results, the full Green–Lagrange nonlinear model could predict the nonlinear behavior of plates efficiently and accurately, whereas other approximations produce considerable inaccuracies in the case of thick plates subjected to large rotations and deflections.

Nonlinear buckling and postbuckling of sandwich FGM cylindrical shells reinforced by spiral stiffeners under torsion loads in thermal environment

The main purpose of this paper is the investigation of the nonlinear torsional buckling and postbuckling of sandwich functionally graded cylindrical shells with the von Karman large deflection nonlinearities under thermal effect by an analytical method. The shell skin is reinforced by stringer, circular ring and spiral stiffeners, the material properties of shell skin and stiffeners are assumed to vary continuously through the thickness. The very large effect of spiral stiffeners on the buckling load-carrying capacity of a cylindrical shell in comparison with orthogonal stiffeners is clearly proved in numerical investigations. Based on the Donnell shell theory and the improved smeared stiffener technique for both thermal and mechanical terms of spiral stiffeners, the equilibrium equations of the shell are established in this paper. By using the Galerkin method, the postbuckling curves and critical buckling loads are obtained. The effects of temperature change, stiffeners, material and dimensional parameters on the nonlinear torsional buckling and postbuckling of shell are numerically analyzed.

Solving Post-Buckling Characteristic of Thermal-Resistance Films Attached to Glass Façade via an Optimization Method

A novel thermal-resistance film is proposed, and its thermal post-buckling behavior is studied by using the meshless method. The Von Kármán plate model together with the nonlocal theory is adopted for the formulation of the problem. The post-buckling curve of the graphene sheet (GS) film is obtained. The effect of the parameters, such as nonlocal parameters, geometric size, geometric shape and elastic medium, on the thermal post-buckling response of the GSs film is examined. It is found that it is important to design proper geometric size and geometric shape and control the radius of tensionless area. This study provides theoretical guidance for fire resistance design of thermal-resistance films attached to glass façade.

NURBS-based postbuckling analysis of functionally graded carbon nanotube-reinforced composite shells

An investigation into the postbuckling and geometrically nonlinear behaviors of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shells is carried out in this study. The discrete nonlinear equation system is established based on non-uniform rational B-Spline (NURBS) basis functions and the first-order shear deformation shell theory (FSDT). The nonlinearity of shells is formed in the Total Lagrangian approach considering the von Karman assumption. The incremental solutions are obtained by using a modified Riks method. In the present formulation, the rule of mixture is used to estimate the effective material properties of FG-CNTRC shells. Effects of CNTs distribution, volume fraction and CNTs orientation on the postbuckling behavior of FG-CNTRC shells are particularly investigated. Exact geometries of shells are modeled by using NURBS interpolation. Several verifications are given to show the high reliability of the proposed formulation. Especially, some complex postbuckling curves of FG-CNTRC panels and cylinders are first provided that could be useful for future references.

Nonlinear buckling and post-buckling analysis of imperfect porous plates under mechanical loads

The nonlinear buckling and post-buckling response of imperfect porous plates is investigated analytically in this paper. The porous materials with elastic moduli are assumed to vary through the thickness of the plate according to two different distribution types. Governing equations are derived based on the classical shell theory taking into account Von Karman nonlinearity and initial geometrical imperfection. Explicit relations of load–deflection curves for rectangular porous plates are determined by applying stress function and Galerkin’s method. The accuracy of present theoretical formulation is verified by comparing it with available results in the literature. The effects of varying porosity distribution, porosity coefficient, boundary condition and imperfection on post-buckling behavior of the porous plate are studied in detail. A parametric study is carried out to investigate the effects of varying porosity distribution, porosity coefficient, boundary condition and imperfection on post-buckling behavior of the porous plate. The results show that the critical buckling loads decrease with increasing porosity coefficient and the post-buckling curves for nonlinear symmetric porosity distribution are always higher than those for nonlinear non-symmetric porosity.