In-plane and shear buckling analysis of FG-CNTRC annular sector plates based on the third-order shear deformation theory using a numerical approach

The buckling analysis of thick composite annular sector plates reinforced with functionally graded carbon-nanotubes (CNTs) is presented under in-plane and shear loadings based on the higher-order shear deformation theory. It is considered that the plate is resting on the Pasternak-type elastic foundation. The overall material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are estimated through the micromechanical model. The governing equations are derived on the basis of the higher-order shear deformation plate theory, and the quadratic form of the energy functional of the system is presented. An efficient numerical method is presented in the context of variational formulation to obtain the discretized version of stability equations. The validation of the present study is demonstrated through comparisons with the results available in the literature and then comprehensive numerical results are given to investigate the impacts of model parameters on the stability of CNT-reinforced composite annular sector plates.     

Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method

This paper presents the free vibration and buckling analyses of functionally graded carbon nanotube-reinforced (FG-CNTR) laminated non-rectangular plates, i.e., quadrilateral and skew plates, using a four-nodded straight-sided transformation method. At first, the related equations of motion and buckling of quadrilateral plate have been given, and then, these equations are transformed from the irregular physical domain into a square computational domain using the geometric transformation formulation via discrete singular convolution (DSC). The discretization of these equations is obtained via two-different regularized kernel, i.e., regularized Shannon’s delta (RSD) and Lagrange-delta sequence (LDS) kernels in conjunctions with the discrete singular convolution numerical integration. Convergence and accuracy of the present DSC transformation are verified via existing literature results for different cases. Detailed numerical solutions are performed, and obtained parametric results are presented to show the effects of carbon nanotube (CNT) volume fraction, CNT distribution pattern, geometry of skew and quadrilateral plate, lamination layup, skew and corner angle, thickness-to-length ratio on the vibration, and buckling analyses of FG-CNTR-laminated composite non-rectangular plates with different boundary conditions. Some detailed results related to critical buckling and frequency of FG-CNTR non-rectangular plates have been reported which can serve as benchmark solutions for future investigations.

     

Thermal buckling analysis of temperature-dependent FG-CNTRC quadrilateral plates

The main objective of the present study is to analyze the thermal buckling of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) quadrilateral plates. Functionally graded patterns are introduced for the distribution of the carbon nanotubes (CNTs) through the thickness direction of the plate. The effective material properties of nanocomposite plate reinforced by CNTs are considered to be temperature-dependent (TD) and estimated using the micromechanical model. By the use of minimum total potential energy principle and based on the first-order shear deformation theory of plates, the stability equations are obtained. In order to use the generalized differential quadrature (GDQ) method and solve the stability equations, the irregular domain of quadrilateral plate is transformed into regular computational domain employing the mapping technique. The efficiency and accuracy of the proposed approach are first validated. Then, a comprehensive parametric study is presented to examine the effects of model parameters on the thermal buckling of FG-CNTRC quadrilateral plates. The results indicate that considering temperature dependency of the material properties plays an important role in the stability of the FG-CNTRC quadrilateral plates subjected to thermal loading.     

Analysis of axially temperature-dependent functionally graded carbon nanotube reinforced composite plates

This article presents a comprehensive analysis to investigate the static buckling stability and static deflection of axially single-walled (SW) functionally graded (FG) carbon nanotubes reinforced composite (CNTRC) plates with temperature-dependent material properties and graded by different functions, for the first time. The distribution of the carbon nanotubes is described by two functions, one for the x-direction CNTs distribution (CNTRC-x plate) and another for z-direction CNTs distribution (CNTRC-z plate). The graduation functions of CNTs are unidirectional (UD CNTRC), FG-X CNTRC, FG-O CNTRC, and FG-V CNTRC. The extended rule of mixture and the molecular dynamics simulations are exploited to evaluate the equivalent mechanical properties of FG-CNTRC plate. Equilibrium equations are formulated using principal of Hamilton and solved analytically using Galerkin method to cover various boundary conditions. New higher order shear deformation theory is proposed. The numerical results gained by the proposed solution are verified by comparing with those of published ones. Numerical results present influences of gradation function, inhomogeneity parameters, aspect ratio, thickness ratio, boundary conditions and temperature on the static buckling and deflection of FG-CNTRC plate using modified higher order shear theories.

     

Static stability analysis of agglomerated multi-scale hybrid nanocomposites via a refined theory

Present paper is proposed to capture the influences of carbon nanotubes’ agglomeration on the stability behaviors of multi-scale hybrid nanocomposite beams within the frameworks of refined higher order beam theories for the first time. In this research, a mixture of macroscale and nanoscale fillers will be utilized to be dispersed in an initial matrix to possess a multi-scale hybrid nanocomposite. The equivalent material properties are seemed to be calculated coupling the Eshelby–Mori–Tanaka model with the rule of the mixture to consider the effects of carbon nanotubes inside the probably generated clusters while finding the mechanical properties of such novel hybrid nanocomposites. Furthermore, an energy-based approach is implemented to obtain the governing equations of the problem utilizing a refined higher order beam theorem. Next, the derived equations will be solved in the framework of Galerkin’s well-known analytical method to reach the critical buckling load. It is worth mentioning that influence of various boundary conditions is included, too. Once the validity of presented results is proven, a set of numerical examples are presented to explain how each variant can affect the structure’s stability endurance.

     

Optimum stacking sequence design of composite laminates for maximum buckling load capacity using parameter-less optimization algorithms

This paper presents a comparative study of the application of parameter-less meta-heuristic algorithms in optimum stacking sequence design of com of composite laminates for maximum buckling load capacity. Here, JAYA algorithm, along with Salp Swarm Algorithm, Colliding Bodies Optimization, Grey Wolf Optimizer, and Genetic Algorithm with standard setting and self-adaptive version are implemented to the problem of composite laminates with 64 graphite/epoxy plies with conventional ply angles, under several bi-axial cases and panel aspect ratios. Optimization objective is to maximize the buckling load of symmetric and balanced laminated plate. Statistical analysis are performed for six cases and the results are compared in terms of mean, standard deviation, the coefficient of variation, best and worst solutions, accompanied by the percentage of the independent runs that found the global optimum \(\left( {{R_{{\text{op}}}}} \right)\) and near global optimum \(\left( {{R_{{\text{no}}}}} \right)\). The Kruskal–Wallis nonparametric test is also utilized to make further confidence in the examinations. Numerical results show the high capability of the JAYA algorithm for maximizing the buckling capacity of composite plates.

     

Buckling and postbuckling of circular plates containing concentric penny-shaped delaminations

Buckling and postbuckling analyses of circular laminated composite plates with delaminations are presented. An axisymmetric finite element model based on a layer-wise laminated composite plate theory is developed to formulate the problem. Geometric nonlinearity in the sense of von Kármán and imperfections in the form of initial global deflection and initial delamination openings are included. A simple contact algorithm which precludes the physically inadmissible overlapping between delaminated surfaces is proposed and incorporated into the analysis.

Numerical results are obtained addressing the effects of the initial imperfections, the number of delaminations and their sizes on the critical buckling load and buckling mode shapes as well as postbuckling responses.     

Size dependency in nonlinear instability of smart magneto-electro-elastic cylindrical composite nanopanels based upon nonlocal strain gradient elasticity

In the present article, a new size-dependent panel model is established incorporating the both hardening-stiffness and softening-stiffness small scale effects jointly with electrostatics and magnetostatics to study analytically the buckling and postbuckling behavior of smart magneto-electro-elastic (MEE) composite nanopanels under combination of axial compression, external electric and magnetic potentials. To this end, the nonlocal strain gradient elasticity theory in conjunction with the Maxwell equations is applied to the classical panel theory to develop a more comprehensive size-dependent panel model including simultaneously the both nonlocality and strain gradient size dependency. With the aid of the virtual work’s principle, the size-dependent differential equations of the problem are derived. The attained non-classical governing differential equations are solved analytically by means an improved perturbation technique within the framework of the boundary layer theory of shell buckling. Explicit analytical expressions associated with the nonlinear axial stability equilibrium paths of the electromagnetic actuated smart MEE composite nanopanels including nonlocality and strain gradient micro-size dependency are proposed. It is displayed that the nonlocal size effect leads to reduce the buckling stiffness, while the strain gradient size dependency causes to enhance it. Moreover, it is found that by applying a negative electric field as well as positive magnetic field, the influences of the nonlocal and strain gradient size effects on the critical buckling load of an axially loaded MEE composite nanopanel are more significant.

     

FE-FD model for magneto-elastic buckling of ferromagnetic plates

A cantilever beam-plate, of magnetically soft material, inserted with its middle surface normal to a uniform magnetic field will buckle when the magnetic field reaches a critical value. The system of differential equations derived in this study are based on the Moon-Pao's model. The basic difference lies in the abandonment of the assumption that the magnetic couple acting on an element of the plate is proportional to the rotation of the middle surface of the plate. This assumption is employed in previous studies and leads to linear differential equations that is similar to the buckling problem of compressed rods or plates. Consequently, the complexity of the problem is reduced considerably. The derivation in this paper shows that the magnetoelastic buckling problem of the beam-plate is nonlinear. The solution to the system of differential equations is derived by coupling a finite element model for the magnetic field and a finite difference model for the cantilever beam-plate. The critical magnetic field, derived in this study is about 20% smaller than the experimental one, while theoretical results from previous works are about 180% larger than the experimental ones.     

Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model

This paper aims to investigate the size scale effect on the buckling and post-buckling of single-walled carbon nanotube (SWCNT) rested on nonlinear elastic foundations using energy-equivalent model (EEM). CNTs are modelled as a beam with higher order shear deformation to consider a shear effect and eliminate the shear correction factor, which appeared in Timoshenko and missed in Euler–Bernoulli beam theories. Energy-equivalent model is proposed to bridge the chemical energy between atoms with mechanical strain energy of beam structure. Therefore, Young’s and shear moduli and Poisson’s ratio for zigzag (n, 0), and armchair (n, n) carbon nanotubes (CNTs) are presented as functions of orientation and force constants. Conservation energy principle is exploited to derive governing equations of motion in terms of primary displacement variable. The differential–integral quadrature method (DIQM) is exploited to discretize the problem in spatial domain and transformed the integro-differential equilibrium equations to algebraic equations. The static problem is solved for critical buckling loads and the post-buckling deformation as a function of applied axial load, CNT length, orientations and elastic foundation parameters. Numerical results show that effects of chirality angle, boundary conditions, tube length and elastic foundation constants on buckling and post-buckling behaviors of armchair and zigzag CNTs are significant. This model is helpful especially in mechanical design of NEMS manufactured from CNTs.

     

Effects of random system properties on the thermal buckling analysis of laminated composite plates

This paper examines the effect of random system properties on thermal buckling load of laminated composite plates under uniform temperature rise having temperature dependent properties using HSDT. The system properties such as material properties, thermal expansion coefficients and thickness of the laminate are modeled as independent random variables. A C⁰ finite element is used for deriving the eigenvalue problem. A Taylor series based first-order perturbation technique is used to handle the randomness in the system properties. Second-order statistics of the thermal buckling load are obtained. The results are validated with those available in the literature and Monte Carlo simulation.     

Natural frequency and buckling optimization of laminated hybrid composite plates using genetic algorithm and simulated annealing

In this study, genetic algorithm and simulated annealing are used to maximize natural frequency and buckling loads of simply supported hybrid composite plates. The aim of the study is to use two different techniques of optimization on the frequency and buckling optimization of composite plates, and compare the techniques for their effectiveness. The composite plate is made of carbon/epoxy and glass/epoxy hybrid plies, and assumed to be symmetric and balanced. The effect of hybridization is investigated using both techniques. The buckling problem has many maxima in the vicinity of local maxima. The best configurations are identified for different plate aspect ratios. The performance of both techniques is compared in terms of number of function evaluations as well as the capability of finding best configurations.     

Buckling analysis of laminated composite plates with an elliptical/circular cutout using FEM

In this study, a buckling analysis was carried out of a woven–glass–polyester laminated composite plate with an circular/elliptical hole, numerically. In the analysis, finite element method (FEM) was applied to perform parametric studies on various plates based on the shape and position of the elliptical hole. This study addressed the effects of an elliptical/circular cutout on the buckling load of square composite plates. The laminated composite plates were arranged as symmetric cross-ply [(0°/90°)₂]_s and angle-ply [(15°/−75°)₂]_s, [(30°/−60°)₂]_s, [(45°/−45°)₂]_s. The results show that buckling loads are decreased by increasing both c/a and b/a ratios. The increasing of hole positioned angle cause to decrease of buckling loads. Additionally, the cross-ply composite plate is stronger than all other analyzed angle-ply laminated plates.     

On the stability of plates under combined compression and in-plane bending

The conventional stability analysis of plates under combined compression and in-plane bending is based on the assumption that the plate is free to move laterally and, hence, the restraints imposed by the attached elements against this motion are ignored. The paper explores the influence of these restraints on the plate under this type of loading. The unloaded edges are assumed to be partially restrained against in-plane translation while remaining straight and the distributions of the resulting forces acting on the plate are shown. The stability analysis is done numerically using the Galerkin method and various strategies that economize the numerical implementation are presented. The results are obtained showing the variation of the buckling load, from free edge translation to fully restrained, for simply supported and clamped unloaded edges for various plate aspect ratios and stress gradient coefficients. An apparent decrease in the buckling load is observed due to these destabilizing forces acting in the plate and changes in the buckling mode are observed by increasing the intensity of the lateral restraint. A comparison is made between the buckling loads predicted from various formulae in stability standards based on free edge translation and the values derived from the present investigation. A difference of about 34% in the predicted buckling load and different buckling load were found.     

An incremental galerkin method for plates and stiffened plates

In order to perform a detailed analysis of large deflection behavior of a rectangular plate or stiffened plate, an efficient semi-analytical method is developed. First, incremental forms of the governing differential equations of plates and stiffened plates with initial deflection are derived. These equations are linearized and may be easily solved. Secondly, these equations are solved for each load increment by the Galerkin method with a special consideration of simply supported boundaries.A procedure of equilibrium correction at intermediate load steps is presented such that good accuracy of the solution may be maintained with larger load steps.This method is successfully applied to plates with initial deflection subjected to in-plane as well as out-of-plane loads to obtain the whole histories of the behavior of these plates. Application of this method to stiffened plates with initial deflection is also presented. Comparisons of results obtained by this method with those obtained by other methods are made and the validity of the method is demonstrated.This incremental version of the Galerkin method is found to be extremely advantageous in certain types of plate and stiffened plate problems. These types are identified and the efficiency of the method is demonstrated.     

Post-buckling behaviour of moderately thick cylindrically orthotropic annular plates

A finite element formulation including the effects of shear deformation and cylindrically orthotropic material properties is described for studying the post-buckling behaviour of annular plates. Numerical results for the buckling load parameter and ratios of nonlinear load parameter to buckling load parameter for various values of orthotropic properties, thicknesses and radii ratios of the plates are presented.     

Post-buckling behavior of thin steel plates using computational models

Thin plates loaded in plane will buckle at very small loads, and due to unavoidable out-of-plane imperfections, the theoretical buckling load cannot be observed experimentally. If the plate is adequately supported along its boundaries, it will be able to carry a much higher load than the theoretical buckling load.Computational models can be used to study the post buckling behaviour of thin plate structures up to failure. Failure of such structures is usually due to large out-of-plane deflections, yielding, and rupture. Therefore, the computational model should include the effects of geometric and material nonlinearities. In this paper, the nonlinear finite element analysis program NONSAP and ANSR-III were modified and used in the analysis. Since these programs did not include the suitable elements and material properties to conduct the subject study, new elements and new material properties were added to the programs. In particular, a thin shell element was added and the solution routines were modified to improve its accuracy and efficiency.The modified programs were used on a Super Computer to calculate the post buckling strength of stiffened and unstiffened plates subjected to uniaxial compression, and plates subjected to in plane bending or shear. Crippling of plates subjected to in-plane or eccentric edge compressive loads was also examined. The results from the computational models were compared with test results and reasonable agreements were obtained. A computational model was developed for a multi-story thin steel plate shear wall subjected to cyclic loading and the results from the model were compared with experimental results, and again agreement was achieved.     

Shear flexible element based on coupled displacement field for large deflection analysis of laminated plates

A shear deformable theory accounting for the transverse-shear (in the sense of Reissner-Mindlin’s thick plate theory) and large deflections (in the sense of von Karman theory) is employed in the construction of variational statement. A four-node, lock-free, shear-flexible rectangular plate element based on the coupled displacement field is developed in this paper to carry out the large deflection analysis. The displacement field of the element is derived by making use of the linearized equations of static equilibrium. A bi-cubic polynomial distribution is assumed for the transverse displacement ‘w’. The field distribution for the in-plane displacements (u,v) and plate normal rotations (θ_x, θ_y) and twist (θ_xy) is derived using equilibrium of composite strips parallel to the plate edges. The displacement fields so derived are coupled through material couplings. The transverse shear strain fields of the proposed element do not contain inconsistent terms, so that the element predicts even shear-rigid bending accurately.The element is validated for a series of numerical problems and results for deflections and stresses are presented for rectangular composite plates with various boundary conditions, loading and lay-ups. The influence of the sign of the loading on the deflection of unsymmetrically laminated plates, in the large deflection regime is also investigated.     

Asymmetric thermal buckling of temperature dependent annular FGM plates on a partial elastic foundation

In this investigation, the asymmetrical buckling behaviour of FGM annular plates resting on partial Winkler-type elastic foundation under uniform temperature elevation is investigated. Material properties of the plate are assumed to be temperature dependent. Each property of the plate is graded across the thickness direction using a power law function. First order shear deformation plate theory and von Kármán type of geometrical nonlinearity are used to obtain the equilibrium equations and the associated boundary conditions. Prebuckling deformations and stresses of the plate are obtained considering the deflection-less conditions. Only plates which are clamped on both inner and outer edges are considered. Applying the adjacent equilibrium criterion, the linearised stability equations are obtained. The governing equations are divided into two sets. The first set, which is associated with the in-contact region and the second set which is related to contact-less region. The resulting equations are solved using a hybrid method, including the analytical trigonometric functions through the circumferential direction and generalised differential quadratures method through the radial direction. The resulting system of eigenvalue problem is solved iteratively to obtain the critical conditions of the plate, the associated circumferential mode number and buckled shape of the plate. Benchmark results are given in tabular and graphical presentations dealing with critical buckling temperature and buckled shape of the plate. Numerical results are given to explore the effects of elastic foundation, foundation radius, plate thickness, plate hole size, and power law index of the graded plate. It is shown that, stiffness foundation, and radius of foundation may change the buckled shape of the plate in both circumferential and radial directions. Furthermore, as the stiffness of the foundation or radius of foundation increases, critical buckling temperature of the plate enhances.     

Exact solutions for dynamic analysis of composite plates with distributed piezoelectric layers

Exact solutions are presented for analyzing dynamics of composite plates with piezoelectric layers bonded at the top and the bottom surfaces. The expressions for mechanical displacements, stresses, electric displacements and potential are derived from constitutive relations and field equations for the piezoelectric medium under applied surface traction and electric potential. The procedure is illustrated with a simply supported symmetric cross-ply (0°/90°/0°) graphite–epoxy composite plate covered with piezoelectric material polyvinylidene fluoride (PVDF). Results are in good agreement with those obtained from finite element model.