Search

Curved nanobeams are one of the essential components in manufacturing nano-electromechanical systems needing nonlinear stability design. In the current investigation, the nonlinear buckling characteristics of functionally graded porous reinforced curved (FGPRC) nanobeams having different degrees of curvature are analyzed by counting the higher-order gradients of the classical strain tensor as well as nonlocal-type interatomic interactions. In this regard, two independent length-scale constants within the framework of the nonlocal strain gradient theory (NSGT) of continuum elasticity are taken into account. Via employing the promising low computational cost and geometrically adaptable method of isogeometric collocation, various branches of NSGT-based equilibrium graphs of FGPRC nanobeams are plotted relevant to each considered degree of curvature. It is extrapolated that the quantity of graphene platelet (GPL) weight fraction has a negligible influence on the significance of nonlocal-type of interatomic size dependency as well as the strain gradient kind of small-scale effect in the value of the maximum deflections or lateral loads at the detected limit points, especially attributed to a higher degree of curvature. Besides, it can be remarked that, in the FGPRC nanobeam owning a small degree of curvature, lessening the quantity of porosity index results in an increment in the significance of the nonlocal-type of interatomic size dependency as well as the strain gradient kind of small-scale effect on the maximum deflection at both upper and lower limit points. However, in the FGPRC nanobeams owning medium and large degrees of curvature, it gets lesser at the upper limit point, but becomes higher at the lower limit one. [ABSTRACT FROM AUTHOR]

The Hencky bar-net model (HBM) has been shown to be a physical structural representation of the finite difference plate model (FDM). This paper extends the HBM to tackle buckling problems of rectangular plates under in-plane non-uniform loads. This class of problems requires a pre-buckling stress analysis involving plane-stress elasticity equations in order to establish the non-uniform stress distributions in the plate domain for input loads to the HBM. The converged results given by HBM compare well with existing reference solutions; thereby verifying the model. The HBM is also used to generate some new buckling solutions for rectangular plates under various point loads and partially distributed loads. [ABSTRACT FROM AUTHOR]

The paper is concerned with the sensitivity analysis of structural responses in context of linear and non-linear stability phenomena like buckling and snapping. The structural analysis covering these stability phenomena is summarised. Design sensitivity information for a solid shell finite element is derived. The mixed formulation is based on the Hu-Washizu variational functional. Geometrical non-linearities are taken into account with linear elastic material behaviour. Sensitivities are derived analytically for responses of linear and non-linear buckling analysis with discrete finite element matrices. Numerical examples demonstrate the shape optimisation maximising the smallest eigenvalue of the linear buckling analysis and the directly computed critical load scales at bifurcation and limit points of non-linear buckling analysis, respectively. Analytically derived gradients are verified using the finite difference approach. [ABSTRACT FROM AUTHOR]

In this paper, we give a targeted review of the state of the art in the study of planar elastic beams in large deformations, also in the presence of geometric nonlinearities. The main scope of this work is to present the different methods of analysis available for describing the possible equilibrium forms and the motions of elastic beams. For the sake of completeness, we start by giving an overview of the nonlinear theories introduced for approaching this argument and then we account for the variational principles and deformation energies introduced for modelling beams undergoing large deformations and displacements. We then consider different kinds of loads treated in the literature and the corresponding induced beam deformations. We conclude by accounting for the available analysis for stability and some considerations about problems where live loads are applied, as well as by describing some relevant numerical methods of use in the applications we have in mind. The selection criterion for the reviewed papers is dictated by the need to study large deformations and the dynamics of pantographic sheets. (Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc R Soc A 2016; 472(2185): 20150790), dell'Isola et al. (Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Z Angew Math Phys 2015; 66(6): 3473–3498), Turco et al. (Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Z Angew Math Phys 2016; 67(4): 1–28)]. [ABSTRACT FROM AUTHOR]

This paper deals with extending the domain of applicability of a recently developed Generalised Beam Theory (GBT) formulation intended to perform elastic linear buckling analyses of thin-walled members (i) exhibiting arbitrary flat-walled cross-sections (including those combining closed cells and open branches), and (ii) acted by general loadings. These loadings, which include transverse forces acting away from the member shear centre axis, are termed “general” in the sense that they may involve the presence of pre-buckling stress distributions associated with any possible combination of all the stress tensor membrane components ( σ xx , σ ss and τ xs , for a plane stress state), including cell shear flows – therefore, all the relevant geometrically non-linear effects need to be taken into consideration. After briefly presenting the main concepts and procedures involved in the development and implementation of the above GBT formulation, this same formulation is employed to analyse the buckling behaviour of beams with different types of cross-section geometry (containing closed cells) and exhibiting different loading and support conditions. In particular, they consist of (i) a RHS cantilever acted by two tip point loads, (ii) a closed-flange I-section simply supported beam subjected to a uniformly distributed load and (iii) a two-cell RHS section cantilever acted by tip transverse forces and couples. In all cases, the loads are applied both along the shear centre axis and also along axes parallel to it and located at the beam top and bottom surfaces. The results presented and discussed, which consist of pre-buckling stress fields, buckling curves and buckling mode shapes, are obtained by means of the newly released code GBTul 2.0 and validated by means of the comparison with shell finite element values obtained with the code Ansys . [ABSTRACT FROM AUTHOR]

Buckling mode interaction of elastic systems in the presence of initial imperfections is well known to have a detrimental effect on the response of a wide range of structural systems. This has been demonstrated mostly analytically for simple models, assuming small displacements, thus obtaining results with questionable validity in the post-buckling range. In order to acquire additional insight into this issue, in the first part of the present paper, two different versions of the well-known 2-DOF Augusti model, whose independent buckling modes are both stable, are studied analytically without any simplifying assumptions with respect to the magnitude of deformation, in order to accurately demonstrate the coupling phenomena in the presence of imperfections in the pre- and post-buckling range. Depending on the nature of the structure's rotational springs, its post-buckling equilibrium path may be either stable or unstable. Afterward, the elastic response of two examples of laced built-up columns is illustrated numerically, one characterized by interaction between in-plane global and local buckling and the other by in- and out-of-plane global buckling, featuring similar response to that of the two 2-DOF models, respectively, thus demonstrating occurrence of such behavior in actual structural systems. [ABSTRACT FROM AUTHOR]

In this study, a finite element method (FEM) based on the size dependent nonlocal integral elasticity theory is implemented for buckling analysis of nanoscaled beams with various boundary conditions. The method is based on the principle of total potential energy. The variations of buckling load with respect to the scaling effect parameter and to the length-to-thickness ratio are investigated. Furthermore, the effect of attenuation function type on the buckling load is examined. The results are compared with the corresponding solutions of governing stability equations which are derived in the context of nonlocal differential elasticity theory. It is found that the small scale coefficient has a noticeable effect on the buckling load of nanobeams. [ABSTRACT FROM AUTHOR]

In this paper, the local buckling behaviour of profiled skin sheets resting on tensionless elastic Winkler foundations under in-plane shear loadings is studied. The profiled sheets are modelled as thin orthotropic plates with the buckling behaviour being expressed through a group of nonlinear partial differential equations. For very long plates with both ends clamped, the buckling mode is composed of a series of periodically repeating buckling waves and hence an infinite plate model with only one buckle wave is effective to predict the buckling behaviour. The infinite orthotropic plate is further simplified to a one-dimensional mechanical model by assuming a lateral buckling mode function. After solving the governing equations of the one-dimensional model in both contact and non-contact regions, shear buckling coefficients of the system and the related buckling modes are obtained. Fitted formulae for the contact shear buckling coefficients in terms of relative foundation stiffness and skin profile parameters are developed. The analytical solutions are verified through examining extreme cases from previous studies and a series of finite element (FE) models in ABAQUS. Finally, a practical example is given to show the efficiency of the developed method. [ABSTRACT FROM AUTHOR]

By using a three-dimensional (3D) space-frame-like model, a molecular mechanics (MM) approach is proposed for determination of the buckling loads, effective Young's modulus and Poisson's ratio of the nanosheets, using a proper unit cell. The governing equations are derived based on the 3D kinematics of deformations and the principle of minimum total potential energy. The unit-cell-based results are employed for the space-frame-like finite element model of the nanosheet. The nonlinear MM equations are solved by representing bonds of the boron nitride nanosheet (BNNS) by beam elements to extract the local characteristics. These properties are employed in modelling of the nanosheet, as a space-frame-like finite element structure. The force field constants are chosen according to the Morse, AMBER, UFF and DREIDING models to determine the buckling strength, and effective Poisson's ratio and in-plane rigidity of the whole graphene and BNNSs. Silicon Carbide nanosheets are analysed based on different force constants. These results are concordant with the results available in the literature. The comparisons reveal that the DREIDING force field usually gives the most accurate predictions. [ABSTRACT FROM AUTHOR]

In this study, effects of the presence of vacancy defects in a hexagonal nanosheet on Young's modulus, effective Poisson's ratio, buckling loads and buckling modes, regardless of its constituent atoms, have been studied. Explicit expressions are proposed in order to define these characteristics considering a defect distribution term as a modifying parameter. Molecular structural mechanics concepts and FEM simulation are utilized in order to obtain these expressions and results. Different sizes and shapes of defects as well as random distribution of vacancies have been considered. The results for perfect Boron Nitride, Silicon Carbide and graphene nanosheet as well as defected Boron Nitride nanosheets are in a good agreement with those available in literature. Linear degradation behavior of Young's modulus and linear increase of effective Poisson's ratio in terms of defects distribution are observed in obtained results. A second order behavior is also observed in decreasing buckling load in terms of increasing vacancy distribution. Moreover, buckling mode characteristics due to the percentage of defects distribution has been investigated. [ABSTRACT FROM AUTHOR]

In this article, an analytical approach is presented to analyze static stability of defective annular graphene sheet. Due to production process and constrains conditions, graphene sheet may be opposed to structural defect. Some of the defects can be modeled as an eccentric hole. The graphene sheet is elastically restrained at the bottom surface. Nonlocal thin plate theory as well as the translational addition theorem are employed to solve the problem. The stability and accuracy of results are examined by the literature and the finite element analyses. Effects of eccentricity of defects, nonlocality, Winkler and Pasternak foundation parameters and various boundary conditions on the critical buckling load of an annular graphene sheet are investigated. It is observed that the eccentricity and size of defects have significant effect on the critical load. [ABSTRACT FROM AUTHOR]

The buckling behaviour of thin shell structures under load has been a persistent challenge to engineering designers and researchers over many decades. In this article I consider two unusual experimental studies on the buckling of thin-walled elastic cylindrical shells, each of which sheds intriguing light on the buckling phenomena. The classical theory of buckling of thin cylindrical shells under axial compression predicts that the buckling stress will be proportional to t/R– the ratio of thickness to radius – other things being equal. But collected results of experimental studies from many laboratories, when plotted on log–log scales, show clearly that the buckling stress is actually proportional to (t/R)1.5, with the measured buckling stresses being scattered through a factor of about 4 for shells with R/t > 200. Such scatter is commonly judged to be in accord with Koiter's theory of 'imperfection sensitivity'. But that theory lays no claim to an understanding of the empirical 1.5-power law. I claim that a key to this situation is the experimental performance of some small-scale open-topped silicone rubber shells, buckling under their own weight, which clearly demonstrates a 1.5-power law, but with very little scatter. The buckling mode of these shells involves almost entirely inextensional deformation, with a single small dimple growing near the base, separated from the rest of the shell by a narrow boundary layer that accounts for almost all of the dimple's elastic strain energy. A straightforward, simple analysis of the mechanics of the dimple is consistent with the experimental 1.5-power law. As noted above, experimental buckling loads of shells that are closed at both ends also show the empirical 1.5-power law, but now with significant statistical scatter. A second aim of the paper is to throw light on that phenomenon. I venture to attribute it to the effect of the boundary conditions of the shell. I adduce support for this view from experimental observations on the buckling of a shell with special, frictional end-fittings. That feature produces significantly higher collapse loads, and with much smaller scatter, than for corresponding shells with fixed boundaries; and it permits striking pre-buckled deformations to occur, of a kind not previously noted. It will be appreciated that neither of the two parts of this article depends on the widely accepted theory of imperfection-sensitivity; hence my choice of title. It is a pleasure for me to submit this article to a special publication in honour of Michael Rotter, with whom I have discussed matters of this sort through three decades. [ABSTRACT FROM AUTHOR]

The lateral buckling problem for continuous welded rail tracks under thermal loading has been well researched and is known to involve a localisation buckling phenomenon where only a limited region of the track buckles. In this paper, a sandwich column model is formulated to model the rail track-tie structure as sandwich column. The constitutive relations for the thermally induced stresses and finite strain are based on a hyperelastic constitutive model. A hyperbolic function for the nonlinearity of the axial and lateral resistance between the tracks and ballast was used. A critical track length is proposed beyond which the localisation is unaffected. Increasing the track length or changing the boundary conditions does not influence the localised buckling behaviour for track lengths beyond the critical length. The use of fasteners with large rotational stiffness substantially increases the lateral stability of the track-tie structure. Nonlinear numerical solutions are compared with the results in the literature. The variation in the axial compressive force within the localisation zone in the rail is studied and discussed. Parametric studies are performed on examples to identify the influence of track imperfections on the localised lateral buckling and safe temperature change. [ABSTRACT FROM AUTHOR]

The effect of distortional deformation on the elastic lateral buckling of thin-walled box beam elements under combined bending and axial forces is investigated in this paper. For the purpose, an analytical model is developed for the stability of laterally unrestrained box beams according to higher order theory. Ritz and Galerkin's methods are applied in order to discretize the governing equilibrium equations and then the buckling loads are obtained by requiring the singularity of the tangential stiffness matrix. The different solutions are discussed and then compared to the finite element simulation using Abaqus software where shell elements are used in the mesh process. The numerical results reveal that classical stability solutions as those adopted in Eurocode 3 overestimate the real lateral buckling resistance of thin-walled box beam members, particularly for the ones with high ratios between the height and the thickness of the cross-section. Numerical study of incidence of compressive forces on lateral buckling resistance of thin-walled box beam is investigated. [ABSTRACT FROM AUTHOR]

Abstract: Presented in this paper is analysis of buckling of double-orthotropic nanoplates based on nonlocal elasticity theory. It is assumed that two nanoplates are bonded by an internal elastic medium and surrounded by external elastic foundation. Three characteristic types of buckling are considered. Governing equations are derived based on nonlocal theory, while the expressions for the buckling load are given in explicit form for a nanoplate with all edges simply supported. Explained in detail is the influence of small scale coefficient, aspect ratio, and stiffness of internal elastic media and external elastic foundation, on the nondimensional buckling load. [Copyright &y& Elsevier]

Abstract: The present study is concerned with the elastic/plastic buckling of thin rectangular plates under various loads and boundary conditions. The in-plane loads are placed uniformly and linearly varying in the uniaxial compression and biaxial compression/tension. The equilibrium and stability equations are derived and analyses are carried out based on two theories of plasticity, i.e. deformation theory (DT) and incremental theory (IT). The elastic/plastic behavior of plates is described by the Ramberg–Osgood model. Generalized Differential Quadrature (GDQ) discretization technique is used to solve the buckling of plate equation. To examine accuracy of the present formulation and procedure, several convergence and comparison studies are investigated and new results are presented. The differences between the IT and DT results increase by increasing loading parameter in linearly varying in-plane loading. Some new consequences are achieved regarding the validation range of two theories. Furthermore, effects of aspect, thickness to length and loading ratios, boundary condition, type of plasticity theory and linearly varying in-plane loading on the buckling coefficient are discussed. Contour plots of buckling mode shapes for various loading parameters are also illustrated. [Copyright &y& Elsevier]

Abstract: A continuum based approach is presented for buckling analysis of single layer graphene sheets with skew shape. The straight-sided quadrilateral graphene field is mapped into a square domain in the computational space using a four-node graphene element. Then, the governing equations and boundary conditions of the graphene are transformed from the physical domain into a square computational domain by using the geometric transformation. Some numerical examples related to buckling loads of skew shaped graphene are presented for different geometric parameters. Results related to buckling loads of the single layer graphene with skew shape have been presented which can serve as benchmark solutions for future investigations. [Copyright &y& Elsevier]

Abstract: The paper studies and provides solutions to the elastic buckling of whole channel sections including flanges and lips where the sections are loaded in pure shear parallel with the web. The elastic buckling analyses of thin-walled channel sections in pure shear have been implemented by means of the Spline Finite Strip Method (SFSM) to determine the elastic buckling loads (V cr ) of the sections. The shear buckling coefficients (k v ) of the web for design of a section are back-calculated from the shear buckling loads. The main variables are the flange widths, member lengths and lip sizes. The boundary conditions at two end sections are simply supported. The results of the analyses plotted in the format of interaction charts are given as design guidelines for designers to predict the elastic buckling shear coefficient (k v ) without using the Spline Finite Strip Method (SFSM) software. Typical buckling modes at different flange widths, member lengths and lip sizes are also included in this paper. The modes include local web and flange buckling, section distortional buckling and section twisting. [Copyright &y& Elsevier]

Abstract: Elastic stability of fibers in fiber-reinforced composite materials subject to compressive loading is studied. The transversal buckling mode is considered, and two limiting cases, the dilute and non-dilute composites are analyzed. In the case of a non-dilute composite, the cylindrical model and the lubrication approximation are applied. The original problem is reduced to a problem of stability of a rod on elastic foundation. Through the solution of this problem a simple formula for the buckling load is obtained. In the case of a dilute composite, the solution of a problem of stability of a compressed rod in elastic plane is used. On the basis of the obtained solutions in two limiting cases the interpolation formulae are derived. These formulae describe buckling of fiber in the fiber-reinforced composite for any value of the fiber volume fraction. Comparison with known numerical and experimental results is carried out, and the sufficient accuracy of the derived formulae is demonstrated. [Copyright &y& Elsevier]

Highlights • Post-buckling analysis of annular sector plate subjected to in-plane shear load at upper surface. • Governing equations are based on three dimensional theory of elasticity. • Non-linear Green strain tensor is used. • Due to weak coupling between shear loads and resulting twisting moments, post-buckling of simply supported FGM plate is of bifurcation-type buckling. Abstract In this paper, shear post-buckling analysis of functionally graded annular sector plates is considered. In-plane shear loads have been applied to either radial, circumferential, or all edges of annular sector plates. Moreover, post-buckling of annular sector plates subjected to in-plane shear load at upper/lower surfaces is investigated for the first time. Material properties are graded asymmetrically in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents while the Poisson's ratio is assumed to be constant. The governing equations are based on three dimensional theory of elasticity in conjunction with non-linear Green strain tensor instead of the approximate plate theories and Von Karman assumptions. The governing equations are developed based on the principle of virtual work and solved based on graded finite element method. Non-linear equilibrium equations are solved based on iterative Newton–Raphson method. The effects of material gradient exponent, different sector angles, thickness ratio and four different boundary conditions on the post-buckling behavior of FGM annular sector plates have been investigated. Results denote that due to weak coupling between applied shear loads and the resulting twisting moments, shear post-buckling behavior of simply supported FGM plates is of the bifurcation-type buckling following a stable post-buckling path. [ABSTRACT FROM AUTHOR]

This article deals with the study of buckling behaviour of multilayered skew magneto-electro-elastic plate under uniaxial and biaxial in-plane loadings. The skew edges of the skew magneto-electro-elastic plate are obtained by transforming the local skew coordinate to the global using a transformation matrix. The displacement fields corresponding to the first-order shear deformation theory along with constitutive equations of magneto-electro-elastic materials are used to develop a finite element model. The finite element model encompasses the coupling between electric, magnetic and elastic fields. The in-plane stress distribution within the skew magneto-electro-elastic plate due to the enacted force is considered to be equivalent to the applied in-plane compressive loads in the pre-buckling range. This stress distribution is used to derive the potential energy functional of the skew magneto-electro-elastic plate. The non-dimensional critical buckling load is attained from the solution of the allied linear eigenvalue problem. Influence of skew angle, stacking sequence, span-to-thickness ratio, aspect ratio and boundary condition on the critical buckling load and their corresponding mode shapes is investigated. [ABSTRACT FROM AUTHOR]

This paper presents an experimental study on the out-of-plane inelastic buckling strength of fixed circular steel arches under symmetric and non-symmetric loading. A test loading arrangement that allows for lateral deflections to develop freely under vertical loading is described. A finite element (FE) model consisting of the tested steel arch and the loading system is established for carrying out supplementary numerical investigation on the inelastic out-of-plane buckling strength of the fixed steel arches. The FE numerical model is validated by the experimental results. From the experimental results and supplementary FE investigation, it is found that the out-of-plane inelastic buckling strength of fixed steel arches is influenced significantly by the magnitude and distribution of initial out-of-plane geometric imperfections, as well as the out-of-plane elastic buckling modes and the in-plane loading patterns. It is also found that the out-of-plane buckling strength of a fixed steel arch under non-symmetric loading is lower than that under symmetric loading. Based on the experimental and FE results, a lower bound interaction equation is developed for predicting the out-of-plane inelastic buckling strength in the design of fixed circular steel arches against their out-of-plane failure. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper, a three-dimensional elasticity approach is employed to investigate buckling of heterogeneous functionally graded plates under biaxial compression, shear, tension-compression, and shear-compression load conditions. In this regard, a formulation that employs a full compatible three-dimensional Hermitian element with 168 degrees of freedom and guarantees continuity of the strain and stress components is used. It is known that all of the available famous commercial finite element softwares and the proposed series solutions satisfy continuity conditions of the displacement rather than the stress components. Buckling occurrence is detected based on checking both the instability onset and equilibrium criteria. Results are extracted based on a Galerkin-type orthogonality. Therefore, they are more accurate than those obtained based on the traditional Ritz method. The presented three-dimensional finite element analysis and the extracted results are quite new. A vast variety of results including results of biaxial compression, compression-tension, shear, and shear-compression load cases is considered and discussed in detail. [Copyright &y& Elsevier]

Highlights: [•] Shear and torsional rigidities of trusses are theoretically derived. [•] Out-of-plane buckling loads of pin-ended circular truss arches are derived. [•] Effect of shear deformation on buckling loads of truss arches is included. [Copyright &y& Elsevier]

Abstract: In this paper, skew and rhombic isotropic plates subjected to in-plane loadings are analyzed using the element-free Galerkin method. Inelasticity effect is included in the buckling analysis while plates are thin thickness-tapered type. The governing differential equation for a plate in plastic range of response is numerically solved using the Galerkin method. The shape functions are constructed using the moving least squares (MLS) approximation and the essential boundary conditions are introduced into the formulation through the use of the Lagrange multiplier method and the orthogonal transformation techniques. The Stowell theory for the plastic buckling of flat skew plates with variable thicknesses is used. The inelastic analysis is based on the Ramberg–Osgood representation of the stress–strain curve which is used in the deformation theory of plasticity. Using this method the initial inelastic local buckling of skew plates with or without intermediate line supports is studied. Stiffness and geometric matrices are formulated by weak form of the Galerkin method. Finally, the inelastic local buckling loads of these plates are obtained and the results are compared with known solutions in the literature. [Copyright &y& Elsevier]

Abstract: Arches are often supported by other structural elements which provide elastic types of restraint at both of their ends and these elastic restraints usually have different stiffnesses. This paper presents a nonlinear elastic in-plane buckling and postbuckling analysis of pin-ended shallow circular arches having elastic rotational end restraints of unequal stiffness when subjected to a uniform radial load. An energy method is used to derive the differential equations of equilibrium, and the differential equations of buckling equilibrium. The nonlinear equilibrium equation between the external load and the axial force is also established. Analytical solutions for the nonlinear in-plane behaviour and buckling loads of the arches are derived. It is found that the effects of the unequal stiffness of the rotational end restraints on the nonlinear behaviour and buckling of the arches are significant. The nonlinear buckling loads increase with an increase of the stiffness of rotational end restraints. It is also found that the pin-ended arch with elastic rotational end restraints of unequal stiffness may buckle in a limit point instability mode, but they cannot buckle in a bifurcation mode. This is different from pin-ended arches with equal rotational end restraints, which can buckle in either a limit point instability or a bifurcation mode. It is demonstrated that there exists a specific geometric parameter (as defined in the paper) that defines a switch between arches and beams curved in elevation. Comparisons with finite element results show that the analytical solutions can accurately predict the nonlinear behaviour and buckling loads of pin-ended shallow circular arches with elastic rotational end restraints of unequal stiffness. The analytical solutions are shown to reduce to those of pin-ended arches with equal rotational end restraints or without rotational end restraints, fixed arches, and pinned-fixed arches. [Copyright &y& Elsevier]

Abstract: We study a variation on the classical problem of the buckling of an elastica. The elastica models a nanoscale sheet that interacts with a rigid substrate by intermolecular forces. We formulate a buckling problem in which the sheet is perpendicular to the substrate and a load is applied to the edge of the sheet further from the substrate. Our study is motivated by problems in nanomechanics such as the bending of a graphene sheet interacting with a rigid substrate by van der Waals forces. After identifying a trivial branch, we combine computation and analysis to determine the stability and bifurcations of solutions along this branch. We also study the boundary-layer problem that arises if the length of the sheet is large compared to the characteristic length over which the van der Waals interaction is significant. [Copyright &y& Elsevier]

Abstract: An investigation on the use of the Finite Strip Method (FSM) to calculate elastic buckling loads of perforated cold formed storage rack columns is presented. Nowadays, this calculation can be accurately performed by means of the Finite Element Method (FEM), because the effect of perforations can be explicitly considered in the analysis. However, the FSM is preferred in cold-formed steel design since it is implemented in much convenient and easy to use software. The problem with FSM is that holes cannot be easily modeled. In this paper, the concept of the reduced thickness of the perforated strip is applied to take into account their effect. A formulation is presented for the reduced thickness that has been calibrated with loads obtained in eigen-buckling FEM analyses. Its accuracy has been verified carrying out analyses on real rack columns with different end conditions. [Copyright &y& Elsevier]

Abstract: Inelastic buckling and postbuckling behavior of stocky plates under combined shear and in-plane bending stresses are investigated and compared to slender plates. Aluminum and steel plates having various slenderness ratios are modeled and analyzed by means of (i) numerical nonlinear finite element method and (ii) theoretical p-Ritz energy method; and both results are compared to the classic interaction equation. It is observed that whereas in slender plates, elastic buckling occurs prior to the material''s proportional limit load, stocky plates buckle in an inelastic way within the post-yield stage. In contrast to slender plates with considerable postbuckling reserves, the buckling of stocky plates is immediately followed by softening. In addition, it is shown that the classic interaction equation overestimates buckling loads; and therefore, a modified equation that can safely be applied to both stocky and slender plates is proposed. [Copyright &y& Elsevier]

Abstract: The buckling of nanostructures including as a nanobeam, nanorod, and nanotube in a temperature field is investigated based on the non-local elasticity field theory with non-linear strain gradients first proposed by Eringen. New higher-order governing differential equations both in transverse and axial direction for buckling of such nanostructures are derived based on the exact variational principle approach with corresponding higher-order non-local boundary conditions. Based on these new governing equations and boundary conditions, new analytical solutions for some practical examples on buckling of nanostructures are presented and analyzed in detail. Subsequently, the effects of non-local nanoscale and temperature change on critical buckling load are analyzed and discussed. It is observed that those factors have great influence on the critical buckling load of the nanostructures. In particular, the non-local stress very much affects the stiffness of nanostructures and the critical buckling load is significantly increased in the presence of non-local stress. The paper concludes that at low and room temperature the critical buckling load of nanostructures increases with increasing temperature change, while at high temperature the critical buckling load decreases with increasing temperature change. A critical temperature change which causes buckling without external load is also derived and discussed. [Copyright &y& Elsevier]

Abstract: The in-plane elastic buckling of a steel column with load-dependent supports under thermal loading is investigated. Two elastic rotational springs at the column ends are used to model the restraints which are provided by adjacent structural members or elastic foundations. The temperature is assumed to be linearly distributed across the section. Based on a nonlinear strain–displacement relationship, both the equilibrium and buckling equations are obtained by using the energy method. Then the limits for different buckling modes and the critical temperature of columns with different cases are studied. The results show that the proposed analytical solution can be used to predict the critical temperature for elastic buckling. The effect of thermal loading on the buckling of steel columns is significant. Furthermore, the thermal gradient plays a positive role in improving the stability of columns, and the effect of thermal gradients decreases while decreasing the modified slenderness ratios of columns. It can also be found that rotational restraints can significantly affect the column elastic buckling loads. Increasing the initial stiffness coefficient α or the stiffening rate β of thermal restraints will increase the critical temperature. [Copyright &y& Elsevier]

Abstract: A closed-form solution is obtained to determine the buckling and post-buckling behavior of elastically restrained composite panels under compressive loading. The approach allows to study the response of stiffened panels undergoing local buckling modes, taking into account the restraints provided by the stiffeners to the rotation of the skin edges. The panels are modeled as thin plates referring to Marguerre type equations together with classical lamination theory. The equations are written in non-dimensional form, allowing for the study of a wide class of orthotropic laminates. The problem is formulated in terms of out of plane displacement, represented with a single-mode approximation, and Airy stress function. The compatibility equation is solved exactly, while the method of Galerkin is applied to impose the equilibrium. The buckling load, the out of plane displacement at different load levels, and the post-buckling stiffness are derived and compared with finite element analyses, revealing good accuracy. Sensitivity analyses are also performed obtaining design charts. [Copyright &y& Elsevier]

Abstract: The buckling behaviors as well as the load transfer mechanisms between the shells of double-walled nanotubes (DWNTs) are investigated through a series of molecular dynamics simulations. When buckling occurs, the strain energy of cross-linked (defective) DWNTs undergoes a modest transition indicating that energy gathered in one shell is shared by the other through the inter-shell crosslinks. It is confirmed that the compressive stress applied to the outer shell can be efficiently transferred to the inner shell through the covalent bonds between shells resulting in a uniform load distribution. The existence of inter-shell crosslinks leads to dramatic decreases of the sustainable loads and modest decreases of the Young’s modulus. The computational studies imply that the crosslinks can improve the loading conditions and reduce strain energy in carbon nanotubes based nanocomposites and nanoelectromechanical systems. [Copyright &y& Elsevier]

Abstract: Axial buckling analysis of double-walled Boron Nitride nanotubes (DWBNNTs) embedded in an elastic medium under combined electro-thermo-mechanical loadings is presented in this article. Virtual displacement method based on nonlocal cylindrical piezoelasticity continuum shell theory is employed to derive the equilibrium equations. Boron Nitride nanotube (BNNT) is assumed to be surrounded by a bundle of carbon nanotubes (CNTs) as elastic medium for reinforcement. The elastic medium is simulated as Winkler–Pasternak foundation, and adjacent layers interactions are assumed to have been coupled by van der Walls (vdW) force evaluated based on the Lennard–Jones model. The effects of parameters such as electric and thermal loads, elastic medium and small scale are investigated on the buckling behavior of the DWBNNTs. The electric field and its direction are found to have affected the magnitude of the critical buckling load. Moreover, an analysis is carried out to estimate the nonlocal critical electro-thermo-mechanical load for the axial buckling of embedded DWBNNTs. [Copyright &y& Elsevier]

Abstract: A mathematical model for a slip-buckling problem has been proposed and its exact solution has been found for the analysis of materially inelastic two-layer composite columns with non-linear interface compliance. The mathematical model has been carried out to evaluate exact critical buckling loads. It has been demonstrated mathematically exactly, that exact critical buckling loads are influenced by the initial stiffness, and hence on linear portion of the interface force-slip relationship. Besides, it has been shown that material inelasticity can reduce the critical buckling loads significantly and that the interlayer stiffness has an important effect on the transition between the elastic and inelastic buckling. [Copyright &y& Elsevier]

The elastic buckling behavior of quadrilateral single-layered graphene sheets (SLGS) under bi-axial compression is studied employing nonlocal continuum mechanics. Small-scale effects are taken into consideration. The principle of virtual work is employed to derive the governing equations. The Galerkin method in conjunction with the natural coordinates of the nanoplate is used as a basis for the analysis. The buckling load of skew, rhombic, trapezoidal, and rectangular nanoplates considering various geometrical parameters are obtained. It is shown that nonlocal effects are very important in arbitrary quadrilateral graphene sheets and their inclusion results in smaller buckling loads. Also the effects of geometrical parameters such as aspect ratio, angle, and mode number on the buckling load decrease when scale coefficient increases, for all arbitrary quadrilateral SLGS. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a comprehensive study of the lateral compressive response of hexagonal honeycomb panels from the initial elastic regime to a fully crushed state. Expanded aluminum alloy honeycomb panels with a cell size of 9.53mm, a relative density of 0.026, and a height of 15.9mm are laterally compressed quasi statically between rigid platens under displacement control. The cells buckle elastically and collapse at a higher stress due to inelastic action. Deformation then first localizes at mid-height and the cells crush by progressive formation of folds; associated with each fold family is a stress undulation. The response densifies when the whole panel height is consumed by folds. The buckling and crushing events are simulated numerically using finite element models involving periodic domains of a single or several characteristic cells. The models idealize the microstructure as hexagonal, with double walls in one direction. The nonlinear behavior is initiated by elastic buckling while inelastic collapse that leads to the localization observed in the experiments occurs at a significantly higher load. The collapse stress is found to be mildly sensitive to various problem imperfections. The subsequent folding can be reproduced numerically using periodic domains but requires a fine mesh capable of capturing the complexity of the folds. The calculated crushing response is shown to better resemble measured ones when a 4×4 cell domain is used. However, the average crushing stress can be captured with engineering accuracy even from a single cell domain. [ABSTRACT FROM AUTHOR]

Abstract: In the present work differential transformation method (DTM) is used to predict the buckling behaviour of single walled carbon nanotube (SWCNT) on Winkler foundation under various boundary conditions. Four different boundary conditions namely clamped–clamped, simply supported, clamped hinged and clamped free are used to study the critical buckling loads. Effects of (i) size of SWCNT (ii) nonlocal parameter and (iii) Winkler elastic modulus on nonlocal critical buckling loads are being investigated and discussed. The DTM is implemented for the nonlocal SWCNT analyses and this yields results with high degree of accuracy. Further, present method can be applied to linear and nonlinear problems. [ABSTRACT FROM AUTHOR]

Abstract: Conventionally, when a circular arch is subjected to uniform radial loading, the difficulty in performing an accurate prebuckling analysis is usually circumvented by considering that the uniform radial loading produces axial compression, but no bending, in the arch. However, the transverse loading may produce bending as well as axial compression in the arch, particularly in shallow arches, and the arch may deform considerably prior to buckling. Treating the prebuckling stresses in the arch as being trivial is therefore questionable, and the prebuckling analysis should be carried out with caution. This paper investigates the prebuckling response of the classical analysis and its effects on determining the out-of-plane and in-plane elastic buckling loads of circular arches that are subjected to uniform radial loading. It is found that the current solutions based on the classical prebuckling analysis overestimate the in-plane buckling loads of shallow arches and underestimate the out-of-plane buckling loads of shallow arches significantly. Proper prebuckling analyses for out-of-plane and in-plane buckling are presented and new analytical solutions for the out-of-plane and in-plane buckling loads of arches are derived accounting for both axial compressive and bending actions. [Copyright &y& Elsevier]

Abstract: Web core panels, foam-filled sandwich panels with interior webs, are a structurally efficient option for transverse load bearing applications. In web core panels, the interaction between the webs and core material can have a substantial impact on web shear buckling strength and is a key element of lightweight structural design. The present work is an investigation of web buckling behavior in web core panels under a distributed load. To solve this problem, web shear buckling was analyzed for the case of pure shear loading with foam support, and this analytic model was extended to the case of panels with a transverse distributed load. The webs are modeled as simply supported plates resting on a Pasternak elastic foundation. To that end, a buckling model for plates on a Pasternak foundation is presented, along with closed-form approximations of the solution for square and infinitely long plates. An accurate model for the foundation constants is developed using energy methods. Applicability of the plate buckling model to web core panels with transverse loads is presented via a finite element study. In panels, the slenderness and spacing of the webs have a slight effect on the boundary conditions between the webs and face sheets. The effect is relatively small, however, and the model presented in this work underpredicts buckling strength by less than 25%. The model in this work is thus a reasonable approach to the practical design of web core panels. [Copyright &y& Elsevier]

The buckling problem of annular plate with only one edge (either inner or outer edge) constrained is formulated and presented in this paper. The boundary condition is simply supported inner edge - free outer edge. The results are compared between the FEM results obtained using ABAQUS and results reported in literature. The results shows good agreement for critical buckling load, mode shapes and stress distributions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, variational iteration method (VIM) is applied to the problem of determination of critical buckling loads for Euler columns with constant and variable cross-sections. VIM is a powerful method for the solution of nonlinear ordinary and partial differential equations and integral equations. Hence it is a suitable approach for the analysis of engineering problems where an exact solution is difficult to obtain. This study presents the application of VIM to various buckling cases and results are produced for columns with different support conditions and with different variation of cross-sections. The results obtained are accurate which show that variational iteration method is a very efficient technique in the analysis of elastic stability problems. [Copyright &y& Elsevier]

Abstract: Higher order shear deformation theory (HSDT) is reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived. The developed equations of motion have been applied to study buckling characteristics of nanoplates such as graphene sheets. Navier''s approach has been used to solve the governing equations for all edges simply supported boundary conditions. Analytical solutions for critical buckling loads of the graphene sheets are presented. Nonlocal elasticity theories are employed to bring out the small scale effect on the critical buckling load of graphene sheets. Effects of (i) nonlocal parameter, (ii) length, (iii) thickness of the graphene sheets and (iv) higher order shear deformation theory on the critical buckling load have been investigated. The theoretical development as well as numerical solutions presented herein should serve as reference for nonlocal theories as applied to the stability analysis of nanoplates and nanoshells. [Copyright &y& Elsevier]

Abstract: A mathematical model for slip-buckling has been proposed and its analytical solution has been found for the analysis of layered and geometrically perfect composite columns with inter-layer slip between the layers. The analytical study has been carried out to evaluate exact critical forces and to compare them to those in the literature. Particular emphasis has been placed on the influence of interface compliance on decreasing the bifurcation loads. For this purpose, a preliminary parametric study has been performed by which the influence of various material and geometric parameters on buckling forces have been investigated. [Copyright &y& Elsevier]

The article focuses on a study on the elastic buckling behavior of perforated plates subjected to linearly varying in-plane loading. It notes the use of the multipurpose finite element software program ANSYS in the study. It discusses the results of the study using ANSYS, including the variations of the buckling load ratios for a number of plates with different loading cases. It also notes the effect of a circular hole located out of the plate's critical zone on the buckling load.

Abstract: Buckling response of orthotropic single layered graphene sheet (SLGS) is investigated using the nonlocal elasticity theory. Two opposite edges of the plate are subjected to linearly varying normal stresses. Small scale effects are taken into consideration. The nonlocal theory of Eringen and the equilibrium equations of a rectangular plate are employed to derive the governing equations. Differential quadrature method (DQM) has been used to solve the governing equations for various boundary conditions. To verify the accuracy of the present results, a power series (PS) solution is also developed. DQM results are successfully verified with those of the PS method. It is shown that the nonlocal effects play a prominent role in the stability behavior of orthotropic nanoplates. Furthermore, for the case of pure in-plane bending, the nonlocal effects are relatively more than other cases (other load factors) and the difference in the effect of small scale between this case and other cases is significant even for larger lengths. [Copyright &y& Elsevier]

An analytical approach for studying the elastic stability of thin rectangular plates under arbitrary boundary conditions is presented. Because the solution is given in closed-form, the approach can be regarded as "exact" under the Kirchhoff-Love assumption. The proposed procedure allows us to obtain the buckling load and modal displacements that do not depend on the number of elements adopted in the numerical discretization using, say, the finite element method. Due to the fact that the longitudinal variation of the displacements is taken into account, the two-dimensional model established for the plate is considered "complete." Such an approach overcomes the shortcomings of conventional modeling presented in the literature. In order to demonstrate the generality of the proposed approach, several examples are prepared and the results obtained are compared with finite element and analytical solutions existing elsewhere. [ABSTRACT FROM AUTHOR]

Abstract: Three-dimensional buckling analysis of the functionally graded plates has not been investigated under non-uniform in-plane compressive loads so far. The only available work in this field has been performed based on the first-order shear deformation theory, employing the simple rule of mixtures. In the present paper, a non-linear three-dimensional energy-based elasticity analysis is developed for buckling investigation of functionally graded plates subjected to non-uniform in-plane compressions. The comparative studies performed in the present paper reveal that the rule of mixtures is generally inappropriate for determination of the effective material properties. For this reason, a micromechanics-based model is used instead. In contrast to the common displacement-based numerical or semi-analytical elasticity analyses (with C 0-continuity), present formulations are C 2-continuous due to using the proposed 3D cubic B-spline element. Buckling analysis is accomplished through a two-step procedure wherein the prebuckling stresses are determined first. A non-linear weighted-residuals-based finite element solution is used. The buckling load associated with each of the adopted 8 types of the load distribution patterns is detected based on a generalized geometric stiffness concept. Various and comprehensive parametric studies are accomplished and various boundary conditions are considered to extract more precise conclusions. [Copyright &y& Elsevier]

Highlights: [•] A fixed shallow arch was used to study the equilibrium path of the bistable strut. [•] The virtual work principle was adopted to establish nonlinear equilibrium equations. [•] The critical load for symmetric snap-through buckling was obtained. [•] The post-buckling behavior of the arch was also investigated. [Copyright &y& Elsevier]

Abstract: This study presents the analytical, numerical, and experimental results of thin hemispherical metal shells into the plastic buckling range illustrating the importance of geometry changes on the buckling load. The hemispherical shell is rigidly supported around the base circumference against horizontal and vertical translation and the load is vertically applied by a rigid cylindrical boss at the apex. Kinematics stages of initial buckling and subsequent propagation of plastic deformation for rigid-perfectly plastic shells are formulated on the basis of Drucker–Shield''s limited interaction yield condition. The effect of the radius of the boss, used to apply the loading, on the initial and subsequent collapse load is studied. In the numerical model, the material is assumed to be isotropic and linear elastic perfectly plastic without strain hardening obeying the Tresca or Von Mises yield criterion. Both axisymmmetric and 3D models are implemented in the numerical work to verify the presence of non-symmetric deformation modes in the case of thin shells. In the end, the results of the analytical solution are compared and verified with the numerical results using ABAQUS software and experimental findings. Good agreement is observed between the load–deflection curves obtained using three different approaches. A secondary bifurcation point is detected in thin shells in which the deformation degenerates from symmetric to non-symmetric behavior. The bifurcation point depends on the (R/t 0) ratio and the material parameters. [Copyright &y& Elsevier]