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Buckling of a simple supported and cantilever nano‐rod and under action of self‐weight was modeled, taking into account the surface effect according to two models of surface elasticity – Gurtin–Murdoch and Steigmann–Ogden. The size effect under the buckling was studied by the Ritz method using two kinematic hypotheses, known as the rod theories of Euler–Bernoulli and Timoshenko. The simulation results were compared depending on the accepted surface elasticity model and the kinematic hypothesis. [ABSTRACT FROM AUTHOR]

The present research aims at determining the axial buckling load of stiffened multilayer cylindrical shell panels made of functionally graded graphene-reinforced composites (FG-GPL RCs). Rings and stringers are applied as stiffening tools for shell panels, whose elastic properties are determined according to the Halpin–Tsai relations. The virtual work principle and finite element approach are implemented here, according to a first-order shear deformation theory (FSDT) and Lekhnitskii smeared stiffener approach, in order to determine the governing equations of the stability problem. Four different dispersions of nanofillers are assumed in the thickness direction, including the FG-X, FG-A, FG-O, and UD distributions. A large systematic investigation considers the effect of different geometric and material parameters on the buckling loads and mode shapes of the stiffened FG-GPL RC cylindrical shell panel, primarily the dispersion and weight fractions of the nanofiller, the number of rings and stringers, and the boundary conditions, with useful insights for design purposes. [ABSTRACT FROM AUTHOR]

The quest for safe lower bounds to the elastic buckling of axially loaded circular cylindrical shells has exercised researchers for the past 100 years. Recent work bringing together the capabilities of nonlinear numerical simulation, interpreted within the context of extended linear classical theory, has come close to achieving this goal of defining safe lower bounds. This paper briefly summarizes some of the important predictions emerging from previous work and presents new simulation results that confirm these earlier predictions. In particular, we show that for a specified maximum amplitude of the most sensitive, eigenmode-based geometric imperfections, normalized with respect to the shell thickness, lower bounds to the buckling loads remain constant beyond a well-defined value of the Batdorf parameter. Furthermore, we demonstrate how this convenient means of presenting the imperfection-sensitive buckling loads can be reinterpreted to develop practical design curves which provide safe, but not overly conservative, design loads for monocoque cylinders with a given maximum permitted tolerance of geometric imperfection. Hence, once the allowable manufacturing tolerance is specified during design or is measured post-manufacturing, the greatest expected knockdown factor for a shell of any geometry is defined. With the recent research interest in localized imperfections, we also attempt to reconcile their relation to the more classical, periodic, and eigenmode-based imperfections. Overall, this paper provides analytical and computational arguments that motivate a shift in focus in defect-tolerant design of thin-walled cylinders--away from the knockdown experienced for a specific geometric imperfection and towards the worst possible knockdown expected for a specified manufacturing tolerance. [ABSTRACT FROM AUTHOR]

Although it is often asserted that, in view of their reduced length, axially compressible beams have a higher buckling load than their inextensible counterparts, a detailed analysis demonstrates that this is not necessarily the case. The argument to arrive at this conclusion is made in terms of relatively straightforward concepts of elasticity and structural mechanics. It is shown that for certain classes of materials, the reduced prebuckling length is more than compensated for by a softening of the elastic response, leading to a reduction of the Euler critical load. [ABSTRACT FROM AUTHOR]

This paper investigates the size-dependent vibration and buckling behavior of nanoplates described by the nonlocal Kirchhoff model analytically. The nonlocal and local constitutive relations, governing equations and boundary conditions are comprehensively discussed. A shifted governing equation in terms of local stress resultants with associated local boundary conditions is chosen in the current work. The direct separation of variables method is formulated based on the Hamiltonian dual form of the governing differential equation to address the eigenvalue problems of nonlocal rectangular plates with each edge either simply supported or clamped. The closed-form eigen-solutions are acquired for the first time for the cases where two adjacent clamped edges exist. The validity and accuracy of the present approach are verified by comparison with published analytical and numerical results. Parametric studies are performed to investigate the influences of nonlocal parameter and boundary conditions on the size-dependency of natural frequencies and critical buckling loads of nanoplates. [ABSTRACT FROM AUTHOR]

We investigate the effect of defect geometry in dictating the sensitivity of the critical buckling conditions of spherical shells under external pressure loading. Specifically, we perform a comparative study between shells containing dimpled (inward) versus bumpy (outward) Gaussian defects. The former has become the standard shape in many recent shell-buckling studies, whereas the latter has remained mostly unexplored. We employ finite-element simulations, which were validated previously against experiments, to compute the knockdown factors for the two cases while systematically exploring the parameter space of the defect geometry. For the same magnitudes of the amplitude and angular width of the defect, we find that shells containing bumpy defects consistently exhibit significantly higher knockdown factors than shells with the more classic dimpled defects. Furthermore, the relationship of the knockdown as a function of the amplitude and the width of the defect is qualitatively different between the two cases, which also exhibit distinct post-buckling behavior. A speculative interpretation of the results is provided based on the qualitative differences in the mean-curvature profiles of the two cases. [ABSTRACT FROM AUTHOR]

This study investigates the size-dependent buckling behavior of Timoshenko nanobeams resting on Winkler-Pasternak foundations in a non-uniform thermal environment. A non-uniform temperature distribution is established through nonlocal heat conduction. Subsequently, the equivalent thermal load due to the obtained temperature distribution and boundary constraints is derived using the governing equations of the axial thermal deformation of the beams based on the stress-driven nonlocal elastic model. To obtain the critical buckling load of the nanobeams, the quadrature element method is used to numerically resolve the eigenvalue problem. In the numerical simulation section, we have presented a series of examples to analyze the effects of length-to-height ratios, nonlocal scale parameters, and Winkler-Pasternak foundation parameters on the buckling loads of the nanobeams under various boundary conditions. Moreover, we examine the effect of comprehensively considering both the elastic and thermal nonlocality on the thermal loads and finally mechanical buckling loads. [ABSTRACT FROM AUTHOR]

Long glass fiber-reinforced polypropylene (LGFR-PP) composite structures with stiffeners are important substitutes for metal parts for vehicle lightweighting; a good understanding of the buckling characteristics of LGFR-PP stiffeners would provide an important reference for engineering design. The current work is therefore intended to study the buckling characteristics of different cross-sectioned LGFR-PP stiffeners under axial compression via experimental and theoretical analysis. Firstly, LGFR-PP stiffeners with semicircular, rectangular, and trapeziform cross-sections were compressed at the axial direction using a universal testing machine to obtain the buckling process data. Then, the elasticity stability theory modified according to the experimental results was derived to estimate the buckling resistance of LGFR-PP stiffeners in different designs. The test results showed that the LGFR-PP stiffeners possessed a flexible–torsional bulking instability mode under axial compression, the LGFR-PP stiffeners with a semicircular cross-section had higher compression buckling resistance, and the rectangular and trapeziform cross-sectioned stiffeners had better rigidity. The theoretical analysis showed that the modified elasticity stability theory could generally predict the buckling resistance of LGFR-PP stiffeners under axial compression. [ABSTRACT FROM AUTHOR]

Buckling analysis of functionally graded piezoelectric nanoshell is studied in this paper based on the higher-order shear and normal deformation theory and accounting thickness stretching effect. The nanoshell is subjected to axial load, applied electric potential and thermal loads. Thickness stretching effect is accounted in the analysis based on higher-order shear and normal deformation theory. Small scale effects are accounted based on the Eringen nonlocal elasticity theory. The Navier solution is used for the buckling analysis of the cylindrical nanoshell with simply-supported boundary conditions. The accuracy and trueness of the present paper is justified using comparison with literature. The importance of the present analysis and corresponding results is justified using presentation of results with and without thickness stretching effect. A large parametric analysis is presented to investigate the influence of significant parameters such as dimensionless small scale parameter, length to radius ratio, thickness to radius ratio, temperature rising and applied electric voltage on critical buckling axial loads. One can conclude that the critical buckling axial loads are decreased with increase of small scale parameters and applied electric potential. [ABSTRACT FROM AUTHOR]

The analysis of the post-buckling behavior of elastic structures still needs the resolution of a body of non-linear differential equations according to equilibrium equations. Also, the yielding and buckling factors are important in the procedure of designing members under forces such as axial or conjoint axial force and bending moment. In such a way that if the length of the member is too long or the member is thin, before the yielding, buckling will occur in the member, and it is necessary to check and control the member for possible buckling. The present work deals with the stability analysis of elastic columns with variable cross-sections under concentrated end load and proposes a simplified approach to the evaluation of the critical buckling force of columns according to the assumptions of the Elastica theory. In this paper, the power series are used to simplify the equations. The numerical issues of the critical buckling force are presented for prismatic and non-prismatic columns subjected to end force, and the effectiveness of this approach is verified for buckling analysis of tapered columns, and the rate of accuracy is assessed. The elastic buckling force of elastic structures shows that the introduced model is computationally extremely efficient with the details presented in general. This paper should be a basic reference to compare the results with other research. [ABSTRACT FROM AUTHOR]

This study presents a comparative analysis of the structural performance of two innovative steel beam–column connections, namely a self-centering pinned connection with friction damper (SC-PC-FD) and a heat-treated beam section (HBS). The findings indicate that the SC-PC-FD connection exhibits stable, flag-shaped behavior, while the HBS connection can withstand applied loadings up to a rotation of 6% without any occurrence of lateral–torsional buckling. Upon comparison of these connections, it is evident that the SC-PC-FD connection can eliminate residual drifts and provide higher ductility up to a rotation of 7%, while maintaining the main members within the elastic range. [ABSTRACT FROM AUTHOR]

The size-dependent buckling problem of cracked micro- and nanocantilevers, which have many applications as sensors and actuators, is studied by the stress-driven nonlocal theory of elasticity and Bernoulli–Euler beam model. The presence of the crack is modeled by assuming that the sections at the left and right sides of the crack are connected by a rotational spring. The compliance of the spring, which relates the slope discontinuity and the bending moment at the cracked cross section, is related to the crack length using the method of energy consideration and the theory of fracture mechanics. The buckling equations of the left and right sections are solved separately, and the variationally consistent and constitutive boundary and continuity conditions are imposed to close the problem. Novel insightful results are presented about the effects of the crack length and location, and the nonlocality on the critical loads and mode shapes, also for higher modes of buckling. The results of the present model converge to those of the intact nanocantilevers when the crack length goes to zero and to those of the large-scale cracked cantilever beams when the nonlocal parameter vanishes. [ABSTRACT FROM AUTHOR]

In the present work, the small size effects on stability properties of perforated microbeams under various types of deformable boundary conditions are studied considering the Fourier sine series solution procedure and a mathematical procedure known as Stokes' transformation for the first time. The main benefit of the present method is that, in addition to considering both the gradient elasticity and the size effects, the kinematic boundary conditions are modeled by two elastic springs as deformable boundary conditions. The deformable boundary conditions and corresponding stability equation are described using the classical principle which are then used to construct a linear system of equations. Afterward, an eigenvalue problem is adopted to obtain critical buckling loads. The correctness and accuracy of the present model are demonstrated by comparing results with those available from other works in the literature. Moreover, a numerical problem is solved and presented in detail to show the influences of the perforation properties, geometrical, and the variation of small-scale parameters and foundation parameters on the stability behavior of the microbeams. In addition, according to the best knowledge of the authors, there is no study in the literature that examines the buckling behavior of perforated microbeams on elastic foundation with the gradient elasticity theory. [ABSTRACT FROM AUTHOR]

Geometrical imperfections are ubiquitous in load-bearing structures, including beams, columns, and shells. Fabrication processes of structural members most often create geometrical imperfections of random size and shape, which lead to non-deterministic load-carrying capacity. This study investigates the statistics of the buckling load of a beam with a random initial imperfection profile that rests on a nonlinear elastic foundation. The geometrical imperfection is represented by a zero-mean Gaussian random field, generated using the Karhunen-Loève expansion. The spatial distribution of the random imperfection is characterized by the probability distribution of the local imperfection magnitude and a spatial autocorrelation function. A finite-difference scheme is used to solve the governing equilibrium equation for a given initial imperfection profile, from which the buckling load is determined. Through a set of Monte Carlo simulations, the mean and variance of the buckling load are determined. The simulations reveal the influence of different length scales on the statistics of the buckling load, including the beam length and the autocorrelation length of the geometrical imperfection. The size effects predicted with the simplified model have implications for reliability-based structural design. [ABSTRACT FROM AUTHOR]

Closed form solutions for the flexural-torsional buckling of elastic beam-columns may only be obtained for simple end boundary conditions, and the case of uniform bending and compression. Moment gradient cases need approximate analytical or numerical methods to be used. Investigations presented in this paper deal with the analytical energy method applied for any asymmetric transverse loading case that produces a moment gradient. Part I of this paper is devoted entirely to the theoretical investigations into the energy based out-of-plane stability formulation and its general solution. For the convenience of calculations, the load and the resulting moment diagram are presented as a superposition of two components, namely the symmetric and antisymmetric ones. The basic form of a non-classical energy equation is developed. It appears to be a function dependent upon the products of the prebuckling displacements (knowfrom the prebuckling analysis) and the postbuckling deformation state components (unknowns enabling the formulation of the stability eigenproblem according to the linear buckling analysis). Firstly, the buckling state solution is sought by presenting the basic form of the non-classical energy equation in several variants being dependent upon the approximation of the major axis stress resultant MY and the buckling minor axis stress resultant MZ. The following are considered: the classical energy equation leading to the linear eigenproblem analysis (LEA), its variant leading to the quadratic eigenproblem analysis (QEA) and the other non-classical energy equation forms leading to nonlinear eigenproblem analyses (NEA). The novel forms are those for which the stability equation becomes dependent only upon the twist rotation and its derivatives. Such a refinement is allowed for by using the second order out-of-plane bending differential equation through which the minor axis curvature shape is directly related to the twist rotation shape. Secondly, the effect of coupling of the in-plane and out-of-plane buckling forms is taken into consideration by introducing approximate second order bending relationships. The accuracy of the classical energy method of solving FTB problems is expected to be improved for both H- and I-section beam-columns. The outcomes of research presented in this part are utilized in Part II. [ABSTRACT FROM AUTHOR]

Size-dependent functionally graded material thin plate buckling and post-buckling problems are considered using the framework of the MCST (Modified Couple Stress Theory). Based on modified couple stress theory and power law, the post-buckling deflection and critical buckling load of simply supported functionally graded material thin plate are derived using Hamilton's minimum potential energy principle. The analysis compares the simulation results of linear buckling and nonlinear buckling. Innovatively, a power-law distribution with scale effects is considered. The influences of scale effect parameters l and power-law index parameters k on buckling displacement, load, and strain energy of plates have been investigated. In this article, it is found that the critical buckling displacement, critical buckling load, and buckling strain energy increase with increases in the power-law index parameters k. The membrane energy decreases as the power-law index parameter increases. If the upper and lower layers are swapped, the opposite result is obtained. In comparison, the scale effect parameter is more influential than the power-law exponent. The critical buckling displacement in the x-direction is not affected by scale effects. The critical buckling load, the membrane energy, and buckling strain energy increase as the scale effect parameter increases. Scale effects increase material stiffness compared with traditional theory, and the power-law index parameters affect FGM properties such as elastic modulus, Poisson's ratio, density, etc. Both scale effects parameters and power-law index parameters have important effects on the mechanical behavior of materials. [ABSTRACT FROM AUTHOR]

• A new exact buckling model based on MGE for thin-walled microbeams is proposed. • Besides the classical flexural–torsional coupling, two new coupling effects are found. • The first new coupling effect is the higher-order flexure–flexure coupling. • The second new coupling effect is the higher-order flexural–torsional coupling. More and more thin-walled micro-beams, including the I-section thin-walled micro-beams, are applied in MEMS. The buckling of a thin-walled micro-beam exhibits some different phenomena from that of a large scale thin-walled beam, such as the size effect, and the coupling effects caused by strain gradient. To investigate the influence of the coupling effects and size effect on the buckling of thin-walled micro-beams, a new buckling model is proposed. Based on modified gradient elasticity (MGE), the governing equations and boundary conditions of the new model are derived from the variational principle. The governing equations and boundary conditions can be simplified to the classical Vlasov's theory, the MGE Bernoulli–Euler beam model, MGE Bernoulli–Euler beam model considering bi-directional flexure respectively. Solved by the feature expansion method, the size effects of the thin-walled micro-beam can be captured by the new model, that is the buckling load increases with internal length scales. Besides the classical flexural–torsional coupling effect, two new coupling effects, such as the higher-order flexure–flexure coupling and the higher-order flexural–torsional coupling, are also captured by the new model. These two new high-order coupling effects of the new model reduce the critical buckling load of thin-walled micro-beams. The higher-order flexural–flexural coupling effect is far greater than the other two coupling effects, and the higher-order flexural–flexural coupling effect is on the same order of magnitude as the effect of MGE (size effect), with the former being negative and the latter positive. The size effect and higher-order coupling effect cannot be ignored and need to be considered carefully for the buckling of thin-walled micro-beams. [Display omitted] [ABSTRACT FROM AUTHOR]

A buckling model of Timoshenko micro-beam with local thickness defects is established based on a modified gradient elasticity. By introducing the local thickness defects function of the micro-beam, the variable coefficient differential equations of the buckling problem are obtained with the variational principle. Combining the eigensolution series of the complete micro-beam with the Galerkin method, we obtain the critical load and buckling modes of the micro-beam with defects. The results show that the depth and location of the defect are the main factors affecting the critical load, and the combined effect of boundary conditions and defects can significantly change the buckling mode of the micro-beam. The effect of defect location on buckling is related to the axial gradient of the rotation angle, and defects should be avoided at the maximum axial gradient of the rotation angle. The model and method are also applicable to the static deformation and vibration of the micro-beam. [ABSTRACT FROM AUTHOR]

• Semi-analytical method to study agglomerated CNT composite cylindrical panel considering non-local elasticity is presented. • Buckling and free vibration behaviours of agglomerated CNT nano composite panel under non-uniform edge load are presented. • Effects of non-local parameter, nature of agglomeration and non-uniform edge loadings are analysed. • The results obtained using the present approach are in good agreement with the results available in literature. A semi analytical nonlocal elasticity model to analyze the effect of non-uniform edge loads on static stability and free vibration characteristics of agglomerated carbon nanotubes (CNTs) reinforced nano cylindrical panels are presented. Effective material properties of the agglomerated CNT reinforced composite are obtained using a two-parameter micro-mechanics model while Eringen's non-local theory is used to account the size effect. Sinusoidal shear deformation theory is adopted to analyze the buckling and vibration parameters using Galerkin's approach. The accuracy of the proposed model is presented first by comparing the results in the literature. Then a comprehensive study is carried out to analyze the influence of various degrees of agglomeration (complete, partial), nature of edge load, and non-local effects on the buckling and free vibration response of CNT reinforced nano cylindrical panel. The results revealed that non-local size effect leads to a reduction in stiffness and thus reduces buckling and dynamic characteristics. Moreover, it is observed that critical buckling load varies with type of in plane load and reduction in natural frequency is different for different in plane loading conditions. [ABSTRACT FROM AUTHOR]

In this investigation, a size-dependent numerical solution methodology is devised to analyze nonlinear buckling and postbuckling of cylindrical microsized shells made of checkerboard randomly reinforced nanocomposites subjected to a combination of axial and lateral compressions. To accomplish this purpose, the modified couple stress elasticity continuum is formulated within the third-order shear flexible shell model. Using a probabilistic-based homogenization plan in conjunction with the Monte-Carlo simulation, the effective mechanical parameters of the randomly reinforced nanocomposites are captured. The established size-dependent problem is then numerically solved via using the moving Kriging meshfree technique having the ability to enforce the required boundary conditions straightly at the associated nodes without using any type of penalty technique. By tracing the nonlinear stability paths, it is revealed that for the both axial dominated and lateral dominated loading cases, the stiffening feature related to the rotation gradient tensor causes that the microshell endures higher shortening before the buckling phenomenon occurs. In addition, it is found that by increasing the length to width ratio of graphene nanofillers, the effect of combination of axial or lateral load increases a bit. [ABSTRACT FROM AUTHOR]

To accurately capture size effect and geometrically nonlinear effect on thermal post-buckling of micro-beams, a new theoretical analysis model is presented. The model is based on modified gradient elasticity and von Kármán geometrically nonlinear theory. Considering the thermal effect, the governing equations of thermal post-buckling are derived by the principle of minimum total potential energy. The complete information including post-buckling bifurcation, configuration, amplitude of hinged–hinged and clamped–clamped micro-beams under different temperature rises is obtained by analytical solution. Comprehensive discussions are represented for size effect and geometrically nonlinear effect on thermal post-buckling of micro-beams. It confirms that taking into account the effect of geometric nonlinearity results in smaller post-buckling amplitude and the buckling resistance increase with the decrease in beam size. Both size effect of the critical buckling temperature rises and the post-buckling amplitude can be captured. Compared with other models, the ability of this model to predict the thermal buckling behaviors is improved. [ABSTRACT FROM AUTHOR]

The need to improve on analysis and design of structural members for overall weight, material and financial economy is indispensable. This paper presents the results of post-buckling analysis of an axially loaded prismatic column. The theory of elasticity was employed to formulate the total potential energy functional for a prismatic column in the post buckling regime. The derived functional was minimized with respect to deflection function to obtain the governing equation of equilibrium. The governing equation was then solved to obtain the deflection equation. Again, the functional was minimized with respect to the coefficient of the obtained deflection function to obtain the formula for calculating the post-buckling load of the prismatic column. A numerical example was given to demonstrate the applicability of the present formulation. The prismatic column was analyzed within the post buckling regime to determine the stiffness coefficients. It was shown that as the deflection-thickness (Δ/t) ratio increased, the combined buckling stress increased geometrically. It was found that the critical buckling load obtained from the present method and those obtained from literature showed consistency. It was also found that at the buckled state of the prismatic column, the yield strength of the column material has not been attained and deflection is very small. This gives room for the column to take on more loads before it fails either by excessive deflection or by stress exceeding the material strength. [ABSTRACT FROM AUTHOR]

Motivated by the observations of snap-through phenomena in pre-stressed strips and curved shells, we numerically investigate the snapping of a pre-buckled hemispherical gridshell under apex load indentation. Our experimentally validated numerical framework on elastic gridshell simulation combines two components: (i) discrete elastic rods method, for the geometrically nonlinear description of one-dimensional rods, and (ii) a naive penalty-based energy functional, to perform the non-deviation condition between two rods at joint. An initially planar grid of slender rods can be actuated into a three-dimensional hemispherical shape by loading its extremities through a prescribed path, known as buckling-induced assembly; next, this pre-buckled structure can suddenly change its bending direction at some threshold points when compressing its apex to the other side. We find that the hemispherical gridshell can undergo snap-through buckling through two different paths based on two different apex loading conditions. The structural rigidity increases as the number of rods in the gridshell structure becomes denser, which emphasizes the mechanically nonlocal property in hollow grids, in contrast to the local response of continuum shells. The findings may bridge the gap among rods, grids, knits, and shells, for a fundamental understanding of a group of thin elastic structures, and inspire the design of novel micro-electro-mechanical systems and functional metamaterials. [ABSTRACT FROM AUTHOR]

The sensitivity of critical buckling load and critical stress concerning different geometrical and physical parameters of Euler{Bernoulli nanobeams with defects is studied. Eringen's nonlocal theory of elasticity is used for the determination of critical buckling load for stepped nanobeams subjected to axial loads for different support conditions. An analytical approach to study the impact of discontinuities and boundary conditions on the critical buckling load and critical stress of nanobeams has been developed. Critical buckling loads of stepped nanobeams are defined under the condition that the nanoelements are weakened with stable crack-like defects. Simply supported, clamped and cantilever nanobeams with steps and cracks are investigated in this article. The presented results are compared with the other available results and are found to be in a close agreement. [ABSTRACT FROM AUTHOR]

In this paper, we present an analytical method based on equivalent continuum homogenization for the global buckling analysis of the general grid lattice cylindrical structures. In the proposed scheme, grid structures with arbitrary cell geometries can be analyzed by obtaining their effective cell stiffness based on force and moment analysis of the struts. The grid structures are assumed to be composed of curvilinear axial, circumferential, and oblique unidirectional composite ribs. To evaluate the results of the presented analytical method, a parametric finite element code is derived to generate the desired geometry of the grid structures and their buckling loads are obtained and compared with the analytical method. The effects of various parameters, including the number of ribs, their thickness and elastic properties, and helical angle of the oblique ribs are studied for hexagonal, triangular, and mixed grid shells. The results are compared with the available data published in the literature with a good agreement. [ABSTRACT FROM AUTHOR]

The need to improve on analysis and design of structural members for overall weight, material and financial economy is indispensable. This paper presents the results of post-buckling analysis of an axially loaded prismatic column. The theory of elasticity was employed to formulate the total potential energy functional for a prismatic column in the post buckling regime. The derived functional was minimized with respect to deflection function to obtain the governing equation of equilibrium. The governing equation was then solved to obtain the deflection equation. Again, the functional was minimized with respect to the coefficient of the obtained deflection function to obtain the formula for calculating the post-buckling load of the prismatic column. A numerical example was given to demonstrate the applicability of the present formulation. The prismatic column was analyzed within the post buckling regime to determine the stiffness coefficients. It was shown that as the deflection-thickness (Δ/t) ratio increased, the combined buckling stress increased geometrically. It was found that the critical buckling load obtained from the present method and those obtained from literature showed consistency. It was also found that at the buckled state of the prismatic column, the yield strength of the column material has not been attained and deflection is very small. This gives room for the column to take on more loads before it fails either by excessive deflection or by stress exceeding the material strength. [ABSTRACT FROM AUTHOR]

A mathematical model for nonlocal vibration and buckling of embedded two-dimensional (2D) decagonal quasicrystal (QC) layered nanoplates is proposed. The Pasternak-type foundation is used to simulate the interaction between the nanoplates and the elastic medium. The exact solutions of the nonlocal vibration frequency and buckling critical load of the 2D decagonal QC layered nanoplates are obtained by solving the eigensystem and using the propagator matrix method. The present three-dimensional (3D) exact solution can predict correctly the nature frequencies and critical loads of the nanoplates as compared with previous thin-plate and medium-thick-plate theories. Numerical examples are provided to display the effects of the quasiperiodic direction, length-to-width ratio, thickness of the nanoplates, nonlocal parameter, stacking sequence, and medium elasticity on the vibration frequency and critical buckling load of the 2D decagonal QC nanoplates. The results show that the effects of the quasiperiodic direction on the vibration frequency and critical buckling load depend on the length-to-width ratio of the nanoplates. The thickness of the nanoplate and the elasticity of the surrounding medium can be adjusted for optimal frequency and critical buckling load of the nanoplate. This feature is useful since the frequency and critical buckling load of the 2D decagonal QCs as coating materials of plate structures can now be tuned as one desire. [ABSTRACT FROM AUTHOR]

This study presents the size-dependent nonlinear thermal postbuckling characteristics of a porous functionally graded material (PFGM) microplate with a central cutout with various shapes using isogeometric numerical technique incorporating non-uniform rational B-splines. To construct the proposed non-classical plate model, the nonlocal strain gradient continuum elasticity is adopted on the basis of a hybrid quasi-three-dimensional (3D) plate theory under through-thickness deformation conditions by only four variables. By taking a refined power-law function into account in conjunction with the Touloukian scheme, the temperature-porosity-dependent material properties are extracted. With the aid of the assembled isogeometric-based finite element formulations, nonlocal strain gradient thermal postbuckling curves are acquired for various boundary conditions as well as geometrical and material parameters. It is portrayed that for both size dependency types, by going deeper in the thermal postbuckling domain, gaps among equilibrium curves associated with various small scale parameter values get lower, which indicates that the pronounce of size effects reduces by going deeper in the thermal postbuckling regime. Moreover, we observe that the central cutout effect on the temperature rise associated with the thermal postbuckling behavior in the presence of the effect of strain gradient size and absence of nonlocality is stronger compared with the case including nonlocality in absence of the strain gradient small scale effect. [ABSTRACT FROM AUTHOR]

[Display omitted] • Buckling model of an active plate is established based on the odd elasticity theory. • Anomalous tensile buckling of odd elastic plates is investigated. • Novel form of active buckling has been identified. • Active buckling is concluded to only occur in the form of chiral deformation. Buckling of thin-walled structures such as plates and shells is a consequence of in-plane stress being released through out-of-plane displacements. Generally, thin-walled structures that are subjected to tension or zero stress conditions remain stable. In this article, the anomalous tension buckling and stress-free active buckling of odd elastic plates are reported, which are a novel instability caused by odd elastic effects. The latter can only occur in the form of left- or right-handed chiral deformation and it does not involve external loads or internal active stress in a critical state. We demonstrate that the chiral rotation angle deformation is responsible for the active buckling of the plates, because the energy required for instability can be obtained based on the odd elastic effect. These findings can serve as an interpretation in a novel way for the occurrence of surface morphologies with biological activities, as well as provide references for buckling designs and applications of active structures. [ABSTRACT FROM AUTHOR]

Biologically active shells typically exhibit complex mechanical behaviors, with their morphogenesis involving rich symmetry breaking events and non-conservative biological forces. In this article, an accurate buckling solution model of active cylindrical shells is established based on the odd elasticity theory. The tension buckling, non-reciprocal torsional buckling and stress-free active buckling of odd elastic cylindrical shells are reported, which are an anomalous instability caused by odd elastic effects. The result shows that the anomalous buckling of cylindrical shells can only occur in the form of chiral deformation, because the energy required for instability can be obtained based on the odd elastic effects. • A new odd elastic stability model of cylindrical shell is established. • Anomalous tensile buckling and non-reciprocal torsional buckling are investigated. • Novel form of active buckling has been identified. [ABSTRACT FROM AUTHOR]

The buckling behavior of a nonuniform single-walled carbon nanotube (SWCNT), subjected to axially compressive load, is studied using the nonlocal elasticity theory. The differential transformation method (DTM) has been used to obtain the nonlocal buckling loads of the nonuniform SWCNT under various boundary conditions, namely simply supported, fixed–fixed, and fixed-simply supported. The nanotube's nonlocal buckling load increases significantly with an increase in the tip's diameter; however, it decreases substantially with increasing the small-scale parameter for both uniform and nonuniform SWCNTs. The results obtained from the DTM agree well with those reported in the literature for uniform SWCNTs. The accuracy of the results revealed that DTM is useful and convenient for investigating the buckling behavior of nonuniform CNTs with small-scale effects for various boundary conditions compared to other analytical methods. This work would provide helpful insights into the design of nonuniform CNT-based devices. [ABSTRACT FROM AUTHOR]

This paper investigates the macroscopic behaviour of an axially loaded discrete granular system from a stability perspective. The granular system comprises uniform grains that are elastically connected with some bending and shear interactions and confined by some elastic supports. This structural system can then be classified as a discrete repetitive system, a lattice elastic model or a Cosserat chain model. It is shown that this Cosserat chain model is exactly tantamount to the finite difference formulation of a shear-deformable Timoshenko column in interaction with a Winkler foundation. The buckling of the discrete column with pinned ends is first analytically investigated through the resolution of a finite difference equation. The solution is compared to a nonlocal approach derived by continualizing the discrete problem. The approximated Timoshenko nonlocal approach appears to be efficient with respect to the reference lattice problem and highlights some specific scale effects. This scale effect is related to the grain size with respect to the total length of the Cosserat chain. Finally, the paper shows the key role played by the shear interaction in the instabilities of granular structural system, especially when the bending interaction can be neglected. [ABSTRACT FROM AUTHOR]

This paper is concerned with postbuckling behaviors of nanorods subjected to an end concentrated load. One end of the nanorod is clamped while the other end is fixed to a support that can slide in the slot. The governing equation is developed from static equilibrium and geometrical conditions by using the exact curvature corresponding to the elastica theory. The nonlocal elasticity, the effect of surface stress, and their combined effects are taken into account in Euler-Bernoulli beam theory. Differential equations in this problem can be solved numerically by using the shooting-optimization technique for the postbuckling loads and the buckled configurations. The results show that nanorods with the nonlocal elasticity effect undergo increasingly large deformation while the effect of surface stress in combination with nonlocal elasticity decreases the deflection of nanorods under the same postbuckling load. [ABSTRACT FROM AUTHOR]

The nonlinear buckling and postbuckling characteristics of cylindrical nanoshells embedded in elastic foundations are investigated based on a strain-consistent elastic shell model including the both surface tension and the induced residual stress. For this purpose, in contrast to the previous models on the basis of the Gurtin-Murdoch elasticity theory, a new non-classical shell model based on Ru's surface elasticity theory is developed in which the non-strain displacement gradient terms are eliminated from the surface stress-strain relations. Using the virtual work's principle, the governing differential equations incorporating surface effects are derived. After that, the size-dependent governing equations are deduced to a boundary layer type problem which is subsequently solved through employing a two-stepped singular perturbation technique. It is revealed that because the edge supports of nanoshells are movable, before applying the axial compression, surface effects lead to an initial shortening due to induced residual strains, but the terms related to the residual strain and initial surface tension vanish in the size-dependent nonlinear governing equations. As a result, it is observed that before applying the axial compressive load, the surface effects cause an initial end-shortening for very thin nanoshells and these effects quickly diminish by increasing the shell thickness. Communicated by Krzysztof Kamil Żur [ABSTRACT FROM AUTHOR]

Tensegrities are prestressable trusses that have been proven to support various load distributions with minimum mass. This article presents a novel efficient method for designing lightweight tensegrities under local and global failure constraints. Local failure includes buckling and material yielding of individual members in the tensegrity. Global failure refers to global buckling of the tensegrity, where it loses stability without undergoing local failure at its individual members. The formulation and numerical approach to determine the critical global buckling forces and mode shapes of tensegrities with arbitrary shape and topology are first provided. Next, the design method considering local and global failure is presented, which starts with the local sizing of the member areas of the given tensegrity for the prevention of local failure. The method then determines the dominant failure mode by comparing the external forces and the critical global buckling force of the locally sized structure. If the critical global buckling force is larger than the external force, the dominant mode is a local failure and the locally sized design is returned as the minimum mass design. Conversely, if global failure is the dominant mode, different global reinforcement approaches are applied to raise the critical buckling force of the structure until it matches the external force, preventing global buckling. These reinforcement approaches include increasing the areas of the members and increasing the prestress in the tensegrity. Representative examples are provided to demonstrate the effectiveness of the design method considering box and T-bar tensegrities. [ABSTRACT FROM AUTHOR]

In nature, hair-like whiskers are used to detect surrounding information, such as surface texture and air flow field. The detection requires a comprehensive understanding of the relationship between whisker deformation and the contact force. With a whisker being modeled as a slender beam, the contact problem cannot be solved by small deformation beam theory and thus requires a new mechanical model to build up the relationship between whisker deformation and the contact force. In this work, the contact problem between a whisker and a round obstacle is solved, considering three factors: large deformation of the whisker, size of the obstacle, and frictional effect of the interface. Force and energy histories during the contact are analyzed under two motion modes: translation and rotation. Results show that the rotational mode is preferred in nature, because rotation of a whisker over an obstacle requires less energy for frictional dissipation. In addition, there are two types of detachment during the slip between the whisker and the obstacle. The detachment types are dependent on the whisker's length and can be explained by the buckling theory of a slender beam. [ABSTRACT FROM AUTHOR]

Buckling of axially loaded cantilever nanobeams with intermediate support have been studied in the current study. Higher order size dependent strain gradient theory has been utilized to capture the scale effect in nano dimension. Minimum total potential energy formulation has been used in modeling of nanobeam. Approximate Ritz method has been applied to the energy formulation for obtaining critical buckling loads. Position of the intermediate support has been varied and its effect on the critical buckling load has been investigated in the analysis. Mode shapes in critical buckling loads have been shown for various intermediate support positions. Present results could be useful in design of carbon nanotube resonators. [ABSTRACT FROM AUTHOR]

In the paper, the size dependent buckling analysis of hybrid organic/inorganic nanobeam with I cross section is investigated. Eringen's nonlocal elasticity theory is used to take the size effect into consideration. Comparative buckling loads of nanobeams for first ten modes is plotted in figure using Euler-Bernoulli theory and Eringen's nonlocal elasticity theory. Two different size parameter is used. It is clearly demonstrated that the size effect can be neglected for first modes while it is unneglectable for higher modes. Simply supported case in investigated. The advantages of I-cross section are discussed. [ABSTRACT FROM AUTHOR]

This paper surveys techniques for estimating the modulus of elasticity of 3D printed parts through free vibration tests. The study involves, in the context of common criteria of International Standards, the comparison between different procedures of recording and analyzing the vibratory motion using high precision equipment (proximity sensor connected to professional vibration analyzer) and cheap optical techniques (common high-speed camera and video tracking software). In both procedures, the input motion in the samples is carried out by a sudden low-energy impact. Frequency response functions, natural frequencies, and modes shapes are obtained from the experimental data, to calculate the elasticity modulus of specimens based on appropriate mathematical models and/or standards. The Taguchi method is employed to construct the experiment and the analysis of variance is carried out for evaluating the influence of several parameters of the additive manufacturing process. The elastic properties obtained from the test specimens are then employed in the calculation (by means of finite element procedures) of the dynamic behavior of 3D printed assembled thin walled structures (e.g., U-beam) and contrasted with their corresponding experimental values. [ABSTRACT FROM AUTHOR]

Investigating the compression properties of randomly ordered fiber networks experimentally is difficult which has resulted in ongoing disputes as to the mechanisms controlling the compression strength in such materials. In this work, we investigated compression properties of randomly oriented fiber networks with a special emphasis on cellulose products such as paperboard. We numerically reconstructed the conditions of the short span compression test widely used to quantify the compression strength of paperboard. We found that the phenomenological failure mode of such networks is elasto-plastic buckling. The x-shaped failure mode observed in physical experiments appears when test specimen restraints are included in the model. The most significant improvements to sheet strength can be obtained by improving the elastic properties while the strain to failure is increased most by an improvement of the plastic yield and hardening properties of individual fibers. Bond breaks were confirmed to have a smaller influence on the overall response. Fiber level microscopic buckling was investigated in depth, providing quantitative estimates of the fraction of mass likely to buckle at the microscopic level. The analysis indicated that only a low to moderate number of load carrying fibers can be expected to buckle. The inherent strength reserve in non-ordered fiber networks was investigated by introducing hinge mechanisms throughout the network, and the effect was shown to be small for a small to moderate number of hinges. [ABSTRACT FROM AUTHOR]

Size-dependent buckling of compressed Bernoulli-Euler nano-beams is investigated by stress-driven nonlocal continuum mechanics. The nonlocal elastic strain is obtained by convoluting the stress field with a suitable smoothing kernel. Incremental equilibrium equations are established by a standard perturbation technique. Higher-order constitutive boundary conditions are naturally inferred by the stress-driven nonlocal integral convolution, equipped with the special bi-exponential kernel. Buckling loads of compressed nano-beams, with kinematic boundary constraints of applicative interest, are numerically calculated and compared with those obtained by the theory of strain gradient elasticity. [ABSTRACT FROM AUTHOR]

We present the elastic buckling problem of moderately thick and thick beams as a boundary value problem of the classical mathematical theory of elasticity. The study considered homogeneous, isotropic, linear elastic beams. Small deformation assumptions were used together with kinematic, constitutive relations and the differential equations of equilibrium to obtain the governing field equations as a fourth order non-homogeneous ordinary differential equation (ODE) when both axial, compressive and transverse loads are considered, and a fourth order homogeneous ODE when only axial compressive force is considered. Using trial function method, the homogeneous ODE is solved in general for any end support conditions to obtain a general solution for the buckled beam in terms of four unknown constants of integration. The boundary conditions corresponding to the four cases of end support conditions considered were used to obtain the characteristic buckling equations, which were expanded to obtain transcendental equations with an infinite number of roots in each case, thus yielding an infinite number of buckling loads. The least root of the transcendental equations was used to obtain the critical buckling load, which was found to depend on the ratio h/l and the Poisson's ratio, µ. Critical buckling loads for each end support condition was calculated and tabulated. The results show that for each end support condition, as h/l < 0.02, the critical buckling load coefficient obtained was approximately equal to the critical buckling load coefficient of Euler - Bernoulli beam. As h/l > 0.02, which is the threshold for thin beams, the critical buckling load is found to be much smaller than the critical buckling load obtained from Euler - Bernoulli theory. It is thus concluded that the shear deformable theory is necessary for a more realistic analysis of the critical load buckling capacities of moderately thick, and thick beams for safety in their design. [ABSTRACT FROM AUTHOR]

An infinitely long elastic plate/layer is under uniaxial compression with its long dimension held by adhesion to a flat rigid base without friction. A prescribed length of the plate/layer is free of adhesion. This configuration is similar to a pre-stressed elastic film for which buckling of an unbonded section is a necessary, but not sufficient, condition for delamination. For that configuration, buckling occurs at the Euler buckling load of a fixed-fixed plate. Although the present study does not include friction or tangential interface stresses, the onset of buckling should be similar for these two cases. For the case of an elastic plate, a cohesive zone is used and it is found that the fixed-fixed buckling load is not attained except for extremely large values of a cohesive zone parameter. For realistic values, the buckling load is about half of that value. For the situation of an elastic layer with adhesion (without a cohesive zone), the buckling load approaches the fixed-fixed value only for very large values of the ratio of the unbonded length to the thickness. [ABSTRACT FROM AUTHOR]

We present a weak form implementation of the nonlinear axisymmetric shell equations. This implementation is suitable to study the nonlinear deformations of axisymmetric shells, with the capability of considering a general mid-surface shape, non-homogeneous (axisymmetric) mechanical properties and thickness variations. Moreover, given that the weak balance equations are arrived to naturally, any external load that can be expressed in terms of an energy potential can, therefore, be easily included and modeled. We validate our approach with existing results from the literature, in a variety of settings, including buckling of imperfect spherical shells, indentation of spherical and ellipsoidal shells, and geometry-induced rigidity (GIR) of pressurized ellipsoidal shells. Whereas the fundamental basis of our approach is classic and well established, from a methodological view point, we hope that this brief note will be of both technical and pedagogical value to the growing and dynamic community that is revisiting these canonical but still challenging class of problems in shell mechanics. [ABSTRACT FROM AUTHOR]

Buckling and elastic stability study of vertical well aligned ZnO nanorods grown on Si substrate and ZnO nanotubes etched from the same nanorods was done quantitatively by nanoindentation technique. The critical load, modulus of elasticity, and flexibility of the ZnO nanorods and nanotubes were observed and we compared these properties for the two nanostructures. It was observed that critical load of nanorods (2890 μN) was approximately five times larger than the critical load of the nanotubes (687 μN). It was also observed that ZnO nanotubes were approximately five times more flexible (0.32 nm/μN) than the nanorods (0.064 nm/μN). We also calculated the buckling energies of the ZnO nanotubes and nanorods from the force displacement curves. The ratio of the buckling energies was also close to unity due to the increase/decrease of five times for one parameter (critical load) and increase/decrease of five times for the other parameter (displacement) of the two samples. We calculated critical load, critical stress, strain, and Young modulus of elasticity of single ZnO nanorod and nanotube. The high flexibility of the nanotubes and high elasticity of the ZnO nanorods can be used to enhance the efficiency of piezoelectric nanodevices. We used the Euler buckling model and shell cylindrical model for the analysis of the mechanical properties of ZnO nanotubes and nanorods. [ABSTRACT FROM AUTHOR]

The thermal buckling of an armchair double-walled carbon nanotube (DWCNT), which is subjected to axial compression due to temperature rise, is derived and analyzed based on Timoshenko beam model, including transverse shear deformation and rotary inertia. According to the analysis, the effect of van der Waals force between the nanotubes on the critical buckling temperature of mode 1 of armchair DWCNT is significant, especially for smaller diameter nanotubes. The van der Waals force makes the DWCNT stiffer and increases the buckling temperature. In addition, the effect of shear deformation and rotary inertia on the buckling temperature is more obvious for the higher-order modes. The critical buckling temperature ratio of a Timoshenko beam to a Euler beam for the armchair DWCNT significantly decreases with increasing the diameter and mode number. Therefore, for the higher-order modes, the Timoshenko beam model is able to predict the critical buckling temperature of larger diameter DWCNT. [ABSTRACT FROM AUTHOR]

In this paper, flexural vibrations of cracked micro- and nanobeams are studied. The model is based on the theory of nonlocal elasticity applied to Euler–Bernouilli beams. The cracked-beam model is established using a proper modification of the classical cracked-beam theory consisting of dividing the cracked element into two segments connected by a rotational spring located at the cracked section. This model promotes a discontinuity in bending slope, which is proportional to the second derivative of the displacements. Frequency equations of cracked nanobeams with some typical boundary conditions are derived and the natural frequencies for different crack positions, crack lengths, and nonlocal length parameters are calculated. The results are compared with those corresponding to the classical local model, emphasizing the differences occurring when the nonlocal effects are significant. [ABSTRACT FROM AUTHOR]

A stiff thin film on a heated compliant substrate may buckle when the system is cooled due to the thermal expansion mismatch between the film and substrate. Highly ordered and disordered herringbone patterns (wavy structures) then emerge as the system continues to cool. We have established an analytic approach to study one-dimensional, checkerboard, and ordered herringbone buckling patterns. The analytical approach gives the buckle wave length and amplitude in terms of the thin film and substrate elastic properties, thin film thickness, and the thermal strain. It is shown that the herringbone mode has the lowest energy, which explains why this mode is frequently observed in experiments. These classes of materials might be interesting as a route to high performance electronics with full, two-dimensional stretchability. [ABSTRACT FROM AUTHOR]